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I! NEWS NOTES AND RESEARCH BRIEFS g. Giant Horn Simulates Sound of Saturn Firing The Wyle Laboratories Acoustic Fatigue Test Facility, activated on 2 February 1963 adjacent to the George C. Marshall Space Flight Center at Huntsville, Alabama, emits the loudest sustained noise ever produced by man. The 167- decibel roar, which shattered a quarter-inch sheet of safety plate glass in its "ribbon-cutting" cere- mony, is a simulation of the noise anticipated from the firing of the first-stage of the Saturn C-5 booster, which will launch the Apollo spacecraft toward the moon. The source is a giant loud- speaker, which will be able to fur- nish information on the levels of durability necessary for compo- nents of the Saturn and for launch-site, ground-support equip- ment to withstand the tremendous pressure waves created by the 7
89 million-pound-thrust blast-off. Acoustic fatigue testing is rela- tively new to the aerospace indus- try. Little was known--or even ex- pected-of the role sound would play in the conquest of space. But its effects proved to be destructive in many early missile and rocket firings. The sounds created by high-velocity gases escaping from the rocket exhausts and their im- pact against the launching pad are of such intensity that the pres- sure waves can actually break steel plates. The C-5's cluster of F-1 engines comprises the most power- ful rocket yet devised. Therefore, it can be assumed that this tremen- dous power plant will create acous- tic problems that could cause dis- aster in the Saturn program if they are not solved in advance of the actual launchings. The Wyle horn measuring 48 feet end to end (see photograph) is capable of producing sounds rated at 167 decibels, the sound- pressure level expected from the Saturn's blast. The faintest audible sound to a person with sensitive hearing is zero decibels (see Table I). As sound increases it is measured logarithmically to 180 decibels, equivalent to the noise of one million roaring auto- mobile engines. Above this level the vibrations caused by sound are measured as shock waves and are so intense they sometimes become visible, producing an effect similar to heat waves rising from an as- phalt pavement under a hot sun. Examples of this can be seen at the leading edge of a supersonic aircraft's wing or emanating from a nucelar explosion. ]t is these waves "dropping off" a jet air- plane as it passes Mach 1 that cause sonic booms. Soft, relatively pliable objects, such as the human body, have the ability to "give" under the on- slaught of sound waves, although T.~BIJE I. Typical sound-pressure levels for various acoustic sources. Sound-pressure level (decibels re 0.0002 dynes/square centimeter) Source -167- -160- -150- -140- -130- -120- -110- -100- -90- -80- -70- -60- -50- -40- --0-- Maximum near rocket engine Maximum near field 10 000-pound thrust jet engine 75-piece orchestra\ Pipe organ J Peak revolutions/minute levels Small aircraft engine Large chipping hammer BB~ tuba Peak revolutions/minute levels Blaring radio Centrifugal ventilating fan (13 000 cubic feet/minute) Vane-axial ventilating fan (1500 cubic feet/minute) Voice shouting Turbojet airlines landing @500 foot altitude Voice conversational level Voice--very soft whisper Minimum audible sound 34 SOUND Volume 2, Number 3 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28Giant horn of new high-lntensity acoustic test facility at Wyle Laboratories, Huntville, Alabama, is 48 feet long and con- structed of reinforced concrete up to and more than a foot thick. Special siren at left, feeding into small end of horn, simulates sound of rocket engines. At maximum output, horn emits loudest sustained noise ever produced by man. the maximum sound levels which can be tolerated have not been ac- curately determined. Coincident with the Saturn testing, Wyle is planning extensive research into the human factors involved in such sound levels. One of the psycho- logical effects to be studied is that, although no pain is experienced, time seems interminable under ex- treme noise conditions, causing the human being to want subcon- sciously to get away from the noise source. An example of this was dis- covered among the deck personnel aboard a jet aircraft carrier. Although the three astronauts aboard the Apollo spacecraft will not be subjected to the pressures of the blast-off sound, it is, in the final analysis, their presence which establishes the need for extreme re- liability in the Saturn program. In America's space program, noth- ing takes priority over human life. Discounting the fact that giant Saturn boosters cost many times the amount of money involved in the testing of unmanned rockets and missiles, the Saturn, with its human cargo, must perform flaw- lessly. Whereas soft materials can ab- sorb and dissipate the effects of high-intensity sound, metals, glass, and other firm materials tend to vibrate in resonance. When a cer- tain level of vibration is reached, the material can no longer sustain the stresses and, as a result, frac- tures. Through this type of fatigue- testing, researchers will discover which materials and structural de- signs will qualify for Saturn ap- plication. Obviously, the noise created by the Wyle horn must be kept under control since it approaches 170 decibels, which is 300 million times the lowest audible sound. Wyle en- gineers therefore built the walls of the horn and its chambers of one- foot-thick concrete. At the large, or "bell," end of the horn, they installed sound-absorbing fiber- glass wedges to muffle the blast, after it had passed by the test specimen. The horn then was en- tirely enclosed in a blockhouse built of acoustic-absorbing mate- rials. The result is that conversa- tions can be held in normal voices just outside the blockhouse during tests. Wyle engineers, wearing ear protection, will even be able to work inside the blockhouse during most of the horn's operations. Sound in the test chamber is pro- duced by a siren with a variable aperture created by four closely spaced spinning disks with open- ings which coincide at random, permitting blasts of compressed air to pass through to create "white noise." A regular loud- SOUND May-June 1963 35 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28(Top) Control room is insulated to the degree that engineers need not wear ear protectors, though giant horn is only a few feet away. Test specimen can be oh- served through windows at left. (Right) Wyle Laboratories manager, William Brooks, in center, and Wyle engineers, Robert Jeffries and Dan Bozich, at right, inspect four-foot-square test panel of one- fourth-inch safety plate glass shattered by sound waves during demonstration of new high-intensity sound generator. speaker, of the type found in a radio, would be impractical, inas- much as, in the size required, the voice coil would exert forces that would destroy the cone. The acous- tic facility was built by Wyle at a cost of $250 000. Engineers who operate the system rate it as the most versatile acoustic test facility in existence. Although it was de- signed primarily for the Saturn program, construction is such that the horn is adaptable to any an- ticipated aerospace acoustic test requirement. 9 U. S. Sonics Acquires Underwater Test Facility U. S. Sonics, Inc., of Cambridge, Massachusetts, advanced acoustic materials research and develop- ment company, has acquired one of the largest underwater test sites in New England, a mile square lake with a 40-foot depth near Bo]- ton, Massachusetts. It is now build- ing a floating laboratory equipped with electronic and acoustic test equipment to facilitate develop- ment of antisubmarine warfare de vices for the Navy, and develop- ment of new underwater sensing devices for commercial use. The free-field test facility is expected to provide a number of advantages over standard tank- testing equipment. Without rever- beration from tank walls, engineers will be able to determine more pre- cisely the sensitivity of Navy hy- drophones. Under outdoor condi- tions, such as icy water, the effects of temperature variations on trans- ducers can be determined. The company will shortly begin testing devices such as sonar heads, depth sounders, and underwater distance indicators as well as Navy hydrophones in the new facility. 9 36 SOUND Volume 2, Number 3 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28Transistor Microphone In 1957, Warren P. Mason of Bell Telephone Laboratories sug- gested that the piezoresistive prop- erties of some semiconductors could be used for converting mechanical or acoustic pressure into electrical signals and vice versa. Such de- vices are widely used by industry to obtain very sensitive strain and pressure measurements. They are also useful in picking up and gen- erating acoustic signals in air, sea water, and solids; in detecting earth tremors ; in phonograph pick- ups; as roughness indicators; and as gauges for measuring tension, compression, acceleration, pressure, shear force, and torque. Since then, it has been shown that thin p-n junctions (Esaki diodes) can also be used as very sensitive hydrostatic pressure transducers, and models made of silicon, germanium, gallium-arse- nide, or gallium-antimonide tun- nel diodes have been built. An ex- perimental semiconductor micro- phone was made by F. P. Burns of Bell Laboratories. This micro- phone produced a power output only one ten-millionth that of a typical carbon microphone. Research on transistor micro- phones has been carried forward independently and concurrently by Bell Laboratories and Raytheon Company. On 21 August 1962, at the West- ern Electronic Show and Conven- tion, Los Angeles, W. Rindner and R. Nelson of Raytheon described a semiconductor strain transducer based on the sensitivity of shallow p-n junctions to suitably applied anisotropic stress. The Bell Laboratories transistor microphone was described first by M. E. Sikorski and P. Andreatch on 29 August 1962 at The Ameri- can Physical Society Meeting in Seattle. (A summary of the talk had been sent to the Bulletin of The American Physical Society on 19 June.) During the last week of Septem- ber, Dr. Rindner and R. Nelson of Raytheon disclosed to the news- papers that they were working on a similar device---a very small transistor microphone of high sen- sitivity. They described the device in the October issue of the Pro- ceedings of the IRE. (Their article was sent to the IRE Proceedings before 30 July 1962.) The Bell Laboratories transistor microphone is described in the Oc- tober 1962 issue of The Review of Scientific Instruments in a paper by 5I. E. Sikorski, P. Andreatch, A. Grieco, and H. Christensen. The paper was sent to the RSI on 15 August 1962. Both the Raytheon and the Bell Laboratories devices were described in detail on 27 October at the IRE 1962 Electron Devices Meeting in Washington, D. C. The Bell Laboratories transistor microphone has the following char- acteristics as compared with a typ- ical carbon-granule microphone and an earlier semiconductor mi- crophone that made use of the piezoresistive effect. The transistor microphone is more sensitive (approximately four times on the voltage basis) and gives a higher signal-to-noise ratio (54 decibels) than the carbon mi- crophone. Harmonic distortion is less than 3% at 1000 cycles per second and a sound pressure of 3 dynes/centimeter ~. Its fre- quency response is limited by the design of the diaphragm and the stress-transmitting system. The idea that a transistor could be used as a pressure transducer is not new. It was previously pro- posed that, were pressure to be ap- plied to the point contact of a point-contact transistor, the point would deform, varying the area of its contact with the emitter sur- face. Thus, contact resistance would vary in proportion to the applied stress. Such a transducer was not developed because its efficiency was not very high. Bell Laboratories' microphone uses a junction transistor. Pressure applied to a point on the surface of the emitter is passed through the emitter region and across the two p-n junctions of the transistor. At each junction, there is a thin region of high resistivity called a depletion layer. (The difference in Fermi levels of the negative- and positive-type materials at an inter- face gives rise to an internal elec- tric field that sweeps or "depletes" a thin region free of mobile charge carriers, thereby increasing its re- sistance.) The stress across the de- pletion layer changes its resistance and, therefore, the current flowing in the transistor changes. This modulation of the current may be due to various mechanisms. For example, at strains of 1000 microinches per inch, current changes can be explained by the effect of piezoresistance. However, at higher strains, a change in the energy gap of the semiconductor material can be the dominant fac- tor. There is also a possibility that at large strains recombination cen- ters may be created in the semi- conductor material under the in- denter. This would affect the life- time of the carriers. In any case, the flow of current across the junctions is affected. Since the emitter-base junction is nearer the point of pressure than the collector-base junction, the stress on it is greater and the con- duction through it is affected to a larger extent. The change in the resistance of the emitter-base junc- tion is analogous to the application of a signal to the base-emitter cir- cuit in a transistor amplifier: an amplified signal in the collector- emitter circuit is obtained. Microphone Sensitivity (in millivolts ~c revolutions/ minute for a 1 dyne/centi- meter 2 pressure at 1 kilocycle/second Efficiency ac power out'~ dc power in ] (at 10 dyne/centimeter 2 pressure) Transistor microphone Carbon-granule microphone Piezoresistive microphone 63 millivo!ts 16 millivolts 2 X 10 ~ millivolts 11% 1XlO-l% 1.4XlO 6% SOUND May-June 1963 37 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28The electrical output is com- paratively large. For example, in the experimental transistor micro- phone, a sound pressure of 1 dyne/centimeter 2 at a frequency of 1 kilocycle/second resulted in a voltage signal of 63 millivolts (revolutions/minute). The noise (in the frequency range of 600 to 4000 cycles/second) accompany- ing the 63-millivoIts signal was 0.12 millivolts; thus the signal-to-noise ratio was 54 decibels. Although the signal was measured at an im- pedance of 30 000 ohms, the above figures have been reduced to 100 ohms impedance level for compari- son purposes. The experimental transistor mike was made from a Western Elec- tric 20C NPN transistor; however, other types of commercially avail- able junction transistors--such as diffused-base, double-diffused mesa, microalloy-diffused, epitaxial mesa, double-diffused planar, epitaxial planar, surface barrier, and mul- tiple-junction transistors--can be used. Some of these transistors have comparatively large emitter and collector regions to which stress can be applied. The transistor of the microphone is a common, diffused-base tran- sistor in which the collector region is n-type silicon. The base is a p-type film that was diffused onto the collector. The emitter region, which was diffused onto the base, is n type. Metal films provide electrical contacts to the base layer and emit- ter. The transistor wafer is bonded to a metal base, which serves as the collector contact. The tran- sistor is biased with the emitter lead grounded. Stress is applied to the emitter surface through a l-rail radius sapphire stylus ; however, any hard metal such as molybdenum, tung- sten, or steel, a hard ceramic, or hard crystal such as diamond could be used. In the experimental model, the diaphragm that collects the acous- tic energy and funnels it to the sty- lus is of standard design. Smaller diaphragms can also be used; how- ever, sensitivity will be reduced. Sensitivity can be increased by de- creasing the thickness of the emit- ter region or by using a germa- nium instead of a silicon transistor. Bell scientists have also sug- gested even more effective meth- ods for stressing the transistor junctions. For example, the tran- sistor can be affixed directly to a diaphragm without need of a sty- lus. Or, a deformable semiconduc- tor sheet or wafer may be used as a diaphragm and the transistor junctions diffused directly into or epitaxially grown onto it. Various other modifications and extensions of this invention are apparent. For example, with ap- propriate circuitry, the transistor transducer can be used as a pres- sure-responsive switch. As the pres- sure reaches a critical point and changes the voltage across the junction, a compensating or cor- rective circuit can be activated that will trigger an alarm signal. Also, transistors made of piezo- electric materials, such as GaAs, CdS, or CdSe, could be used, which might result in even greater sen- sitivity. 9 OTS Bibliography Gives over 500 References on Shock and Vibration Environment More than 500 references on shock and vibration environment including references on space en- vironmental conditions and testing methods are presented in an Armed Services Technical Information Agency bibliography just released to science and industry through the Office of Technical Services, U. S. Department of Commerce. W. L. Hercules, Shock and Vibra- tion Environment A Report Bib- liography. (Armed Forces Techni- cal Information Agency, August 1962.) Pp. 135. (Order AD 277 392 from OTS, U. S. Department of Commerce, Washington 25, D. C.) Price $3.00. More than 500 reports of re- search and conference proceedings on shock and vibration environ- ment have been abstracted and cited in this bibliography. The ref- erences are grouped within these broad topic areas: mechanical shock and vibration; application to particular fields such as space technology, naval engineering., mil- itary equipment, test facilities ; and associated environments. Un- der associated environments are included references to space en- vironmental conditions, radiation effects on electronic equipment, and radiation effects on fuel and organic materials. Entries are arranged alphabeti- cally by subject area. Within each subject area, reports published by Department of Defense contractors are listed alphabetically by source, contract, and date; military orig- inated reports are arranged by source and title. 9 Korfund Dynamics Corporation Provides "Silence" for U. S. Scientists Korfund Dynamics Corporation, Westbury, New York, is doing its part to help the United States land a man on the moon by providing the quiet atmosphere in which the nation's space scientists can work undisturbed. Korfund has recently completed its contract to design and fabricate four huge high-velocity "silenc- ers" for the air-distribution sys- tems at the George C. Marshall Space Flight Center, Huntsville, Alabama. These silencers are in- tended for insertion into the duct- ing system in order to quiet the roar of the high-pressure blowers which force a total of 73 000 cubic feet of air per minute through four large ducts. The silencers, tubular in design with a baffle located con- centrically within each of them, are ten feet long and forty-four inches in diameter--among the largest ever built. They were de- signed to cut the noise level of air under high pressure from 100 decibels down to 60 decibels. To give an idea of the significance of this reduction, 110 decibels is the level of noise encountered in a heavy machine shop operation, where prolonged exposure will def- initely affect hearing. Sixty deci- bels is the level of the noise en- countered in a quiet office and is 38 SOUND Volume 2, Number 3 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28somewhat less than the level of noise of normal conversation. This significant reduction in noise level is being achieved with a four-inch layer of a high density, noneroding type of sound-absorb- ing material, specially developed by Korfund Dynamics. The outside steel casing of the silencers are specially constructed to withstand the extremely high pressures gen- erated by the air conditioning sys- tem's huge blowers. As a part of the same project, Korfund is also supplying a special acoustical hous- ing for the blowers which pumps out a total of 200 000 cubic feet of air per minute. This enclosure contains the airborne noise within its confines, thereby reducing the noise level of areas that are ad- jacent to the blowers. This housing, 69 feet long by 14 feet wide by 11 feet high, is constructed of Noise- guard acoustical panels. In this instance, the noise level will be re- duced to approximately 70 decibels. Korfund's Noiseguard acousti- cal products are of metal-clad con- struction with fire-retardant acous- tical inner cores which provide for sound absorption as well as the reduction in the transmission of sound from one area to another. In addition to their use in air-condi- tioning systems for offices, theaters, hospitals, and other places, where excessive noise may pose a serious problem, Noiseguard panels are used in the construction of audio- metric rooms, and industrial and business machinery enclosures. The silencing of air involves the reduction of four major types of noises. These are: (1) mechanical vibration in the fan or motor, (2) noise in the fan due to the motion of high velocity air, (3) duct rum- ble, and (4) grille noise. In the older, heavy concrete structures, where air conditioning and other mechanical equipment were set in the basement rather far away from occupants, the effects of vibration and noise were fairly well con- tained. Today's buildings are made of lighter weight, conductive mate- rials, principally steel and alumi- num, which transmit vibration and sound readily. Also, mechanical equipment is now placed on roofs and intermediate floors, very close to building occupants. This can make for serious noise problems. Moreover, the general noise level of office buildings has been raised substantially by the introduction of more persons per square foot of area, accounting machines, com- puters, and other devices which re- quire special equipment to abate noise and vibration. 9 Dawe Instruments on Display: London Physical Society Exhibition 1963 Dawe Instruments has again had a range of newly developed instru- ments accepted for display at the 1963 London Physical Society Ex- hibition. Production on all the new instruments is planned for 1963 and all the following instruments are being demonstrated on the stand. Type 1109 Visigauge 14. The Visigauge 14 employs an ultrasonic resonance method to measure the thickness of a wide range of mate- rials from one side of the material. The resonance method is particu- larly suitable for thin materials in the range 0.005 to 2 inches, and an accuracy of thickness measure- ment to 0.1% can be achieved. The particular feature of this new de- sign is the innnersion testing' facility, which permits thickness measurements to be taken without physical contact between the trans- ducer and the material under test. Coupling for the ultrasonic waves can be provided either by a water column or by immersion of trans- ducer and test piece in a tank of water. The obvious advantages are that there is no transducer wear, con- tinuous measurement can be made on flow-line production, and con- sistently uniform coupling permits very high accuracy of both visual and recorded measurements. Type 1419 Octave-Band Sound- Level Meter. This instrument is a combination of an accurate sound- level meter and high-stability oc- tave-band filter, enabling sound levels in the range 24 to 140 deci- bels to be directly measured and analyzed. The equipment is fully transis- torized and the sound-level-meter section incorporates the three weighting networks A, B, and C recommended in IEC and BS spec- ifications for sound-level meters. The filter section covers the range 90 cycles/second to 5.6 kilocycles/ second in six filter steps, each of one octave, and center frequencies of 125, 250, and 500 cycles/second, 1, 2 and 4 kilocycles/second, sup- plemented by low-pass setting with cutoff at 90 cycles/second and highpass setting with cutoff at 5.6 kilocycles/second. Type 1463 One-Third Octave- Band Filter. This compact light- weight filter is intended for use in conjunction with the Dawc se- ries 1400 sound-level meters for one-third octave-band analysis of sound spectra. It meets and ex- ceeds tile requirements of the pro- posed IEC specification for one- third-octave bandwidth filters. The filter covers the frequency range 25 cycles/second to 22.4 kilocycles/ second in thirty steps of one-third octave bandwidth, with normalized minimum insertion loss and mid- band frequencies equally spaced within each of the three decades of the frequency spectrum. Typi- cally the attenuation exceeds 50 decibels one octave away from the midband frequency. 9 U. S. Atomic Energy Commission Research Reports in Acoustics The Omce of Technical Services of the U. S. Department of Com- merce (Washington 25, D. C.) an- nounces the availability of the fol- lowing research reports of interest to acousticians : NYO-9586. "Applications of Ultra- sonic Energy. Ultrasonic Casting of Ceramic and Cermet Slips." November 1961. Pp. 47. $1.25. NY0-9587. "Ultrasonic Filling of Tubular Cladding with Ceramic Fuel Powders." November 1961. Pp. 4O. $1. NYO-10007. "Ultrasonic Hot-Press- ing of Metals and Ceramics." De- cember 1961. Pp. 27. $.50. HW-70638. "Ultrasonic Testing of Heavy-Walled Zircaloy Tubing." August 1961. Pp. 40. $1. * SOUND May-June 1963 39 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28Suggested Solutions to Aircraft Noise Given in Recent OTS Report Some possible answers to the problem of noisy aircraft and the reactions of a community to the recurring noise of aircraft opera- tions are offered in a research re- port now available through the Of- fice of Technical Services (OTS), U. S. Department of Commerce. This report, and two others relat- ing to aircraft noise, may be ob- tained from OTS. The three re- ports are discussed below. There are more than 150 active Air Force bases in the United States, the first report notes, and "the majority of these bases have noise problems." A. Pitrasanta et al., Guide for the Analysis and Solution of Air-Base Noise Problems. (Bolt Beranek and Newman Inc. for the U. S. Air Force, November 1961.) Pp. 163. (Order AD 278 688 from OTS, U. S. Department of Commerce, Washington 25, D. C.) Price : $3.00. Eleven possible answers to the problem of aircraft noise originat- ing at a military airfield are given in this report which presents procedures for analyzing air-base noise problems and the reactions of the nearby community. One of the few ways that the noise ex- posure from take-off operations can be reduced in nearby communities, the report says, is to alter the flight path during take-off to avoid popu- lated areas as much as possible. Because of their noise potential, jet-powered aircraft receive special attention in this study but the find- ings may be applied to piston-en- gined, propeller-driven planes as well. Three recommendations be- lieved to be of particular value are to re-orient aircraft run-ups (of engines) so that the noise extend- ing back along the exhaust path is turned toward another direction, changes runway utilization both as to time and to the choice of run- way and to use runup noise sup- pressors. The latter are of several different types, some absorbing part of the sound and others di- verting a large part of the noise skyward. Other methods presented are: Increase the distance between the noise source (aircraft) and the lo- cation affected; use buildings, hilly terrain, or other shielding struc- tures to decrease the noise; and construct buildings, near the air- base or on it, so as to absorb sound. Proper windows in the buildings are particularly important. Welden Clark, Reaction to Air- craft Noise. (Bolt Beranek and Newman Inc. for the U. S. Air Force, November 1961.) Pp. 138. (Order AD 278 622 from OTS, U. S. Department of Commerce, Washington 25, D.C.) Price : $3.00. One of several conclusions reached in this study, to analyze and evaluate the reaction of per- sons to aircraft noise, states that no single prediction procedure is adequate for estimating the reac- tion of nearby residents. Moderate fear of air crashes on or near homes appeared to be a factor to many of individuals questioned. Reasons prompting the study are: "Noise intrusion evokes nega- tive feelings," the report explains, "and in extreme cases overt ac- tions against the aircraft opera- tors." A series of socio-psychologi- eal interview studies were made of people in the vicinity of airports including prediction of community reaction, prediction of reaction to noise inside airfield office buildings, and a review of other noise studies. An interesting finding estab- lished by the researchers was that important differences may exist in the attitudes of individuals to air- craft (or other noise) and that dif- ferences may exist in the noise stimulus as well. W. E. Clark, Noise from Aircraft Operations. (Bolt Beranek and Newman Inc., for the U. S. Air Force, November 1961.) Pp. 124. (Order AD 278 625 from OTS, U. S. Department of Commerce, Washington 25, D.C.) Price : $2.75. Purpose of this work is to pre- sent a generalized and organized collection of information obtained during the course of air-base noise studies. It is intended to be inter- mediate in complexity between simple handbooks and complex, specific studies. Information from other than Air Force sources in- cludes the U. S. Navy, U. S. Army, and Port of New York Authority studies on airport-noise problems. The report is organized into three main areas: noise-source characteristics of aircraft, charac- teristics of aircraft operations, and propagation of aircraft noise. Pro- cedures are presented for making engineering estimates of noise due to aircraft operations. 9 ITT Calibration Seminars Engineers and technicians throughout the nation concerned with vibration measurements are indicating an awareness for the necessity of special training in proper calibration of their accel- erometers and other instruments. This is evidenced by the attend- ance at a series of seminars, en- titled "Calibration of Vibration and Shock Pickups," held in San Fernando, California by the In- dustrial Products Division of the International Telephone and Tele- graph Corporation (ITT). The first five-day lecture and laboratory seminar was from 7 January to 11 January 1963. Other sessions were held from 11 Febru- ary to 15 February 1963 and from 4 March to 8 March 1963. Most of the students attending the ITT seminars were standards and calibration personnel con- nected with aerospace industries, or with military or Government research facilities. Also repre- sented were environmental test en- gineers who calibrate their own instrumentation. In addition to aerospace organizations, men at- tending were drawn from the elec- tronics, automotive, naval ship- building, and other industries. Further information concerning these seminars may be obtained from Wayne Tustin, seminar con- sultant to ITT, 15191 Bledsoe Street, San Fernando. * 40 SOUND Volume 2, Number 3 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28NBS Standard Frequency Broadcasts Unchanged for 1963 During 1963 the standard fre- quency broadcasts of stations WWV, WWVH, WWVL, and WWVB of the National Bureau of Standards (NBS) (U. S. Depart- ment of Commerce) will continue to be offset from Ephemeris Time by 13 parts in one billion. This off- set--which follows the current rec- ommendation of the International Bureau of Time, Paris, France--is so slight that it cannot be detected by ordinary radio receivers. How- ever, it could be significant to lab- oratories and industries making highly precise time or frequency measurements. This could be of im- portance, for example, in very pre- cise work in acoustics. The standard frequency and time signals broadcast by NBS are regulated by cesium atomic stand- ards. The cesium atoms "tick" 9,192,631,770 times during an ephemeris second which is the in- ternational basis for scientific time. A time scale, UT2, on which we base the time of day, is determined by the period of the earth's rota- tion on its axis, which we subdivide into hours, minutes, and seconds. As the period of the earth's rota- tion is not constant--in fact, varies from day to day--the length of the second as given by the atomic standards does not exactly coincide with the second determined from the earth's rotation. Therefore, the broadcast signals are offset from the scientific scale as much as is necessary to keep them in close agerement with UT2. During 1962 the International Bureau of Time consulted observa- tories throughout the world and compared astronomical and atomic measurements of time. From these data was determined the average variation from Ephemeris Time ex- pected in the speed of the earth's rotation during 1963. Their find- ings indicate a difference of about 13 parts in a billion--the same value as was used in 1962. The cor- rect frequency on the ephemeric scale can be determined by adding 13 parts in a billion to the fre- quency signal as received. 9 Donald W. Kuester Naval Ordnance Laboratory Senior Scientist Lost with U.S.S. Thresher A Senior Scientist and Division Chief of the Acoustics and Elec- tronics Division, V. S. Naval Ord- nance Laboratory, White Oak, Maryland, Donald W. Kuester, 41, was aboard the nuclear-powered submarine, U.S.S. THRESHER, lost at sea with all hands, Wednes- day, April 10. Kuester, formerly of West Hy- attsville, Maryland, had boarded the submarine on April 8, in Ports- mouth, New Hampshire, as an elec- trical engineer. He was to perform various tests and recording meas- urements in connection with a new, classified acoustics material he had recently invented and which was installed on board the THRESHER. He had been involved in under- water acoustics and weapon re- search and development since com- ing to the Laboratory in 1943 and held patents for two other inven- tions for which the Laboratory re- cently recognized him. These in- clude a low-frequency hydrophone for receiving underwater signals and a low-frequency transducer for sending underwater signals. In 20 years at the Naval Ord- nance Laboratory, Kuester re- ceived the Secretary of the Navy's Meritorious Civilian Certificate, a Superior Accomplishment Award and several Certificates of Com- mendation. He received his B. S. degree in Electrical Engineering from Iowa State University (1943), and was an associate mem- ber of the IEEE and a member of Tau Beta Pi and Eta Kappa Nu. Survivors include his wife and three children. 9 llie cancer nobody lalks about lakes more lives in this country than any other type of cancer. Because so many people ignore its symptoms. Or hope they will "go away." Or expect to do something "tomorrow." In short, they avoid the one thing that will help-seeing their doctors. For cancer of the colon and rectum can be cured in 3 out of 4 patients when discovered early and treated properly. Its danger signs-change in bowel habits or unusual bleeding-call for prompt medical examination. It may not be cancer, but only a physician will know. Every adult man and woman can have life-saving protection from cancer of the colon and rectum. An annual health checkup, including digital and proctoscopic examinations, can detect this cancer before any symptoms appear. Call your local American Cancer Society Unit for more information and material on this subject. AMER CAN CANCER This space contributed by the publisher SOUND May-June 1963 41 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.113.76.6 On: Tue, 25 Nov 2014 16:11:28

1.1777031.pdf

Optical Absorption Edge in GaAs and Its Dependence on Electric Field T. S. Moss Citation: J. Appl. Phys. 32, 2136 (1961); doi: 10.1063/1.1777031 View online: http://dx.doi.org/10.1063/1.1777031 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v32/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 19 Mar 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2136 PALIK, TEITLER, AND WALLIS TABLE II. Effective mass ratios obtained by Faraday and Voigt effects at liquid-nitrogen temperature. Combination Faraday Voigt of Voigt and Carrier effect effect Faraday Material concentration m*/m m*/m effects InSb 4.0XI016 0.019 InSb 2.0X 10" 0.023 InAs 7XI015 0.026 InAs 1 X 1017 0.030 0.031 GaAs 4.3XI016 0.076 0.071 available. Consequently, for a new sample with N known, the Faraday effect would usually be measured first as it produces the largest rotation. However, if N is not known, the two experiments will yield both N and m*. ACKNOWLEDGMENTS We wish to thank G. S. Picus and J. R. Stevenson for contributions to portions of work presented in this paper. We benefited from discussions with E. Burstein, F. Stern, and R. Toupin. Samples were kindly provided by the National Bureau of Standards, Naval Ordnance Laboratory, R.C.A. Research Laboratory, Services Electronics Research Laboratory, and Texas Instru ments, Inc. JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32. NO. 10 OCTOBER, 1961 Optical Absorption Edge in GaAs and Its Dependence on Electric Field T. S. Moss Royal Aircraft Establishment, Farnborough, Hants, Rngland Values of absorption constant covering the range 1 cm-l to 104 cm-l have been derived from transmission measurements made on single-crystal gallium arsenide. The absorption edge is very steep up to ",4000 em-I, where there is a knee beyond which the absorption increases relatively slowly with photon energy. The energy bands have been calculated using Kane's theory. From these a theoretical absorption curve has been ob tained which shows very good agreement with the experimental data. Using semi-insulating material, it has been possible to measure the shift of the edge with applied electric field. The effect is small (",200-J' ev shift for 5000-v/cm field) but is in good agreement with theory. INTRODUCTION GALLIUM arsenide is one of the group III-V inter metallic semiconductors which were first studied by Welkerl and his colleagues. It has many interesting properties, and is currently of much technological interest for making various solid-state devices, par ticularly transistors, parameteric diodes, and tunnel diodes. It shows promise of being the best solar battery material,2 and for this application in particular it is necessary to have detailed information on the optical absorption up to high levels. Early measurements of absorption in the neighborhood of the edge have been published/ but they reached an absorption level of only 100 cm-1• Some of this work has already been described in a recent pUblication.4 EXPERIMENTAL DETAILS The material used for the study of the absorption edge was pure single-crystal GaAs containing ...... 3 X 1016 cm-3 excess electrons. Plane parallel samples were prepared by grinding with silicon carbide and polishing with diamond paste. Specimens from 1 cm down to 7-J.L I H. Welker, Z. Naturforsch. 7a, 744 (1952); 8a, 248 (1953). 2 T. S. Moss, Solid State Elect~onics 2, 222 (1961). 3 F. von Oswald and R. Schade, Z. Naturforsch. 9a, 611 (1954). 4 T. S. Moss and T. D. Hawkins, Infrared Phys.l, 111 (1961). thickness were used. The thickness of the thin speci mens was found by measuring interference fringes in the 5 to lS-Jlo waveband. For the measurements of the edge shift produced by an electric field, samples of very high resistance GaAs were obtained. This material contains about the same density of impurities as the above, but the free carrier density is only 107 cm-3 free electrons.5 Specimens of this material were prepared in the same way except that they were etched after polishing, since this was found to give a considerable increase in specimen resistance. It was possible to prepare fairly thick specimens of resistance > 1011 ~, and to use applied voltages up to 10 kv. In order to avoid heating effects, the specimens were immersed in a liquid. Ligroin proved convenient for this purpose. The radiation was provided by a tungsten lamp and a Leiss double-prism monochromator. The prisms were flint glass and a resolution of 3 X 10-3 ev was used on the steepest part of the absorption curve. On the flatter part of the curve, at high K levels, the resolution was 6X 10-3 ev. Great care was taken to ensure spectral purity, and measurements could be made with insertion losses of up to 104: 1 on the flatter part of the absorption 6 J. W. Allen, Nature 187, 403 (1960); W. R. Harding, C. Hilsum, M. E. Moncaster, D. C. Northrop, and O. Simpson, ibid., 405 (1960). Downloaded 19 Mar 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsOPTICAL ABSORPTION EDGE IN GaAs 2137 curve. The detector system was an infrared photo-,."'-----------------;r---, multiplier followed by an 800-cps amplifier. ABSORPTION RESULTS The measured values of absorption coefficient (K) are plotted in Fig. 1, from which it will be seen that the main part of the edge is very steep, K rising rapidly from 4 to 4000 em-I. Over this range the edge is ex ponential-as has been observed for many other materials6-with a slope "" 100 ev-1• The steepness of the edge indicates that no phonon-assisted transitions are involved, i.e., the absorption edge is due to the onset of vertical transitions. CALCULATION OF ENERGY BANDS AND ABSORPTION COEFFICIENT The E-k curves have been computed using Kane's theory.7 They are given by k2= [2m/E(E-G) (E+Ll) (G+2Ll/3)] /[(E+2Ll/3)G(G+Ll)], a.u.8 (1) where G is the energy gap, Ll the spin-orbit splitting, m/ is the electron mass at the bottom of the conduction band, and E refers to the conduction band (Ee), the light-hole band (E2), or the split-off valence band (Ea). The heavy-hole band (E1) is assumed to be a simple, parabolic band corresponding to a mass m/=0.68 mo.9 ~ 10' ~------+.Jj-------+------ i ~ ~ .. z ... u ~ loll--------J4..l------J-------j ... o v z o t o. '" ., tX~ERIMEHT~l RESUI.TS. ---THEORETICAL CURVE ~IO~--~---+------+---------j ., .., I'~ 1'4 1'5 P HOT 0 N ENE R" Y (e v) FIG. 1. Absorption in gallium arsenide. H. 6 T. S. Moss, Optical Properties oj Semiconductors (Butterworths Scientific Publications Ltd., London, and Academic Press Inc., New York, 1959, 1961), pp. 39, 86. 1 E. O. Kane, J. Phys. Chern. Solids 1, 249 (1957). 8 In a.u., m=e=k/2.".= 1. 9 H. Ehrenreich, Phys. Rev. 120, 1951 (1960). ,.~ ~ > .!!. > I!/ .~ 01'~~========::~~~~~~~~====J E, -0 FIG. 2. E-k curves for gallium arsenide. The curves have been computed using the parameters Eg= 1.4 ev, Ll=0.33 ev,IO and mc*=O.072 mo,l! and are plotted in Fig. 2. The conduction band is slightly non parabolic, the effective mass increasing with energy. Using the expression for effective mass which occurs in Faraday effect or dispersion experiments,6 namely, m*=h2(kdk/ dE) F, (2) where the subscript F means the value at the Fermi level, the electron mass should increase to m*=0.082 mo at 0.1 ev above the bottom of the conduction band, i.e., for 3 X 1018 em-a carriers. The masses of the light-hole and split-off valence bands are given by the slopes as k~ 0, namely m2*=0.085 mo, ma*=O.25 mo . The absorption coefficient can be calculated directly from the E-k curves.7 Assuming direct, vertical transi tions, we have K= (4r/cnhv) Lj Mrpj, a.u., (3) where n is the refractive index and the summation is over the three valence bands (j= 1,2,3). 10 R. Braunstein, J. Phys. Chern. Solids 8, 280 (1959). 11 T. S. Moss and A. K. Walton, Proe. Phys. Soc. (London) 74, 131 (1959). Downloaded 19 Mar 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions213,11 T. S. MOSS 100 /1,. ~" 0 t4' / / e so ) • 0 / 0 I 5 ) 2 r>. ~ z ::> ~ c: ~ 10 at: 6 ...J c: Z <!I ii) V l Z 5 10 .... PPLIEO 'IIOL.T .... GE (kV Tms) FIG. 3. Dependence of absorption edge on electric field. The optical matrix element is Ml= (2P2j3){ (AcCj+CcAj)2+ (AcBj-BcAj)2}, (4) where the bracketed term-which is always near unity -is computed for each k value from the coefficients A=kp(E+2N3)jN, B=2t(E-G)t1j3N, C= (E-G) (E+2N3)jN, whereN isa normalizing factor such thatA2+ B2+C2= 1, and P=E(E+t1)j2(E+2t1j3)mc*. For the parabolic El band, Al = 0, Bl = 1, and C 1 = O. The density of states Pi is given in terms of the slopes of the E-k curves by (5) The absorption coefficient plotted from Eq. (3) is shown in Fig. 1. The agreement with the experimental data is seen to be very good, particularly at short wave lengths where the calculated curve does not depend on the use of any arbitrary constants or adjustable parame ters. In the neighborhood of the absorption edge, the fit has been improved by sliding the curve slightly side ways. This process gives a value for the energy gap at 292°K of Eg= 1.41 ev, which is considered to be some what more accurate than the value of 1.4 ev assumed in the analysis. The above absorption theory is essentially that for a perfect GaAs lattice, with n~ pe;turbati~~s ~~e to thermal vibrations or crystallme megulantles. The presence of these could well explain the slope of the edge observed under the experimental conditions . SHIFT OF EDGE WITH ELECTRIC FIELD As the shift of the edge with electric field is quite small, it was esse,ntial to use very high fields; this necessitated using high resistance specimens in order to prevent heating. The best material obtained had a specific resistance> 1080 cm (when in complete .dark ness) and using this, it was possible to make speCImens of resistance > 101lQ. As the temperature dependence of the edge in GaAs is fairly large (dEj dT= 500 l1-ev;oC), the measurements were always made with specimens in a liquid bath. Measurements with dc electric fields and chopped radiation were inconclusive; therefore, a system using steady radiation and ac fields was developed. It was assumed that the effect would be independent of the sign of the field (F), and would be proportional to F2. The frequency used for the field, therefore, was made half that of the amplifying system, namely 400 cps. (Subsequent comparisons made with 800 cps and 400-cps fields confirmed this hypothesis, the effect with 800 cps being many times smaller than with 400 cps.) The dependence of the observed signal at optimum wavelength on electric field is shown in Fig. 3, from which it will be seen that the points lie well on a line of slope 2 for rms voltages up to 4 kv. The reason for the tendency to saturate at the highest fields used is not completely understood; it is probably due in part to deterioration in the waveform from the amplifier used to supply the field when working near its maximum output. The spectral dependence of the field-induced signal is shown in Fig. 4. The response peaks sharply at the wavelength where the zero-field transmission curve is steepest, having a width at half-amplitude which is <kT. As Fig. 4 shows, its shape and position are vir tually identical with the derivative of the transmissions, thus proving that the signal observed is due to a pure shift of the absorption edge by the applied field. The absolute magnitude of this shift can be determined from the relative magnitudes of the field-induced signal and the differential transmission when they are measured under comparable conditions. In the experiments, the optical conditions were identical for the two measure ments but it was convenient to use different waveforms, namely, sinusoidal for the applied voltage, and square wave chopping for the zero-field transmission. The 12 R. H. Paramenter, Phys. Rev. 97, 587 (1955). Downloaded 19 Mar 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsOPTICAL ABSORPTION EDGE 11'\ GaAs 2139 OJ ... Z ::J .. It • ... • z ., -t:,....b- SIGNAL DUE TO FIELD (4I<Vrml) __ TRANSMITTID SIGNAL IN ZlRO ,IILD - - - DIFFIR!.NTIAL Of ZERO FIILO TRANSMISSION. SPECIMEN TH"KNESS sse .... O~--~I,3~g~----~I.+3e~----~ ... ~~~----~I.~3bL.e~Y~ PHOTON EH~R"Y FIG. 4. Spectral dependence of change in transmission due to an electric field. latter signal must, therefore, be reduced by 7l'/4: 1 to obtain the equivalent signal for sinusoidal chopping. For 3 kv rms at 400 cps applied to a specimen 8.8 mm long, the observed shift was IJ.E= 198±2-,uev. The peak-to-peak field at 800 cps in this case is F=3(2)!/0.88=4.83 kv/cm. Thus, ~E/p2=8,SX1O-16 ev per (v/m)2. (6) It has been shown theoretically by Franz13 that at low absorption levels, for an exponential absorption edge defined by A =Ao expa(w-wo), (7) 13 W. Franz, Z. Naturforsch. 13a, 484 (1958). the displacement is such that the edge becomes A = Ao expa(w-wo+a2e2p2/12Izm*); the shift is then or (8) lz~w/ep2=cx2e/12m* ev per (v/m)2. (9) The effective mass in this equation should presumably be the reduced mass of the electron-hole pair, i.e., 1/m*= 1/mc+1/ml, or m*=0.06Smo. (10) From Fig. 1, a=98 ev-r, so that from Eq. (9) the expected shift is ~E/p2=9.3X1O-16 ev per (v/cm)2, (11) This is less than 10% greater than the observed shift [Eq. (6)J, so that the agreement between theory and experiment can be considered quite satisfactory. The experiment, therefore, may be a useful way of obtaining fairly accurate values of effective masses in high-resis tance materials for which, at present, no other reasonable method is available. CONCLUSIONS Measurements have been made of the absorption coefficient of single-crystal GaAs in the neighborhood of the absorption edge. The results obtained agree well in absolute magnitude with values calculated from the E-k curves which have been computed for this material. A shift of the absorption edge with electric field has been observed. The good agreement found between the measured shift and theory indicates that this might be a useful method of measuring effective mass in insula tors or near-insulators. ACKNOWLEDGMENTS Thanks are due to D. H. Roberts of the Plessey Company, C. Hilsum of S.E.R.L., and D. J. Dowling of Mining and Chemical Products, for providing the gallium arsenide used in this work, and to T. D. F. Hawkins for assistance with the measurements. Downloaded 19 Mar 2013 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1702660.pdf

Gas—Solid Suspensions at High Temperatures S. L. Soo Citation: Journal of Applied Physics 34, 1689 (1963); doi: 10.1063/1.1702660 View online: http://dx.doi.org/10.1063/1.1702660 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High temperature reactor system for study of ultrafast gassolid reactions Rev. Sci. Instrum. 64, 1989 (1993); 10.1063/1.1143987 Anomalous diffusion of momentum in a dilute gas–solid suspension Phys. Fluids A 4, 1337 (1992); 10.1063/1.858411 Kinetic theory for a monodisperse gas–solid suspension Phys. Fluids A 2, 1711 (1990); 10.1063/1.857698 Comments on GasSolid Suspensions at High Temperatures J. Appl. Phys. 35, 2550 (1964); 10.1063/1.1702902 Erratum: Gas—Solid Suspensions at High Temperatures J. Appl. Phys. 34, 3644 (1963); 10.1063/1.1729288 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:38JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 6 JUNE 1963 Gas-Solid Suspensions at High Temperatures* S. L. Soo University of Illinois, Urbana, Illinois (Received 10 December 1962) The study consists of thermal electrification of a gas-solid suspension and its electrical conductivity at high temperatures. Possibilities of enhancing ionization and removal of electrons of a high temperature gas by solid particles were considered. INTRODUCTION A STUDY of equilibrium between thermionic emis- sion from solid particles and space charges of the phases in a gas-solid suspension is made. Hereafter, we refer to this reaction as thermal electrification. It is shown that at magnitudes of temperatures of 103 OK, ionization of the gaseous phase can be neglected, and thermionic emission as opposed by space charges is the major mechanism, and that the time to reach equi librium is extremely short. In this paper we consider thermal electrification of a gas-solid system with regard to the following items: (1) Effect of material and energy gap of the solid phase, (2) initial charge on the solid particles, (3) rate processes, (4) electrical conductivity of the suspension. 1. EQUILIBRIUM IN THERMAL ELECTRIFICATION Ionization of carbon particles at high temperatures was considered by Smith, who used the analogy of ionization of a gas.! In the following, the difference between thermal electrification and ionization of a solid particle surrounded by an electron cloud is clarified. First, let us consider a static system of a single solid particle in a finite volume. Due to continuous thermionic emission, equilibrium is unobtainable whenever a solid particle· exists in an infinite medium which consists of a vacuum or a gas. Equilibrium state is attainable, however, when a solid particle is in a finite volume that is evacuated or gas filled, because at equilibrium it is as likely for an electron to escape from the field of a solid particle electron cloud assembly as it is for a free electron to be attracted into the assembly. the case where there is negligible evaporation of the solid material. The potential V around a solid particle is given by: . where r is measured from the center of the spherical particle of radius a, Zp is the charge on the solid particle e is the electronic charge, E is the permittivity, and n: is the electron density around it. Assuming Maxwellian velocity distribution of these electrons, their density is given by2: ne(r)=nea exp( -eV /kT), (1.2) where k is the Boltzmann constant and T is the tem perature of the system. nea is just the density of elec trons at radius a, which is given by equating the thermionic current density to the current density of electrons in a cavity3: nea=2(27rmkT/h2)J exp[ -(lI'+eVa)/kT] =n,s exp[ -1I'/kT], (1.3) (the first equality applies to a metal, the latter defini tion makes the relation general) where a= Zp& / 4'/1'wkT, the ratio of electrostatic to kinetic energy; h is the Planck constant, m is the electronic mass, II' is the thermionic potential energy, and ne. is the density of conduction electrons in the solid phase; for a metal at high temperatures,4 nes~(total electron concentration in the solid)Xexp(-EF/kT), where Ep is the energy of the Fermi level; for an insulator, the number of free electrons are fewer and its distribution follows Max wellian and no further qualification on the above defini tion needs be made. We consider the case of a solid particle of radius a in a spherical gas volume of radius R, with the whole system at a given temperature. The inside wall of the container is taken as a pure geometric surface. The phenomena thus include equilibrium of thermal elec- 2 E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill Book t'fi . f h l'd . 1 d Company, Inc., New York, 1938), pp. 75,83. n catlOn 0 t e so 1 partlc e an first degree thermal 3 D. ter Haar, Elements of Statistical Mechanics (Rinehart and ionization of the gas. We further restrict ourselves to Company, New York, 1956), p. 246. ____ 4 The number of electrons per unit volume of solid is approxi- * This work was sponsored by Project SQUID which is sup-mated semiclassically by ported by the U. S. Office of Naval Research, Department of the J Navy, under contract U. S. Nonr-3623(S-6) NR-098-038. Repro- (8n-jh3) exp(EF!kT) exp( -E/kT)p2dp duction in full or in P3:rt is permitted for any use of the United States Government. ThIS paper was prepared from Project SQUID· = 2 (21rmkT /h2)J exp(EF/kT), ~eports ILL-5-P and ILL-7-P, March and June 1962. Revision where. is the kinetic energy and p is the momentum.6 mcludes Report ILL-lO-P March 1963. 6 L. D. Landau and E. M. Lifschitz Statistical Physics. 1 F. T. Smith, Proceedi~gs of the Third Conference on Carbon (Addison-Wesley Publishing Company, I~c., Reading, Massa- 1957 (Pergamon Press, Inc., New York, 1959), pp. 419-424. ' chusetts, 1958). 1689 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:381690 S. L. 500 For convenience in the derivation, we denote r* =r/ a, and p=4rr~r* (ne-nz)r*2dr*/n ea= (Ne-Nz)/neaa3, (1.4) where nz is the ion density at r*; nz is produced by gas atoms only; N e and N z are the total number of elec trons and ions inside radius r*, and R* PR=411' ~ (ne-nz)r*2dr*/nea=Zp/neaa3, (1.5) where R*=R/a, and P*=P/PR. Equation (1.2) becomes (ne-n.)/n ea= PR(dp* / dr*)/ 4rrr*2. (1.6) Various parts of Eq. (1.6) can be dealt with according to the following considerations: (1) The total charge on a solid particle is given by: (1. 7) where nCH is the density of conduction electrons in the solid when there is no thermionic emission (or when the solid particle is grounded). Combining Eqs. (1.7), (1.3), and (1.5) gives: (411/ PR)a= ({3.-3a) exp[ -qJ/kTJ, (1.8) FIG. 1. Distributions around a solid particle in argon of R* = 10 (I-JL zirconia particle at 3000oK, no= 1()20 1m3, Zp=206,a = 2.27). for singly charged ions and nza=nzR exp(-a). (1.15) From Eqs. (1.9) and (1.11), for {3o=noIPa2/ekT, we get (nea/nzR) = ({3,/{30)2 exp[ -(2qJ/kT)-aJpK p + ({3J (3o) exp[ -qJ/kT]' (1.16) Substitution of Eqs. (1.8) and (1.16) into Eq. (1.6) gives (dp*/r*2dr*)={3,A exp[ -a+a(l-p*)/r*J/a where {3,=n Cbe2a2/ ekT, and nea= [ncs-(3Zp/4rra3)J exp[ -qJ/kT]' -(3o exp[ -a(l-p*)/r*J/a[B exp( -a)+ 1J, (1.17) (1.9) where (2) ne and n. are related by ionization equilibrium at R where V =0 and dV /dr=O, and n/ (neR+nzR) = pK p/ (neR+nzR+n), (1.10) where n is the neutral atoms left at the equilibnum condition, nzR=nz(R), neR=ne(R), p is the pressure, Kp is the equilibrium constant, and n+nzR=nO, the original number of gas atoms per unit volume. We get nzR= nO(neR+nO)/[neR(pK p+ l)+noJ (1.11) the latter approximation is for pKp»l, nO»neR, and5 pKp= (g/2gz) (211'mkT/h2)!(nO+neR) exp(I/kT), where g and gz are the statistical weights of the ground states of the atom and the ion, I is the ionization po tential of the first degree of ionization. The extension to higher degrees of ionization is a matter of detail. (3) Since there is no net loss of electrons, Eq. (1.2) gives: ne=nea exp[ -a+a(1-p*)/r*J, and neR=nea exp( -a), nz= nzR exp[ -a(l-p*)/r*J, (1.12) (1.13) (1.14) A =exp( -qJ/kT), and B=nC8 exp[(I/kT)- (qJ/kT)J(g/2g.)(211'mkT/h 2)!; with the boundary conditions: r*=1, p*=O; r*=R*, p*=1. Integration of ~q. (1.17) with a trial value of a to satisfy the boundary condition and to get equilibrium value of a is straightforward. It is to be noted that at T=O (103), the second term on the right hand side of Eq. (1.17) is extremely small. As an example, we take a 1-~ zirconia6 particle in argon of no= 102°/m3, R*= 10, all at 30000K (this ap proxima tes a gas-solid suspension of n p = 2 X 1015/ m3 • qJ=3.4 eV, 1=15.756 eV (5), gz=6, g= 1. Here we have A = 2.04XH)-6, B=2.66X104s• Integration by trial values of a gives a=2.27 and Zp=206, nea=4.62 XlOls/m3, neR=4.76XlO17/ m3, while nzR,,-,lo-2°/m3 (that is, negligible effect of ionization of the gas. Without the solid particle, nz=ne= 1.8X lo-5/m3). The distributions are shown in Fig. 1. Since the extent of ionization is extremely small, Fig. 1 also applies to the case of a finite vacuum. The curves include: the total electrons included in r* around a solid particle, p* =Ne/Zp; the distribution with respect to the radius, 6 J. D. Cobine, Gaseous Conductors (Dover Publication, Inc., New York, 1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:38GAS-SOLID SUSPENSIONS AT HIGH TEMPERATURES 1691 TABLE 1. Examples of thermal electrification of suspension. Points in Fig. 2 A B T, oK 2500 2500 np, /m3 1016 lOIS d, micron 1 0.1 Z p, holes/particle 442 18 n., /m3 4.42 X 1017 1.8X 1019 Pp, kg/m3 1.3 1.3 dp* I dr* = d (N el Z p) I dr*; the potential distribution, V* =47rt:a V I Zpe; the field, dV* Idr*= d(47rfa2V I Zpe)ldr; the density, n*=nelnea. It is seen that the solid particle is surrounded by an electron cloud of diminishing density, bound by the distributed potential. Ionization of this system occurs when part of the above electron cloud is set free (moved to infinity theoretically). This can be accomplished by an external electric field greater than dV I dr at a given radius, by a magnetic field to produce spin of electrons around the solid particle and attendant drift, or by scattering of gas atoms such as in a turbulent suspen sion. In general, thermal electrification is not identical to ionization of solid particles. From the trend of the field in Fig. 1 it is seen that more than 80% of the bound electrons can be easily set free by disturbances such as turbulence. 2. GAS-SOLID SUSPENSION When applied to a uniform gas-solid suspension of np solid particles per unit volume, the volume ratio of gas to solid is (R*3-1). When applied to a solid of density pp and particle cloud density Pp, R*3", (nppplpp) + 1. (2.1) The mass ratio of solid to gas (subscript g) is (M pi M 0)'" ppl pgR*3. (2.2) The above applies to a uniform suspension for large R* without external field or turbulent field. In a turbulent gas-solid suspension, a sufficient amount of emitted electrons becomes free due to scat tering. The field around a solid particle has to be dealt with statistically, based on average free-electron den sity ne, given by ne=nes exp[ -CPelkTJ, (2.3) where CPo is the equivalent thermionic potential energy. Summing over a field which is much greater than each solid particle-electron cloud system, 5 Zpe2 f n,e2 CP.= cP+-+ :E --d (volume) 4?rr volume 4?rfr. B C D 2500 3000 3000 1017 1015 lOIS 1 1 0.1 180 387 16.1 1.8X 1019 3.87X1017 1.61 X 1019 130 2.7 2.7 where ns is the space-charge density at r. from the solid particle under consideration, and Ro is the Debye Huckel length given by summing over locations i, or R02= (t::E npiZpi)-lkTfe2> (np)-l. (2.5) i Combining Eqs. (1.7), (2.3), (2.4), and the condition of charge neutrality within Ro, that is, (2.6) we get a exp[(cp/kT)+aJ~ncse2/47rmpakT, (2.7) for np(47ra3/3) exp[(cplkT)+aJ»l, which is the usual case of interest. In Eq. (2.7) the magnitude nCB depends on whether the solid particle is a metal or insulator. It is noted that (cplkT)/a>3 in the latter case, there fore, the effect of thermionic emission is not usually negligible. For the example in the previous section, a"-'2.2 was obtained from Eq. (2.7). Hence, the order of magnitude of thermal electrification was not altered by this approximation. Take the case of liquid aluminum particles (cp=3.57 v) ncs'" 1028/m3, we have the situation shown in Table I. Hence, small particles produce greater numbers of free electrons for a given mass of solids. (Items A, B, B.) For the case of zirconia particles, ncs remains to be estimated3 (2.8) where Af is the energy gap ('" 10 eV for insulators, '" 1 eV for insulators at high temperatures and depending on impurities). At Af'" 1 eV, ncs'" 1026/m3• In this case, we have in Fig. 2 the information represented by points C and D in Table I. Hence, for nonmetals, less emission is expected, but not in direct proportion to ncs. Ac tually, similar order of magnitude of electron concen tration is expected, although less in amount. It should be noted that thermal electrification de pends on the temperature of the solid particles pri marily. In a gas-solid system such as that at the exit of a rocket, the temperature of the solid particles could be much higher than that of the gas phase.7 The above examples also show that, in a gas-solid (or gas-liquid) suspension with a distribution in size 7 S. L. Soo, A.1.Ch.E. ]. 7, 384 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:381692 S. L. sao 20 ... ... .. ~ v ~ V 10 / / / 1'/ '-A / / / ICft / / / / '// / 2 / I I o'a / I I I / / / oil 'j 1 I II I I I 5 J J L I I 2 I J II frs 10 15 i i I Iff 10" 10' 10' "ell .,.. ,..ak T "p FIG. 2. Equilibrium thermal electrification. of particles, an estimate can be made on charges on particles of various sizes and the average electron density. 3. REMOVAL OF FREE ELECTRONS BY POSITIVELY CHARGED SOLID PARTICLES Due to the nature of the equivalent thermionic po tential, ionization or thermal electrification can be suppressed or enhanced by the initial charge on the solid particles prior to their release in the gas phase. This suggests the possibility of removing electrons from the jet of a rocket, or the boundary layer at re-entry, thus reducing the interference of the gas jet to radio communication. The initial (subscript i) charge on a solid particle can be represented by: (3.1) Zpi is taken as positive for positively charged solid particles. Taking the general case of introducing charged solid particles into a monatomic ionized gas with ion concentration nr, the resultant electron concentration (initially equal to nr) is: (3.2) Substitution of Eqs. (2.3) and (2.4), and (3.1) into Eq. (3.2) gives, for the same condition of Eq. (2.7), the resultant electron concentration as: ( nr ZPi) n.'" nC8 -npV P -V P Equation (3.3) shows that, to suppress ionization or to remove electrons in the gas phase, an insulator is pre ferred to a metal for the solid particles (lower nrs). Large thermionic potential of solid material and low inlet particle temperature [although effect of cooling is not included in Eq. (3.3)J are also desirable. The above relation gives a conservative estimate of the extent of removal of electrons because recombination at the surface of the solid is neglected here for simplicity . We take, for our example, an ionized gas of n[= 1017, to which l015/m3 of 1-J.L zirconia particles (1.3 kg/m3) charged initially to Zpi is released. The result of calcu lations is shown in Fig. 3, assuming a constant tem perature of 3000°K, It is interesting to note that as long as the solid particles are well (uniformly) dis persed, relatively small charge to mass ratio of solid particles can produce significant modification to electron concentration. The value of n. for initially uncharged solid particles is much higher than in the gas alone because the ions in the gas initially reduces the space charge to which solid particles emit electrons. The reverse case of initially negatively charged solid particles is also interesting in that it enhances ioniza tion in a significant manner. The trend as shown in Fig. 3 probably is the reason for the anomaly of experimental result of water injec tion into a rocket jet, which increases the electron con centration in some cases and decreases it in others.s The charge-to-mass ratio required in the above ex ample is well within the reach of corona charging.9.10 The latter requires the particles to be not too small. Increase of electron concentration by one order of .2 £!l !!IM:~ARGE I 2 I Icr. r'E~~ - ...J u ",I i= ~:;ilf 1~le ~1",15 :z:i i i ZpI 2000 1000 200 100 FIG. 3. Effect of initial charge of solid particles on an ionized gas (data based on example of const temp. of 3000oK, n,i = 1017/ m3, l-~ zirconia particles, np= lou/m3, charged to Zpi). 8 W. W. Balwanz, "Ionization Phenomena," paper presented at the annual meeting of Project SQUID, Univ. of Virginia, Charlottesville, Virginia, 28 February 1962. 9 D. M. Tombs, "Seed Sorting," La Physique des Forces Elec trostatiques (Centre National de la Recherche Scientifique, Paris, 1961), pp. 392-402. 10 A. T. Murphy, F. T. Adler, G. W. Penney, Trans. AlEE 78, 318-326 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:38GAS-SOLID SUSPENSIONS AT HIGH TEMPERATURES 1693 magnitude due to solid particles was observed in MHD generator experiments with combustion gas, which was not recognized as thermal electrification, however. 4. RATE OF ELECTRIFICATION The general case involving rate of thermionic emis sion is more complicated. The thermionic current density would then be affected by the equivalent work function given byEq. (2.4) which becomes time dependent at nonequilibrium. In treating an ionized system as in this case, the conventional magnetohydrodynamic procedures will not be adequate. This is in the sense that different stream lines of gaseous phase and the particle phasell render the usual charge neutrality consideration entirely in adequate. Redistribution of charges by local states of unbalance will render charge to mass ratio of particles and local electron concentration dependent on the dynamics of fluid motion; variables further include local solid particle concentration, local charge density, and rate of emission and electron removal. The non equilibrium in a gas-solid suspension, be sides that in momentum, energy (or temperature of phases), and mass (when chemical reaction is con sidered) transport,1 should include charge transport (a form of chemical process in the general sense) be tween the phases. The rate of electrification before equilibrium is reached can be determined from the electrons escaping at a finite rate in analogy to the loss of planetary atmosphere, with velocity Vc at the surface of the solid particle given by mvN2=Z pe2/47rea. The maximum rate of escape per unit area is given by2: riec= [2/ (7r)lJnea[m/ (2kT)lJ3 1'" 1"/2 X v3 exp( -mv2/2kT)dv sinO cos8dO Vc 0 =nea(kT/27rm)1(1+a) exp(-a). (4.1) The rate of thermal electrification is therefore: dZp/ dt= 47ra2riec = (47reT/e2) ({38-3a) exp[ -(cp/kT)-aJ X (1+a) (kT/27rm)t, (4.2) or, da/dt*= ({3.-3a) (l+a) exp(-a)/{3., (4.3) where t*= {3.(kT /27rma2)1 exp( -cp/kT)t. Integration of Eq. (4.3) gives t*= -'- exp( -1) [exp(y)/yJdy ( (3)[ j(1+a) {3.+3 a 1 j-(,3,I3)+a ] -exp({3./3) -{38/3 [exp(y)/yJdy. (4.4) ---- 11 S. L. Soo, Proceedings of Third Congress of Chemical Engineers, London (Bartholomew Press, Dorking, 1962), p. A40. 4 --7 f! . "" FIG. 4. Thermal electrification as a rate process. In the case of a metal at T=30000K, {38,,,7X106'{3. is slightly smaller for insulators, but, in general, (3.» 1. Hence, Eq. (4.4) reduces to (Ha) t*"'exp(-l) ~ [exp(y)/yJdy. (4.5) Equation (4.5) is plotted as shown in Fig. 4. The value of t* for a metal is given by: t*=texp(-cp/kT) X (47rmkTe2a/eh3). At 30000K and cp",3.5 eV, t*=5 X lOto t. Figure 4 shows that the initial stage of thermal electrification occurs at an extremely fast rate. In the above example, the value of t*"'SX 1010 t. However, it is interesting to note that it takes 10106 sec for a 1 fJ. metallic particle to lose 90% of its electrons in an infinite vacuum. The fraction of electron loss is given by 3a/{38' It is also interesting to determine the heat removed from the solid particle due to thermal electrification, in addition to other energy transports. Where there is no loss of energy due to convection and radiation, each electron emitted requires an excitation of cp eV. The temperature change of a solid particle due to emission is given by: mpcp(dT/dt) = -cp(dZp/dt) = -(47reakcp/e3)d(aT)/dt, (4.6) where mp is the mass of the solid particle and Cp is its specific heat. Equation (4.6) integrates to, for change from state 1 to state 2, (4.7) where K= (mpcpe2/47rwkcp) = (e2a2cppp/3ekcp), where Pp is the density of the solid material. For most solids, K", lOS, hence, the temperature drop due to thermal electrification alone is negligible within the time of most experiments. Equation (4.7) may be considered as the adiabatic equation of thermal electrification. 5. RATE OF REMOVAL OF ELECTRONS BY CHARGED SOLID PARTICLES The rate of removal of free electrons is, of course, an important quantity. The rate is affected by at least two phenomena: one is the rate of dispersing solid particles in a given volume of a gas; another is the rate of removal of electrons by collision with solid [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:381694 S. L. SOO particles. When applied to dispersion through a viscous medium, the equation of spherically symmetric motion should be written as: vdv =(~)2 Mp _~ CDpv2a2, (5.1) dr mp 47r~r2 2 mp where v is the radial (r) velocity, (q/mp) is the charge to-mass ratio, M p is the total mass of solid particles, CD is the drag coefficient of the particles, p is the density of the gas, mp is the mass of a particle. The terminal velocity of the particles without the last term on the right hand side of the above equation was given as: which gives the decay of ne with time for sufficient npZpi to begin with according to Eq. (3.3). For Zpi in the example in Fig. 3, for removal of 90% of initial concentration by Zpi= 2000, (npZpi)taenpt) tanh t "'0.95 8 (mE)! (5.11) or t'" 10-8 sec. Hence, the rate of dispersion of solid particles is the controlling factor in removal of electrons by charged solid particles. 6. ELECTRICAL CONDUCTIVITY OF A SUSPENSION (5.2) WITH THERMAL ELECTRIFICATION R being the initial radius, which gives the total time for particles charged to 0.1 C/kg to take 10-5 sec to be dispersed from a radius of 1 cm to 1 m. Take the motion at relatively high speed such that CD is nearly constant, Eq. (5.1) is integrable in simple form or ( q )2 M [( 1 1) r e{3r' I r ] v2= --p ----{3e-{3r' -, dr' , mp 2rE R r -r I R (5.3) where f3= (r)(CD/mp)a2= (3CDP/4ap p), (5.4) and the latter integral (logarithmic integral) is given by reference 12; this term modifies the above rate of dispersion by the viscous force. For CD'" 1, p/ Pp'" 10-8, f3"'0.06, V is reduced by 5% from that given by Eq. (5.2). The phenomenon of capture of electrons by posi tively charged particles is complicated. The order of magnitude of the event can be seen from a simple model of collision based on the geometrical cross section of the solid particles. The rate of capture can be ap proximated by: (t is the time) dn./ dt= inpra2neC, and the mean speed C2"'2(Ee/m)A. The mean field intensity is approximated by E", (Zpe/rE) (np) I, and mean free path A give:;} by A ........ 1/ (npra2) (5.5) (5.6) (5.7) (5.8) The rate of capture is, by substitution of the above: dn./dt= [denpl/4(mE)l]ne(npZpi-ne)t. (5.9) This equation is readily integrated to: (S.lO) It is interesting to investigate the electrical con ductivity of the above mixture consisting of charged solid particles (of micron or submicron range), elec trons (due to thermal electrification alone), and the gas atoms of the suspending gas. It is seen that the cross section for collision between electrons (subscripts e below) and charged solid particles (subscripts p below) with Coulomb (C) interaction far exceeds that between, say, helium atoms (subscript a) and electrons interact ing with an inverse fifth power relation. Due to large Debye shielding distance in this case, combination of effects of diffuse scattering and space charge should lead to lower electrical conductivity than in an ionized gas of similar electron concentration. The method of calculation of electrical conductivity of a gas consisting of ions, electrons and atoms as presented in reference 13, and reduced by Cann,14 can be simplifi.ed for the present case as: (1= (3/16) (2r/mkT)t(e2/qep), (6.1) where the reference cross section qep is given by: qep=r(e2Z p/8rkTE)2l n(ad)ep (6.2) and the cutoff parameter in C interactions is given by: (aa)ep= l+4(12rEkT/Z pe2)2(€kT/2e2ne). (6.3) Thus, for 0.1 Ji. zirconia particles at lOl8/ma in helium at 30000K and 1 atm (3X lQ24/m3) with n.= 1019/m3, the electrical conductivity of the mixture is 2.8 mho/m and is nearly independent of the pressure of the sus pending gas. This is in comparison to a combustion gas at 30000K seeded with, say, 1% potassium whose electrical conductivity is nearly 60 mho/m at 1 atm,15 but is reduced considerably as the pressure increases_ 12 E. Jahnke and F. Emde, Tables of Functions (Dover Publi cations, Inc., New York, 1945), p. 2. 13 S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, New York ~~. ' 14 G. L. Cann, "Energy Transfer Processes in a Partially Ionized Gas," Memo No. 61, California Institute of Technology Army Ordinance Contracts No. DA-04-495-0rb-1960 and 323i (1961). 16 R. G. Deissler, NASA-TN D-680 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:38GAS-SOLID SUSPENSIONS AT HIGH TEMPERATURES 1695 10" / ~ 1:- V IOrr 2000 2200 2400 2600 2800 3000 Tp. OK FIG. 5. Electron concentration due to thermal electrification by zirconia particles (d=O.l J.', np= 1018). For the electron concentration of the above gas-solid mixture as given in Fig. 5 at different temperatures, curve A in Fig. 6 was calculated. From the point of view of C scattering of electrons by solid particles alone, a large reference cross section suggests higher electrical conductivity as the charge on solid particles decreases. In general, (6.4) and (6.5) In the case illustrated, a'>!, and therefore the elec trical conductivity increases with decrease in tempera ture from the range considered. However, for the given pressure of the suspending gas, as the influence of C scattering decreases, the elastic scattering of gas atoms becomes more significant. The electrical conductivity may then be approximated by: (6.6) with (6.7) Equation (6.6) gives the electrical conductivity as shown by curve B in Fig. 5. This suggests an optimum electrical conductivity in a medium due to thermal electrification at an intermediate temperature below the boiling point of the solid particles. It is known that in an electric field E, a spherical dielectric particle such as zirconia will polarize with the surface charge density of 3EE cose, where e is measured from the direction of the field.ls It can be shown that for an 0.1-~ particle, the chance of polari zation with one electron is no more than 10-4 for a field of 100 V /m while for the above example of 0.1-~ zir conia, the total charge is 10 holes per particle (or a charge to mass ratio 0.32 C/kg) j therefore, no signifi cant effect on thermal electrification can be expected from polarization of solid particles. 16 J. D. Jackson, Electrodynamics (John Wiley & Sons, Inc., New York, 1962), p. 461. For submicron particles in a high-temperature me dium, one can also expect sufficient partition of energy in the rotational degrees of freedom. For O.l-~ zirconia particles at 30000K the mass is 5 X 10-8 kg and the moment of inertia amounts to (t)XIQ-32 kg-m2, the root mean squared angular velocity (w2)1 amounts to . 3 X 106 rad/sec for each degree of freedom (partition of mean translational energy gives rise to a velocity of no more than 10-1 m/ sec). The energy of the solid is still due to its temperature; its ratio to that of kinetic partition is mpNo/M, where mp is the mass of each particle, No is the Avogadro number, M is the molec ular weight (this ratio is 2.5X 107 for 0.1-~ zirconia particles). The rotational random motion further elimi nates the effect of polarization of a particle in a field. 7. DISCUSSION The above study shows that besides the interesting features of a gas-solid system as presented earlier/·ll thermal electrifIcation is an important phenomenon when dealing with solid propellant rockets and MHD propulsion and generating systems. In the intermediate range of temperature where solid or liquid particles can exist, thermal electrifica tion is a signifIcant contributor to electron concentra tion in a gas-solid system. Thermal electrification can be controlled by the initial charge on the solid particles: positive charge on solid particles can suppress ioniza tion of a hot gas, while electrically neutral solid par ticles or negative charge on solid particles initially can promote ionization at a given temperature. Important application here is the removal of electrons from an ionized gas by positively charged particles. It has also been shown that, other conditions being the same, insulators are useful for electron removal, and metals are useful to promote ionization. The interesting feature of electrical conductivity of FIG. 6. Electrical conduc tivity due to thermal e1ec- lIE trification and comparison a to combustion gas seeded i with potassium. .. 10 1 01 J / I 1 I ~- "Ru(Q-I IATM. "\. / "- ~ /B(lA™I " [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:381696 S. L. SOO a gas-solid suspension has been demonstrated. It is seen that thermal electrification alone cannot contribute to high electrical conductivity of a gas-solid suspen sion. However, solid particles such as due to ash or soot from combustion or deliberately introduced par ticles of controlled size contribute favorably to MHD generation. Presence of solid particles in plasma MHD . accelerators in general would reduce the performance of such thrust producing devicesP Presence of the \7 Note: Patents pending concerning deionization and MHD generation with gas-solid systems. solid particles, in general, increases the electron con centration in an ionized gas. Even at moderate tem peratures (20000K) where the gas phase is not ionized, presence of solid particles and its concentration of electric charge produces large acceleration of free elec trons in the gas, and thus may produce electromag netic radiation of high frequency.8 ACKNOWLEDGMENT The author wishes to thank Professor E. A. Jackson for discussion with him. JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 6 JUNE 1963 Ferrimagnetic Resonance Linewidth in Dense Polycrystalline Ferrites P. E. SEIDEN* IBM Research Center, Yorktown Heights, New York AND J. G. GRUNBERG Centre d'Etudes Nucleaires de Grenoble, Grenoble, France (Received 30 March 1962) The linewidth of polycrystalline ferrites has been calculated from the dipole narrowing theory including both anisotropy and porosity effects. It is shown that porosity broadening is successfully accounted for and that porosity is the dominant line broadener even in very dense ferrites. The temperature dependence of the linewidth for samples of yttrium iron garnet and manganese ferrite are in good agreement with the predictions of the theory. IN this paper we want to consider the contributions to the linewidth in dense polycrystalline ferrites. It has previously been recognized that the porosity of the material is a determining factor in materials whose densities are low with respect to the theoretical den sity; however, it has been assumed that for dense samples porosity effects are negligible and the crystal line anisotropy of the material determines the linewidth. We show here that for a majority of common ferrites one must still consider porosity broadening to be a large and even a dominant factor in determining the polycrystalline linewidth in samples having porosities of less than 1%. For the majority of dense ferrites except those wit.h very low saturation magnetizations one finds that the mean square fluctuation field (H F2), which is respon sible for both the anisotropy and porosity broadening effects in the material, is small compared to the satura tion magnetization. This means that dipole narrowing effects are important1-a and therefore we calculate line- * Part of this work was done under a National Science Founda tion Postdoctoral Fellowship at the Institute Fourier of the Uni versity of Grenoble, Grenoble, France. 1 S. Geschwind and A. M. Clogston, Phys. Rev. 108, 49 (1957). 2 E. Schlomann, J. Phys. Chern. Solids 6, 242 (1958). 3 P. E. Seiden, C. F. Kooi, and J. M. Katz, J. App!. Phys. 31, 1291 (1960). widths including this effect. This problem has been considered by both Geschwind and Clogston1 and by Sch16mann.2 They find that when 4trM»(H F) one ob tains a linewidth narrower than that expected from the simple independent particle inhomogeneity broaden ing mode1.4 Their calculations give the dipole narrowed linewidth as1 (1) where 4trM is the saturation magnetization and J is a shape factor which for the spherical samples considered here is Ho being the resonant magnetic field. We consider the fluctuation fields as arising from two sources, the first being the crystalline anisotropy. The mean square fluctuation field due to anisotropy has been calculated by Schlomann.2 He finds that (3) 4 E. Schlomann, Proc. Coni. on Magnetism and Magnetic Materials (1956), p. 600. Raytheon Research Division Technical Report R-15 (15 September 1956). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.240.225.44 On: Sat, 20 Dec 2014 15:18:38

1.1735965.pdf

Theory of Tunneling Evan O. Kane Citation: J. Appl. Phys. 32, 83 (1961); doi: 10.1063/1.1735965 View online: http://dx.doi.org/10.1063/1.1735965 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v32/i1 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJ 0 (; R N ,\ L 0 F ,\ P P L I Ii n P H Y S T C S VOLUME 32. NITMRER I J ,\ N 11 ,\ R V. I I) Ii I Theory of Tunneling EVAN n. KANE Semiconductor Materials Department, Hughes Research Laboratories, Malibu, California (Received June 6, 1960) The theory of "direct" and "phonon-assisted" tunneling is reviewed. Theoretical I-V characteristics are calculated using the constant field model. Generalizations to non constant field and more coml"llicateJ bln:J structure models are discussed briefly. I. HAMILTONIAN AND WAVE FUNCTIONS THE theory of tunneling has been investigated by a number of authors,! most recently by Franz,2 Keldysh,3 the author,4 and Price.5 The problem is most simply discussed by making the assumption of a con stant field. The Hamiltonian for an electron may then be written4 The Hamiltonian is written using a Bloch function basis; En (k) is the band energy, F the constant force on the electron taken to be in the x direction, and X nn' is the interband matrix element of the position operator. 6 E is the eigenvalue, or total energy of the state. In the more familiar position representation the wave func tion is rf>(r) = L cxn(k)if;nk(r), (2) n,k where the if;nk(r) are Bloch functions. If we ignore the term containing X nn', a complete set of eigenfunctions of the Hamiltonian may be written down I i ikX } qn(k)=NntexPIF 0 [E-En(k')]dkx' Xo(ky-kyo)o(kz-kzo). (3) The "perpendicular" components of k, kyo, and kzo are constants of the motion. N n is a normalization factor. The functions qn(k) are "localized" to a single band, since we have neglected the interband coupling X nn'. If this term is now included using ordinary time-depend ent perturbation theory, a Zener tunneling between bands results.4 1 C. Zener, Proc. Roy. Soc. (London) 145, 523 (1934); W. V. Houston, Phys. Rev. 57, 184 (1940); G. H. Wannier, Resumes Intern. Congr. Sol. State Phys., Brussels 89 (1958); Phys. Rev. 100,1227 (1955); 101, 1835 (1956); P. Feuer, ibid. 88, 92 (1952). 2 W. Franz, International Conference on Semiconductors, Gar misch-Partenkirchen, 1956) (Interscience Publishers, Inc., New York, 1958), p. 317. Z. Naturforsch. 14a, 415 (1959). 3 L. V. Keldysh, Soviet Phys.-JETP 6(33), 763 (1958); 34(7), 665 (1958). 4 E. O. Kane, J. Phys. Chern. Solids 12, 181 (1960). 5 P. J. Price and J. M. Radcliffe, IBM J. Research Develop. 3, 364 (1959). 6 E. N. Adams, J. Chern. Phys. 21, 2013 (1953). The "tunneling functions" qn(k), are most easily understood with the use of Eq. (2) and an application of the method of stationary phase.4 The method is valid when the motion is semiclassical, in this case when the electric field is small. Stationary phase gives results reminiscent of the WKB approximation; in fact, the WKB equations can be derived by the method of stationary phase, using Feynman's wave propagation formulation of quantum mechanics.7 The "tunneling functions" may be written Qn(r) = L qn(k)if;nk(r) k ~Nn!(Lx) (27rF/ JEn)! un(k,r) 27r Jkx (V)t XexPij fX kx'dX'+kYOY+kzoz} (4) (5) un(k,r) is the periodic part of the Bloch function. Lx is the length of the junction in the x direction and V is the volume. In Eq. (4), kx is understood to be a function of x through the energy conservation relation, Eq. (5). The solutions of Eq. (5) in the forbidden band involve imaginary kx so that the exponential phase factor in Eq. (4) leads to attenuation. When k is real, ±k are both solutions for a given x and Qn(r) should be written as a sum of two terms. Qn(r) is then a standing rather than a running wave. 83 The semiclassical nature of the approach is evident in Eq. (5). Quantum mechanically, kx and x cannot be simultaneously determined, hence Eq. (5) is only mean ingful in the classical limit. II. DIRECT TUNNELING "Direct" tunneling can occur between two extrema located at the same point in k space. The "forbidden" gap is bridged by proceeding along the imaginary k axis as shown in Fig. 1. The two bands come together at a branch point kB, on the imaginary axis. One may think of the electron as penetrating the forbidden gap along the imaginary k axis and making a "smooth" transition into the other band at the branch point. 7 R. P. Feynman, Revs. Modern Phys. 20, 367 (1948). Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsEVAN O. Kl\NE , , ~i}c conduction band valence bond k FIG. 1. E vs k illustrating "direct" and phonon assisted ("indirect") tunneling. The dashed lines in the forbidden gap represent imaginary k bands, thought of as perpendicular to the plane of the paper. The branch point kB is the point of stationary phase for direct tunneling. k, and k,' are the points of stationary phase for indirect tunneling. They are the points between which the phonon scattering takes place . .More rigorously, we use the wave functions of Eg. (3) and treat the last term in the Hamiltonian of Eg. (1) as a perturbation. This treatment is carried out in the work cited in footnote 4. The results are most easily adapted to the tunnel diode problem by determining the transmission coefficient T of an electron striking the junction barrier. In the constant field case, the electron cycles repeatedly through the Brillouin zone. The period to is given by to=IiK/F. (6) K is the width of the zone in the x direction, assumed to be a principal vector. The transmission coefficient is given by T=wto, (7) where w is the transition probability per unit time, computed in the work cited in footnote 4. The result for T may be written 11"2 1 fX' T=-exp -2 Kdx; K=ik 9 Xl 11"2 r-1I"m*!EG1} T=-exp\ exp{ -2Ei/Ei} 9 2V1hF El =h2(kl+k z2)/2m* Ei =V1hF /1I"m*~EG!. (8) (9) (10) (11) In Eg. (8), Xl and X2 are the classical turning points (see Fig. 2). Except for the factor 11"2/9, the transmission is just what would be intuitively expected for a "smooth" transition through the branch point. K is to be determined from Eg. (5), the transition from con duction band to valence band occurring at the branch point. The factor r/9 does not actually disagree with the WKB picture, since the coefficient (iJE/iJk)-! in Eq. (4) is zero at the branch point. Consequently, the simple matching approach is indeterminate. The effect of "perpendicular energy" is to further reduce the transmission, since the tunneling gap is larger. "Perpendicular momentum" is conserved in direct tunneling. To obtain the tunneling current for the tunnel diode, we use the simple model of a diode shown in Fig. 2. We compute the incident current per unit area in the energy range dr~xdEi v_v +E' E -~2k2/2 L'.J-~x 1, x-flt x mx (12) (13) By taking m* as isotropic and equal for the nand p sides, and by using Eqs. (9) and (13), we obtain the tunnel current per unit area, it: We have used Eq. (12) to give E, Ei as the variables of integration rather than Ex, Ei. The limits of integration are determined by the conditions O~Ei~El (15) O~El~E2, where El and E2 are the electron energy measured from the nand p band edge, respectively, as shown in Fig. 2. hand fz are occupancy factors. The limits on E are given by the band edges. The integral over Ei can be carried out immediately with the result 't= em* exp 1-1I"m*!Eo!} (Ei) J 18h3 I 2V1hF 2 Xf [h(E 1)-h(E2)][1-exp(-2E s/Ei)]dE, (15') where Es is the smaller of El, E2• We now give complete formulas for T=OoK, so that h, fz are step functions.8 The symbols are as shown in Fig. 3. All quantities are positive except D and V which are positive for forward bias and negative for reverse bias. D= f [l-exp( -2Es/Ei)]dE, eV~t min-t max D=eV+(Er/2){exp( -2t n/Ei)+exp( -2t P/Ei) (16) -2 exp[(eV -t n-t p)/Ei]}, (17) 8 Similar expressions have recently been derived by R. Stratton. Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsTHEORY OF TUNNELING RS rUlil1-rmA.X~e V ::;~Iflin, rlllax- S1I1ilt D=eV -(Ed2){cxp{2(eV -tmin)/E\} -cXp(-2tmin/E 1)}, (18) t rnin:SeV:St max-tmin (19) S max-t min:S e V:S S min D=eV + (El/2){2 cxp[(eV -S n-S p)/ E1J -cxp[2(e V -t n)/E\]-exp[2(e V -t p)/E1]}, (20) tmin, rmax-tmin::;eV::;rmax D=t min+(Ed2){2 exp[(eV -t n-S p)/E1J -exp[2(eV -S max)/E1J-1}, (21) D= (t n+t p-eV) +El{exp[(eV-tn-sp)/El]-l}. (22) S min and t max refer to the smaller and larger of r n, S p, respectively. The composite pieces of the characteristic join continuously with continuous slope so that the transition from one region to another is not very marked. Eq. (19) shows that for r max> 2s min, D has a "£lat topped" form as a function of voltage. However, F also varies with voltage which affects D slightly through El and which also affects the main tunneling attenua tion factor. For forward bias, F decreases so the I-V characteristic will "droop." Consider the case E1«eV, r min. The exponential terms in Eqs. (17)-(22) may then be ignored. The "D" factor has a trapezoidal form as a function of voltage as shown in Fig. 3. (The corners of the trapezoid are rounded over a range et:.V,,-,El in conformity with the continuous joining property mentioned above.) The opposite limiting case is E1»smin, eV. The ex ponentials may then be expanded to lowest nonvanish ing order. These expansions are calculated below. FIG. 2. Constant field model of a p-n junction with illustration of symbols. o FIG. 3. Effective .density of states" D" ys voltage for "direct" or "indirect" tunneling with El very small. eV:Ss min-S max D= (tE1-l)[2(t n-t p)2+4(t n+t p)eV -2(eV)2], (17') Smin-Smax:SeV:SSmin, Smax-Smin D= (eVjEl)[2tmin-eV] (18') D~s min2jEl (19') S max-S min:S eV:St min D= (!E1-l) [2 (tn+Sp)eV -3(eV)2- (tn-Sp)2] (20') S min; S max-S min:S e V:SS max D= CtE1-l)[2S minL 2s max2+4t nS p -4eVCSmin-Smax)-2(eV)2] (21') t max:S e V:S S max+t min D= (!E1-l) (Sn+Sp-eV)2. (22') The maximum value of D for forward bias occurs for eV=Smin when Smax~2Smin and for eV=(Sn+Sp)/3 when S max:S 2s min. The quantity D is shown in Fig. 4 for the case Sp=Sn and smax=3s min. We now give a few numerical estimates. For an abrupt junction the width W is given by (23) where K is the dielectric constant, nand p are the majority carrier concentrations, and V is the potential difference between the nand p sides. By taking n=p=1019jcc, K=16, and V=1 v, we compute W= 1.9X1Q-s cm. By defining an average F by WF= eV, we have F=5.3X105 ev/cm. By taking mn*=O.04m, mp*=0.4m, and EG= 1 ev we find E1=0.033 ev; Sn=0.42 ev; Sp=0.042 ev; jmax= 3X 105 exp( -15) ampsjcm 2. An increase in the forward bias by 0.05 v would reduce F by 2.5% and cause a decrease of a factor 1.5 in the Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions86 EVAN O. KANE D o .2 .4 . 6 o .4 .8 1.2 -.. .. , , .8 , , , , , , , , 1.0 \ , \ \ \ \ 1.2 \ \ \ , ----DIRECT --INDIRECT , , , , I . , .. .... ... ...... -VOLTAGE; eV/~n (a) - - -DIRECT --INDIRECT ~max.· 3~min. . . , , , , , , , \ , \ \ , , , , .. .. .. 1.6 2.0 2.4 2.8 3.6 eV/~min. (b) 2.0 4.0 FIG. 4. Effective density of states "D~ vs forward voltage for "direct" and "indirect" tunneling with El very large. (ak p=t n; (b) tmax=3tmin. exponential factor. Hence in this case the I-V charac teristic would not be very "flat-topped." The negative resistance in I vs V would result from the tunneling attenuation factor rather than the density of states. To minimize the negative resistance it may be desirable to eliminate the "flat-topped" region of D by doping so that tn=tp. In the above example we have used m*=mn* in the tunneling attenuation factor. The very much larger value of mp does not affect the tunneling which is governed by the light hole mass, mpZ (mpz=mn in the approximation we are using). The heavy valence mass is here principally important in determining the Fermi level. The large value of t n in the preceding example is related to the low n-type mass. If t n exceeds the band gap, the energy bands cannot actually be uncrossed. This circumstance can occur in lnSb and, perhaps, in other III-V's as well, although a large mass conduction minimum lying less than Ea above the (000) minimum will tend to prevent it. Negative resistances will still be found, because of decreasing barrier transmission with increasing forward bias. III. INDIRECT TUNNELING The "indirect" type of tunneling will occur when the energy band extrema are not located at the same point in k space. This situation is illustrated in Fig. 1. The tunneling states extend into the forbidden gap as before but do not meet at a common value of k. Tunnel ing can occur if the momentum difference between the states is supplied by a scatterer such as phonons or impurities. The phonon case has been calculated by Keldysh3 and by Price.5 We adapt Keldysh's result to the tunnel diode by determining a transmission coefficient as in the direct case. [ [-4(2mrx)! ] X (n+l) exp 3hF (EaOFhw)! { -4(2mrx)! }] +n exp 3hF (Ea±hw)! (24) E 1 = JiF j2V2mrxt Eat (25) Ji2 (kiy-kOiy)2 Ji2(kiz-kOiz)2 Eli = + . (26) 2miy 2miz n is the phonon occupation number. kOI and k02 are the k vectors of the extrema. The upper signs in Eq. (24) are for n to p tunneling (forward bias), the lower signs for p to n tunneling (reverse bias). The reason for the difference may be seen qualitatively by noticing that for phonon emission with forward bias the classical turning points are closer together than for phonon absorption, while for reverse bias the opposite is true. In M2'U, 'U is the volume. M is the phonon matrix element for scattering from one extremum to the other at the imaginary k vector kxs for which the scattering takes place. M(kol+k xs; k02+kxs) kxs= -i(2mrx)!Ea!jlt (27) (28) Eq. (28) for kX8 is given by the method of stationary phase. The x component of phonon wave vector is in the vicinity of kOl:&-k02x. Since kx is not a constant of the motion, a range of phonons with different x momen tum can take the electron from a given initial state to a given final state. These phonons have been summed over in deriving Eq. (24). On the other hand, Eq. (24) refers to a transition between an electronic state of definite perpendicular momentum in band 1 to an Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsTHEORY OF TUNNELING 87 electronic state of definite perpendicular momentum in band 2. The perpendicular momentum of the phonon is then exactly the difference of the perpendicular momenta of the two states. T should be summed over phonon polarizations and over the optical and acoustical phonon branches. Account should also be taken of the multiple conduction minima and the orientation of the mass ellipsoid with respect to the field. In our calcula tions we ignore all but one ellipsoid which we take to have the optimum alignment for tunneling, namely, with the transverse (low) mass in the junction field direction. The parabolic approximation for the energy bands was used in deriving Eq. (24). In computing the tunnel current, we integrate T over all possible states of differing Eu into which transi tions can be made. This gives the total transmission factor for a single electron from side 1. The total T is then multiplied by the differential incident current [Eq. (13)J from side 1 and integrated again as in the direct tunneling case. We require the density of "perpendicular" states on side 2, P12 dn12= P12dE12 P12= A (mY2mz2)~/27r-h2. (29) (30) A is the area of the junction. Using Eqs. (13), (24), and (30), we obtain a triple integral of the form f [j1(E1)-fz(E2)J Xexp[ -(Ell+El2)/ElJdEdElldEl2. (31) The integrals over Eli can be performed immediately, giving (El)2f [h(E 1)-fz(E2)J[1-exp(-E 1/El)J X[1-exp( -E2/El)JdE. (32) EI and E2 are the energies of the state measured from the respective band edges as shown in Fig. 2. The final result may be written e(El)2(mxlmylmzlmZ2my2mz2)tM2'U j { [-4(2mrx)! ] XD (n+l) exp 3liF (EGTIiw)! r -4(2mrx)! } } +n eXPl 3liF (EG±liw)! D= f [h(E 1)-fz(R2)J[1-exp(-EJ/E 1)J X[l-exp( -E2/E1)JdR. (33) (34) D is evaluated below for the case T=OoK so that the distribution functions h, 12 are step functions. In deriving the formulas, we have ignored the phonon energy. All quantities are positive except D and V which are positive for forward bias and negative for reverse bias: eV:::;frnin [ (eV-f -f )] D=eV 1+exp E: p +E1{ exp( -f n/El)+exp( -f iEl) -exp[(eV -f n)/EIJ-exp[(eV -f p)/ElJ} (35) f min:::; eV:::;f max D=f min{1+eXP[(eV -f n-f p)/ElJ} +El{eXp( -f min/El)+eXP[(eV -t n-fp)/El]-l -exp[(eV-fmax)/ElJ} (36) fmax:::;eV:::;fn+fp D= (f n+f p-eV){l +exp[(eV -f n-f p)/ElJ} +2El{exp[(eV -f n-f p)/E1J-l}. (37) As in the direct case, the composite pieces join together with continuous value and continuous slope. In deter mining the I-V characteristic, account should also be taken of the field dependence on forward bias which most strongly affects the exponential tunneling factor. Increasing forward bias causes the tunneling factor to decrease. Consider the two limiting cases of the above formulas. For the case El«f n, f p, leV I the exponential terms may all be ignored and a trapezoidal form for "D" results as in the direct case (see Fig. 3). The "corners" of the trapezoid are rounded over a voltage range el1 V",E l. In the opposite limiting case E1»f n, f p, leV I the exponentials may be expanded to the lowest non vanishing order with the results (35') (36') (37') The maximum in the forward characteristic comes at (38) The maximum value of D is The "indirect" D factors have a stronger voltage de pendence than the "direct" D factors because of the Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions88 EVAN O. KANE extra "perpendicular" degrees of freedom. No "flat topped" region of D vs V exists in the "indirect" case. Instead the D factor decreases linearly with voltage in the region t min <eV <t max· The "trapezoidal" limiting case is shown in Fig. 3. The case of Eqs. (35')-(37') is shown in Fig. 4 for tmin=rmax and rmax=3 min o We now give a few numerical estimates, roughly applicable to germanium. n= p= 1019/cc; K= 16; eV=EG=0.78 ev; W=1.7XlO-6cm; F=4.7X105ev/cm; tn=0.031j tp=0.048 ev. We estimate M2'U from the magnitude of indirect optical absorption to be M2'U=4.3X10-49 erg2 cm3.9 [Optical absorption gives M2'U for scattering from the (000) to the (111) conduction band minimum. What we really need is M2'U for scattering between extrema at k values near the middle of the forbidden band. Use of the above value is obviously extremely rough.] The current should be summed over all ellipsoids. We assume that just one ellipsoid is most favorably aligned for tunneling and take mxl = myl = 0.08m; mzl = 1.58m; mx2= my2= mz2= 0.04m. Very roughly, we set D= t n3/MiJ~2 and compute E1=0.031 ev; j=3 exp( -17) amp/cm2• The factor in front of the exponent is observed to be ~4X 103 smaller than the coefficient in the "direct" case previously computed. For III-V's the phonons may be relatively more important, because the polar character gives a larger scattering matrix element.lO Although we have ignored the phonon energy in deriving expressions for D, it can be important. At low temperature, no "indirect" current caused by phonons can flow until the forward bias is 2 the phonon energy. The experimental observation of phonon energies has provided the most conclusive proof of the importance of phonons in tunneling processes and has given as well very accurate measurements of the phonon energies themselves.u The maximum phonon assisted current densities (exp~1) are considerably less than maximum observed current densities which implies that phonons are not important in these cases. This statement is proved more directly by experimental observation of the phonons. The impurities giving high current densities, P and As 9 This value corrects the estimate given in footnote 12. 10 H. Ehrenreich, J. Phys. Chern. Solids 2, 131 (1957). 11 N. Holonyak, Jr., 1. A. Lesk, R. N. Hall, J. J. Tiemann, and H. Ehrenreich, Phys. Rev. Letters 3, 167 (1959). More extensive data, as yet unpublished, were reported hy R. N. Hall, Bull. Am. Phys. Soc. Ser. II 5, 38 (1960) and at the January, 1960 Meeting of the American Physical Society, the West Coast Tunnel Diode Symposium of the AlEE-IRE; L. Esaki and Y. Miyahara, Solid State Electronics 1, 13, (1960). 4-_---....,.L! 3 FIG. 5. Tunneling with energy loss mechanism can lead to "excess" current. Electron one is scattered by coulomb-coulomb interaction into state three. Excess energy is given to electron two which is scattered into state four. in germanium, are also those for which the phonons are observed to make a small contribution to the total current,u The contrast between "direct" and "indirect" tun neling has been observed in the reverse characteristic of germanium.12 Because "direct" tunneling has a much larger prefactor than phonon assisted tunneling, an abrupt rise in tunneling current is noted when the reverse bias is sufficient for electrons from the valence band to tunnel into the (000) conduction band minImum. IV. GENERALIZATIONS The foregoing theory contains a number of approxi mations. Some of these can be improved upon fairly easily while others are more difficult. We give a rather brief discussion of some of these approximations. A. Band Structure The "direct" tunneling calculations were made on the basis of the "two band" model which applies fairly well to InSb but less accurately to the other III-V's and germanium. This approximation can be improved on by going to the "three band" model where the "k· p" interactions between the conduction band, the low mass valence band, and the spin-orbit split off valence band are all treated exactly.13 The formulas of the work cited in footnote 13 may be analytically con tinued into the complex plane to give the necessary E vs k for the tunneling calculation. Using Eq. (4) of the work in footnote 13 we obtain the secular equation for E vs k as E3+E2(il-EG)-E( k2P2+~il2+ilEG) -jilk2P2=O. (40) 12 J. V. Morgan and E. O. Kane, Phys. Rev. Letters 3, 466 (1959). 13 E. O. Kane, J. Phys. Chern. Solids 1, 249 (1957). Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsTHEORY OF TUNNELING 89 ~ is the spin-orbit splitting. P may be determined from the relation J>2( 2 1) Hmn*)-l=3 EG + EG+~ . (41) In Eq. (41) 11m is neglected as small compared to l/m*. In Eqs. (40) and (41) higher bands are ignored. In germanium, higher bands contribute about 10% to l/mn*. A plot of E vs k2 is given in Fig. 6. The method can be generalized to "n" bands if the information on band gaps and momentum matrix elements is available. Strictly speaking, the three band approximation is a four band approximation, since the heavy mass band should also be considered. However, the heavy mass band does not interact via the k·p perturbation with the other three bands. It will have a branch point con nection with some higher band but may be ignored for most purposes in tunneling calculations. Indirect effects of the heavy mass band such as determination of the Fermi level will, of course, be important. In calculating the tunneling current with the new band structure, the tunneling attenuation factor of Eq. (8) must be recomputed with the use of Eq. (5). A new value of El must be determined from the relation : -2 -2 E in eV conduction band EG = .9 eV fj, = .3 eV (a) E in eV split off band conduction band E = .9 eV JO. : 87 eV (b) (42) FIG. 6. R vs ,,2 for "three hand" model showing connection between conduction band and low mass band at branch point (subscripted B). In addition, the values of Din Eqs. (17)-(22) must be recalculated, since the light hole mass no longer equals the electron mass. The more general expressions for D are given in the Appendix in Eqs. (17A)-(22A). The current is then em* { i jt=--exp -2 18h3 x (43) In the indirect case, present calculations have as sumed parabolic bands. This assumption is rather in accurate in germanium. To improve on the calculation, we must first note that the scattering between minima may be thought of as taking place at the point of stationary phase, k., Xs which is given by En (ks) = En' (k/)±hw (44) ks-ks'=rr (45) (46) In Eq. (46) we have generalized on Eq. (5) by writing the potential energy as V (x). (J' is the separation of the energy extrema in k space. The F in the prefactor of Eq. (24) is to be calculated at the point of stationary phase x •. The value of mxl and mx2 in Eq. (24) must be calculated at the point k •. The tunneling attenuation factor must be recalculated as in the direct case using more accurate En(k) in Eq. (5). In the integral f~i Kdx, the transition from band n to band n' occurs at the point ks, x8• El must also be recalculated as in the direct case. No change in D is required. With these corrections, Eq. (33) may be used to compute the current. It should be noted that there are two types of quanti ties appearing in Eq. (33). The first type, which pertains to the scattering during the act of tunneling, comes from Eq. (24) and is to be evaluated at the point of stationary phase. The second type of quantity comes from bound ary conditions imposed by the field free material on either side of the junction and is to be evaluated in the field free region. In the case of germanium, the principal nonparabolic effects in indirect tunneling will come from the valence band. The use of Eq. (40) should considerably improve the calculation. The point of stationary phase ks will actually occur above the branch point so that the electron will go from the (000) light mass valence band into the (000) conduction band and then be scattered into the (111) conduction band at the point of stationary phase k.,. This case is shown in Fig. 1. The transition through the branch point will introduce a factor 7r2/9 as in the direct tunneling case. Between the branch point ku and the point of stationary phase k. the F"Ck) in Eq. (47) should be that of the (000) conduction band. Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions90 EVAN O. KANE FIG. 7. Forward tunnel diode characteristic showing "excess" current. B. Variable Field If the derivatives of the field can be ignored, the preceding work can be generalized to nonconstant field by simply replacing Eq. (5) by (47) in all calculations. This generalization is important, since in forward biased tunnel diodes the field is far from constant. If more general band structure is used, the use of a nonconstant field presents no further difficulty, providing we make the approximation that we can use an average field in determining D. If this is not permissible, the integrals in Eq. (14) and Eq. (31) must also be evaluated numerically. C. Field Effect on Band Structure The electric field modifies the band structure through the term Xnn, in Eq. (1). This effect is discussed in the work in footnote 4. In a material lacking inversion symmetry a finite X nn may exist leading to a band shift linear in F. This effect may be of either sign, but is probably too small to be of importance. The quadratic shift may be larger. Here bands effectively "repel" each other so that gaps increase at high fields. Calculations for InSb and germanium indicate that the quadratic shift is not too important for the fields of interest in tunnel diodes. D. Other Effects Franz2 has estimated the effect of electron-hole attraction on tunneling. The effect is not important in the materials of present interest in tunnel diodeB. Typical tunnel diode fields are so large that the approximation of stationary phase is becoming in accurate. In this case a direct numerical evaluation of Eq. (7) in the work in footnote 4 may be in order. If the approximation of stationary phase becomes bad, then the high degree of cancellation which makes numerical approaches difficult will largely disappear. No theory yet exists for computing the effects due to derivatives of the field. Discontinuities in the field such as that at the junction boundary may be treated by matching. The matching approach is most easily handled when the accurate wave functions in k space can be converted into equations in r space by the method of stationary phase as was done in Eq. (3). V. EXCESS CURRENT "Excess" current is the name given to the diode current occuring for voltages where the energy bands have been "uncrossed" so that energy conserving tun neling processes are no longer possible (see Fig. 7). At high temperature or low doping, the steeply rising part of the characteristic at high voltages is caused by internal thermionic emission in accord with the familiar rectifier equation. At low temperature or high doping a steeply rising characteristic remains which cannot be attributed to thermionic emission. The possible mechanisms leading to excess current may be divided into two classes: that which requires states in the forbidden gap and that which does not. A. No Gap States If gap states are ignored, "excess" current can flow only if the electron can lose energy in the act of tun neling. The possible energy loss mechanisms are: photons, phonons, electrons, or plasmons. 1. Photons The photon mechanism has been calculated by the author and by Pricel4 and has been found to be too small to be important. 2. Phonons Since typical "excess" current biases are much larger than phonon energies, energy loss by phonons must be of the "multiphonon" type. In germanium, these processes are most probably unimportant. In silicon, however, two phonon processes have been identified, both in opticap5 and tunnel diodell work. These observa tions suggest that three or even higher numbers of phonons may be observable in silicon. Also, phonons have been observed in direct materialsll where none are needed. However, it seems doubtful that mUltiphonon transitions can be responsible for "excess" currents comparable to "normal" tunnel currents. 14 P. J. Price (unpublished). . 15 B. N. Brockhouse, J. Phys. Chern. Solids 8/ 400 (1959). Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsTHEORY OF TllNNELING 91 3. Electrons and Plasmons The electron-electron energy loss mechanism can be described as follows: Two electrons tunnel into the junction region; one drops into a tunneling hole and gives up its excess energy to the other electron (see Fig. 5). This mechanism has been calculated and has been found to be too small to explain the "excess" currents which are usually observed.16 The plasmon mechanism has not yet been calculated. Typical plasma energies are large (",0.1 ev), so that if this process is important it might be identified by its energy threshold. Plasmon and electron-electron interactions are both mitigated by the fact that electrons are excluded from the junction region. B. Gap States It seems likely that most excess current is in some way associated with the presence of gap states. In an impure material there is no sharp band edge; the density of states "tails off" into the forbidden band in a manner which is not quantitatively understood at present. In addition, imperfections such as dislocations will introduce states into the forbidden gap. Disloca tions surrounded by precipitated material would be particularly important sources of gap states. Some correlation of "excess" current with dislocations has been observed at reverse biasesY "Excess" current mechanisms involving gap states contain two essential ingredients, either of which may turn out to be rate limiting. The first ingredient is the rate of transport through the states; the second ingredient is rate of energy loss. The injected carriers ultimately end up at the Fermi level on the p side having lost an energy equal to e V where V is the forward bias. If the transport rate through the gap states is high, conduction may proceed by this mechanism and the energy loss mechanisms may not be rate limiting. If the transport rate through the gap states is low, they will not conduct current and the current flow will depend on the rate at which the electron can lose energy and descend into a higher mobility state. The "mobility" of gap states would be expected to be a minimum near the center of the gap. Energy loss through gap states may occur through a succession of first order phonon emissions. The energy loss mechanism would then be aided at high tempera ture, probably proportional to T. If energy loss is rate limiting, some correlation be tween "excess" current and carrier lifetime may be expected. 16 E. O. Kane (unpublished). 17 A. Goetzberger and W. Shockley, Bull. Am. Phys. Soc. Ser. II 4, 411 (1959). APPENDIX In the evaluation of the density of states factor "D" for direct tunneling [Eqs. (17)-(22)J, equal masses were assumed for electrons and light mass holes. ~'e now generalize these formulas to the case of unequal masses. The analog of Eq. (16) may be written D= f [1-exp( -4rsRs/.E\)JdE rs=ms/m,,+mp (16A) e V < (r8!:8-rl!:l)/r8 E~{ exp( -4rn!:n/E~) exp( -4rp!: p/El ) D=eV+- +----- 4 rn rp (17A) r.!:. and rl!:l are, respectively, the smaller and larger of r n!: n, r p!: po All quantities are positive except V and D (r.!:8-rl!:I)/r.<eV <!: min; (rl!:l-rs!:.)/rl D= eV + (Ed4rmin){ exp( -4rmin!: min/El) -exp[4Tmin(eV -!: min)J}. (18A) Note :The subscript min refers to the minimum of !: n, !: p; the subscript s refers to the minimum of rn!: n, T pI p. !:min<eV < (rl!;l-rs!;.)/rl D=!: min+ (El/4rmin)[exp( -4rmin!: min/E l) -lJ, (19A) 1 --exp[4rn(eV -!; n)/ElJ Tn -~ exp[4rp(e V -!; P)/ElJ}, (20A) rp El{ 1 1 D=!; min+--exp[4rnTp(eV-!;n-!: p)/ElJ-- 4 rnrp rmin !:rnax <e V <!: max+!:min (22A) Downloaded 15 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1735110.pdf

Dislocation Acceptor Levels in Germanium Rolf K. Mueller Citation: Journal of Applied Physics 30, 2015 (1959); doi: 10.1063/1.1735110 View online: http://dx.doi.org/10.1063/1.1735110 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Impurity effects on the generation and velocity of dislocations in Ge J. Appl. Phys. 109, 113502 (2011); 10.1063/1.3592226 Spontaneous Terahertz Electroluminescence under Impact Ionization of a Shallow Acceptor in Ge AIP Conf. Proc. 893, 209 (2007); 10.1063/1.2729842 Magnetophotoluminescence of neutral acceptor states in InSb Appl. Phys. Lett. 80, 2332 (2002); 10.1063/1.1461056 Thermal activation of dislocation array formation Appl. Phys. Lett. 79, 2387 (2001); 10.1063/1.1408599 Characterization of dislocations in germanium substrates induced by mechanical stress Appl. Phys. Lett. 73, 1068 (1998); 10.1063/1.122086 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Thu, 27 Nov 2014 11:43:25LETTERS TO THE EDITOR 2015 Dislocation Acceptor Levels in Germanium ROLF K. MUELLER General Mills, Incorporated, Minneapolis, Minnesota (Received July 27, 1959) LOGAN, Pearson, and Kleinmannl reported in a recent paper measurements of the carrier depletion in plastically de formed germanium and derived from these measurements using Read's dislocation model' for the acceptor levels, connected with the dislocations, a value of 0.17 to 0.20 ev below the edge of the conduction band. Conductance and capacitance measurements on germanium bicrystals with low angles of misfit (4°_6°),-5 lead to a value of the acceptor level connected with the grain boundaries of O.06±2 ev above the valence band. This value agrees also with unpublished data obtained with 1 ° boundaries currently under investigation. Since in 4° tilt boundaries the dislocations are already 60 A apart one should expect that the acceptor levels manifest in these boundaries are of the same nature as those connected with random dislocations. Since the number of dislocations introduced into a germanium specimen by plastic bending is known only in its order of magni tudel•6 this number provides an additional adjustable parameter besides the position of the acceptor level in any theoretical model used to describe the observed phenomena. One can expect there fore that widely different models can be successfully adjusted to the experimental data yielding different best fitting values for the acceptor level. If one chooses as proposed by Read' single, non interacting acceptor sites to describe the dangling bonds along the dislocation, the acceptor level has to be placed in the upper half of the energy gap in order to account for the experimental data.I.6 If one assumes that the electrons in the closely spaced dangling bonds (each able to accommodate up to two electrons) interact to form a narrow band of electron states along the dislocation,' the following discussion shows that one has to place this band in the lower half of the energy gap in order to account for the same data. The carrier depletion data alone are therefore insufficient to ascertain a definite position of the dislocation level. They provide, however, the possibility of evaluating the relative merits of the two competing highly simplified dislocation models if the dis location level can be determined independently. Figure 1 in which the results of the following discussion are summarized shows that the carrier depletion data are in good agreement with acceptor levels observed in grain boundaries if the dislocation band model, briefly described in the following paragraphs, is used for the interpretation of the data. O.B 00 0 O. 6 --~ ~ ib ~ 4 F ~ r----- a --~ --2 ~~ O. 0 o 50 100 150 200 250 300 FIG. 1. The relative carrier depletion E. The circles are experimental points obtained by Logan, Pearson, and Kleinmann.l Curve a. Theoretical relation given by Logan, Pearson, and Kleinmann 1 based on Read's dis location model with the two available parameters E, and N adjusted for best fit (E, =0.179 ev below conduction band, N =8No where No is the number of dislocations calculated from observed bending radius). Curve b. Theoretical relation based on dislocation band model and the energy level E2 =0.06 ev above valence band observed in grain boundaries. N adjusted to N =3No. We assume with Shockley7 and Read' that the observed acceptor sites are connected with the dangling bonds along the dislocation. Since the dangling bond sites on the dislocation of interest are closely spaced (3-8 A)I we shall assume that they interact to form a band of electron states along the dislocation with 2N, states per centimeter of dislocation where Nt is the number of dangling bonds per centimeter. The dislocation as a whole is electrically neutral if the band is half-filled. Assuming the energy level observed on grain boundaries, the electrons in this band are tightly bound to the dislocation, which implies that the radial extension of the electron cloud in the dislocation band is of the order of the extension of the electron cloud in a bond orbital, i.e., a",,3 A. We can therefore for the purpose of this discussion treat the dislocation as a cylinder of radius "a" which carries a homogeneously distributed net charge of ql1nt coul/cm if Nt+l1nt electrons per cm are present in the band. In order to avoid the introduction of an undetermined parameter which measures the density of states in the dislocation band we replace the band approximately by a single level characterized by the ionization energy Ei of an electron from a neutral dislocation. Since the filling level J of this band deviates even for the highest observed net charges relatively little from j = 1, this approximation is not only valid for a narrow band but also for wider bands with a sufficiently high density of states around Ei. Following Read's' analysis one finds for the equilibrium condi tion of a dislocation with the host crystal (1) where 1\* is the electrostatic work necessary to bring one electron from the undisturbed bulk to the dislocation charged to its equilibrium value. EF is the Fermi energy in the bulk measured upward from the valence band and E, is the position of the dislocation level for a neutral dislocation measured upward from the valence band. (2) where Ey is the gap energy. Equation (1), though formally identical with Read's expression, is different in two respects. E* in our case is simply given by E*= -q</>o (3) where </>0 is the electrostatic potential difference between the undisturbed bulk and the homogeneously charged dislocation at equilibrium. E* differs from Read's case by the rearrangement energy which is characteristic for a localized charge model. Furthermore, the term kT In(l/ j-1) is in our case a small correction term which can be written in the form kT In[(l/ j) -1]",,2kT(l1nt/N,) (4) where l1nt is the number of excess electrons in the band above the Nt electrons present in a neutral dislocation. Since l1nt/Nt is, even for the highest charge accumulation, of the order of lO-', kT In(l/ j-l) can be neglected against the temperature dependent part of EF. We find therefore for the eqUilibrium potential of the charged dislocation cylinder against the undisturbed bulk in good approximation (5) which gives in the temperature range above .'lOOK where practically all donors are ionized , No -q</>o=Eg o-E2-OlT-kT In-'-CN-d---N=-:-a) (6) where Ego is the gap energy at OK, Ego=0.78 eV,8 Ol is the tem perature coefficient for the gap energy at constant pressure, 0l=4.4Xl0-4 ev;oK,· and No is the effective number of states in the conduction band,· N, = 2 X lO15TI. The charged dislocation is surrounded by a space charge cylinder of radius R. Replacing the space charge due to ionized donors by a homogeneously distributed space charge of equal average density and disregarding the hole population in the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Thu, 27 Nov 2014 11:43:252016 LETTERS TO THE EDITOR neighborhood of the dislocation, one finds, neglecting (aiR)' against unity q(NrN a) (7) where K is the dielectric constant in germanium. Neglecting the hole population around the dislocation is, in our case with the low lying level, more serious than in Read's case. The solution of Poisson's equation which includes the contribution of the hole inversion layer shows, however, that due to the small volume of the region in which the hole density is appreciable, Eq. (7) is, with the adopted value of E2=0.06 ev, a good approximation up to temperatures of about 250oK. We can therefore use the relation of Eq. (7) for our further discussion. The electron depletion on per centimeter length of dislocation is given by (8) The quantity on which is experimentally determined in electron depletion experiments is not identical with !ln" the number of excess electrons in the dislocation, but differs from !ln, by the hole contribution to the total space charge in the space-charge cylinder which is small for sufficiently low temperatures. The total relative electron depletion G due to the introduction of N dislocations pr em' is given by G= onX~=N7rR2. (Nil-N a) (9) Equation (7) shows that R', and therefore G, is insensitive to variations (compatible with our model) of the value of the parameter "a." We choose for our numerical calculations a=3 A. A change of "a" to 6 A changes the G versus T characteristic only insignificantly. One can further derive from Eqs. (6) and (7) that in the higher temperature regime dR2/dT depends only logarithmically on (fluo-E 2) which shows that the slope of the G versus T charac teristic depends, for our model, essentially on the bulk properties and the number N of dislocations present. Since only the order of magnitude of N is determined directly in Logan's et at. experi ments, N has to be found from the observed maximum value of G. G should according to both models show a plateau below 200K. Disregarding, with Logan et al., the lowest temperature points which do not fit into this picture, one estimates Gmax""0.56. For T -> 0, Eq. (5) reduces to (10) where Ed=O.Ol ev is the donor ionization energy and E2=0.06 the dislocation level determined from grain boundaries. From Eqs. (10), (9), and (7) one finds for N (11) Simplifications in X-Ray Diffraction Line Breadth Analysis P. S. RUDMAN Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel (Received July 23, 1959) THE pure diffraction breadth is a much employed measure of crystallite size and state of strain. However, the observed breadth is always somewhat greater than the desired pure breadth because of instrumental effects. Stokes1 has solved the problem of correcting for the instrumental broadening for the general case of arbitrary line profiles by the Fourier integral method. The usefulness of this solution is however diminished by its tedious ness. It will be shown here that by introducing the generally non restrictive assumption that the pure diffraction line profile is a symmetric function, the observed line profile and the broadening profile being arbitrary functions, the labor involved in the Stokes method can be halved. It will further be shown that the final step of the Stokes method, the synthesis of the pure diffraction profile, is unnecessary so that actual calculation time can be reduced by a factor of about }, thus making the method competitive with the generally used rougher approximations2 even from the labor viewpoint. We let f(x) be the pure diffraction profile, hex) be the observed line profile and g(x) be the instrumental broadening profile, the last being observable as the diffraction profile from "perfect" material under identical experimental conditions. Jones2 has shown that the functions are related by the convolution integral: h(xl = i: f(y)g(x-y)dy. If we form the image function: h(-x)= i:f(z)g(-x-z)dz, use the assumed symmetry f( -y) = j(y), and make the change of variable y= -z we obtain that: h( -x) = i:j(y)g[ -(x-y)]dy, and hence: where ;',= II (x) +h( -x) and g,= g(x)+g( -x) are even functions. Thus in the Stokes method where it would normally be necessary to express hex), g(x), and f(x) as complete Fourier integrals including both sin and cos terms, the symmetry of j(x) allows the convolution integral to be formulated in terms of even functions only and hence only cos analysis is required. The actual operations involved are two cos analyses: lI(z) = (z",)-'i: h(x) cos27rxzdx, where No is the value of N expected from the bending radius. This and value has to be compared with the one determined by Logan et al. using Read's model (12) With N determined we have no additional free parameter left to adjust our theoretical G versus T relation to the experimental data. The good agreement shown in Fig. 1 is therefore a strong indication for the validity of our model which allows a unified description of the phenomena observed on random dislocations and low angle grain boundaries. 1 Logan, Pearson, and Kleinrnann, J. Appl. Phys. 30, 885 (1959). 'W. T. Read, Jr., Phil. Mag. 45, 775 (1954). 3 R. Mueller, Report on 18th Annual Conference on Physical Electronics, Massachusetts Institute of Technology (1957), p. 33. 4 R. Mueller, J. Appl. Phys. 30, 546 (1959). 'R. Mueller, J. Chern. Phys. Solids 8, 157 (1959). 6 Pearson, Read, and Morin, Phys. Rev. 93, 666 (1954). 7 W. Shockley, Phys. Rev. 91, 228 (1953). • F. S. Morin and J. P. Maita, Phys. Rev. 96, 28 (1954). "H. Brooks, Advances in Electronics and Electron Phys. 7, 120 (1955). and a cos synthesis: f(x) = (211-)-li)1(z)/G(z) cosz",xzdz. But this final synthesis does not require explicit solution if only the integral breadth is required. The integral breadth is defined as !3=f-: f(x)dx/ fma, where in practice fmax= f(O). Thus we obtain without Fourier synthesis: frO) = (27r)-IJ: II (zl/G(z)dz and i>(x)dx= i1j(x1dx= (27r)-li)1(z)/G(z)J:~ cosz",xzdxdz, limL ....... 00 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Thu, 27 Nov 2014 11:43:25

1.3062556.pdf

Nuclear sizes and density distributions Kamal K. Seth Citation: Physics Today 11, 5, 24 (1958); doi: 10.1063/1.3062556 View online: http://dx.doi.org/10.1063/1.3062556 View Table of Contents: http://physicstoday.scitation.org/toc/pto/11/5 Published by the American Institute of Physicsa By Kamal K. Sethconference Hi report NUCLEAR SIZES and DENSITY DISTRIBUTIONS |"N recent years a large number of conferences some- -*- what vaguely devoted to nuclear structure have been held both within and without the United States. While conferences, especially those abroad, are more than welcome, quite often such conferences tend to be too diffuse in their objective and more or less repetitive in their subject matter. The International Conference on Nuclear Sizes and Density Distributions held at Stan- ford University, Stanford, Calif., on December 17, 18, and 19 was, by definition, above this criticism. It con- cerned itself with a topic which had never before been the exclusive subject of any conference, and the man- ner in which it was organized by Dr. Hofstadter and his colleagues on the steering committee left no room for vagueness or confusion. The conference was jointly sponsored by the Na- tional Science Foundation and Stanford University, with the cooperation of the Office of Naval Research and the US Air Force Office of Scientific Research. It met each day for morning and afternoon sessions, each of three hours duration, thus allowing ample time for the presentation and discussion of twenty-two invited pa- pers and about ten short contributions. The precise defi- nition of the domain of the conference allowed a sys- tematic development of its deliberations. The subject of nuclear sizes was vigorously examined from all pos- sible angles. The variety of these approaches, which ranged from the classic atomic to those "a week old", might at first suggest unending confusion; however, it can be safely said that the conference succeeded in re- solving quite a few discrepancies, and reconciling many more differences. Of course no conference could be of lasting value if it did not pose twice as many questions as it answered. The Stanford meeting was no exception in this respect. It is extremely difficult to summarize a good confer- ence; it is impossible to summarize Bethe's summary of the conference. Yet this is exactly what I must try to do here. Currently engaged in neutron physics research at Duke University, Kamal Kishore Seth was born in 1933 in Lucknow, India. He re- ceived his bachelor's degree (1951) and his master's degree in physics (1954) from Lucknow University. He came to the US in 1954 and was awarded the PhD in physics by the University of Pittsburgh in 1957.The problem of nuclear sizes has been defined in terms of three fundamental properties of all nuclei— they all have a mass, a charge, and a field of spe- cifically nuclear force. The determination of the radial and angular variations in the distributions of these three quantities provide one of the most reliable ways of un- derstanding nuclear structure as well as some insight into the structure of nucleons. Only the nuclei which correspond to closed shell numbers of protons and/or neutrons are known to be spherical, others are more or less deformed. In the first approximation, however, we may consider a typical nucleus as spherical. The sim- plest conjecture about its size, on assuming a uniform density, is that the nuclear volume is proportional to the total number of nucleons (A) contained, i.e., the radius Ro = r^A1/'*. But uniform density is quantum mechanically impossible; the nuclei must have a sur- face region over which the density falls slowly from its central value to zero. Accurate experiments have, as a matter of fact, shown that this is the case. Most of the data is now analyzed in terms of such a distribution (Fig. 1), which is characterized by a parameter r1 = R-LA-1/3 where Ry is the "halfway radius", i.e., the radius at which the density reaches half its central value, and t, a parameter specifying the "surface thick- ness", i.e., the distance in which density falls from 90 percent to 10 percent of its value at the center. The questions that now arise are: (a) Is such a picture of nuclear density borne out by all experiments? If so, what are the values of rx and t? Fig. 1. One of the com- monly used rounded edge distributions, RADIAL DISTANCE , RVariously called Saxon or Fermi distribution, its equivalent square well radius Ro is larger than the half-density radius Ri. 24 PHYSICS TODAY(b) Are the values of these parameters different for the three distributions (matter, charge, and potential) that one can measure? (c) Are neutrons and protons point particles, or do they have a structure also? The theoretician has his share of questions too: (a) Is nuclear radius just an "operational concept" or does it have a physical meaning—how can it be ex- plained in terms of the general problem of nuclear saturation, compressibility, etc.? (b) How and why are the different values of r1 and t for the three distributions related? (c) How does nucleon structure arise—can it be ex- plained with the help of meson theory? The purpose of the Stanford Conference was to bring together the latest experimental information on the sub- ject and the most recent developments in the theoreti- cal understanding of nuclear structure in order to an- swer these questions. TT7HEN a projectile which interacts strongly with * » all the nucleons in a nucleus is employed, the in- teraction can always be described in terms of the opti- cal model, in which a complex average potential re- places the actual nucleus. Such projectiles are neutrons, protons, alpha particles, and w mesons, and the optical model analysis of experiments with them was reviewed in detail by Fernbach, Glassgold, Rasmussen, Seth, and Cool. Fernbach and Glassgold reviewed the data on the total and differential cross section of neutrons and pro- tons, respectively, and concluded that a diffuse edge potential with a halfway radius parameter rx = 1.2S ± 0.0S fermis (1 fermi = 10"13 cm) fits all the exist- ing data up to. 1.3 Bev. Fernbach, in a historical de- velopment of the optical model, emphasized the im- portance of spin-orbit potential being included in the interpretation of high-energy data. Glassgold discussed the energy dependence of the optical model parameters, which is very clearly brought out by proton elastic scattering experiments, and stressed the basic difficulty of optical model analyses. Such analyses always tend to yield the value of a combination of potential depth with the radius (V0Rn, n = 2 at low energy, 2 ^ n ^ 4 at higher energy). Thus it is difficult to arrive at a unique value of the radius parameter. Rasmussen surveyed the alpha-particle scattering as well as alpha-decay experiments. The interpretation of these experiments is particularly difficult because of the relative insensitivity of the a particle to the potential value in the inside of the nucleus. However, recent analyses yield results in essential agreement with neu- tron and proton experiments. Seth pointed out the rather pronounced effect of nuclear deformations in the very low-energy (1 ev to 100 kev) neutron experi- ments, and presented experimental evidence for defor- mation of medium heavy and heavy nuclei. No theo- retical justification was forthcoming for the very large (^ 35 times the value estimated by the conventional theory) value of the spin-orbit potential required bySession on nuclear surface; E. Teller of Berkeley (presid- ing) and L. Wilets of the Institute for Advanced Study. E. Segre (Berkeley) ... an interpretation of large proton-antiproton cross sections . . . A question from the floor by L. B. Okun (USSR) the experimental data at high energy. Only direct po- larization experiments can be expected to settle this question. Similarly, though theoretical considerations favor the idea that absorption should take place pre- dominantly on the nuclear surface (where the Pauli principle inhibits capture less than it does deep inside the nucleus), the experimental data which has been suc- MAY 1958Delegation from the USSR: Dzhele- puv, Blokhintsev, Okun, Nikitin.Conversation with F. Bloch. A. de-Shalit (Weizmann In- stitute) and F. Villars (MIT). cessfully analyzed in terms of an imaginary potential confined to the nuclear surface is still too meager for one to say how successful this interpretation is. How- ever, it appears that the optical model is undergoing sophistication at such a rapid rate that before long it will qualify as a full-fledged theory. Cool presented the results of optical model analysis of if absorption cross sections at energies near 1 Bev. At these energies the -w meson wavelength is so small compared to the nuclear size that it essentially sees the nuclear matter distribution as modified by the range of pion-nucleon interaction. Cool's results agree with a rounded-edge optical potential with rx — 1.14 fermis. When the effect of the finite range of interaction is un- folded Cool essentially gets the measure of nuclear matter distribution. However, a more direct measure of matter radius can only be obtained when a very weakly interacting projectile is used. Leiss reported on the coherent photoproduction of neutral pions in car- bon. These experiments were done at energies 0-70 Mev above the TT° production threshold so that elastic production was the major contribution. Since the 7r° production cross section is almost the same for protons and neutrons, these experiments essentially measure av- erage nuclear matter distribution. Mention was also made of Jones' suggestion that the A"~ meson capture by heavy nuclei in the nuclear emulsions, giving 2" mesons and TT* mesons, might be used not only to de- termine matter radius but also to look into the ques- tion of neutron excess on the nuclear surface where the absorption occurs. The analyses of Cool, Leiss, and Jones essentially bear out the hypothesis that, within the limits of experimental error, the nuclear matter dis- tribution has the same extent as nuclear charge dis- tribution. Can this be interpreted to mean that neu- tron and proton distributions in a nucleus have the same extent ? Opinion on this question was divided and nobody was prepared to commit himself. For the dif- ference rn - rp numbers like — (0.3 ± 0.3) and + (0.8 ± 0.8) fermi were quoted and it is not surprising that few attached any real significance to them. A rather interesting report came from Segre and Chew on the interpretation of abnormally large proton- antiproton cross sections. A few months back the only way of interpreting these results seemed to be in terms of an interaction radius which was disconcertingly large. Using a model in which the strong repulsive core of the phenomenological two-nucleon potential was replaced by an infinite sink (in which every antiproton was annihilated) Chew has been able to account for thelarge experimental cross sections at an antiproton en- ergy of 140 Mev in a simple manner. The application of the optical model by Glassgold to antiproton inter- action with heavy nuclei leads to r0 = 1.3 fermis. SO far I have dwelt only on the potential and mat- ter distributions in the nucleus. There are numer- ous ways of looking at the nuclear charge distribution, but the field is unquestionably dominated by the high- energy electron scattering experiments. Though Hof- stadter likes to call them "nonprecision experiments", these are probably the most definitive experiments in the whole field of nuclear sizes. These experiments are too well known to require any elaboration here. How- ever, the final results of experiments with a large num- ber of elements may be mentioned. Ravenhall summa- rized the up-to-date situation on the interpretation of these experiments. It is found that the nuclear charge distribution is not uniform throughout. In the central region it is more or less uniform (the accuracy of the present experiments cannot distinguish between slight modifications in the central charge density), but on the surface it gradually tapers off. The half-density radius is R-i = 1.07A1/3 fermis and the surface thickness is t ~ 2.5 fermis. Alternatively the radius of the equiva- lent uniform model is Ro ~ 1.07^11/3 + 0.7 fermis, which gives i?0 = 1.35A1/3 for the very light nuclei and Ro — 1.18 for the very heavy nuclei.* Amongst other methods of determining nuclear charge distribution, Henley discussed results obtained by studying the transitions between levels of jn-mesic atoms. The energies of these transitions are very strongly dependent on the finite nuclear charge dis- tribution, the ^-meson orbits being very close to the nucleus because of the meson's heavier mass. These ex- periments yield values of r0 in excellent agreement with those obtained by electron scattering. Kofoed-Hansen discussed the coulomb energy difference between mirror nuclei. Hitherto these measurements led to values of r0 as large as 1.4S fermis. However, when one takes into account the fact that the mirror nuclei differ in the single nucleon which is one of the outermost, it is found that ro= 1.28 ± 0.05 fermis. It may be noted that electron scattering gives almost the same value of * Elton reported on more accurate semiempirical formulae for both i?i and Ro, and they might be mentioned here as an illustration of the complicated nature of seemingly simple things. Ri = 1.1214V" — O.97OA-1'* fermis, t = 2.53 ± 0.06 fermis, or Ro = l.UU1/* + 2.4264-V — 6.6HA-* fermis. Of course these describe the experimental results better. 26 PHYSICS TODAYPROTON NEUTRONFig. 2. Two of the vari- ous possible states of neutrons and protons. The "bare" proton is denoted by the filled circle and the "bare" neutron by the open circle. r0 in this region of atomic weights. Similar improve- ments in agreement with the results of the electron scattering experiments were reported by Breit and Brix, who discussed isotope shifts, Shacklett, who discussed x-ray fine structure, and Jaccarino, who reported on magnetic hyperfine splitting. These experiments are not easy to interpret but it is encouraging to note that as knowledge of the corrections that must be applied in interpreting the data is increasing, the results are tend- ing to be in better and better agreement with electron scattering experiments. The subject of nuclear shapes was excellently reviewed by Temmer, who described the complementary nature of the experiments based on the effects of nuclear deformations on the electron cloud around the nucleus, and experiments which determine nuclear deformations by observing the effects they give rise to in the nucleus itself, e.g., the rotational spectra. The major triumph, upset, or sensation (it depends on whether you are an experimentalist, theorist, or a newspaperman) of the electron scattering experiments is provided by the results for the proton and the neu- tron. Experiments done over a large range of energies and for a number of different angles of scattering lead Hofstadter and his co-workers (Yearian and Bumiller) to the conclusion that both the charge and the mag- netic moment scattering of the electrons by protons is very different from what would be predicted on assum- ing that the proton is a point particle. Stated in the physicist's jargon, the form factor for both charge and magnetic moment scattering by a proton has a value different from unity (< 1). This implies that the pro- ton charge and magnetic moment are both distributed over a finite volume. Hofstadter, in fact, finds the root- mean-square radii rcharge = rmag. mom. = 0.8 fermi for the proton. Moreover, the magnetic moment dis- tribution of the neutron is found to have the same radius. Since the net charge of a neutron is zero, the determination of the charge distribution of a neutron is much more difficult. However, whether it turns out to be concentrated positive in the center and diffuse negative on the outside, or identically zero everywhere, it poses problems. In order to appreciate this and the consequent distress of Goldberger, Chew, and other theorists present at the conference, let us look into what structure theory would expect protons and neu- trons to have. The suggestion of a structure of the nucleons comes directly from the fact that they have magnetic mo- ments (fly = + 2.79 ran, /% = — 1.91 nm) which can- not be explained in terms of their over-all charge alone.27 The charge of the proton accounts only for one unit of its magnetic moment, so that we are left with an anomalous magnetic moment which is equal and oppo- site for the proton and the neutron (^ ± 1.85 nm). The explanation for this is traditionally given in terms of the weak coupling meson theory. Here one postulates that part of the time a physical nucleon is made of a "bare nucleon" core with a meson cloud enveloping it (Fig. 2). The nucleon core is by definition without ex- tension, while the meson cloud has a finite size (of the order of h//xc = 1.4 fermis). The motion of the charged meson cloud gives rise to the anomalous magnetic mo- ments. The electron-neutron interaction, which was discussed in detail by Foldy, provides the main principal evidence for the idea that the charge core of the neutron is al- most a point. However, Hofstadter's experiments claim that the proton core is 0.8 fermi in radius. If there is a charge symmetry, the "bare neutron" should have the same extension also. This would mean that somehow the picture of nuclear structure drawn from the elec- tron-neutron interaction experiments is fallacious. One hates to say so, because that is the picture the theory also predicts, and that is the picture which explains anomalous magnetic moments rather well. If, however, both Hofstadter's and the electron-neutron interaction experiments are being correctly interpreted, one would have to re-examine one of the long cherished ideas of physics, namely, charge symmetry. On the other hand there is always the possibility that we are not inter- preting the electron scattering experiments correctly. The present interpretation, however, is based on the fundamentals of electrodynamics. A modification would have to be basic in nature, and rather sensational, like postulating that the electron has a finite size, or that electrodynamics breaks down at small distances (.--'1 fermi). In this connection it was pointed out by Blokhintsev (USSR) that Tamm has recently postulated that one need not be so radical as to revise these basic concepts. If only one considers that the •jr-meson cloud around the core dissociates continuously into nucleon-antinu- cleon pairs, and the antinucleons annihilate with the bare nucleon in the center, then the net result is a core which is extended as far as the meson cloud (Fig. 3). This viewpoint would not require that there be a mirror symmetry between the charge distributions of the proton and the neutron and would therefore pre- serve the result of the electron-neutron interaction as well as electron scattering experiments. 0 j •> • + O n-MESON NUCLEON ANTINUCLEON Fig. 3. Illustrating Tamm hypothesis: annihilation of antinucleon N (of each N -\-N pair produced) with the central "bare" nucleon is denoted by arrows, with distribution of bare nu- cleons left behind through- nut the volume. MAY 195828 There are objections to this explanation and it ap- pears that more elaborate and accurate experiments on the electron scattering by neutrons will have to be done in order to determine higher moments of the neutron's charge distribution. At the moment, a phenomenologi- cal model like Schiff's is the best one can do. As to the validity of the electrodynamics at small distances, Okun pointed out that it has been proposed by Pomeranchuk that scattering of polarized electrons from polarized protons be studied. This must be considered a rather distant prospect. The theoretical aspects of nuclear matter, the sur- face, saturation, and compressibility problems were the subject of papers presented by Wilets, Brueckner, and Watson. Wilets reported on the phenomenological self- consistent statistical considerations developed by him- self, Swiatecki, Brueckner, and others. According to this treatment there is a fundamental nonlinear rela- tionship between nuclear matter density and the aver- age potential to which it gives rise. This causes a rela- tive extension of the potential distribution beyond the matter distribution, and, when the "finite range of nu- clear force" effect is folded in, completely accounts for the difference between the potential radius (ra = 1.25 fermis) and charge radius (rx = 1.07 fermis). Since the nonlinearity does not exist for the potential as seen by a meson, this also explains the lower radius obtained by Cool (Vj = 1.14 fermis). Wilets' theory also shows how coulomb repulsion, which tries to push protons on towards the nuclear surface, and the symmetry energy, which tries to push neutrons to the surface, balance each other, so that there is hardly any net excess of neutrons over the protons on the surface. This explains the experimental results for rn — rp already mentioned. Brueckner and Watson took up the subject of nuclear saturation from the point of view of the many-body problem. The numerical results of these theories as ap- plied to finite size nuclei are so far not available. Green discussed the information about nuclear structure that one obtains from the study of nuclear masses. His re- vised mass formula yields results in good agreement with electron scattering experiments and Wilets' theory. THE conference concluded in a joint session with the American Physical Society, presided over by Dr. K. K. Darrow, in the music auditorium of Stan- ford University. A capacity crowd heard Bethe de- liver the closing talk, a masterly summary of the de- liberations of the conference—the excellence of which can be savored only by listening to him or reading the transcript which is to be published along with other contributions made at the conference in the Reviews of Modern Physics. (We wish to take this opportunity to thank Dr. E. U. Condon for this new trend in RMP which enables those who cannot attend such confer- ences to keep from falling behind the latest develop- ments in physics.) It is a pity that in any attempt at reporting on such a conference, one of the best parts invariably remains unreported. These are the discussions across the dinnertable, in bathrooms and lounges, and in private sessions between little groups of two or three during coffee breaks. It is impossible to report on these because sometimes they consist of frank opinions and pro- jected thoughts on which people would rather remain unquoted, and sometimes detailed discussions of mi- nutiae, which are out of place in such a report. How- ever, I feel that it would not be out of place if I took this opportunity of stating how stimulating and refresh- ing these personal contacts are. It would be unjust if I gave an exclusively academic picture of the conference, for it had its social high- lights too. The traditional banquet was held on the 17th at Rickey's Studio Inn, after a cocktail party. The banquet itself was much more relaxed and infor- mal than any I have attended in a long time. Much of the credit for this must be given to the dozen foreign speakers who tried to outperform each other in saying thank you to America and thank you to Hofstadter and his colleagues. In his after-dinner talk, Condon reminisced aloud about the good old days of physics, when quantum mechanics was embryonic, and nuclear physics only a young upstart. The best physicists of today were only graduate students, or fresh PhD's, and tended to congregate at the universities of Munich and Gottingen. Condon gave a picturesque description of the plight of these young men (which included Condon, Bethe, Oppenheimer, Rabi, and others), as they labored to keep pace with physics which was entering the awk- ward age of adolescence, an age when it grew very fast, and was most difficult to understand. The uninhibited laughter which greeted Condon's jokes made one of the foreign physicists comment that he had "never seen physicists who looked and acted more unlike physi- cists than those present that evening!" On the 19th the Blochs, Chodorows, Schiffs, and Hofstadters said thank you to the participants of the conference by being hosts at another cocktail party at Hofstadter's home. If you are impressed by statistics, here are some: Registration showed two hundred and thirty-seven physicists in attendance, but a conservative estimate of the over-all number is about four hundred. This included delegates from Australia, Canada, Denmark, England, France, Germany, Holland, Israel, Italy, Switzerland, and the USSR. (The Russian delegation was headed by Dr. D. I. Blokhintsev and comprised of Drs. D. I. Blokhintsev, V. P. Dzhelepov, S. Nikitin, and L. B. Okun.) The delegates were lodged mostly in the Stern Hall Dormitory where the arrangements were perfect. The travel arrangements and other personal conveniences were expertly taken care of by the Physics Department, and tours of the high-energy and microwave labora- tories were conducted a number of times. After this conference it can hardly be said that "nu- clear radius" is merely an operational definition. It is beginning to have a rather well-defined physical mean- ing now. The nucleons; well, they are still a different story. PHYSICS TODAY

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Physics Calculations George I. Sackheim Jacques Romain , Citation: Physics Today 14, 4, 62 (1961); doi: 10.1063/1.3057513 View online: http://dx.doi.org/10.1063/1.3057513 View Table of Contents: http://physicstoday.scitation.org/toc/pto/14/4 Published by the American Institute of PhysicsNASA—GODDARD SPACE FLIGHT CENTER experimental physicists and engineers The Planetary Atmospheres Laboratory of the Goddard Space Flight Center offers stimulating and professionally rewarding positions for versatile and experimental physicists and engineers. Duties include planning and execution of rocket and satellite experiments to measure atmospheric pressures, densities, temperatures, winds, and composition, including neutral particles, ions, and free-radicals. Results of these measurements will be used to describe the physics of the upper atmosphere. Appropriate general problems in physics, electronics, mechanics, and aerodynamics are involved; examples of specific topics are vacuum physics, neutral particle and ion mass spectrometry, light scattering, and molecular beam phenomena. These positions require a Ph.D. degree in physics or engineering, or a Masters degree and suitable experience. For additional information, address your inquiry to: N. W. Spencer, Head, Planetary Atmospheres NASA Goddard Space Flight Center, Greenbelt, Maryland (Suburb of Washington, D.C.) National Aeronautics and Space Administrationof which the editor is a member, and under the spon- sorship of the Office of Ordnance Research of the US Army. It contains fifteen papers reviewing recent theo- retical and experimental work on liquids and solids sub- ject to high and in many cases impulsive stresses. Such stresses are propagated through materials in various ways, depending on the stress intensity and on the elastic properties of the substance in question. The classical theory of elastic radiation is adequate in some cases, but in general one faces extra complexity in the guise of viscoelastic and anelastic behavior in various combinations. Fortunately the necessary mathematical techniques are pretty well understood and numerical solutions can be obtained in many interesting and prac- tically significant cases by the use of computers. The symposium reported here covered considerable scope and included references to stress waves produc- ing fracture in solids, seismic pulses in layered media, photoelastic methods for studying stress propagation, the dispersion of surface waves in solids, measurement of dynamic elastic properties, and other related matters. The treatment throughout concentrates on the macro- scopic behavior of the material under stress and there is little or no attention to the connection between this and the internal constitution. Thus there is no discus- sion of relaxation behavior in terms of lattice proper- ties and the like. Nevertheless the solid- and liquid- state physicist will find here much background material of value to him in his fundamental investigations. The survey of recent research results in a broad field through the publication of symposium papers as in the volume under review obviously has both advantages and disadvantages. On the good side can be reckoned the usefulness of having under one pair of covers a collec- tion of very readable, up-to-date articles which other- wise might be scattered through a number of periodi- cals. On the other side, it must be admitted that the scheme does not lend itself to a completely coherent and well-organized presentation such as one can get in an account prepared by a single well-qualified authority. There appears to be no solution to this problem, though the authorities on communication and information the- ory are doubtless giving it some attention. Physics Calculations. By George I. Sackheim. 267 pp. The Macmillan Co., New York, 1960. Paperbound $3.50. Reviewed by Jacques Romain, University of Elisabethville, Katanga. >TpHE purpose of this book is to help the student -I solve problems by showing him how to conquer the two main difficulties usually encountered: how to link the theoretical principles with the different types of problems, and how to handle the units so that the answer appears with the proper unit. The points where difficulties might arise are duly stressed. Each of the many short sections into which the book is divided deals with a definite topic. In each section the basic principles are recalled in a few precise words, the mks and English units of each quantity are denned 62 PHYSICS TODAY63 Transmission of Information By ROBERT M. FANO, Massachusetts Institute of Tech- nology. Offers an excellent introduction to coding theory or information theory, and approaches the subject from an en- gineering point of view. (An M.I.T. Press Book.) 1961. Approx. 350 pages. Prob. $7.50. Elements of Nuclear Engineering By GLENN MURPHY, Iowa State University of Science and Technology. Presents a general survey of radiation, fission, fusion, and other nuclear transformations, with indi- cations of how these transformations may be exploited in industrial and engineering applications. 1961. Approx. 224 pages. Prob. $7.50.* The Fermi Surface Proceedings of an International Conference Held in Cooperstown, New York, August 22-24, 1960 Edited by W. A. HARRISON and M. B. WEBB, General Electric Research Laboratory, Schenectady, N. Y. Includes general discussions and detailed information on the size and shape of Fermi surfaces. 1961. 356 pages. $10.00. The Physical Principles of Astronautics Fundamentals of Dynamical Astronomy and Space Flight By ARTHUR I. BERMAN, Rensselaer Polytechnic Insti- tute. A concise, thorough exposition of the basic principles of astronautics. A large number of practical examples are included. 1961. Approx. 360pages. Prob. $9.25* Quantum Mechanics By EUGEN MERZBACHER, University of North Caro- Una. Presents as complete as possible a treatment of modern quantum mechanics and its application to simple physical systems. 1961. Approx. 580 pages. Prob. $11.00.* Introduction to Geometry By H. S. M. COXETER, University of Toronto. A lively, rigorous presentation of the subject. 1961. Approx. 384 pages. Prob. $9.75.Fundamentals of Modern Physics By ROBERT MARTIN EISBERG, University of Minne- sota. Contains an integrated presentation of the historical development of quantum mechanics and its applications, and uses the theory to evolve a much more mature discussion of atoms and nuclei than is usual in modern physics texts. 1961. 729 pages. Prob. $10.50. Boundary and Eigenvalue Problems in Mathematical Physics By HANS SAGAN, University of Idaho. Develops the theory of orthogonal functions, Fourier Series and Eigen- values from boundary value problems in mathematical physics. 1961. Approx. 416 pages. Prob. $9.50* Plasmas and Controlled Fusion By DAVID J. ROSE and MELVILLE CLARK, JR., both of Massachusetts Institute of Technology. Stresses principles rather than applications and experiments of a limited in- terest, and presents the material as a unified, detailed whole. 1961. In press. Radioactive Wastes: Their Treatment and Disposal Edited by J. C. COLLINS, University of Manchester. Covers in detail the implications of radioactivity for water supply and waste water disposal as well as the problems of disposing of radioactive solid wastes and radioactive gases. In Press. Viscoelastic Properties of Polymers By JOHN D. FERRY, University of Wisconsin. Care- fully expands a discussion of the phenomenological theory of viscoelasticity followed by the presentation of a wide variety of experimental methods and a critical appraisal of their applicability to polymeric materials of different character- istics. 1961. 482 pages. $15.00. Progress in Dielectrics—Volume III Edited by J. B. BIRKS, Manchester University; American Editor: JOHN HART. Co-ordinates current knowledge of dielectric phenomena, materials, and techniques, and reviews recent progress. In Press. ' Textbook edition also available for college adoption.Send for examination copies. JOHN WILEY & SONS, Inc. 440 Park Avenue South, New York 16, N.Y. April 196164 (a table of conversion from mks to cgs units is avail- able in an appendix), and the author provides detailed solutions of some simple problems. The section is then concluded with numerous problems, both in metric and English units, with the answer to every other problem given (why not to all of them? The habit of giving answers for only odd-numbered problems seems less justified here than anywhere else). The whole field of general introductory physics is covered in this book: mechanics, heat, electricity and magnetism, sound, light, and some atomic physics. The general level is that of a first course, but a few non- elementary topics are included (e.g., reverberation of sound in a room). Four-place logarithms and trigono- metric tables make the book self-contained. Such a book should be the proper place to accustom students to the use of standard symbols for the units, but nt is used for newton, KW for kilowatt, and the like, and even LB and FT are used for lb and ft in the drawings. Plasma Physics. By J. G. Linhart. 278 pp. (North- Holland, Amsterdam) Interscience Publishers, Inc., New York, 1960. $7.00. Reviewed by D. J. Rose, Massachu- setts Institute of Technology. LINHART'S book is not wholly good, and not wholly ' bad. If you are planning a one-semester course in plasma theory, you should keep it in mind. It is more satisfactory for that purpose than any of the smaller monographs on the subject and gives a somewhat broader coverage. However, watch carefully for typo- graphical errors not included in the errata and for am- biguous figures. The treatise is strictly theoretical; the last two parts (Chaps. 7 and 8), y-clept Applications, is a brief summary of principles (of controlled fusion, electromagnetic energy generation, MHD conversion, propulsion, and energy storage). Little is said about the recent developments of the 6A^-dimensional Liouville formulation, giving purported solutions for the plasma particle correlations. Lack of such intensive study might seem like fishing in a prepared bucket because it's simpler, rather than out in the ocean where the fish really come from. Such an appearance, while partly correct, omits the fact that assembly of the appropriate theoretical tackle for such an expedition would occupy a whole book this size. Thus Linhart chooses his mate- rial well enough for a short treatise, bringing the reader up to about the level of the Fokker-Planck equation. Complete plasma theory may share with its alchemical offspring—controlled fusion devices—the property that neither will come in small sizes. Particularly good are Linhart's analyses of individual particle motions. The fluid description starts off with the Liouville equation, conveniently explained, with the relativistic and non- relativistic Boltzmann equations, and velocity averages derived as consequences. Strangely enough, the Fokker- Planck equation is not built directly on this foundation. but is developed much later under the title of Collision and Relaxation Processes. One fifth of the book isgiven over to waves and instabilities. Under this topic, Linhart packs in considerable information, particularly about the oscillations of a plasma cylinder. A very use- ful bibilography of some 200 pertinent papers and IS books is listed at the end. Physique et Technique des Tubes electroniques. Volume 1, Elements de Technique due Vide, 214 pp., 1958, 29 NF; Volume 2, Theorie et Fabrication des Tubes, 427 pp., 1960, 58 NF. By R. Champeix. Dunod, Paris. Reviewed by L. Marton, National Bureau of Standards. IT would be misleading to the readers of this journal if I were to classify the two volumes of Physique et Technique des Tubes electroniques as books on physics. They are books on a technology used by physicists, and while they may be quite useful for them, they have not been written for the physicist but, as reflected by the organization of the book, for the technician. Both, the author and Professor Boutry, who wrote the preface, emphasize that the books have been written for the instruction of students at the Ecole Nationale de Radio- technique and at the Ecole Franchise de Radio-Electri- cite. The present volumes indicate clearly that the instruction at these institutions is not at the university level. Wherever physics background is needed for the understanding of the phenomena some of the mathe- matical background is given without elaborate proof. Most of the treatment is at the technician's level. Both volumes contain sets of problems (without solu- tions) for the student, which are probably the best fea- ture of the book. I believe that a person who is familiar with the subject may find some interesting information in these two volumes, but for a beginner I would not recommend the use of either of these volumes unless it is supple- mented by ample other reading material. Finite Difference Equations. By H. Levy and F. Lessman. 278 pp. Pitman Publishing Corp., New York, 1959. $9.25. Reviewed by Herman Feshbach, Massachu- setts Institute of Technology. THIS book forms an excellent introduction into a subject which is unfortunately often completely missing from the mathematical background of most physicists. Difference equations can be treated in a manner quite analogous to the procedures employed to solve differential equations; and once the fundamental background is developed (Chapters 1-3), linear differ- ence equations with constant coefficients, with variable coefficients, eigenvalue problems as well as partial dif- ference equations can be discussed (Chapters 4 and 8). The solution of nonlinear first-order equations is con- sidered in Chapter 5. Chapter 7 deals with applications which are unfortunately quite uninteresting. There seems to be no treatment of the use of continued frac- tions. Many problems are given, together with some of the answers. PHYSICS TODAY

1.1730619.pdf

Dependence of C13Proton Spin Coupling Constants on s Character of the Bond J. N. Shoolery Citation: The Journal of Chemical Physics 31, 1427 (1959); doi: 10.1063/1.1730619 View online: http://dx.doi.org/10.1063/1.1730619 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/31/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Signs of Spin—Spin Coupling Constants between Methyl Protons and Ring Fluorine Nuclei in Fluorotoluene Derivatives. Further Evidence for a Positive Hyperfine Interaction in the C–F Bond J. Chem. Phys. 47, 5037 (1967); 10.1063/1.1701756 Variation of the S Character and of the Average Excitation Energy in the NMR 13C–H Coupling Constants J. Chem. Phys. 47, 3082 (1967); 10.1063/1.1712342 Nuclear Quadrupole Coupling Constants and the Ionic Character of Covalent Bonds J. Chem. Phys. 44, 4036 (1966); 10.1063/1.1726569 Deuterium Isotope Effect in Proton—13C Coupling Constants J. Chem. Phys. 42, 3724 (1965); 10.1063/1.1695788 Bond Characters and Nuclear Quadrupole Coupling Constants of Halogen Molecules J. Chem. Phys. 30, 598 (1959); 10.1063/1.1730007 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.207.120.173 On: Fri, 21 Nov 2014 18:28:53LETTERS TO THE EDITOR 1427 of the neutral fragment R. This may indicate that the competing process becomes more probable in the sequence R = CHg, I, Br. Electron transfer between an ion A-and a neutral molecule B can take place only if the electron affinity of B is higher or equal to that of A provided that A is not formed with significant vibrational energy. The occurence of the charge transfer processes listed in the table shows that EA(S02) > EA(SO), EA(S02) > EA(CsH.N02) and EA(N0 2»EA(O). In the case of sulfur dioxide and nitrogen dioxide the parent ions S02-and N02-are not observed as primary ions even at low electron energies. The capture of a thermal elec tron apparently leads to a high-lying vibrational level of S02-and N02-. These ions must rapidly decompose by the reverse process since no third body is present to take up the vibrational energy. As a result of the electron transfer process low-lying levels of S02- and N02-are reached so that the potential energy is below that of the ground vibrational level of the neutral molecules. The ions are therefore stable with respect to spontaneous ejection of the attached electron. In the transfer processes the excess energy EA(B) EA(A) probably appears as vibrational energy of both A and B. The results show that certain negative ion-molecule reactions can occur with very high cross sections and that they therefore cannot have any activation energy. Negative ions are generally formed less frequently than positive ions when ionizing radiation is absorbed in material. However, in systems containing atoms of high electron affinities negative ions may play an important role as intermediates in radiation chemical reactions. It is obvious from the present results that negative ion-molecule reactions can frequently occur in these cases. Finally the observation of charge trans fer reactions between negative ions and molecules may become important for studies of the electron affinities of molecules where the negative ions cannot be formed directly by electron bombardment. * Th.is :vork is supported, in part, by the U. S. Atomic Energy CommiSSIOn. I D. P. Stevenson and D. O. Schissler, J. Chern. Phys. 23, 1353 (1955); 29, 282 (1958). 2 Field, Franklin, and Lampe, J. Am. Chern. Soc. 79, 2419 (1957) . 3 R. F. Pottie and W. H. Hamill, J. Phys. Chern. 63, 877 (1959). • Eyring, Hirschfelder, and Taylor, J. Chern. Phys. 4, 479 (1936) . • G. Gioumousis and D. P. Stevenson, J. Chern. Phys. 29, 294 (1958) . 6 T. R. Hagness and R. W. Harkness, Phys. Rev. 32, 784 (1928) . 7 O. Rosenbaum and H. Neuert, Z. Naturforsch. 9A, 990 (1954) . 8 Lampe, Field, and Franklin, J. Am. Chern. Soc. 79, 6132 (1957) . Dependence of Cia-Proton Spin Coupling Constants on s Character of the Bond J. N. SHOOLERY Varian Associates, Palo Alto, California (Received July 13, 1959) ARECENT observation of the various multiplet splittings in the high resolution NMR spectrum of methyl acetylene! has yielded the value 248 cps for the =C!3H coupling constant. This measurement, along with values previously obtained for singly bonded and olefinic carbon atoms, permits an interesting test of the relative importance of the contact term and all other terms in the expression for the electron-spin coupling of the proton magnetic moment with other nuclear moments. 2,g The energy of interaction between two nuclei X and N' with spins IN and IN' is generally written with J NN' being made up of terms involving magnetic dipolar interactions between electrons in non-s orbitals and nuclear moments, and a term which is proportional to the Fermi4 contact interaction between s electrons and nuclear spins. If we consider only the coupling which arises from the contact term we expect that it will depend upon the square of the coefficient of the 2s wave function in the LCAO description of the hybrid orbitals characteristic of singly, doubly, and triply bonded carbon; i.e., 1/4, 1/3, and 1/2 the coupling which would be expected if the bond to carbon were formed exclusively with the 2s orbital. Table I lists the observed CI3_H coupling constants for a number of compounds of the type described above. These J values have been plotted against the s character of the corresponding bond type in Fig. 1. The linearity of the plot and the absence of an appreci able intercept strongly supports the estimate that the contact term dominates the coupling. TABLE 1. C13-H coupling constants and bond types. C(CH3). Si(CH3). Cyclohexene Benzene Compound o / CHa-C (aldehyde proton) '" H Methyl acetylene (acetylenic proton) Hybridization lc13_H (cps) Sp3 120 Sp3 120 Sp2 170 Sp2 159 Sp2 174 sp 248" a See reference 1. All other 'dlues from P. C. Lauterhur, J. Chern. Phys. 26, 2!7 (1957). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.207.120.173 On: Fri, 21 Nov 2014 18:28:531428 LETTERS TO THE EDITOR JC'~H CCpS) 300 20e) 100 o I .25 .333 .50 PERCENTs CHARACTER FIG. 1. Dependence of JC13_H on s character. Electronegative atoms bonded to the carbon would be expected to lower the electron density about the carbon nucleus and to decrease the screening. The coupling will then increase as the cube of the effective nuclear charge seen by the s electron. Points corres ponding to halogen or oxygen substituted hydro carbons invariably fall above the line in Fig. 1. It is interesting to note that if we estimate the coupling for 100% s character to be four times that for spa bonds, i.e., 480 cps, and take account of other fac tors we can compute the coupling for the H2 molecule. Replacing the CIa with a proton increases the coupling by the ratio of the magnetic moments, 3.98, while decreasing the effective charge· from 3.25 to 1.0 intro duces the factor (0.308) a. A factor (2\1'2)2 must also be introduced to take account of the normalizing factors for 1s and 2s orbitals. The result, JH_H(calc) =330 cps is in surprisingly good agreement with the value of 280 cps observed by Carr and Purcell,6 and lends further support to the dominance of the contact term. Similarly, good agreement is obtained for coupling constants to other first row elements. '0bservation in the laboratories of Varian Associates Palo Alto, California (to be published). ' 2 N. F. Ramsey, Phys. Rev. 91, 303 (1953). 3 H. M. McConnell, J. Chern. Phys. 24, 460 (1956). 4 E. Fermi, Z. Physik 60, 320 (1930). 5 Screening constants taken from Quantum Chemistry by Eyring, Walter, and Kimball (John Wiley & Sons, Inc., 'New York), p. 163. 6 H. Y. Carr and E. M. Purcell, Phys. Rev. 88, 415 (1952). Note added in proof: The linear relationship of Jc13_H and s character has also been observed by Muller and Pritchard, J. Chern. Phys. 31, 768 (1959). Signs of the Proton Spin-Spin Coupling Constants in Pure Ethyl Alcohol* P. T. NARASIMHAN AND MAX T. ROGERS Kedzie Chemical Laboratory, Michigan State University, East Lansing, Michigan (Received June 12, 1959) RECENTL YI-a much interest has centered around the relative signs of the spin-spin coupling con stants in the NMR spectrum of a molecule and in this connection we have examined the possibility of determining the relative sign of the two coupling con stants in pure ethyl alcohol. The Hamiltonian4 •• for this system may be written where the subscripts A, B, C refer to the CHa, CH2, and OH group protons, respectively. VA, VB, Vc refer to the common resonance frequencies of the protons of the A, B, and C groups. JAB and J BC (cps) are the two spin-spin coupling constants; J AC apparently is zero.4 From the experimental high-resolution proton resonance spectrum of pure ethanol at 30.5 Mc Arnold has ob tained the values of the two coupling constants as well as the internal chemical shifts OI=VA-VB and 02= VB-VC' To determine the relative signs of JAB and J BC one has to calculate the NMR spectrum of this compound for both the case where the constants have like signs and the case where they have opposite signs and com pare these with the experimental spectrum. Anderson· and Arnold 4 employed perturbation theory to calculate line frequencies and relative intensities. A third-order treatment of frequencies and a first-order treatment of intensities yielded a theoretical spectrum in reasonable agreement with the experimental spectrum. However, they apparently made no attempt to determine the relative signs of the coupling constants. We have therefore made a calculation of the theoretical spectrum of ethyl alcohol treating it as an AaB2C system (Bern stein, Pople, and Schneider6) and solving the secular equation directly without recourse to the approxima tions necessarily involved in perturbation calculations. The zero-order spin eigenfunctions for the system were built up by forming the products of the symmetry functions7 of A (D3h) and B(Dcn h) groups with the two spin functions of C. The secular determinant could be factored according to the different allowed values of the total spin ~I z of the system and the energy levels and stationary state eigenfunctions were obtained in the usual manner following the evaluation of the various matrix elements. The NMR transition frequencies and relative intensities could then be obtained in accordance with well-known selection rules. The above general scheme of computations has been successfully pro- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.207.120.173 On: Fri, 21 Nov 2014 18:28:53

1.1777137.pdf

Physical Chemistry of Compound Semiconductors Jerome S. Prener Citation: Journal of Applied Physics 33, 434 (1962); doi: 10.1063/1.1777137 View online: http://dx.doi.org/10.1063/1.1777137 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Semiconductor heterojunctions at the Conference on the Physics and Chemistry of Semiconductor Interfaces: A device physicist’s perspective J. Vac. Sci. Technol. B 11, 1354 (1993); 10.1116/1.586940 Twenty years of semiconductor surface and interface structure determination and prediction: The role of the annual conferences on the physics and chemistry of semiconductor interfaces J. Vac. Sci. Technol. B 11, 1336 (1993); 10.1116/1.586938 Structural chemistry of the cleavage faces of compound semiconductors J. Vac. Sci. Technol. B 1, 732 (1983); 10.1116/1.582682 Physics and chemistry of semiconductor interfaces: Some future directions J. Vac. Sci. Technol. 21, 643 (1982); 10.1116/1.571805 Physics of compound semiconductor interfaces: A historical perspective J. Vac. Sci. Technol. 16, 1108 (1979); 10.1116/1.570169 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 10:57:14JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 33, NO.1 JANUARY, 1962 Physical Chemistry of Compound Semiconductors JEROME S. PRENER General Electric Research Laboratory, Schenectady, New York The problem of point-defect equilibria in compound semiconductors is considered. It is shown that the e~pe~ted ?o~or or acceptor. properties of a defect in a compouud are entirely independent of the type of bmdmg (lOme or covalent)}n}he compound. From general thermodynamic arguments it is shown that the num~er of degrees of freedom o~ a compound in internal equilibrium is one more than the number of chemical constItuents of the compound mdependent of the number and nature of the defects it contains. The conse quences of this are discussed and it is shown how mass action laws result, describing internal reactions among the defects. A~ example from the literat~re is ~resented to illustrate the methods of setting up and solving t~ese mass actIOn laws and how the solutIOns mIght be compared with experimental results. Finally associa- tIon between oppositely charged defects in solids is discussed. ' I. INTRODUCTION THE physical chemistry of compound semi conductors obviously includes a large area of investigation, but this paper is concerned primarily with one aspect of this subject; the problem of point defect equilibria in solid compounds. These defects are impurities, charge carriers, the so-called "native defects" such as vacancies, interstitials, and misplaced host crystal atoms, and associates of the atomic defects. It is the presence of the native defects in a compound which result in a property obviously not possessed by an element; namely the ability to exist as a single phase over a range of composition. Since many investigations have indicated that these native defects as well as impurities can behave in the solid as singly or multiply ionizable donors or acceptors, it is evident that the control of stoichiometry becomes as important as the control of impurity content in this class of compound semiconductors. In this paper we first consider the symbols used to represent point defects in solids and then present the thermodynamics of defect equilibria in solids. Of primary importance will be the result giving the number of degrees of freedom of a compound containing defects since this tells us the number of intensive variables whose values must be fixed in order that the compound have fixed defect concentrations. The relationships between the concentrations of the defects and the values of these intensive variables also follow,from the thermodynamic treatment. Finally, association of charged defects are discussed. The material presented in this paper is a composite of the work of many people to whom reference is not always made. Very complete lists of references and more detailed discussions of many of the points made in this paper can be found in references 1 through 4. 1 F. A. Kroger and H. J. Vink, Solid-State Physics edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1956), Vol. 3, pp. 307-435. 2 F. A. Kroger, F. H. Stieltjes, and H. J. Vink, Philips Research Repts. 14,557 (1959). 3 R. F. Brebrick, J. Phys. Chern. Solids 4, 190 (1958) . • W. Schottky, Halbleiterprobleme, edited by W. Schottky . (Friedrich Vieweg und Sohn, Braunschweig, Germany, 1958), Vol. 4, pp. 235-281. The pioneer work in the field of defect equilibria in solids was published thirty years ago by Wagner and Schottky .. II. SYMBOLS FOR AND PROPERTIES OF ATOMIC DEFECTS The symbols adopted by Kroger and Vink1 are the ones that will be used here. Thus in the binary com pound MaX b vacancies are represented by V m and V x interstitials by Mi and Xi and misplaced atoms by Mx and Xm and finally, Mm, Xx, and Vi represent the constituents of the compound at their normal lattice sites and empty interstitial sites. Impurities are repre sented by their chemical symbol and the site they occupy (e.g., CUzn for Cu impurity at a Zn site in ZnS). With regard to the use of these symbols, several comments should be made. First, since the ratio of lattice sites alb as well as the ratio of interstitial to lattice sites are fixed, the various defects represented by the above symbols cannot be added or taken from the lattice independently. For example V m cannot be added to the lattice without removing M m or adding V x simultaneously. For this reason chemical potentials cannot be assigned to these defects as represented by the symbols but this does not lead to any difficulty in the thermodynamic formulation of the problem of defects in solid compounds.2 The second point concerns the charge of the defect. Except in the case of extreme ionic compounds, the charge on a particular atom of a compound is not well defined. It is for this reason that the "atomic symbolism" is used in which only the effective charge of the defect is indicated; the con stituents of the compound having by definition zero effective charges. The details of the charge distribution near a defect are needed neither for the thermodynamic formulation of the problem, nor in the qualitative arguments leading to a decision as to whether a particu lar defect can act as a donor or acceptor. An example will be used to illustrate the concept of effective charge of a defect and to illustrate how consideration of different types of binding in a compound leads to the 6 C. Wagner and W. Schottky, Z. physik. Chern. (Leipzig) B11, 163 (1931). 434 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 10:57:14P H Y SIC ALe HEM 1ST R Y 0 F COM PO U N D S E M leo N DUe TOR S 435 same conclusions regarding the donor or acceptor nature of a defect. In the II-VI compound ZnS each Zn is tetrahedrally surrounded by four S and vice versa. We consider first that ZnS is an ionic compound composed of Zn++ and S= ions. If we remove a neutral sulfur atom from the solid into the gas phase, the defect left behind is ob viously neutral, V.o, and the two electrons left behind can be considered as being trapped in the vicinity of the defect. These can be removed one at a time into the conduction band by thermal or optical ionization giving first V.+ and then V.++ and conduction electrons. In the simplest model due to Bethe,6 the binding energies of the electrons to V.+ and V.++ are equal to those in a hydrogen atom or helium ion in a polarizable medium whose static dielectric constant is that of ZnS (about 8). It is obvious from the above discussion that a V.o can act as a double donor and later in this paper evidence will be presented for such behavior in ZnS. If we now consider the other extreme of covalent binding then ZnS is made up of Zn=( .. ·3d104s4p3) and S++(·· . 3s3p3) , each with the tetrahedral Sp3 electron configuration.1 Thus four covalent bonds are formed by Zn and S to neighboring atoms. Incidentally, the electron distributions will be strongly enhanced in the region of the S and this is equivalent to an ionic contri bution to the bonding. The removal of a neutral S(3s23p4) to the gas phase removes six of the eight bonding electrons leaving two electrons, as before, in the vicinity of the V.o which is formed. Thus the atomic notation using effective charges yields the same results for the nature of the defect formed independent of the type of binding of the compound. By similar arguments, a zinc vacancy, V Zno can bind two holes and behave as a double acceptor, ionization of which gives V Zn-and V Zn=. Analogously an impurity such as Cu substituting for Zn or As substituting for S can behave as a singly ionizable acceptor whereas Al substituting for S or Cl for Zn can behave as a singly ionizable donor.8 III. DEGREES OF FREEDOM OF A COMPOUND In this section the number of degrees of freedom will be determined for a compound containing any number of chemical constituents and defects. This will give the number of intensive variables which a free to be altered independently and arbitrarily when the solid is in a state of internal equilibrium. For this purpose we consider a solid compound containing Q different chemical constituents A 1, A 2, "', A Q, and containing a total of S different types of neutral and charged defects (constituent atoms at interstitial sites, vacancies, misplaced constituent atoms, associates, electrons and holes) and constituent atoms on lattice 6 H. Bethe, MIT Radiation Laboratory Rept. No. 43-12(1942). 7 C. A. Coulson, Valence (Oxford University Press, London, 1952), p. 263. 8 J. S. Prener and F. E. Williams, J. Electrochem. Soc. 103, 342 (1956). sites. We denote by n(Di) the number of defects or constituent atoms of type Di in the solid. Since the ratios of the number of different sites, both lattice and interstitial are fixed for a particular structure and since electrical neutrality of the crystal as a whole must be preserved, there will be some total number R of "ratio of site" and neutrality restrictions relating the n(D)'s each of the type: s :E aj>-n(Di)=Oj j=1, 2, "', R. (1) i=I The a's may be positive, negative, or zero. Accordingly the number of independent composition variables are reduced from 5 in number to V = (5 -R) and will be represented by N 1, N 2, "', Ny. These are the com ponents of the solid phase and the Gibbs free energy may be written as: y G= :E Jl.iNi• ;=1 (2) The JI.;'S are the chemical potentials given by: (3) The condition for internal equilibrium of a closed phaes is that the change in the Gibbs free energy be zero for any infinitesimal process occurring at constant tem perature and pressure. Therefore: y (dG)r. P. closed phase=O= L Jl.idNi, (4) i=1 smce y :E dJ.l.;N.=O i=l under the stated conditions. If N (A j) are the total number of atoms of the constituent Ai at lattice and interstitial sites, then the requirement for a closed phase is met by Q relations of the type: y dN(Aj)=O= :E bjidNij j=l, 2, "', Q. (5) i=l The Q Eqs. (5) and Eq. (4) give a-set of (Q+l) equa tions in the V variations dNi. This leads to a set of (V -Q) relations among the chemical potentials: y :E CjiJl.i=Oj j=l, 2, "', (V-Q). (6) i=1 In a solid phase, the number of independent intensive variables can generally be represented by T, p, and (V -1) ratios of the V components j the number of independent intensive variables are (V + 1). However, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 10:57:14436 JEROME S. PRENER for a compound at equilibrium, Eq. (6) gives a set. of (V -Q) relations between int.ensive variables, the chemical potentials. Hence the number of intensive variables which remain independent and can therefore be altered arbitrarily (i.e., the number of degrees of freedom) when the compound is in a state of internal equilibrium is (V + 1) -(V -Q) = Q+ 1. This is an important result, for it tells us that for a binary com pound (Q=2), the number of degrees of freedom is three. If we fix the temperature and pressure and the chemical potential of one of the constituents, say by contact with an external gaseous phase containing the constituent at a given partial pressure, then the concen trations of the various defects in the solid are fixed. For a ternary compound or a binary compound containing an impurity (e.g., ZnS: Cu), if we fix the temperature, pressure, the concentration of copper in the solid and the chemical potential of sulfur, then again the concen tration of all other defects are fixed. In Sec. V, we give the results of some experimental work on this system. The set of Eqs. (6) can be expressed in terms of the concentrations of the various defects, when we ascertain how the chemical potential depend on T, p and the composition of the solid. No details will be given here for obtaining this dependence by statistical methods since they can be found in many of the references cited.2-5•9 The parameters involved in the f..L/s that are characteristic of the solid, are in the first place deter mined by the electronic structure of the solid, namely; the thermal band gap, the thermal ionization energies of donor and acceptor levels and the effective density of states in the conduction and valence bands. The other energy parameters are the cohesive energy of the ideal solid relative to the state of infinite dispersion of the atoms in their lowest states, and the energies required to create the various defects in the solid. Finally there are thermal entropy, and pressure volume terms. IV. MASS ACTION EXPRESSIONS FOR DEFECT EQUILIBRIA When the concentration of defects is small [n(Di) «number of lattice sites lVli that can be occupied by DiJ, and when the Fermi level is sufficiently far removed from the band edges so that classical statistics is applicable to the electrons and holes in their respective bands, then the set of equations (6) become mass-action law expressions involving the concentration of defects.2,3 The IIi's can be related to the n's by the set of S equations: v n(Di)= L dijNj; i=l, 2, "', S (7) j=! the d's being chosen so that the R equations (1) are 9 H. Reiss,]. Chern. Phys, 21, 1209 (1953). fulfilled. From this seL of equations we gel s f..Lj= L djiHDi); j = 1, 2, "', V. (8) i=l The quantities HDi), discussed in great detail in reference 2, and called by the authors "virtual thermo dynamic potentials," are defined by: ac* HD,)=---, an(Di) (9) where C*=G for sets of n's obeying the R restriction equations (1). Under the conditions stated at the beginning of this section, the es can be written as When the defect is an electron or a hole, ;Vli becomes the effective density of states in the bands. The quanti ties ~O(Di) are functions of temperature and pressure and contain the parameters characteristic of the compound. If these expressions for HDi) are put into Eq. (8), then the set of V -Q equations (6) become mass action law expressions, in which the equilibrium constants are functions of temperature and pressure. Actually for condensed phases, the pressure dependence of the equilibrium constants is negligible, except at extremely high pressures, and is usually omitted from consideration. V. AN EXAMPLE OF DEFECT EQUILIBRIA IN SOLIDS A very simple example involving an impurity and a vacancy will be given to illustrate the method outlined in the previous sections and some experimental data will be presented.lo Copper in small concentrations (10-4 to 10-2 mol percent) substituting for Zn in ZnS gives rise to a singly ionizable acceptor level. The presence of CUD (nonionized acceptor) is shown by absorption bands in the infrared due to hole transitions from CUD to valence band and subsequent infrared luminescence by the reverse process. The intensity of the infrared lumi nescence is proportional to the concentration of CUD and was used as a measure of this concentration. The total copper concentration in the solid was fixed and the material was equilibrated at a fixed temperature in contact with a gaseous phase containing sulfur at various fixed pressures from 7X 10-5 to 30 atm. Under the conditions of the experiment only the following defects were considered: CUD, Cu-, h+, and V8++. Applying the methods of the previous sections, we find that the conditions for internal equilibrium of the 10 E. F. Apple and J. S. Prener, J. Phys. Chern. Solids 13, 81 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 10:57:14P H Y SIC ALe HEM 1ST R Y 0 F COM P 0 U N D S E 1\1 leo ~ Due TOR S 437 solid phase is given by: (11) This then gives the required mass action expression: where the brackets denote concentrations in number per unit volume. This is the mass action expression for the reaction describing the thermal ionization of the Cu acceptor: (13) Furthermore it can be easily found from (5), (7), and (8) that the chemical potential of sulfur in the solid is given by J.L(S) = HS.)-HV.++)+2Hh+). (14) For a gas phase containing primarily S2 molecules: When the solid and gas phases are in equilibrium, these two are equal and we get the mass action expression for the formation of ionized sulfur vacancies: P(S2)![V.++ J/[h+ J2= K2(T,p) (16) o ZnS(e-h+)2~ -7 ZnSl_~(V.++);(e-)2~+-S2(gas). (17) 2 The constant concentration of copper in the solid is expressed by [CUOJ+[Cu-J=[CuJ (total). (18) Equations (12), (17), and (18) and the neutrality condition [Cu-J=2[V.++J+[h+J give a set of four equations for the four unknowns in terms of the four intensive variables T, p, [CuJ (total) and P(S2). It is generally not possible to get solutions of these equa tions in analytic form. The graphical method of Kroger and Vink1 can be used to good advantage however. It can be shown, using these methods, that there is a region of low-sulfur pressure in which [CUO] and [h+ ]ap(S2)t and [Cu-] and [V.++] are constant. In this region the incorporation of Cu may be represented by the equation: o ZnCU6S1+~ -7 ZnCu6-(V.++)o/2 S1+o/2+-S2(gas) 4 and the ZnS is nonstoichiometric. This region IS followed by one at higher sulfur pressures in which [CUO], [Cu-], and [h+] are constant. In this region, the incorporation of Cu may be represented by: ZnCu~Sl+O -7 ZnCuo-ho+S1+o. After equilibration at the high temperature, the material was quenched rapidly to room temperature. It is generally assumed that the atomic equilibria are frozen in at their high temperature values, but free charge carriers redistribute themselves among the available levels. We could therefore expect that the infrared emission intensity, measured at -196°C, should be proportional to p (52)1 at low pressures and become constant at higher pressures. The experimental data do indeed show a t power dependence from 7 X 10-5 atm to about 10-1 atm and a leveling off above this pressure. References to the very large number of studies of defect equilibria in solids can be found in reference 1. VI. ASSOCIATION OF DEFECTS So far we have not considered the association of defects. Defects with opposite effective charges interact coulombically and will therefore not be randomly distributed in the solid. These may be treated within the framework already outlined if one considers the interaction energy only of those oppositely charged defects at near neighbor sites.2,1l Thus charged defects are divided into two groups; associated pairs at nearest neighbor sites and isolated defects. Another approach to the problem of association in solids in the case when only two types of oppositely charged defects are involved, is to apply the methods used in aqueous solution of electrolytes,J2 taking into account that charged defects occupy lattice sites. The statistical analysis leads to an expressionl3 for the fraction ai of pairs separated by the distance ri i ai=AcZi[exp(+qlq2/Dr ikT)][exp(-c L Zj)J, (19) i=1 where c is the concentration of the oppositely charged defects, Zi the number of sites at ri, ql, q2 are the effec tive charges on the two types of defects considered, D the static dielectric constant of the 'solid, and T is the temperature below which diffusion over interimpurity distances does not occur. A is a normalization constant. Calculations using this equation indicate that associa tion into nearest-neighbor pairs is expected to be appreciable. This is particularly true when one or both of the defects is doubly charged. ZnS containing any one of the donor impurities AI, Ga, CI, Br, or I exhibits an intense blue emission band. There is evidence14,15 that this is due to a recombination between a conduction electron and a hole trapped at an acceptor level resulting from the associated pair formed between 11 A. B. Lidiard, Phys. Rev. 101, 1427 (1956). 12 R. M. Fuoss, Trans. Faraday Soc. 30, 967 (1934); H. Reiss, J. Chern. Phys. 25, 400 (1956). 13 J. S. Prener, J. Chern. Phys. 25, 1294 (1956). 14 J. S. Prener and D. J. Weil, J. Electrochern. Soc. 106 409 (1959). ' 15 P. H. Kasai and Y. Otorno, Phys. Rev. Letters 7,17 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 10:57:14438 JEROME S. PRENER V Zn= and ])+ (D being any of the donor impurities). Such an associated pair will have an effective single negative charge and can trap a hole to give a neutral species, hence it may be considered an acceptor. The binding energy of the hole to the associated pair should be different depending on whether the donor impurity occupies one of the sulfur sites adjacent to the vacancy (CI, Br, I) or one of the next nearest-neighbor Zn sites (Al, Ga). This effect has been observed experimentally in slight differences in the luminescent spectra. These differences are in accord with calculations based on a simple Bethe-type model.14 More recently EPR studies of this system has led to further confirmation of the model involving an associated pair.15 VII. SUMMARY We have discussed in a general way the expected donor or acceptor characteristics of defects in com pounds. Thermodynamic analysis leads to the result that the number of degrees of freedom of a compound containing Q-chemical constituents is Q+ 1 independent of the number of defects. Further, under certain limiting conditions very often realized in experimental studies of defects in compounds, mass action expressions result from the analysis. These relate the concentration of defects to the values of the independent intensive variables of the system. Finally association of defects was considered, and it was indicated that this is frequently a very important effect in solids. JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 33, NO.1 JANUARY, 1962 Kinetics and Equilibria Involving Copper and Oxygen in Germanium c. S. FULLER AND K. B. WOLFSTIRN Bell Telephone Laboratories, Murray Hill, New Jersey Germanium solutions supersaturated with respect to both oxygen and copper have been investigated in the range 300-500°C by means of conductivity and Hall effect measurements. Kinetics results indicate that the initial rates of disappearance of holes is second order in both the Cu and the 0 concentrations. The failure of the hole mobility to increase with degree of reaction suggests the formation of an ion cluster. Determinations of ionization energy during reaction show changes in the level scheme of Cu to occur and confirm previous work on the ionization properties of donors produced by oxygen. A tentative model is proposed consisting of an initial cluster of two Cu and four 0 atoms on which further oxygen reactions take place. The diffusion of 0 :ygen is found to be accelerated by the presence of Cu. I. INTRODUCTION REACTIONS between impurity atoms in crystals are of interest not only for the information they furnish on diffusion, but also because they offer an opportunity to examine the electrical behaviors of complex solutes in crystals. Kinetics and equilibrium measurements are very useful in the investigation of such reactions. As in chemical reactions taking place in organic liquids or water, one mu~t. ~etermine the c?n centrations (or preferably the actIVItIes) of the reactmg species as functions of time at a series of temperatures and concentrations. The problem is much more difficult in solids, however, because of the greater influence of the medium-in this case, the host lattice---on the reactions. The measurement of the concentrations of the impurity atoms themselves also presents special problems. Finally, since the solutes are ~enera~ly charged, the possibility of interaction to form IOn paIrs must be a first consideration. It is convenient to distinguish different reactions depending upon the kinds of solutes taking part. Re actions may occur between (1) neutral atoms, such as between oxygen atoms in Si and Ge, (2) ions and atoms to form charged products, for example, Li+ ions with oxygen in Si,1 and (3) oppositely charged ions, such as in ion pair formation.2 In the reactions of oxygen in Si or Ge, there is evi dence that atoms of oxygen migrate through the crystal lattices to form molecular aggregates of two, three, and four oxygen atoms which, in the presence of one another, acquire donor properties.3 The process is similar to that which occurs in the formation of Guinier-Preston zones or in general precipitation in metal alloys.4 The com bination of two oxygens, referred to later as the "02 donor," in the case of Ge, has been found to have a donor ionization energy of 0.2 ev from the conduction band.3 The "04 donor," comprising four oxygen atoms, shows a donor level at 0.017 ev from the conduction band. No level corresponding to an 03 compound has yet been identified. Additional unidentified donor levels resulting from oxygen in both Si and Ge have, however, 1 E. M. Pell in Solid-State Physics in Electronic Telecommunica· tions, edited by M. Desirant and J. L. Michiels (Academic Press, Inc., New York, 1960), Vol. I, p. 261. 2 H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J. 35, 535 (1956). 3 c. S. Fuller and F. H. Doleiden, J. Phys. Chem. Solids 19, 251 (1961) and references therein. 4 This similarity was pointed out to one of the authors by Professor A. G. Guy of the University of Florida. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 10:57:14

1.1733361.pdf

Field Desorption of Barium from Tungsten H. Utsugi and R. Gomer Citation: The Journal of Chemical Physics 37, 1706 (1962); doi: 10.1063/1.1733361 View online: http://dx.doi.org/10.1063/1.1733361 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/37/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Isothermal ramped fielddesorption of benzene from tungsten J. Vac. Sci. Technol. 16, 868 (1979); 10.1116/1.570103 Field desorption of helium and neon from tungsten and iridum J. Vac. Sci. Technol. 12, 210 (1975); 10.1116/1.568717 Field Desorption of Thorium from a FieldEvaporated Tungsten Surface J. Appl. Phys. 36, 2656 (1965); 10.1063/1.1714554 Field Desorption of Carbon Monoxide from Tungsten J. Chem. Phys. 39, 2813 (1963); 10.1063/1.1734111 Field Desorption of Cesium from Tungsten J. Chem. Phys. 37, 1720 (1962); 10.1063/1.1733362 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36THE JOURNAL OF CHEMICAL PHYSICS VOLUME 37, NUMBER 8 OCTOBER 15, 1962 Field Desorption of Barium from Tungsten H. UTSUGI* AND R. GOMERt Institute for the Study of M elals and Department of Chemistry, University of Chicago, Chicago 37, Illinois (Received May 16, 1962) The field desorption of Ba has been studied over a range of fields and coverages in an attempt to determine the mechanism of desorption and the charge of the desorbing species. It was found that adsorption seems to be polar but not ionic and that desorp tion of Ba ++ occurs under our conditions. If this interpretation is correct ",~60 13 for the polarizability of adsorbed Ba results, close to the value for the free atom. The accessible field-tempera ture range is restricted by the onset of field-dependent surface diffusion, which was also investigated. It was found that positive fields have more effect on the activation energy Ed and pre- DESORPTION from metal surfaces under the action of very high electric fields was discovered by Muller! for electropositive metals and subsequently found by him2 and others3 to extend to all types of adsorbates, as well as to the substrate metal lattice itself.2 Muller postulated direct ionic evaporation to account for his observations. A theory based on the deformation of the relevant potential energy curves by the field was advanced by Gomer4 and recently extended by Gomer and Swanson5 to a more detailed treatment of the rate constant of field desorption, par ticularly for the case of electronegative adsorbates. As pointed out previously,4.5 field desorption is not only of intrinsic interest but can shed considerable light on the nature of the adsorbed state. In the case of electro negative adsorbates this may consist of a fairly direct determination of the shape of the potential energy curve of adsorption; in the case of electropositive adsorbates it may be possible to decide between co valent and ionic adsorption, and to find out how well the interaction of an ion with a metal surface can be described by an image potential. Recent developments in energy conversion lend added interest to systems of this kind. The first study of field desorption of Ba from tungsten was carried out by Muller2 who determined the field necessary for desorption in a fixed time of 3 sec as a function of temperature. MUller concluded that the desorbing species was Ba+ +; his results at low coverage were later reinterpreted by Gomer4 who concluded that desorption occurred as Ba+ + at high fields and * Present address: Dept. of Applied Science, Faculty of Engi- neering, Tohoku University, Sendai, Japan. t Alfred P. Sloan Fellow. 1 E. W. Miiller, Naturwiss. 29, 533 (1941). 2 (a) E. W. Miiller, Phys. Rev. 102, 618 (1956) j (b) E. W. Miiller, Adv. in Electronics and Electron Phys. XIII, 102 (1960); (c) E. W. Miiller and R. D. Young, J. App!. Phys. 32, 2425 (1961) . 3 M. G. Inghram and R. Gomer, Z. Naturforsch. lOa, 863 (1955) . 4 R. Gomer, J. Chern. Phys. 31, 341 (1959). 6 R. Gomer and L. W. Swanson, J. Chern. Phys. (to be pub lished). exponential term of the diffusion coefficient than negative ones. While a t ",[?2 dependence of Ed is compatible with the limited data obtained for negative fields, no simple behavior was found for positive ones, Ed going through a minimum of 0, accompanied by a drastic reduction in pre-exponential term. A tentative ex planation in terms of a compensation effect is advanced. Values for the zero field heat of adsorption were obtained over a wide coverage interval and agree well with those of Moore and Allison where overlap occurs. low temperatures and as Ba+ at low fields and high temperatures. This interpretation of Muller's data also indicated abnormally low-frequency factors in the rate constants. In view of these suggestive results and the fact that it is impossible to determine frequency factors and activation energies independently from rate constants determined at only one temperature, it was decided to obtain these quantities by measuring desorption rates at various fixed fields as a function of temperature. In order to have relevant thermal data the activation energy of desorption at zero field was also determined over a wide coverage range. EXPERIMENTAL The field emission tubes and tip assemblies used were conventiona1.6 The latter were supplied with potential leads for temperature measurement and control. The Ba source consisted of an electrically heatable hairpin of Fe-clad Ba wire,7 with a notch cut into the Fe cladding at the apex. To prevent Ba deposition on the cooler portions of the tip assembly, e.g., the po tential leads, the Ba source was surrounded by a conical glass shield with a small hole at the apex. As indicated in Fig. 1 the Ba source was mounted perpendicular to the tip axis so that only one side of the tip received a Ba deposit. This made it possible to work either with unilateral deposits, or to permit equiliza tion by surface diffusion. The sources were prepared by spotwelding the Ba wire to its electrical leads and separately outgassing this entire assembly in high vacuum. The Fe cladding was then notched, the shield mounted, and the completed source assembly rapidly installed in the tube which was then baked out on a conventional, all glass high vacuum line. High vacuum after seal-off was insured by depositing molybdenum and/or tantalum films in a connecting getter bulb. In operation tubes were im- 6 R. Gomer, Field Emission and Field Ionization (Harvard University Press, Cambridge, Massachusetts, 1961), Appendix I. 7 Obtained from the Kemet Company. An analysis kindly sup plied by Kemet indicated an impurity content of 0.1-1% Sr and no other impurities above 0.01 %. 1706 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1707 mersed in liquid nitrogen. Under these conditions clean W patterns could be maintained for many hours, indi cating pressures of adsorbable gases < 10-11 mm Hg. After slight aging, sources produced gas-free Ba as indicated by the fact that no gas contamination could be noted after Ba deposition, either on the still Ba free portion of the tip or on any portion of it after thermal or field desoption. Since the desorption temperatures and fields for Ba are below that of most common con taminant gases, this proves their absence. It was oc casionally noted that bursts of H2, identified by its characteristic emission pattern and desorption tempera ture from tungsten,S were released by sources after they had been performing satisfactorily for varying periods. It is believed that this resulted from the opening of H2 pockets in the Ba, although the phe nomenon was not further investigated. The H2 so pro duced could be gettered with Ta. The electrical equipment used has been described in detail previously.6 A 30-kV beta dc power supply mod ified by the inclusion of a reversing switch was used to supply desorption alld electron emission voltages. A 50-MQ precision high-voltage resistance tapped at 500 Q was used with a millivolt potentiometer for voltage measurements. Electron emission currents were meas ured with a vibrating reed or a Keithley electrometer. Tip temperatures were controlled with a servo device similar to one described previously. 6 DESCRIPTION OF MEASUREMENTS AND RESULTS Coverage Detenninations Before describing desorption experiments it will be useful to indicate how the values of coverage 8 were arrived at. No absolute determinations were attempted; FIG. 1. Schematic diagram of field emission tube for field desorption of Ba from tung sten. Ta, Mo, getter filaments; TA tip assembly; Ba iron clad Ba wire; Sh shield for Ba source; G guard electrode, equipotential with screen, S; C conducting coating on glass 4.J[ FIG. 2. Plot of average work <j>(eV) function (thermionic) vs cov- erage for Ba on W. Solid line 3~' refers to data of Moore and I Allison,IO dotted line to data of Becker.u 20 o 05 1.0 8 1.5 all values are based on work function measurements obtained from Fowler-Nordheim plots. These were then combined with the data of Becker9 and of Moore and Allison,I° which are shown in Fig. 2. The former are relative values, corresponding to thermionic work functions obtained when successive equal but unknown Ba doses were evaporated onto a W filament. The latter represent absolute values deter mined by a radio-tracer technique, but again refer to thermionic work functions. It is seen that the agree ment between the two sets of data is very good. In order to use these results for field-emission work functions one must rely on the substantial equality of thermionic and field emission contact potentials. In all cases where reliable data exist for comparison this equality has been found to hold very well. In the present case we were able to duplicate closely the minimum in the curve of Fig. 2, obtaining cf>min = 2.08 eV. It is therefore felt that the present method of estimating coverages yields reliable average values. As will be seen later there is some emission anisotropy at most coverages so that the () values require some interpretation when referred to a given situation. Diffusion It is interesting to know whether one is dealing with a mobile or immobile film in any desorption experiment. This knowledge is particularly important in field desorption, which occurs only from the region of high field at the tip, so that diffusion into this zone may vitiate desorption measurements. The situation is further complicated by the fact, first noted by Drech- envelope; PL potential leads, 1§~~~~~ on tip, and Ba source assem- =\ blies. c sler,ll that the mobility of highly polarizable adsor bates is enhanced by the applied field, so that there may be a region on the shank where the field is too low for desorption but high enough for appreciable diffu sion. It was therefore attempted to see whether diffu- 8 R. Gomer, R. Wortman, and R. Lundy, J. Chern. Phys. 26, 1147 (1957). 9 J. A. Becker, Trans. Faraday Soc. 28, 151 (1932). 10 G. E. Moore and H. W. Allison, J. Chern. Phys. 23, 1609 (1955) . 11 M. Drechsler, Field Emission Symposium 1958, and private communications. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36170B H. UTSUGI AND R. GOMER FIG. 3. Field emission pattern from field desorbed tip and Ba covered shank, prepared by applying a field of F= 1.85 V I A, at 77°K to an assembly with original ¢=3.51 eV, 11=0.24. sion could have affected our results. This was done by ascertaining (a) whether there was appreciable diffu sion into the tip zone during the time of a desorption experiment and (b) by attempting to determine whether there was appreciable transport in the tip zone itself. Measurements of the first type were carried out as follows. Ba was evaporated onto the tip, spread by heating, and the desired initial coverage {} obtained by further heating, i.e., partial thermal desorption. The tip was then allowed to cool and complete field desorp tion carried out at 77°K, resulting in a clean tip and a Ba covered shank (Fig 3). The tip assembly was then sUbjected to the fields and temperatures used in the actual field desorption experiments, and the ap pearance of Ba in the tip zone looked for (Fig. 4). Negative results indicated that diffusion into the desorption zone either did not occur at all, or in such a way that all the infiowing Ba was desorbed in times much less than those required for the actual field desorption from {}i to {}f. Experiments of type (b) were performed by using a unilateral deposit and looking FIG. 4. Tip prepared as in Fig. 3 under conditions where diffu sion does interfere with desorption: T=400oK, F=0.69 VIA. FIG. 5. Unilateral Ba deposit. for diffusion into the initially clean portions of the tip, under various combinations of field and temperature (Figs. 5-7). In this way the permissible range of fields and temperatures for desorption experiments was ascertained, as indicated by the lowest F and highest T values in Table III At lower fields diffusion into and over the tip did occur when it was attempted to apply temperatures leading to desorption in times of 100- 1000 sec, so that the upper limit of activation energies of field desorption accessible by the present method was limited to at most O.B eV. These results are unable to differentiate between total absence of diffusion and diffusion (from a limited zone) so rapid as not to be rate controlling. Since it was of interest to know this and also to have an idea of the mean diffusion length within the desorption zone itself, the diffusion coefficient D and activation energy of diffusion Ed were determined as a function of applied positive and negative fields. This was done by starting with a reproducible unilateral Ba deposit and deter mining the time required for diffusion to proceed to various stages as a function of T at a given field. The endpoints chosen were the first appearance of Ba just beyond the (initially clean) central 110 face and the appearance of Ba on 123. The initial and endpoints are shown in Figs. B~lO. The results obtained for Ed are show in Fig. 11 and Table I. The range was limited in the case of positive fields by the onset of desorption, and in the case of negative ones by excessive electron FIG. 6. Tip of Fig. 5 under conditions where diffusion predominates over desorption: F= 0.61 V / A, T= 425°K. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1709 FIG. 7. Tip prepared as in Fig. 5 under conditions where desorp tion predominates over diffusion: T= 77°K, F=0.89 V / A. FIG. 8. Field emission pattern corresponding to starting point of sur face diffusion experi ments for Ba on W. Tip prepared by evaporating Ba with tip at 77°K, fol lowed by heating to 484°K without applied field for 10 sec to smooth the deposit. FIG. 9. Pattern of Fig. 3 after heating to 484°K for 200 sec with al.0sitive field of F=0.15 V / . This stage was taken as the endpoint for diffusion over the 110 plane. FIG. 10. Tip of Fig. 4 after additional heating under the same condi tions for 300 sec. This stage was taken as the end-point for diffusion over the 123 planes. FIG. 11. Activation energy of surface diffusion Ed and pre-exponential term of diffu sion coefficient A as function of applied field F. Fields are posi tive in all cases except (x) points which correspond to negative fields. Full points correspond to diffusion over 123, empty ones to diffusion over 110. -2 [-r-Tl -4 --I « \ j' :s -6 \ -I \ -8 -..!~-~- -x \. I I 0 \~ 1 -10 r-" 1 I " 1 '.-..... lot --T 08 0.6 0.4 0.2 emission to the values shown. It is seen that positive fields have a greater effect on Ed and A than negative ones. The values of the pre-exponential term A)hown in Fig. 11 were calculated by using the relations (1) where x is the mean distance traversed by the diffusate, determined directly from the field emission pattern and the known tip radius, and t the diffusion time. Thermal Desorption In order to have values of the zero field heat of ad sorption Ha the activation energy of thermal desorption was determined at various coverages. If, as seems reasonable, there is no activation energy of adsorption these values may be equated with Ha. The experimental TABLE 1. Summary of surface diffusion results. The column T indicates the temperature range over which diffusion rates were measured. The pre-exponential term A is determined from Eq. (1) and the values of activation energy Ed. The figures in pa rentheses (110, 123) are Miller indices of ;the pertinent diffusion region. 469-540 0.0 0.41 7.50 0.83 3.8 425-497 +0.15 0.22 9.4 0.44 7.5 382-481 +0.26 0.14 -10.0 0.32 8.4 330-400 +0.33 0.10 -10.0 0.22 9.0 274-370 +0.50 0.0 -11.7 0.07 -11.2 77-191 +0.60 0.05 9.9 0.005 -11.2 133-163 +0.75 0.12 7.7 0.04 -11.1 372-432 -0.14 0.36 8.4 234-266 --0.28 0.27 7.3 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:361710 H. UTSUGI AND R. GOMER ,0 g 80 ...J 60 o o o 02 04 0.6 0.8 10 1.2 e FIG. 12. Heat of adsorption H a and pre exponential term Vo of thermal desorp tion rate constant as function of average coverage (J for Ba on thermally annealed tungsten . procedure consisted of dosing the tip with Ba, spreading the dose by diffusion and heating until the desired initial coverage had been attained by partial desorp tion. The time t required to reach the final coverage Of was then determined as a function of temperature T. Initial and endpoints were first established by work function measurements and subsequently determined by emission at fixed voltages. Activation energies were determined from the slopes of the corresponding Arrhenius plots. For reasons indicated in the discus sion, the form log tiT vs liT was preferred to log t vs liT. 'Vhile the value of the pre-exponential term of the desorption rate constant is sensitive to the form chosen, the value of the activation energy is unaffected well within the experimental error. The results are summarized in Fig. 12 and in Table II. Since mobility was high at the desorption temperatures the results represent average values over the crystal surface. As pointed out previously,S such averages are weighted heavily in favor of the highest value of Ha occurring in a TABLE II. Summary of thermal desorption results. Initial and final work functions <p and coverages (J are indicated by subscripts i and j, respectively. Po is based on Eg. (18) with the last term included. T column indicates temperature range for desorption in 60-600 sec. <Pi eV (Ji <PJ cV OJ Q eV log Vo TOK 4.20 0.08 4.50 0.00 3.77 12.8 1350-1520 3.29 0.29 3.79 0.17 3.77 12.3 1270-1367 2.67 0.55 3.62 0.21 3.71 11.4 1267-1386 2.32 0.72 3.01 0.37 3.39 10.5 1163-1298 2.05 1.00 2.54 0.58 2.86 9.8 1100-1240 2.16 0.87 2.46 0.63 2.82 9.3 1040-1165 2.20 0.82 2.36 0.70 2.44 8.8 930-1046 2.06 1.00 2.21 0.80 2.08 19.7 790-856 2.07 1.00 2.17 0.86 1.96 6.9 943-1040 2.12 1.14 2.08 1.00 1.94 10.0 708-774 2.18 1.23 2.11 1.12 1.96 8.2 800-880 given coverage interval. Since desorption occurs at temperatures where fIeld evaporated tungsten tips begin to rearrange to the thermally annealed form, measurements were restricted a priori to the latter. It is seen that our values of Ha agree well with Moore and Allison'slO value at low coverage and approach the binding energy of Ba at high coverage. Field Desorption Field desorption from thermally annealed tips was carried out at a number of coverages in essentially the same way as thermal desorption, except for the pre sence of the applied positive field. The data were analyzed as in the thermal case to yield values of the activation energy Q and a pre-exponential term B. In addition, field desorption was also carried out from field-evaporated W tips. These were prepared by ap plying very high fields (4-5 V I A) at 300o-S00oK FIG. 1.3. Field-emission pattern of field evaporated tungsten tip. until the electron-emission pattern typical of the field evaporated end form, shown in Fig. 13, was obtained. Tips were not checked for atomic perfection by ion microscopy, but reliance was placed on the corre spondence of the electron to the ion emission pat terns. Attempts to cover the central 110 plane of a field evaporated tip with a uniform Ba deposit at low coverage failed, probably because of the high mobility of Ba on this plane and the fact that its high work function makes detection at low coverage very difficult. These experiments were therefore restricted to the 100 and 211 regions, and carried out by evaporating very small doses of Ba onto the tip while the latter was kept at 260oK. In this way diffusion did not occur, but a smooth, nongranular Ba deposit resulted, indi cating the absence of crystallites. Endpoints (complete desorption) were determined visually in this case by comparing the appearance of the initially Ba covered region with that of the corresponding Ba free region. The fIeld desorption results are summarized in Table III. Initial and endpoints arc shown in Figs. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1711 T ABLE III. Summary of field desorption results. Coverage and work function ranges for each set indicated in the table. Sets 1,2,3 refer to thermally annealed tungsten substrates, and desorption mainly from the vicinals of 110, as shown in Figs. 14-19. Set 4 refers to desorption from the 211 region of a field evaporated tip (Fig. 13) at low coverage. The cf> and 0 values shown in brackets are averages which do not correspond to the desorption region where cf> was much higher. Set 5 corresponds to desorption from the 100 region (Fig. 20) of a field evaporated W tip. In this case the cf> value listed corresponds closely to the actual desorption region. B values are based on Eq. (18) with the last term in cluded. T column lists temperature range used in finding activa tion energies Q. FI'/A Q eV cf>i=4.3-4.4 curve 1: Oi=0.03-0.05 0.89 0.00 0.83 0.15 0.76 0.27 0.71 0.37 0.68 0.48 cf>i= [3.43J curve 2: Oi=[0.25J 1.06 0 1.02 0 0.985 0 0.885 0.27 0.949 0.017 0.876 0.19 0.841 0.46 0.803 0.49 0.845 0.50 0.809 0.47 0.762 0.65 0.762 0.67 0.784 0.61 0.757 0.63 0.694 0.87 cf>i=[2.78J curve 3: Iii = [0.46J 1.19 0.006 1.17 0.08 1.10 0.25 1.00 0.59 1.03 0.49 1.10 0.24 1.05 0.36 cf>i=[4.30J curve 4: Oi=[0.05J 0.950 0.0 0.867 0.17 0.806 0.26 0.745 0.44 0.682 0.65 cf>i=4.35 curve 5: Oi=0.03 1.25 0.007 1.21 0.14 1.12 0.28 1.06 0.38 1.01 0.54 roK cf>j~4.5 Oj~O.O 216-276 206-259 216-277 347-458 384-458 cf>f= [3 .9J Of=[0.15J 77-120 77-120 106-130 205-245 122-164 152-180 189-205 209-230 230-256 218-257 249-276 290-330 250-296 234-259 348-378 cf>r=[2.97J °r=[0.38] 77-108 105-148 143-162 199-220 170-186 144-166 178-204 cf>f~[4.5J Or~O 77-140 180-224 208-250 250-284 262-290 cf>f=4.50 Of=O.O 77-106 130-145 193-329 265-320 325-373 log B -4.1 0.1 2.7 1.8 2.8 -4.5 -4.0 -4.6 1.9 -2.9 1.9 7.7 7.2 6.5 6.0 8.3 7.1 7.2 8.9 8.0 -4.5 -1.5 4.1 9.7 9.5 3.6 5.1 -3.7 1.1 2.8 5.4 8.7 -3.7 2.4 3.0 3.7 4.9 FIG. 14. Emission pattern of tip dosed with Ba, then heated to 14100K for 30 sec. cf>=4.20 eV, 0= 0.08. This is the starting coverage for set 1. 14-20, except for desorption from 211 on field evapo rated tips, where contrast was too low for effective photography. It is estimated that activation energies could be determined to ±5% to 10% in all cases. Fields were determined from Fowler-Nordheim plots on clean W tips, so that average values could be determined to better than 1 %. Since these refer to electron emission they may deviate considerably from the absolute values of locally effective fields. Relative field strengths on the same region are not affected by this consider ation. DISCUSSION Surface Diffusion Although the principal subject of this paper is field desorption the diffusion results obtained in this con nection are interesting in their own right and deserve discussion. We consider first the results at zero field. Our values of Ed are somewhat higher than the corre sponding ones obtained by Drechsler,12 who found, at 8= 1, 0.2 eV for 110 and 0.65-0.67 for 123, while our corresponding values are 0.4 and 0.8 eV, respectively. The difference is probably due to the fact that Drechsler used an applied negative, (i.e., electron emission field) of the order of 0.1-0.3 VIA and to the fact that diffu sion at high coverage may involve various coopera tive effects, particularly on the smooth regions of the surface. However, our values for the 110 plane are FIG. 15. Pattern cor responding to endpoint for set 1. cf>~4.5, O~O. 12 M. Drechsler, Z. Elektrochem. 58, 340 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:361712 H. UTSUGI AND R. GOMER FIG. 16. Pattern from tip dosed with Ba then heated to 13000K for 130 sec 4>=3.29 eV, 0=0.29. This was the starting point for set 2. subject to some uncertainty since they are based on the assumption that the appearance of Ba just beyond this plane can be equated with the endpoint of diffusion over 110, while diffusion over a region with higher Ed may have contributed. If the pre-exponential term A is equated with A = va2, where v is a jump frequency and a is a jump length ,,-,3A, values of V123= lOll secr and VllO= 108 secr are obtained. The former is es sentially normal while the latter is quite low. This suggests that our value on 110 may in fact have mixed in diffusion with a higher Ed. If the values Ha=3.7 eV on 123 and Ha=2.0 eV on 110 are used our results lead to ratios EdIHa"-'0.2 in both cases. These assign ments of Ha are speculative, and based on the assump tion that the high value, obtained from thermal de sorption at low coverage, can be equated with adsorp tion on the tigh t binding regions of the tip while the low value, obtained at high coverage can be as signed to binding on the smoother, 110-like regions. This argument ignores decreases in Ha with coverage due to ad-ad interactions and thus underestimates it on 110. Consequently, the true value of EdlHa on that plane is likely to be less than 0.2. The values fall in the "normal" range in any case. Figure 11 and Table I show that applied positive fiel4s cause Ed to fall off to zero at F=O.S and 0.6 V I A for 110 and 123, respectively, and then to increase again. The values of A show similar behavior, and go through a minimum. It was first pointed out by Drechslerll that an effect on diffusion is to be expected from the fact that the ad-particle may experience a higher field in the activated state, which may corre spond to a more exposed position. However, a simple FIG. 17. Pattern cor responding to endpoint of set 2, obtained in this case by applying a posi tive field of 1.10 V / A for 60 sec at n°K. De sorption has occurred mainly around 110. The work function and aver age coverage on 111 and around 100 is the same as that in Fig. 16. FIG. 18. Emission pat tern from tip dosed with Ba and heated to 110soK for 60 sec. 4>= 2.7 eV, 0=0.53. This represents the starting point for set 3. field dipole (P) or polarization interaction should lead to decreases in Ed of the form P. Fr (1-F21 Fr) or taFl[l- (F21 FrF], where Fr is the field in the activated position and F2 the field in the normal position. It will be seen from Fig. 11 that our results cannot be fittted in this way. Furthermore, this model fails to account for the remarkable behavior of A. These results must be contrasted to those for negative fields: A is relatively unaffected while the Ed values can be reconciled with a taP dependence. These observations suggest that the assumption of constant polarizability and a fixed ratio between the field ex perienced in the potential minimum and the saddle point is inadequate. It is probably more correct to regard interaction between substrate and adsorbate as a metallic-like bonding, so that the adsorbate becomes more or less a part of the metal. In this case the elec trostatic energy of the system in its various configura tions will still depend on P but the effective polariza bility will be a complicated function of average electron concentration at the adsorbate, and will vary with its position. Consequently, the effect on the activation energy should be given by an expression like LlEd= tarFr2[l- (adar) (F22IFr) 2]+ Pl.F1-P2.F2, (2) with ar and a2 themselves functions of field and posi tion. Since the effective polarizabilities depend on the electron concentration at the adsorbate, they will also be different for positive and negative fields; in addition FIG. 19. Endpoint for set 3, obtained in this case by applying a field of 1.2 V / A for 60 sec at nOK. Desorption oc curs mainly around the central 110 face, but has also taken place on the still bright (i.e., Ba covered) regions of the tip where now 4>= 2.97 and 0=0.40. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1713 the P.F contribution will change sign, so that the difference in t!.Ed for positive and negative fields can be accounted for. As the field is increased the erstwhile saddle point for diffusion may become depressed below the erstwhile minimum, i.e., an inversion of the po tential structure may occur, thus accounting for the fact that Ed goes through a minimum as F is increased. It is still necessary to explain the behavior of the pre exponential term with field. If there is a choice of reaction paths, as is almost surely the case on any real surface, the system will select that one which maximizes the total rate. At zero field where diffusion occurs at relatively high temperatures, this may not be the path of minimum activation energy, if the latter involves too unfavorable a pre-exponential term. At the lower temperatures where diffusion occurs in the presence of applied fields, a different, intrinsically lower energy path may be more economical, even if this in volves a decrease in pre-exponential term. As the field is increased and the temperature falls, the reaction path may be selected more and more on the basis of activa tion energy rather than entropy. At fields where the potential structure has been inverted and the activation energy rises again, this behavior may reverse itself. The hypothesis is illustrated in Fig. 21. It is obvious that this explanation must be considered tentative and that phenomena of considerable complexity are in volved. If diffusion were accompanied by field desorp tion the apparent Ed and the apparent A would both be decreased. It is conceivable that this occurs at the higher fields but is improbable at low ones, and fails in any case to account for the increases in Ed and A beyond the minimum values. Effect of Diffusion on Desorption Before discussing the desorption experiments proper it is necessary to investigate the conditions under which they will yield meaningful results. As already pointed out, field desorption is complicated by the fact that diffusion into the desorption zone may occur either from the entire Ba covered shank (if the temperature is high enough), or from a limited region just beyond the desorption zone, where the field may be sufficient FIG. 20. Pattern from field evaporated tungsten tip, after Ba deposition in lower 100 region. Tip heated to 3500K for 60 sec without applied field. <1>=4.35 eV, 8=0.04. This rep resents the starting point for set 5. The endpoint is obtained when the lower 100 region is Ba free and is shown in Fig. 13. FIG. 21. Hypothetical potential energy scheme to explain effect of positive fields on surface diffusion behavior of Ba on W. Solid curves represent adsorption, each minimum corresponding to a equilibrium adsorption site. Path from one minimum to the next along solid curves represents "normal" diffusion path. Dotted curve represents possible path corresponding to changed location of the saddle point configuration for diffusion. As the field increases from left to right, (F=O at far left) the curves are deformed in such a way that the activation energy is lowered to zero. At even higher fields the erstwhile saddle point is now the point of lowest energy. to enhance diffusion, but too low to cause desorption. A detailed analysis, taking into account the field de pendence of D and the diffusion potential is difficult but the following simple treatment delineates various cases adequately. Let us assume that the diffusion zone just behind the desorption region (tip zone) can be characterized by a width d. Then diffusion will not be rate controlling in two limiting cases. (1) Diffusion is very rapid relative to desorption. Under these conditions d will be determined by the tip-shank geometry and the applied field. If every adatom in the diffusion zone can reach the desorption zone in a time short compared to its mean lifetime T in the latter, diffusion will not be rate controlling. This condition becomes (3) or (3a) where kl~l/T is the field desorption rate constant. In this case d is likely to be of the order r, the tip radius, under field desorption conditions. (2) Diffusion is very slow compared to desorption. Under these conditions desorption will be controlling if the number of adatoms desorbed in a given time inter val is much larger than the number entering the de sorption zone, or if 71'r20kl»-27rr(O/kT)D gradJl, (4) where it is assumed that the coverage is uniform in the desorption zone and continuous across its boundary where the gradient of the chemical potential Jl must be evaluated. Since Jl=JlO+kT InO-texP-P.F, (5) "VJl= kT ao _(exFaF +paF)a{3, (6) o ax a{3 a{3 ax where ex is the polarizability of the adatoms, P their dipole moment, F the local field, and {3 the polar angle measured from the tip apex. Since a{3/ax~l/r (7) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:361714 H. UTSUGI AND R. GOMER FIG. 22. Schematic potential energy diagram for ionic adsorp tion (a) in the absence of, (b) in the presence of an applied field. The neutral curve is labeled M+A, the ionic M-+A+. H. heat of adsorption with respect to the neutral free adsorbate A; Q acti vation energy of desorption in the presence of a field; I ionization potential of A, cf> work function of metal M; applied potential is:Fx. we have on combining Eqs. (4), (6), and (7) (8) where k?= [aFo2(F /Fo)+ p.Fo]a(F)/Fo/a{3 (9) - rkT and Fo is the field at the tip apex. When Eqs. (4) and (8) are valid, d is approximately given by (D/kl)t so that (10) The condition of validity for desorption as the rate con trolling step then becomes (11) If it is assumed that the polarizability of adsorbed Ba is essentially that of the free atoms,13 £¥",70A3; if r",10-5 em; if F/Fo"'0.5 and a(F/Fo)/a{3=0.4 at the boundary,14 a value of k2= 2.8X 1013Fo2/T results, with Fo in volts/ A. If l/kl is replaced by the experimentally observed desorption time, which was adjusted to range from 60-600 sec in most cases, we obtain from Eqs. (3) and (11) (12a) or D«1O-9 (12b) as the conditions of validity for desorption experi ments. That is to say, D= 10-8 to 10-9 cm2/sec is the excluded range of D values where, under our experi mental conditions, diffusion interferes with desorption. If our results for the diffusion coefficient can be extrapolated to the region where field desorption occurs, values of ",10-11 cm2/sec result, indicating that de- 13 M. Drechsler, Naturwiss. 43, 52 (1956); M. Drechsler and E. W. Miiller, Z. Physik 132, 195 (1952). 14 W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, J. App!. Phys. 24,575 (1953). sorption occurred from an effectively immobile layer. However, the diffusion behavior is seen to be so complex that such an extrapolation cannot be trusted very much. It is quite possible for instance, that in sets 2 and 3 where coverage was high some diffusion into the central 110 plane preceded desorption, or that mo bility within the dark desorption zone was high. Field Desorption The adsorption of electropositive atoms for which I -cp is small but positive can be described in terms of potential energy diagrams as follows. At large dis tances the ground state consists of neutral metal sub strate (M) and neutral adsorbate (A), and the lowest ionic state, M-+A+, formed by ionizing A and letting the electron enter the metal at the Fermi level /J., lies above it by It-cpq, where It is the ionization energy required to produce the ion of charge qe. As the surface adsorbate distance x is decreased, the ionic curve can be described by an image potential Vim= (qeF/4x (=3.6q2/X in eV-angstrom units). As x decreases even further the level of the adsorbate from which ionization is being considered will broaden if it falls within the metal band, i.e., if I -qcp is not too large. The concept of a broadened A leveP5 is a short-hand way of saying that those allowed states of the total system with energies not too different from that of the unperturbed A level will have larger amplitudes at A than anywhere else. More correctly, we should speak of an A band; the latter may be wholly adsorbed into the main band at the equilibrium separation Xo and in that case adsorption will be purely metallic. The half-width r of the A band is related to the tunneling time T by (13) and thus increases as the potential barrier between the metal and the adsorbate becomes thinner with decreas ing x. If the image potential raises the A band wholly above the Fermi level /J., as x decreases, the ionic state will be that of lowest energy at distances less than x= 3.6q2/ (I -qcp+ r), (14) and the M+A curve which ceases to have meaning beyond this point will merge with the ionic one in this region, as shown in Fig. 22(a). The potential curve will remain pure ionic until the broadening of the A band and/or the effect of repulsive terms specifically resulting from the charge on A+ again take it partially below /J.. If this occurs only when x<xo, the ground state will still be ionic at Xo. An applied positive field will then deform the ground state curve as shown in Fig. 22 (b). The highest point on the desorption path will be the Schottky saddle and de- 15 R. W. Gurney, Phys. Rev. 47,479 (1935). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1715 sorption will occur by vibrational activation to this point with a rate constate given by kd=1l exp( -Q/kT), (15) where Il is the vibration frequency along the desorp tion coordinate and the activation energy Q repre sents the energy difference between the minimum and maximum of the curve in Fig. 22 (b). Since there is some field penetration into the metal when an ex ternal field is applied, the potential of the surface will be approximately F / g and the surface of zero potential will have been pushed into the metal by a distance g-1, where g-l is a Debye-Fermi screening length, of the order of 0.5 A, which can be found from (16) where me is the effective electron mass and p the normal electron density. Consequently the Schottky saddle is to be found as the maximum of the potential -Fq(x+g-1) -(qeF/4(x+g-1) which leads to the ordinary result -(q3rJlF)!( = -3.8qtF~ in eV-angstrom units) relative to the field free zero of the M-+A+ curve. The minimum of the ionic curve will be shifted from the field free value by qF(~XO+g-l), where ~xo represents the shift that would occur with field even if g-l = O. The activation energy of desorption will therefore be ( 17) where xo is the effective position of the minimum and (18) Ha being the zero field heat of adsorption with respect to the neutral atom. If the A band lies partly below /J. its states will be filled to that height. If A + approaches the surface rapid electron exchange will occur at distances where the A level loses its sharpness, and beyond this point the ionic state loses its meaning as a stationary quantum state, merging with the M+A curve into a polar 8, FIG. 23. Proposed potential energy diagram for Ba adsorption in the (a) absence of and (b) presence of an applied field, (here F=O.7S VIA). Pure ionic curves are labeled Ba+, Ba++, respec tively, polar curve is labeled Ba. Dotted lines represent virtual states. It is seen that the polar curve can lie below the ionic ones in the region of maximum binding. "~ 06 \(4) &\;1'. (I) \ Q(eV) \ \ A A "\\ 0.2 O.B FIG. 24. Activation energy of field desorptoni Q vs F!. Numbers refer to sets listed in Table III. ground state. This situation is depicted in Fig. 23 (a). The presence of an applied field will be to lift the A band above /J. as x increases, and consequently the potential curve will be deformed as shown in Fig. 23 (b). If the applied field raises the A band wholly above /J. for x~xo, i.e., if M-+A+ becomes the ground state in the presence of the field, the previously described situation and Eqs. (15) and (17) apply. If the (now polarized) fIeld-free ground state remains that of lowest energy in the vicinity of Xo but if the Schottky saddle occurs at distances where the A band already lies wholly above /J., it will represent the maximum of the desorption path and the activation energy will be given by Here aa is the effective polarizability in the adsorbed state, ai the ionic polarizability, and P the dipole moment formed by the adsorbate and its electrical image at -Xo. The terms !aqF2+!P.F. represent the electrostatic energy by which the system is lowered when the adsorbate is at Xo. Strictly speaking, this should be written as 1XO 1° -00 Fxpdx --co Fxpdx, where the first integral is to be taken in the presence and the second in the absence of the adsorbate. Since the charge in the region of A depends on the potential there, this expression will have the form if not the strict meaning of a polarization and dipole interaction. The fields to be used in Eqs. (17) and (19) are those existing at the surface after the adsorbate has become ionized. If the A band has merged completely with the conduction band of the metal near Xo the mechanism of This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:361716 H. UTSUGI AND R. GOMER TABLE lV. Summary of polarization, valence, and heats of adsorption values computed from field desorption. q.PP is based on Eq. (13) and Fig. 24, while the other results are based on Eqs. (17), (19), and (20) and Figs. 26 and 27. "'c, "'i polarizabilities of absorbate and free ion, respectively. q=l Set <Pi <P, Qo Ha ac-ai Q. no. (eV) (eV) q.PP (eV) (eV) (A3) (eV) 4.35 4.50 1.01 3.4 2.6 -8.7 8.0 2 3.43 [3.90J 1.29 4.0 2.2 -20 8.8 3 2.78 [2.97J 1.41 1.41 1.9 -20 10.1 4 [4.3J [4.5J 1.05 3.7 3.0 -14 8.3 5 4.35 4.5 1.05 4.2 3.5 -10 9.7 desorption will be essentially identical to that just described for the broadened, partially submerged A level. However, the P.F term should probably be omitted, the adsorbate being essentially a part of the substrate and experiencing a different field at Xo from that existing there after A has been desorbed. It should therefore be possible to determine the charge of the desorbing species and the mechanism of desorption from the field dependence of Q. We start by plotting Q vs F~ as shown in Fig. 24. It is seen that reasonably good straight lines result, but that their slopes yield values of q increasing from 1 to 1.5 as () increases. (Table IV) The mere existence of an integral value of q under some conditions is, of course no guarantee that this re presents the actual charge of the departing ion, as indicated by the results for es, where an apparent q=0.8 is found.16 If Eq. (17) is tested by plotting Q+3.8qiF! vs F, negative values of Xo are obtained with q= 1 for all sets, while q= 2 leads to xo= 1.8-2.0 A for sets 1, 4, 5 (low coverage) and xo= 1.4-1.5 A for sets 2 and 3 (high coverage). If the adsorbed species were Ba+ but came off as Ba+ + the xo values listed above would have to be doubled (if polarization can be neglected) and are then seen to be implausibly large. For q= 2 throughout, extrapolation leads to the Qo and Ha values listed in Table IV. The latter are un reasonably small, even if allowance is made for the fact that the nominal work functions used for sets 1, 2, 3, 4 may have been too low and that the thermal Ha values may refer to regions where binding is stronger than those from which field desorption occurred. However the Xo values for q= 2 are not unreasonable since rBa++= 1.4 A and g-l=O.S A so that xO=rBa+++ g-l= 2.0 A, which is close to the values found. Hence adsorption as Ba+ + cannot be ruled out completely in the presence of an applied field, although the Ha values make this improbable. If the effective dipole moment per ada tom is calculated from the data of 16 H. Utsugi and R. Gomer, J. Chern. Phys. 37, 1720 (1962), following article. based on Eq. (20) based on Eq. (17) q=2 with xo=2A with q=2 UBa+ Ha Uc-Ui Qo Ha -UBa++ Qo Ha Xo (eV) (A3) (eV) (eV) (A3) (eV) (eV) (A) 1.9 63 7.4 1.3 36 6.5 0.4 2.0 0.5 44 8.1 -0.2 19 7.9 -0.6 1.4 0.6 23 8.6 -0.9 16 8.6 -0.9 1.4 2.2 56 6.8 0.7 27 7.1 1.0 1.8 3.5 33 8.8 2.6 16 7.8 1.7 1.8 Moore and Allison1o on the basis P= A¢>/27rA~oO a value of 4.4 D is obtained at low coverage, and smaller ones as () increases. If the extrapolation to low coverage can be trusted the dipole moment is too small even at low () for adsorption as Ba+ let alone Ba+ +, in the absence of applied fields, even at the lowest coverage where the work function is most favorable for ionic adsorption. We attempt next to apply Eq. (19) with q= 1 or q=2. Figure 25 shows plots of Q+3.8q!pt-P.P vs p2 for q= 1 and q= 2. While better fits are obtained for q= 2 the insufficient accuracy and limited range of the data do not permit the exclusion of q= 1 on this basis alone. The slopes of the resultant curves lead to values for aa-ai which are listed in Table IV. It is seen that q= 1 leads to negative aa-ai implying higher polariza bility in the ionic than in the adsorbed state. This result is intuitively not very appealing and hard to reconcile with the diffusion results. If q= 2, aa-ai= 60 A3 at low coverage (except for set 5), in close agree ment with the value obtained by Drechsler and Muller for free Ba,13 and suggests that the effective polariza bility of electropositive adsorbates is close to that of the free atoms. As () increases the apparent value of aa drops, probably because of depolarization effects within the adlayer. The low value of 30 A3 obtained for set 5 at low coverage may be due to some screening of the adsorbate by the local substrate geometry. As already mentioned, it is conceivable that, at low coverage, adsorption occurs as Ba+ and desorption as Ba+ +. If polarization is included the activation energy of desorption should give given by where xo is the effective equilibrium distance of the adsorbed Ba+ ion. Figure 26 shows a plot of Q+3.8(2)!P!-x oF, vs p2 with xu= 2 A. The data cannot be particularly well represented in this way; the polarizabilities obtained on this basis are listed in Table IV; it is seen that values of aBa+ -aBa++ ranging from 16-40 A3 are ob tained. In particular it should be noted that the high This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1717 value refers to low 0 while the low values refer to high o. Since there is little doubt that adsorption cannot occur as Ba+ except at very low coverage this would indicate that this interpretation is unlikely to be correct. Extrapolation of the curves in Figs. 25 and 26 to p2=0 permits a determination of Qo and hence Ha if rf> can be correctly assigned. Table IV lists the results for all cases. It is seen that the Ha values are considerably lower than the corresponding thermal desorption values at the same nominal o. The reasons for this are probably the following: (1) Desorption occurred mainly from the vicinals of 110 (except for case 5) so that the rf> values, based on rf>ll'=4.5 are too low. This is probably a major effect, as indicated by the relatively good agreement between thermal and field desorption Ha values in case 5 where the assignment of rf>ll'=4.5 is appropriate. In the case of desorption as Ba+ + it must be noted that (21) where II and 12 are the first and second ionization potentials of Ba so that any errors in rf> appear doubled in the resultant Ha values. If, in the extreme case, a value of rf>w=5.95 eV, i.e., that of the 110 planel7 is used the Ha values for curves 1, 2, 3, and 4 would be increased by 1.5 eV for q= 1 and by 3 eV for q= 2. It is therefore obvious that the agreement in Table IV could have been made to look much better by plausible adjustments of rf>w. 11.0 -q "2 100 f-- ~ a 4.0 3.0 -",Ai 14;7 II) q "I (3) (2) (5)~ (4)~ II)~ 0.2 0.6 1.0 14 F2 IV jA)2 - - - FIG. 25. Plots of QeXDt1+3.8q!Fi-P.F. (with P=O.45 1-1) vs F' for Ba field desorption. Numbers refer to sets listed in Table III. Upper curves refer to q=2, lower ones to q= 1. 17 R. D. Young and E. W. Muller, J. Appl. Phys. 33, 91 (1962) and previous papers there referred to. 5 FIG. 26. Plots of QexDt.+3.8 2'F!-2F vs F' for Ba field desorp tion. Numbers refer to sets listed in Table III. (2) Field anisotropies may have resulted in the inversion of the potential structure discussed in con nection with the diffusion experiments, so that the field desorption values of Ha may be too small by Ed, the activation energy of surface diffuson. (3) There is no obvious one-to-one correspondence between the average Ha values obtained in thermal desorption and those obtained by field desorption since the respective desorption regions need not coincide and field desorption may have occurred from an immobile layer. In particular field desorption is most likely from high work function regions, in view of Eqs. (18) and (19), and since these are atomically smoothest, they are also likely to have smaller Ha values than the atomically rougher, low work function regions from which thermal desorption occurs (or better, whose H a values will be the ones measured in thermal desorption experiments) . It is seen from Table IV that the disagreement be tween thermal and field desorption values is least for q= 1, and worst for the assumption of Ba++ adsorption and desorption. The values obtained in the latter case do, in fact, constitute strong evidence against ionic adsorption. The measurements of desorption rate and its tem perature variation also permit a determination of the pre-exponential term B of the rate constant. If Q changes appreciably during a desorption experiment the desorption time t will yield, in the limit of large I:!.O information on the smallest rate constant involved.18 If n=aQ/ao is positive, this will be ki the rate constant corresponding to the initial coverage Oi. If n is negative it will be k" the rate constant corresponding to Of. The relations are ki=B exp( -Qi/kT) = (kT/Oi \n\)/I, (22a) kj=B exp( -Qt/kT) = (kT/Oj \n\)/t, (22b) ----- 18 L. W. Swanson and R. Gomer, J. Chern. Phys. (to be pub lished). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:361718 H. UTSUGI AND R. GOMER 100 =--, L 6 a l- ~ 6, 6, CD I ~ 0 g 2.0 r-• • ~ ... 0 I -" ~ 6, 6, • • • • 6, • 0 0 0 0.4 Q -eV 0 •• • • • 100 20. 3=6, 40• 5 = 0 • FIG. 27. Plot of logarithm of preexponential term B of field desorption rate constant vs. Q for Ba field desorption. Numbers refer to sets listed in Table III. Log B computed on the basis of Eq. (24) including the term log (kT/();n). respectively. In the case of thermal desorption n is just the slope of the Ha vs 0 plot and seen to be negative, so that Eq. (22b) applies and the experimental heats approach Ha(O,). In the case of field desorption of electroposi tive adsorb a tes (23) and thus positive since act>/ao is negative and exceeds aHa/aO. Consequently in the limit of large I:J,.(j the Arrhenius plots will yield Qi and B=QjkT-lnt+ln(kT/Oin) (24) while in the limit of small f::,.O the last term must be omitted and the subscript (i) becomes superfluous. Figure 27 shows a plot of 10gB vs Q based on Eq. (24) with the last term included. It is seen from Eq. (24) that its inclusion leads to B values smaller by approximately 102• Even in the most favorable case, i.e., if the term In(kT IOn) can be omitted, B decreases markedly with decreasing Q in all cases. Compensation effects of this kind are observed in a large variety of activated processes and suggest the existence of multiple reaction paths, among which the system chooses the one which optimizes the rate under a given set of conditions. It is known from ion micro scopy that field anisotropies exist on the atomic scale even on atomically almost perfect surfaces. In the present case this may mean that the desorption path is a relatively tortuous one to take advantage of these anisotropies Our interpretation of the results indicates the desorption of Ba+ + from a polar but not ionic ground state to be most likely. If the mechanism we have outlined is correct, this involves an essentially adiabatic smooth depletion of the electron population of the s band of Ba as it moves from the surface. In the field and temperature range involved the energetics cer-tainly favor formation of Ba+ +, as indicated in Fig. 23 (b). It is possible however that the s band of Ba is broad only for removal of the first electron, i.e., that the second is more tightly bound and must be removed by tunneling from an almost sharp level on Ba+. If this were the case a relatively slow transition from the adiabatic curve (which would lead to Ba+ desorp tion but with higher activation energy) to the Ba+ + curve might occur where these "intersect." It is possi ble that the decrease in pre-exponential term is partly due to this effect. If the present interpretation of the desorption results is correct it permits some deductions about the shape and location of the potential energy curve of adsorp tion. If, as we have assumed, the maximum on the desorption path is given by a Schottky saddle, the latter must always lie to the right, or coincide with the ground state potential curve. The location of the Schottky saddle is given by Xs= (3.6q/F)!-g-t, (25) in angstroms for F in volts/A. For Q=O we therefore find that xo+g-IS2.4-2.7 .A. in the various cases, if desorption occurred as Ba+ +. If the desorbing species had .been Ba+ this value would have been XO+g-IS 1. 7 A, which is implausibly small. The fact that the potential curve of adsorption may reach its minimum at relatively large values of x helps to explain why the adsorbed state is not purely ionic, since Vim may be too small at these distances to lower a pure ionic state below the actual one, i.e., cannot raise the s band of Ba completely above p.. At distances where an image poten tial would lower the ionic below the actual groundstate, repulsive forces may predominate, because of the large size of Ba and its ions, and the s band of Ba may be too broad for pure ionic adsorption. Thus the absence of ionic adsorption is evidence for the breakdown of the image law near xo. Comparison with Muller's Results Muller determined the field required to cause desorption from the edge of the 110 plane in 3 sec as a 1.1 I 10! _.t! 0.9 '<{ " > 0.8 0.7 0.6 o T OK FIG. 28. Plot of Fl vs T for the Ba field desorption data of Miiller2 at low cov erage. Some of the corresponding data of set 1 ofthe present paper extrapolated into this region are shown as the black points. For interpre tation, see text. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36FIELD DESORPTION OF BARIUM FROM TUNGSTEN 1719 function of temperature. If it is assumed that kl=B exp( -Q/kT) =t secI, (26) Q=kT In3B, (27) and Fl= (Qo/3.8q~) -(kT In3B/3.8 ql). (28) When Muller's data are plotted on this basis the curve of Fig. 28 results, which led Gomer4 to postulate de sorption of Ba+ on the high T and Ba+ + on the low T limb. If our data for the comparable coverage range 0=0.05, set 1, are extrapolated for a given F to yield the value of T required for desorption in 3 sec the superimposed black points on Fig. 28 result, indicating essential experimental agreement between Muller's data and ours in the region where they overlap. Since our experiments indicate diffusion into the tip zone at higher temperature and lower fields, they were not extended into the region corresponding to Ba+ on Fig. 28. It is likely that Muller's results in this region are valid but correspond to desorption with mobility, i.e., to the case given by Eq. (3), since his desorption times were quite short. Valid results by our (Arrhenius plot) method in this region would require special techniques not currently available. Summary It may be useful to summarize at this point the salient experimental results and their interpretation. It was found that field desorption conforms approxi mately to a simple image law but that nonintegral ionic valences are obtained in this way. When the effect of the field on the adsorption minimum at Xo is taken into account the assumption of Ba+ adsorption overcorrects (if Ba+ is the desorbing species), or leads to excessive Xo values and low Ha values (if Ba+ + is the desorbing species), while the assumption of Ba+ + adsorption (with Ba+ + as the desorbing species) leads to exces sively small Ha values. This suggests that adsorption is polar rather than ionic, so that correction terms to the activation energy of desorption arise from polariza tion. The charge of the desorbing ion can be determined in principle from the fit of the experimental data to various models. In practice, the limited range of fields and activation energies does not permit an unequivocal answer on this basis alone. An interpretation in terms of Ba+ desorption leads to the conclusion that the effective polarizability of the adsorbed state is smaller than that of the ion. An interpretation in terms of Ba+ + desorp tion indicates an effective polarizability closer to that of the free atom. This interpretation is more appealing and it is therefore probable, though not proved that the de sorbing species under our conditions was Ba+ +. The pre-exponential terms of the desorption rate constant as well as of the surface diffusion coefficients show a marked dependence on activation energy when the latter is altered by the application of high fields. This suggests the existence of multiple reaction paths, and perhaps in the former case a fairly slow electronic transition to the final desorbing state. While these experiments are necessarily incomplete, in large part because of severe restrictions on the readily accessible T -F region for significant experi ments, they indicate the type of information and conclusions to be gained from field desorption studies. ACKNOWLEDGMENT We wish to thank the California Research Corpora tion and the National Science Foundation for financial support of this work. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Sat, 22 Nov 2014 07:30:36

1.1734265.pdf

Electron Spin Resonance Spectrum of XeF in γIrradiated Xenon Tetrafluoride J. R. Morton and W. E. Falconer Citation: The Journal of Chemical Physics 39, 427 (1963); doi: 10.1063/1.1734265 View online: http://dx.doi.org/10.1063/1.1734265 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/39/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electron Spin Resonance of γIrradiated Crystals of Bromoacetic Acid J. Chem. Phys. 46, 1539 (1967); 10.1063/1.1840894 Electron Spin Resonance of γIrradiated Fluoroalcohols J. Chem. Phys. 43, 2914 (1965); 10.1063/1.1697237 Electron Spin Resonance of γIrradiated Sodium Citrate Pentahydrate and Pentadeuterate J. Chem. Phys. 43, 1996 (1965); 10.1063/1.1697065 Electron Spin Resonance of a γIrradiated Single Crystal of Trifluoroacetamide J. Chem. Phys. 37, 1357 (1962); 10.1063/1.1733285 Electron Spin Resonance of γIrradiated Glycylglycine J. Chem. Phys. 35, 117 (1961); 10.1063/1.1731877 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.198.30.43 On: Wed, 05 Nov 2014 10:12:08SPECTRA OF METAL CHELATE COMPOUNDS. IX 427 between these two groups. Thus it is concluded from the shift of As that the Pt-S bond is slightly weaker in the ethyl than in the methyl derivative. Previously Chatt et at.1 concluded that the contribu tion of lb, IIb, and lIIb to the total structure is about THE JOURNAL OF CHEMICAL PHYSICS equal in N-alkyl and "\ N-dialkyldithiocarbamato com plexes. The present infrared study suggests, however, that the contribution of lb is larger than the others even in dithiocarbamato complexes and it increases appreciably in N-alkyl derivatives. VOLUME 39, NUMBER 2 15 JULY 1963 Electron Spin Resonance Spectrum of XeF in 'Y-Irradiated Xenon Tetrafluoride* J. R. MORTON AND W. E. FALCONER Division of Applied Chemistry, National Research Council, Ottawa, Canada (Received 5 April 1963) The radical XeF has been detected by means of electron spin resonance in a single crystal of XeF4 'Y-irra diated at 77°K. The following hyperfine interaction constants were obtained for magnetic field directions parallel and perpendicular to the Xe-F bond: F1I19= 2649 Mc, F .1.19=540 Mc, Xel1129= 2368 Mc, and Xe.l.129= 1224 Mc. The unpaired electron occupies a fI orbital, and it was sbown from parameters derived from the respective atomic wavefunctions that the orbital is predominantly F 2p and Xe 5p in character. The ex perimentally determined g values for XeF were gll=1.974O, g.l.=2.1251. Departures from the free-spin value are interpreted in terms of interaction between the orbital ground state and excited states of the molecule. INTRODUCTION THE discovery of the first stable compounds of xenon1,2 has excited considerable interest in the chemistry of the rare gases. The existence of such compounds was suggested by Pauling,3 but his sug gestion remained unsubstantiated by an unfortunate failure to prepare a chloride or fluoride of xenon by Yost and Kaye.4 Thirty years later Bartlett1 prepared xenon hexafluoroplatinate and this synthesis was fol lowed by that of xenon difluoride,5,6 tetrafluoride,2 and hexafluoride,7-9 The crystal structures of the difluoridelO,ll and tetrafluoride10,12-14 have been de termined. Since single crystals of the three fluorides of xenon can be grown with moderate ease, it seemed of interest to investigate the electron spin resonance (ESR) * N.R.C. No. 7449. 1 N. Bartlett, Proc. Chern. Soc. 1962, 218. 2 H. H. Claassen, H. Selig, and J. G. Maim, J. Am. Chern. Soc. 84,3593 (1962). 3 L. C. Pauling, quoted in J. Am. Chern. Soc. 55, 3890 (1933). 4 D. M. Yost and A. L. Kaye, J. Am. Chern. Soc. 55, 3890 (1933) . 5 J. L. Weeks, C. L. Chernick, and M. S. Matheson, Ref. 2, p.4612. 6 D. F. Smith, J. Chern. Phys. 38, 270 (1963). 7 J. G. MaIm, I. Sheft, and C. L. Chernick, J. Am. Chern. Soc. 85, 110 (1963). 8 E. E. Weaver, B. Weinstock, and C. P. Knop, Ref. 7, p. 111. 9 F. B. Dudley, G. Gard, and G. H. Cady, Inorg. Chern. 2, 228 (1963). 10 S. Siegel and E. Gebert, Ref. 7, p. 240. 11 H. A. Levy and P. A. Agron, Ref. 7, p. 241. 12 J. A. Ibers and W. C. Hamilton, Science 139, 106 (1963). 13 J. H. Burns, J. Phys. Chern. 67, 536 (1963). 14 D. H. Templeton, A. Zalkin, J. D. Forrester, and S. M. Williamson, Ref. 7, p. 242. spectra of a radiation-damaged single crystal of XeF4, in the hope that a paramagnetic fragment would be trapped in the lattice. The ESR method is especially powerful when applied to the study of oriented species trapped in a crystal lattice, and can yield information on both the s and the p-character of the orbital of the unpaired electron. The trapped radical XeF was de tected,15 and in the present paper its ESR spectra are discussed in detail. The hyperfine interaction of the P9, Xel3l, and Xe129 nuclei are analyzed in terms of second-order theory and the results are related to parameters derived from the respective atomic wave functions. EXPERIMENTAL Xenon tetrafluoride was prepared by a method similar to that described by Claassen, Selig, and Malm.2 A 400-cm3 Monel-K pressure vessel was charged at room temperature with t-atm xenon and i-atm fluorine. After heating to 400°C for two h the product was dis tilled into a small Pyrex and quartz vacuum system. The single stopcock, which isolated the system from the pumps, had a Teflon barreL The xenon tetra fluoride was purified by resublimation and finally was distilled into quartz tubes which were then sealed off. A mass-spectroscopic analysis confirmed16 the presence of XeF4, since peaks corresponding to XeFn+ (n=O, 1, 2, 3, 4) were detected. Single crystals of XeF4 were grown at approximately 7°C inside the sealed tubes by careful sublimation down a temperature gradient. 15 W. E. Falconer and J. R. Morton, Proc. Chern. Soc. 1963, 95. 16 F. P. Lossing (private communication). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.198.30.43 On: Wed, 05 Nov 2014 10:12:08428 J. R. MORTON AND W. E. FALCONER FIG. 1. Diagram of XeF4 crystal, identifying the crystallographic a, b, c axis system and the x, y, z axis system used in the text. Many of the crystals (including the one used in the ESR experiments) had the form shown in Fig. 1, and were identified17 as the low-density form of XeF4. This modification of XeF4 is monoclinic with14 a= 5.050, b= 5.922, c= 5.771 A, (3= 99.6°. Unit-cell parameters determined17 from several crystals of this type were in good (0.5%) agreement with this data. Figure 1 establishes the relationship between the crystallographic a, b, c axis system and the x, y, z axis system to be used later. The single crystal used in the ESR experiments was approximately 3 mm long and was ,,-irradiated at 77 oK. The irradiation was carried out by the Com mercial Products Division, Atomic Energy of Canada Limited, using an 1l00-Ci C060 source (Gammacell 220). The total radiation dose was 5 Mrad. On ir radiation the crystal turned blue and exhibited an ESR spectrum.15 Warmup experiments on a powder sample showed that the blue color and the ESR spec trum diminished rapidly in intensity at approximately 140°K. The ESR spectra of the sample crystal were obtained with a Varian V-4502 X-band spectrometer having 100-kc modulation. All spectra were obtained at 77°K using the V-4546 quartz Dewar accessory filled with liquid nitrogen. The irradiated crystal was placed successively in quartz sample tubes specially designed to hold the crystal with one of the axes x, y, or z (see Fig. 1) vertical. The magnetic field of the spectrometer explored the yz, XZ, and xy planes, re spectively. The crystal was transferred from tube to tube in a dry box previously flushed with dry nitrogen. The crystal was kept under liquid nitrogen except for the few tenths of a second when it fell out of one tube into another. No reirradiation was necessary even after several such transfers. A quartz rod sealed to the top of the sample tube enabled the crystal to be rotated about the vertical axis. The rod connected to a small single-circle goniometer attached to the V-4553 wave guide bend. The strength of the magnetic field at resonance was measured with the aid of an NMR proble (ff1 and Li7) placed coaxially with the para magnetic sample in the magnet gap. The frequency of the NMR marginal oscillator was counted with a Computer Measurements Company model 731B fre quency converter and 707B frequency-period counter. 17 L. D. Calvert (private communication). The measurement of the magnetic field was to within ±0.5 G. The microwave frequency was measured to within ±0.1 Mc using a Hewlett-Packard model 540B transfer oscillator in conjunction with the C.M.C. model 732B frequency converter and the 707B fre quency-period counter. Sufficient microwave power was available from the 20-db coupler in the V-4500-40 X-band microwave bridge to enable this measurement to be made. ANALYSIS OF THE SPECTRA General Features It was observed that within the accuracy of align ment of the crystal in the magnetic field (±3°), the ESR spectra of the irradiated XeF4 crystal were highly anisotropic in the magnetic field direction except when the field explored the xy plane. The spectrum15 for the orientation H parallel to x was virtually indis tinguishable from that obtained for H parallel to y, but these spectra differed from that obtained when H was parallel to z. The spectrum for the orientation H parallel to z was unique and is reproduced in Fig. 2. It was apparent from the spectra that the paramag netic fragment XeF was present in the lattice of the irradiated crystal, and that all the Xe-F bonds were aligned parallel to the longitudinal axis of the crystal. Analysis of the spectrum shown in Fig. 2 in terms of the different isotopes of xenon is indicated. The strongest lines18 in the spectrum are due to XeF radicals con taining xenon isotopes of zero nuclear spin, i.e., the even mass numbers 124 through 136. The zero-spin isotopes constitute 52.5% of naturally occurring xenon and cannot be distinguished from each other by ESR. The symbol Xe132 hereafter represents all zero-spin isotopes of xenon, of which mass 132 is the most abundant. The splitting between the lines assigned to Xe132F is due to a hyperfine interaction with the p9 nucleus (spin 1= t). The xenon nucleus of mass 129 (abundance 26.2%) also has spin I=t, and accord ingly Xe129F contributes a four-line pattern to the spectrum. Finally, Xe131 (21.2%, I =~) in Xe131F contributes eight lines to the spectrum. Although no accurate relative intensity measurements were under taken, it is obvious that the individual line intensities are in accordance with the isotopic distribution in naturally occurring xenon. Taking into account the xenon nuclear spin, individual lines in the spectrum due, respectively, to Xe132F, Xe129F, and Xe131F should have intensities approximately in the ratios 8: 2: 1. As soon as it became apparent that the XeF radi cals were all aligned parallel to the longitudinal axis of the crystal, the spectra for those crystal orientations such that this axis was either parallel or perpendicular 18 The doublet structure of the lines in Fig. 2 is due to hyperfine interaction with a Fl" nucleus on a neighboring XeF4 molecule. This splitting is never fully resolved, but reaches a maximum of 10 G for this orientation. It is ignored hereafter. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.198.30.43 On: Wed, 05 Nov 2014 10:12:08ELEcTRON SPIN RESONANCE OF XeF 429 2.5 3.0 3.5 4.0 kiloGauss v = 9070 Me/sec I I I X~31F I I I I I X 129 e F 132 Xe F etc. FIG. 2. First derivative ESR spectrum of XeF in a -y-irradiated single crystal of XeF. (H parallel to z). The doublet structure of the lines is due to a near-neighbor interac tion with p9 on an adjacent XeF. molecule. to the magnetic field became of special interest. The spectra for H parallel to z and for H perpendicular to z are respectively called the "parallel" and the "per pendicular" spectra. This designation refers both to the orientation of the longitudinal axis of the crystal, and to the orientation of the Xe-F bonds in the mag netic field. The field strengths of the centers of all the hyperfine lines were measured for both the "parallel" and "perpendicular" (H parallel to x) orientations, and at the same time the microwave frequency was T ABLE I. ESR field strengths (gauss) of XeF radicals in -y irradiated XeF. for directions' of H parallel and perpendicular to the Xe-F bond. Species M1(F), M1(Xe) Parallel Perpendicularb Xel32F H 2792.6±0.5 2915.1±0.5 -! 3752.0 3098.9 Xel29F +!, H 2343.2 2671. 7 H, -! 3210.2 (3095)· _1. H 3308.3 2848.9 2, _1. -! 4165.4 3273.6 2, Xe131F H, H 2403.7 2719.8 H, H 2655.8 2810.8 H, 1 2911.4 (2915) c -. H, -! 3171. 7 3035.8 -i, H 3368.4 2935.1 -i, H 3617.2 3069.5 -.1 _.1 2, 2 3868.9 3162.9 -j, -! 4127.5 (3275)0 Microwave frequency (Mc) 9049.5 9046.2 a The error is ±3°, b H was parallel to x. e Overlapped, error ±5 G. also determined. The mean results of several such measurements are collected in Table I. The field strength measurements are accurate to ±O.S G, and the microwave frequency did not deviate more than ±O.S Mc during the measurements on a particular orientation. Determination of Hyperfine Interaction and g Tensors It will be seen at once from Table I that the hyper fine spliUings of the nuclei p9, Xe!29, and Xe!3! are exceedingly large. For this reason the electron spin cannot be assumed to be quantized along the magnetic field direction, and terms of second order in the hyper fine interactions of the nuclei with the unpaired electron must be retained in the Hamiltonian. For Xe!32F the hyperfine interaction is that of a single nucleus (P9) of spin I=!. The application of second order theory to such a case has been developed!9.20 for P3! in the P03= radical. With the magnetic field along the z-axis the simplified Hamiltonian 3C= {3S,gzzH ,-"II ,H ,+ SxFxJx+ SlIFlIyIlI+ S,F "I. may be used, where {3 is the Bohr magneton and "I the magnetogyric ratio of the p9 nucleus. It is assumed that the axis-system x, y, z diagonalizes both the g tensor g and the p9 hyperfine interaction tensor F. If Sx and Sy are nonzero, the electron spin is not quantized along z, and it can be shown with the aid of electron and nuclear spin raising and lowering operators that a mixing of certain electron spin states occurs. Off diagonal terms in the energy matrix cause shifts in the energy levels, the perturbations being of the order of F2/411 where II is the microwave frequency. If the spectrum is observed at constant microwave fre- 19 A. Horsfield, J. R. Morton, and D. H. Whiffen, Mol. Phys. 4, 475 (1961). 20 M. W. Hanna and L. J. Altman, J. Chem. Phys. 36, 1788 (1962). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.198.30.43 On: Wed, 05 Nov 2014 10:12:08430 J. R. MORTON AND W. E. FALCONER TABLE II. Principal values of the hyperfine interaction tensorsa and g tensorsb of the radical XeF. Species Xell XeJ. Xel32F Xe129F 2368 1224 Xel3lF 701 a Units are Me, errors ±1O Me. b Errors ±O.OOO8. FII 2649 2637 2653 quency it may be shown21 that F ... gil 540 1.9740 526 1.9740 1.9740 gJ. 2.1251 2.1264 p=g.J3H.-+ !F .. + (F.",2+F lly2)/Sg .. j3Hz- for the low-field line at H.-G, and v= gzzj3Hz+ -iF •• + (F.,,?+F yi)/Sgzzj3H z+ for t~e high-field line at Hz+ G. The corresponding ~quatlOns for H parallel to x or yare obtained by mak Ing the necessary changes in the subscripts. In the present case (Xe132F) axial symmetry about the z direction may be safely assumed, so that F II = F II and F.,,,= Fyy= F.I.. The four equations which were solved iteratively for FI II F.I., gil and g.l. are for the "parallel" orientation, ' VII=gllj3H II± =F !FII + F.l.2/4gllj3HII±, and for the "perpendicular" orientation, P.I.= g.l.j3H.I.± =F !F.I.+(F II2 + F.l.2)/Sg.l.j3H.I.±. With the aid of these equations and the data in Table I the principal values of the g tensor g and the FI9 hyperfine tensor F for the radical Xe132F were de termined. These data appear in the first row of Table II. The hyperfine interaction in Xe129F is that of two nuclei of spin 1= t, resulting in a four-line contribu tion to the spectrum. The second-order correction out lin~ above was applied first to the p9 hyperfine inter actIOn, then to the Xe129 interaction, and this process was repeated until the iteration converged. The results of this calculation also appear in Table II. The data for Xel29F is quite independent of that for Xe132F but the agreement between the respective principal values of the FI9 hyperfine tensor and the g tensor is reasonably good. The nucleus Xe131 has spin I = ~, and so an eight-line pattern was contributed by Xe131F, taking into account also the spin I=t of FI9. To first order the four lines arising from hyperfine interaction with the Xe131 nucleus should be equally spaced in the magnetic fie~d .. Taken to. second order, however, the M1(Xe) = ~2 hnes are displaced downfield by 8 G, and the ±~ hnes by 0.58 G, where 8'-' .. Xe2/g/Jv. Having corrected for the second-order perturbation the values of Xelll3l. FII, and gil (Table II) were calculated from the 21 In the .equ~t~ons .which follow the "{I.H. term has been neg lected for simpliCIty; It was retained in the actual calculation. "parallel" spectrum. It will be noted that within the ~xperimental error the value of XeIl129/XeII181 is 3.375, In accordance with the ratio of their respective mag netogyric ratios. It was not possible to determine the p~rameters Xe.l.l81, F.I., and g.l. from the "perpen dicular" spectrum because three of the eight lines due to Xe13IF were overlapped by lines arising from the other isotopic species (see Table I). However, these parameters would not have represented any new in formation. . The ~rincipal values of the FI9 and Xel29 hyperfine InteractlOn tensors may be resolved into isotropic components CAF, Axe) and anisotropic components (BF, Bxe). The parameters A and B are defined such that the principal values of a tensor possessing cylindri cal symmetry are A+2B parallel to the unique direc tion, and A - B perpendicular to this direction. With the values from Table II of FII, F.I. for XeI32F and X~lh X,=.1. for Xel29F the respective isotropic and alllsotropiC components were determined: AF=1243 Mc, Ax.= 1605 Mc, BF=703 Mc, Bxe=3S2 Mc. DISCUSSION Nature of the XeF Radical It is apparent that XeF is a u-electron radical' that is,. the unpaired electron occupies an orbital poss~ssing aXIal symmetry about the internuclear (z) axis. This orbital can be described by a combination (with various coefficients) of atomic orbitals of the same symmetry, for example F 2s, 2pz and Xe 5s, 5p •. Since SCF wave functions of both the n= 2 shell of fluorine22 and the n= 5 shell of xenon23 are available it was possible to use the spectroscopic parameters A and B to estimate the contribution of the various atomic orbitals to the molecular u-orbital. If inner-shell polarization can be neglected the iso tropic parameter A is a measure of the co;tribution of the valence s atomic orbital to the molecular orbital occupied by the unpaired electron. Thus AF, being the Fermi "contact" interaction of the unpaired elec tron with the pR nucleus, is a measure of the spin density at that nucleus. Furthermore, F(2s)f2(0) = 11.97 a.u. corresponds24 to a pure 2s hyperfine inter action with FI9 of [S7rg/J'Y/3h)p2(0) or 47 900 Mc. The observe~ value of A~, 1243 Mc, indicates an F 2s spin pop~latlOn of 2.6% If such a figure has any meaning III VIew of the possibility of Is polarization. Similarly, the value23 of Xe(5s)f2(0) is 26.71 a.u. corresponding to a pure 5s hyperfine interaction with Xel29 of 33 030 Mc. In the absence of inner shell polarization the ob- 22 E. Clementi, C. C. J. Roothaan, and M. Yoshimine Phys Rev. 127, 1618 (1962). ' . 23 D. F. Mayers, University of Oxford (private communication). 241.. R. Morton, J. R. Rowlands, and D. H. Whiffen, National PhYSical Laboratory Report No. BPR13. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.198.30.43 On: Wed, 05 Nov 2014 10:12:08ELECTRON SPIN RESONANCE OF XeF 431 TABLE III. Comparison of experimentally determined isotropic (A) and anisotropic (B) tensor parameters with the corresponding one-electron parameters derived from atomic wavefunetions. [87rgi3"Y /3h] %nsspin Nucleus Aobs X.pn8(0) population F19(n=2) 1243 47900 Me 3 Xe129(n=5) 1605 33 030 Me 5 served Axe, 1605 Mc, would correspond to 4.9% Xe Ss spin population. The traceless part of the hyperfine interaction tensor, represented by the parameter B, arises from the inter action between the electron spin and nuclear dipoles. If the unpaired electron is in a pure p orbital the inter action depends on the value of (r-3 )np for the atom and orbital in question. Thus for the fluorine 2p or bita124 (r-3)2p is 7.546 a.u. and the pure 2p hyperfine interaction with p9 is [2gi1I'/ShJ(r-3)2p or 1515 Mc for directions of H perpendicular to the p orbital di rection. The experimentally determined value of BF, 703 Mc, is therefore indicative of approximately 47% F (2p) character in the orbital of the unpaired electron. For Xe (Sp), the value23 of (r-3)5p is 17.825 a.u. and [2gi1I'/ShJ(r-3).p for Xe129 is 1052 Mc. Compared with this, the observed BXe of 382 Mc corresponds to 36% Xe Sp character for the orbital of the unpaired elec tron. The results of these calculations are summarized in Table III. In the absence of nd wavefunctions for xenon, no estimate can be made of the contribution of such or bitals to the experimentally determined Bxe. However, the above discussion indicates that a reasonable de scription of the u-orbital of the unpaired electron in XeF is possible in terms of sand p wavefunctions only. The g-Tensor g The departures of the principal g-values of XeF from 2.0023 (free spin) must be associated with a spin orbit interaction between the ground u-state and excited 7r-states of the molecule. A hypothetical term scheme may be constructed for XeF using fluorine 2p and xenon Sp atomic orbitals, the ground-state configuration being written ... ; (U2PF+USPXe, ul)2; (7r2PF+7rSPXe, 7rl.2)4; (7r2PF-7rSPXe, 7r3.4)4; (u2PF uSPXe, (2)t, or, more briefly, U12; 7rli; 7r3.44; U21. The degeneracy between 7rl.2 and between 7r3.4 may be lifted by the crystalline field, but this possibility is ignored in the following discussion. The transitions and [2gi3-y/5h] %np spin Nucleus Bobs X( r-'}np population f19(n=2) 703 1515 Mc 47 Xe129(n=5) 382 1052 Me 36 are responsible for the g shifts, since both excited states are connected to the ground state by the spin-orbit interaction. If the energy difference between the states connected by the spin-orbit interaction is E Cln-t, and the spin-orbit coupling constant is A cm-\ it can be shown25•26 that and ~gJ.= gJ. -go ~ -2a2Aj E, where a2 is the fractional p character of the orbital of the unpaired electron, and go is the free-spin g value, 2.0023. Of course the exact values of the (AI E)'s are not known but they will be of the order -0.05, so that ~gll is predicted small and negative, whereas ~gJ. should be larger, but positive. The experimentally determined g tensor corresponds to ~gll = -0.028 and ~gJ.= +0.123, in general agreement with these considerations. SUMMARY Electron spin resonance measurements on a l'-ir radiated single crystal of xenon tetrafluoride have yielded information on the trapped radical XeF. The highly anisotropic p9 and Xe129 hyperfine interactions indicate that the unpaired electron occupies an anti bonding u orbital of chiefly F 2p and Xe Sp character. Deviations from the free-spin g-value are consistent with this interpretation. ACKNOWLEDGMENTS The authors are grateful to Dr. L. D. Calvert for his painstaking x-ray analysis of the crystals, and also to Dr. F. P. Lossing for mass-spectrometric measure ments. Thanks are also extended to Dr. D. F. Mayers, who allowed the authors to use his values of Xe lV5.(0) and Xe (r-3 )5p in advance of publication, and also to Dr. E. Whalley who gave valuable advice on the fluorine experimentation. 25T. G. Castner and W. Kanzig, J. Phys. Chern. Solids 3,178 (1957). 26 T. Inui, S. Harasawa, and Y. Obata, J. Phys. Soc. Japan 11, 612 (1956). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.198.30.43 On: Wed, 05 Nov 2014 10:12:08

1.1728609.pdf

Nuclear Resonance in Ferromagnetic Alloys Toshimoto Kushida, A. H. Silver, Yoshitaka Koi, and Akira Tsujimura Citation: Journal of Applied Physics 33, 1079 (1962); doi: 10.1063/1.1728609 View online: http://dx.doi.org/10.1063/1.1728609 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nuclear magnetic resonance in a ferromagnet–semiconductor heterostructure Appl. Phys. Lett. 83, 3335 (2003); 10.1063/1.1620685 Magnetic properties and ferromagnetic resonance in amorphous Gd alloys J. Appl. Phys. 51, 561 (1980); 10.1063/1.327362 Nuclear ferromagnetism AIP Conf. Proc. 24, 772 (1975); 10.1063/1.30283 Nuclear Magnetic Resonance in PalladiumSilver Alloys J. Appl. Phys. 39, 553 (1968); 10.1063/1.2163515 Nuclear Resonance in Ferromagnetic Cobalt J. Appl. Phys. 31, S205 (1960); 10.1063/1.1984666 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Fri, 19 Dec 2014 21:53:18JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 33. NO.3 MARCH, 1962 de Haas van Alphen Effect in Zinc Manganese Alloys* F. T. HEDGCOCK AND W. B. MUIRt The Franklin Institute Laboratories, Philadelphia, Pennsylvania Observations of the long period de Haas van Alphen effect in pure zinc and in a zinc alloy containing 0.01 at.% manganese have been made to determine whether the Fermi surface of the alloy changes in the tem perature region where the zinc manganese alloy exhibits an electrical resistance minimum. On the basis of a nearly free electron interpretation it has been shown that: (i) \Vithin the experimental error the Fermi sur face of pure zinc and the 0.01 at. % zinc manganese alloy are identical. (ii) The number of electrons added or subtracted from the conduction band on alloying cannot be greater than 1 electron per impurity atom. (iii) If the resistance minimum is due to a change in the density of states in the conduction band, this change is less than 0.0015%. (iv) If the ionic state of the manganese ions changes in the temperature range of the resistance minimum, then less than half of the ions are involved. In order to obtain a consistent interpreta tion of the variation of the amplitude of the de Haas van Alphen oscillations in the alloy it was necessary to assume that the collision damping (Dingle) factor varies with magnetic field. A simple extension of the Schmitt scattering model [R. W. Schmitt, Phys. Rev. 103,83 (1956)J would predict this behavior both for the relaxation time derived from magnetoresistance and the de Haas van Alphen effect. * To be submitted for publication in full to The Physical Review. t Submitted as partial fulfillment of the requirement for the Ph.D. degree at the University of Ottawa, Ottawa, Canada. JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 33. NO.3 MARCH, 1962 Nuclear Resonance in Ferromagnetic Alloys TOSHIMoTo KUSHIDA AND A. H. SILVER Scientific Laboratory, Ford Motor Company, Dearborn, Michigan AND Y OSHITAKA KOI AND AKIRA TSUJIMURA Faculty of Engineering, Tokushima University, Tokushima, Japan The internal fields Hi in ferromagnetic alloys were measured at both the solvent and the solute nuclei at magnetic and nonmagnetic atoms using NMR techniques. C059 resonances in Co-rich alloys (Co-Fe, -Ni, -Cu, -Cr, -Mn, -AI) have fine structures which depend on the kind and the concentration of impurity metals. Similar structures are found for the Fe·7 resonances in Fe-rich alloys. These structures are tentatively in terpreted as caused by the anisotropy of Hi at the nearest neighbors. The C059 resonance is also observed in Fe-rich Co-Fe alloys, and it is found that Hi seen at the Co site is lower by about 50 koe than that at the Fe site. The CO·9 line width is about 400 kc for (Fe+ 1 % Co) alloy. Both the temperature and the pressure dependences of the CO·9 frequency were measured. Hi at CU63 and CuG5 was measured in Co-rich Co-Cu and in Fe-rich Fe-Cu ferromagnetic alloys. The magnitude of these internal fields at Cu and their observed pressure dependence are, at least, not inconsistent with the contention that the observed Hi is mainly produced by the 4s conduction-electron polarization, although Hi at the Cu nucleus, 217.7 koe for Fe-Cu alloy and 157.5 koe for Co-Cu alloy, is considerably higher than the usual theoretical prediction. INTRODUCTION SINCE Portis and Gossard! discovered a strong nuclear resonance signal of C059 in ferromagnetic fcc cobalt metal, nuclear magnetic resonance (NMR) technique has been used to investigate the internal field Hi which is the magnetic field seen at the nucleus in many ferromagnetic materials. High accuracy of this method enables us to measure the pressure dependence of Hi2,3 as well as to measure its temperature dependence very precisely. The inhomo geneity of Hi inside the samples caused by either alloy- 1 A. M. Portis and A. C. Gossard, J. Appl. Phys. 31, 205S (1960). 2 Y. Koi, A. Tsujimura, and T. Kushida, J. Phys. Soc. Japan 15, 2100 (1960). 3 G. B. Benedek and J. Armstrong, J. Appl. Phys. 32, 1065 (1961 ). int,5 or by mechanical defects5,6 has been observed as an increase in line width or as additional lines. Thereby more detailed information about the distribution of Hi can be obtained than from a single averaged value of Hi which is obtained from a low-temperature-specific heat measurement,1 although if the inhomogeneity exceeds a certain amount, the lines are smeared out and unable to be observed. The present article will deal with Hi measured at the 4 Y. Koi, A. Tsujimura, T. Hihara, and T. Kushida, ]. Phys. Soc. Japan 16,574 (1961); R. Street, D. S. Rodbell, and W. L. Roth, Phys. Rev. 121,84 (1961). 5 R. C. LaForce, S. F. Ravitz, and G. F. Dav, Phvs. Rev. Letters 6, 226 (1961). . . 6 W. A. Hardy, J. Appl. Phys. 32, 122S (1961). 7 V. Arp, D. Edmonds, and R. Petersen, Phys. Rev. Letters 3, 212 (1959). 1079 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Fri, 19 Dec 2014 21:53:181080 KUSHIDA, SILVER, KOI, AND TSUJIMURA nucleus of both magnetic and non-magnetic impurity metal atoms in ferromagnets, and the distribution of Hi among solvent ferromagnetic atoms. EXPERIMENTAL METHODS NMR in the alloys has been measured using a mar ginal-oscillator spectrometer in zero external field. The spectrometer is frequency modulated and is operated either regeneratively or superregeneratively. The samples were powders of dilute alloys prepared from high purity metals in an induction furnace. The pressure dependence of the resonance frequency has been measured in a steel bomb in conjunction with a modified Bridgman press8 in the range 1 to 8000 kg/cm2• EXPERIMENTAL RESULTS AND DISCUSSION Hi at Magnetic Solute Atoms in Ferromagnets The NMR line of C059 in iron-rich Fe-Co alloy was observed as a function of temperature9 and pressure. The resonance frequency in (Fe+ 1 % Co) was measured from liquid N2 temperature to 650oK. The resonance frequency at OaK, 1'(0), was extrapolated from these values using a T~ law, as 289.2 Mc, corresponding to Hi(O) equal to 289.7 koe. This value is about 30 koe smaller than the value deduced from specific-heat meas urements/ which is much less accurate than NMR measuremen ts. It is noted that although Hi(O) seen at Co in Fe is about 50 koe lower than that at Fe, 339 koe, it is appreciably higher than that at Co nuclei in any of fcc and hcp Co metals. Hi(T)/H;(O) at the Co nucleus decreases with in creasing temperature more rapidly than that at the Fe nucleus. Recently a similar discrepancy between the temperature dependences of Ni61 and C059 resonance frequencies in Ni-rich Ni-Co alloy was noted by Bennett and Streever,lO although the Co frequency drops off more slowly with temperature than the Ni resonance frequency. It is interesting to see whether this discrepancy between the temperature dependences of the Fe and the Co resonance frequencies in Fe-Co alloy is caused simply by an implicit effect of temperature via thermal expansion or by an explicit temperature effect, namely, to see whether or not this discrepancy still exists at constant volume. Measurement of the pressure dependence of the resonance frequency makes this point clear. Since the pressure dependence of the Fe57 frequency in pure Fe has been measured by Benedek and Armstrong3 and this pressure dependence is not expected to change 8 T. Fuke, J. Phys. Soc. Japan 16, 266 (1961). . 9 Y. Koi, A. Tsujimura, T. Hihara, and T. Kushlda, J. Phys. Soc Japan 16, 1040 (1961). IOL. H. Bennett and R. L. Streever, Jr., J. Appl. Phys. 33,1093 (1962), this issue. One of the a~thors! T. K., is gra::ef~l to Dr. Bennett for sending the manuscnpt pnor to the publIcatIOn. TABLE 1. The pressure dependence of the internal field Hi at Co·, nucleus in iron-rich Fe-Co alloy, Cu's nucleus in iron-rich Fe-Cu alloy, and Fe67 in pure iron. Nucleus Co" in Fe Cu·3 in Fe Fc57 in Fe a See reference 3. dlnHddP (kg/cm2)-1 +1.6XI0-7 -3.0XlO-7 -1.6X 10-78 appreciably in our alloy (Fe+ 1 % Co), only the Co resonance frequency in this alloy was measured as a function of pressure. The Co resonance frequency in creases linearly with increasing pressure. On the other hand, the Fe resonance frequency decreases with pres sure. The results are shown in Table 1. The pressure dependence of Cu6:J NMR frequency in Fe, which will be mentioned in the next section, is also shown. Using the pressure dependence of Hi in conjunction with the values of compressibility and thermal expan sion coefficient, the temperature dependence of the in ternal fIeld Hi(T) at constant pressure can be converted into that at constant volume.!! The results are shown in Fig. 1 in a reduced scale, namely as Hi(T)/Hi(O) versus T/Tc, where Tc is the Curie temperature of each ferromagnet. There is a definite discrepancy between Hi(T)/Hi(O) at Co and Fe even at constant volume. This difference in explicit temperature dependence may suggest that the numbers of localized d electrons at Co atom may be slightly increased at higher temperature, though a unique interpretation is very difficult as will be shown later. If the similar discrepancy between Hi(T) at Ni and Co in the Ni-rich Ni-Co alloy found by Bennett and StreeverlO is also predominantly due to an explicit temperature effect, we might speculate that the explicit temperature effect tends to push the d electrons at Fe atoms into adjacent Co atoms and to expel the Co d elec trons into surrounding Ni atoms. The linewidth of the Co resonance was measured as a function of Co concentration from 0.5 to 5%. The width for 0.5% sample is 370 kc and increases with the increasing concentration to 900 kc for the 5% alloy. The center frequency does not change with concentra tion within the experimental error. According to MarshalF2 the internal field at the nucleus in iron group ferromagnets arises mainly from the contact interaction of the nucleus with the 4s elec trons and with the inner core electrons. The contribution from the 4s electrons consists of two parts. The first part is due to the 4s conduction electrons polariz:d by the spins of the 3d electrons, and the second part 15 due to some mixing of the 4s wave function into the 3d band. 11 T. Kushida, G. B. Benedek, and N. Bloembergen, Phys. Rev. 104, 1364 (1956). 12 W. Marshall, Phys. Rev. 110, 1280 (1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Fri, 19 Dec 2014 21:53:18N U C LEA R RES 0 NAN C E IN FER ROM A G NET T CAL I. 0 Y S 1081 The magnitude of Hi due to this mechanism is ex pressed by lJi= -(8'IT/3),u I if; (0) 12np, p= p+2Sa2/n, (1) where I if;(0)j2 is the average probability density of 4s electrons at nucleus, n is the number of conduction electrons per atom, p is their polarization, S is the mean spin per atom, and a2 is an average of the amount of 4s wave function mixed into the 3d wave function. Both parts produce a positive field, i.e., a field parallel to the direction of the magnetization, since the electron g value is negative. On the other hand, the inner core s electrons polarized by the spin of the 3d electrons produce a negative field. The actual field has a negative sign. In the case of dilute ferromagnetic alloys, the main internal field at a solute atom nucleus may come from the following sources: (1) 4s electron polarization at the solute atom. The 4s electrons are polarized by the 3d electrons of the solvent atoms as well as those of the solute atom itself. The internal field from this origin is expressed by Eq. (1). (2) Transfer of 3d electrons between the solvent and the solute atoms on alloying. The change in the number of 3d electrons affects both the 4s electron polarization and the core polarization at the solute atom. (3) Other contributions from the 3d electrons of the solvent atoms surrounding the solute atom. For instance, the core s electrons at the solute atom could be polarized by the 3d spins of the sur rounding atoms directlyl3 and/or indirectly through the 3d shell of the solute atom. Hj (TJ 0.91--------------'i~---_; Hj (OJ I Fe57 in Pure Fe at Const. P 2 Fe57jn Pure Fe al ConsI.V 3 3 Co59jn(Fe+ I%CoJal ConsI.P 4 Co59jn(Fe + I%CoJ 01 ConsI.V 0.80'----'0.-1 -0-'-.2--0 ..... 3--0'-.4-0-'.-5 -0-'-.6--0"".7--' T/Tc FIG. 1. Temperature dependence of the internal field Hi(T) at Fe57 and Co" in iron both at constant pressure and at constant volume in a reduced scale. 13 A. J. F. Boyle, D. St. P. Bunbury, and C. Edwards, Phys. Rev. Letters 5, 553 (1960). TABLE II. The internal field IIi at the Cu nucleus in iron and in cubic cobalt. The data for pure iron and cobalt are included for comparison. Resonance Internal Temperature frequency field Nucleus DC Me koe CU63 in Fe 240.0 0 212.7 Cu6• in Fe 257.1 CU63 in Co 177.9 9 157.5 CU65 in Co 190.7 Fe" in Fe 0 46.65 330.5" C059 in Co 0 213.2 213.5b II. See reference 3. b See reference 1. These interactions make a unique interpretation of the origins of the internal field at the ferromagnetic impurities in ferromagnet& very difficult.l4 Hi at a Nonmagnetic Solute Atom in Ferromagnets The measurement of the internal field at nonmagnetic solute atoms, however, could eliminate some of the causes mentioned above and would help to elucidate the various contributions to Hi in ferromagnets. For instance, a proper choice of nonmagnetic atoms as a field probe could minimize the sources (2) and (3). One could hope to measure the magnitude of conduction electron polarization as in the case of Knight shift ex periments in nonmagnetic metals. Hi at relatively heavy nonmagnetic metals in ferromagnets have been measured using different methods.l3,l5,l6 The internal field caused by (3), how ever, seems to make an estimate of conduction electron polarization difficult.l 3 For instance, the relative magni tudes of Hi at Sn1l9 measured by means of Mossbauer effect in Fe, Co, and Ni give inconsistent values with those expected from conduction electron polarization picture.l3 As a part of a systematic study of Hi at non magnetic metal atoms in ferromagnets, Hi at the Cu nucleus in iron and cubic cobalt has been measured using NMR techniques. Results of the measurements are shown in Table II together with data for pure iron3 and cubic cobaltl for comparison. Unfortunately the sign of the field has not yet been determined. The pressure dependence of Hi at Cu in iron has also been measured, and the results are given in Table 1. The resonance frequency of CU63 decreases linearly with in creasing pressure. It is noted that the internal fields at the Cu atoms 14 G. K. Wertheim, J. App!. Phys. 32, 110S (1961). 15 C. T. Wei, C. H. Cheng, and P. A. Beck, Phys. Rev. 122, 1129 (1961 ). 16 B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Soviet Phys.-JETP 11, 261 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Fri, 19 Dec 2014 21:53:181082 I\: (I S II r [)\, S I J. \. E R, I\: 01, A I\ \l T S II J I 1\1 (T R :\ in iron and in cobalt are comparable in magnitude (~70%) with the fields in pure iron and in cobalt· the ratio of the fields at the eu nucleus in iron and in c~balt i~ nearly.equal to the ratio of the saturation magnetiza tIOns of Iron and cobalt; and the pressure dependence of the field at the Cu nucleus in iron is more negative than that at the Fe in iron. These results suggest that Hi at Cu in these ferro magnets is mainly produced by the conduction electron polarization in ferromagnets. Namely, the mechanism (1) mentioned in the previous section seems to be the dominant one, where the second part of the effective polarization p in Eq. (1), may not be important. Since [f(O) [2 is practically unchangedl3 for Fe and Co, the :al~e of. Hi arising from the conduction electron polar Iza~IOn .IS roughly proportional to the saturation mag n~tlzatlOn. F~rt~ermore, H, at Cu in Co agrees roughly with our prehmmary measurement using Al27 as a field I2robe in fcc CoP This is in agreement with the model ~p. The disturbances caused by the mechanisms (2) and (3) are expected to be much smaller in this case. One of the possible objections to this model is that the magnitude of Hi itself is appreciably larger than the usual theoretical estimatesl2; but it could be within the uncertainty in the theoretical estimations,18,19 if we assume that these measured values have the same sign. The second mechanism (2)20 is, at least, not likely to be the dominant source of the measured Hi. If (2) were the main effect, we may expect d InH;/dP for Hi at Cu to be positive or at least equal to that at Fe. The ob served pressure dependence is, however, in disagreement with this expectation. Furthermore, there seems to be no positive evidence for the Cu impurity in Fe possessing a localized magnetic moment. . The third mechanism (3) cannot be the main source of the field. If it were the dominant term, the same mechanism would play an important role in pure ferro magnets also. This makes it difficult to understand the experimental results that H, at Cu is roughly 70% of that at solvent nuclei, since there is a large additional (subtractive) core-polarization field as expected from the localized magnetic moment at ferromagnetic atoms. In addition, if this assumption is true, the presence of a Cu atom in Co metal will greatly affect the field at Co atoms surrounding the Cu atom. The observed change 4 however, is less than 1%. ' Although the model that the conduction electron polarization is mainly responsible for the observed Hi is not inconsistent with the present experimental results, a more systematic investigation using different non magnetic atoms as the field probes and a determination 17 The preliminary value of Al27 NMR frequency in fcc Co (Co+ 1 % AI) is 189.1 Mc, which corresponds to 170.4 koe. 18 R. E. Watson and A. J. Freeman, J. App!. Phys. 32 118S (1961). ' '9 D. A. Goodings and V. Heine, Phys. Rev. Letters 5 370 (1960). ' 20 W. M. Lomer and W. Marshall, Phil. Mag. 3, 185 (1958). of the sign of Hi at these atoms are highly desirable in order to clarify this problem. Hi at Solvent Atoms in Ferromagnets Internal fields at solvent ferromagnetic atoms which have been disturbed by the presence of impurity atoms can also be observed using NMR techniques. Co reso nance lines in fcc Co-rich alloys have been observed in Co-Fe, Co-~i, Co-.Cu, C?-AI, Co-Mn, and Co-Cr alloys as a function of Impunty concentration. 4 Essentially the same results for Co-Fe and Co-Ni alloys, with so~ewhat better re~olution probably because of higher punty of the constituent metals, have been reported b'y La Force et af.O Similar but less systematic observa tIOns have been made for the Fe resonance in Fe-rich alloys.9,21 The alloy NMR lines generally have structures which are less pronounced in highly-doped specimens: Resolved satellites are observed for cubic Co-Fe and Co-Ni, whereas for the other Co-rich alloys investigated the lower-frequency tails (some of which have fine structures) spread more strongly than the higher frequency tails upon alloying. Since the quadrupole broadening of Co lines in the alloys can be estimated to be of the order of 1 MC,4 the observed structures of the lines essentially describe the distribution of the internal field ~n t~e alloys. !he sh5ft of the center of gravity of the lme IS compatible With the average Hi in Co-Fe and Co-~i alloys observed by means of a low-tempera ture speCIfic-heat measurement/ although the intensity of the far wings in the NMR lines is difficult to meas ure because of the uncertainty in the base line. .Fe57 NMR lines in (Fe+0.9% Cr) and (Fe+0.5% N.I) all?ys. ha:e longer tails at the higher frequency wmgs, mdlCatmg poorly resolved satellites. Since the ~e57 nucleus has no nuclear quadrupole moment, the lme shape change upon alloying has a magnetic origin. Some of the satellite lines observed in very dilute Co alloys have been identified as produced by the presence of stacking faults5,6 and of the hcp phase.2,5,6 These lines are also observed in pure Co samples, and their relative intensities depend on the metallurgical history of the samples. The other satellites in the alloys are produced by the spacial distribution of internal field around the im purity. However, the assignment of each satellite to a particular neighboring site around the impurity is am biguous at present. It is known that the disturbance of the electronic structure in metals caused by an impurity is confined to the immediate neighborhood of the impurity.22-24 The spacial distribution of Hi around the impurity is divided into two parts: (1) a radial distribution, i.e., Hi 21 J. I. Budnick, L. J. Bruner, E. L. Boyd, and R. J. Blume, Bull. Am. Phys. Soc. 5, 491 (1960). 22 J. Friedel, Nuovo cimento, Supp!. 7, 287 (1958). 23 N. Bloembergen and T. J. Rowland, Acta Met. 1, 731 (1953). 24 K. Yosida, Phys. Rev. 106, 893 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Fri, 19 Dec 2014 21:53:18NUCLEAR RESONANCE IN FERROMAGNETIC ALLOYS 1083 varies sharply with the distance, Til between the impurity and the pertinent nucleus j; and (2) an angular distribution, i.e., Hi at the neighboring atoms with the same distance Tj from the impurity might not be equal and could depend on the angle 8 between the direction of the magnetization and the radius vector of the position of the pertinent nucleus. In the case of the NMR lines in the nonmagnetic or the paramagnetic alloys, the hyperfine field in these metals, Knight shift, has been successfully interpreted in terms of (1) and (2).25,26 The density of conduction electrons p(r) about the impurity is modulated as 25,26 Here j! and nl are Bessel and Neumann functions, 01 is a scattering phase shift, and km is the wave number at the Fermi level. The oscillatory nature of Eq. (2) pre dicts the presence of satellites in the NMR lines in these metals.25 Usually the linewidth of each component smears out the expected structure. When the impurity atom has a magnetic moment, the polarization of the conduction electron is also modulated though a spin dependent interaction between the magnetic electrons at the impurity and s-band electrons. The internal field at the neighboring atoms will be doubly modulated in this case.24•26 In the case of ferromagnetic alloys an impurity will affect the internal field at its neighbors in a more com plicated manner. The transfer of 3d electrons between the impurity and the neighboring atoms will strongly affect the value of Hi at the neighbors as well as at the impurity. The 3d electrons at the' impurity would directly polarize the inner core of the neighboring atoms. H. at the nearest neighbors may have a strong angular 26 A. Blandin and E. Daniel, J. Phys. Chern. Solids 10, 126 (1959). 26 D. L. Weinberg and N. Bloembergen, J. Phys. Chern. Solids 15,240 (1960). dependence as welL A classical dipole field from the impurity and anisotropic indirect-coupling26 through the conduction electron may produce this angular de pendence in H ,. A crude estimate of this effect at the nearest neighbors gives about the same order of mag nitude as the observed satellite separation.27 Since the total intensity of the satellite lines in Co+ 1 % Fe or Co+ 1 % Ni alloys is of the order of 10% of the intensity of the entire spectral line and the relative intensity among the satellites are essentially independent of the impurity concentration, it is sug gested that the satellite lines are mainly caused by the anisotropy of Hi at the nearest neighbors, the number of the nearest neighbors being 12 in a fcc structure. The radial distribution of Hi and its anisotropy at the further neighbors may produce the broadening of the component lines, although some of the fine structure could come from the radial part. CONCLUSION The internal field Hi in ferromagnetic dilute alloys was measured at both the solvent and the solute metal nuclei using NMR techniques. Hi at Co in Fe-rich Fe-Co alloy lies about midway between H. at Fe-in-Fe and Co-in-Co. Hi at Cu was measured in Co-rich Co-Cu and in Fe-rich Fe-Cu alloys. The magnitude of these internal fields at Cu and their pressure dependence are, at least, not inconsistent with the contention that the observed Hi is mainly produced by the 4s conduction electron polarization. The structures usually observed in the solute-atom NMR lines are tentatively inter preted as caused by the anisotropy of Hi at the nearest neighbors. ACKNOWLEDGMENTS The authors would like to express their appreciation to Professors A. M. Portis, T. Nagamiya, and K. Yosida for valuable discussions. 27 The difference between Hi at Co in fcc Co and in hcp Co has been explained in terms of the dipole field.28 28 Y. Koi, A. Tsujimura, T. Hihara, and T. Kushida, Report of International Conference of Magnetism in Japan (1961), to be published. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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1.1729824.pdf

Co60 γRadiationInduced Point Defects in Bi2Te3 M. J. Smith Citation: Journal of Applied Physics 34, 2879 (1963); doi: 10.1063/1.1729824 View online: http://dx.doi.org/10.1063/1.1729824 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in First-principles study of native point defects in Bi2Se3 AIP Advances 3, 052105 (2013); 10.1063/1.4804439 Large thermal conductivity decrease in point defective Bi2Te3 bulk materials and superlattices J. Appl. Phys. 113, 013506 (2013); 10.1063/1.4772783 Magneto-resistance up to 60 Tesla in topological insulator Bi2Te3 thin films Appl. Phys. Lett. 101, 202403 (2012); 10.1063/1.4766739 Defects in the Compound Bi2Te3 caused by Irradiation with Protons J. Appl. Phys. 38, 2417 (1967); 10.1063/1.1709917 The OH Yield in the Co60 γ Radiolysis of HNO3 J. Chem. Phys. 35, 936 (1961); 10.1063/1.1701241 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:11JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 9 SEPTEMBER 1963 Co60 y-Radiation-Induced Point Defects in BbTea M. J. SMITH Solid State Division, Oak Ridge National Laboratory, * Oak Ridge, Tennessee (Received 6 March 1963) The effects of extended C060 'Y radiation upon the electrical resistivity, Hall coefficient, and magneto resistance of Bi2Tea have been examined. C060 'Y radiation causes an increase in, the Hall coefficient in n-type Bi.Tea and a decrease in p type. For 'Y-ray exposures in the range of 1018 photons cm-<l, the apparent carrier removal rate is ",10--1 carriers per Coso photon. Thermal annealing of radiation-induced damage was also investigated. The results may be most consistently analyzed in terms of a model which consists of radiation induced tellurium vacancies and interlaminar clusters of Te interstitials. The effects of the radiation-induced point defects are discussed. Evidence of an effect on impurity band conduction at low temperatures in n-type Bi2Tea is observed. INTRODUCTION BISMUTH telluride crystallizes in a rhombohedral lattice, although, for convenience it may be in dexed as a hexagonal structure.! The crystal lattice of the compound is constructed of layers (or leaves2) of atoms lying perpendicular to the major axis of the unit rhombohedral cell. Each layer is composed of atomic sublayers arranged in the sequence TecBi-Te1cBi-TeI. The superscripts are used to distinguish types of differently bonded tellurium atoms.2 The bonding between tellurium atoms in adjacent layers is assumed to be of the van der Waals typea indicating an electron density of effectivelv zero between these bonds. Bi-Tell bonds are assumed' to be fully covalent, while there is some degree of ionicity in the Bi-TeI bonds.2 It is assumed that excess Bi atoms occupy Tell sites and behave as single ionized acceptors, whereas excess Te atoms substitute for Bi atoms and act as single ionized donors.4 These donor and acceptor levels lie in the conduction and valence bands, respectively, so that the defects are ionized at all temperatures.5 Airapetiants et al.6 propose that electron conduction proceeds along Bi layers, and holes move within the Te sublattice. Schultz et aU conclude that bismuth and tellurium vacancies (V Bi, V Te) are electrically active defects and are acceptor and donor types, respectively; whereas, interstitials are effectively neutral. Most theoretical treatments of the band structure of Bi2Tea have assumed that only one current carrier type is present. While this simplifying assumption gives qualitative explanation of most of the phenomena, the real picture of conduction in BhTea is suspected to be * Oak ~idge National Laboratory is operated by Union Carbide Corporation for the U. S. Atomic Energy Commission. 1 M. H. Francombe, Brit. J. App!. Phys. 9, 415 (1958). 2 J. R. Drabble and C. H. L. Goodman, J. Phys. Chern. Solids 5,142 (1958). 3 J. Black, E. Conwell, L. Seigle, and C. W. Spenc.er, J. Phys. Chern. Solids 2,240 (1957). 4 C. B. Satterthwaite and R. W. Ure, Jr., Phys. Rev. 108, 1164 (1957). 5 B. Yates, J. Electron. 6,26 (1959). • S. V. Airapetiants, B. A. Efimove, T. S. Slavilskaia, L. S. Stil'bans, and L. M. Sysoeva, Zh. Tekhn. Fiz. (to be published). 7 J. M. Schultz, J. P. McHugh, and W. A, Tiller, Scientific Paper No. 929-8901-Pl, Westinghouse Research Laboratories (1961). considerably more complex.5,s,9 Drabble and Wolfe10 suggested a "six tilted ellipsoids" model for both the conduction and valence bands. Assuming that only one current carrier type is present, they derive 12 indepen dent galvanomagnetic terms and show that the dimen sionless factors P12a2/Pl1{Jijkl (where the notation is the same as that of Ref. 11) are equal to a constant: P1232/ PllPijkZ=B2(R/ p)2p/6.p=KijkZ. (1) This relation was substantiated for n-type material over a limited range of dopingY However, the results could not be repeated by Goldsmid8 on highly doped n-type crystals. Since most of the electrical and thermal properties of BizTea are extremely sensitive to constitutent-element ratios,4 it is difficult to obtain quantitatively repro ducible data from one specimen to the next. Conse quently, the models which have been proposed for Bi2Tea have been based, for the most part, on qualita tive trends. In order to study the influence of defects on the electronic behavior of extrinsic BizTea, it is desirable to measure the properties of a particular specimen, change its properties by some controlled method of defect introduction without altering stoichiometric proportions, and then quantitatively evaluate the changes. It has been shown in groups IV and III-V semicon ductors that C060 'Y radiation produces point defects which result in a shift of the Fermi level.12 The small, total amount of lattice damage introduced by 'Y radi ation in these semiconductors does not generally alter lattice parameters, bonding strength, effective mass, or over-all band structure. The damage is due primarily to vacancies and interstitials which result in additional doping levels, trapping centers, and variation in current . carrier mobility. 8 H. J. Goldsmid, J. App!. Phys. 32, 2198 (1961). 91. Ya. Korenblit, Fiz. Tverd. Tela 2, 3083 (1961) [English trans!': Soviet Phys.-Solid State 2, 2737 (1961)]. !O J. R. Drabble and R. Wolfe, Proc. Phys. Soc. (Lonoon) 69 1101 (1956). ' 11 J. R. Drabble, R. D. Groves, and R. Wolfe, Proe. PhI'S. So('. (London) 71,430 (1958). ' 12 D. S. Billington and J. H. Crawford, Jr., Radiation Damage in Solids (Princeton University Press, Princeton, New Jersey, 1961). 2879 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:112880 M. ]. SMITH This investigation of radiation damage in p-and n-type Bi2Tea was undertaken with the hope that the results can be used to contribute to the existing knowl ledge concerning point defects in Bi2Tea. The literature contains a limited amount of informa tion on the effects of radiation upon the electrical properties of V-VI compoundsP-17 Artman and Goland1a have studied the effects of a total dose of '" lOla C060 'Y photons at near dry ice temperature on Bi2Tea. The electrical resistivity and thermoelectric power increased in n-type polycrystalline BbTea under these conditions; whereas in p-type specimens, the thermoelectric power increased and the resistivity decreased. The effects annealed with a half-life of 29 h in n'-type specimens. Levy14 observed a decrease in the thermoelectric power of polycrystalline, n-type BhTea after 100-A-h dosage of 2-MeV electrons from a Van de Graaff accelerator. Work on the effects of fast and thermal neutrons upon p-and n-type BbTea has also been reported. 15--17 However, analyses neglected the effects of the high-energy 'Y radiation produced by the Cd used to shield against thermal neutrons.18 EXPERIMENTAL Crystals were prepared from 99.999% pure bismuth and tellurium obtained from the American Smelting and Refining Company. Thirty gram samples of the correct composition ratios were weighed to an accuracy of ±1XIo-4 g. Current carrier type and resistivity were controlled by the excess or deficit of tellurium in the melt.4 The samples were sealed in 9-mm Vycor tubes under a vacuum of '" 10-5 mm of Hg. To obtain an ingot of Bi2Tea, the capsules were dropped through a Bridgman furnace at a rate of '" 1.3 cm h-1• The cap sules were broken and the ingots placed in reactor-grade graphite crucibles. The crucibles and ingots were re sealed in Vycor capsules. The capsules were held above the melting point of Bi2Tea in a Bridgman furnace for 24 h, in order to provide thorough mixing in the liquid phase. The capsules were then lowered through a temperature gradient of ",80°C cm-1 at a rate of 0.14 cm h-1• Single-crystal specimens were usually obtained, although frequently the ends of the ingot were poly crystalline. The cleavage planes (0015) were parallel to the growth direction within 20. Single crystals of both p-and n-type specimens with a range of electrical conductivity were cleaved from the ingots. The samples for electrical measurements were 13 R, A, Artman and A, N, Goland, Bull. Am, Phys, Soc. 5, 168 (1960), 14 R, A. Levy, BulL Am. Phys. Soc. 5, 168 (1960). 14 R. A. Levy, Bull. Am. Phys. Soc. 5, 168 (1960), 151-C. Corelli, R. T. Frost, and F. A. White, Bull. Am. Phys. Soc. 5, 168 (1960). 16 M. Balicki, J. c. Corelli, and R. T. Frost, Metallurgy of J(lemental and Compound Semiconductors (Interscience Publishers, Inc., New York, 1961), Vo!' 12. 17 R. A. Artman, Bull. Am. Phys. Soc. 7, 187 (1962). 18 J. W. Cleland, R. F. Bass, and J. H. Crawford, Jr., J. Appl. Phys. 33,2906 (1962). cut with "dog-ears" electrical leads by a Glenite ultra sonic cutter. Specimen measurements were ",0.OSXO.20 XO.6S cm. Electrical leads were soldered in place with No. 4300-R liquid flux and No. 4300 Bi solder obtained from the American Brazing Alloys Corporation. Since cutting and preparing of the specimens introduced defects, it was necessary to anneal the specimens at 50°C for a period of ",24 h prior to measurement of preirradiation properties. Measurements were to be made on each specimen before and after radiation in a C060 'Y source. Therefore, to avoid annealing of the radiation damage upon re soldering the leads, the leads were soldered to the specimen before beginning the experiment, the specimen and leads were secured to a specimen mount, the measurements taken, and the entire specimen mount assembly was placed in the 'Y source for irradiation. Since percentage changes in properties upon radiation were the main points of interest, this technique in creased the accuracy of the measurements by eliminat ing errors in measurements of specimen dimensions. The Co60'Y source which was employed has a photon flux of 1016 cm-2 h-1 and an ambient temperature of ",45°C. Resistivity and Hall measurements, parallel to the cleavage plane, were made by employing the constant temperature apparatus described by Ure.19 This appara tus gives good temperature control and avoids thermal gradients, thereby nullifying thermoelectric voltages. An aluminum specimen mount was used which fits snugly into the specimen chamber indicated in Ref. 19. The Ure apparatus was used also for thermal anneal ing of the specimens before and after radiations. Post anneal electrical properties were measured at -196° and -123°C and recorded vs time of anneal. Annealing temperatures ranged from room temperature to 130°C. Anneal data were taken before and after radiations to insure that the observed changes in properties upon annealing were due only to radiation effects. The measurements of Drabble et aZ.n were repeated, in part, on irradiated BbTea primarily to determine the validity of Eq. (1). Because of experimental difficulties discussed above, data were restricted to those which could be obtained with the current parallel to the cleavage plane. As a consequence only Kllll and K1l33 of Eq. (1) were obtained. RESULTS C060 'Y radiation produces two independent effects upon the electrical properties of Bi2Tea. The first effect is seen after small doses ('" 1016 photons cro-2) of 'Y and x radiation, and is not significantly dependent upon the energy of the photon or specimen temperature during irradiation. The second and most important effect appears after extended exposures (> 1018 photons cm-2) and is considered to be associated with lattice damage. 19 R. 'V. Ure, Jr., Rev. Sci. Insle. 28, 836 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:11CO'; 0 'Y -RAn I A T ION -INn IT C ED POI N T DE FEe T SIN R i, T (,,, 2SS1 TABLE 1. Properties of Bi2Te, after varying exposures to C060 'Y radiation. Temperature of measurement: -196°C. 4> RJ R2 pXlO-3 J P2XlO-' Specimen h cm'/C cm'/C O-cm O-cm An 0 2.72 2.72 0.563 0.563 I 2 2.53 2.72 0.530 0.566 94 2.91 3.09 0.572 0.608 1 191 3.02 3.25 0.610 0.630 291 3.1.J. 3.34 0.630 0.661 Bn 0 2.38 0.305 j 100 2.70 0.327 200 2.92 0.344 300 3.00 0.357 0 0.305 100 0.327 200 0.344 300 0.357 A. 0 3.73 3.73 1.40 1.40 1 2 3.92 3.73 1.46 1.35 102 3.10 2.87 1.28 1.14 Bp 0 4.66 4.66 1.93 1.93 1 99 5.25 4.55 2.06 1.97 103 5.26 4.56 2.06 1.91 298 4.42 3.69 1.72 1.70 "8 =arc SinC~~!J For reasons given, the low 'Y-fiux effect is described as electronic. The electronic effect decreases the magnitude of the Hall coefficient in n-type and increases it in p-type Bi2Tea and is observed in all specimens. It anneals at room temperature in 10 to 24 h. The magni tude of the electronic effect is dependent upon the initial characteristics of the crystal and is relatively insensitive to the radiation history of the crystal. Table I gives changes in the Hall coefficient and resistivity of several illustrative specimens as a function of radiation histories. The values tabulated under the heading R1-R2 give the Hall coefficient after the specified time in the source minus the coefficient after a 24-h anneal at room temperature. Hence, this column gives the changes in the Hall coefficient which are due only to the elec tronic effect. The temperature of measurement of the data of Table I is -196°C. For C060 'Y-ray exposures > 1018 photons cm-2, the second, or lattice damage, effect appears. This effect reduces the magnitudes of the Hall coefficient and electrical resistivity in p-type and increases them in n-type Bi2Tea. Magnetoresistance parallel to the cleav age planes increases in n-type Bi2Tea with extended 'Y radiation. Room temperature annealing is negligible after a period of 100 h. The values tabulated in columns R2, P2, and t.p/ p of Table I indicate the changes in the Hall coefficient, resistivity, and magnetoresistance due to the second effect. These values were all recorded after sufficient time had elapsed for complete anneal of the low 'Y-fiux, electronic effect. At annealing temperatures greater than 50°C, re covery of the high 'Y-fiux effect is observed. When post annealing properties are obtained at -196°C, approach to preirradiation properties occurs in two stages. At low annealing temperatures or short times, the Hall coeffi-R,-R2 P,-P2 R/,?X106 cm3/C X 10-' O-cm cm/V-sec t:.p/p Ki;kl OS 8.58 90° -0.19 -00.36 8.50 j -0.18 -0.036 8.35 -0.23 -0.020 8.19 -0.20 -0.031 7.64 25.6 20.92 1.046 90° 25.4 22.94 1.070 1 24.7 23.41 1.108 23.5 22.30 1.139 6.70 3.27 00 6.68 3.68 1 6.67 3.89 6.65 3.82 90° +0.19 +0.11 1 +0.23 +0.14 1.25 90° +0.70 +0.09 1.17 1 +0.70 +0.15 1.25 +0.73 +0.02 1.28 cient increases in p-type and decreases in n-type BbTea with time; whereas, the resistivity changes only slightly. This process is reversible in the following sense: If the specimen is held at room temperature for "-'12 h after a period of anneal during the first stage, the Hall coeffi- cient returns to the postirradiation value. At higher anneal temperatures or long times, the recovery process TABLE II. Thermal anneal of C060 'Y-radiation damage in specimen Ap. (Req=3.730 cm'/C, peq= 1.40 O-cm, time in source: 102 h, temperature of measurement: -196°C.) Anneal Anneal temperature time I/R P ( (OC) • (103 sec) C/em' (lO-'O-em) (%) ---.------ - 50 0 0.3483 1.14 100 I 9 0.3451 1.14 96 18 0.3371 1.15 86 35 0.3323 1.16 80 52 0.3258 1.16 72 67 0.3218 1.15 67 78 0.3194 1.16 64 96 0.3210 1.16 66 113 0.3178 1.17 62 130 0.3162 1.17 60 72 0 0.3389 1.14 100 9 0.3247 1.14 80 18 0.3141 1.15 65 30 0.3070 1.16 55 43 0.3070 1.17 55 69 0.3021 1.22 48 94 0.2957 1.23 39 113 0.2957 1.26 39 123 0.2915 1.26 33 100 0 0.3194 1.19 100 j 9 0.2964 1.26 40 18 0.2830 1.28 21 30 0.2773 1.34 13 38 0.2738 1.37 8 44 0.2716 1.36 5 54 0.2702 1.39 3 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:112~82 1\[. .1. S 1\1 IT H .23.5 .-____ 4-•• ~~ ____ ~ FIG. 1. Effect of Coso l' radiation upon R/,J of Bi2Tea. Temperature of measurement: -196°C. becomes irreversible. The ratio of the Hall coefficient to resistivity, which usually defines the Hall mobility, increases slightly during the second stage. Table II shows the annealing behavior of the Hall coefficient and resistivity of a typical specimen which had received 1018 C060 'Y photons. Post anneal data are taken at -196°C. If data are taken at temperatures above -123°C after anneal at temperatures greater than 50°C, only one anneal step is observed. Post anneal data at -123°C follow the same general trends as the second anneal process of Table II. DISCUSSION When p-and n-type BbTe3 are exposed to light doses of "I or x radiation, an electronic effect appears which causes an increase in the ratio R/ p. This condition is unstable at room temperature. Since this effect is in dependent of the energy of radiation and saturates after very small doses, it is considered to be purely electronic. The changes in the electrical properties and the re covery of these changes are in qualitative agreement with the low "I-exposure results reported by Artman et alY and the preliminary results of Smith.20 Extended 'Y radiation produces a more stable effect upon p-and n-type Bi2Tes which appears to be associ ated with lattice damage. The observable changes in the resistivity and galvanomagnetic properties after 1018 photons cm-2 of C060 'Y radiation are surprisingly large for the high number of current carriers (1017 to 1019 cm-a) present in the samples. Significant changes in the Hall coefficient of Ge cannot be seen after 1018 photons cm-2 if the number of carriers significantly exceeds 1015 cm-3.12 If the simple relation, n=1/eRH (2) for the number of current carriers as a function Hall 20 M. J. Smith, Solid State Div. Ann. Progr. Rept., 31 August 1962, ORNL-3364, p. 91. coefficient is assumed, the apparent C060 'Y radiation carrier removal rate in Bi2Tes is '" 10-1 current carrier per C060 'Y photon in the range of 1018 photons cm-2• The galvanomagnetic factors of Eq. (1) are tabulated in Table I as a function of C060'Y exposure. It is evident that K is not constant but is a slowly varying function of lattice damage. A reasonable condition for the con tradiction of Eq. (1) is the existence of more than one current carrier type at low temperatures in Bi2Tes. Multiple carrier type conduction was also proposed by Yates5 who suggested impurity band conduction as a model for explaining the temperature dependence of the electrical properties of n-type Bi2Tea. If one can assume only one n-and one p-lype im purity band which is capable of conduction, the general expression for the Hall coefficient is given by: R/ p2= e(rlnMI2-r2PM22+rsniJ1.l-r4PiJ.L42), (3) where the subscript i refers to impurity band. The first two terms correspond to conduction and valence band conduction, respectively. We assume that radiation damage modulates one or more of the quantities in Eq. (3) and that the isothermal expression H¢)=A¢ (4) is valid; here ~ is the physical quantity which is modu lated, A is a constant, and ¢ is the total integrated C060 'Y flux. It now becomes informative to plot experimental values of R/ p2 vs ¢2 and ¢. Figures 1 and 2 give these plots of data taken at -196° and -123°C, respectively, for an n-type crystal. It is seen that R/ p2 may be ex pressed by: at low temperatures j whereas, R/p2=B'-C'¢ (5) (6) at higher temperatures, where Band C are constants. Assuming r of Eq. (3) remains relatively constant and Eq. (4) is correct, one may conject from Eqs. (3), (5), ,----- -~---~ ---, I ' 1.30 \ 25 ! • I'i ' ~ 1.20 L _____ ~ ____ ~_~ _____ ._----"~-----i o 50 100 150 200 250 300 350 400 .. Ih I FIG. 2. Effect of Coso l' radiation upon R/,J of Bi2Tea. Temperature of measurement: -123°C. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:11C 060 'Y -R A D I A T I 0 i\' -I N Due E D POI N T D E FEe T SIN B i 2 T e 3 2883 and (6) that at low temperatures, where impurity con duction is significant, radiation damage affects pri marily mobilities, JJ.3 and JJ.4 of Eq. (3). However, at higher temperatures, where conduction occurs primarily via the conduction and/or the valence band, the concen tration of carriers is altered by irradiation. The quantity R/ p2 is relatively constant in the case of the p-type specimen. This phenomenon suggests that the impurity bonding effect is not so important in p-type material. Until a more sophisticated model for impurity band conduction in Bi2Tea is available, a simple explanation for this discrepancy in the case of p-type material can not be offered. With the use of the foregoing model, which appears to be appropriate at least for n-type BbTea, some conclu sions may be drawn from the anneal data. Two stages of anneal are observed when the post-annealing meas urements are made at -196°C; whereas, only one is observed if the data are taken above -123°C. It appears that the first stage involves reduction of the disturbance to the impurity band; whereas, the second is associated with recovery of carrier concentration and annihilation of radiation damage. The anneal stages seem to be first-order processes. In a first-order anneal process, the fraction of defects remaining after t seconds of anneal at temperature T is given by: f=A exp(-kt), (7) where k is defined as the rate constant. If the reciprocal of the Hall coefficient is proportional to the defect concentration,I2 the fraction of defects remaining is WO ....::~--,-------,-----,-----r-----,--------, 20 8=t =--r-----' • T=60 DC T=50 °C 4 -.-------l..---+-----'~__c=~-_1___--_+---- 10 12 f (sec) FIG. 3. Fraction of defects remaining in specimen Ap vs time of anneal. (A constant correction factor has been subtracted from each point. See text.) TEMPERATURE (OC) 120 100 80 60 40 163 ~t:=±=:t~~--~--~:::!f:±! ---=::"l--t:::-:-~-=-tl, ~: ---=-==_ ~_ == __ ~~~ r---~±-:..:~t1 :------;--+-----l---i 5 ---~---+-- r----- 2 f--T--:~-t _~_I 4,,66 "---_-'---_-"--_--L_-----' __ L-.--'-------' 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 1000/r (OK) FIG. 4. Rate constant vs lOOOIT for the secondary anneal process. defined by: f= (1/Req-1/R)/(l/Req-1/R i), (8) where Req is the equilibrium or preirradiation Hall coefficient, R is the Hall coefficient following anneal, and Ri is the postirradiation Hall coefficient. If the recovery process has a unique activation energy, the rate con stant k is given by: k=koexp(-E/kT), (9) where E is the activation energy for annihilation of the defect. Figure 3 gives a plot of the fraction of defects remain ing in the specimen of Table II after anneal. Since preparation of the specimens introduce annealable defects, the postirradiation annealing often resulted in a greater recovery than expected from the radiation induced property change. Hence, it was necessary in many instances to adjust Req of Eq. (8) by a constant correction factor of ",5%, or less, in order to obtain the exponential behavior of Eq. (7). When the total inte grated 'Y flux significantly exceeds 1018 photons, the anneal curves begin to deviate from Eq. (7). Neverthe less, the electrical properties continue to approach the same post anneal values after long anneal times. The elevated temperature anneal process evidently involves annihilation of a charged defect. The activation energy obtained from Fig. 4 and Eq. (9) for the annihi lation of the defect is 0.9±0.05 eV. Kuliev and Abdul laev21 report an activation energy of 2.18 eV for the diffusion of Se in Bi2Sea. Since Te and Se are similar in size and electronic configuration, it is reasonable to assume that the activation energy for the diffusion of 21 A. A. Kuliev and G. D. Abdullaev, Fiz. Tverd. Tela I, 603 (1959) [English trans!': Soviet Phys.-Solid State 1,545 (1959)]. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:112884 :VI. J. S MIT H TEMPe;RATURE (oC) 3 x 10'3 r-'C,0-----, __ ',OO-,-_---,-8,O_..,- __ 6rO_----,_----,,40 5 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 'OOO/r (OK) FIG. 5. Rate constant vs 1000/T for the initial anneal process. Te in Bi2Tea is similar to that for the diffusion of Se in Bi2Sea. The activation energy for diffusion Ed is the sum of the vacancy formation energy Ej and the vacancy motion energy Em, i.e., (10) If it is assumed that22 (11) the results of Ref. 21 indicate that Em~ 1 eV for the energy of motion of Te vacancies in Bi2Tea. This value is in reasonable accord with the activation energy of O.9±O.05 eV obtained from the slope of the curve of Fig. 4. When postirradiation data are taken at -196°C after low temperatures of anneal, the Hall coefficient anneals reversibly in temperature with an activation energy obtained from Fig. 5 of O.7±O.1 eV. If dusters of interlaminar, defect atoms result in a buckling of the 22 C. E. Birchenal, Atom Movements (American Society for Metals, Cleveland, Ohio, 1951). la ttice such as has been observed in graphi te, 12 one would expect that the reversible anneal process might well be dissociation of dusters of interstitial Te atoms at anneal temperatures. On cooling to room temperature, the Te atoms reassociate. Higher thermal energies are neces sary for dissociation, diffusion, and annihilation of the Te vacancy-interstitial defect. It is significant that the lattice buckling phenomenon does not affect the elec trical properties when the property measurements are made above -123°C which is the temperature range in which the impurity band contribution should be negli gible. However, preliminary measurements indicate that thermal phonons are scat tered over the entire temperature range. Hence, it appears that Te inter stitials resulting in the buckling of the lattice scatter only impurity carriers. CONCLUSIONS It would appear, therefore, that the model for 'Y radiation damage which most consistently agrees with the experimental results presented above consists of, at least in part, a radiation-induced Te vacancy which is an electrically active defect in both p-and n-type BhTea and interlaminar dusters of interstitial Te atoms. The rate of anneal of radiation damage is controlled by the motion of the Te vacancies. Radiation disturbance of the electrical properties is acceptor type and reduces the apparent Hall mobility only slightly at room tem perature. It may be concluded from the data that more than one type of carrier is present in n-type Bi2Tea at low temperatures and that radiation induced inter laminar clusters of Te atoms alter the carrier mobility in the impurity band. ACKNOWLEDGMENTS The author is grateful to O. L. Curtis, Jr., and J. H. Crawford, Jr., of this laboratory for their consultations and suggestions in the interpretations of the results of this research program. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 06:26:11

1.1725637.pdf

Paramagnetic Resonance of Mn++ in NaN3, KN3, and RbN3 Gerard J. King and Brian S. Miller Citation: The Journal of Chemical Physics 41, 28 (1964); doi: 10.1063/1.1725637 View online: http://dx.doi.org/10.1063/1.1725637 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/41/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A rigidion model of the structure, vibrational, and elastic properties of the tetragonal alkaliazide crystals KN3, RbN3, and CsN3 J. Chem. Phys. 96, 1735 (1992); 10.1063/1.462128 Neutron Investigation of the Spin Dynamics in Paramagnetic RbMnF3 J. Appl. Phys. 42, 1378 (1971); 10.1063/1.1660257 Nuclear Magnetic Resonance in RbMnF3 J. Chem. Phys. 42, 3806 (1965); 10.1063/1.1695841 Antiferromagnetic Resonance in Cubic RbMnF3 J. Appl. Phys. 34, 1036 (1963); 10.1063/1.1729359 Paramagnetic Resonance of Color Centers in NaNO3 J. Chem. Phys. 23, 1967 (1955); 10.1063/1.1740630 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4428 JAMES A. IBERS values are approximately 10% higher than the corre sponding ones for KHF2 and KDF2 (Table I). For the NaHF2-NaDF2 systeml a careful study of the isotope effect was carried out by superposition of high-order reflections from both substances on the same Weissen berg film. With the use of CoKal and CoKa2 on the 0.0.15 reflection it was found that at 27°C CH-Cn= 0.0137±0.0016 A; with MoKal and MoKa2 on the 220 reflection it was found that aH is equal to an to within 0.001 A. Because of the three-layer repeat in sodium acid fluoridel these data lead, on the assumption of localization of isotopic changes, to These differences in frequencies and in isotope effect between the sodium and potassium salts undoubtedly arise from differences in the bifluoride ion environment. The potential function used here, since it is based on the assumption of an isolated ion, cannot account for such environmental effects. Nevertheless, the simple function proposed here should be useful in the inter pretation of bifluoride ion spectra, and the collection of reliable infrared and Raman data on the NaHF2- NaDF2 and KHF2-KDF2 systems is to be encouraged. ACKNOWLEDGMENT (F-H-F)- (F-D-F)=0.0046±0.0005 A. This assumption is made more credible by the absence of an isotope effect in the a direction. THE JOURNAL OF CHEMICAL PHYSICS It is a pleasure to thank M. Wolfsberg for his interest, for some very helpful discussions, and for criticism of the manuscript. VOLUME 41, NUMBER 1 1 JULY 1964 Paramagnetic Resonance of Mn ++ in NaN 3, KN 3, and RbN 3 GERARD J. KING AND BRlAN S. MILLER Basic Research Laboratory, USAERDL, Ft. Belvoir, Virginia (Received 4 November 1963) The paramagnetic resonance of Mn++ in single crystals of NaNa, KNa, and RbNa is studied at 9.1 Gc/sec as a function of crystal orientation in the magnetic field. In KN3 and RbN3 containing Mn++ the crystalline electric field is the resultant of a large axial field in the [001] direction plus a small cubic field. The g values for Mn++ in KN3 are gil = 1.9961±0.OO05, g.l= 1.9878±0.OOSOj and for Mn++ in RbN3 gil =2.000S±0.OOOS, g.l=1.9971±0.00SO. The axial electric field parameter D is -S34±3.0 G for KNa and -278±3.0 G for RbNa at 2SoC. The cubic field parameter ao is lO±O.S G for KN3 and 8.7±0.S G for RbNa. The Mn++ hyperfine coupling constants A and Bare -89.7 and -91.1±0.S G, respectively, in KNa. In RbN3, A and Bare -88.0 and -88.9±0.S G, respectively. The large magnitudes of D, A, and B allow the forbidden LlM=±I, Llm=±1 transitions to be intense. Two inequivalent sites result from the displacement of the Mn++ from a cation site toward a bound nearest-neighbor cation vacancy. For an unheated crystal of NaN3, the main Mn++ resonance is a single broad line at g= 1.9S±0.01. Heat ing the NaN3 crystal changes this broad line into mUltiple sets of 30-line spectra. At 2SoC these sets of 30-line spectra decay slowly and the original broad line regrows. A similar effect previously found for Mn++ in NaCl has been attributed to mobility of Mn++-cation vacancy complexes. One type of Mn++ spectrum in NaN3 is due to the vacancy-associated complex and another type is due to the dissociated or excited complex. For both types of spectra gO =g.l= 2.001±O.OO2 and A ""B=87±1.0 G. For the dis sociated Mn++ complex state the spectrum has axial symmetry about the c axis and D= -240 G. For the vacancy-paired Mn++ spectrum an additional rhombic distortion occurs, and D= -265 and E= +S7 G. The variation of linewidth with temperature is used to show that the KNa spectra result from charge com pensation by vacancy pairing. Additional effects produced by vacancy jumping are noted. Differences between the high-temperature properties of Mn++ in NaNa and KNa are related to cation size effects. A low temperature line broadening of the Mn++ spectrum in KNa and RbNa is reported, and the similarity to the Mn++ resonances in solution-grown KCI and KBr is noted. I. INTRODUCTION WHEN Mn+ + ions are substituted in an alkali ion site in alkali halide crystals, the ion charge dis parity results in the generation of an alkali ion vacancy to preserve electrical neutrality. For the proper thermal conditions, the alkali ion vacancy will associate directly with the Mn+ + ion and a complex defect center will form. The existence and the properties of these Mn+ + complexes have been established and studied by many authors, but strongest evidence has been presented by Breckenridge,! by Schneider and Caffyn,2 and later, by Watkins.3 1 R. G. Breckenridge, J. Chem. Phys. 18, 913 (1950); Imper fections in Nearly Perfect Crystals (John Wiley & Sons, Inc., New York, 19S2), p. 219. 2 E. E. Schneider and J. E. Caffyn, Report on the Bristol Con ference on Defects in Crystalline Solids, 1954 (The Physical So ciety, London, 19S5) , p. 74. 3 G. D. Watkins, Phys. Rev. 113, 79, 91 (1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44ESR OF Mn++ IN NaN., KN., AND RbNa 29 Breckenridge l investigated the electrical effects of impurity divalent cations in alkali halides and attrib uted the temperature-dependent, resonant electrical loss effects to the defect structure generated for charge compensation of the divalent ion. Watkinsa used para magnetic resonance experiments to study Mn+ + in alkali chlorides. He was successful in interpreting the spectra by attributing the orientational properties of the electric field splittings of the resonance patterns to the alkali ion vacancy sites associated with the Mn+ +. It was shown that a number of energetically equivalent positions could be assumed by the associated vacancy in the cubic lattice near the Mn+ +. This site degeneracy was removed by the field interactions in the paramag netic resonance experiment, producing extremely com plex sets of spectra. By relating the symmetry proper ties of the electric field splittings of the spectra with the symmetry of the host lattice site, Watkins was able to assign the different spectra to specific configurations of the Mn+ + and its paired vacancy. Watkinsa also studied the thermal effects on the line widths of the paramagnetic resonance spectra. Ther mally promoted jumping of the vacancy from one neighboring cation site to another can produce a Stark modulation of the paramagnetic resonance spectra if lifetime conditions are favorable. This was observed as a lifetime broadening of the resonance lines as temper ature was increased above 200°C. Relaxation times were computed for these jump processes from linewidth measurements. Watkins determined the relaxation times for the same process in separate dielectric loss measure ments and obtained a direct correlation with the para magnetic resonance results. It is our intent in this paper to present some para magnetic resonance experiments which indicate that a similar type of Mn+ +-alkali-ion-vacancy complex exists in a series of compounds related to alkali halides, namely, the alkali azides. The azide ion is usually considered as a halogenoid 4 with respect to chemical behavior. In ionic materials, the azide ion is a linear symmetric tri atomic chain of nitrogen atoms which are covalently bonded together. Potassium, rubidium, and cesium azides are highly ionic and have body-centered tetrago nal crystal structure. The structural similarity of alkali halides and alkali azides was noted by Hendricks and Pauling.5 The deformation of the CsCl structure to the body-centered tetragonal structure is in such a manner as to accommodate the nonspherical shape of the azide ion itself. Recent optical studies have shown that alkali azides have exciton levels and band-to-band transitions6 which are strikingly similar to those of alkali halides. An additional optical absorption band group around 2225 A has been attributed to a low-lying excited level of the azide ion. In sodium azide, which has rhombo- 4 L. F. Audrieth, Chern. Revs. 15, 169 (1934). is. B. Hendricks and L. Pauling, J. Am. Chern. Soc. 47,2904 (1925). 6 S. K. Deb, J. Chern. Phys. 35,2122 (1961). hedral or hexagonal structure, irradiation produces elec tron spin resonance absorptions and an optical band, both attributable to an F center.7,8 Other electron spin resonances found in sodium and potassium azides after irradiation have been attributed to nitrogen or nitrogen fragments.9-l2 The alkali azides exhibit behavior attributable to general lattice behavior in some cases and to specific properties of the azide ion in some cases. The general lattice behavior is similar to that observed in the alkali halides. To study this latter behavior, an attempt is made here to extend the Mn+ + doping experiments to the alkali azides for purposes of comparison with Mn+ + -doped alkali halides. II. EXPERIMENTAL Single crystals of NaNa, KNa, or RbNa were grown at 25°C by slow evaporation of 200 ml of aqueous solu tion in polyethylene containers. Doping with Mn+ + was usually accomplished by adding milligram quanti ties of MnCb or MnS04 to the growth solution. Crystals used in these experiments had, typically, a manganese concentration of 1 X 10-4 mole fraction as determined by colorimetric analysis after appropriate chemical treat ment. The body-centered tetragonal5 crystals of KNa and RbNa were clear, colorless, and showed a well developed bipyramidal form. The tetragonal c axis ([OOlJ direction) ran between the peaks of the bipyra mid, and the equivalent [110J; [IlOJ axes formed the edges of the common basal plane. Crystals containing lO-a mole % Mn+ + were found to have strongly re tarded growth in the c-axis direction. Orientations were checked by x-ray analysis and optical goniometry. Sodium azide crystals were found to grow in plates with major surfaces perpendicular to the c axis. Addi tion of Mn+ + to the growth solution in this case pro moted growth in the c-axis direction. Sodium azide has hexagona15 crystal structure, with the hexagonal plane lying in the developed crystal face for temperatures above 19°C.la Paramagnetic resonance measurements were made using a Varian V-4500 X-band spectrometer with 100- kc/sec modulation and a V-4531 multipurpose cavity. Magnetic fields were measured by proton resonance, and microwave frequencies were determined by a Hewlett-Packard 540B counting system. The crystals were glued to vertical glass rods which connected to a pointer for indicating angular orientation. 7 F. F. Carlson, G. J. King, and B. S. Miller, J. Chern. Phys. 33,1266 (1960). 8 G. J. King, B. S. Miller, F. F. Carlson, and R. C. McMillan, J. Chern. Phys. 35,1442 (1961). 9 A. J. Shuskus, C. G. Young, O. R. Gilliam, and P. W. Levy, J. Chern. Phys. 33, 622 (1960). 10 D. W. Wylie, A. J. Shuskus, C. G. Young, O. R. Gilliam, and P. W. Levy, Phys. Rev. 125, 451 (1962). 11 G. J. King, F. F. Carlson, B. S. Miller, and R. C. McMillan, J. Chern. Phys. 34, 1499 (1961). 12 F. F. Carlson, J. Chern. Phys. 39, 1206 (1963). 13 B. S. Miller and G. J. King, J. Chern. Phys. 39, 2779 (1963). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4430 G. J. KING AND B. S. MILLER o \ -\ \. \ \ \ 1 IS \ \ \. \ \ 30 " " '" 45 '" r / 60 1 75 90 \ \ J \ / \ J J/ 1\ ./ VI \/ ~ II /\ j:1 v /] r---.../ IV /-, '..l\ V ij~ -" '\ 1 / / "\ 1 .I L V I r-FIG. 1. Fine structure lines RNa:Mn++. Magnetic field position of the centers of the sextets as a function of angle () between c axis and magnetic field (in gauss) for Mn++ in RNa. The solid lines are computed from Formula (1) using the value of D derived from the ()=o spectrum. ® Experiment; -theory. 1.200 2200 3200 4200 5200 The end of the rod was ground off square to give a reference plane for the crystal. The magnet remained fixed while the crystal was rotated by turning the glass rod around its vertical axis. The cavity was a TE102 type with the broad faces parallel to the pole faces of the magnet. The entry point for the glass-rod crystal mount was at the center point of the narrow 02 face, which was horizontal. The crystals of KNa: Mn+ + and RbNa:Mn++ were glued to the end of the rod in two positions. In one position, the c axis was perpendicular to the cylindrical axis of the rod, with a [110J edge of the crystal contacting the square end of the rod. For this orientation, rotation of the rod around its vertical axis displayed the spectra as a function of the angle (J between the c axis and the dc magnetic field. In the other position, the crystal was mounted with the c axis aligned with the vertical axis of the glass rod. Rotation of the rod around its vertical axis displayed the spectrum as a function of the angle if; between the [110J direction and the magnetic field. In connection with the growth habit of KNa and RbNa, it was found that all of our crystals, pure or doped, grew with basal plane edges which were [110J, [110J crystal directions. This represents the results of x-ray analysis and optical goniometry, as well as the results from paramagnetic resonance. This same result has been obtained by others.14 One reference 15 states 14 R. B. Horst, J. H. Anderson, and D. E. Milligan, Proceedings of the Tenth Basic Research Group Symposium, 1961 (unpub lished). 16 B. L. Evans, A. D. Yoffe, and Peter Gray, Chern. Revs. 59, 515, 525 (1959). that crystals of KNa "may be" grown in another habit, but we have not found this effect in our crystals. For high-temperature measurements, the use of glue was eliminated by cutting and shaping crystals to fit closely into quartz tubes, which were then evacuated and sealed off. III. DESCRIPTION OF RESULTS FOR KNa AND RbNa The discussion in this section pertains to the spectra of Mn+ + in KNa and RbNa. The spectra observed in NaNa show special features which warrant separate discussion. The results for NaNa: Mn+ + lend strong TABLE 1. The spin-Hamiltonian parameters· for Mn++ in RNa and RbNa. KNa gil = 1. 9961±O.OO05 gJ.= 1. 9878±O.OO50 A = (-)89. 7±O.5 Gb B=(-)91.1±O.5G D=-534±3.0 G a=+10±O.5 G RbNa gil =2.0005±O.OOO5 gJ. = 1. 9971±O.0050 A = (-)88.0±O.5 Gb B = ( - ) 88 . 9±O .5 G D=-278±3.0 G a=+8.7±O.S G Temperature = 25°C • Preliminary values were reported byG. J. King, R. C.McMillan, B. S.Mil Ier, and F. F. CarIson, BuH. Am. Phys. Soc. 7, 449 (1962); and by G. J. King and B. S. Miller, ibid. 8, 344 (1963). b The sign for A and B for manganese is assumed negative, and the signs for the other parameters are derived on this basis. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44ESR OF Mn++ IN NaNa, KNa, AND RbN 3 31 support to the interpretation of results for Mn+ + in both KNa and RbNa. The spectra for Mn+ + in both KNa and RbNa show axial symmetry with the axial field parameter D larger than A, the hyperfine coupling constant. The axial field in both cases is aligned with the tetragonal [OOlJ or c axis. There is an additional cubic field splitting which is very much smaller than D but which is also symmetric about the [OOlJ axis. For the strong magnetic field case, the axis of quantization is taken along the direction of magnetization and the solution for the resonance line positions for electric axial and hyperfine interactions are, to the second orderl6-18: Mf--+M -1, ~m=O H=Ho-D(M -!) [3(gINg.l.2) cos2{j-1J + (D2/2Ho)[4S(S+1) -24M(M -1) -9J X (gIl2g.l.2/g4) sin2{j cos2{j-(D2/8Ho)[2S(S+1) -6M(M -1) -3J(g.l.2/ g4) sin4{j-Km -(A2/2Ho) X[I(I+1)-m 2+m(2M-1)J gauss. (1) The experimental results are given in Table I and Fig. 1. For the orientational studies shown in the figures, each crystal was rotated about a vertical [110J axis, which was thus perpendicular to the dc magnetic field. This rotation displays the very large variation in the field position of the resonance lines as a function of the angle (j between the magnetic field direction and the c axis direction, which is also the main symmetry axis. The signs of D and A were obtained by observing the second order effects in line spacing after Bleaney and Ingram,16 and by assuming A is negative.a,17 Spectrum of Mn++ in KN a In Fig. 2 (I) the lowest-field sextet is the M = -~ -! transition, and the one at highest field is the M = +~+! transition. The m= +! line is the highest field line in each sextet. Each of the five sets is a single fine-structure line which is split into six lines by the electron-nuclear interaction. The cross-over of the fine structure lines observed around 54° is typical of the strong magnetic field case for predominantly axial sym metry. The normal resonance lines in Fig. 2(1) are the ~M=±l, ~m=O. For angles other than {j=0° or 90°, the ~M=±l, ~m=±l and ~M=±1, ~m=±2 trans itions are allowed. For a large value of D, particularly for KN a: Mn+ +, the so-called "forbidden" ~m = ± 1, ±2 lines will be intense. The ratio of the intensity of the (M, m+1f--+M-1, m) and (M, mf--+M-l, m+l) 16 B. Bleaney and D, J. E. Ingram, Proc. Roy. Soc. (London) A205,336 (1951). 17 W. Low, Paramagnetic Resonance in Solids (Academic Press Inc., New York, 1960). 18 F. K. Hurd, M. Sacks, and W. D. Hershberger, Phys. Rev. 93,373 (1954). transitions to that of the ~m=O transition is19 [3D sin2{j/4g(jHJ2{ 1 +[S(S+ 1) /3M(M -1) Jl2 X[I(I+l) -m2+m]. (2) This formula is expected to be accurate for small values of {j if D is large, since it is obtained by perturbation theory and D is assumed very small compared to g{3H. The agreement with the Bleaney-Rubins theory is good at small (j, and a more thorough study of the data by computer techniques is currently in preparation. Some general observations can be made. As predicted by the computer calculations of Bleaney and Rubins the ~m= ± 1 and ~m = ± 2 do exceed the" normal" ~m = 0 nuclear lines for intermediate orientation [see especially Figs. 2(11) and 2(111)]. For {j<5° the ~m=±1 doub lets are well resolved within the electronic !f--+-! sextet. They are symmetrically disposed between the ~m=O lines and have the intensity ratio 5:8:9:8:5 for m= -!, 3 lId 3 t' 1 Th . -2, -2, 2, an 2, respec lve y. e asymmetry m doublet spacing found for Mn+ + in A120a does not occur here; as noted by Folen,20 this may be related to the negative sign of D. For angles greater than (j= 5°, the ~m= ±2 transitions begin to interfere and resolution is lost. In the ±~±! electronic transitions, four of the ~m = ± 1 nuclear lines fall outside the main ~m = 0 transition, two on the high field side and two more on the low field side. In Fig. 2(11) this effect is observable on the extreme right. The line positions agree well with the work of Friedman and LOW.21 The +~+t trans ition shows particularly well the ~m= ± 1 lines for {j= 7.5°, as shown in Fig. 3. The positions of the three doublets and the four single lines are correct for an S=!, I=! atom. The clutter introduced by the intens ity and closeness of the ~m= ±2 and ~m= ±3 lines for (j near 45° is sufficient to prevent identification of the individual lines. As may be seen in Fig. 2 [extreme left in (III), (IV), and (V) J, the ~M = ±2 electronic half-field transitions also have appreciable intensity in the KNa: Mn+ + spectrum. We note that the center set of lines for KNa: Mn+ + in Fig. 2(1), appears diminished in intensity. The lines in the center set are superimposed on a broad resonance at g= 2, and the power is shared. Spectrum of Mn++ in RbN a The spectrum for Mn+ + in RbNa is similar to the KNa case. The axial symmetry around the crystal c axis is quite apparent (see Fig. 4). The main transitions are ~M=±l, ~m=O. The ~m=±l nuclear transition lines are easily seen, but are not nearly so intense as in KN a: Mn+ +. This result is expected for the smaller 19 B. Bleaney and R. Rubins, Proc. Phys. Soc. (London) 77 103 (1961). ' 20 V. Folen, Phys. Rev. 125, 1581 (1962). 21 E. Friedman and W. Low, Phys. Rev. 120, 408 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4432 G. J. KING AND B. S. MILLER , II III IV V H-> FIG. 2. (I) Spectrum of Mn++ in KN3 at maximum splitting (8=0°). Marker is at g=2.0005±OJ)OO3. (II) Spectrum for 8=22.5°. The breakup of the i ..... -i and -i--! fine structure groups is caused by the electron-nuclear inter action of Bleaney and Rubins (see text). At the extreme right, a doublet occurring outside the related sextet is seen. (III) Spectrum for 8=45°. For this orientation the electronic M -M -2 transitions are strong and interfere on the left with the -! ..... -! fine-structure transition. (IV) Spectrum for 8=67.5°. This spectrum occurs after the cross-over point. The tiny lines on the extreme left are the M-M-2 electronic lines. (V) Spectrum for 8=90°. The large shift in the position of the l--l set caused by the large value of D is apparent from the marker position. Two of the lines are exactly superposed and hence appear as a single clipped line. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44ESR OF Mn++ IN NaNa, KNa, AND RbNa 33 H- FIG. 3. The M=!-M=! transition group of KNa:Mn++. The six main lines are the normal tJ.m=O electronic lines. The 10 small lines are the electronic transitions accompanied by a nuclear transition (tJ.m= ±1). The angle () is 7.5°. At larger angles the tJ.m = ±2 lines are clearly visible, also. The line intensity for the tJ.m = ± 1 lines is in substantial agreement with the Bleaney Rubins theory. value of D in RbN3:Mn+ + and from Formula (2). Here, smallness of D prevents easy identification of the line structure due to Am~O transitions. The usual Am= ± 1 doublets appear symmetrically between the Am= 0 lines in the +~-! electronic transition for small values of O. In addition to the usual features of the KN3:Mn++ spectrum, the central set of +!~-! electronic lines in the RbN3:Mn++ spectrum is badly distorted at room temperature. This latter effect is being investigated further. Cubic Splitting A pure axial field splitting for a paramagnetic ion in a host crystal is not a complete physical description, and one usually observes a secondary cubic field for a tetragonal host. In the case of KN 3: Mn+ + the value of a, the cubic field parameter, can be obtained from line shifts in the 0=0° spectrum, as described by Bleaney and Ingram.I6 It turns out that this procedure is valid for KN3:Mn++ and RbN3:Mn++, but the actual spectra are complicated by the unique" cubic" struc tural effects. Measurement of the angular variation of fine-structure splitting due to the cubic field can be done a b FIG. 4. The spectrum of Mn++ in RbNa at 25°C. The value of () is 0° in a and 90° in b. The intensity of the central set is anoma lous. At somewhat lower temperatures, the relative intensity of the sets approaches the theoretical value of 5: 8: 9: 8: 5. IjI (Degrees) FIG. 5. "Cubic" line doubling for KN3:Mn++. For the unsplit spectrum the appearance is identical to Fig. 2 (V). Each single line in that figure appears to double every 45° of rotation of angle >/to We show here the fit of Formula (9) to the observed data. The magnetic field is measured from an arbitrarily fixed point, and the M=-i-M=-!, m=-i line is used here. accurately only in the position for 0= 90°. Thus, if one rotates the crystal in the 0= 90° position in such a manner that the angle if; (see Sec. II) is described, then the large variations in fine structure due to D, and the complication due to the AM = ± 1, Am~O lines are both eliminated. The expected result of such a rotational study is a fourfold variation in the fine structure which can be described by the first-order Kronig-Boukamp calculation. 17.22 When the orientational studies were performed it was found that there were two apparent cubic sites present, and this gave the effect of a small line splitting of 15 G for every 45 deg of the angle 1/;. A related line doubling for Mn+ + in calcite has been found by Kikuchi and Matarrese23 (KM); we will use their techniques for analysis here. Analysis of the data showed that a fairly good fit was obtained by two separate cos41/; terms shifted roughly 50° (see Fig. 5). The source of this doubling became apparent from the x-ray data.5 If the Mn+ +interacted with the nearest nitrogen atom in each azide ion (see Fig. 6) then local symmetry would consist of a" cube" which has the top twisted some 49°38' from the bottom. If we were to untwist this" cube" its dimensions would be 3.3X3.3X3.5 A. We will assume that the Mn+ + interacts with the nearest nitrogen atom on each of the eight nearest- FIG. 6. Projection on (001) plane of the Mn++ ion site. When the solid-line square is the upper one, this is an A site. In a B site, the dashed-line square is the upper one. ["0] 22 R. De L. Kronig and C. J. Bouwkamp, Physica 6,290 (1939). 23 C. Kikuchi and L. M. Matarrese, J. Chern. Phys. 33, 601 (1960). These authors are referred to as KM in the text. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4434 G. J. KING AND B. S. MILLER ®@ 0 ... Mnt+ K' N3 FIG. 7. The structure of KN •. Manf!a nese replaces any K ion but the com pensating vacancy locates only directly above or below the Mn++. The [001] direction is vertical in the figure. neighbor azide ions, as the data suggest. In order to understand how the twist angle gives the effect of two "cubic" sites, it is useful to analvze the effect in a manner similar to that of KM. The line doubling ob served by KM for Mn+ + in calcite was the result of two inequivalent sites. For KNs:Mn++, in an undistorted state, there are two geometrically inequivalent sites. In Fig. 7, the c axis is vertical. If Mn+ + ion is inserted in an upper K ion site, then the interaction with the ei?ht nearest nitrogen atoms is shown in projection in FIg. 6. The upper (solid-line) net of four nitrogen atoms is twisted 24 °49' clockwise from [001]. The lower square net of four nitrogen atoms (dashed lines) which interact with the Mn+ + is twisted 24°49' counterclockwise. For the K ion site directly beneath the one we have been consid~ring, the rotation of the square nets of nitrogen atoms IS reversed. We call the first site an A site and the second, a B site. We can show that if these two sites are undistorted, then the A and B sites are entirely equivalent in the paramagnetic resonance experiment even though they are not geometrically equivalent. If the Mn+ + ions are displaced from the K ion sites along the c axis direction, the A and B sites are no longer equivalent in the resonance experiment and we can account for the observed results accurately. The mech anism we propose for the distortion is the attraction of the Mn+ + for a positive-ion vacancy in the nearest neighbor site. This nearest-neighbor site is always in the c-axis direction for KNa and RbNa as a result of the body-centered tetragonal structure. The require ment for ~ posi~iv~-ion vacancy is based upon charge ~ompensatlOn prInClples. That the charge compensation IS produced by a vacancy and not by a negative impurity ion is discussed in Sec. IV. We first show the necessity for including distortion effects by solving the undistorted case where the Mn+ + is exactly in the K ion site and no vacancy or other mechanism is included. We will then use the simple result t? calculate the distorted case, assuming the calculatlOn of the crystal field matrix elements is still valid for small distortions. Using the Mn+ + as the origin and [OOlJ as the polar axis, the potential due to the interaction with the nearest end nitrogen atom in each of the neighboring azide ions is given by24 V = a20Y20+ a4oY4o+.a44Y44+a4-4Y 4-;" where the Y1m are spherical harmomcs. The selectlOn of the particular terms for 24 B. Bleaneyand K. W. H. Stevens in Rept Progr Phys 16 108 (1953). . . . t the potential is discussed by Bleaney and Stevens24 and by Sachs.25 The crystal parameters a44 and a4-4 contain the information about the angle of twist. We follow the analysis of KM and write the potential here as V = a20Y20+a40[Y40+b4Y 44+b-4Y4-4], (3) where b±4= a4±4/ a40 = '2;RrSP4;f:4(cosaj) exp(Ti4~j)/L:RrsP40(coSaj) J j (4) and ~j is the azimuthal angular position of the lattice p_oint j and aj i~ the colatitude of the point j. Both P40(C05aj) and P44(C05aj) are even functions of COsaj and b4 = b-4 = P 44 (cosao) cos4~o/ P40 (cosao) , where 0'.0 is the colatitude of anyone of the eight lattice points and 2~o is the twist angle (see Fig. 6). This means that b±4 is the same for both the A and B sites and hence the two sites are identical. No crystal field doublets can arise from Mn+ + exactly in a cation site. 1£ we assume a distortion along the c axis, which is the only one permitted by the observed axial symmetry of the spectrum, then the A and B sites are inequivalent and four possible sites arise since the vacancy can be above or below the A or B sites (see Fig. 8). The resonance experiment can distinguish only two different sites although higher order effects may produce line bro~dening as a result of the four sites. The physical baSIS for the generation of inequivalent sites is that the displaced Mn+ + reacts more strongly with the nearer s~uare net of nitrogen atoms. The orientational proper ties of the stronger interaction are dominant in the reso?an~e experiment. For Site A (or B), an upward c-aXIS dIsplacement yields a strong interaction which is t:visted 2~o from that obtained by a downward c-axis dIsplacement. Information concerning the displacements is contained in both a40 and a4±4. In calculating the matrix elements of the potential (3) one can use the operator equivalent method of Stevens or the tensor operator formalism of Racah. Suitable operator forms and evaluated matrix elements are provided by KM. 0-MnH [~~-vacancy FIG. 8. The four sites created by the displacement of the Mn++ along the c axis. A 1 and B2 are equivalent, as are A2 and B1. The result of the distortions is a pre dominant clockwise rotation for one class and a predominant counterclockwise rotation for the other. Two sites are thus resolved in the resonance experiment. The solid and the dashed lines repre sent edge views of the squares in Fig. 6. 2Ii M. Sachs, Solid State Theory (McGraw-Hill Book Company Illc./ New York,1963). • This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44ESR OF Mn++ IN NaNa, KNa, AND RbNa 35 In order to give the usual meaning to the spin Hamiltonian parameter ao, the "cubic" part of the crystal field expression needs to be multiplied by an appropriate constant. This is determined by requiring that when all distortions are eliminated and the twist angle is reduced to zero, then the crystal field reduces to to that of a simple cube. The equation for the crystal field can be written in operator form in the notation of KM: V=D[SzLtS(S+1) J +-l5ao[To(4)+b4T4(4)+b--4T --4(4)J. (5) The second term in (5) has been multiplied by n, so that solution of the secular equation yields three twofold degenerate levels at E=-tD+ao, E=!D-!ao±[(2D+ao)2+¥ao2Ib412J!. When D= 0 and 2{3o= 0 then I b4 12= -h for a cube,24 and one obtains a quartet energy level at E= ao and a doublet at E= -2ao. This is the usual result for a cubic material and an 6Sl state ion, as indicated by Bethe.26 The ao term here is the same one used by KM, and ao= a-F in the notation of Bleaney and Trenam.27 The sign of ao is positive as we can show from the experi mental data, in agreement with the theory of Watanabe.28 For our solution of the problem we have taken the z axis along [OOlJ in the crystal electric field calcula tion, since this is the fourfold symmetry axis. We treat the Zeeman problem as a strong magnetic field case and thus the axis of quantization must lie in the direc tion of the magnetic field in the isotropic case. The perturbation procedure requires that all terms be eval uated using the coordinate system which has its z axis aligned with the magnetic field. Because the experi mental equipment is different, it is not immediately obvious that the Euler angles 8 and I/; of KM are the same as ours. This is the case, however, and we measure 8 as the angle between the magnetic field and the crys tal c axis, which is the fourfold axis. We must remount the crystal to display the pure I/; variation. It is the I/; variation which allows accurate measurement of the "cubic" effect. We rotate the crystal coordinate system into the magnetic coordinate system and, in doing so, we obtain the transformed spherical harmonics y Im(8., 1/;.) = Lamm' (8,1/;, cp) Y 1m' (8.',1/;.'). m' All terms have same meaning as in the KM calculations, and (8.',1/;.') and (8.,1/;.) are the spherical coordinates in the new and old coordinate systems, respectively. For 26 H. A. Bethe, Ann. Physik 3, 133 (1929) (English trans!.: Consultants Bureau, Inc., New York.) 'n B. Bleaney and R. S. Trenam, Proc. Roy. Soc. (London) A223,1 (1954). 28 H. Watanabe, Progr. Theoret. Phys. (Kyoto) 18,405 (1957). a first-order perturbation, m' = O. This is sufficient here, since we are dealing with a relatively small energy term. The amo(l) are given by KM, and Y2o(8.,I/;.) = (2/5)tP20(cosO) Y20(8.',I/;.') , Y4o(8., 1/;.) = (2/9) IP40( cosO) Y40(8.', 1/;.') , Y44(8e, 1/;.) = (2/9) !P4--4 (cos8) exp[i4I/;JY 40(8.', 1/;.') , Y 4--4(8.,1/;.) = (2/9) tP4--4 ( cosO) exp[ -i41/;J Y40(8.', 1/;.') , where P1m(cosO) are normalized associated Legendre polynomials. A solution to the energy equation JC=g{3H·S+AI·S+ V, where g and A are assumed isotropic, is given by the diagonal term for site A 1 (or B2) : (M, ml JC 1M, m)(Al)=g{3HM+AMm +DaOO(2) (MI Y20'l M)+ (2/9)tnaO[P40(COsO) +b4(AI) exp( i41/;) P4--4( cos8) +b_PI) exp( -i41/;)P4--4(cos8) J (MI Y40' 1M). (6) Sites A 1 and B2 are equivalent, and sites" A2 and B1 are also equivalent. For site A 1 (or B2), . (M, ml JC 1M, m )(AIL (M -1, ml JC 1M -1, m )(AI) =g{3H+Am + DaOO(2l[ (MI Y20' IM)-(M -11 Y20' 1M -1) J +nao[P40(cos8) + 105/(8!)! sin40\ b4(AI) exp( i41/;) +bjAI) exp( -i41/;) I J X[(MI Y4o'l M)-(M-11 Y40'l M-1)J. (7) If we consider the c axis to be held in a vertical position (8=90°) and the crystal rotated about this axis, the angle y, is described. The only variation in the spectrum is produced by the last term in (7) . We call this term C(Al). Referring to Fig. 8, Formula (4) can be expressed as P4±4(cosa) exp(±i4{30) b±4= - -P 40 ( cosa) + [ R/ R'J5 P 40 ( cosa') P4±4 (cosa') exp (=Fi4{30) + - -P40( cosa') +[R' / RJ6P40C cosa) = S exp(±i4{30) + S' exp(=Fi4{30), (8) where a and R refer to the upper square and a' and R' refer to the lower square in all four cases. Using this definition of Sand S' one can write for the last term of Formula (7) for M = -~-! C(A 1) =nao\P40( cos8) +[2/(8 !) lJP44 (cos8) XeS cos4(1/;+{30) + s' cos4(1/;-{30) Jl (¥), (9) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4436 G. J. KING AND B. S. MILLER q o FIG. 9. Mixed spectra of Mn++ in NaN3 at 25°C. Spectra in b are found for a crystal wit~ no th:rmal tr~atment: The ~~o~d line results from exchange mteractlOn of hIgh-densIty Mn m defects. Heating the same crystal to 200°C, followed by quen~h ing to 25°C disperses the Mn++ throughout the crystal, leavmg sharp lines ~s in a. The strongest 30-line spectrum is the Mn++ in the Type II spectrum. The six small lines at each end are from one of the Type III sets of va.cancy-compensated Mn++ io~s. The spectra in a revert to those m b after several days at 25 C. In strument gain in b is 10 times the gain in a. for the equivalent A 1 and B2 sites. The expression C(A2) for the equivalent A2 and B1 sites is the same as (9) except Sand S' are interchanged. The terms Sand S' and f30 were used as adjustable parameters to fit the formulas for C~A1) and C(.;t-2) to the experimental points. The partlcular set of hnes chosen were the M = -~-!, m= -! lines and the formulas are multiplied by the value of the matrix ele ments for this transition (+¥). The fitting of the formulas was accomplished by trial and error. A dis placement of the Mn+ + away from its cation site toward the vacancy was assumed and the angles and distances in the formulas for Sand S' were then com puted using known x-ray data. The best-fit angle f30 was found to be 23°30', which is close to the undistorted value of 24°49' from x-ray data. The best-fit distortion was found to be 0.26 A, which is about 7% of the cation to-cation spacing in the c direction. The distortion for the similar case of Mn+ + in NaCl is about 10%. Both values here are entirely reasonable. The agreement of the theory with observed line splittings for 8= 90° is shown in Fig. 5. The angle if; is measured between the [110J direction (an edge of the basal plane) and the dc magnetic field. The parameter ao is obtained from a separate experiment in which the 8= 0° spectrum is displayed. The maximum line separation of C(A1) and C(A2) in Fig. 5 is about 14.5 G. The same result is obtained for the M = !~! group. The absolute value of the matrix elements in (9) for the M=~! and M=-~-! groups is ¥, and the maximum line separation should be tx 14.5= 11.6 G. The high field (for 8= 71/2)M = _!!~_J!. group maximum line separation is 14 G but 2 2 •• 15G the low field M =~! maximum separatIOn IS on y . In addition, the M = !~! group lines are 1.5 times broader than the other lines for this orientation. On the basis of the first-order perturbation calculation, there should be no line separation in the M = !~-! group. We observe, however, that there is a line broaden~ng having the same orientational propert~es as the l~ne separations of the other groups. The hne broademng corresponds to a 2-G separation of the centers of the lines. The discrepancies noted here are likely the result of the use of perturbation treatment which requires the Zeeman energy to be much larger than the crystal field energy. The large magnetic field separation of the ±~±! groups may introduce magnetic field-depend ent differences. IV. Mn++ IN SODIUM AZIDE The most remarkable feature of the resonance spec trum of Mn+ + in NaNa is its similarity in behavior to Mn+ + in N aCl. When grown slowly from aqueous solution29 or from the melt,23 the Mn+ + spectrum in NaCl crystals is primarily a single broad line with rudimentary fine structure. Heating the crystal to around 250°C, followed by quenching, produces sets of characteristic 30-line Mn+ + spectra. The cubic structure of NaCI allows vacancy-type charge compensation to be energetically equivalent when the vacancy is in any one of the six nearest-neighbor cation sites. The para magnetic resonance crystal field splittings are sensitive to the direction between the vacancy and the manganese. There are at least two other inequivalent sets of spectra observed whenever one set is observed at maximum splitting, and there is a 90° rotation around a crystal axis separating the maximum splittings for each set. We have observed similar effects for Mn+ + in NaN3. The crystals, when first grown from solution, show. a single broad resonance (Fig. 9) with an apparently dIS ordered line structure superimposed. The similar broad line has been attributed, in NaCl, to precipitation of vacancy-Mn+ + complexes in sites where the density of Mn+ + is sufficient to cause broadening by dipole-dipole and exchange effects. When NaN3:Mn++ is heated to 150°C the broad line vanishes and multiple sets of 30-line spectra occur which are semistable at 25~C. This effect is qualitatively the same as that occurnng in NaCl and is shown in Fig. 9. The positive-ion vacancy-Mn+ + pair has high mobility in the lattice, and heating followed by quenching causes the pairs to be trapped in low concentration throughout the crystal. The spectra are then sharp-lined. The Mn+ + spectra in NaNa, like those in NaCl, decay slowly at .25°C and return to the original broad-line spectrum III several days. Charge Compensation The sharp-line spectra of Mn+ + in NaN3 show the symmetry of the hexagonal structure. Sodium azide has a layer-type structure (see drawings in Ref. 5) and the symmetry in the metal plane is trigonal. The c axis 29 K. Morigaki, M. Fujimoto, and J. Itoh, J. Phys. Soc. Japan 13, 1174 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44ESR OF Mn++ IN NaNa, KN8, AND RbN 8 37 ([00.1J) is normal to the metal and the azide planes. Rotational studies performed with the c axis perpendic ular to the de magnetic field show that a different set of lines reaches a maximum splitting every 60°. The vacancy can form a pair with equal binding energy in anyone of the six nearest-neighbor cation positions, which all are in the metal plane. The resonance experi ment distinguishes only three sets of 30-line spectra for a predominant axial field splitting. This coordination in NaNa should be compared with the coordination in KNa and RbNa. In these two compounds, there are only two possible nearest-neighbor cation sites, one on either side of the Mn+ + in the c-axis direction. These two positions allow the best charge compensation to be by a vacancy mechanism; since they are equivalent in axial symmetry, we see only one axial30-line spectrum. It is useful to note that the symmetry of the KNa and RbNa spectra seems to require a vacancy for charge compensation of the Mn+ +. The inclusion of a com pensating doubly charged negative ion would require it to be in a positive-ion site, to satisfy the observed symmetry of the spectra. Vacancy-Mn+ + complexes in alkali chlorides can exist in excited states in which, for instance, the vacancy can be removed to the next-nearest cation site. In fact, the ground state separation of the complex need not be the closest separation permitted by the geometry.3o.a1 For NaNa:Mn++, determinations of the relation of the crystal axes to the symmetry of the spectra indicate that vacancy compensation does occur for the nearest site, but an excited state also exists in which the vacancy and Mn+ + are dissociated. Spin Hamiltonian The dominant 30-line spectrum shown in Fig. 9(a) is not from vacancy-compensated Mn+ + but from the dissociated vacancy-Mn+ + complex. This differentia tion is easy to make, since the line shape is roughly independent of temperature in the range 20°-300°C. The line shapes for the vacancy-compensated Mn+ + spectra are very sensitive to temperature and broaden in a characteristic manner (see Sec. V). These two types of spectra are in thermal equilibrium, and at tempera tures above 180°C the vacancy-compensated spectra convert into the uncompensated spectrum. For NaCl the high-temperature spectrum is a single set of six lines, in accord with the cubic structure. For NaNa, the high temperature spectrum also shows the undistorted crystal symmetry. Following the work of Watkins,a we call the high temperature spectrum Type II and the vacancy compensated spectrum Type III. Figure 9(b) corre sponds to Watkins' Type I spectrum. The relationship of the Type II and Type III spectra in NaNa is obvious from the values of the parameters in the spin Hamiltonians. In both cases we use the 30 M. P. Tosi and G. Airoldi, Nuovo Cimento 8, 584 (1958). 31 F. Bassani and F. G. Fumi, Nuovo Cimento 11, 274 (1954). II> SO II> ~ I ::a:: 20 l-e .i. ... 10 z ::::; J ~ TEMP. ·C FIG. 10. The linewidth of the m= -! line in the M = -i M = -! transition of KN3:Mn++. The solid line is the result of Watkins3 for manganese in NaCl and the circles are the experi mental results for manganese in KN3. 0 KN3; -NaC!. crystal c axis as the major symmetry axis, and the crystal field is described by JC= DS12+ E( Sa2-S22) , where the subscript 1 refers to the c axis, 2 refers to the trigonal axis direction between the vacancy and the manganese, and the 3 axis is perpendicular to both the 1 and 2 axes. For the Type II, or dissociated defect, the value of D is -240 G, E=O and g=2.00i±0.002. For the Type III spectrum, the vacancy produces a rhombic distortion, and D= -265 G, E= +57 G, and g= 2.00i±0.OO2. In both cases A, the hyperfine coupling constant, is 87 G. The signs of the spin-Hamiltonian parameters are determined by observation of the second order effects in the spacing of the hyperfine lines. The Type I spectrum in NaNa is caused by dipole dipole and exchange effects resulting from the high density of Mn+ + ions. This line is always strong in heavily doped samples and always absent in weakly doped samples. The central g value is 1.95±O.01 and the linewidth (peak-to-peak on the derivative curve) is 240 G. We find no evidence of a defect related to the Type IV spectrum found by Watkins in NaCl. This spectrum in NaCI is thought to be due to Mn+ + compensated by a divalent impurity anion. In order for the observed symmetry of the type II spectrum to be conserved, the position of such a divalent anion would be along a c axis direction from the Mn+ +. The nearest anion site in this direction is 7.5 A away from the Mn+ + site. In addition the anion site is coordinated with six close-in sodium ions, making charge compensation from this source very unlikely. Other Effects When NaNa is heated above 300°C the crystal de composes and the remaining powder is dark blue. Dis ordered Mn+ + spectra are evident, and at g= 2 a strong sharp line appears. This line is the typical resonance of conduction electrons in sodium colloids.a2 This result is confirmed by cooling the material through the melting 32 G. J. King, B. S. Miller, F. F. Carlson, and R. C. McMillan, J. Chern. Phys. 32, 940 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4438 G. ]. KING AND B. S. MILLER point of sodium and observing the linewidth change33 in resonance. This behavior is not typical of pure sodium azide, since ultraviolet or x irradiation is neces sary, in addition to heating, to produce the colloid resonance. There is an additional complication in the Mn+ + resonance in NaNa which we have not observed in KN3 and RbN3. We have reported a crystal structure trans formationl3 in NaNa which occurs at 19± 1°C. The observations were made using optical, x-ray, and para magnetic resonance data. The transformation involves a reduction of symmetry in the lattice from rhombo hedral (or hexagonal) to orthorhombic. The major transformation occurs sharply at 19°C, but secondary changes in the Mn+ + resonance are observed down to about lOoC. At the present time we have no definite interpretation of the effect, but the appearance of domain structure in our optical studies suggests ferro-or antiferroelectric phenomena. Crystals of sodium azide which are multidomained show red and blue bands when cooled below 19°C, even though the crystals are clear and water-white above 19°C. V. LINEWIDTH MEASUREMENTS-THERMAL EFFECTS The linewidths of the resolved Mn+ + spectra in alkali chlorides are generally around 9 G wide (peak to peak on the derivative curve) at room temperature. a For NaNa and KNa, the linewidths and line shapes observed at room temperature are identical to those observed in alkali chlorides. Heavily doped KNa crystals show some line broadening and line asymmetry which may be instrumental. The linewidths for RbNa are slightly greater than for KNa or NaNa. For Mn+ +, the linewidths in both the alkali chlorides and alkali azides are unusually broad. Watkins attributes this condition in the alkali chlorides to possible covalency effects. It seems possible that a similar situation exists for the alkali azides. High-Temperature Effects The Type III (resolved 30-line Mn+ +) spectra in N aCI show a special line shape change and increase in linewidth as the temperature is increased. Each of the 30 lines changes from roughly Gaussian to Lorentzian shape at elevated temperatures. Watkins demonstrates that such an effect can be produced by vacancy hopping if the vacancy is largely responsible for the electric-field splitting of the resonance. Increasing temperature in creases the rate of vacancy hopping, and if the hopping rate becomes comparable to the low-temperature or natural-resonance linewidth then lifetime broadening can occur. This result follows because the jumping of the vacancy modulates the Stark splitting of the reso nance. In principle, another effect can be observed. The central M = -~M = +t set of lines in the Mn+ + spectra are superimposed as a result of the like g values. 33 R. C. McMillan, G. ]. King, B. S. Miller, and F. F. Carlson, J. Phys. Chern. Solids 23, 1379 (1962). These lines are affected only in the second order by modulation of D. As a result it is expected that the other sextets will decrease in apparent intensity com pared to the central -~+t set. Watkins looked for such an effect and reached the conclusion that his Type II spectrum was not the result of such a process but represented the dissociated Mn+ + in a pure cubic site. In KNa and RbNa we can resolve only one pure axial spectrum, if we ignore the cubic doubling due to the azide ions. If we observe the spectrum in KNa at 0=0°, we get the pattern of Fig. 2(1). As the temperature of the sample is increased the line shapes of the outer sextets change to Lorentzian and the linewidths in crease in a manner which is numerically identical to NaCI:Mn++ (see Fig. 10). The results of simple theory indicate that line broad ening of the Mn+ +-vacancy complex in KNa should be the same as found for NaCl. The formula for the dis sociation34 temperature To characteristic of the centers is given by kTo=e2/ksTo, where k. is the static dielectric constant and To is the metal-to-metal lattice spacing. For an undistorted site, the value of k.To for NaCl at 20° is 22.4 X, using k.= 5.62. No value for the static dielectric constant is available for KNa, but using the c axis separation of 3.5 X and k= 6.85 at 50 Me/sec one gets kTo= 24.2 X. For comparison, if we compute kTo for NaCl using the audio-frequency dielectric con stant of 6.12, then we obtain the value 24.4. The agree ment of the data on the linewidth increase and the'disso ciation temperature of the two defects is significant. The central -t~t set, which is relatively unaffected by D, becomes the dominant feature of the spectrum ab()ve 250°C, and above 300°C there is difficulty in measuring any other than the central set. In order for the effect to occur, the site for the Mn+ + must appear to be approximately cubic after the vacancy hops far enough away. The shortest jump that a cation vacancy can take in KNa is 3.5 X along the c axis. Such a jump will separate the Mn+ + by 7 X from the vacancy, and at this distance its influence is likely to be reduced sufficiently so as to leave the Mn+ + in the" cubic" site referred to in Sec. III. There is an essential difference between the KNa:Mn++ spectrum and the NaCl:Mn++ or the NaNa:Mn++ spectra. For KNa, but not for NaNa or NaCI, the total intensity of the 30-line spectrum de creases as an irreversible rate process at high tempera tures, and the effect can be noted as low as 275°C. If the sample is heated for 1 h at 330°C, which is only 20 deg below the melting point, a single, intense line of 70 G width (peak to peak on the derivative) appears at 25°C, and no other resonances are seen in any strength. This behavior is nearly identical to the behavior observed by Forrester and Schneider35 for KBr, KCI, and KI salts, also grown from aqueous solu- M F. Seitz, Rev. Mod. Phys. 26, 7 (1954). 36 P. A. Forrester and E. E. Schneider, Proc. Roy. Soc. (London) B69, 833 (1956). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44ESR OF Mn++ IN NaNa, KN3, AND RbN 3 39 tion. RbNs shows the same general behavior as KNa, but the development of the high-temperature, single line spectrum occurs rapidly around 300°C. The de velopment of the line broadening in the 30-line spectrum begins strongly at 250°C, and bleaching of these lines is more rapid than in KNs. The single-line, high-tem perature spectrum is about 65 G wide at room tem perature and is stable. The single-line, high-temperature spectrum is surely from manganese concentrated in defects, since the shape is Lorentzian and obviously exchange narrowed. Forrester and Schneiders5 point out that manganese fits rather well into the NaCllattice but that manganese is too small to fit into potassium halides without lattice distortion. Apparently we are seeing the same effect here. When the manganese becomes thermally excited and wanders into a defect site, it is permanently trapped, since it requires special accommodation to re-enter the lattice. Perhaps this difficulty would not occur if one could heat the crystal to high temperatures, but the instability of the azide materials prevents this. In any event, trapping of the manganese on grain boundaries, cracks, or other defects would produce clusters of manganese, and this would account for all the observed effects. One further observation should be made on KNs and RbNs high temperature spectra. There is sometimes superimposed on the exchange-narrowed single line a much weaker (0.5% of the total intensity) 30-line spectrum. This spectrum has the general appearance of vacancy-compensated spectra, but the lack of variation of line width with temperature indicates that it prob ably is an impurity-compensated spectrum. In contrast to the behavior of KNs and RbNs, the Mn+ + spectrum in NaNa at high temperature can be observed in undiminished intensity up to 320°C. In this temperature range, gross thermal decomposition and fracture of the crystals occur. Fragments of the crystal of NaNa:Mn++, when recovered and remounted, still show the Type II and Type III spectra. In addition to the Type II and Type III spectra there is the sharp (9 G wide) sodium colloid line referred to in Sec. VI. The coagulated-manganese resonance is always stable in NaNa at 25°C, but we have not observed any trace of coagulation at high temperature or immediately after quenching. Low-Temperature Effects In addition to the high-temperature line-broadening effects due to vacancy hopping, there is a remarkable low-temperature line broadening in KNa and RbNs. A minimum in linewidth for KNa actually occurs at 115°C. The line broadening is so strong that the spectrum of KNa:Mn++ nearly vanishes at -180°C. A possibly related effect was observed by Fo.rrester and Schneider35 for Mn+ + in solution-grown KBr, KCl, and KI. The effect is apparently not observed in melt-grown crystals of potassium halides. The NaN a: Mn+ + spectrum does not broaden at -180°C, although the crystals were solution grown. NaCI:Mn++ grown from the melt or from solution also does not show a line-broadened spec trum at -180°C. We must qualify our statement con cerning NaNa somewhat, since it undegoes a structural transformation1S at 19°C. The orientation of the ob served spectra change when the structure is transformed, but our measurements indicate that the same type of spectra persist at -180°C. There are several possible explanations for this broad ening effect, including exchange interactions, dipole dipole effects, and the electric field effects mentioned by Forrester and Schneider. Electric field effects can enter in two ways; directly, through the fine structure splitting D, and indirectly through the forbidden19 hyperfine lines. If, at low temperature, there are slight distortions of the Mn+ + in some direction other than the c-axis direction, such a broadening might occur. In such an event the four sites (Sec. III) become inequiv alent. In addition, the hyperfine effects will tend to become quite pronounced under these conditions. The variation of D with temperature is large and is given within ±3% for KNa by I D 1=553-0.76T G (-180° to +250°C), where T is the temperature in centigrade degrees and D is in gauss. This very powerful field induces the .:lm~O nuclear transitions which even for small values of the angle 0 are larger in intensity than the lines correspond ing to .:lm= 0 nuclear transitions. The fine structure is observed at 0=0 in order to avoid powerful broadening due to the forbidden transitions. Any distortion of the Mn+ + in a direction not along the c axis will change the effective value of O. For a large D value this will admit the forbidden transitions with the .:lm=O nuclear transitions and, in addition, will shift the field positions of the .:lm= 0 resonance lines. At -180°C, where I D I is 680 G, the effect is stronger than at 25°C, where I D I is 534 G. The reason the nonaxial distortion be comes apparent only at low temperature is probably related to the fact that the vacancy jumping involves the exchange of position of the Mn+ + and its paired cation vacancy. As noted by Watkins,S this special type of jumping produces line narrowing in the usual sense of motional narrowing. We have a minimum in our linewidth for KNa which occurs at 115°C, and it is expected that such an effect would become apparent before high temperature line broadening takes over. It is likely that such vacancy-manganese interchange prevents the distortion from occurring or averages out its effect. The decrease of the vacancy-manganese inter change rate at low temperature allows the distortion to have a net effect. For KNa:Mn+ + the "cubic" doubling, which measures the amount of distortion of the Mn+ + ion from the center of the cation site, is temperature dependent. We hope to correlate the observations with a specific distortion in the lattice. As a preliminary matter, however, we must determine by x-ray analysis This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:4440 G. J. KING AND B. S. MILLER whether KNa and RbNs undergo structural transforma tions at low temperature similar to the one found in NaNa. VI. CONCLUSIONS Paramagnetic resonance data show that manganese enters the KNa and RbNa structures in the Mn+ + state. The existence of two inequivalent sites in KNa and RbNs results from the displacement of the Mn+ + from a cation site. This displacement results from the attrac tion to a nearest-neighbor cation vacancy. The tetrag onal structure of KNa and RbNa permits only two Mn+ + spectra to occur. The spectra of Mn++ in NaNs show a remarkable similarity to those of Mn+ + in NaCl. The same mobility and coagulation effects are noted for both materials and multiple sets of Mn+ +-cation-vacancy complexes are also observed. An excited state of the Mn+ +-cation vacancy complex is observed in NaNs, just as in NaCl. Vacancy hopping, which produces lifetime broaden ing of the resonance lines, is observed in KNa, RbN~, THE JOURNAL OF CHEMICAL PHYSICS and NaNa. The line broadening in KNa and NaCI is numerically equal, in agreement (with simple theory. The spectra of Mn+ + in NaNa shows the same high and low-temperature behavior as the spectra of Mn+ + in solution-grown or melt-grown NaCl. The Mn+ + spectra in. KNa and RbNa show the same high-and low-temperature behavior as Mn+ + spectra in solution grown KCI, KBr and KI. This behavior is completely different from that observed in melt-grown Mn+ +-doped KCl. Mn+ + in NaNs, KNa, and RbNa exhibits most of the paramagnetic resonance properties shown by Mn+ + in alkali chlorides that are also grown from aqueous solu tion. Where differences exist, reasonable explanations for these differences can be proposed. ACKNOWLEDGMENTS The authors wish to thank Dr. Z. V. Harvalik, Director of the Basic Research Laboratory, for his interest in this work. It is a pleasure to also acknowledge the assistance and encouragement of Dr. H. J. Mueller. VOLUME 41, NUMBER 1 1 JULY 1964 Band Shape of the OH Stretching Vibration in Aliphatic Alcohols. Evidence for the Occurrence of an Intramolecular Interaction * ELEANOR L. SAIER, LAUREN R. COUSINS, AND MICHAEL R. BASILA Gulf Research &-Development Company, Pittsburgh, Pennsylvania (Received 14 February 1964) The band shape of the OH stretching vibration in a series of aliphatic alcohols has been investigated. The asymmetric shape which occurs in the majority of alcohols is due to an overlapping band on the low frequency side of the major band. The asymmetry, which is concentration independent, is shown to occur in the deuterated species and in the first overtone vibration as well. The temperature dependence of the minor band has been investigated and a negative !!.H found which suggests that the minor band is due to an intramolecular interaction. A model is proposed in which the hydrogen of a CH group at the 'Y position interacts with the lone pair electrons at the hydroxyl oxygen atom. The model qualitatively predicts the correlation between the formation constant and the number of available 'Y-CH groups which is experi mentally observed. INTRODUCTION THE band shape of the OH stretching vibration in saturated alcohols has been known to be asymmetric for a number of years. Phenols, on the other hand, exhibit a symmetric band shape. The asymmetrical shape in the saturated alcohols is produced by the occurrence of a weak band which overlaps the major band on the low-frequency side. In the aliphatic alcohols, there are a few exceptions such as methanol, ethanol, and I-butanol which have symmetric band shapes; but by far, the majority are asymmetric. It has been shown that the overtone is asymmetric as well as the fundamental and that this asymmetry * Presented at the Symposium on Molecular Structure and Spectroscopy, The Ohio State University, Columbus, Ohio, June 1963. is also present in the comparable vibration of the deu terated species.I·2 The asymmetry is also known to be concentration independenU-6 Several attempts to iden tify this minor band have been made. Fermi resonance between the fundamental OH stretching vibration and an overtone or combination tone was suggested as a possible explanation by Flynn el al.a; however, the majority of workersl•2•4•5 have favored the "conforma- l R. Piccolini and S. Winstein, Tetrahedron Letters No. 13, 4 (1959) . 2 F. Dalton, G. D. Meakins, J. H. Robinson, and W. Zaharia, J. Chern. Soc. 1962, 1566. 3 T. D. Flynn, R. L. Werner, and B. M. Graham, Australian J. Chern. 12, 575 (1959). 4 M. Oki and H. Iwamura, Bull. Chern. Soc. 32, 567, 950 (1959) . i P. Arnaud and Y. Armand, Compt. Rend. 253, 1426, 1547 (1961); 255,1718 (1962). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.222.12 On: Thu, 27 Nov 2014 12:16:44

1.1729133.pdf

XRay Analysis of Stacking Fault Structures in Epitaxially Grown Silicon G. H. Schwuttke and V. Sils Citation: Journal of Applied Physics 34, 3127 (1963); doi: 10.1063/1.1729133 View online: http://dx.doi.org/10.1063/1.1729133 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in XRay Diffraction Analyses and Etch Patterns of Faults in Epitaxial Silicon J. Appl. Phys. 35, 3061 (1964); 10.1063/1.1713173 Erratum: XRay Analysis of Stacking Fault Structures in Epitaxially Grown Silicon J. Appl. Phys. 34, 3420 (1963); 10.1063/1.1729219 ORIGIN OF STACKING FAULT IN EPITAXIALLY GROWN SILICON Appl. Phys. Lett. 3, 158 (1963); 10.1063/1.1753912 Erratum: Structure and Origin of Stacking Faults in Epitaxial Silicon J. Appl. Phys. 34, 3153 (1963); 10.1063/1.1729152 Structure and Origin of Stacking Faults in Epitaxial Silicon J. Appl. Phys. 34, 406 (1963); 10.1063/1.1702622 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:28J 0 URN A L 0 F ,\ I' P LIE D P H Y SIC S VOLUME .34, "UMBER 10 OCTOBER 1'11>3 X-Ray Analysis of Stacking Fault Structures in Epitaxially Grown Silicon G. H. SCHWUTTKE*t General Telephone E:f Hlectronics Laboratories, {nc., Bayside, Ne1<J York AND V. SILS Sylvania lolectric Products Inc., Woburn, Massachusetts (Received 1 March 1963; in final form 25 May 1963) X-ray diffraction microscopy measurements were made on epitaxial silicon. The diffraction contrast produced by different stacking fault combinations was investigated. Fault structures were deduced from changes in diffraction contrast that occur in topographs recorded by dilIerent Bragg reflections. It was found that diffraction contrast of single-bend and multihend faults is in agreement with the occurrence of stair-rod dislocations at each bend. Diffraction contrast due to line defects is not consistent with the idea of simple stacking faults. The x-ray measurements indicate that the line defects are also traces of bent stacking faults. The bend must be parallel to the substrate-layer interface. It is concluded that line defect:; are generated by a stress relief mechanism. INTRODUCTION CR YSTALL()(~RAPHIC imperfections in epitaxial silicon have recently been investigated by optical microscopy/ by transmission electron microscopy,2.3 and by x-ray diffraction microscopy.4.5 ['sually, these defects are revealed by microscopic inspection of the layer surface after etching. Different etching procedures have been described in the literature.l,6 A typical etch pattern of an n-type layer grown on an n-type substrate of (111) orientation is seen in Fig. 1. This micrograph shows an epitaxial layer surface after etching for one hour in copper nitrate etch.6,7 All fault combinations known to occur on epitaxial silicon are present. The defect s can be classified as single lines, open triangles, closed triangles, and other more complicated arrange ments. In the photomicrograph of Fig. 1 single lines are seen at position 3 b, open triangles at position 1 c, closed triangles at position 4 d, and a complex arrange ment at position 2 a. By now it is well established that etch patterns 011 * Research supported by the U. S. Air Jiorce Cambridge Re search Laboratories. t Present address: IBM Corporation, Poughkeepsie, New York. l T. B. Light, Metallurl!.Y of Semiconductor Jlaterials, AIME :Vletallurgical Society Conference, Los Angeles, August 1961 (Interscience Publishers, Inc., ="iew York, 1962), Vo!' 15, p. 137. 2 O. Haase, Jl etallur{!,y o( Semiconductor Jl aterials, AIME :\fetallurgical Society Conference, Los Angeles, August 1961 (Interscience Publishers, Inc., New York, 1962), Vo!' 15, p. 159. 3 H. J. Queisser, R. H. Finch. and ]. Washburn. J. App!. Phys. 33, 1536 (1962). 'G. H. Schwuttke, Semiconductor Symposium, ECS, Detroit. Michigan, October 1961; 19th Annual Diffraction Conference, Pittsburgh, Pennsylvania. ="iovember 1961; J. App!. Phys. 33. 1:;38 (1962). 5 G. H. Schwuttke and V. Sils, Semiconductor Symposium. ECS, Los Angeles, May 1962. 6 R. Giang and E. S. \;radja, Jletallurgy of Semiconductor JIaterials AIME Metallurgical Society Conference, Los Angeles, August 1961 (Interscience Publishers, Inc., New York, 1962), Va!. 15, p. 27; H. J. Beatty, R. Giang, and J. G. Kren, "Vapor Phase Growth of Silicon," Report No.7, Contract No. DA 36-039-SC- 1;7395. epitaxial silicon are t races of stacking faults. '.9 Possible fault structures connected with these traces have been deduced from electron diffraction microscopy measure ments.8•9 The results are listed in Table 1. It is noted that a single line could be the trace of a simple stacking fault bounded by two Shockley partials and lying in 4 3 2 a FIG. 1. Optical photomicrograph of an (111) epitaxial silicon surface showing etch traces of stacking fault structures. The crystal area is 3 mmX4 mm. 6 R. H. Finch, H. J. Queisser, J. WashLurn, and G. Thomas, J. 7 Copper nitrate etch consists of 30-ml hydrofluoric acid, 15-ml Apr!. Phys. 34,406 (1963). nitric acid, l.l-g copper nitrate O.l-ml bromine, and 450-ml water. 9 G. R. Booker ann R. Stickler, J. Appl. Phys. 33, J2R1 (1962). 3127 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:283128 G. H. SCHWUTTKE AND V. SILS TABLE 1. Etch traces and possible fault structures on (111) epitaxial silicon layers. Burgers Fault Form Dislocation vector ----------A-B Line 2 Partials ia(112) at A,B /1 Open 2 Partials ~a(1l2) C triangle at A,B; ka(llO) "'B stair rod at C A B/"'C Closed 3 Stair rods Aa(llO, triangle at A,B,C /~'" ~ultibenrl Stair rods at ka(IIOl every bend ========= ~-"===-=cc-..cCC= anyone of the octahedral planes. More complex faults occur if a simple stacking fault bends from one octa hedral plane into another. An open triangle is thus con nected with one stair-rod dislocation located at the bend and two partials terminating the fault, while closed figures contain stair-rod dislocations at each bend. In this paper we report x-ray diffraction microscopy measurements made on epitaxial silicon. The diffraction contrast produced by the different fault combinations was investigated. Fault structures were deduced from changes in diff:action contrast observed in topographs recorded by dIfferent Bragg reflections. It was found that the diffraction contrast of single-bent and multi bent faults is in agreement with the presence of stair-rod dislocations at each bend. Diffraction contrast due to line ~efects is not consistent with the idea of a simple stackmg fault. The x-ray measurements indicate that line defects are also traces of bent stacking faults. The bend must be parallel to the interface plane. SAMPLE PREPARATION The epitaxial layers were prepared by SiCl4 vapor decomposition at 1265°C in a hydrogen atmosphere.lO T~n-and thirty-micron layers were grown in a multi s~lce furnace on. the (111) face of 100-1-' slices of As-doped smgle-crystal SIlicon ranging in resistivity from 0.01 to 0.?06.Q-cm. The substrate surface was prepared by gnndmg and finally polishing with Linde alumina abrasive type A. For the x-ray investigations the sub strate was etched down as close as possible to the inter face without penetrating the interface . . The experiments were conducted by x-ray diffraction mIcr~scopy usi~g the extinction contrast technique as descnbed prevIOusly.11,12 Mo-radiation was used at 50 k V and 20 mAo The micrographs were recorded on Ilford nuclear plates type GS, emulsion 50 I-' thick. MEASUREMENTS Triangle defects can be thought of as tetrahedrons with one corner at the interface. For the following dis- :~ H. C. Theurer, J. Electrochem. Soc. 108, 649 (1961). 12 A. R. Lang, J. App!. Phys. 30, 1748 (1959); 29, 597l(1958). G. H. Schwuttke, J. Electrochem. Soc, 109, 27 (1962). cussion it is convenient to use the geometry sketched in Fig. 2, which shows a tetrahedron opened up into the (111) layer surface. The tetrahedron has been drawn in such a way that the position of the triangle ABC cor responds to the position of any triangle fault in Fig. 1. The triangle sides AB, AC, CB are thus parallel to the directions of the line defects in Fig. 1. Since D designates the corner of the tetrahedron, the lines AD, BD, and CD can be used to describe the location of stair-rod dislocations. Defect analysis by x-ray diffraction microscopy is based on fault vector determination. Burgers vector directions of total dislocations are rather conveniently determined by this techniqueY Any main Bragg reflec tion is normally picked up with ease, and therefore the use of the criterion o no contrast cos1: (gb)= 1 max contrast (1) permits complete determination of the fault vector direction. The procedure is as follows: The x-ray image of the specimen is recorded by using different reflections until the reflection is found for which the dislocation concerned is out of contrast and the one for which the contrast appears to be maximum. The diffraction vector g is perpendicular to the Burgers vector if the reflection is out of contrast and parallel to the Burgers vector when the dislocation shows maximum contrast. This fixes the direction of the Burgers vector; its magnitUde cannot be found in this way, but must be deduced from consideration of the crystal structure. We have found that criterion (1) is also true for p.artial dislocations; Burgers vectors of partial disloca tIOns are therefore determined the same way as for total dislocations. A stacking fault is characterized by its fault vector R. Stacking faults are in contrast for g. R~0.13 For D o AS = [T 1 oj At = [TO IJ Cit· [011] FIG. 2. Sketch of fault tetrahedron opened up into the (111) plane. 13 K. Kohra and M. Yoshimatsu, J. Phvs Soc JaT}an 17 1041 (1962). _. . t , , [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:28X-RAY ANALYSIS OF STACKING FAULT STRUCTURES 3129 TABLE II. Values of cos 4: (gb) for dislocations and principal reflections in silicon. Plane Direction cos 4: (gb) for different g h,k,l of b 112 121 211 il0 iOl oli 111 lli ill 1il 112 1 1 0 -!~ !~ 0 ~v2 -1v2 -1v'1 -2 -2 121 -! I 1 -i~ 0 -!~ 0 -iv'1 -1v2 ~v2 -2 211 1 1 !~ !v'3 0 0 -1v2 }v'1 -1v2 -2 -2 (111) ilO 0 -!v'J !VJ I 1 1 0 0 1(6)1 -1(6)1 2 2 iOl -!VJ 0 !~ ! 1 0 -1(6)1 1(6)1 0 -. 1 Oli !VJ -!VJ 0 ! -2 I 0 1(6)! 0 -1(6)1 III 0 0 0 0 0 0 ~ 1 1 3 3 112 1 1 0 !vJ -l~ 1\>2 0 1v:2 1vl -3 Ii 12i 1 2 -!v3 -1~ -!~ -iv'1 0 -~v2 1vl " 3 (IIi) iti -t; !VJ !~ 1VJ "":1v2 0 1v'2 -!Vl 101 -i 1 1 -! 0 1 1(6)! 0 0 !(6)! -. -2 11i N:2 -iv2 -1v'1 0 -l(6)1 l(6)! 1 1 1 3 -, -. il2 2 !~ -!VJ !VJ -1v'2 1v'2 0 -~v'1 -3 i21 5 2 -tVJ 1~ -!VJ -1v2 -~v'1 0 1v2 -. -"3 (i II) 211 1 1 1 -t~ -!VJ 0 iV:2 1v2 0 *,,2 6 .. -, 1 110 !v3 -tv3 -tv:} 0 -l[ ! t(6)! t(6)! 0 0 il1 -1v2 -lv2 ~v'1 -!(6)! t(6Jl 0 1 1 -, -3 tii 1 -1VJ -!~ lVJ -!v2 1v2 -~y2 0 • 121 1 !VJ -3 (til) iiI 1 2 AvJ " :l 011 -tVJ -tyJ !VJ 1 ..- iii -!v'1 NZ -1v2 -!(6)! x-ray diffraction microscopy of stacking faults in thin films, the stacking fault area is relatively small. Its contribution to the diffraction contrast is negligible. Values for cos1: (gb) for the different fault vectors are listed in Table II for all strong reflections. RESULTS Experimentally, it is found that the contrast of line defects is strongly reflection-dependent while single and multibend faults produce diffraction contrast that varies only very little with the recording reflection. This can be seen in the topographs shown in Figs. 3 and 4. These topographs represent the 112 and the 110 x-ray images of the crystal shown in Fig. 1. Practically every fault visible on the surface of this crystal can be cor related with a diffraction contrast spot in the 112 topo graphs. (A correlation is conveniently done by preparing an overlay of Fig. 1 that can be superimposed on the different topographs.) The slight variation in contrast which is observed for single-and multibend faults is in agreement with the presence of stair-rod dislocations at each bend. Stair-rod dislocations in a triangle defect should have their axis in AD, BD, or CD along (110) directions (Fig. 2). The corresponding Burgers vectors would be,6'Y= (a/6)[011], 'Ya= (a/6)[101], and 0:{3= (a/6)[liO]. Table II shows that, for such a dislocation configuration, at least two dislocations out of three are always in contrast for any listed reflection. An open triangle also contains three dislocations. If the fault, for instance, bends from the 0 t~ ~v2 !v2 !v2 0 !VJ -ty3 -1v2 -~v'1 !v2 0 1 ..- 0 0 !(6)! 0 !(6)1 0 -t(6)! 1 1 1 -3 -, (lil) plane into the (111) plane the stair-rod dislocation would be in CD along the direction [110]. The Burgers vector of CD is a,6= (a/6)[liO]. The other two partials lie in the (lil) plane, respectively, (111) with Burgers vectors in these two planes along (112) directions. Here again, two dislocations out of three are always in con trast for any listed reflection. Line defects produce diffraction contrast that is strongly reflection-dependent. This is very clearly visible in the topographs shown in Figs. 3 and 4. In the Figs. 3(a), 3(b), and 3(c) it can be noticed that lines in AB orientation produce more contrast for the 112 re flection than those of AC and AB orientation, while defects in AC orientation produce more diffraction con trast for the 121 reflection than lines in AB and CB orientation, and defects CB appear stronger in the 211 reflection than those of AB and CB orientation. For certain reflections line defects produce no contrast at all. This is shown in the topographs of Fig. 4. In the ilo topograph of Fig. 4(a) all lines of AB orientation are out of contrast, therefore faults of AB orientation must have their fault vectors parallel to the (110) plane. In the 101 topograph of Fig. 4(b) lines of AC orientation are out of contrast, and therefore the fault vectors must lie in the (101) plane. Finally, in the 011 topograph of Fig. 4(c) lines of CB orientation are out of contrast, indicating fault vectors parallel to the (Oli) plane. Obviously, line defects have strong fault vector com ponents active in the (112) direction, which is perpen dicular to the line direction. This indicates that line defects are not traces of simple stacking faults lying in [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:283130 4 2 a 4 3 2 a b C (h) (a) d (i. H. S C H W LJ T T K E f\ N V \'. S I L S o (el FIG. 3. 112 diffraction topographs of epitaxial silicon laye[ shawn in Fig. 1. Jhe crystal area is_the same as in Fig. 1. (a) 112 topograph. (b) 121 topograph. (c) 211 tapograph. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:28X-RAY ..... :-.;ALySrS OF STACKING FAULT STRCCTURES 3131 4 3 2 o laJ (b) FIG. 4. 110 topographs of epitaxial silicon layer shown in Fig. 1. Ca) i10 topograph, (b) iOt topograph, (c) oli topograph. (c) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:283132 G. H. SCHWUTTKE ;\.\il) V. SILS LAYER EPITAXIAL LAYER SUBSTRATE FIG. 5. Sketch of line fault structure deduced from x-ray measurements. one octahedral plane and bounded by two Shockley partials. There are apparently two sets of vectors that satisfy the cos<}:: (gb) = 0 criterion for zero contrast; for lines in the AB direction they are {(a/3)[111J, (a/6)[112J), and {(aj3)[111J, (ai6)[112J}. Each of these pairs describes an extended dislocation of the Shockley Frank type, which is sessile. In the diamond lattice the Shockley-Frank extended dislocation is not stable. HornstraI4 points out that the Frank partial may dis sociate into two partials: (al 3)[111 J~(a/6)[11OJ+ (ai6)[112J or (a/3)[11IJ~(a/6)[110J+ (a/6)[112} (2) Taking this reaction into account, the contrast criterion should be satisfied by the three vectors: (a/6)[11OJ, (a/6)[112J, and (a/6)[112]. For g= [llOJ one obtains cos<}:: ([~ lOJ[llOJ) {O cos<}:: ([11OJ[112J)= 0 (3) cos<}:: ([1 lOJ[112J) 0 and with the help of Table II it can be shown that this set of fault vectors also gives maximum contrast for the 112 reflection. Since (a/6)[110J is the Burgers vector of a stair-rod dislocation, it follows that the line defect must contain one bend. In Fig. 5 the fault has been drawn with an acu te bend; an obtuse bend is also possible. The bend is parallel to the substrate-layer interface. Evidently, this fault is also sessile, which is in good agreement with the electron microscopic observation that none of the de fects has ever been observed to move. To date, faults of this lype have escaped detection by the electron microscope. Samples for transmission elec tron microscopy are prepared by etching off the lower part.8,9 What is left is the top section of the epitaxial 1·' J. Hamstra, J. Phys. Chern. Solids 5, 129 (1958). layer; line faults appear, therefore, as simple stacking faults. DISCUSSION Surface damage and surface contamination prior to the deposition have been considered to be the main causes of fault formation in epitaxial silicon. Finch et af.k have shown that a change of stacking sequence can be caused by oxide patches on the substrate surface. They have presented evidence that impurity deposits, in particular oxide films on the substrate surface, nucleate stacking faults that grow into the layer along { 111} planes. Booker and Stickler9 propose that first a stacking fault forms on the (111) plane parallel to the substrate surface. As a result, a small area grows on the (111) plane which is crystallographically mismatched with respect to the surrounding areas. Mismatch boundaries form when these areas come together. If the layer grows, the boundaries propagate as stacking faults on {111} planes. The shape of the mismatch boundary determines the geometrical form of the defect. Both explanations assume that fault formation begins at the substrate-layer interface. This is certainly true for closed figures, because the dimensions of triangles can be used to measure the layer thickness quite accur ately.1 No such conclusion can be reached for line de fects. Their size can vary considerably, showing that they may also originate in the layer itself. 2\1echanical damage and! or impurity contamination of the surface can certainly initiate fault formation. Triangle and multibend faults are probable growth defects in the sense tha t they are initiated by the occur rence of mismatch boundaries. Line defects form by a different mechanism than triangle or multibend faults. This is already suggested by the difference in fault geometry (bend parallel to the layer surface) and also by their spontaneous nucleation throughout the layer volume. A mechanism leading to line faults could, for FIG. 6. Optical micrograph of an etched epitaxial layer surface. The layer was grown on a slightly stressed substrate. Line faults are visible along (110) direction. Crystal area ~1 mmX 1.2 mm. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:28x· !{ ,\ Y .\ X A. L Y SIS 0 F S T.\ C K I \ C F.\ t' L T S T J< li C T U RES 3133 instance, become active if localized stresses arise in the layer due to clustering of impurities that have become incorporated during the deposition process. Another possible source could be the lattice mismatch at the substrate-layer interface caused by impurity concen tration gradients between substrate and layer. A rather severe lattice mismatch, for instance, occurs if a boron doped layer is grown on an arsenic-doped substra te. The mismatch is more critical for t he higher doping con cen t ra t ions. Stresses in thin films caused by impurity concentra tion gradients are relieved by the generation of dis locati;ll1s.l,j,ln Stress-induced dislocations lie in the tilm surface and have a positive Burgers vect or edge C0111- ponent in the surface.11i Dislocations in silicon providing the necessary stress relief have been identified by x-ray diffraction microscopy as edge dislocations with {Om} glide planeP It is interesting to note that this pure edge dislocation cannot split with a stacking fault parallel to its {OOl} glide plane. If it splits a partial with (a 3) [111 J Burgers vector is formed which is not stable and consequently dissociates into two partials according to the reaction (2).].1 For a dislocation in the .18 direction t he following reaction may occur: (a 2)[110J:;::=:(a/6)[112J+ (a 6)[11OJ+ (a 6)[112]. (-1-) The result of this dissociation is exactly the defect described in Fig. S. The Burgers vector-squared meas ure of energy ~hows practically no change in energy, therefore the dislocation t ha tact ually forms is deter IIlined by the local stress situation. That in silicon, faults of the type described in Fig. S are a rather efficient means of stress relief is supported by experimental evidence. Figure 6 shows part of an epitaxial layer surface after etching for one hour in copper nitrate etch.' The layer is II-type, and was grown to 10 J.l thick on an II-type substrate. During growth the substrate was submitted to a slight stress. The etching reveals line defects aligned along the (110) slip direc tions. The defects are of the type shown in Fig. 5; they produce diffraction contrast that is zero for (l10) and maximum for (112;. Theobservation t hat in boron-eloped layers line defects prevail, if fault nucleation centers on the substrate sur face have been eliminated, is also in good agreement with the assumption that line defects are caused by a stress relief mechanism. Figure 7 shows two topographs of a boron-doped layer, -t £2-C111 resistivity, grown on an II-type substrate. Figure 7(a) is the 112 reflection and Fig. 7(b) the 110 reflection. The defects are deiinitely line faults (Fig. 5). Two things can be noted: the defects appear to be aligned in (110) directions and, in addition, If, H. J. Queisser, J. AppL Ph:,s. 32, 1776 (1961). 16 S. Prussin, J AppJ. Phys. 32, 1876 (1961). 17 G. H. Schwuttke and H. J. Queisser, J. :\ppJ. Phys. 33, 15~() (1962) . la) (b) I,'IG. 7. X-ray tupographs of uoron-doped epitaxial layer gruwn 011 a fault free substrate surface. The faults are line defects. Crystal area -~ mmX~ mm. taJ 112 topograph (h) 110 lopograph. there exists a tendency to cluster. This is in good agree ment with the experimental observation that boron clusters in silicon and t ha t these clusters cause micro st rains. I' \\'e have also observed that for conventional doping levels, stresses that arise at the interface due to lattice mismatch, do not interfere with the formation of layers of good perfection. As a matter of fact, slight boron doping seems to make the growths of good layers easier. For higher boron doping the growth of fault-free layers becomes more and more difficult, in agreement with the rising lattice mismatch at the interface. I' G, H. Schwultke, Bull .. \m. Phys. Soc. 8,64 (1963); ], AppJ. 1'h:'5. 34, 1662 (1963). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:283134 G. H. SCHWUTTKE AND V. SILS ACKNOWLEDGMENTS The authors are indebted to E. D. Jungbluth and R. J. Modena for their help in performing the measure-ments. The research reported here was sponsored in part by the U. S. Air Force Cambridge Research Laboratories under Contract AF 19(604)7313. JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 10 OCTOBER 1963 Colloid Absorption Band in CsBr* RICHARD E. JENSEN Arizona State University, Tempe, Arizona (Received 29 April 1963) An optical extinction band in colored CsBr at 1050 mil is attributed to colloidal cesium precipitates. The peak wavelength may be closely predicted by assuming the extinction is due to a plasma resonance of con duction electrons in the cesium particles. The band is relatively insensitive to temperature change. The bandwidth decreases as the particle size increases. The shape of the extinction curve is distorted at low temperatures, indicating a distortion of the particle shape. The band resists bleaching. RECENT studies1.2 on the optical absorption bands in irradiated and additively colored CsBr have shown a broad absorption band at about 1050 m,u which is relatively insensitive to variations in temperature. Evidence is presented here that this band is due to colloidal cesium in CsBr. A narrow, temperature-sensitive band1.3 has also been observed in this spectral region. Lynch! has sug gested it may be an M band. It is unstable at room temperature in the dark. The experimental curves pre sented here were obtained from samples irradiated with x rays at room temperature and were bleached with strong light in the spectral region from about SSO to 3000 m,u. No evidence of the M band is seen in these samples. Hampe4 has suggested that the absorption of colloidal metal particles in a dielectric medium is due to a plasma resonance of the conduction electrons. For spherical particles, the plasma resonance occurs at an angular frequency determined by (1) where n= density of electrons in the metal, q= electron charge, m = electron mass, Em = permittivity of the sur rounding medium. On the basis of the Bohm-Pines5 theory, one might expect that if !two< (EF-Ec), with EF the energy on the Fermi surface and Ee the energy on the bottom of the conduction band and ft Planck's constant divided by h, n might be less than one elec tron per atom and m would be an effective mass, or that if !two> (EF-E c), n would be one electron per atom * This work was supported by the U. S. Atomic Energy Commission. 1 David W. Lynch, Phys. Rev. 127, 1537 (1962). 2 H. Rabin and James H. Schulman, Phys. Rev. 125, 1584 (1962). 3 P. Avakian and A. Smakula, Phys. Rev. 120, 2007 (1960). 4 Wilfried Hampe, Z. Physik 152, 476 (1958). 6 D. Pines and D. Bohm, Phys. Rev. 85, 338 (1952). and m the electronic mass. However, Gossick6 has noted that a number of observed colloid bands in alkali halides agree approximately with Eq. (1) taking n as one electron per atom, and m the electronic mass, al though (EF-Ec) for the bulk metal exceeds !two. For cesium particles in CsBr, the plasma resonance cor responds to a wavelength of 1060 m,u assuming 1 elec tron per atom for n and the free mass for m. This com pares remarkably well with the observed wavelength of 1OS0 m,u (Fig. 1). Examination of the CsBr crystals with a dark-field microscope for direct evidence of the cesium particles has not been successful. The scattering cross section of the particles is a maximum at resonance which is in the infrared. In the visible region scattering is due to Thomson scattering and depends on the particle size. The absence of visible scattering from the particles sug gests that linear dimensions of the particles should be ~ 'w C II °0'5 a .2 a. o o~~~~~~~~~~o~o~--~--~--~ Waveleng th em pI FIG. 1. Extinction spectrum of colloid band in cesium bromide. Solid curve taken at room temperature. Dotted curve taken at liquid-nitrogen temperature. • B. R. Gossick, J. App!. Phys. 31, 650, (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 160.36.178.25 On: Mon, 22 Dec 2014 08:41:28

1.1735630.pdf

Effect of Temperature on Photovoltaic Solar Energy Conversion Joseph J. Wysocki and Paul Rappaport Citation: Journal of Applied Physics 31, 571 (1960); doi: 10.1063/1.1735630 View online: http://dx.doi.org/10.1063/1.1735630 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/3?ver=pdfcov Published by the AIP Publishing [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42JOURNAL OF APPLIED PHYSICS VOLUME 31, NUMBER 3 MARCH, 1960 Effect of Temperature on Photovoltaic Solar Energy Conversion* JOSEPH J. WYSOCKI AND PAUL RAPPAPORT RCA Laboratories, Princetun, New Jersey (Received August 3, 1959) Photovoltaic solar energy conversion is investigated theoretically over a temperature range of 0-400°C using semiconductor materials with band gaps varying from 0.7 to 2.4 ev. Three cases are considered. In Case I, the junction current is the ideal current. In Case I I, the junction current is the ideal plus a re combination current; and in Case I I I, a recombination current. The best conversion performance is obtained for the ideal current; the worst, for the recombination current. The maximum conversion efficiency occurs in materials with higher band gap as the temperature is increased. GaAs is close to the optimum material for temperatures below 200°C. Experimental measurements are presented on Si, GaAs, and CdS cells. The measurements on Si and GaAs agree with theoretical expectations as far as the gross behavior is concerned. The CdS cell behaves anomalously as if it were made from a material with band gap of 1.1 ev. g TABLE OF SYMBOLS concentration gradient in a diffused junction absorption coefficient in a semiconductor diffusion constant for electrons diffusion constant for holes Dn[1 + (Dp/ Dn)!J2, effective diffusion constant energy gap of a semiconductor Fermi energy level energy level of traps intrinsic energy level permittivity of semiconductor generation rate of electron-hole pairs per cm3 per sec I. short-circuit current density Ij junction current density I L load current density Imp load current density at maximum power 10 reverse saturation current of a p-n junction k Boltzmann's constant K dielectric constant of semiconductor Ln diffusion length of electrons Lp diffusion length of holes l thickness of absorbing semiconductor mp+ hole effective mass for density of states mn+ electron effective mass for density of states nph(Eo) number of photons with energy greater than the band gap Eo 1/ solar conversion efficiency N A number of net acceptor impurities per unit volume N D number of net donor impurities per unit volume N. geometrical mean of the number of states in the conduction and valence bands ni intrinsic carrier density ¢ barrier height in a p-n junction P mp maximum power output of a solar converter Q collection efficiency of p-n junction Rmp load resistance at maximum power * This work was supported by the U. S. Army Signal Research and Development Laboratory, Fort Monmouth, New Jersey. r T reflection coefficient of front face of solar con- verter temperature in OK electron lifetime hole lifetime electron lifetime in material in which all the traps are empty Tpo hole lifetime in material in which all the traps are full voltage at maximum power width of the depletion region q/kT INTRODUCTION A SEMICONDUCTOR photovoltaic cell converts . solar energy directly into electrical energy by means of a p-n junction. Incident photons with energies greater than the band gap of the semiconductor create electrons and holes which are separated by the junction. A potential is thus created across the junction, and energy can be delivered to a resistive load.1-12 The factors which make the conversion process temperature dependent are introduced by the properties of the semi conductor and the behavior of p-n junctions. This temperature dependence is the subject of the present paper. The discussion will be concerned with solar energy conversion. However, the conclusions are equally applicable to the conversion of other forms of ionizing radiation. 1 Early history: V. K. Zworykin and E. G. Ramberg, Photo electricity and Its Applications (John Wiley~& Sons. Inc., New York, 1949). 2 K. Lehovec, Phys. Rev. 74, 463 (1948). 3 R. Cummerow, Phys. Rev. 95, 16 (1954). 4 R. Cummerow, Phys. Rev. 95, 561 (1954). 5 Reynolds, Leies, Antes, and Marburger, Phys. Rev. 96, 533 (1954). 6 E. Rittner, Phys. Rev. 96, 1708 (1954). 7 Chapin, Fuller, and Pearson, J. AppJ. Phys. 25, 676 (1954). 8 W. Pfann and W. van Roosbroeck, J. App!. Phys. 25, 1422 (1954). 9 M. Prince, J. App!. Phys. 26, 534 (1955). 10 Jenny, Loferski, and Rappaport, Phys. Rev. 101, 1208 (1956). 11 J. J. Loferski, J. App!. Phys. 27, 777 (1956). 12 Rappaport, Loferski, and Linder, RCA Rev. 17, 100 (1956). 571 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42572 ]. ]. WYSOCKI AND P. RAPPAPORT IS! Iii (II IL• Is-[i 121 IJ• l:Iole"-I) THEORY 1. General FIG. 1. Equivalent cir cuit of photovoltaic solar converter. The equivalent circuit of a photovoltaic cellll•12 is shown in Fig. 1. Series resistances and shunt conduct ances are assumed negligible. The junction current Ij is related to the junction voltage V by an equation of the form, (1) where the summation sign is used to indicate that more than one mechanism may determine the junction be havior; i.e., the total current may be the sum of the ideal junction current,13 a recombination current,14 and a leakage current,16 all of which can have a voltage 10 10-6 0.6 0.8 1.0 1.2 v (Y I FIG. 2. Junction cur rent liS voltage. 13 W. Shockley, Electrons and Holes in Semiconductors (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1950), p. 314. 14 Sah, Noyce, and Shockley, Proc. lnst. Radio Engrs. 45, 1228 (1957). 15 M. Cutler and H. Bath, Proc. lnst. Radio Engrs. 45, 39 (1957). dependence of the type shown under certain conditions. The short-circuit current I. is related to the input radiation by the following equationll I.=Q(1-r)(1-e-a1)qnph(Eg). (2) When the p-n junction is suitably located and the life times are sufficiently great, I. can be express as16 (3) 2. Effect of Temperature on 1. The temperature dependence of the short-circuit current arises primarily with the diffusion lengths, which can be expressed as L= (Dr)l. (4) 1000 .---..,.---.---y--,---, 900 800 700 600 400 300 200 100 T. ·C FIG. 3. V malt liS T, N as parameter. Since the temperature dependence of the diffusion constant is. 1'-1, the net effect on L is small. It was assumed that the temperature dependence of the life time is determined by the single-level recombination statistics of Hall, Shockley, and Read17,18 who show the lifetime in the n-type region to be [ (ET-EF)] r=rpo l+exp kT ' (5) and (6) 16 R. Gremmelmaier, Proc. lnst. Radio Engrs. 46, 1045 (1958). 17 W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952). IS R. Hall, Phys. Rev. 87, 387 (1952). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42PHOTOVOLTAIC SOLAR ENERGY CONVERSION 573 in the p-type region. The temperature dependence of the lifetime arises from the exponential terms in the fore going equations. If the difference between EF and ET is large compared to kT, the exponential terms in Eq. (5) and (6) are negligible, and they remain negligible as the temperature is increased until EF is within a few kT of ET• The temperature at which the lifetime increases depends upon the doping and the trap level. For con venience in performing the calculations, the trap level will be located at the intrinsic Fermi level. The lifetime will thus be approximately constant in the temperature range in which solar energy conversion will be con sidered for material with a doping level of 1017 per cm3• Another factor in I. to consider is the generation rate g. The generation rate increases slightly with temper- 1000 ,---...,---.,-- GoAs 900 N = 10" /em! 800 700 600 ~ FIG. 4. V max vs T, case -" 500 as parameter. } 400 300 200 100 o L--~O-~IO~0~~2~0~0-~3~00~-4~00 T I °C ature. This increase is due to the decrease in band gap, and consequent increase in the number of photons which are effective in creating electron-hole pairs. The effect is small in the temperature range considered, however, and will be neglected. It is concluded that I. is not a rapidly varying" function of T for heavily doped material, and its temperature dependence will be neglected hereafter. 3. Effect of Temperature on Ij The junction current, as has been indicated, can be determined by several mechanisms. The leakage mecha nism will not be considered in what follows because leakage through surface channels gives rise to equations similar in form to those obtained with the recombination 2.o,---..,.------,.-----,--,---...,..-----, CASE I 1.8 N 'lo'T/eM' 1.6 1.4 1.2 ~ 1.0 > " ... :: 0.8 InP 0.6 0.4 0.2 FIG. 5. Vmux vs temperature. model. Its effect can be inferred, therefore, from the results obtained with the recombination model. The following two mechanisms will he considered in detail. a. Ideal Junction Current This current arises from carriers which flow over the junction barrier; it depends upon voltage in the follow- 72 64 56 48 alE E " -: 40 E 32 24 16 Go 5, CdS CASE I N = 101T/em, 8L---~0-~10~0~~2~0-0-~3~00--4~00 T,OC FlO. 6. Imp vs temperature. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42574 J. J. WYSOCKI AND P. RAPPAPORT 32 28 24 20 ~ 16 J:" 12 8 4 00.2 ing manner13: G. 1.0 Cdli AISb CASE I Y( 70%GoAs N-10"/c"'- 30% GoPI 1.4 1.8 Eg (evl Z(50%GoAs- 50%GoPI 2.2 2.6 3.0 FIG. 7. Efficiency us energy gap. (7) (8) The behavior predicted by Eqs. (7) and (8) is well known. The magnitude of 10 and Ij is determined primarily by the band gap of the semiconductor through its effect on the square of the intrinsic density n/'. The coefficient 10 increases exponentially with temperature resulting in an exponential increase in Ii' 32.---.----.---,----,---,----,---. 28 24 20 ~ 16 12 8 4 GoAs InP CASE II N.I017/cml Si yt\CdTe '( (70% GaAs-30% GaP) Vi --, / "- "-CdS "-\//', 'T·298·K , .... __ ...... 373 0 ". ,/ / - - - - 423· I _----473· / '. // .---.--;;i:: / ...... ' 1/ \/ ...... / ~.~. /' .....-/ Ii '\~:/ / td J....-r 673· 0.6 1.0· 1.4 1.8 2.2 2.6 3.0 E9 (evl FIG. 8. Efficiency l'S energy gap. b. Recombination current The recombination current arises from carriers which recombine in the depletion region from centers existing in the forbidden gap.14 For single-level centers, the current is determined by integrating the Shockley Read expression for the recombination rate17 over the depletion region, leading to the following equation14: Ii qn; 2 sinh(qV/2kT)f(b) ---w . (TpOTnO)! (¢-V) (q/kT) (9) f(b) is a slowly varying function of the voltage, trap level; lifetimes, and barrier height. The magnitude of the current varies as ni, instead of nl consequently, it is determined by half the band gap. The current increases exponentially with temperature with an activation 28 24 20 ~16 J:" 12 8 4 00.2 GoA. AISb InP CelT. Y(70%GoAs-30"I.GoP) /11\1'1'" __ Z(50%GoA.· v' v "\.. 50% GoP I Si / ""CdS Ge / / "- "\..T'298.K / -"- 0.6 1.0 \ ",--'_"373· 1/' ,/ .... 423' \ ",/ :/ - - - 473· V _-523· /"'V/: __ ~_573. ;1::1--.... 673" 1.4 1.8 2.2 2.6 3.0 Eg (evl FI~. 9. Efficiency '/IS energy gap. energy which depends upon the location of the trap level. 4. Effect of Temperature on Solar Energy Conversion The temperature effects already considered were incorporated into the equations governing photovoltaic energy conversion, and conversion performance was calculated as a function of temperature. The calcula tions were performed on an IBM 650 Digital Computer. Three cases were considered. In Case I, Ij was the ideal junction current. Con sequently the equations for photovoltaic energy con version could be put into the following closed form: (10) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42PHOTOVOLTAIC SOLAR ENERGY CONVERSION 575 TABLE 1. Semiconductor parameters used in calculations. (Note: estimated and state-of-the-art values.) Semiconductor I" cm'lv-sec 1'0 cm'lv-sec TO -Tn (seC) mn+ Ge 3000 13S0 10-6 O.SS Si 710 360 10-7 LOS InP 4000 100 10-8 O.OS GaAs SOOO 400 10-8 0.06 CdTe 300 30 10-8 LOS AISb 710 360 10-8 LOS Y (70% GaAs-30% GaP) 200 20 10-8 LOS Z (SO% GaAs-SO% GaP) 200 20 10-8 LOS CdS 200 20 10-8 LOS 1 Rmp , XIoe'~Vmp (11) (12) !mp=X V mpe)..Vmv!o, (13) XVmp2 !. ( 10) '11 1+-1 +X V mp 1.35 I.' (14) for a power input of 135 mw / cm2• ! 0 is given by Eq. (8). The junction current in Case II was the sum of the ideal junction current and a recombination current. sinh(XV/2) 1;= Io(e)..v -1)+1 n feb), (15) X(t/>-V) where f': dz f(b) = ; <I Z2+2bZ+1 b=exp( _ X;); Z1.2= e'f)"/2(~V); FIG. 10. V max TiS tem perature-silicon solar cell. 700 600 500 ;; E 400 200 100 (16) (17) (18) LEGEND N' 10"/. CMS x-Si "I DATA 0L---0~--1~00~~2~0~0~3~00 T ,DC m.+ 0.36 0.60 0.60 O.SO 0.60 0.60 0.60 0.60 0.60 and n. No. [. cm' (3000K) (ma/cm') E. (ev) S.3XlO12 16 S$ 0.83-4X1o- 4 T 1.1 X 1010 12 5S 1.2-3.SX1O-4 T SX107 11 SO 1.39-4.6X1o-4 T 9.2X106 11 4S l.S-SXlO-4 T 1.2X 107 12 42 1.S7-4X10- 4 T 1.7 X 106 10 40 1.67-4X 10-4 T 3.7X104 12 30 1.9-4 X 10-4 T 3.1XlQ2 12 24 2.1-4X10-4 T 1.2X1O-1 12 14 2.S2-4X 10-4 T In 2qn; --[(12e/qa)(t/>- V)Jl. (T pOTnO)! (19) 10 is again given by Eq. (8). The energy conversion equation could not be put into closed form for this case. In Case III, I; was a recombination current given by (20) with In' (21) The energy conversion equations could again be put into closed form yielding equations similar to Eqs. (10) (14) with the exceptions that 10 is replaced by In' and X is divided by 2. 5. Assumptions and Values Used in Calculations Semiconductors with band gaps ranging from 0.7 to 2.4 ev were studied over a temperature range of 0-400°C. The generated current I. was determined by solar conditions outside the atmosphere where the solar power density is 135 mw/cm2• The number of photons FIG. 11. P!I. 'liS tempera ture for silicon solar cell. 0.6 0.5 0.4 ;; :.,,0.3 .... Q. 0.2 0.1 o a 100 200 T, DC [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42576 J. J. WYSOCKI AND P. RAPPAPORT 1.9 1.6 1.4 1.2 1.0 2: O.B " .. :l 0.6 0.4 0.2 0 ~ N = 1017 /CM' X'CdS#1 , CASE m .~ FOR SILICON , ., ., ), I'., , , o 100 200 300 400 T "C FIG. 12. V max vs temperature for cadmium sulfide. effective in creating hole-electron pairs was taken from published curves. The following assumptions were made to simplify and to facilitate the calculations. The collection efficiency was taken as unity. Losses due to reflection, leakage conductance, and series resistance were assumed negli gible. The electron and hole lifetimes, and impurity concentrations in the nand p regions were assumed equal. Traps were located at the intrinsic energy level and a concentration gradient of 4X 1019 per cm4 was assumed' for diffused junctions. Table I lists the values used in the calculations for the various semiconductor parameters. These values are state-of-the-art values in some cases, and in others estimates of the actual values. RESULTS The results of the calculations are summarized in Figs. 2 through 10. Figure 2 illustrates the forward current-voltage characteristic of a GaAs p-n junction for the different cases assumed in the calculations. Two sets of curves are shown-one for a temperature of 25°C and the other for 300°C. Figure 3 shows V max of a GaAs solar converter as a function of doping for Case I. The curves indicate that the highest value of V max and the lowest rate of decrease with temperature is obtained at the higher doping TABLE II. Characteristics of Cells. Room temperature Cell Area (em') efficiency Silicon 1.7 10% GaAs 0.2 3.8 CdS 0.385 3.6 levels. The effect of the different!j on V max of a GaAs junction doped to 1017 per cms is depicted in Fig. 4. It is obvious that best performance is obtained with Case I operation. Figure 5 is a plot of V max vs T for the semi conductors investigated for Case I and a doping level of 1017 per ems. The slopes of the lines are all roughly the same-approximately 2 mv;oC. The maximum voltage is greater, of course, for higher band gap values. A composi te curve of Imp vs T for Case I and a doping level of 1017 per cm3 is shown in Fig. 6. Imp approaches !I. as the temperature is increased. This asymptotic behavior is a consequence of the fact that junction resistance approaches zero as the temperature is increased. It is therefore not apparent in the higher band gap materials over the temperature range studied. Composite curves of efficiency and power vs bandgap are shown in Figs. 7 to 9. Case I is considered in Fig. 7. 1.6 1.4 1.2 1.0 > O.B Go 11. ... '"I"' 0.6 0.4 0.2 - 0 CASE m ,"'"'' " 0 100 200 LEGEND N= 1017/cM3 X-Cd S '" I 300 400 roc FIG. 13. Pmp/J. vs temperature for cadmium sulfide. The material with optimum efficiency at room tempera ture is GaAsY The optimum shifts to higher bandgaps as the temperature is increased in agreement with the results of Halsted.19 Figure 8 shows the efficiency for Case II, while Fig. 9 is a similar plot for Case III. The optimum band gap is roughly the same for all cases, however the efficiency is much less for Case III as compared to Case I. COMPARISON WITH EXPERIMENT Measurements were made on three cells. Table II specifies the area and room temperature efficiency of each cell. The silicon cell was a commercial unit made by Hoffman Semiconductor Corporation and the CdS cell was kindly furnished by Reynolds of WADC. 19 R. Halsted, J. Appl. Phys. 28, 1131 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42PHOTOVOLTAIC SOLAR ENERGY CONVERSION 577 During measurement, the cells were mounted in a small furnace with a glass window to allow the cells to be irradiated. A Chromel-Alumel thermocouple was attached to the cells to measure the temperature, and dry helium was pumped through the system to main tain a standard atmosphere. A light source supplied an input power density of 100 mw/cm2 to the cells. The results of the measurements are plotted in Figs. 10 to 16 together with theoretical curves for Case I and Case III at a doping level of 1017 per cma• Figure 10 is a plot of V max vs T for silicon. The experimental points fall between the theoretical curves, and the rate of decrease agrees well with the theoretical value. Similar agreement is found in Fig. 11 where (P mp/ I.) is plotted vs T. . The CdS curves are shown in Figs. 12 and 13. The experimental points fall below the theoretical curves; in fact, they lie roughly on the theoretical curves for Si, Case III. .6 .5 .2 .1 °2~--.~4----.6=---~.8~~I~.O~-7I.~2--~I.4~~L·6 I (mAl FIG. 14. Current-voltage characteristics of gallium arsenide cell vs temperature. The curves for GaAs are shown in Figs. 14 to Hi. Figure 14 is a plot of the i-v characteristics of the GaAs cell in the light with temperature as a parameter. The figure illustrates that I. is substantially independent of T as assumed in the calculations. The experimental curves in Figs. 15 and 16 are somewhat below the theoretical curves but not to as great an extent as in the case of CdS. The theoretical curves for Si, Case III, are also shown for comparison. The temperature measurements on these cells agree quite well with theory as far as the rates of decrease with temperature is ~oncerned. The lack of agreement in absolute values in some cases can be ascribed to the use of nonoptimilm cells. One feature of the measure~ ments is the behavior of the CdS cell which corroborates the results found in spectral analyses of similar cells where appreciable absorption is found at wavelengths 1000r---v----r---,----,---, 900 BOO "700 -600 g >~500 400 300 200 CASE \ \ \ \ \ \ m,Si~ \ t.EGENO N'IO'/CM! a-GAM 27 #3A o 100 200 300 400 T,oC FIG. 15. Maximum voltage 1)$ temperature for gallium arsenide solar cell. greater than that corresponding to the band edge.20 The spectral response and the temperature behavior of the solar cell indicate that the CdS cell is behaving as if it were made from a material whose band gap is closer to that of Si than that of CdS. CONCLUSIONS The optimum conversion performance is obtained when the junction current is the ideaL current. A deg- }'IG. 16, Pmp/I. vs. temperature for GaAs solar cell. 0.9 r--n---,-----y---, 0.8 0.7 0.6 0.5 ~~, :::: 0.4 0. E 0. 0.3 0.2 0.1 GAM 27 #3A o 100 200 300 T, be 20 D. C. Reynolds, "The photovoltaic effect in CdS c~stals," Trans. Conf. Use of Solar Energy (Tucson, Arizona, 1955). . [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42578 J. J. WYSOCKI AND P. RAPPAPORT radation in performance results when the current is influenaed by recombination or leakage through surface channels. The conversion performance of any given semiconductor improves as the doping level is increased. Practical limits may be set by the onset of degeneracy. The optimum material for solar energy conversion is a function of temperature. As the temperature is increased, the maximum efficiency shifts to materials with a larger band gap. For temperatures below 200°C, the band gap of GaAs is close to the optimum band gap. Temperature measurements on a commercial Si cell agree well with theory. Measurements on experimental GaAs and CdS cells indicate the following. The GaAs cell was not optimum. Losses which were not considered in the analysis led to lower values of V max and P mp/ I. than expected. The CdS cell, on the other hand, behaved as if it were made from a material with lower band gap. ACKNOWLEDGMENTS The authors wish to acknowledge the suggestions and discussions of Joseph J. Loferski. The calculations were performed by Paul Rygg and Sherwood Skillman. JOURNAL OF APPLIED PHYSICS VOLUME 31. NUMBER 3 MARCH. 1960 Penetration of Rotating Shaped Charges SAMPOORAN SINGH Defence Science Laboratory, Ministry of Defence, New Delhi, India (Received September 10, 1959) This paper presents an attempt to correlate theoretically the depth of penetration and the angular velocity of the liner in a rotating shaped charge. Each element of the rotating liner imparts an angular velocity to the corresponding jet element, and this results in a continuous increase of the cross-sectional area of the jet element as it travels in space and a corresponding decrease in the depth of penetration. In order to check the theory, numerical evaluations have been carried out in case of standard M9A1 steel cones. The theoretical results seem to explain the scanty published experimental data of the rotating shaped charges. and is given by the expression and "1= CAL'(Ri)2 AL Rt (2) (3) IT is well established that when a shaped charge rotates about its axis, there is loss of penetration. Kerr cell photographsl and x-ray flash photographs2 of rotating shaped charges about their axes show that rotation brings about an increase of the cross-sectional area of the jet. The theory of penetration by rotating !lhaped charges was developed by Singh3-6 and the basic equation of the theory is AP=EAL(:~y[ 1-p~V2 { (::Y+1}1 (1) where AL' represents the length of an element of the jet that is just formed from a finite element in the slant surface of the liner, C an "elongation" constant, Rj the where AP is the depth penetration, E an empirical con stant, AL the length of a jet element at the instant when it strikes a target, p the density of the liner material, Pt the density of the target material, V the mean velocity of the jet element, and the quantity u is the difference between two quantities, Ut and Uj, which represent the resistance of the target and the jet, respectively, to the plastic deformation required by the penetration process. The term 'Y takes into consideration all corrections for discontinuities within the jet element, i.e. the breakup of the jet into particles and the waver of the jet element due to imperfections in charge or liner; 1 L. E. Simon, German Research in World War II (John Wiley & Sons, Inc., New York, 1947), p. 119. 2 R. Schall and G. Thorner, Proceedings of the Second Inter national Congress on High-Speed Photography (Dunod, Paris, 1954). 3 S. Singh, Proc. Natl. Inst. Sci. India 19, 665 (1953). , S. Singh, J. Sci. Ind. Research (India) 14B, 669 (1955). 6 S. Singh, Proc. Phys. Soc. (London) 71, 508 (1958). I.OO"":~---------------. " z 0.8 ~ ~ a: z 0.6 ~ Cl. " 0.4 z ~ b a: 0.2 Cl. 0 0 50 100 150 200 250 300 REVOLUTIONS PER SECOND (R PS) FIG. 1. Ratio of the depth of penetration by rotating charges/ depth of penetration by unrotating charges at 7.62 cm standoff distance in mild steel targets as a function of the speed of rotation of the standard M9A1 steel linear in the standard C.I.T. labora tory charge. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 140.180.128.86 On: Sun, 12 Jan 2014 22:22:42

1.3057461.pdf

Lev Davydovich Landau: Winner of the second Fritz London Award J. R. Pellam Citation: Physics Today 14, 3, 42 (1961); doi: 10.1063/1.3057461 View online: http://dx.doi.org/10.1063/1.3057461 View Table of Contents: http://physicstoday.scitation.org/toc/pto/14/3 Published by the American Institute of PhysicsLEV DAVYDOVICH LANDAU WINNER OF THE SECOND FRITZ LONDON AWARD An address presented at the 7th International Con- ference on Low Temperature Physics (Toronto, Aug. 29 to Sept. 3, 1960) on the occasion of the 2nd Fritz London Award ceremony. Dr. Landau was unable to attend. By J. R. Pellam Photo by L. Aiwa IHAVE been asked by the Committee for the Second Fritz London Award to give an account of the life and work of this eminent recipient of the Award, Lev Davydovich Landau. I was very honored that I had been asked to undertake this task but felt rather overwhelmed by the responsibility it entailed. Because Landau has contributed to so many fields of physics, an award could have been made to him at any one of several conferences in any one of several fields. The main problem, I found, was to limit myself primarily to Landau's work in the field of low-temperature phys- ics for which this Award is made. My own work in this field has been so strongly influenced by these significant contributions that I, like so many of us similarly influ- enced, feel that I do know him, although I have never met him personally. A considerable wealth of material is available de- scribing Landau's work in the many fields of physics to which he has contributed. The following outline of John R. Pellam is professor of physics at the California Institute of Technology, Pasadena, Calif.Landau's career is drawn from two articles }>2 published in Soviet scientific journals commemorating his fiftieth birthday, which he kindly arranged to have fall two years before winning the Fritz London Award. Lev Davydovich Landau was born on January 22, 1908, in Baku, the capital of Azerbaijan on the Caspian Sea. His father was an engineer; his mother a doctor. His mathematical talents were apparent at a very early age and he can scarcely remember not being able to differentiate and integrate. At the age of fourteen he entered Baku University, from which he transferred two years later to the University of Leningrad, where he completed his studies in 1927 at the age of nineteen. Scientific writing did not await the completion of his studies, however, for he published twice during each of his last two school years. He developed an active inter- est in the new science of quantum mechanics, and at the age of nineteen introduced the concept of the den- sity matrix for energy which is now so widely used in 'Soviet Physics—JETP 7, 1 (1958). 2Uspekhi Fizicheskikh Nauk 64, 616 (1958). 42 PHYSICS TODAY43 quantum mechanics. His active scientific research career began in the Leningrad Physicotechnical Institute where he stayed from 1927 to 1929, working on the theory of the magnetic electron and on quantum electrodynamics. In 1929 he was sent abroad and spent a year and a half as guest of the Danes, the Germans, the Swiss, the Dutch, and the English. Of particular importance to Landau's development was his work at the Institute in Copenhagen during this period, and he considers him- self a student of Niels Bohr. (At Bohr's invitation, Landau was in Copenhagen again in 1933 and in 1934, participating in theoretical conferences.) OOME measure of his personality can be gained by ^ the following quotations from letters which I have received from two physicists associated with Landau during this period. The first is from Professor Niels Bohr, his teacher: It is a great pleasure indeed to learn that the Fritz London Award will be presented to Landau. Of course we all here share in the appreciation of Landau's great work and have vivid remembrances from the time about thirty years ago when he joined our group in Copenhagen. From the very beginning we got a deep impression of his power to penetrate into the root of physical problems and his strong views on all aspects of human life, which gave rise to many discussions. In the booklet which was published at my seventieth birthday, Rosenfeld has given a vivid picture of the stir at the Institute caused by the paper of Landau and Peierls on the measurability of field quantities, which eventually gave rise to a long treatise by Rosen- feld and myself. Also from our visits to Russia before the war my wife and I have many treasured remem- brances of Landau's personal attachment and his striv- ing for promoting mathematical physical research in Russia, in which he since has had so great success. In the years after the war we have constantly hoped to see Landau here again, but so far he has not been able to come. However, my son Aage and several of the other members of the Institute have, on visits to Rus- sia, met and spoken with Landau and not only learned about the admiration in which he naturally is held by his colleagues, but in him found the same warm and enthusiastic personality, which we all here hold in so deep affection. The other letter is from Professor Edward Teller, a contemporary of Landau: I met Landau in Leipzig in 1930 and later I spent some time with him in Copenhagen in 1934. My most vivid visual memory of him is the red coat he wore in Copenhagen. Mrs. Bohr teased him that he was wear- ing precisely the correct outfit for a postman. You will understand the somewhat strange circumstances that I would have forgotten about the red coats of the Copenhagen postmen except for this incident. I liked Landau very much and learned from him a great deal of physics. He enjoyed making statements calculated to shock members of the bourgeois society. While we were both in Copenhagen I married. He approved of my choice (and played tennis with my wife). He also asked both of us how long we intended to stay married. When we told him that our plans were definitely for a ratherlong duration and, in fact, We had given no thought to terminating the marriage, he expressed most strong disapproval and argued that only a capitalistic society could induce its members to spoil a basically good thing by exaggerating it to this extent. In Copenhagen Lan- dau had many arguments with James Franck about religion. He considered his religious belief incredibly outmoded for a scientist and expressed himself in im- moderate terms both in the presence and absence of Franck. Franck always laughed at him. It was very nice that when Landau left Copenhagen he made a very special point to say good-bye to Franck. It was quite clear that if he meant what he said about Franck, he did mean it in rather a peculiar way and, in fact, he meant perhaps the opposite of what he said. I continue to have a great deal of affection for Landau and I am glad that he is getting the Fritz London Award; he fully deserves it. One should remember that Landau was very young at this time; he may have mellowed some since. During this period abroad there occurred the first step which represented a transition of his interests and was destined to confront him with the major problems of low-temperature physics. The interesting pattern which had dominated his previous work provided the ammunition for tackling new problems. This became a cumulative process. At the age of 22 he developed the theory of "Landau diamagnetism" of metals, showing that a degenerate ideal electron gas possessed a dia- magnetic susceptibility equal to h the paramagnetic susceptibility. Some years later (1937-38) this led to the explanation of the de Haas-van Alphen effect. In this very case of diamagnetism, the proficiency which in his early years Landau had developed in manipulat- ing Fermi systems has been basic to his latest theory predicting "zero sound" in liquid helium-3, involving distortions of the Fermi surface. Landau's ease of han- dling this situation is quite understandable considering the mastery of Fermi systems which he gained thirty years earlier. HIS return to Leningrad was of short duration, for at the age of 24 he went to Kharkov to head the theoretical section of the Physicotechnical Institute (1932-37), where versatility both in achievement and outlook began to appear. His publications during the first year at Kharkov range from a paper "On the The- ory of Stars" to a paper "On the Theory of Energy Transfer in Collisions". The latter characterizes a Lan- dau specialty: the solution of difficult theoretical prob- lems by brilliant mathematical flank attacks. The same methods have held him in good stead—his mastery of collision problems reached a peak in 1949 when he con- sidered roton-roton and roton-phonon collisions (with Khalatnikov) to predict (correctly) the attenuation of second-sound waves. Landau's convictions that independent creative work in any field of theoretical physics must begin with a sufficiently deep mastery of all its branches took root at Kharkov, where he developed the special program widely known among his physics students as the "theo- March 196144 retical minimum'. Here also he began to accumulate a following among students, of whom the best known in low-temperature physics include Lifshitz and Pome- ranchuk. His versatility is illustrated by quoting the titles of the papers which he wrote during his last two years at Kharkov: Theory of Photo-emf in Semiconductors, Theory of Monomolecular Reactions, Theory of Sound Dispersion (with E. Teller), Kinetic Equation of the Coulomb Effect, Properties of Metals at Very Low Tempera- tures, Scattering of Light by Light, Theory of Phase Transitions. All these were published in 1935. In 1936 he published: The Kinetic Equation for the Case of Coulomb Inter- action, Absorption of Sound in Solids, Theory of Phase Transitions, Theory of Superconductivity, Statistical Model of Nuclei, Scattering of X-Rays by Crystals Near the Curie Point, Scattering of X-Rays by Crystals with Variable Structure, Origin of Stellar Energy. Of deeper consequence to the field of low-tempera- ture physics, however, was a direction of interest which he developed at Kharkov and continued after moving to Moscow, during the organization of the P. L. Kapitza Institute for Physical Problems. Landau's attention to diamagnetism proved transitional between quantum me- chanics and the theory of metals. Besides explaining the de Haas-van Alphen effect, Landau's applications of thermodynamics to electronic systems at low tempera- tures included the following: 1. He introduced the concept of antiferromagnetic or- dering as a new thermodynamic phase; 2. He developed the thermodynamic theory of mag- netic domains (with Lifshitz), providing a founda- tion for theories of magnetic permeability and reso- nance of ferromagnetics; 3. He studied phase transitions and determined the pro- found relation between transitions of the second or- der and variation of symmetry of the system. He gave a detailed thermodynamic theory of the behav- ior of systems near the transition point; 4. He studied the intermediate state of superconductors and proposed a theory of laminar structure of super- conductors. Also during this Kharkov period, Landau started the series of now well-known monographs on theoretical physics. FT was only natural upon his arrival at Moscow in -*• 1937, where he was appointed head of the theoreti- cal section of the Institute for Physical Problems, that his interests turned to the subject of superfluidity which was then being investigated experimentally by Kapitza himself. This marks an all-out assault by Landau on pure low-temperature physics, and under his attack the major problem of the nature of the helium II phase of liquid helium-4 soon withered. This work was close to the well-known interests of Fritz London, who solved the problem using another approach. The crux of Lan-dau's cracking the helium problem (published in 1941) was his ability to deduce semiempirically the energy spectrum 3 of the Bose excitations in this liquid. The shape of the now well-known curve of energy versus momentum for such quasi-particles included a valley occurring at an energy height (equivalent kT) of 8 — 10 °K. Such a spectrum permitted these quasi-particles to exist in equilibrium at this level, and these, following a suggestion by I. Tamm, Landau named "rotons". The energy gap, A, inherent to these rotons, permits the existence of superfluidity. As a consequence of Landau's interpretation of super- fluidity, he was able to predict independently the exist- ence of the "second-sound" mode of wave propagation in liquid helium II (independently, because Tisza some- what earlier had predicted second sound on the basis of Fritz London's approach). Two aspects of Landau's manner of handling the second-sound problem are particularly noteworthy, in that they may also bear on his most recent predictions of "zero sound" in liquid helium-3: 1. Landau's presentation shows certain detachment from the problems of experimental generation and de- tection of second sound. Early efforts by Shalnikov and Sokolov before the war were unrewarding because they attempted to detect second sound using standard acous- tical methods. In fact, the problem was clarified by a subsequent publication by Lifshitz, who pointed out the essential thermal nature of second sound. On the basis of this prescription, Peshkov observed second sound experimentally in 1944. 2. In the same 1941 paper, Landau correctly pre- dicted the magnitude of the velocity of second sound in the vicinity of absolute zero as CJ/A/3, where cx is velocity of ordinary sound. He produced this result only after complicated mathematical acrobatics, and one wonders how much faith could possibly be placed in such a conclusion. Landau's own faith in his result was eloquently expressed, in a 1949 Letter to the Editor of The Physical Review defending his theory: ... I have no doubt whatever that at tempera- tures of 1.0—1.1 °K the second-sound velocity will have a minimum and will increase with the further decrease in temperature. This follows from the thermodynamic quantities in helium II calcu- lated by me. Who could be so certain? This clearly demonstrates Landau's extraordinary physical intuition. Despite the intricate mathematics he recognized the situation at absolute zero, not as an extrapolation, but as an end position for buttressing the results. Thermodynamic complications dissolved as T —> 0 °K. With only phonons of first sound present, the root-mean-square velocity component along any particular propagation direction of any more subtle propagation could occur only 1/V3 as fast. This was perhaps Landau's ace-in- the-hole and private little joke besides. We will later 3 A purely quantum-mechanical derivation of this spectrum has been achieved recently by Feynman. PHYSICS TODAYMODEL 31 Detects currents to 10 >7 ampere. Provided with ten ranges, separate input-preampli- fier unit. VIBRATING REED MODEL 32 Designed specially for radioisotope studies. Single-unit design, provided with four ranges. Outstanding instruments for precise, reliable measurement of extremely small charges, currents and voltages. Several models are available to serve a variety of applications including radioisotope assays, ion current measurements, pH determinations, and solid-state studies. MODEL 36 Offers exceptional response, DC to 10 cps, sensitivity and stability for small currents to 1015 ampere originating in a high impedance source. INSTRUMENTS APPLIED PHYSICS CORPORATION 2724 South Peck Road, Monrovia, California46 recall these two facets in connection with the theory of "zero sound" in liquid helium-3, and how they may bear on this subject. FT is quite out of the question to consider all aspects -*- of Landau's accomplishments. Typical of his versa- tility is a series of five papers published in 1945 con- cerning shock waves at large distances from their place of origin, and related subjects. (This work was car- ried out under the Engineering Committee of the Red Army.) Then, in 1946, papers appeared on oscillations of plasmas, which, it is stated, "received specially large notice recently in connection with the study of the properties of plasmas". A large amount of work in this field has been carried out recently by a group under A. E. Akhasier in Kharkov. During the late 1940's Landau devoted his efforts to a whole gamut of activities. Efforts in the field of low- temperature physics consisted primarily of further ap- plications of his spectrum of excitations in liquid helium to examining various kinetic processes. This included viscosity, thermal conductivity, and attenuation of second-sound waves (with Khalatnikov). In recent years his efforts have included a series of papers (with A. A. Abrikosov and M. I. Khalatnikov) on quantum electrodynamics. During the period when nonconserva- tion of parity in weak interactions had been proposed by Lee and Yang, but before experimental verification, Landau proposed the hypothesis of the conservation of combined parity. He transferred his attention to the fact that nonconservation of parity does not, without fail, require violation of the properties of symmetry of space, if it is assumed that also <charge conjugation> is not conserved simultaneously but the product of these quantities, named by him "combined parity", is con- served. This puts definite restrictions on the general hy- pothesis of conservation of parity. He predicted the polarization of the neutrino, as did Lee and Yang, who did not however connect it with the principle of com- bined parity. He also discussed the polarization of /? particles. The theory of "zero sound" in liquid helium-3 may quite possibly develop into Landau's greatest contribu- tion to low-temperature physics. This combines Lan- dau's talents in the fields of diamagnetism and of the properties of quantum liquids. Essentially it is a treat- ment of oscillations of the surface of the Fermi sea, and Landau is quite at home navigating waves on the Fermi sea. As in his successful approach to the helium-4 prob- lem, Landau considers not the individual particle mo- tion, but instead the collective motion of particles, i.e., the "elementary excitations" or quasi-particles. Also, as in the case of his second-sound predictions, the precise nature of "zero sound" in the sense of the experimental techniques for generation or detection is not discussed; at least, this is the case for the experimentalist who is speaking! The ubiquitous V? shows itself again, and, as before, I feel sure that it carries more physical sig- nificance than the limiting form of a complicated for-mula. But here the velocity of "zero sound" equals the velocity (cx) times v3, rather than (cx) divided by \5. Probably this is the key to the reason Landau has named this mode of propagation "zero sound" rather than "third sound", for example. It evidently repre- sents 4 a turning back of the crank to arrive at an even more elementary excitation than first sound! The scientific accomplishments of Lev Davydovich Landau have received due recognition within his own country. In 1946 he was elected an active member of the Academy of Sciences of the USSR. He has been awarded the Stalin prize three times (once in 1941 for his theory of liquid helium and work on phase transi- tions). Outside his own country, Landau has been elected to membership in the Danish and the Dutch Academies of Sciences; he has recently been elected a foreign member of the Royal Society of London and of the US National Academy of Sciences. He has pub- lished well over a hundred papers in more than a dozen scientific journals, and is the author or coauthor of a total of ten books. I will conclude with two ex- cerpts from the JETP article x written on the occasion of his fiftieth birthday, which to me appear particu- larly appropriate: It is not without significance that at the weekly seminar which Lev Davydovich conducts at the Insti- tute for Physical Problems, reports are presented not only on theoretical researches but also on the results of experimental work on the most varied problems in physics. Participants in the seminar are repeatedly amazed to see Lev Davydovich show equal enthusiasm and thorough knowledge in discussing, for example, the energy spectrum of the electrons in silicon, directly after dealing with the properties of the so-called "strange" particles. . . . The breadth of Lev Davydovich's grasp of con- temporary physics is even more convincingly shown by the course of theoretical physics which he has writ- ten together with E. M. Lifshitz. Taken together, these books are a fundamental trea- tise on theoretical physics. In originality of exposition and broad grasp of the material they are unprecedented in the whole world-wide literature of physics, and so have attained wide popularity not only in this country but also abroad. The contribution for which theoretical physics is in- debted to Lev Davydovich is not exhausted by his own scientific writings. We have already spoken of another side of his activity—his founding of a broad school of Soviet theorists. His inextinguishable enthusiasm for science, his acute criticism, his talent and clarity of thought attract many young people to Lev Davydovich. The number of those, both young and mature scien- tists, who turn to Dau (as his pupils and associates have come to call him) is very large. Lev Davydovich's criticism is hot and merciless, but behind this outer sharpness is hidden devotion to high scientific princi- ples and a great human heart and human kindness. Equally sincere is his wish to aid the success of others with his criticism, and equally warm is his expression of approval. 4 Zero sound appears distinguished from first sound primarily as a distortion, rather than a displacement, of the Fermi surface. PHYSICS TODAY

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Spin Densities in Organic Free Radicals Thomas H. Brown, D. H. Anderson, and H. S. Gutowsky Citation: The Journal of Chemical Physics 33, 720 (1960); doi: 10.1063/1.1731246 View online: http://dx.doi.org/10.1063/1.1731246 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/33/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Local spin density theory of free radicals: Nitroxides J. Chem. Phys. 80, 4286 (1984); 10.1063/1.447259 Spin Susceptibility of Organic Free Radicals J. Chem. Phys. 51, 1256 (1969); 10.1063/1.1672136 Erratum : Spin Densities in the Cyclohexadienyl Free Radical J. Chem. Phys. 38, 1023 (1963); 10.1063/1.1733749 Spin Densities in the Cyclohexadienyl Free Radical J. Chem. Phys. 37, 1094 (1962); 10.1063/1.1733219 Spin Densities in the Perinaphthenyl Free Radical J. Chem. Phys. 28, 51 (1958); 10.1063/1.1744078 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43THE JOURNAL OF CHEMICAL PHYSICS VOLUME 33, NUMBER 3 SEPTEMBER,1960 Spin Densities in Organic Free Radicals* THOMAS H. BROWN, t D. H. ANDERSON,t A:!.'D H. S. GUTOWSKY Noyes Chemical LavoraJory, University of Illinois, Urbana, Illinois (Received March 30, 1960) The proton hyperfine constants obtained from proton magnetic resonance spectra of polycrystalline free radicals and from valence bond calculations of 'lI"-orbital spin densities are compared for the free radicals a ,a'-diphenyl-,B-picryl hydrazyl, and tris-p-chlorophenylaminium perchlorate. Two lines are observed in the proton magnetic resonance spectra, one shifted upfield and one downfield from the normal resonance fre quency. These are assigned to the ortha and para protons, and the meta protons of the free radicals, re spectively. Though the agreement between theory and experiment is not quantitative for the hyperfine constants, the theoretical and experimental ratios of hyperfine constants agree within experimental error. This serves to verify the existence of positive and negative spin densities and the signs of Rome of the rela tionships involved. I. INTRODUCTION THE nuclear spin-electron spin coupling in organic free radicals is an important source of detailed information concerning molecular electronic structure. Usually, the nuclear hyperfine coupling constant ai is obtained by observing the splitting of the electron magnetic resonance spectrum.1,2 The experimental value is then compared with the results of calculations for various theoretical models of the free radical. In this paper we are concerned with proton hyperfine con stants obtained from observations of the nuclear magnetic resonance shifts in the solid free radicals a, a'-diphenyl-,B-picryl hydrazyl (DPPH) and tris-p chlorophenylaminium perchlorate (TPPAP). A brief account has been givenS of such observations and pre liminary valence-bond calculations which demon strated the existence of both positive and negative proton coupling constants in DPPH. Similar observa tions are reported here for TPPAP and the details of the calculations are presented for both substances. In these substances, we consider the values of ai to be determined very largely by the contact term4 (1) where ['l'(0) Jl is the density of the unpaired electron at the ith nucleus, and the g and ,B are the g factors and magnetons for the electron and nucleus. The cal culations of the coupling constants are conveniently expressed in terms of the spin densities5-8 Pi at the * Acknowledgment is made to the donors of the Petroleum Re search Fund, administered by the American Chemical Society, and to the Office of Naval Research and E. 1. du Pont de Nemours and Company for partial support of this research. t Postdoctoral Fellow, Division of General Medical Sciences, United States Public Health Service. :j: Now at Sandia Corporation, Albuquerque, New Mexico. 1 C. Kikuchi and V. W. Cohen, Phys. Rev. 93, 394 (1954). 2 C. A. Hutchinson, R. C. Pastor, and A. G. Kowalsky, J. Chelll. Phys. 20, 534 (1952). 3 H. S. Gutowsky, H. Kusumoto, T. H. Brown and D, H. Ander son, J. Chern. Phys. 30, 860 (1959). Independent experimental evidence for the negative sign in Eq. (2) and for negative spin densities on carbon atoms has been obtained in several studies; see footnote reference 42 and prior work cited there. 4 E. Fermi, Z. Physik 60, 320 (1930). various nuclei in a radical. For protons attached to carbon atoms in an aromatic ring, the proton hyperfine splitting results from a q-7r exchange interaction,5.9-n for which McConnell5-7 has given an approximate semiempirical relation between the 7r-electron spin density at the carbon and aH. This relation, aH= -22.Spc, (2) where aH is the splitting in gauss of the electron reso nance, is approximately independent of the substituents in the aromatic system. We will calculate PC from the wave function and then use Eq. (2) to obtain values of aH for comparison with experiment. Simple valence bond and molecular orbital calcula tions have been attempted for a number of free radi calsa,6,s-ls; however, the agreement with experiment varies considerably from radical to radical. Two general types of radicals have been distinguished. These are the even-alternate radicals, of which the aromatic hydro carbon negative ionsl7,18 are examples, and the odd alternate radicals such as DPPH,19,20 perinaphthyl,21 a H. M. McConnell, J. Chern. Phys. 24, 632,764 (1956). 6 H. M. McConnell and H. H. Dearman, J. Chern. Phys. 28, 51 (1958) . 7 H. M. McConnell and D. B. Chesnut, J. Chern. Phys. 28, 107 (1958); 27, 984 (1957). 8 H. S. Jarrett and G. J. Sloan, J. Chern. Phys, 22,1783 (1954), 9 B. Venkataraman and G. K. Fraenkel, J. Chern. Phys. 24,737 (1956) . 10 R. Bersohn, J. Chern. Phys. 24, 1066 (1956); Arch. sci, (Geneva) 11,12 (1958). 11 S. I. Weissman, J. Chern. Phys. 25, 890 (1956). 12 P. Brovetto and S. Ferroni, Nuovo cimento 5, 142 (1957). 13 H. M. McConnell, J. Chern. Phys. 28, 1188 (1958); 29, 244 (1958) . 14 D. B. Chesnut, J. Chern. Phys. 29,43 (1958). IS G. J. Hoijtink, Mol. Phys. I, 157 (1958). 16 A. D. McLachlan, Mol. Phys. 1,233 (1958). 17 R. L. Ward and S. I. Weissman, J. Am. Chern. Soc. 76, 3612 (1954). 18 R. C. Pastor and J. Turkevich, J. Chern. Phys, 23, 1731 (1955). 19 A. N. Holden, C. Kittel, F. R, Merritt, and W. A. Yager, Phys. Rev. 77,147 (1950). 20 C. H. Townes and J. Turkevich, Phys. Rev. 77, 148 (1950). 21 P. B. Sogo, M. Nakazaki, and M. Calvin, J. Chern. Phys. 26, 1343 (1957). 720 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43SPIN DENSITIES IN ORGANIC FREE RADICALS 721 and the triphenyl aminium salts.22.23 For radicals of the first type, the results of molecular orbita15.24 and valence bond calculations25 agree with experiment and with each other, though not always to the extent that one might wish. However, for odd-alternate radicals, simple molecular orbital calculations yield positive Spill densi ties of reasonable magnitude or zero values depending upon the position in the molecule.6•10 The valence-bond calculations, on the other hand, yield both positive and negative spin densities, all of reasonable magnitude, with the negative spin densities occurring at those positions which were zero in the molecular orbital case.5-7.10.12.13 Inasmuch as we present experimental evidence for the existence of negative spin densities, we employ a valence-bond approach in our calculations. II. EXPERIMENTAL PROCEDURE AND RESULTS The experimental magnitudes and signs of the proton hyperfine constants reported here were determined by observing the proton magnetic resonance spectra of solid molecular free radicals. The magnitudes of the hyperfine constants in free radicals are usually obtained from the hyperfine splittings observed in electron mag netic resonance spectra. The energy levels involved are given by the expression E=g-/3Jloms- LgJ3NHomi+ Laige{3emsmi, (3) i i where ai in the hyperfine interaction term is here the splitting in gauss of the electron magnetic resonance by the ith nucleus.26 The selection rules for electron magnetic resonance are ~ms = ± 1 and ~mi = 0; and for nuclear magnetic resonance, ~ms=O and ~mi= ±1. The symmetry of the hyperfine interaction in the plus and minus values of ms and mi makes it impossible to determine the sign of ai from the splittings and transi tion frequenCIes. Changing the sign of aN simply inter changes the hyperfine components within the NMR spectrum and also within the electron resonance. However, at thermal equilibrium, the upper energy levels are labeled by their decreasing population, and the order of the levels does depend upon the sign of aN. This difference produces small differences in the in tensities of the hyperfine components of both the elec tron and nuclear magnetic resonances, which in prin ciple could be used to establish the sign of aN. Un fortunately, the differences in intensity produced by changing the sign of aN are of the order of one part in 10:; and too small for practical use except at very low temperatures. A more readily observed effect results3.7 when the hyperfine structure is averaged out by some dynamic process, such as a short electron Tl or by electron exchange as in our experiments on solid DPPH 22 R. S. Codrington, J. D. Olds, and H. C. Torrey, Phys. Rev. 95,607 A (1954). 23 R. 1. Walter, J. Am. Chern. Soc. 77, 5999 (1955). 24 E. de Boer, J. Chern. Phys. 25, 190 (1956). 25 T. H. Brown, H. S. Gutowsky and J. C. Schug (to be puh lished). 26 Absolute values are used for g" gN, fI" and fiN. and TPPAP. In this case, the hyperfine splitting term becomes (4) where (mS)i is the average value of ms in radicals with nuclear spin states defined by mi. The value of (ms)i is determined by the populations of the ms= + ! and ms= -t states for which the i nuclear spins have speci fic values mi; in particular we find that (ms)i= LNs,ms/LN s,""'.J-ge!3Jlo/4kT. (5) ms ms The NSi are the populations of radicals in which the electron and nuclear spin states are ms and mi. The last step in Eq. (5) makes the reasonable approximations that kT» 1 ge{3eHo I» 1 gi{3NHo I, 1 aige!3e I. Upon sub stituting Eqs. (4) and (5) in Eq. (3), we find that the ~E corresponding to nuclear magnetic resonance, i.e., Ami=±1, ~ms=O, is given as AE= gJ3NHo+aige{3c(ge/3 eHo/4kT). (6) The net effect of the averaged hyperfine interaction is thus a displacement of the ith nuclear resonance by an amount and direction depending upon the sign and magnitude of ai. The effect is similar to the Knight shift27 produced by the conduction electrons in metals and also to the dynamic averaging of other types of NMR splittings and shifts.28 The averaging out of the hyperfine splitting occurs when either the electron exchange time T or the electron Tl is short enough that its reciprocal exceeds aige!3e. In our experiments, the proton resonance was observed with a fixed frequency spectrometer,29 in which case Eq. (6) predicts a differ ence, ~H=H*-Ho= -ai('Yel'lH) (ge{3eHo/4kT), (7) between H*, the magnetic field required for the reso nance of the ith group of protons in the radical, and Ho, the resonance field for protons not subjected to any hyperfine interaction. The 'Yare the magnetogyric ratios g{3. Of the two free radicals considered here, the experi mental results for a,a'-{3-picryl hydrazyl (DPPH) have been reported previously.3 For tris-p-chloro phenylaminium perchlorate23 (TPPAP), the derivative of the proton resonance was observed29 at 77°K in a polycrystalline sample at 26.90 Mc. The integrated line shape, as given in Fig. 1, can be resolved into two components with the same intensity but different widths, one shifted upfield by 3.90 gauss and the other downfield by 2.30 gauss from the proton resonance in a diagmagnetic reference sample. For this particular radical, there are only two types of protons, the ortho 27 W. D. Knight, Phys. Rev. 76,1259 (1949). 28 H. S. Gutowsky, D. W. McCall and C. P. Slichter, J. Chern. Phys. 21, 279 (1953); H. S. Gutowsky and A. Saika, ibid. 21, 1688 (1953). 29 H. S. Gutowsky, L. H. Meyer and R. E. McClure, Rev. Sci. Instr. 24, 644 (1953). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43722 BROWN, ANDERSON, AND GUTOWSKY and meta ring protons, and the assignments are simpler than in the case of DPPH. In the latter, under the same conditions as for TPPAP, the four proton resonance components expected were only partially resolved into two components of unequal widths but having the same intensity, one shifted upfield by 2.95 gauss, the other downfield by 1.75 gauss.3 The assignments are based upon the results of the theoretical calculation discussed in the next section. III. CALCULATION OF SPIN DENSITIES A typical structure for each of the two free radicals considered is given by Formulas (I) and (II), respec tively. The calculations DPPH o NO, I I :N-N-~-N02 I .. ~- I o NC, (I) TPPAP CI I o I CI-( )-~+-( )-Cl (II) of the ground-state wave functions are quite similar to those reported previously for triphenyl methyU2 For our two radicals, only the mono-phenyl aminyl fragment need be considered in detail. The rules de veloped by Pauling and Wheland30 can then be used to extend the results to the biphenyl and triphenyl cases. For the mono-phenyl aminyl fragment, the three types of structures which enter into the ground state are Formulas (III)-(V): . o 0 .0 I II II N N N (III) (IV) (V) ao L. Pauling and G. Wheland, J. Chern. Phys. 1, 362 (1933). ~It$-'N~ r H H J3 -10 • 0 H-tio-GAUSS FIG. 1. The proton magnetic resonance absorption observed at 26.90 Mc in polycrystalline tris-p-chlorophenylaminium per chlorate (TPPAP) at 77°K. H* is the magnetic field applied to the sample while Ho (~631O gauss) is that at which the proton resonance occurred in a diamagnetic reference, corrected for bulk magnetic susceptibility differences. The wave function is the linear combination given by Eq. (8), (8) where 1/;N represents the contribution of structures where the unpaired electron is on the nitrogen atom, 1/;p on the para position, and 1/;0 on the ortho positions. It is important to note that, within the framework of simple valence bond theory, it is not possible to draw structures which have the unpaired electron on the meta ring positions. By using simple valence-bond theory, which con siders only near-neighbor interactions, and simple overlap, the coefficients Cl, C2, and C3 may be obtained. From these, using the spin-density operator given by McConnell,1 (PA) ij= (1/;i I PA I 1/;j)= (1/ Sz) (1/;i I L~A(k)Skz l1/;j), k (9) the spin densities in the various orbitals may be ob tained. In Eq. (9), (PA) ij represents the contribution to the 7r-orbital spin density at atom A from structures i and j. The total 7r-orbital spin density at atom A, PA, is then obtained by the appropriate use of Eq. (8). The values of the spin densities given by Eq. (9) can be obtained by modification of the Pauling rules con cerning the coefficients of exchange integrals from super positions diagrams.6•l2 We now consider in detail the calculations and results obtained for the two free radi cals. A.DPPH The usual structural formula for DPPH is Formula (I). However, even though the unpaired electron is allotted to the nitrogen adjacent to the picryl ring, it is This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43SPIN DENSITIES IN ORGANIC FREE RADICALS 723 clear that, in order for the radical to be stable, the unpaired electron must be considerably delocalized over both sets of ring systems. Stability of this type is evidenced in the contrast between the triphenyl methyl radical and the considerably more reactive phenyl methyl radical. In addition, electron magnetic resonance studies2 indicate large spin densities on both the central nitrogen atoms of DPPH. The only plausible way in which the unpaired electron can migrate onto the phenyl rings and onto the other nitrogen is to assume that an ionic structure contributes to the molecular wave function. Therefore, we repre sent the ground state of DPPH as a superposition of the two structures, Formulas (VI) and (VII): .+ rhN-N-P rhN-N-P, (VI) (VII) where cf> is a phenyl ring and P a picryl ring. The coordinate pair of electrons on each nitrogen atom has been drawn in for clarity. The fact that (VI) allows migration only onto the picryl ring and (VII) only onto the phenyl rings, suggests that the calcula tion may be decomposed into two parts involving the monophenyl aminyl (MPA) and the biphenyl aminium (BPA) fragments. As a further simplification, the nitro groups on the picryl ring as well as the positive charge on the aminium nitrogen will not be considered. The problem has thus become one of calculating the un paired electron spin densities of MPA and BPA, Formulas (VIII) and (IX). Each fragment will be assigned O.S electron, which assumes MPA o I N (VIII) BPA 00 ""/ N (IX) that (VIII) and (IX) make approximately equal contributions to the ground-state wave function. This assumption will be considered in the discussion section. TABLE 1. Electron spin densities at the nitrogen and carbon nuclei in the monophenyl aminyl (MP A) and biphenyl aminium (BPA) fragments. The values were calculated via Eqs. (8) and (9) using 1.2 for the ratio « of the carbon-nitrogen to the carbon-carbon exchange integral." PN PIC po Pm MPA +0.300 -0.160 +0.185 -0.100 +0.190 BPA +0.266 -0.129 +0.131 -0.060 +0.122 & PIC is the spin density at the carbon to which the nitrogen is attached; Po. Pml and pp are the spin densities at the carbons ortM, meta, and para to the I-carbon. TABLE II. Electron spin densities at the nitrogen and carbon nuclei in tris-p-chlorophenylaminium perchlorate as a function of «, the ratio of the carbon-nitrogen and carbon-carbon exchange integrals." p""a 0.5 0.8 1.0 1.2 1.5 2.2 PN +0.800 +0.646 +0.562 +0.484 +0.430 +0.331 PIC -0.191 -0.224 -0.228 -0.219 -0.227 -0.215 po +0.135 +0.181 +0.200 +0.213 +0.225 +0.239 Pm -0.076 -0.096 -0.104 -0.109 -0.113 -0.116 pp +0.140 +0.171 +0.182 +0.189 +0.193 +0.192 Po/Pm-1.78 -1.89 -1.93 -1.96 -1.99 -2.06 " PIC is the spin density at the carbon to which the nitrogen is attached; Po, Pm, and pp are the spin densities at the carbons orllw, meta, and para to the I-carbon. The absolute values of the energy levels are not re quired; all that is needed to solve the secular equation which leads to the ground-state wave function is the ratio a of the carbon-nitrogen to the carbon-carbon exchange integral. By using a value of 1.2 for this ratio, which is based on an analysis of bond energies, the following values are obtained for the coefficients in Eq. (8): MPA BPA C1 -0.334 -0.134 C2 -0.254 -0.117 C3 -0.280 -0.138. The effect of varying the ratio of the exchange integrals will be discussed in more detail in the following section. By combining Eqs. (8) and (9), the spin densities at the various nitrogen and carbon nuclei can be ex pressed in terms of the coefficients C1, C2, and C3. The re sults for the two fragments, using the numerical values of C1, C2, and C3 given above for a= 1.2, are listed in Table VI A comparison of these results with experiment will be given in the concluding section. B. TPPAP The calculation of the spin densities in TPPAP, of which a typical structure is (II), proceeds in a fashion similar to the calculations for DPPH and triphenyl methyJ.l2 The preliminary calculations for TPPAP neglect the effect of the chlorine substituents.32 Because of the uncertainty in the value of the carbon-nitrogen exchange integral involving the charged nitrogen and because the assignment of the proton shifts ob served for this radical is unambiguous, the calculations were performed using several values of a, the ratio of the carbon-nitrogen to carbon-carbon exchange in- 31 The results for DPPH are slightly different from those re ported earlier in footnote reference 3. The details of the ground state wave function calculation and the expressions for the spin densities in DPPH may be found elsewhere, David H. Anderson, Ph.D. thesis, University of Illinois, 1959. The formulas for PIC on pp. 90 and 114 are in error, which changes slightly the numerical results reported therein. 32 This assumption seems to be a reasonable first approximation in light of the relatively small effect chlorine has on the proton hyperfine constants in substituted semiquinones, as shown by the experimental results of B. Venkataraman, B. G. Segal, and G. K. Fraenkel, J. Chern. Phys. 30, 1006 (1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43724 BROWN, ANDERSON, AND GUTOWSKY TABLE III. Experimental and theoretical values for !he p:oton hyperfine splitting constant in DPPH and TPPAP, In Units of gauss splitting of the electron spin resonance. Radical Type of proton Theory··b ExptC ortho (phenyl) -2.94 -1.63 para (phenyl) -2.74 -1.63 meta (phenyl) +1.55 +0.97 meta (picryl) +2.55 +0.97 (ortho) Av/(meta) Av -1.82 -1.68 DPPH TPPAP ortho -4.77 -2.16 meta +2.29 +1.27 (orlho)/(meta) -1.96 -1.70 • See footnote reference 31. b Calculated fora= 1.2; see Table II and footnote reference 38 for other values for TPPAP. C The values for DPPH are from the two proton resonance lines resolved, each of which presumably has two unresolved components. tegral. The results are listed in Table II. It should be pointed out that in the calculations reported here for both radicals the rules used to obtain the spin densities at the various carbon positions were modified when considering the central nitrogen atoms.33 IV. DISCUSSION By means of Eq. (7), values for the proton hyperfine constant aH, including the sign, can be obtained from the proton shifts observed in the solid free radicals. Also, Eq. (2) can be used to calculate aH from the theoretical spin densities at the carbons. Table III lists such theoretical values, for a= 1.2, as well as the experi mental values from the proton shifts. The assignments for the experimental results were made by comparing the experimental values with the relative magnitudes of the theoretical results. A.DPPH The results in Table III indicate that the proton hyperfine constants are negative at the ortho and para positions and positive at the meta positions. Unfor tunately, the theoretical values of aH for DPPH are based upon such an approximate model that better agreement between the experimental and theoretical magnitudes probably cannot be expected. It should also be noted that the dipolar contribution to the proton shift was neglected in the calculation. However, even though the magnitudes of aH depend upon good values for exchange integrals, relative electron affinities of the aromatic rings, and other quantities which are difficult obtain, the resulting errors in aH tend to be system atic, as shown in Table II. Consequently, it is better to compare the ratios of the splitting constants than their magnitudes. The· proton resonance observed upfield is assigned to the unresolved absorption of the four ortho and two para protons and that downfield, to the four meta (phenyl) and two meta (picryl) protons. The ratio of 33 T. H. Brown and J. C. Schug (unpublished results). these shifts is -1.68. To obtain a comparable theoreti cal value, we take the weighted averages of the corre sponding aH and obtain a theoretical ratio of -1.82. The agreement between these ratios is excellent inas much as the difference between the two is within the intrinsic error of decomposing the experimental curve into components. The approximate agreement between the experimental and theoretical aH, and the quantita tive agreement between the experimental and theoreti cal ratios confirm the existence of positive and negative spin densities. The results actually give us more information since the theoretical values leave little doubt that the ortho and para protons have shifts which are larger in magni tude than those of the meta protons. The experiments demonstrate that the upfield proton shift is larger than the downfield. Therefore, the upfield shift is assigned to the ortho and para protons. However, in view of Eq. (7), aH for the upfield protons must be negative. But the approximate valence bond treatment shows that the ortho and para carbon spin densities are positive, and hence the sign of all is the opposite of the spin density on the adjacent carbon, as given in Eq. (2). The ratio of the ortho and para spin densities are relatively in sensitive to the parameters in the calculation, so this conclusion seems unambiguous. Now let us return to our rather arbitrary assignment of O.S electron, each, to the MPA and BPA fragments used in the theoretical treatment. The actual electron distribution can be represented to a better approxima tion by alloting a different fraction of the unpaired electron to each fragment. However, if the fractions differed very much from O.S, the agreement between the experimental and theoretical ratios of all for the ortho and meta protons would suffer,34 indicating that the assignment of O.S electron to each fragment is a good approximation. A final point concerns the relative electron spin densi ties on the nitrogen atoms. Our theoretical results summarized in Table I give an electron spin density which is 13% greater at the nitrogen adjacent to the picryl ring than at the nitrogen adjacent to the phenyl rings. Recent detailed studies of the hyperfine splittings by the two central nitrogen atoms in solutions of DPPH indicate that they are not equal.35 The ratio of the two nitrogen hyperfine constants was found to be 0.82± 0.01. Because of the different bonding of these two atoms and the effects of the nitro groups on the picryl rina one would expect the ratio of the nitrogen hyper fin:'constants to differ somewhat from the ratio of the 7T-orbital spin densities given here. In addition, the contributions to the nitrogen splittings from spin densities on adjacent atoms are not known, but the 34 The theoretical ratio can be fitted to the observed ratio of -1.68 by assigning different fractions of an electron to the two fragments; this gives 0.525 and 0.475 electrons to MPA and BPA, respectively. 35 R. M. Deal and W. S. Koski, J. Chern. Phys. 31,1138 (1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43SPIN DENSITIES IN ORGANIC FREE RADICALS 725 agreement of the theoretical ratio of the nitrogen spin densities (PN lPN' = 0.89) 36 with the ratio of experimental hyperfine constants, 0.82, is, however, encouraging. The experimental results obtained so far do not indicate which nitrogen atom has the greater hyperfine con~ stant.35 Recently, both the isotropic and anisotropic contributions to the nitrogen hyperfine constants in DPPH have been investigated in experiments37 on dilute single crystals containing N15. There is, however, an ambiguity in the interpretation of the hyperfine tensor such that there are two choices for the isotropic and anisotropic contributions to aN', i.e., for the ,B-nitrogen, and two values for the ratio aNI aN'. Studies in liquid solution of the N15 DPPH are needed, apparently, to resolve the question, or else detailed calculations of the hyperfine constants from the ?r-orbital spin densi~ ties. B. TPPAP With this free radical, as in the case of DPPH, the quantitative agreement between the theoretical and experimental values of the proton hyperfine constant is not too good. However, the ratio of the theoretical hyperfine constants as given in Table III agrees well with experiment and can, in fact, be fitted by varying the parameter a as shown in Table II.3s Though it is probably not meaningful to fit the theoretical value of the ratio to experiment, the agreement between theory and experiment supports the existence of both positive and negative spin densities. s In addition, it is concluded that the sign of aH is opposite to the sign of the ob ~erved proton shift and of the theoretical spin densities at the carbons.3 These conclusions are the same as those reached in the case of DPPH and should hold quite generally for this type of free radical. In contrast to the case of DPPH, the comparison between theory and experiment for TPPAP can be made without considering unresolved lines and average values of shifts, except insofar as they might arise from nonequivalence of protons in the crystal unit cell. Within the existing framework of the calculation, one might first consider the value of the ratio of exchange in tegrals a, discussed previously. It is seen, however, from Table II that, though a has a small effect on the spin densities at the ortho and meta positions, it can not provide the factor of two which is necessary for quanti tative agreement between theory and experiment. However, the spin density on the nitrogen atom is quite sensitive to the value of a. It is possible that '6 A ratio PN/PN,=0.93 was obtained for DPPH by R. Bersohn, iootnote reference 10, using a molecular orbital calculation. This result, however, suffers from the difficulties mentioned in the introduction. '" R. W. Holmberg, R. Livingston and W. T. Smith, J. Chern. Phys. 33, 540 (1960). as The effect of varying a in the DPPH calculations gives re sults similar to those in Table II (see footnote reference 31). A more complete calculation using all of the canonical structures for the phenyl aminium fragment gives agreement between the theoretical and experimental ratios for a value of a of 1.0; T. H. Brown and J. C. Schug (unpublished calculations). comparison of theoretical values for aN obtained from the spin densities given here, with experimental re sults39 will serve to determine the best value of a. In this way values for some of the integrals occurring in the valence bond theory may possibly be obtained from good experimental values of hyperfine constants. Though the measurements of the proton shifts in free radicals give the sign of the hyperfine constant as well as a value for its magnitude, a more accurate method for determining the latter is the measurement of elec tron magnetic resonance splittings. A comparison of the results of the two methods would be useful in the present case in determining the extent of the dipolar contribution to the proton shifts in the solid free radi cals and as a further check on the results of the theoret ical calculations. Unfortunately, in the case of both TPPAp39 and DPPH,2,35 no proton splittings of the electron magnetic resonance have been resolved.40 It is clear, however, from the number of nonequivalent magnetic nuclei present, that in DPPH there may be as many as 17 875 transitions. All that is necessary to account for the apparent absence of proton splittings, therefore, is to assume that the protons have several much smaller hyperfine constants than the central nitrogen nuclei. Kaplan,41 in fact, has ascribed the width of the rather broad nitrogen hyperfine lines to un resolved proton lines, using the values given previously for aH.3 In the case of TPPAP, however, one would predict a maximum of only 147 lines in the electron magnetic resonance spectrum, neglecting the chlorine nuclei. The value of aN is fairly large, about 10 gauss,39 and even if the proton spliUings cannot be resolved com pletely, the general shape of the envelope of partially resolved lines is sensitive to the values of all. In par ticular, it may be possible to differentiate between the theoretical and experimental sets of values for aH given in Table III. C. General Comments Further comment is warranted on the relatively poor quantitative agreement between the calculated and experimental spin densities for both radicals, even '9 O. R. Gilliam, R. 1. Walter and V. H. Cohen, J. Chern. Phys. 23, 1540 (1955). 40 Note added in proof. Partial resolution of the proton hyperfine splitting has just been reported by Y. Deguchi U. Chern. Phys. 32, 1584 (1960) 1 for DPPH in carefully purified solvents. A continuous series of fairly regular splittings of about 0.45 gauss is observed in addition to the much larger splittings of 10 gauss produced by the N14 nuclei. The intensity distribution leads us to suggest that each nitrogen hyperfine component is split by the protons into a series of lines extending over a range of about 15 gauss, certainly no less than the NI4 splitting of 10 gauss. The ex tent to which the protons should split each NI4 hyperfine compo nent is given by 2;n 1 an I. The values for an obtained from our observations of the proton shifts in solid DPPH are given in Table III. They lead to a value for 2;n 1 an 1 of 15.6 gauss which agrees with our interpretation of the ESR spectrum reported by Deguchi. 41 J. 1. Kaplan; (private communication). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.251.14.35 On: Thu, 18 Dec 2014 13:02:43726 BROWN, ANDERSON, AND GUTOWSKY though ratios of the densities at the various ring carbons agree reasonably well. There is some evidence33 that valence-bond structures in addition to those considered here and in earlier treatments of odd alternant radicals may contribute appreciably to the ground state of DPPH and TPPAP. The extent to which they also contribute to the spin densities is presently being investigated. Also, it should be re membered that in the calculation of the hyperfine coupling and proton shifts we neglected the anisotropic contribution, which arises mainly from the electron spin, nuclear-spin dipolar interaction.42 The contribution can be of the same order42 as the isotropic interaction which we also treat. However, in polycrystalline samples such as we used, the anisotropic component of the hyperfine interaction tensor would produce a distribution of proton resonance shifts and hence broaden the proton resonance, and perhaps shift its center of gravity as well. It is difficult to determine experimentally the extent of such broadening, because of the partial resolution of the proton resonance com ponents in DPPH and TPPAP. However, the second moments of the two proton resonance components for TPPAP, in Fig. 1, suggest that any "anisotropy broadening" is a modest fraction « t) of the isotopic shift. Moreover, the fact that the proton resonance components are symmetrical indicates that any aniso tropy is itself "symmetrical" about the average shift, which must, therefore, be that from the isotropic part of the hyperfine splitting. To be sure, these arguments are qualitative and 42 H. M. McConnell, C. Heller, T. Cole, and R. W. Fessenden, J. Am. Chern. Soc. 82, 766 (1960). We are indebted to a kindly referee who pointed out the relevance of this work to ours. subject to considerable experimental uncertainty. They are nonetheless compatible with the detailed treatment by McConnell et al.42 of the proton hyperfine splitting tensor in the CH (C02Hh radical in a single crystal of malonic acid. In any event, the existence of "aniso tropy broadening" of the proton resonance in TPP AP is supported by the upfield proton line, in Fig. 1, which is not only shifted more than the downfield line but also is appreciably broader. Part of this greater breadth can arise from larger proton-proton dipolar broadening, which would be expected for the ortho protons. In addition, the proton spin-lattice relaxation times are short enough to broaden the proton resonances ap preciably; attempts to measure them by rf pulse methods indicate that they are of the order of 100 ,",sec. Such relaxation no doubt arises from the large magnetic fields produced at the protons by the elec trons and the resultant broadening would be propor tional to the proton resonance shift. In conclusion, if the crystal structure were known in detail and the proton lines could be completely resolved in experi ments at higher fields and lower temperatures, it should be possible in principle to obtain all three diagonal components of a nuclear hyperfine splitting tensor from a study of the nuclear resonance shifts and absorption line shapes in a polycrystalline free radical. ACKNOWLEDGMENTS We wish to express our thanks to Professor Martin Karplus for helpful discussions and to John Schug for assistance in some of the calculations. Our apprecia ation is extended to Professor R. I. Walter for gener ously supplying the samples of the tris-p-chlorophenyl aminium perchlorate used in our experiments. This article is copyrighted as indicated in the article. 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1.1728928.pdf

Some Electrical Properties of the Porous Graphite Contact on pType Silicon George G. Harman, Theodore Higier, and Owen L. Meyer Citation: Journal of Applied Physics 33, 2206 (1962); doi: 10.1063/1.1728928 View online: http://dx.doi.org/10.1063/1.1728928 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electrical properties of the graphitic carbon contacts on carbon nanotube field effect transistors Appl. Phys. Lett. 101, 033101 (2012); 10.1063/1.4737169 Ageing of aluminum electrical contacts to porous silicon J. Appl. Phys. 85, 583 (1999); 10.1063/1.369394 Comparison of some properties of nanosized silicon clusters in porous glasses Appl. Phys. Lett. 72, 3005 (1998); 10.1063/1.121522 Electric and photoelectric properties of diode structures in porous silicon J. Appl. Phys. 77, 2501 (1995); 10.1063/1.358779 Electrical properties and formation mechanism of porous silicon carbide Appl. Phys. Lett. 65, 2699 (1994); 10.1063/1.112610 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 20:29:272206 G. G. HARMAN AND T. HIGIER mental resistance should be greater than 10 times the bulk resistance. Now the theoretical ratio will be given by q,theory (A2) q,theory This ratio is now set equal to the experimental ratio of resistances at V 2 and V 3. When V 3 is chosen so that V3=3V, a reasonable approximation is to take (<p-Va) = O. By repeating this calculation for a few points an estimate of the error in Sand cf> can be obtained. When both barriers contribute to the surface re sistance, in general the values of S]<p]! and S2<P21 must JOURNAL OF APPLIED PHYSICS be equal within 2 or 3%. In this case, the resistance of the two barriers will be equal, and the value of voltage used in formula 2 will be just t the experimental volt age. In general, when SI differs from S2 and <PI differs from t/!2, four experimental points will be needed to determine these parameters. One other value can be calculated from Eq. (2) when Sand <P are known. This is the voltage V at which Eq. (2) has a maximum. The main variation in Eq. (2) when V is greater than 0.1 v is due to 6.q."" Vexp[1.025S(t/!V/2)iJ. (A3) The maximum of this expression is determined by put ting its derivative equal to zero, which gives 1.025SV(t/!- V /2)-b4. (A4) VOLUME 33. NUMBER 7 JULV 1962 Some Electrical Properties of the Porous Graphite Contact on p-Type Silicon * GEORGE G. HARMAN AND THEODORE HIGIER National Bureau oj Standards, Washington, D. C. AND OWEN L. MEYER Diamond Ordnance Fuze Laboratories, Wasltington, D. C. (Received January 22, 1962) Some unusual properties of the porous graphite contact on p-type silicon are described. Current through the sample reaches a saturation level that is inversely proportional to the amount of adhered water vapor. Other gases such as ammonia, H2S, and HCl modify the shape or amplitude of the saturation current. There is a hysteresis loop in the low voltage region which is similar in appearance to that of a ferroelectric. The general method of measurement can be applied to studying the semiconductor surface as well as the contact phenomena. The possible applications include such devices as current regulators, humidity detectors, and surface-barrier radiation detectors. An electronic band model, which includes a trap-dominated inversion layer, is presented to explain the phenomena. This model also integrates various conflicting theories of metal-semiconductor contacts. IN the course of studying the surface properties of semiconductors, using "dirty contacts," it was found (after a study of work function dependence) that gra phite rubbed onto the surface of p-type silicon produced an essentially ohmic contact.! For ease of application, a water-based paste of graphite was applied to etched,2 p-type silicon. The characteristics were essentially ohmic when the graphite dried. However, this contact was not ohmic when it was in high humidity. Instead the current reached a saturation value that was stable to about 100 V, as shown in Fig. 1 (a). Stable charac teristics with values intermediate between the saturated * Part of this work was sponsored by the Air Force Cambridge Research Center, Bedford, Massachusetts. I G. G. Harman and T. Higier, J. Appl. Phys. 33, 2198 (1962), preceding paper. 2 All samples described in this letter were etched in 90% HNOs + 10% HF for one minute. One sample was etched in CP4 and appeared to have a lower breakdown than the others, but this etching effect was not investigated further. humidity and dry curves of Fig. 1 (a) were achieved by controlling the sample humidity. • It should be pointed out that the experimental curves (Fig. 1) were obtained with two identical contacts back-to-back, the reverse electrode (determined by applied-voltage polarity) controlling the current flow. A single contact was studied by alloying aluminum, for a coiwentional ohmic contact, as the counter electrode. When reverse-biased (positive on graphite), the graphite contact had the same characteristics as in Fig. 1, but it showed injection under forward bias. At low voltage there is a hysteresis loop [Fig. 1(b)J which changes shape and magnitude with the bulk resistivity, the humidity, and the rate of change in applied voltage with respect to time (dv/ dt). The barrier capacity was measured in the hysteresis region and showed an increase with reverse bias, in contrast to tht. usual decrease with depletion layer widening. This [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 20:29:27POROUS GRAPHITE CONTACT ON p-TYPE SILICO;\' 2207 implies that the surface was driven into a strong inver sion layer, with accompanying minority-carrier storage, and therefore that the hysteresis is at least partially the result of charging a nonlinear capacitor. It is probable that the first molecular layer of water adsorbed on the surface will have a fixed orientation.3,4 The additional molecular layers of water can exhibit a field-induced polarization which would contribute to the hysteresis effect similar to that seen in a ferroelectric material [Fig. 1 (b)]. Ammonia is known to create a silicon surface state with properties similar to those of water5 (but irrever sible). To observe this effect, etched samples were coated with a collodial graphite dispersion,6 dried, and then exposed to the vapor of NH40H. The effect was immediate, but the I-V characteristics were basically similar, with some exceptions, to those for water alone. However, a sample so treated, subsequently gave a lower saturation current for a given humidity. The NHa appears to enhance the surface attraction for H20, and Jlat curves of the type in Fig. 1 (a) could be obtained in lower humidities than before NHs treatment, Other gases such as HCI and HzS had pronounced effects upon the shape of the I-V curve. The magnitude and nature of these ambient effects were very dependent upon the silicon substrate. Surfaces which had been sandblasted before applying the electrodes resulted in an essentially ohmic contact, and with an etched sample, the lower the bulk resistivity the higher the saturation current for a given humidity. The effect appeared to be continuous over the entire temperature range (+80° down to 170°C) when the measurements were carried out in a nitrogen-water vapor atmosp!lere. The saturation current decreased slowly with decreasing temperature. The sample re sponse time was measured by observing the current waveform in response to applied dc and ac voltages. The saturation current value was the same for both dc and low frequency ac. The barrier shunt capacity appears to be the limiting factor at high frequency and on a typical sample (contact area ",0.5 cm2) this was great enough to obscure the saturation current region at about 100 kc. Both p-type germanium and SiC showed effects analogous to those of Fig. 1 (a). Fifteen Q-cm Ge 3 W. A. Yager and S. O. Morgan, J. Phys. Chern. 35, 2026 (1931). 4 H. Statz and G. A. deMars, Proceedings oj the International Conference on SaUd State Physics in Electronics and Telecommunica tions (Proceedings of the Brussels Conference), edited by M. Desirant and J. Michiels (Academic Press Inc., Ncw York, 1960), Vol. 1, p. 587. 5 H. Statz, G. deMars, L. Davis, Jr., and A. Adams, Jr., Phys. Rev. 106,455 (1957). 6 Dag dispersion #226 was used in all of the experiments re ported here. It is obtainable from Acheson Colloids, Port Huron, Michigan; other commerical graphite dispersions were tested and all gave the same gencral characteristics but resulted in either erratic, noisy behavior, lowered breakdown voltages, or poor adhesion. However, a reasonably satisfactory electrode can he made hy simply combining distilled water 5---10% silicates (waler glass) and fmely powdered graphite into a paste consistency. Good contact is made 10 the "Dag" by pressing a wire against it. (a) (b) FIG. 1.(a) Current vs voltage (I-V) oscilloscope trace for a graphi~e contact on 140 U-cm p-type silicon at 25°C. Horiwntal curve IS for the contact saturated with water vapor. The origin is at the dot on the right. Scale: 2 V /horizontal division and 0.1 mA/verlical division. Vertical curve shows the ohmic nature of t he same contact after it was desiccated. Scale: 2 V /horizontal division and 2 rnA/vertical division. (b) The 60 cps 1-V hysteresis loop for a 40 n-cm p-ty-pe silicon sample with a graphi e electrode, in air saturated with water vapor at 25°C. The origin is at the dot in the center. Scale: 2 V /horizontal division and 1 rnA/vertical division. required a totally H20 saturated ambient to produce the effect, and displayed only a neglibible hysteresis. The addition of NH3 changed the contact to essentially ohmic, On SiC the effect was observed only when the graphite was actually wet; NHs produced no change in the characteristics and the hysteresis effect dominated the entire curve. Our interpretation of the above phenomena is based on a detailed model for a trap-dominated surface barrier contact7 which can be represented as an extension of a graphical treatment by Johnson.8 To complete the picture one need only consider the possibility that the field in the dielectric (between the surface and metal) can be of opposite polarity to that in the semiconductor space-charge region. This results in the unorthodox band scheme shown in Fig. 2(a), with two dipoles back-to-back. Here, the net surface donor charge (shown in the figure by a distributed-in-energy box-scheme for simplicity) is opposite to that in the metal (or dielectric), 7 O. L. :Vlcycr (to be published). 8 E. O. Johnson, RCA Rev_ 18,525 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 20:29:272208 HARMAN, HIGIER, AND MAYER (a) -N E U lOti ...... ,OIZ (J) !:: z :::> w (b) (!) a:: <t 10" I U o I v,- ~---Q~~~~~~ 1 v.' 10 FIG. 2. (a) An equilibrium model of the energy diagram of gra phite on p-type silicon showing the inversion layer caused by adhered water. Ec and E. denote conduction and valence bands, F is the equilibrium Fermi level, X is the electron affinity, <PM and q, denote work functions of metal and semiconductor, V F-and V R+ denote forward and reverse voltages. (b) Charge balance and equilibrium barrier diagram for the contact of 2(a), where q is the electronic charge and Y MO is the barrier height for QM = 0 or q,M=q, [see Fig. 2(a)]. but so large that it dominates the equilibrium situation. One then writes for the charge neutrality equation (1) where the symbols, with the charge sign as superscript, denote the space charge, surface trap charge, and charge on the metal, respectively. One can reverse the sign of the charge on the metal in Johnson's graphical treat ment9 and show its logarithmic curve bending toward a barrier increase in a mirror image (about Y MO) of the former, as seen in Fig. 2(b). The curves for surface charge and charge on the metal now subtract rather than add, as formerly, yielding a resultant curve which plunges asymptotically to zero where they cross at A. This resultant curve, in tum, gives the equilibrium barrier height Yo by its intersection with the space charge curve at B. The saturation current for the phenomena of Fig. 1 can be descirbed by simple diode theorylO assuming an electrode spacing (by an oxide-impurity complex) thin enough for tunnel transport of the barrier-controlled current. The equation takes the form J=Jpo(eQVlkT-l), (2) where the saturation current is primarily a majority carrier or hole current for contacts of this sort on long lifetime p-type material with negligible minority carrier generation. A Richardson-type emission which, in conjunction with Eq. (2), relates the saturation current density to the equilibrium barrier of Fig. 2(b) follows: (3) where VT denotes the carrier thermal velocity, po the extrinsic doping, and Yo the equilibrium barrier height in kT / q units. Thus by using porous, ohmic contacts on semicon ductors it is possible to study the effects of various ambients by passing current perpendicularly through the surface (analogous to Lehovec's capacitive methodY) rather than parallel, as in the field-effect method. In addition it is demonstrated, both theoretically and experimentally, how it is possible for a single metal semiconductor contact to show complete work function domination of characteristics12 (ohmic when dry), and total surface state domination of characteristics13 (highly nonohmic in various ambients). The possible applications of the above characteristics include such devices as current-regulator diodes, surface-barrier radiation detectors, and humidity detectors. 9 See Fig. 17 in reference 8. Note that all charge and potential signs need to be reversed in Johnson's treatment of an n-type semiconductor in order to fit the present p-type situation. 10 E. Spenke, Electronic Semiconductors (McGraw-Hill Book Company, Inc., New York, 1958), Chap. 4. 11 K. Lehovec, J. Minahan, A. Sloboskoy, and J. Sprague, Twenty-First Annual Conference on Physical Electronics, Cam bridge, Massachusetts, (March 1961). Also see L. M. Terman, Technical Report No. 1655-1 (February 1961), Stanford Electronics Laboratories, Stanford, California, ASTIA No. AD-253926. 12 E. C. Wurst, Jr., and E. H. Borneman, J. Appl. Phys. 28, 235 (1957). 13 J. Bardeen, Phys. Rev. 71, 717 (1947). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.64.175.185 On: Tue, 02 Dec 2014 20:29:27

1.1702641.pdf

DC Electroluminescence in Thin Films of ZnS P. Goldberg and J. W. Nickerson Citation: Journal of Applied Physics 34, 1601 (1963); doi: 10.1063/1.1702641 View online: http://dx.doi.org/10.1063/1.1702641 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lanthanide doping in ZnS and SrS thin-film electroluminescent devices J. Appl. Phys. 86, 6810 (1999); 10.1063/1.371756 Hot electron luminescence in ZnS alternatingcurrent thinfilm electroluminescent devices Appl. Phys. Lett. 63, 231 (1993); 10.1063/1.110350 Electroluminescence efficiency profiles of Mn in ZnS ac thinfilm electroluminescence devices Appl. Phys. Lett. 34, 525 (1979); 10.1063/1.90851 Pulse response of dc electroluminescent ZnS : Mn powdered phosphor J. Appl. Phys. 44, 3191 (1973); 10.1063/1.1662730 Electroluminescence and Photoluminescence of Thin Films of ZnS Doped with RareEarth Metals J. Appl. Phys. 43, 2314 (1972); 10.1063/1.1661497 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:24JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 6 JUNE 1963 DC Electroluminescence in Thin Films of ZnS P. GOLDBERG* AND J. W. NICKERSON General Telephone &-Electronics Laboratories, Inc., Bayside 60, New York (Received 12 November 1962) Electroluminescent thin films of the composition ZnS:Cu,Mn,C1, which are capable of dc excitation, are examined for their optical, chemical, and electrical properties. The emission is dominated by electronic and ionic processes occurring at the anode. A model is proposed for the excitation mechanism which is based on hole injection across a heterojunction composed of ZnS and a copper-rich phase. The mechanism ex plains emission at lower-than-band-gap voltages. The spectrum of the yellow emission of Mn in these films is composed of two closely spaced emission bands and is, therefore, different from the known yellow emission from Mn in ZnS phosphors. THIS paper describes the results of studies on the mechanism of electroluminescence excited in zinc sulfide thin-film structures by dc fields. This problem assumes special importance because of the attainment by Thornton! of field excitation at voltages smaller than the band gap energy divided by the electronic charge. Thornton has reported detectible emission at potentials as low as 2-V dc. He explained this on the basis of forward-biased p-n junctions in the film and minority carrier injection culminating in recombination lumines cence. Thornton has investigated films activated either by Cu or by Cu and Mn and coactivated by Cl, but only in the films containing both Cu and Mn was dc electroluminescence at 2 V mentioned. For this reason we have stressed the ZnS: Cu, Mn, Cl system in our work. Other investigators2-6 have reported on electro luminescence in ZnS films, but not on films activated with both Cu and Mn. We show that p-n junctions in the ZnS phase, as proposed by Thornton, is not likely to be the correct explanation of light generation in ZnS: Cu, Mn, Cl films. Our experimental results support a model of the excitation process which involves the injection of holes across a heterojunction composed of a p-type copper compound phase and a ZnS phosphor phase. EXPERIMENTAL TECHNIQUE: CHEMICAL AND STRUCTURAL PROPERTIES The films were formed by vacuum deposition of ZnS with Cu and Mn activators and CI coactivator on sub strates of glass conductively coated with tin oxide. Sub sequent thermal treatment served to crystallize the films and to incorporate the activator impurities, providing a thin-film phosphor. After completion of activation, a suitable metal coun ter electrode was evaporated over the film, usually in circular areas t in. in diameter, providing a "sandwich" geometry. The nature of the planar contact between the metal electrode and the sulfide film is critical in determining the ability of a given film to emit under the application of dc voltages. Observations on this point are described in more detail in the following section. Schematic views of typical film geometry and electrode arrangement are shown in Fig. 1 (a). In most of the work reported here, film thickness falls in the range of 1 to 2 fJ.. Useful information was also obtained from a gap cell geometry. In this configuration the film was prepared by evaporation and crystallization on a nonconducting glass substrate. The metal electrodes were then evapo rated onto the crystalline film, the electrode separation in this case being 25 fJ.. This arrangement is shown in Fig. 1(b). Strong electroluminescence was obtained in films made by widely divergent techniques of deposition and activation. The differences among these films were mainly in optical quality (i. e., transparency and light scattering) and continuity (i. e., presence or absence of gross defects, pin-holes, etc.) The results to be reported below on electroluminescence and electrical character istics are considered, therefore, to be reasonably general for ZnS: Cu, Mn, CI films and not to be specific effects for a particular film synthesis. The manganese content of the films is like that of bright, yellow-emitting electroluminescent phosphor powders, i. e., 0.3%-1% by weight. The total Cu con centration is somewhat higher than in representative phosphors and falls in the range 0.5%-0.8% by weight. As in powder syntheses, no electroluminescence is ob tained if the copper content of the films is too low. The copper concentration probably exceeds the solubility limit in the film, the excess residing mostly near the outer ZnS surface. Both Cu and Mn are essential if the films are to show appreciable dc conduction and strong * On leave of absence at The Hebrew University, Jerusalem, emission under low-voltage dc excitation. Israel, during academic year 1962-63. 1 W. A. Thornton, J. Electrochem. Soc. 108,636 (1961). Some crystal structure studies have been made of the 2 R. E. Halsted and L. R. Koller, Phys. Rev. 93, 349 (1954). d . d d b tl til' d fil B f 3 N. A. Vlasenko and Iu. A. Popkov, Opt. i Spektroskopiya 8, eposlte an su sequen y crys a lze ms. e ore 81 (1960) [Opt. Spectr. (USSR) (English Transl.) 9,39 (1960)]. crystallization, the ZnS in the films possesses a small 4 D. A. Cusano, General Electric Research Laboratory Report particle size, as revealed by x-ray line broadening, and No. 61-RL-2879G, November 1961. h h . ( h 1 . ) Af h 1 6 W. J. Harper, J. Electrochem. Soc. 109, 103 (1962). sows t e CUbIC sp a ente structure. ter t erma 6 F. A. Schwertz and R. E. Freund, Phys. Rev. 98, 1134 (1955). treatment, appreciable crystal growth is in evidence. 1601 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:241602 P. GOLDBERG AND J. W. NICKERSON 4 EDGE VIEW + 6 EMISSION (a) (b) FIG. 1. Schematic views of electrolu minescent thin film (a) on transparent conductively coated glass; (b) on micro scope slide with gap electrodes. Relative dimensions exagger ated. (1) Sn02 elec trode, (2) ZnS film, (3) metal electrode, (4) contact points, (5) nonconducting area, (6) glass substrate. The crystalline film shows a random distribution of crystallite orientations (still cubic). The surface emittance of the films was measured with a photomultiplier having an S-4 surface sensitivity, cor rected for "eye response" by a Wratten No. 106 filter and calibrated by a standard diffuse tungsten light source obtained from the National Bureau of Standards. RESULTS: ELECTRICAL, OPTICAL, AND LUMINESCENT PROPERTIES The typical dependence of surface brightness L on dc voltage V is shown in Fig. 2 for a film with Sn02 and aluminum electrodes in the sandwich geometry. The strongly nonlinear brightness-voltage characteristic of 5r---~--------------------------~ FILM 505-1 DC VOLTS FIG. 2. Brightness vs voltage for representative ZnS: Cu, Mn, CI film; Sn02 and Al electrodes. The curve is the smoothed version of an X-V recorder tracing. these films cannot be rendered linear by the various L-V formulas previously applied to electroluminescent ZnS. Near linearity is achieved with log L vs V-I as in Fig. 3. The curvature may be influenced by Joule heating. "The current-voltage relationship, shown in Fig. 4, closely resembles a rectifier characteristic. It is most significant that light is emitted only when the voltage on the aluminum is positive, i. e., the reverse direction for the rectifier. This polarity is opposite to that re ported by Cusano.4,7 The differences between our films and his,8 however subtle, are apparently significant enough to produce quite different electroluminescent characteristics. Light can readily be detected in most films of this kind at voltages where the current begins ::i I l lL 5 7 FIG. 3. Log L vs V-I plot for the line in Fig. 2. :; .5 .1 .15 .10 (VOLTS1-1 .05 to rise sharply (cross-hatched range in Fig. 4.) Thorn ton's observations of light emission at 2-V dc have been verified with these films using a refrigerated photo multiplier tube as described by Wiggins and Earley.9 The luminous efficiency is constant for all brightness levels, as can be seen from the plot of brightness vs power in Fig. S. Most efficiencies fall in the range 10-3 to lO-2Im/W. Efficiencies under ac excitation also fall in this range. We have found that all electrode metals tested thus 7 D. A. Cusano, Doctoral dissertation, Rensselaer Poly tech. In stitute (January 1959). 8 The primary differences between Cusano's films and ours are: (1) His use of a vapor phase reaction to synthesize the films, (2) The use of a Ti02 film as a transparent electrode, and (3) His films belong to the phosphor system ZnS:Mn, CI and had only a chemi cally precipitated surface phase of copper sulJide deposited on them, where our phosphor films are compositionally ZnS:Cu, Mn, CI as the copper was diffused in at high temperatures. 9 C. Wiggms and K. Earley, Rev. Sci. Instr. 33,~1057 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:24DC ELECTROLUMINESCENCE IN THIN FILMS OF ZnS 1603 o E c t " u 200 100 I Voltage on 5n02 external reflected illumination. Figure 6(c) shows the emission from the cell, excited by 12S-V dc, to be local ized at the anode. Emission can be generated in gap cells using Al electrodes at potentials of about 20 V; i. e., at average field strengths nearly a factor of five lower than in the sandwich geometry. This result sug- 6 a 10 12 14 (a) RANGE i +--OF -- STRONG EMISSION FILM 434 EMISSION AREA' I. 25cm2 -100 FIG. 4. Current-voltage diagram of ZnS:Cu, Mn, CI film with SnO. and Al electrodes. The characteristic resembles that of a rectifier. Light is emitted only in the "reverse" direction. far can successfully produce electroluminescence in (b) these films provided that the gap cell geometry is employed. The metals include eu, Ag, Mg, AI, and In. In every case the light is localized at the anode. This observation cannot be emphasized too strongly since it forms the basis for our rejection of the existence of emitting p-n junctions in the ZnS phase in favor of a different model of excitation to be discussed subse quently. Figures 6(a) and (b) are micrographs of a gap cell made with aluminum electrodes photographed under 6r-----------------------------------~--__, o FILII 356 CELL AREA. 1.25, .. 2 EFFICIENCY.5xI0-3LPW .2 ... .6 .8 to 1.2 1.4 1.6 POWER (WATTSl FIG. 5. Brightness L vs power showing approximately constant efficiency over complete brightness range. 1.8 (c) FIG. 6. (a) Photograph (reflected light) of unoperated gap cell; 25-p. gap, AI electrodes; (b) same cell after operation showing presence of anodic deposits; (e) same cell under 125-V dc excitation showing emission at anode; some reflected light used to make photograph. Top electrode is anode in (a), (b), and (c). gests that the excitation and emission processes are dominated by the magnitude of the applied voltage and not by the magnitude of the field. In the sandwich electrode construction, not all of the above metals permit electroluminescence even though [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:241604 P. GOLDBERG AND J. W. NICKERSON c ~ 200 I-100 z I&J II: II: ::> u 650 mA 01 12 V EMISSION AREA-1.25cml -100 -200 FIG. 7. Current-voltage diagrams of ZnS:Cu, Mn, Cl films with Cu electrodes with and without SiD between sulfide film and Cu. the film may be electroluminescent with the gap elec trode configuration. Cu, Ag, and In have thus far failed in sandwich geometries. We believe that this results from the formation of excessive short circuits with these metals when used in the sandwich construction. It is likely that all of the metals form some shorting contacts through the film during the metallizing step. In the case of Al and Mg, however, these shorts burn out, and furthermore, with a positive potential on these metals an anodically produced insulating oxide film forms that serves a blocking or current-limiting function. With Cu and Ag, no insulating anodic film can form, and any conducting inhomogeneity initially present or subse quently produced carries all the current. The blocking contact is not needed in gap cells because of the wide interelectrode spacing (2S p,). The effect of an artificial insulating layer at the metal ZnS interface can be seen in the following experiment. A copper metal electrode was evaporated directly onto a typical sulfide film. The film electroded in this way was not electroluminescent and was nearly Ohmic under both polarities. Another representative film was first coated with an insulating film of silicon monoxide and then with copper. This film showed strong electrolumi nescence and nonlinear I-V characteristics similar to those of the aluminized film of Fig. 4. In Fig. 7 the I-V curves for the films, with and without SiO, are shown. The intensity of light depends strongly on the thickness of SiO. Harper5 has extensively discussed the effects of insulating films of this kind in ZnS: Cu, Cl excited by alternating voltages of triangular waveform. Indium does not function in sandwich constructions, probably because of its low-melting point (1SS°C). Molten In soaks through all pin-holes in the film, re sulting in an excessive number of shorts. Indium may melt during In evaporation or during passage of current. After the application of dc voltages, several kinds of electrolytically produced changes in gap cells can readily be seen. For example, with Al gap electrodes, deposits are generated near the anode edge soon after voltage is first applied. The deposits form at voltages below that required for electroluminescence to appear and, with time, develop in thickness to the extent that New ton rings can be detected. The deposits can be seen along the anode in Fig. 6(b). Some additional change also occurs at the ZnSj Al interface apart from these visibly detectible deposits. This change is manifested by a change in the reflectivity of the aluminum over an area which corresponds precisely with the narrow area of the anode that emits light. With Cu electrodes a thin line of increased reflectivity appears at the cathode, although electroluminescence appears at the anode. In all gap cells black deposits bridging the gap form with time. No light appears along these lines, but occasionally a greater emission intensity is observed at the intersection of these lines with the anode edge. Previously we stated that no electroluminescence is observed in the films if the Cu content of the film is too low. This is true in all cases except for gap cells in which Cu metal electrodes are used. Evidently an electrolytic process occurs at the anode which converts the electrode Cu to a chemical form that is useful in generating elec troluminescence, probably a sulfide or oxide of copper. If after the initial operation of a gap cell with Cu elec trodes the polarity is reversed, light appears at the new anode and at other locations in the gap as well. An unusual but reproducible occurrence is the existence of thin lines of emission outside of the area enclosed by the gap, which can grow in size or diminish depending upon the polarity of the applied voltage. These atypical effects have been found only when Cu electrodes are used. When Al is used for electrodes in gap cells, light appears at the anode as described above. Upon reversal of polarity, the light may first appear at the "new" cathode (i. e., the former anode) for a few seconds and then, in a period too short to be resolved by eye, will rapidly extinguish and reappear at the "new" anode. Additional microscopic observations were made with sandwich cells to establish the spatial distribution of light, i. e., to determine if the light is emitted in discrete points and lines as in powder granules10 or in continuous planar array. A magnification of X440 was used which 10 J. F. Waymouth and F. Bitter, Phys. Rev. 95, 941 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:24DC ELECTROLUMINESCENCE IN THIN FILMS OF ZnS 1605 permitted a resolution of about 0.5 p.. It was found that the pattern of light was distinctly more uniform than one usually observes in powder particles. There was a mild nonuniformity in intensity oJ emission which cor responds qualitatively to the pattern one sees under reflected external illumination. There were in addition a few bright points of emission that could not be cor related with any physical heterogeneities in the film. More precise statements than these about the uni formity of light are rendered difficult by limitations imposed by crystal size and substrate roughness. A significant electrical phenomenon has been ob served repeatedly in all electroluminescent films pro duced for this study. Upon the initial application of voltage to a freshly prepared film in the light-producing polarity, high Ohmic currents and no emission are ob served. This is true even if the voltage is 12 V or more. Continued application of the voltage for approximately 30 sec results in a rapid decrease in current, accompanied by the appearance of light in the shape of a crescent at one edge of the circularly electroded areas. The region first showing light is usually that for which the series resistance along the Sn02 electrode is lowest. The cur rent continues to fall and the illuminated area grows until the entire electroded surface is uniformly excited and the current has fallen to a constant value deter mined by the de voltage. This process is called "form ing," a term which is consistent with similar effects in many kinds of area rectifiersll and which also was ob served by Cusan07 in ZnS:Mn, Cl films. After forming, the L-V and J-V curves are as shown in Figs. 2-5. It must be emphasized that this process cannot be due solely to burning out of evaporated metal shorts be cause forming is required even' for the gap cells de scribed above and also because films with sandwich electrodes sometimes require repetitions of the forming processes after a period in the nonexcited state. Also, the forming process may be reversed in a previously operated film by a brief passage of current in the oppo site direction. For example, if after the passage of cur rent in the "no electroluminescence" direction the normal polarity for electroluminescence is reapplied, the forming process proceeds again before electrolumines cence develops to its former level. The bright yellow emission color is similar in visual appearance to the electroluminescence of ZnS: Cu, Mn, CI powder phosphors. Most of the films exhibit qualita tively the same emission color under ion-bombardment excitation as under field excitation. The photolumines cence, however, is quite different. Visually, the in tensity under 3650-A uv excitation is always very weak and more orange in color than either the field-or ion excited emissions. Films with a frosty appearance tend to show more intense photoluminescence than highly transparent ones. The spectra of the characteristic yellow emission from 11 H. K. Henisch, Rectifying Semi-Conductor Contacts (Oxford University Press, New York, 1957), p. 97. 7r-----------~====~~----~ 6 2 O~----------~------~----~--------~~ 5461 (H; line). 5700 5900 6050 WAVELENGTH (A) FIG. 8. Densitometer recordings of spectral energy distribution from a film excited by 11.0 V dc, and from a powder layer excited by 600 V, 400 cps. The film spectrum possesses a double band with peaks at 5700 and 6050 A. three of these films were recorded photographically using a Gaertner quartz spectrograph, model 165. A repre sentative spectrum in the form of a densitometer trace of the photographic negative (Kodak panatomic-X) is shown in Fig. 8 for a film operated by 11.0-V de. For comparison, the spectrum of a yellow-emitting ZnS: Cu, Mn, Cl powder phosphor is included. The powder was excited by 400 cps, 600 V, impressed across a cell 6 mils thick with castor oil as embedment dielectric; the phos phor: oil weight ratio was 2: 1. It is clear that a pro nounced difference exists between the two spectra in that the film spectrum exhibits a double emission band. The band maxima reside near 5700-5750 A and 6050 A. The phosphor emission band falls close to the value of 5850 A found by Bube12 for ZnS: Mn and ZnS: Mn, CI and by Shionoya13 for ZnS: Cu, Mn, Cl, but has a larger half-width than reported by these workers. Thermoelectric power measurements using a hot probe technique did not give a sufficiently strong signal to provide an estimate of the ZnS film conductivity type. AGING CHARACTERISTICS OF DC-EXCITED FILMS The life of the films under dc operation is short, standing presently at tens of hours to half initial bright ness. The contributing factors to poor life are similar to those of conventional powder cells; e. g., humidity and localized heating. The importance of the heating in these films can be judged upon calculation of the density of the power dissipation. A film under 10-V de passes the 12 R. H. Bube, Phys. Rev. 90, 70 (1953). 13 S. Shionoya, Bull. Chern. Soc. Japan 29, 935 (1956). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:241606 P. GOLDBERG AND J. W. NICKERSON (AI ZnS Mn 3.7eV == ---~---T;'ERMI 'LEVEL (8) --------Resonance IRad,at,ve T~ Transition 0u-:~ FIG. 9. Energy band scheme of heterojunction between CuxS and ZnS semiconductor phases, including Cu and Mn !,-ctivator l~vel~: (a) no field applied; and (b) exciting voltage apphed; CuxS SIde IS positive. order of 0.1 A/cm2 of current. For films 1/J. thick (our present case) the power density is therefore about 10 kW/cm3• A film may be effectively cooled and moisture effects minimized by training a strong jet of dry nitrogen on the film during operation. This improves life slightly, but the time to half initial brightness is still of the order of tens of hours for Al-electroded films. The reasons for this may in part be related physically to the forming process and are discussed further below. Some of the brightness lost during an aging experiment can be re generated after passage of current in the opposite di rection for a brief period. DISCUSSION We now offer a mechanism for the generation of light in electroluminescent films of ZnS: Cu, Mn, Cl that is consistent with our observations and with the observa tions of others. The critical facts upon which the mecha nism is based are: (1) the appearance of electrolumines cence in gap cells when metals of diverse character are used (2) the appearance of light exclusively at the anode, (3) the requirement in the films of a min~um amount of Cu, and (4) the occurrence of electrolummes cence at 2-V dc. The mechanism, qualitatively stated, relies upon the formation at the anode of a copper-rich phase. The phase is probably a sulfide or oxide of copper. In the absence of further information we refer to it as CuzS. The CuzS phase forms a heterojunction with the ZnS crystallites of the film. When the dc voltage is applied, holes pass from the CuzS phase (probably p type) into the ZnS phase (probably well-compensate~, but of unknown conductivity type), the holes ultI mately residing at Cu activator sited in the ZnS. These holes combine with conduction electrons, and the re combination energy is transferred to the Mn impurity centers yielding the yellow emission characteristic of the center. Energy transfer is like the resonance transfer proposed by Shionoya to explain sensitized lumin~s cence in ZnS: Cu, Mn, CI phosphors.13 Electroneutrahty is preserved by the entrance of electrons through the nearlv Ohmic Sn02-ZnS interface. The CUzS phase is formed by an anodic process that utilizes the excess Cu in the film or, if there is no excess, the Cu from the Cu anode. The above model is capable of explaining the low-voltage effects reported by Thorn ton and the low efficiencies reported in this work, as shown subsequently. l A schematic presentation of the energy band diagram at the important contact region between the CuzS phase and the zinc sulfide phase is given in Figs. 9(a) and (b). These figures are intended to serve as an aid in discuss ing the proposed electronic transitions and should not be quantitatively interpreted. Even in dealing with a more tractable system (one which employs a hetero- junction between single crystal phases), considerable difficulties arise in constructing a quantitative theory of heterojunctions.14 Figure 9(a) shows the electronic energy in the CuzS ZnS heterojunction before application of the exciting field. Also shown are the activator levels of Cu and for Mn as proposed by Shionoya. When the CuzS is biased positively at a voltage sufficiently high, injection of .holes into the ZnS phase occurs [Fig. 9(b)]. Electrons sImul taneously enter the film from the Ohmic Sn02-ZnS con tact (not shown) to preserve charge neutrality and recombine with trapped holes. The recombination en ergy is transferred to the nearby Mn center which emits its characteristic radiation. At a voltage of the order of V h (neglecting voltage drops outside the heterojunction), holes originating in the CuzS phase are sufficiently energetic to enter the ZnS and electroluminescence occurs. If the Fermi level lies roughly near the middle of the forbidden gap (a reasonable location for well-compensated ZnS phos phors), then electroluminescence shou!d in principle ~e possible at voltages the order of Vh, 1. e., 2 V. In .th~s way we explain Thornton's low-voltage results. A SIml lar scheme has already been applied by Fischer and Masonl5 to the problem of emission from powder phos phors. Their model leans heavily on the physics of field 14 R. L. Anderson, IBM J. Res. Develop. 4, 283 (1960). 1. A. G. Fischer and A. S. Mason, Airforce Cambridge Res. Labs. Rept. Contr. No. AF19(604)8018, 15 February 1962 (RCA LlI:b oratories, Princeton, New Jersey); Electrochem. Soc. ElectrOnICs Div. Abstracts 11, S5 (1962). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:24DC ELECTROLUMINESCENCE IN THIN FILMS OF ZnS 1607 emission processes from pointed conducting lines in the ZnS insulator. Low luminous efficiency is to be expected from this model, since energy is also dissipated in processes not related to light emission. Thus energy is spent in bring ing conduction electrons from the cathode to the hetero junction and to the anode. The observations reported here cannot be reconciled with a p-n mechanism as offered by Thornton. If junc tions were present in the bulk of the ZnS film, then light should exist not only at the anode but throughout the width of the gap. Figure 6(c) shows that this is obvi ously not the case. We now present a possible mechanism of the "form ing" process which explains most of the observed facts in sandwich and gap cells. This mechanism involves the anodic oxidation of the Al electrode following the application of voltage in the electroluminescence-gener ating direction. The formation of the AbOa film between the ZnS and Al constitutes a partially blocking layer which prevents excessive shorting of the ZnS film. The fall in current coincident with the increase in electro luminescence is a direct consequence of the developing insulating layer. Some current passes through the block ing layer, probably by tunneling or by a path associated with pin-holes. The decline in current during forming may also partly result from a segregation of Cu into an anodic deposit from an initial filamentary conducting array through the film bulk. The components of an emitting film, with, e. g., an Al anode and Sn02 cathode, can be summarized as follows; glass/Sn02/ZnS; Cu, Mn, C1/ Cu",S/ AI20a/ AI. The direction of current rectification of a complex structure such as this would be rather difficult to predict without simplifying assumptions. If it is assumed that the main determinants of over-all current flow are Cu",S (p type)/ Al203/ AI, then based on the reasoning applied to artificial blocking layers between semiconductor metal contacts by Dilworth,16 it can be shown that the direction of rectification should actually be as found in Fig. 4. If our reasoning is correct, then the current voltage characteristics of the films have only indirect connection with charge carriers participating in lumines cent processes. In contrast, Thornton heavily stresses the current-voltage functionality in support of his p-n junction mode. The mechanism whereby Mn serves to increase the dc electroluminescence of ZnS; Cu, Cl is a puzzling one. The high concentrations at which Mn is found in ZnS phosphor lattices attests to its presence at substitutional sites. This receives independent confirmation from the structure studies of Skinner17 on the ZnS-MnS system. It is widely held that Mn is present in the Mn+2 valence state, and little effect on Mn on conductivity should be apparent. We present the following speculative answer 16 C. C. Dilworth, Proc. Phys. Soc. (London) 60,315 (1948). 17 B. J. Skinner, Am. Mineral. 46, 1399 (1961). for the increase in dc conductivity (and, thereby, in dc electroluminescence) of ZnS containing Cu and Mn. Following Shionoya,13 we propose that energy contained in the Cu centers in the form of holes in transferred from the Cu centers to the Mn center by a resonance process. In addition, we propose that this transfer is reversible and that the energy can be propagated through the lattice from Mn to Cu as well. The number of lattice sites for one transfer step is of the order of 500 (i. e., about 0.25)1. in ZnS), and thus only four transfer steps are required to propagate a hole produced at the CuxS-ZnS interface through the 1-)1. film to the cathode. Some of the transmitted energy may be involved in radiative processes. At this time we can offer no proof that this mechanism is valid. No explanation can be made of the twin banded emission (Fig. 8) that would not be unduly arbitrary. We do not wish to extend the discussion of excitation structures and mechanisms proposed here to the case of ZnS powder phosphors or films and crystals with different activators. It should be stated, however, that electro luminescence mechanisms hitherto proposed cannot be used to explain the observations reported above. The earlier mechanisms and the reasons for their rejection may be summarized as follows; Impact ionization by electrons. (See Zalm18 for a dis cussion of this mechanism.) Light is emitted at the anode. Impact ionization by electrons accelerated in the high-field barrier region should produce light at the cathode. Also, the observation of emission at lower than-band-gap voltages is damaging evidence against impact ionization. See also the discussion by Henisch19 on this point. p-n junction in the ZnS phase.i,20.2i The films produce light at the anode. Light should appear throughout the gap for this mechanism to hold. Carrier accumulation.22 Emission is observed at the anode, which runs counter to the theory of this mech anism when applied to n-type phosphors. The only means by which carrier accumulation could explain emission at the anode is to have p-type ZnS, which is possible but unlikely. Minority carrier injection at planar metal-semicon ductor contact.19 The films require a Cu-rich phase. Gen erally it is found that metal-film contacts do not produce electroluminescence unless sufficient copper is present. Bi-polar field emission of carriers from filament con ductors embedded in a semiconductor.a This theory is close in some respects to the model presented here. Fischer and Mason have used the heterojunction model 18 P. Zalm, Philips Res. Rept. 11, 353,417 (1956). 19 H. K. Henisch, Electroluminescence (Pergamon Press Inc., New York, 1962), p. 252. 20 D. W. G. Ballentyne, J. Electrochem. Soc. 107, 807 (1960). 21 J. L. Gillson and F. J. Darnell, Phys. Rev. 125, 149 (1962). 22 H. K. Hensich and B. R. Marathe, Proc. Phys. Soc. (London) 76, 782 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:241608 P. GOLDBERG AND J. W. NICKERSON to explain the commonly observed spatial distribution of light in powder particles that occurs in the form of points and lines. In our application of heterojunctions to the problem of dc emission from films we have no need to propose the existence of filamentary conductors. Finally, we suggest that poor operating life of films with Al electrodes is related to the anodic film formed on the metal electrode. While the anodic film serves a positive function in limiting the current through the whole structure, its growth with time ultimately de creases the flow of all carriers, some of which are in volved in luminescence processes. Also the growth of JOURNAL OF APPLIED PHYSICS the AbOa film correspondingly decreases the excitation voltage available at the Cu"S-ZnS heterojunction. ACKNOWLEDGMENTS It is a pleasure to acknowledge helpful discussions with Dr. L. J. Bodi during the course of the research. F. Palilla and F. Avella made numerous valuable sug gestions for improvement of the manuscript. We ex press sincere appreciation to Dr. A. K. Levine who provided encouragement and advice from the outset and to C. O. Creter for skillful aid in the preparation of the films. VOLUME 34, NUMBER 6 JUNE 1963 Microwave Frequency Multiplication by Hot Electrons KARLHEmz SEEGER II. Physikalisches Institut der Universitiit Heidelberg, Germany (Received 20 December 1962) An experimental study is made of microwave frequency multiplication observed in germanium at power levels of several kW. Using a fundamental frequency of 9.4 Gc/sec, a third harmonic yield of about 1 % was found. The data agree with calculations based on the observed deviations from Ohm's law. The anisotropy and the high frequency limit of the multiplication are calculated. INTRODUCTION RECENTLY, the generation of microwaves with frequencies of more than 100 Gc/sec has been successful in various ways.1-6 The largest known peak power of SO W has been obtained at 140 Gc/sec by harmonic generation using a ferrite. In this paper carrier heating in semiconductors is used as a method of generating harmonics. The drift velocity of the carriers rises slower than linearly with increasing field intensity if lattice scattering is domi nant, as has been found for germanium at room tem perature.7 In the case of a sinusoidal field intensity this nonlinearity produces an alternating current, the Fourier components of which are odd harmonics of the fundamental frequency. Even harmonics do not occur since a field reversal causes a current reversal. There fore, the harmonic with maximum intensity is the third. This is different from ferrites where the second har monic is the one with maximum intensity and where the 1 R. W. Roberts, W. P. Ayres, and P. H. Vartanian in Quantum Electronics, edited by C. H. Townes (Columbia University Press, New York, 1960), p. 314. 2 B. B. von Iperen, Philips Tech. Rundschau 21,217 (1959/60). 3 G. Wade, paper presented at the AlEE Winter General Meeting, January 1962. 4 G. Convert, paper presented at the AlEE Winter General Meeting, January 1962. 5 C. S. Gaskell, Proc. IRE 50, 326 (1962). 6 A. S. Risley and I. Kaufman, J. Appl. Phys. 33, 1269 and 1395 (1962). 7 K. Seeger, Phys. Rev. 114,476 (1959). third harmonic has not yet been detected with cer tainty.6 In this way the hot-electron method of generat ing high microwave frequencies may have an advantage over the ferrite method. EXPERIMENTAL ARRANGEMENT Figure 1 shows the block diagram of the experimental arrangement. A pulsed magnetron (JP9-7D) was operated at a frequency of 9.378 Gc/sec. It generated a standing wave in a short-circuited waveguide. By means of two E-H tuners and a variable short, a standing wave maximum was placed at the position of the sample. The microwave power could be varied continuously using an attenuator. The incident and reflected power FIG. 1. Block diagram of the experimental arrangement (dotted: 8-mm waveguide). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Fri, 28 Nov 2014 06:16:24

1.1736063.pdf

Current Flow across Grain Boundaries in nType Germanium. II R. K. Mueller Citation: Journal of Applied Physics 32, 640 (1961); doi: 10.1063/1.1736063 View online: http://dx.doi.org/10.1063/1.1736063 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Tunnel current across linear homocatenated germanium chains J. Appl. Phys. 115, 043701 (2014); 10.1063/1.4863118 Increase in current density for metal contacts to n-germanium by inserting TiO 2 interfacial layer to reduce Schottky barrier height Appl. Phys. Lett. 98, 092113 (2011); 10.1063/1.3562305 Impact of field-enhanced band-traps-band tunneling on the dark current generation in germanium p - i - n photodetector Appl. Phys. Lett. 94, 223515 (2009); 10.1063/1.3151913 Photoinduced current transient spectroscopy of deep defects in n-type ultrapure germanium J. Appl. Phys. 86, 940 (1999); 10.1063/1.370828 Current Flow across Grain Boundaries in nType Germanium. I J. Appl. Phys. 32, 635 (1961); 10.1063/1.1736062 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38JOURNAL OF APPLIED PHYSICS VOLUME 32, NUMBER 4 APRIL, 1961 Current Flow across Grain Boundaries in n-Type Germanium. II R. K. MUELLER J{ecltanical Diz'isivn of General Mills, Inc., Minneapolis, Minnesota (Received Septemher 6, 1960) The current flow across grain boundaries in n-type germanium has been measured in the temperature range of 350-200oK. Precisely oriented bicrystals have been grown for this study with 4, 6, and 25° tilt boundaries and 6° twist boundaries. The experimental data were found to be in good agreement with theo retical calculations given in the accompanying article [R. K. Mueller, J. App!. Phys. 32, 635 (1961)]. The current across the boundary is mainly carried by electrons crossing the barrier, and it increases with decreasing boundary angle. The activation energy was found to be 0.71±0.01 ev for all boundaries. A lower limit of N B ~ 1013 (cm-2 ev-1) was found for the density of boundary states of 6 and 25° boundaries. No significant difference between tilt and twist boundaries has been observed. I. INTRODUCTION IN a previous paper,l referred to as Paper I in the following, a theoretical discussion of the current flow across homogeneous-plane grain boundaries has been given. It is the purpose of the present paper to report experimental results of a study of the current flow across grain boundaries in carefully oriented ger manium bicrystals with different boundary structures and a wide range of donor content. Good agreement between theory and experiment has been found for boundaries with misfit angles of 60 and larger. Devia tions from the behavior expected for homogeneous boundaries have been observed on 40 boundaries. These FIG. 1. Two bicrystals with 4° tilt boundary grown from both ends of a double seed. deviations are interpreted as evidence for the dislocation array structure of these boundaries. II. EXPERIMENTAL DETAILS A. Preparation of Samples The bicrystals for this study were grown in a hori zontal zone melting furnace. The growth speed was ~ in./hr for all bicrystals. The following boundary con figurations have been prepared: (a) Axis of relative rotation parallel to a [100J direction common to both grains; boundary plane incident with the common [l00J axis and located symmetrically between the (011) planes of the two grains. Tilt angles 4, 6, and 25°. (b) Tilt axis as in (a); boundary plane symmetrically between the (001) planes of the two grains. Tilt angle 6°. (c) Axis of relative rotation parallel to a [100J direction common to both grains; boundary plane perpendicular to axis of rotation [i.e., common (100) plane]. Twist angle 6°. The growth direction for the bicrystals with tilt boundaries [configurations (a) and (b)J was the common [100J direction; for bicrystals with the twist boundary it was the mean [OlOJ direction. X-ray analysis of the bicrystals showed that the misalignment of the "common" [100J axis of the two grains was in all cases less than /0 deg. Two 40 bicrystals grown from the two ends of a double seed are shown in Fig. 1. Laue back-reflection pat terns from bicrystals with a 6° tilt boundary [configuration (a)J and a 6° twist boundary [configuration (c)] are shown in Fig. 2. The photographs were obtained from slices cut perpendicu lar to the growth direction with the cut planes normal to the x-ray beam. The x-ray beam straddled the boundary, giving one-half spot for each grain of the hicrystal. The diameter bisecting the Laue spot indi rates the direction oi the boundary on the l'Ut plane. The Laue patterns :ohow that the axis of relative ro- 1 R. K. Mueller, J. AJljll. Phys. 32, 635 (1961), preceding paper. tation is parallel to the beam for the tilt boundary and 640 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38CUR R E N T F L () \ \. T" 11 T Y l' E C E R 1\1.\ :\ It· 1\ I. I I 641 J " • • • • • ,. ,. • • • • ,. t • ,. t t ,. • ,. • • • (a) (b) FIG. 2. Laue back-reflection patterns from bicrystals 'Yith the x-ray beam straddling the boundary. (a) (,0 t,yist boundary; (b) 6° tilt boundary. perpendicular to the beam for the twist boundary, but in bot h cases perpendicular to the intersection of the boundary with the cut plane. From the bicrystab, rods with rectangular cross section were cut such that the grain boundary was per pendicular to the long axis of the rod, and one pair of the faces parallel to t he long axis of t he rod 'Nas a common or mean (100) plane. All samples were etched in CP-l: after cutting to remove the damaged surface layer and finally etched electrolytically in sodium hydroxide. j\licrographs of the etch grooves of a -10 boundary on the two different faces of a typical sample are shown in Fig. 3. The deep etch groove occurs on the common (100) plane. B. Instrumentation The samples were mounted in a glass metal system for measurement. The sample chamber wa,; evacuated by a mercury diffusion pump, and helium as a heal transfer medium could be introduced into the system by controlled diffusion through a heated quartz tube. Preliminary experiments showed that the sample characteristic;; were Hot afrccted Ly the helium atmo~pherc. The voltage currcnt c1taracteri~tic" were mea~ured with a breaker amplifier followed by an x-y recorder. Urea t care was used to keep t he contact polent ials in the measuring circuit at a low Jevel, so that good data could he obt ained wi t h applied biases of the order of mv acros,; the sample. This permitted accurate meas urement of the zero bias conductance across the samples. C. Surface Effects The impedance of the grain boundary especially at lower temperatures is very high. ~urface leakage can therefore alter t he voltage-current characteristic ap preciably.2 Great care was taken to prevent surface effects from interfering with the accuracy of the meas urements. Preliminary experiments showed that on adequately etched samples the voltage-current char acteristic was reproducibJe under subsequent etchings. Intentional mist realment of samples, e.g., a spark discharge in the system, altered the characteristic drastically but the original characteristic could be re stored by subsequent etching. It was further observed (a) (b) F1G. 3. Micrographs of etch groove on 40 sample. (a) Etch groove on common (100) plane. :\lagnilication Ll25X103. (b) Etch groove on mean (011) plane. Magnification 0.75 X 103• 2 R. H. Kingston, J. Appl. Phys. 27, 101 (1956). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38642 R. K. MUELLER TABLE 1. Zero hias conductance Go for 13 bicrystal samples. Donor concentration Configuration and Sample Nd cm-3 boundary angle 4-1-A 2.0.1015 4° tilt 4-4 1.4.1015 Configuration (a) 4-ID-2 1.1.105 6-3-C 1.9.1015 6° tilt 6-5-A 7.0.1015 Configuration (a) 6-7 3.6.1016 100-6-2 1.4.1014 6° tilt 100-6-4 1.0.1014 Configuration (b) 100-6-5 8.5.1013 6-T-1 1.2.1016 6° twist Configuration (c) 25-X 4.8.1015 25° tilt 25-Y 1.3.1015 Configuration (a) 25-ID-1 6.0.1015 that small induced changes of the surface affected the voltage-current characteristic mainly in the quasi saturation region, i.e., for applied voltages large com pared to kT / q, but had little or no effect on the low bias conductance across the sample. This, together with the excellent agreement of the observed zero bias con ductance with theoretical prediction, makes it very unlikely that the observed zero bias conductance is significantly influenced by surface effects. Surface effects, however, may influence the differential con ductance in the quasi-saturation region. This conduc tance, discussed in Sec. III B, was two to three orders of magnitude smaller than the zero bias conductance and therefore much more susceptible to changes from uncontrolled surface effects. Occasionally, relatively large variations of Gsat were observed with no measura ble change in Go. However, values of G.at which were higher than the observed minimum were always ac companied by typical changes in the appearance of the voltage-current characteristic. With the minimum values of Gsat the voltage-current characteristic was symmetrical under voltage reversal, showed a linear voltage current relation over an extended voltage range, and a sharp transition from the low bias to the quasi-saturation region (see Fig. 6). All data discussed below were taken from those "minimum" character istics. The temperature dependence of Gsat showed an activation energy similar to that of Go, i.e., of the order of the gap energy. This rules out an ohmic surface leakage, e.g" an n-type surface channel across the boundary area. It leaves open the possibility of a p-type channel with a current-flow mechanism as described by lVIcWorther and Kingston. s However, the shape of the voltage-current characteristic, the correlation of Gsat with the boundary angles, and the good agreement of the temperature dependence of Gsat/Go with that ex- 3 A. L. McWorther and R. H. Kingston, Proc. I.R.E. 42 1376 (1954). ' Goe<l>olk7' Go mho/cm2 Capture rate Kexpu=----·1O-8 at T=273°K I' T· (1-1'/2) 1.5 ·10-2 7.6 1.2.10-2 0.2 6.1 1.3 .10-2 6.7 5.5.10- 3 2.9 4.6.10- 3 0.3 2.5 3.8 '10-3 2.1 3.8.10-3 2.1 4.4.10- 3 0.3 2.3 4.1.10- 3 2.2 1.9.10- 3 0.6 1.3 (assumed) 1.4.10- 3 0.9 2.2.10- 3 0.6 1.4 3.4.10- 3 2.1 pected for the grain boundary response lead us to believe that the observed behavior is a true grain boundary response. III. RESULTS AND DISCUSSION A. Zero Bias Conductance Dependence on Donor Concentration and Boundary Angle The zero bias conductance Go in mho/cm2 at 273°K for 13 bicrystal samples with crystallographically dif ferent boundaries and widely varying donor concentra tions is given in Table 1. The donor concentration has been determined from capacitance measurements4 and resistance measurements. Both methods agreed within . experimental accuracy. Since the capacitance measure ments give the donor concentration in the immediate neighborhood of the boundary, and the resistance measurements the donor concentration out in the bulk material, the close agreement of these measurements shows that no significant accumulation of the intro duced impurities occurred around the boundaries. The capacitance measurements showed further that the donor concentration on both sides of the boundary was equal within experimental accuracy. It may be seen from Table I that no correlation between the donor concentration and the zero bias conductance exists in the samples investigated. It has been shown in Paper I that only the hole contribution to Go depends on the donor density. The independence of Go from the donor concentration demonstrates there fore that the hole contribution to the current flow across the boundary is small against the electron con tribution, as predicted in Paper I by theoretical considerations. A correlation exists, however, between the zero bias conductance and the boundary angle. Go increases with 4 R. K. Mueller, J. App!. Phys. 30, 546 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38CURRENT FLOW IN n-TYPE GERMANIUM. II 643 decreasing boundary angle. The close agreement of Go for different samples with the same boundary con figuration indicates that the variation of Go with the boundary angle reflects differences in the barrier struc ture of the different boundaries. Assuming a minimum dislocation array for the 4° boundary,O one finds a dis location spacing of 52 A. This is already of the order of magnitude of the width of the barrier region discussed in Paper I, and one has to expect an effect on the current flow from the finite dislocation spacing. This interpre tation of the increase in the current flow for the lower angle boundary as a manifestation of the dislocation spacing in the boundary is corroborated by experimental results on 1 and 2° boundaries currently under investigation. It is of interest to note that the twist boundary (sample 6-T -1) does not show a higher conductance than other 6 and 25° tilt boundaries. The minimum dislocation array for this boundary is a network of two sets of screw-type dislocations with a spacing of 28 A intersecting at right angles.o According to an analysis by Homstra,6 screw dislocations in a diamond lattice should have no dangling bonds, and it was surprising to find no significant difference in the zero bias con ductance between twist and tilt boundaries. Temperature Dependence of Zero Bias Conductance The temperature dependence of Go in the temperature range where the effect of space-charge generation can be neglected has been determined in Paper I as Go= (1-'Y/2)KTe-4>olkT, (1) 10-a'L-_.l.---'-_--'--_--'-_.l...-----JL..>lIOWJ 3.2 3.4 3.6 3.8 40 42 4.4 4.6 10' T- FIG. 4. Temperature dependence of zero bias conductance for samples with various boundary angles. 6 F. C. Frank, Pittsburgh Report, p. 150 (1950). Office of Naval Research (NAVEXOS-P-834). 6 J. Homstra, J. Phys. Chern. Solids 5, 129 (1958). wl~ J1 E E u 10-15 1000 __ T FIG. 5. Temperature dependence of zero bias conductance for lowest-doped sample. where 'Y is the capture rate for electrons at the bound ary,7 K a constant determined in Paper I as 2.2.108 mhorK cm2, q the electronic charge, k the Boltzmann constant, T the absolute temperature, and cf>o the ap parent activation energy: cf>o=Ecm-EF+cT, where Ecm is the edge of the conduction band at the barrier maximum. EF is the Fermi energy and c is the tempera ture coefficient for the energy gap. Figure 4 shows a plot of InGolT vs the reciprocal absolute temperature T for 4, 6, and 25° samples. The close agreement between the activation energies of the different samples in the temperature range 320-2400K is evident in Fig. 4. The activation energy cf>o was found to be 0.71±0.01 ev for all samples studied. The devia tion from this activation energy for lower temperatures (evident in the lowest temperature points of Fig. 4) is because of the increasing relative importance of the space-charge generation process which, as shown in Paper I, becomes for sufficiently low temperatures the determining process for the current flow. The tempera ture at which the effect of space-charge generation becomes of importance increases with decreasing donor concentration. The effect is demonstrated in Fig. 5, which gives the lnGolT vs liT relation for the lowest doped sample 100-6-5, with .LV d= 8.5 .1013 compared to N d> 1010 for the samples in Fig. 4. The "plateau" value TO of the lifetime determined from the zero bias con ductance at lower temperatures for various samples was found to lie in the range from 0.2-2 J.Lsec in agreement with lifetimes determined from optical measurements at 800K on similar grain boundary samples.8 These values are lower than those observed on p-n junctions9 7 R. K. Mueller, J. Phys. Chern. Solids 8, 157 (1959). 8 W. W. Lindernann,:,and R. K. Mueller, J. App!. Phys. 31, 1746 (1960). 9 E. M. Pell, J. App!. Phys. 26, 658 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38644 R. K. MUELLER Sample 6-3-C T= 2730 K .75 .5 15 25 .5 75 Volls- FIG. 6. Voltage-current characteristic for sample 6-3-C. Dashed line represents 1=10 tanh(qV /2kT). in material with bulk lifetime at room temperature similar to our samples (T of the order of 200 (.Lsec). Since TO derived from the present data are character istic of the density of recombination centers in the im mediate neighborhood of the boundary, one can specu late that the relatively low values of TO indicate an in creased density of trapping centers in close proximity to the boundary. In column 6 of Table I the experimental values Kexptl for the constant K are given as determined according to Eq. (1) from the observed zero bias conductance Go, the observed activation energy <1>0= 0.71 ev for all samples, and the observed values7 of ')' except for the 6° twist boundary where a value of ,},=0.6 has been assumed. The agreement with the theoretical value of 2.2.108 for a homogeneous boundary is excellent. The consistently high values of Kexptl for the 4° boundaries are, as discussed above, an indication of the wider dis- location spacing in these boundaries. . B. Voltage-Current Characteristic A typical voltage-current characteristic (sample 6-3-C) is shown in Fig. 6. The theoretical voltage current dependence derived in Paper I, by neglecting the effects of the finite density of states in the boundary band and of the field asymmetry at the barrier top, is: 1 =10 tanh(q Vo/2kT). (2) This dependence is shown in Fig. 6 as a dotted line. The experimental curve transcribed from an x-y re cording coincides with the theoretical curve up to applied voltages of about 4 kT/q but fails to saturate for higher applied voltages. Over a considerable voltage range the current increase is linear with the applied voltage. In this region the differential conductance is Gsat= (1-10)/V, and we can describe the deviation from saturation by a dimensionless parameter p=G.at/G o. The ratio has been measured for several samples as a function of temperature. The experimental results are shown in Fig. 7. The theoretical model discussed in Paper I gives for p the relation (3) where F is the field at the barrier maximum at equi librium, N B the density of states in the boundary band per cm2/ev, K the dielectric constant, and f a factor of the order of unity. The first term in Eq. (3) represents the effect of the change under applied bias of the filling level of the boundary states for which complete de generacy has been assumed; the second term describes the effect of the variation of the image force depression. The electrical field at the barrier top can be deter mined from calculations given by Kingston and NeustadterlO with the observed activation energy in terpreted according to the barrier model given in Paper 1. It follows that F for our parameter values is practically independent of the donor concentration and lies between 3.lOc6.104 v/cm in the temperature range considered. With F, the expression (q/KF)! from Eq. (3) can be determined, and is shown together with p/Nd in Fig. 7. Since Eq. (3) and the electrical field calculated from Kingston and Neustadter expressions are, as discussed in Paper I, only approximately true, no significance can be placed on the coincidence of piN d for sample 6-5-A and the (q/ KF)! relation. What is significant in the results shown in Fig. 7 is the agreement in the order of magnitude between piN d and (q/ KF)! for all 6 and 25° samples and the similarity between observed and cal culated temperature dependence. This agreement indi cates that the observed nonsaturation is the inherent grain boundary response and, if this interpretation is accepted, that the effect of the finite number of bound ary states on the saturation behavior is masked by the effect of the variation of the image force depression. We can derive from Eq. (3), however, a lower limit for the density of states in the boundary band: (4) which is valid whether the image force effect assumed here or any other surface or bulk effect contributes to the observed values of p/Nd• The inequality Eq. (4) gives for the 25 and 6° samples a value of (5) This value is of the same order of magnitude as values derived for the densities of surface states on free surfaces.ll It is interesting to note that here again the 6° twist boundary did not show a behavior significantly different from that of other 6 and 25° boundaries. In spite of their already higher Go values, the 4° samples show consistently larger values of p/ N d than the 6 and 25° samples. This again indicates a wider dis location spacing which gives rise to an additional current contribution in the quasi-saturation region. 10 R. H. Kingston and S. F. Neustadter, J. Appl. Phys. 26 210 0~~. ' 11 J. Bardeen, Semiconductors and Phosphors (Interscience Publishers, Inc., New York, 1958), p. 81. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38CURRENT FLOW IN n-TYPE GERMANIUM. II 645 This will be discussed in a forthcoming paper together with related effects on 1 and 2° boundaries. CONCLUSION It has been shown that the current flow across grain boundaries with boundary angles of 6 and 25° is in excellent quantitative agreement with a theoretical treatment of the current flow across alhomogeneous boundary given in the accompanying paper.! In the temperature range where carrier generation in the space-charge region can be neglected, the current is essentially carried by electrons crossing the barrier. The activation energy characteristic for the height of the potential barrier was found to be 0.71 ev. This high activation energy. interpreted according to the barrier model given in Paper I, implies that the Fermi energy at the boundary lies close to or within the valence band. A lower limit of NB~ 1013(cm-2 ev) for the density of the states in the boundary band has been determined. No significant difference in the behavior of 6° twist and 6° tilt boundaries has been found. This is of interest since the dislocation array model for a twist boundary is an array of intersecting screw-type dislocations, and screw-type dislocations are assumed to have no dangling bonds associated with them.6 The lowest boundary angles considered in this paper were 4° tilt boundaries which showed significant varia tions in their voltage-current characteristics from the 6 and 25° samples. These variations are believed to be 10·" + 4-I·A • 6-T·1 0 6-3-C x 25-X . 6-5·A 101t * 25·Y + + + 01- ." Z Q: 1617 ~ ('YKE) 2 .. ~ & & • * x 0 x 10" 161 • L--.".L-..."..l--~~,..L.."..._'-,--=,~~ 240 250 260 270 280 290 300 310 T °K---. FIG. 7. Ratio p of differential conductance in quasi-saturation region to zero bias conductance over Nd as function of temperature compared with theoretical expression p/Nd~ (q/KF)i. Valid for densities of boundary states large compared to 1013 (cm-2 ev-l). connected with the wider spacing of dislocations III these boundaries. ACKNOWLEDGMENT The author would like to thank Mr. F. Jaeger for his help with the experimental work. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 07:21:38

1.1732988.pdf

ValenceBond Studies of the Dependence upon Substituents of C13–H and Si29–H Coupling Cynthia Juan and H. S. Gutowsky Citation: The Journal of Chemical Physics 37, 2198 (1962); doi: 10.1063/1.1732988 View online: http://dx.doi.org/10.1063/1.1732988 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/37/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in ValenceBond Studies of Contact Nuclear Spin–Spin Coupling. II. LongRange PiElectron Coupling between Protons J. Chem. Phys. 48, 4463 (1968); 10.1063/1.1668014 ValenceBond Studies of Contact Nuclear Spin–Spin Coupling. I. A Truncated Matrix Sum Method J. Chem. Phys. 48, 4458 (1968); 10.1063/1.1668013 Extension of the ValenceBond Description of Nuclear Spin—Spin Coupling J. Chem. Phys. 46, 811 (1967); 10.1063/1.1840745 Consistent Deviations from Nonadditivity of Substituent Effects on 13C–H Coupling Constants J. Chem. Phys. 40, 2413 (1964); 10.1063/1.1725530 Analysis of ValenceBond Wave Functions for LiH J. Chem. Phys. 36, 1814 (1962); 10.1063/1.1701272 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06THE JOURNAL OF CHEMICAL PHYSICS VOLUME 37, KUMBER 10 KOVEMBER 15,1962 Valence-Bond Studies of the Dependence upon Substituents of C13-H and Si29_H Coupling* CYNTHIA JUAN AND H. S. GUTOWSKY Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois (Received June 1, 1962) An interpretation is presented for the additivity of substituent effects on the CILH coupling constant, which has been observed previously in the high-resolution NMR spectra of substituted methanes. Each atom or group X is assigned a characteristic "affinity" for s character in the carbon hybrid orbital of the C-X bond. Distribution of s character among the carbon orbitals in accord with the relative s affinities of the four substituents leads to the observed additivity relation provided that the total s character is conserved. The valence-bond approach used with this model gives a linear relation between the s character of the carbon hybrid orbital involved in the C-H bond (aIt) and the observed OLH coupling constant (JcH=500 aIt). Also, it allows the determination of the s character of the other carbon orbitals. The dependence of the s character of the C-X bond on the elec tronegativity of X is discussed in terms of electron spin and charge correlation. It is noted that the hybridization changes should affect not only JCH but also JHH, which is consistent with I. INTRODUCTION IN earlier work,I-3 the CILH spin-spin coupling constant (JCR) found in high-resolution nuclear magnetic resonance spectra has been related to the parameters used in describing the C-H bond, namely the carbon orbital hybridization and the coefficient A of the ionic term. The mathematical form of the valence bond equation for JCR is such that increasing A de creases JCR. However, the observed trend2 appears to be the opposite; that is, compounds in which A is expected to be large because of electronegative sub stituents have large values of JCR rather than small. Muller2 called attention to this fact and concluded that the dependence of J CR on the s character of the carbon orbital used in forming the C-H bond predominates over the dependence of JCR on the C-H-bond polarity. This is shown to be so in the present paper by a calcula tion of the explicit dependence of JCR on A and s char acter. We find, by means of the valence bond approach used by Karplus and Grant,4 that JCR is relatively insensitive to the value of A, within certain limits, and directly proportional to the s character. This finding is basic to an analysis of the empirical linear additivity of group contributions to JCR in sub- * Acknowledgment is made to the donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. The work was also supported by the Office of Naval Research. 1 M. Karplus and D. M. Grant, Proc. Nat!. Acad. Sci. U. S. 45, 1269 (1959). See also H. S. Gutowsky, D. W. McCall, and C. P. Slichter, J. Chern. Phys. 21, 279 (1953), for an earlier dis cussion of the coupling of directly bonded nuclei and its depend ence upon the perfect pairing structure. 2 N. Muller and D. Pritchard, J. Chern. Phys. 31, 768, 1471 (1959); N. Muller, ibid. 36, 359 (1962). a J. Shoolery, J. Chern. Phys. 31, 1427 (1959). • Karplus and Grant,l have given general expressions for JCH. We proceed from there to illustrate the explicit dependence of Jrm on). and s character. the observed proportionality between JCH (CH,X) and the cis and trans H-H coupling constants in CH2=C13HX. The treatment developed for methanes has been extended to JCH in substituted ethylenes and to the Si2!LH coupling in silanes. For the former, introduction of the s-electron affinities, obtained from the values of JCH(CH 3X) observed for the substituted methanes, leads to the result that JCH(CH,,==CIaHX) = JCH(CH 2=CH 2) +~[JCH(CHaX) -JCH(CH 4)]. Values predicted in this manner are systematically larger than those observed, which implies that there is a small, negative ",-electron contribution of 5 to 10 cps to JCH(CH,,--CI'HX). Such a contribution is com patible with current theories for proton and CIa hyperfine splittings in ESR spectra of free radicals. The Si2!LH coupling constants observed in substituted silanes exhibit large, systematic devia tions from the simple additivity found in the methanes. These deviations are explicable qualitatively in terms of changes in the Si-H-bond polarity. stituted methanes, which is our main concern. Malinow ski5 reported that the CILH coupling constant observed in compounds of the form CHXYZ was expressible to a very good approximation as: (1) where S-x, the contribution of group X to JCR, is defined from experiment by the relations tH=tJCR(CH 4) and (2) This is only one of several equivalent forms6 in which the additivity may be expressed, all stemming from the basic, empirical relationship, JcH(CHXYZ) = JCR(CH3X) + Jcn(CH3Y) (3) Spin-spin coupling constants have been shown to depend mainly on the Fermi contact term for the CILH group.l Furthermore, for coupling between di rectly bonded atoms, deviations from perfect pairing are not importanLI Using these approximations and a simple model, we have been able to derive Eqs. (1) and (3) describing the observed linear additivity of group contributions to CILH coupling constants in the substituted methanes. Also, as mentioned in a pre liminary account of this work,7 the analysis has been extended to JCR in substituted ethylenes and it enables values of JcR(CHr=CI3HX) to be predicted from those 5 E. Malinowski, J. Am. Chern. Soc. 83, 4479 (1961). • We thank Dr. T. H. Brown for discussion leading to this con clusion. One other equivalent form is JCH(CHXYZ) =i"H'+ i"x'+i"y'+i"z', where i"x'=JcH(CH,X) -lJcH(CH.). 7 H. S. Gutowsky and C. S. Juan, J. Am. Chern. Soc. 84, 307 (1962) . 2198 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06C13-H AND SPLH COUPLING 2199 observed for JcH(CHaX). However, the predicted values are systematically somewhat (5 to 10 cps) larger than those observed (150 to 200 cps). If, as seems likely, these discrepancies result from negative 7r-electron contributions to JCH, they may provide useful information on u-rr interactions and hyperfine constants in free radicals. Also, we consider the Si2LH coupling constant J SiR in the substituted silanes, to which the additivity rule does not apply.7 At first we thought this might be due to the use of d electrons in the silicon bonding orbitals. However, since then, a more detailed analysis indicates that the deviations from additivity result largely from changes in polarity of the Si-H bond. Finally, as discussed elsewhere,s if our model is correct for the effect of X upon J CH in CHaX or CH~CHX molecules, it should lead to a better understanding of substituted effects upon JRH, because the latter also is affected by hybridization of orbitals in the C-H bonds. In fact, the hybridization changes which appear to dominate JCH are consistent, at least semiquantitatively, with the observed de pendences of Jci.HH and JtransHH upon the substituent inCHz=CHX. II. VALENCE-BOND FORMULATION FOR SUBSTITUTED METHANES General Treatment The coupling constant J NN' for a pair of nuclei N and N' may be calculated by second-order perturbation theory using the Hamiltonian given by Ramsey.9 The expression for Jmv' consists of several terms; however, in this paper we are concerned only with the Fermi contact term which has been shownl to be the dominant term for the CILH coupling: - 2 (167r{3fi)2 JCH= (JCH) contact = 3ht1E -3- X 'YC'YH ('lto [ La(rkC) lJ(rjH) Sk' Sj [ %). (4) Ie,j The symbols used above have their standard meanings. The perfect pairing approximation may be used for the ground-state wavefunction 'lto since for electron-spin interactions between bonded atoms deviations from perfect pairing are not important,I,1O We use the separated electron-pair wavefunction, 'lto= (8!)-!2) -1)P P[l/tax (1 , 2)l/tbu(3, 4)l/tcz(5, 6) p Xl/tdh(7, 8)], (5) where l/tr(i,j) =ur(i,j) {[a(i){3(j) -(3(i)a(j) ]/V1} (6) 8 H. S. Gutowsky and C. S. Juan, Discussions Faraday Soc. 34, (in press) (1962). 9 N. F. Ramsey, Phys. Rev. 91,303 (1953). 10 For CH,,"I]Io= 1.08 'h+0.OO1 h-0.028 "'3, where "" is the per fect pairing structure, and "'2 and "'3 are nonperfect pairing terms [M. Karplus and D. H. Anderson, J. Chern. Phys. 30, 6 (1959)]. Although "'2 and "'3 account for JHH', they are not important in JCH. and ur(i,j) is taken to be of the valence-bond form with inclusion of ionic terms Ul (1, 2) = 7Jl[ CPa (1) CPx( 2) + CPa (2) CPx( 1) + AaCPa (1) CPa ( 2) +Axf/)x(1)CPx(2)]. (7) In the latter CPa, CPb, CPo, and CPd are carbon atomic or bitals; CPx, cp", CP., and CPh are atomic orbitals on the atoms bonded to the carbon, and 7J is the normalization con stant. The coefficients of the ionic terms are Aa and Ax. Substituting 'lto into Eq. (4) and using the Dirac identity Sk'Sj=t(2P k/-l), in which Pk/ is an oper ator interchanging the spins of electrons k and j, one obtains l' C'YH(167r{3fi)2 JCH= ME -3- 7J2(cf>d [ a(r,e) [CPd) X (CPh [ a(rjH) [ CPh). (8) We assume carbon hybrid orbitals formed from one 2s orbital and three 2p orbitals; e.g., for the C-H bond, CPd=aHs+(1-aH2)!Pa. (9) aH2 is commonly called the S character of the carbon hybrid orbital. The p,/s are linear combinations of px, p", and p. orbitals and are oriented along different directions in space. In general, ax,Y, or Z will depend on the group or atom X, Y, or Z bonded to the carbon. Substituting CPd and CPh= lsH into Eq. (8), one finds that JCH= ('Yc'YH/ME) (\67r/3fi)27J2aH2 [2sc(0) [2 [lsH(O) [2, where 7J-2= {2+(2+ACAH) [aH2S1.2/+ (1-aH2) S1B2p2 +2aH(1-aH2)!S1828S1s2p]+4(AC+AH) ( 10) X [aHS182s+ (1-aH2)!S182p]+Ac2+AH21, (11) 2sc(0) is the 2s wavefunction of carbon evaluated at the carbon nucleus, and lsH(O) is the corresponding quantity for the hydrogen 1s function. S182s and Sls2p are the overlap integrals between the hydrogen 1s atomic orbital and the respective carbon atomic orbitals. In Eq. (11) for 7J-2, AH is much less than AC and can be neglected, because the electronegativity of C is greater than that of H. Also, Eqs. (7) and (11) are symmetrical in AC and AH, so the coefficient of the ionic contribution to the wavefunction is hereafter denoted by AC-H. Equation (10) leads to (12) where A is a collection of constants, and Jo is 500 cps, as determined from the observed valuel,2 of 125 cps for JcH(CH4). This value for Jo is consistent with the valence-bond theory inasmuch as Karplus and Grantl obtained a reasonable value of 0.374 for AC-H, using the same approach, with J CH = 124 cps, in combination This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:062200 C. JUAN AND H. S. GUTOWSKY TABLE 1. The calculated dependence> upon s character of the normalization constant >f(Xc.-H, aa), defined in Eq. (11) for the valence bond function describing the C-H bond. aIt 7]2 aIt >f 0.2400 0.2419 0.3600 0.2395 0.2600 0.2412 0.3800 0.2394 0.2800 0.2407 0.4000 0.2393 0.3000 0.2403 0.4200 0.2394 0.3200 0.2399 0.4400 0.2395 0.3400 0.2397 0.4500 0.2396 • Calculated using Xc_H=O.374 and overlap integrals from reference 11. with an estimate of tl.E and calculations of the overlap integrals from Hartree-Fock functions. We have ob tained virtually identical results using Kotani's tablesIl for the integrals. Dependence of JCH upon tl.E, AC-H and aH2 Although Eq. (12) was derived for JCIl in substituted methanes, it applies in general to directly bonded nuclei for which the (]' electron, Fermi contact term is dominant. Moreover, depending upon the sensitivity of tl.E and rl to substituents, it affords an attractive semiempirical way to obtain the s character of bonding orbitals from coupling constants. For the substituted methanes, or other classes of closely related compounds, one would expect the tl.E pertinent to Eq. (12) to be very nearly constant. This follows from the fact that tl.E is approximately twice the bond energy,l which varies by only a few percent for C-H bonds. Such an easy semiquantitative generalization can not be made for 1'/2 because of its complex dependence on A, a2, and the overlap integrals, which differ with the atomic species in the bond and also with the substi tuents. However, these dependences can be calculated relatively simply and directly. For the C-H bond, 1'/2 (AC_H, aH) was found to be relatively insensitive to the value of aH, as shown in Table I. 1'/2 changes very 0.6 .-------------~~ 0.4 AC'H AM'H A •. H 0.2 O.O~---,~-~:------,..L-_-L._--.J 0.2 0.4 0.6 0.8 1.0 Fr<:. 1. The M-H "bond polarity parameter," AM-H in MR., and Its dependence upon the difference in electronegativities I XM-XH I of M and H. The empirical AM-H values are from reference 1 and the electronegativities from Pauling. !1 M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, Table of Molecular Integrals (Maruzen and Company, Tokyo, 1955). slowly with aH2 and goes through a minimum at aH2= 0.400. The total change in 1'/2 over the entire range of values of aR2 of interest here is only 0.2%. 1'/2 is also relatively insensitive to AC-R as discussed in the follow ing paragraph. Electron-withdrawing substituents are expected to increase electronegativity of the C atom by about 0.1 to 0.2 units according to estimates of effective electro negativities by proton chemical-shift measurements.l2 From the empirical values of AB-H, AC_H, and AN-n given by Karplus and Grant,1 and plotted in Fig. 1/3 it is apparent that an increase in electronegativity of the carbon by 0.2 units would change Ac-n, from 0.374 to about 0.44. The dependence of 1/2 on AC_H, contained in Eq. (11), was demonstrated by Karplus and Grantl and is shown in Fig. 2. From Fig. 2 we find that the 1.1 -11.0 0.9 0.30 0.35 0.40 0.45 FIG. 2. Variation of the normalization constant >f, in units of >f (CH.), with the C-H bond polarity parameter Xc.-H. These values are from reference 1 for aIt=0.25. estimated increase in Ac-R with the most electronega tive substituents decreases 1/2 to about 0.951/2 (CH4) • However, such an increase in Ac-H is accompanied by an increase in Zeff for the 2s and 2p electrons of carbon which leads to a decrease in the overlap inte grals S1s2s, S182P' and to an increase in 1'/2. Therefore, the effects tend to cancel, and even though aH, AC-H, and the overlap integrals all change with the substi tuents, 1/2 is expected to remain about the same for the substituted methanes. This leads to J~500 cps and to the conclusion that JCR is linearly proportional to aR2, the s character of the carbon orbital in the C-H bond. Derivation of the Additivity Relation Two additional assumptions are required to derive the additivity relation observed by Malinowski.6 The first is that one carbon 2s orbital is used in forming the 12 B. Dailey and J. Shoolery, J. Am. Chern. Soc. 77, 3977 (1955). 18 The electronegativity of B is less than that of H. Hence, the coefficient of the <PH (l)<pH (2) term ("-H) is much larger than the coefficient of the <PB(1)4>B(2) term (AB) so that AB is neglected and AH is plotted as AB-H. In the case of C-H and N-H, the electronegativities of C and N are greater than that of H so that AH is neglected and Xc and "-N are plotted as Ac-H and AN-H. The Si-H case is similar to that of B-H. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06C1LH AND Si29_H COUPLING 2201 C-H, C-X, C-Y, and C-Z bonds. In this case, ax, IXY, az, and aH are related by (13) The second concerns the manner in which the 2s orbital is destributed among the four bonds. We present a simple model for this, the physical basis of which is explained later. A substituent may prefer bond formation with a carbon hybrid orbital having more or less s character than the normal Sp3 value of t. If so, substituents may be arranged in their order of "affinity for s character," analogous to the way in which elements are arranged in the electromotive series or electronegativity scale. Let .1x be a measure of the "affinity" of substituent X for carbon 2s character; further, let .1x be measured with respect to H such that .1x is positive if the "s affinity" of X is less than H and negative if greater than H. Consider the four bonds to be four equivalent interconnected potential wells of possibly different depths as shown diagrammatically in Fig. 3. The differ ence between the depths of the wells for X and H is defined as .1x. The 2s character distributes itself among the wells to give a common 2s level, because of their interconnection. Moreover, this common 2s level, and the content of each well, can be obtained very readily via Eq. (13), i.e., by the assumption that the sum of the 2s content of the four wells is unity. In CH4 or CX4 the four wells are all of the same depth so that 2s character is distributed equally among them. In CHaX, since the H wells are deeper than that of X by the amount .1x, then an H well in CHaX has t.1x 2s character more than an H well in CH4. In general, the H well in CHXYZ has [i.1x+t.1y+t.1z] 2s character more than an H well in CH4• Expressed mathematically, this means that for CH3X (14) H H TAllLE II. Substituent parameters Ilx obtained" from values of oa_H coupling constants observed in some CHaX compounds. JCH(CHaX)b CHaX (cps) CXJt Ilx AI2(CHa)6 113 0.226 -0.096 Si(CHa), 118 0.236 -0.056 (CHa)aSiCN 122 0.244 -0.024 CHaC(CH a). 124 0.248 -0.008 CH, 125 0.250 0.000 CHaCOCH, 126 0.252 +0.008 CHaCHa 126 0.252 +0.008 CHaCHCH,Br 126· 0.252 +0.008 CHa<J> 126 0.252 +0.008 CHaCHO 127 0.254 +0.016 CHaCH2Br 128· 0.256 +0.024 CHaCH2Cl 128· 0.256 +0.024 CHaCOOH 130 0.260 +0.040 CHaCHCb 131 • 0.262 +0.048 CHaN (CHah 131 0.262 +0.048 CHaCH21 132· 0.264 +0.056 CBaNHCH, 132 0.264 +0.056 CHaC=CH 132 0.264 +0.056 CHaNH2 133 0.266 +0.064 CHaCCIa 134 0.268 +0.072 CHaCN 136 0.272 +0.088 (CHahNCHO 138 0.276 +0.104 CH,SH 138 0.276 +0.104 (CHahS 138 0.276 +0.104 CBaSOCH, 138 0.276 +0.104 CHaOH 141 0.282 +0.128 CH,O<J> 143 0.286 +0.144 (CH,OhCO 147 0.294 +0.176 CHaF 149 0.298 +0.192 CHaCI 150 0.300 +0.200 CHal 151 0.302 +0.208 CHaBr 152 0.304 +0.216 • ax was calculated using Eq. (14) with values of aH' obtained from JCH(CH,x) by means of Eq. (12). In the latter, Jo was given by JCH(CHJ ~ iJ.=125 cps. b Taken from reference 2 unless otherwise specified. • Taken from reference 5. and for CHXYZ, aH2( CHXYZ) = aH2 (CH4) +t.1x+t.1y+t.1 z. (15) Eliminating .1x, .1y, and .1z from Eq. (15), with Eq. (14) and similar equations for Y and Z, we obtain FIG. 3. A diagram of the model used in deriving the additivity relation for the effects of substituents upon JCH(CHXYZ). The aH2(CHXYZ) =aH2(CHaX) +aH2(CH3Y) vertical lines enclose the four interconnected potential wells for the 2s electrons; these are the four bonding orbitals of the carbon atom. +aH2(CHaZ) -2aH2(CH4)' (16) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:062202 C. JUAN AND H. S. GUTOWSKY TABLE III. A comparison of aH2, in CHXYZ, obtained directly from the observed values of JOH(CHXYZ) by the relation ~= JOH/ Jo and from the additivity of the substituent effects, a~(CHXYZ) =i(l+/lx+/ly+/lz). JOB" aH2(CHXYZ) Differ- CHXYZ (cps) JOH/JOb i(1+/lx+/ly+/lz) ence .pCH2'/> 127 0.254 0.254 0.0 CH2(COOHh 132" 0.264 0.270 +0.006 CH2(CN)2 1450 0.290 0.294 +0.004 CH3C13H21 149 0.298 0.304 +0.006 CH3C13H2Br 151 0.302 0.306 +0.004 (CH3)2CI3HBr 151 0.302 0.308 +0.006 .pCH2CI 152 0.304 0.302 -0.002 .pCH2Br 153 0.306 0.306 0.0 CICH2CH2CI 154 0.308 0.306 -0.002 BrCH 2CH2Br 157 0.314 0.310 -0.004 CH2CICN 161 0.322 0.322 0.0 CH2I2 173 0.346 0.354 +0.008 CH2Br2 178 0.356 0.358 +0.002 CH2CI2 178 0.356 0.350 -0.006 ChCHCHCb 182 0.364 0.362 -0.002 CH2F2 1850 0.370 0.346 -0.024 CHCl 2CN 189 0.378 0.372 -0.006 CHBra 2060 0.412 0.412 0.0 CHCla 209" 0.418 0.400 -0.018 • These observed values were obtained from reference 5, or from references therein, unless otherwise specified. b Calculated using J.=4JOH(CH,) =500 cps. e From reference 2. Equation (16) may be rearranged to give aH2( CHXYZ) = [aH2( CHaX) -jaH2( C~) ] +[aH2(CHaY) -jaH2(C~) ] + [aH2 ( CHaZ) -jaH2( CH,)]. And by means of Eq. (12) this leads to lCH(CHXYZ) =5"x+5"Y+5"z, (1) which is exactly the form of additivity observed by Malinowski." Equation (16) may also be combined with Eq. (12) to give Eq. (3) and to lead to the other forms6.8 of the additivity relation. Discussion The valence bond approach gives a direct relation between the s character of the carbon orbital involved in the C-H bond in substituted me thanes and the C13_ H coupling constant. The s character of the carbon orbital in the C-X bond may also be obtained with the use of Eq. (13) for CHaX, CH2X2, and CHX 3; and in general, according to the model presented here, ax2(CHXYZ) =t(1+Ax+ Ay+Az) -Ax, (17) as shown graphically in Fig. 3. The values of Ax for a number of substituents are given in Table II. These values exhibit an interesting relation between the s electron affinity Ax and the electronegativity of sub stituent X, with the Ax being small for electropositive substituents ( -0.096 for AI) and large for electronega tive (+0.2 for the halogens). In addition, Table III shows that the effects of substituents upon aH2 are addi tive to within an accuracy of 2% for about 20 polysub stituted methanes. The additivity can be expressed in terms of fCH by taking the product of aH2 and fo, with a value of 500 cps for the latter. The model presented here may seem somewhat arbitrary at first glance. But at least one of its main aspects, the relation between Ax and the electronega tivity of X, can be explained qualitatively as a simple consequence of electron correlation effects. Apart from charge correlation, i.e., the tendency for all electrons to keep apart from each other because of Coulombic repulsion, there is spin correlation, which arises as a result of the Pauli principle. Electrons having the same spin have a low probability of being near one another while those having opposed spins have no tendency to keep apart in this way. Consider the C4-ion. Because of charge and spin correlation of the 8 L-shell electrons, a configuration of four pairs arranged tetrahedrally has the highest probability.!' Methane can be pictured as formed from C4-by attaching four protons and it is at equilibrium in a regular tetrahedral configuration. The electron distribution is most conveniently described in terms of Sp3 hybrid orbitals on the carbon. Now suppose we attach two protons and two X+ TABLE IV. Dependence of /lx on electronegativity and number of lone pair electrons of first atom in the substituent X. First atom No. of JCBs range JCB av /lx av Pauling of X compo (cps) (cps) electroneg. AI 113 113 -0.096 1.5 Si 2 118-122 120 -0.040 1.8 H 125 125 0.0 2.1 C 23 124-136 129 +0.032 2.5 N 11 131-139 137 +0.096 3.0 S 4 138-140 138.5 +0.108 2.5 0 8 141-147 144 +0.152 3.5 Halogen 4 149-152 150.5 +0.204 2.2--4.0 • From reference 2; E. Snyder and}. D. Roberts, }. Am. Chern. Soc. 84, 1582 (1962); P. C. Lauterbur, J. Chern. Phys. 26, 217 (1957). 14 P. G. Dickens and J. W. Linnett, Quart. Revs. (London) 11, 291 (1957). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06CILH AND Si'9-H COUPLING 2203 to C4-, where X is a substituent with electronegativity greater than that of hydrogen It is to be expected that the two bonding pairs to X will be "centered" farther from the carbon nucleus than the two bonding pairs to H. Consequently, the four pairs will no longer tend to be oriented in the form of a regular tetrahedron but both spin and charge correlation would cause the angle sub tended by the bonding pairs between C and X to be less than that subtended by the bonding pairs between C and H. The electron distribution can then be described by nonequivalent hybrids with greater s character in the C-H bonds and less s character in the C-X bonds than the equivalent hybrids in methane. Or, in terms of the parameter introduced here, ~x would be positive as observed. On the other hand, for a substituent X whose electro negativity is less than H, the two bonding pairs to H will be "centered" farther from the carbon nucleus than the two bonding pairs to X. Consequently, the electron distribution can be described by hybrids with greater s character in the C-X bonds than in the C-H bonds, or by a negative ~x. The greater the difference between the electronegativity of X and H is, the more enhanced is this effect, i.e., ~x is a larger number. Therefore, ~x is a characteristic of an atom or group X and de pends on the algebraic difference between the electro negativity of X and H.IS This is shown clearly by the summary of ~x values in Table IV. Also, the latter reveals the importance of another factor. When X possesses lone-pair electrons there are interactions between the lone pairs and the electrons of the other substituents and a possibility of multiple bonds, so that ~x depends upon the electron pairs of the substituent as well as upon its electronegativity. This is apparent in Table IV, where it may be seen that for substituents with the same electronegativity ~x tends to be greater for those which have the greater number of lone-pair electrons. A less satisfactory feature of our results is their TABLE V. "Interorbital" angles calculated from aa" and ar for halomethanes, and their comparison with the observed bond angles. x H-C-X angle in CH3X X-C-X angle in CHaX2 Calc. Obs." Diff. Calc. Obs.b Diff. I 102.2° 106.9° 4.7° 100.5° 114.7° 14.2° Br lO1.S 107.3 5.5 99.7 112±2 ",12 CI 102.6 10S.0 5.4 99.7 l11.S 12.1 F 103.0 10S.5 4.5 9S.5 10S.3 9.S • C. Costain, J. Chern. Phys. 29, 864 (1958). b Interatomic Distances, edited by L. E. Sutton (The Chemical Society, Lon don, 1958); R. A. Levy and L. O. Brockway, J. Am. Chern. Soc. 59, 1662 (1937); R. J. Myers and W. D. Gwinn, J. Chern. Phys. 20, 1420 (1952). 16 We are indebted to a referee for pointing out that a similar idea has been advanced by H. A. Bent, Chern. Revs. 61, 275 (1961). relation to observed bond angles. The "interorbital" angles (angles between the symmetry axes of the carbon hybrid orbitals) corresponding to the hybridiza tion parameters obtained from JCH data are compared with observed bond angles (angles between the lines joining the bonded nuclei) in Table V. We note that the H-C-X angles observed for CH3X are less than tetrahedral in spite of the fact that X is bigger than H. This is clearly due to the effect described above. None thelesss, all of the observed H-C-X and X-C-X angles are significantly greater than the angles cal culated from i:¥H2 and i:¥X2, assuming orbital following. This difference seems to increase with increasing size of X and is greater for CH~2 (""12°) than for CH3X (""5°), presumably due to steric interactions. The incompatibility of i:¥H2 hybridization parameters obtained previously from JCH data with the observed bond angles has been discussed by Muller.2 He invoked "bent bonds" to explain differences found for the methyl halides. However, high-resolution microwave studies made recently by Flygare et al.16 rule out bent C-CI bonds in compounds where they were long be lieved to be present. So it seems unlikely that differ ences given in Table V result from "bent bonds." Furthermore, the plot of i:¥H2 vs "interorbital" angles is almost a perfectly straight line, especially in the region of values of i:¥H2 encountered here so that vibrational averaging17 of the C-H coupling constant assuming orbital followingl8 yields the same value as for the "static" equilibrium configuration. So the discrepancies can not be attributed to vibrational averaging effects. Quite apart from the relationship derived here between hybridization and JCH, the bond angles in CH2Cb, 1,1- dichlorocyclopropane and other molecules can not be reconciled with any carbon hybrid orbitals built only from sand p functions. The problem, therefore, is of a more general nature and does not arise simply from the approximations used in the expression for JCR. In closing this discussion of results for the substi tuted methanes, we wish to amplify on our earlier statement7 that a simple molecular orbital formulation gives results equivalent to the valence bond treatment presented above. A two-center molecular orbital of the form, f=Cl(lsH) +c2(2sc) +ca(2p"c), yields essentially the same equation as Eq. (12). The main difference is the normalization constant, which now depends upon the coefficients CI, C2, and Ca. How ever, these coefficients define the amount of carbon 2s orbital involved in and the polarity of the C-H bond; and the normalization constant for the molecular orbital 16 W. H. Flygare, A. Narath, and W. D. Gwinn, J. Chern. Phys. 36, 200 (1962); W. H. Flygare and W. D. Gwinn, ibid. 36, 7S7 (1962) . 17 This was approximated for J.emHHI by H. S. Gutowsky, V. D. Mochel, and B. G. Somers, J. Chern. Phys. 36, 1153 (1962). 18 J. W. Linnett and P. J. Wheatley, Trans. Faraday Soc. 45, 33, 39 (1949). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:062204 C. JUAN AND H. S. GUTOWSKi' FIG. 4. A diagram of the model used in extending the treat ment of substituent effects from methanes to ethylenes (CH.= CI3JIX). The vertical lines enclose the three O'-bonding orbitals of the carbon atom in the CI3HX group. In a planar st2 CI3H3 radical, the three potential wells would be of the same depth and equally occupied by the 2s electrons. LlCH2 and LlCi represent the different s electron affinities of the CH. = CI3 and C13-CI 0' bonds in CH2=CI3HX, which lead to the redistribution, among the three 0' bonds, of the s character indicated by the horizontal lines at the top. equivalent of Eq. (12) is no more sensitive to changes in these parameters than we found in the valence bond analysis. The model leading from Eq. (12) to Eqs. (1), (3) and (15) to (17) is the same for both approaches. Finally, we wish to point out that the valence bond treatment given below in Sec. III for substituted ethylenes and acetylenes may be formulated as well in terms of the two-center molecular orbital. m. SUBSTITUTED ETHYLENES AND ACETYLENES The treatment carried out for the me thanes may be extended (with some reservation) to the (J' electron contribution to JCH in Sp2 and sp hybridized carbon. Equation (10) applies to the latter, subject to the same approximations. Using Ac_H=0.374 and the overlap integralsll appropriate to the C-H bond distances in ethylene and acetylene, we find 'Y}2 for these two com pounds to be 0.987'Y}Z(CH 4) and 0.977'Y}2(CH 4), respec tively. Moreover, 'Y}Zj f:.E for ethylene and acetylene is affected no more by substituent effects, i.e., by changes in aH2, bond polarity and bond strength, than for the methanes. Hence JCH"'SOOaH2 for Sp2 and sp hybridized carbon, as well as for Sp3, except for possible effects of the 7r electrons. Because of this, the model developed for the effects of substituents on Jca(CHXYZ) should apply to ethylenes as well. Calculation of JcH(CH2=CI3HX) from f:.x for CH3X The main difference encountered in extending the treatment of substituent effects in the me thanes to the ethylenes is that there are three 2s wells instead of four. In ethylene itself the first two are identical H wells and the third differs in depth from these two by f:.CH2' where the subscript denotes the doubly bonded group, CHr-. In substituted ethylenes all three wells are of different depths, as shown in Fig. 4. Following our previous arguments that f:.x depends mainly on the difference in electronegativity of X and H, we should expect the same values for f:.x in the ethylenes as in the methanes. The value for f:.CH2 (ethylenes) is taken as equal to f:.CHa (methanes), i.e., +0.008, since the "substituent" CHz in the ethylenes has no direct counterpart in the methanes. For the same reason, the values for f:.CHX, f:.CXY, and f:.CX2 are taken to be equal to f:.CHoX, f:.CHXY, and f:.CHX2 respectively. On this basis the s character for a monosubstituted ethylene is given by aH2(CHr-CI3HX) =t[1+f:.CH.+f:.X] =aH2(CHr-CH2) +tf:.x, (18) which with Eq. (12) gives rise to J cn (CHz=CI3HX) = J CII ( CHr-CH2) +-lUcH(CH 3X) -Jcn(CH 4)]. (19) A comparison of Eq. (18) for CHr-CI3HX with Eq. (14) for CH3X shows that X causes a change in aH2 and in JCH, which in the ethylene is t that in the methane. This is because the substituent effect is spread among four bonds in the Sp3 methanes and only among three in the Sp2 ethylenes. Comparison with Experiment of Predicted Values for aH2 The f:.x values obtained from JCH( CH3X) , listed in Table II, have been used in Eq. (18) to predict an2 for eight substituted ethylenes for which JCH has been reported. The results are listed in the middle of Table VI, while in Fig. 5 the corresponding, observed value TABLE VI. Summary of JCH coupling constants observed in hydrocarbons, with sp2 and sp hybridization of the carbon orbi tals, and "predicted" values& for aH2. Compound JCH (cps) aH' Reference naphthalene 157 sp' b benzene 159 spz 3 mesitylene (2,4,6 protons) 160 Sp2 h (CH3).C=C=CI3H. 166 Sp2 c cyclohexene 170 Sp2 3 ethylene 157±2 0.336 2 CHCI=CH 2 (cis) d 160 0.341 e CHCI=C13H 2 (trans) d 161 0.341 e 1,1 dichloroethylene 166 0.349 e CH2=CI3HCI 195 0.402 e cisCHCI=CHCI 198.5 0.408 e transCHCI CHCI 199.1 0.408 e CCb=CHCI 201.2 0.416 e CH3C=CI3_H 248 sp 2 ",C==CI3-H 251 sp 2 H-C==C-C=C-H 259.4 sp c 8. For those compounds with aH21isted as Sp2 or sp, there is at present no sim ple means of correcting for the substituent effects. For the ethylenes, aH2 was obtained with Eq. (18) using Llx values from Table II for the methanes. b P. C. Lauterbur, J. Chern. Phys. 26, 217 (1957). C E. Snyder and J. D. Roberts, J. Am. Chern. Soc. 84, 1582 (1%2). d CI is cis or trans to the proton to which CI3 is coupled. • E. B. Whipple, W. E. Stewart, G. S. Reddy, and J. H. Goldstein, J. Chern. Phys. 34,2136 (1%1). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06CILH AND Si'9-H COUPLING 2205 for JCH is plotted against the predicted Cl:H2. The straight line in the figure corresponds to JCH= JOCl:H2=500Cl:H2 cps, and the open circles for the ethylenes should fall on this line if the theory is correct and the s electron affinities of substituents are transferrable from aliphatic to olefinic derivatives. It may be seen that the circles are consistently 5 to 10 cps below the theoretical line, suggesting a deficiency in the treatment of the substi tuted ethylenes. Some check on the significance of the deviations is afforded by the other points plotted in Fig. 5. Two sets of points represent the J CH values observed for the five Sp2 and the three sp hydrocarbons listed in Table ~+ I I ! 200 eO JC"'H cps • • • 1,8 150 • • sp C(2 0.4 H 0.5 FIG. 5. A plot of observed J OR values versus predicted values of <x.Jt. The straight line is J OR = 500 OIFf cps, upon which all points would fall if the methods for predicting OIFf were sufficiently accurate. The points at <x.Jt=! and!, i.e., for Sp2 and sp hybridiza tion, are from Table VI with no corrections for the effects of sub stituents upon the hybridization. The open circles are for the sub stituted ethylenes in Table VI, for which OIFf was predicted using the Llx values obtained from substituted methanes. The other points represent the polysubstituted methanes in Table III, for which 0IH2 was predicted by the additivity relation. VI, with Cl:H2 taken to be 1 and !, respectively. In these cases no corrections have been applied to Cl:H2 for the (unknown) substituent effects. The remaining points are for the polysubstituted me thanes in Table III, with the Cl:H2 values being those predicted for additive substituent effects. These points fall very close to the theoretical line, showing graphically the accuracy of the additivity relation for substituted methanes. In some cases the experimental uncertainties in JCH are com parable with the scatter of points from the line. The points for Sp2 and sp compounds, without correction for substituents, exhibit relatively large displacements, both positive and negative, from the line. But the over all scatter of the points is small enough, in spite of the wide range of Cl:H2 and JCH covered, to show that the magnitude of the CILH coupling constant is indeed determined mainly by the s character of the C-H TABLE VII. Si29-H coupling constants observed in SiH. and the silyl halides' and the substituent parameters IX obtained from them. Jsm(obs)a IX SiHaX (cps) (cps) SiH. 202.5 67.5 SiHaF 229 94 SiHaCl 238.1 103.1 SiHaBr 240.5 105.5 SiHaI 240.1 105.1 • E. Ebsworth and J. J. Turner, J. Chern. Phys. 36, 2628 (1962). We wish to thank the authors for making these results available to us prior to publication. bondl9 and that the Fermi contact term is the dominant term in the coupling. 1I"-Electron Contributions to JCH Nonetheless, the fact that the circles for the sub stituted ethylenes fall quite closely along a straight line about 7 cps below the one drawn in Fig. 5, leaves little doubt about there being a real, systematic dis crepancy. Part of the discrepancy could result from our assumption of +0.008 for LlCH2' which enters in each of the predicted Cl:H2 values. However, for this to be the sole factor, LlCH2 would need to be -0.040 and such a negative value seems an unreasonable departure from the Llx values for the methanes. It appears more likely that a 11" electron contribution to JCH is involved in the ethylenes. This can be estimated by the following extension of Karplus' formulation20 for the 11" contribu tion to JHH, namely- (20) where, for ethylene, Ll1l"= 6 eV and aH, the proton hyper fine splitting constant is -65X 106 cps. The Cia hyper fine splitting constant aC may be estimated by the TABLE VIII. Deviations (D) from additivity of the Si-H coupling constants observed a in the di-and tri-halosilanes. JSiH(obs)a JSiH(add)b 15 SiHnX4-n (cps) (cps) (cps) --------_._-- SiH2F2 282 255.5 26.5 SiH2CI, 288 273.7 14.3 SiH2Br2 289 278.5 10.5 SiH2I2 280.5 277.7 2.8 SiHF. 381. 7 282 99.7 SiHC!, 362.9 309.3 53.6 • E. Ebsworth and J. J. Turner, J. Chern. Phys. 36, 2628 (1962). We wish to thank lhe authors for making these results available to us prior to publication. b JSiH(add) =i'x+i'y+i'z, or (n-l)IR+(4-n)IX for SiHnX.--n. 19 Also noted previously by Muller and Pritchard,2 and by Shoolery.a 20 M. Karplus, J. Chern. Phys. 33, 1842 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:062206 C. JUAN AND H. S. GUTOWSKY Karplus-Fraenkel expression,21 which for the CCHz fragment involved here is given by (21) Using their values for the parameters, SC= -12.7, Qcq,c=+14.4, and QCHc=+19.5, one obtains the re sult aC= +40.7 G. ~his. leads to a value for Jcu7f equal to -2.6 cps, whIch IS of the same sign and order of magnitude as the discrepancy. The agreement provides some indirect evidence that the CI3 hyperfine interaction constant is positive in free radicals related to ethylenes. Also, it leads us to suggest that at least some of the spread of the spz and sp points in Fig. 5 reflects differences in JCH>:, and thus in aC, and that a more detailed study of the matter is desirable. A final point on the substituted ethylenes is that the various bond angles inferred from the predicted values of aH2 and ax2 do not agree well with experiment. The data available are limited, but it appears that the differences and the factors involved are very similar to those discussed in Sec. II for the methanes. IV. HALOSILANES SiHnX'_n Examination of the data in Tables VII and VIII shows that the Si2LH coupling constants of substituted silanes can be arranged in the same order as their corresponding methane analogs (except for CH2F 2) . This seems to indicate that in the silanes, the J SiH dependence on ASi-H and s character of the Si-H bond is of the same form as in the methanes. However, as noted previously,1 and as may be seen in Table VIII for the di-and tri-halo silanes, the substituent effects in the silanes deviate by as much as 30% from the additivity rule followed by the methanes. Deviations from Additivity of Substituent Effects The case of the substituted silanes is complicated by the possibility of d hybridization with sand p, which is assumed to be negligible in the case of the methanes owing to the large carbon 2p~3d promotional energy required. The promotional energy from 3p to 3d in Si is much smaller, so that some d character may be ex pected in the Si hybrid orbitals. This complicates matters because it introduces an additional parameter into the expression for JSiH, and our initial reaction7 was that the d electrons might be the cause of the devia tions from additive substituent effects. But further study of the question indicates that the deviations may result in large part from the greater polarizability of Si, compared to C, and the correspondingly larger changes in ASi-H. The deviations I) from additivity for the halosilanes 21 M. Karplus and G. K. Fraenkel, J. Chern. Phys. 35 1312 (1961). ' in Table VIII are found to be systematic in the follow ing ways: (1) They are all positive, that is Jobs = Jadd+O, where Jobs and Jadd are taken to be positive. (2) They occur in the order of the halogens, namely o(SiH 2F2) > I)(SiH 2CI2) > o(SiH2Br2) > o(SiH 2I2) and o(SiHFa) > o(SiHCIs). (3) For a given halogen, the deviation increases with the number of substituents, o (SiHX 3) > I) (SiH2X2) • Changes in ASi-H can account for these three observed trends. Inductive Changes in ASi-H and Their Effect uponJsiH Consider the effect of an electronegative substituent X on AC-H [the coefficient of th e 4>c ( 1) 4>c ( 2) term] vs ~hat on ASi-H [the coefficient of the 4>H(1)4>u(2) term], III a wave function such as that defined by Eq. (7). The main differences between the me thanes and silanes in this respect are: (a) the direction of change of AM-H with increasing electronegativity of substituent X and with increasing number of substituents, and (b) the order of magnitude of the change of AM-H. In both cases the effect is to increase the effective electro negativity X M of the central atom M insofar as the M-H bond is concerned. We expect the effect of the halogens to be in the order F> CI> Br > I (electro negativities 4.0> 3.0> 2.8> 2.5) and, of course, X3> X2> X. This leads to an increase in the electronegativity difference / XC-XH / for the C-H bond (XC>X H) and a decrease in / XSi - XH I for the Si-H bond (XH>X Si). In general, the coefficient AA-B of the ionic term in the perfect pairing wavefunction describing the bond A-B is related to /XA-XB I (e.g., by a relationship such as shown in Fig. 1). Although the exact relation ship is disputable, it is knovm that as / XA -XB I in creases, so does AA_B.22 Thus (a) by the previous discussion, the effect of a more electronegative sub stituent or a greater number of substituents is toward increasing AC-H and decreasing ASi-H, and (b) since the polarizability of the electron cloud of Si is much greater than that of C, the effect of the electronegative substituent(s) on the electron distribution in Si is much more drastic than that on C. Therefore with the same substituent(s), the change in ASi-H is' expected to be of greater magnitude than the change in AC-H. 22 See for example: L. Pauling, The Nature of the Chemical Bond (Cornell UniverSIty Press, New York, 1960), p. 99; B. P. Dailey and C. H. Townes, J. Chern. Phys. 23, 118 (1955)' W. Gordy ibid. 19, 792 (1951). " This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06ClI-H AND Si'9-H COUPLING 2207 The form of J SiR under approximations similar to those used in deriving Eqs. (10) and (11) for the methanes, is23 (22) where 'fJ-2= [2+2 (a2S82+iJ2Sp2+'Y2Si) +4 (aiJS8Sp +iJ'YSpSd+a'YS8Sd) +4Asi-H(aS.+iJSp+'Y Sd) +ASi_H2]. (23) In Eq. (23) a2, iJ2, and '1'2 are, respectively, the s, p, and d character of the Si hybrid orbital for the Si-H bond, and S., Sp, and Sa are the overlap integrals between the Is orbital on H and the respective orbitals on Si. With the use of Eq. (22), the expression for the deviation from additivity, may be written as 8= I A I ['fJ2(SiHnX4-n) -'fJ2(SiH4) Ja2, (24) where A is a collection of constants. From Eq. (23) we see that a decrease in ASi-H leads to a larger 'fJ2. Therefore, the decrease in ASi-H produced by replace ment of a hydrogen with a substituent which is more electronegative than hydrogen, causes 'fJ2(SiHnX4_n) to be greater than 'fJ2(SiH4)' Hence, the deviations 8 are all positive. Furthermore, since by the previous discussion the effect on ASi-H is more drastic as one goes from X = I to X = F and from SiH3X to SiHXa, then the deviations should be in the order, as observed 8(SiH2F2) > 8(SiH2Cb) > 8 (SiH2Br2) > (8(SiH2I2) 8(SiHF 3) > 8(SiHCh), and Therefore, it seems that the deviations of JSiH from additivity are governed by the inductive effect of the electronegative substituents. Discussion Such inductive effects should also occur in the silyl halides, but the evidence on this point is obscure. For the methanes, the .1x values are about +0.20 for all four halogens. However, for the silanes, the "apparent" .1x increases from +0.13 for F to +0.19 for Br and I. This trend is the reverse of what would be predicted on the basis of the proposed inductive effects, if the .1x values would otherwise be about the same for silanes and methanes. The close agreement between the Br 23 This is not to say that the same approximations are good for the silanes. We present this equation to illustrate the dependence of JSiH on ASi-H. Note that 1'8; is a negative quantity. and I .1x values for me thanes and silanes (+0.20 and +0.19) suggests that further comparisons of this sort may be significant. In fact, the deviations from addi tivity in the halomethanes, although small, parallej24 the deviations in the halosilanes, i.e., 8 (CHX 3) > 8( CHzX2) and in the order F> Cl> Br> I (+12, +3, -1, -4, respectively, for CH2X2). But this does not necessarily mean that the factor(s) responsible for the deviations in the halosilanes is the same factor which causes the small deviations in the halomethanes. The latter are in the wrong direction to be explained by the inductive effect on AC-H although the deviations in the silanes are compatible with changes in ASi-H. An accurate theoretical estimate of the inductive effects upon JSiR would require at least a knowledge of all the appropriate constants in Eqs. (22) and (23). However, several of these are not available so a simpli fied treatment was made by neglecting the d electron terms in Eq. (23) for 'fJ2, calculating the remaining overlap integrals,ll and obtaining an equation for 'fJ2(ASi_H, a2). Reference to Fig. (1) indicates that ASi-H(SiH 4) is about 0.35. With this value, and the equation for 'fJ2, it was found that ASi-H would need to be 0.23 in SiH2F2 if 8(SiH 2F2) were entirely due to the inductive effect. Such a change, while large, is not unreasonable. We have not considered the complication of the silane problem by the reasonance effect, i.e., contributions by structures of the form H H I I H-Si X+ H-Si X-. II X- X+ These are expected to be greatest for F and least for I, for it is well known that large atoms form multiple bonds less readily. This effect is probably also present in the me thanes but to a smaller extent, for Si can form d7r bonds. This resonance effect tends to make Si more electropositive, which is opposite to the inductive effect. In addition, the above structures bring up the possi bility of 7r contributions to the Si-H coupling. In view of the consistent trend in the deviations 8 it seems that these two effects largely cancel one another, thus maintaining the trends in 8 which would be ex pected from inductive effects alone. In the foregoing discussions we have ignored alto gether the noncontact contributions (01 and O2), Karplus and Grant1 estimated 01 and O2 to be 2 and 8 cps, respectively, for CH4 and stated that these contri butions are not necessarily small for compounds which deviate from tetrahedral symmetry. The contributions 01 and O2 are extremely difficult to calculate at present, but an approximate relationship has been found by 24 We wish to thank a referee for calling this relation to our attention. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:062208 C. JUAN AND H. S. GUTOWSKY Pople26 between OINN' and the anisotropy of the screen ing constants for Nand N'. An extension of this treat ment to OlCH and OlSiH shows that these contributions to JCH and JSi-H are not additive. As a final point, the deviations from additivity in dicate that such factors as inductive effect on X, con tributions of doubly-bonded ionic structures to the ground state, and non contact terms may very well be responsible for some part of the substituent parameter rx which we have treated here as if it were attributable entirely to hybridization changes. APPENDIX In the derivation of Eq. (8) the following, usual approximation was used: (cf>N(rl) 1 o(rIN") 1 cf>N,(rI) )~ 1 cf>N(O) 12oNN"ON'N", (25) where cf>N(rl) is an atomic orbital centered on nucleus N. For Nand N' separated by two or more bonds, this is a very good approximation which has always been used in calculating JNN', as in the case of H-C-C H'. In order to determine how valid this approximation is in the case of directly bonded nuclei,26 more specifi cally for JCH, let us examine in detail the integral involved in deriving Eq. (8), namely p=t(u(1,2) lo(rlH)o(rze) +o(rZH) O(rlC) 1 u(1, 2) )+PB (26) where PB=! L (uB(1,2) lo(rlH)o(rze) B=X.Y.Z with uH(1, 2) =n[cf>H(1)cf>c(2) +cf>c(1)cf>H(2) +XHcf>H(1)cf>H(2) +Accf>c(1)cf>c(2)], and ux(1, 2) =nx[cf>x(1)cf>c'(2) +cf>c'(1)cf>x(2) +Xxcf>x(1)cf>x(2) +Xc'cf>c' (1)cf>c' (2)], in which cf>c is the carbon hybrid orbital directed toward the hydrogen and cf>c, is the carbon hybrid orbital di rected toward the substituent X. The leading term which arises upon calculating the integral P is po=n21 cf>c(0) 121 cf>H(O) 12=7P£l'H212sc(0) 12IlsH(0) 12. The use of the approximation expressed in Eq. (25) in evaluating P gives rise to only the leading term, and this results in Eq. (to). The other, smaller terms in P are products of integrals of the form, (cf>H(ri) 1 o(rm) 1 cf>H(ri) )=cf>H(O)cf>c(R) , 25 J. A. Pople, Mol. Phys. 1, 216 (1958). 26 We wish to thank Dr. Ralph M. Deal for raising the question which led to this analysis. where R is the equilibrium distance between the C and H nuclei. The complete expression for P is given by p!?12= 1 cf>H(O) 121 cf>c(0) 12+ 1 cf>H(R) 121 cf>c(R) 12 +XH21 cf>H(O) 121 cf>H(R) 1 2+XC2 1 cf>c(0) 121 cf>c(R) 1 ~ +2(1 +XCXH) [cf>H(O) cf>c(R)cf>c(O)cf>u(R)] +2XH[1 cf>H(O) 12cf>c(0)cf>H(R) + 1 cf>u(R) 12cf>c(R)cf>II(O) J +2Xc[1 cf>c(0) 1 2cf>H(0)cf>c(R) + 1 cf>c(R) 12cf>n(R)cf>c(O)] +PB/1)2. The terms in the summation, Eq. (27), which con tribute the most to PB are of the form and 2nx2Xcx2(cf>c,(1) 1 O(rIe) 1 cf>c,(1) ) (cf>c, (2) lo(r2H) X 1 cf>c,(2) ). (cf>x(2) 1 o(rZH) 1 cf>x(2) ) is the density at the H nucleus of an electron in the X orbital, which is vanishingly small, and (cf>c'(2) 1 o(rZH) 1 cf>c'(2) ) is the density at the H nucleus of an electron in the carbon hybrid orbital directed toward X, which is also small and approximately equal to £l'x21 sc(R) 12. Therefore PB, the contribution to P of the orbitals involved in the other bonds, clearly is very small. The hydrogen atom 1s wavefunction gives cf>H(O) = 0.5642 and cf>H(R) =0.0716 in units of aof; and from the Hartree-Fock atomic wavefunctions for carbon we obtain the values 2sc(0) = 1.664 2p~(0) =0 which lead to 2sc(R) = -0.0791 2p~(R) = 0.0788, cf>c(0) = 1.664£l'H cf>c(R) = -0.0791£l'u+0.0788(1-£l'H2)t. The range of £l'H2 of interest here is 0.25 to 0.50. cf>c(R) is equal to 0.0287 for £l'H2=0.25 and decreases mono tonically and reaches zero at £l'H2=0.498. PB also de creases monotonically with increasing £l'H2. Therefore, by using the value of cf>c(R) for £l'H2=0.25, one can find the upper limit for the error incurred by dropping all terms containing cf>N(R) ; i.e., in using the approxima tion expressed by Eq. (25). Using Xc=0.374 and XH=0.01, one obtains PO/1)2= 0.2203 and p/n2=0.2312+PB/n2 in units of ao6• pB/n2 is found to be 0.0005 in CH4• The maximum error is, therefore, 5% or 6 cps. The percentage error decreases monotonically with Q'H2 and goes to a minimum of 0.17% at Q'u2=0.498, where JCH is around 250 cps, i.e., an error of 0.4 cps. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sat, 29 Nov 2014 16:00:06

1.1701266.pdf

Hyperfine Interactions in XIrradiated Magnesium Phosphite Hexahydrate Melvin W. Hanna and Larry J. Altman Citation: The Journal of Chemical Physics 36, 1788 (1962); doi: 10.1063/1.1701266 View online: http://dx.doi.org/10.1063/1.1701266 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/36/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interaction of defects and H in proton-irradiated GaN(Mg, H) J. Appl. Phys. 97, 093517 (2005); 10.1063/1.1883309 A determination of hyperfine constants for the RCONHO radical in randomly oriented samples of x irradiated hydroxamic acids J. Chem. Phys. 77, 4333 (1982); 10.1063/1.444422 ESR of phosphite radicals trapped in xirradiated single crystals of ophosphorylethanolamine J. Chem. Phys. 70, 1667 (1979); 10.1063/1.437681 ENDOR study of xirradiated glycylglycine HCl J. Chem. Phys. 61, 428 (1974); 10.1063/1.1681659 XIrradiation Damage of Sucrose Single Crystal J. Chem. Phys. 37, 202 (1962); 10.1063/1.1732964 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 206.196.184.84 On: Wed, 17 Dec 2014 02:22:44THE JOURNAL OF CHEMICAL PHYSICS VOLUME 36, NUMBER 7 APR I L I, 1 962 Hyperfine Interactions in X-Irradiated Magnesium Phosphite Hexahydrate* MELVIN w. HANNAt AND LARRY J. ALTMAN Gates and CreUin Laboratories of Chemistry,t California Institute of Technology, Pasadena, California (Received September 11, 1961) An analysis of t~e .electron spin resonance spectra of X-irradiated single crystals of MgHP0 3·6H.O has shown that the prlllClpallong-lived paramagnetic species produced by irradiation is a POa fragment. This fragmen.t retains the same orientati~m within the unit cell as the undamaged molecule. The parallel and perpendicular components of the diagonal (electron-spin)-(nuc1ear-spin) coupling dyadic for the phos phorus nucleus are fou~d to be of .the same sign and of magnitude 2210 and 1730 Mc, respectively. These results are compared wlth theoretical values of the distributed dipole and contact hyperfine interactions calculated on the assumption that the unpaired electron is located in an spa hybrid orbital centered on phosphorus. INTRODUCTION IT is now well known that detailed information about the electronic structure of radiation produced para magnetic defects in single crystals can be obtained by studies of the electron spin resonance (ESR) spectra of these defects.1-9 Most of the recent studies in this field have dealt with the hyperfine interactions between an unpaired electron in a 2p-atomic orbital and an adjacent hydrogen atom,l-4.7-9 These interactions can be of two experimentally distinguishable types. In one type, the hydrogen atom is attached to the carbon bearing the unpaired electron and lies in the nodal plane of the 2p-atomic orbital.1•3•4,7-9 In this case the isotropic component of the hyperfine interaction arises from configurational mixing of the orbital containing the unpaired electron with excited states of the sigma bond framework. In the second type the hydrogen atom is attached to an adjacent carbon and the hyper fine splittings arise because of "hyperconjugative" type interactions.2,7 In both of these cases the isotropic hyperfine interactions are between 60-75 Mc, and in the first type the anisotropic hyperfine interaction is ±30 Mc. The anisotropic interactions in the second type are very small. There have been a few studies on fragments in which the unpaired electron was localized in an orbital centered on the magnetic nucleus. In 'Y-irradiated dimethylglyoxime the unpaired electron is reported to be localized in an Sp2 hybrid orbital localized on * Supported in part by the U. S. Public Health Service and the Arthur Amos Noyes Fund. t Arthur Amos Noyes Fellow. Present address, Department of Chemistry, University of Colorado, Boulder, Colorado. t Contribution No. 2504. 1 H. M. McConnell, C. Heller, T. Cole, and R. W. Fessenden, J. Am. Chern. Soc. 82,766 (1960). 2 C. Heller and H. M. McConnell, J. Chern. Phys. 32, 1535 (1960). 3 D. K. Ghosh and D. H. Whiffen, Mol. Phys.2, 285 (1959). • N. M. Atherton and D. H. Whiffen, Mol. Phys. 3, 1 (1960). • D. Pooley and D. H. Whiffen, Mol. Phys. 4, 81 (1961). • I. Miyagawa and W. Gordy, J. Chern, Phys. 30, 1590 (1959). 7 L Miyagawa and W. Gordy, J. Chern, Phys. 32, 255 (1960). • M. Katayama and W. Gordy, J. Chern. Phys. 35,117 (1961). 9 D. V. G. L. Narasimha Rao and W. Gordy, J. Chem. Phys. 35,362 (1961). nitrogen,6 and in the case of X-irradiated malonic acid the principal values of the C13 hyperfine interactions have been evaluated.l° An interesting characteristic of all of the radiation produced fragments studied so far, with the possible exception of dimethylglyoxime, is that the paramagnetic fragment has a different hybridization than the un damaged molecule. This holds true in spite of the fact that the main skeleton of the fragment retains the same orientation as the undamaged molecule, within the accuracy of the ESR experiment. Thus, in the case of undamaged malonic acid, HOOCCH 2COOH, the central carbon atom is tetrahedrally (Sp3) hybridized, but it has been conclusively shown! that in the radiation produced fragment, HOOCCHCOOH, the unpaired electron occupies a pure p-type atomic orbital. Further, the (l' hydrogen atom has apparently moved up into the plane defined by the two carbon-carbon bonds. It is the purpose of this paper to present the first complete analysis of hyperfine interactions with a pal nucleus. Also, in this study it will be shown that the paramagnetic fragment not only retains the same orientation as the undamaged molecule, but also retains approximately the Sp3 hybridization. Since the unpaired electron is localized in an Sp3 hybrid orbital (see below) there is a direct means of getting unpaired spin density at the magnetic nucleus. The isotropic hyperfine inter actions reported in this work are about 30 times larger than the interactions mentioned above where the odd electron density at the nucleus arises from a more indirect type of interaction. EXPERIMENTAL Single crystals of MgHP03·6H 20 were grown by slow evaporation of aqueous solutions. The crystal that was used in the ESR study was in the form of a hemimorphic trigonal pyramid approximately 3 mm high and 3 mm across the base. The crystal structure has been determined by Corbridge.ll The dimensions of the hexagonal unit cell are [aJ=8.88 A, [cJ=9.10 A. 10 T. Cole and C. Heller, J. Chern. Phys. 34, 1085 (1961). liD. E. C. Corbridge, Acta. Cryst. 9, 991 (1956). 1788 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 206.196.184.84 On: Wed, 17 Dec 2014 02:22:44x -I R R A D I ATE D MAG N E S I U MPH 0 S PHI T E HEX A H Y D RAT E 1789 The unit cell, of space group symmetry R-3, contains three molecules. In this unit cell there is a threefold symmetry axis parallel to the c axis of the crystal and the magnesium atom and the phosphorus-hydrogen bond lie along this threefold axis. A single crystal was X-irradiated for approximately 1 hr from a copper target x-ray tube operating at 45 kv and 14 mao The crystal was approximately 3 em from the target. The ESR spectra were taken on a Varian V-4500, X-band spectrometer employing lOO-kc field modulation. The crystal was mounted on a special goniometer head designed to fit in the microwave cavity which allowed rotation of the crystal about an axis perpendicular to the magnetic field. Illustrative spectra are shown in Fig. 1. The co ordinate system used in these figures is fixed in the molecule and is defined as follows: The z axis lies along the threefold symmetry axis. The x, and y axes are defined as any two orthogonal axes perpendicular to the z axis. For convenience the origin was placed at the phosphorus nucleus. As can be seen from Fig. 1 (a), the ESR spectrum of X-irradiated MgHPO ao6H20 consists of a doublet with an extremely large separation between the two components. In addition, at some orientations of the crystal small satellite lines were observed on either side of the main hyperfine line [Fig. 1 (b)]. At high gain the space between the two lines from the paragmagnetic fragment contained a complex spectrum due to a small quantity of Mn2+ impurity in the crystal. This spectrum was present even in the absence of irradiation and was similar to that reported by Low for other Mn2+ doped Mg2+ host crystals.l2 The three-line spectrum of peroxylamine disulfonate JJL t-'1.------WIo,',II------Jlr (0) JJ1 1/1 Ho (b) FIG. 1. (a) ESR spectrum of POa with the applied magnetic field, along the z axis. The center three lines are due to the per oxylamine disulfonate standard. The other weak lines are due to a Mn2+ impurity. (b) ESR spectrum with the applied field at 45° to II axis showing the satellite lines on either side of the main hyperfine component. 12 W. Low, Phys. Rev. 105, 793 (1957). in aqueous solution was used for scan and g-value calibration purposes. THEORETICAL The spin Hamiltonian that can be used to interpret the ESR spectra of oriented radicals is (1) Here X. is the Zeeman coupling of the electronic and nuclear magnetic moments to the externally applied field, Ho, and XM includes the combined (electron-spin) (nuclear-spin) Fermi contact and dipolar interac tions.l,l3 If a coordinate system is chosen such that the dipolar-coupling dyadic in XM is diagonal this Hamiltonian can be written In Eq. (2) I p.1 and Pp are the electron and pal nuclear resonance frequencies and A, B, and C are the principal values of the hyperfine coupling dyadic. In the case that the unpaired electron distribution pos sesses an axis of cylindrical symmetry B = C and the spin Hamiltonian then contains only two parameters. For the magnetic field along the axis of cylindrical symmetry (z axis) Eq. (2) can be written in the alternate form X=h I p.1 S.-hppl.+hASzI.+!hB[S-rI_+SJ+], (3) where S+, 1+ and S_, 1_ are the electron and nuclear spin raising and lowering operators, respectively.14,l5 For the "strong field" case, i.e., I p.1 »A, B, the eigen energies, spin eigenfunctions, electron resonance transi tion frequencies and transition probabilities are simple extensions of the results obtained for the two spin system in x-irradiated malonic acid.l In the present study, however, the approximation that I p.1 »A, B does not hold, and the calculation of the electron resonance spectra for this "intermediate field" case is a little more involved. Of the zero-order spin functions (4) only 1/11° and 1/14° are eigenfunctions of the Hamiltonian (3). The eigenenergies for the case I/Ilo=a(e) a(p), 1/12o=a(e){3(p) , I/Il={3(e)a(p) , 1/14o={3(e){3(p) , (4) of Ho along the z axis are, therefore, El=!h I p.1 -!hpp+thA, E+=!h[(1 p.1 +Pp)2+B2Jl_thA, E_= -!h[(1 P81 +pp)2+B2]i-ihA, E4= -!h I p.1 +!hpp+thA, 13 S. M. Blinder, J. Chern. Phys. 33, 748 (1960). 14 H. M. McConnell (unpublished notes). (5) 16 K. D. Bowers and J. Owen, Repts. Progr. in Phys. 18, 304 (1955) . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 206.196.184.84 On: Wed, 17 Dec 2014 02:22:441790 M. W. HANNA AND L. J. ALTMAN " 0-0 3-0 6-0 9-0 12 a Me x 10-' a 12 {a} • 9-0 6-0 3-0 ---'I'="'O .....L.~e --Mco;o'O'""'-.""-~e:-'-~,o-- (b) FIG. 2. Angular dependence of the hypcrfine splittings in POa• (a) Applied field in 2X quadrant. (b) Applied field in xy quadrant. and the corresponding eigenfunctions are '/II= V;lo, where V;+= cosf},N+ sin8if;3°, V;-= -sin8if;2°+ cosfh/;ao, V;4= V;40, (6) cOtf}=B/\[(! v.l +vp)2+B2]L(1 vel +vp]l (7) at high values of the applied magnetic field 1 vel »B, and 0-.0 giving the usual high field eigenfunctions. The perpendicular electron resonance transition frequencies are (E1-E+)h-1=HI vel -vp)+tA -!e(1 Ve 1 +vp)2+B2]!, (8a) (E1-E_)h-I=HI Pe I -vp)+tA +t[(lve I +vll)2+B2J!, (8b) (E+-E 4)h-1 HIVe l-vp)-tA +K(I Ve I +vll)2+B2]t, (8c) (E4-E_)h-1= -HI Ve j -vp)+tA +t[(jVe I +vp)2+B2Ji. (8d) The electron resonance transition intensities are proportional to the absolute square of the matrix elements, I (V;.\,u..L lV;i) 12, where,u..L= -go! J3! S.LiS the electron spin magnetic moment operator for a component of S perpendicular to H. Thus, I (v;+ I J.l.J. I Ytl) 19o2 I J3 12 sin28, ! (V;-\ J.l..L \V;l) 12= tgo2 \ {3 12 cos20, I (Yt4! J.l..L ! Yt+) !2= tgo2\ {3\2 cos20, I (Yt4! J.l...L 1 Yt-) 12=tgo2\ {3\2 sin20, I (Yt4 ! J.l..L 1V;1) (Yt+ I JLJ.!V;-) 12= o. (9) There is one allowed parallel transition, Yt-->1f;+, with frequency (E+-E_)h-1=[(1 Ve! +vp)2+B2Ji (10) and intensity proportional to (11) For the applied field perpendicular to the z axis the spin Hamiltonian can be written in the form JC=th Iv. 1 Sx-thvpIx+hBSxIx+lh(A+B) ,(S+,L'+S_'I+') +ih(A -B) (S+,I/+S_'L') (12) and the calculation of the electron resonance transition frequencies and intensities follows the same procedure as above. The problem is somewhat simplified if one constructs new spin functions a' and (3' which are eigenfunctions of the operators S" and I x, and also uses the corresponding spin raising and lowering operators S+" I+' and S_', 1_'. In this case the zero order spin functions v;lo', and Ytr are also connected by nonvanishing matrix elements so that two 2X2 secular equations must be solved. The problem of calculating the energy levels and the dipole intensities for a P03 fragment for a general orientation of the intermediate field is, in principle, a straightforward extension of the methods described above. The calculations are tedious, however, since every zero-order spin function is connected to every other zero-order spin function by a nonvanishing matrix element. This means that a full 4X 4 secular equation must be solved for every orientation of the applied intermediate field to obtain the eigenvalues. RESULTS AND DISCUSSION The doublet hyperfine pattern observed in irradiated MgHP03'6H20 is indicative of the interaction between the unpaired electron and a single nucleus of spin t. The observed hyperfine splittings are too large to be due to interaction with a proton, consequently the ESR doublet must arise from the interactions of the unpaired electron with a PSl nucleus. The variation in the doublet separation as the applied magnetic field is rotated in the zx and xy planes is shown in Fig. 2. The largest hyperfine splitting (hfs) is observed with the applied field along the z axis, and the smallest hfs is observed with Ho in the xy plane. There is no change in the hfs as Ho is rotated in the xy plane [Fig. 2 (b) ]. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 206.196.184.84 On: Wed, 17 Dec 2014 02:22:44x -I R R A D I ATE D MAG N E S I U MPH 0 S PHI T E HEX A H Y D RAT E 1791 These results confirm the fact that the unpaired spin distribution in the parmagnetic fragment retains at least a threefold symmetry axis. In the undamaged molecule, this threefold symmetry axis lies along the phosphorus-hydrogen bond. Thus, the paramagnetic fragment apparently results from homopolar cleavage of the P-H bond and has the formula· P03• A. Hypedine Interactions In this study the parameters of interest are the following: I v. I =8300-10500 Mc, vp=5-6 Mc, A 221O±30 Me, B= 1730±30 Mc. Since the pal resonance frequency is considerably smaller than the uncertainty in A and B it will be neglected in the calculations which follow. Using the above values, the angle () in Eqs. (6) and (7) is 5°-6°. The only transitions having ap preciable dipole intensity will be those whose intensity depends on cos2(}, i.e., 1/I-~1, and 1/Ir+1/I+. The 1/1+---'>1/11, and 1/1-~4 transitions will be approximately 165 times weaker.l6 The allowed parallel transition 1/1-~4 should be observable, but suitable apparatus was not available for detecting it. With Ho along the z axis the ESR spectra will, therefore, consist of a strong doublet whose transition frequencies are given by Eqs. (Sb), (Sc). This doublet will be separated by approximately the frequency A, and the center of gravity of the spectrum will be at B2[ 1 1 ] Hillel++ 1 lie 1-)+8 Iv.l+ + \11.1-. (13) In this equation I lie 1+ and I v. 1-are the electron resonance frequencies at the field strength of the high and the low field line respectively.17 With Ho perpen dicular to the z axis the doublet will be separated by approximately the frequency B, and the center of gravity will be at I lie 1_)+h[(A+B)2+(A-B)2] .[_1 +_1 ] 1 v. 1+ I V. 1-. (14) The values of A and B given above were obtained by measuring the hyperfine splittings for these two orientations of Ho.1s The quantities A and B are of special interest because they give information about the electronic structure of the paramagnetic fragment. These parameters are a sum of two parts-the isotropic or Fermi contact interaction aP and the anisotropic or dipolar inter actions Ad and Bd• Since the trace of the dipolar 16 It is impossible to observe these transitions in a constant frequency experiment. 17 The authors are grateful to Dr. David Whiffen and Mr. John Morton of the National Physical Laboratory, Middlesex, England for much constructive criticism on this point. 18 In the intermediate field case a small correction must be made to the observed splittings to get A and B but in the present work this correction is less than the experimental error and has been neglected. t------l 15 Gauss FIG. 3. ESR spectrum of the high field hyperfine line at high microwave power showing the two sets of satellite lines. coupling dyadic must be zerol•13 aP=t I A+2B I = 1890±30 Me, and Ad= +320±30 Mc, Bd= -160±30 Mc. Calculations of Ad using the distributed dipole formulal,19 for an odd electron localized in a p orbital centered on the magnetic nucleus show that Ad is positive.20 Since only the p part of the Sp3 orbital contributes to Ad, the sign of Ad for an Sp3 hybrid should be the same as the sign for a pure p orbital. In Eq. 15, per) is the spin density distribution function21 and 0' is the polar angle which the vector from the nucleus to the unpaired electron makes in the xyz coordinate system. The iso tropic coupling constant aP must, therefore, also be positive. This is in accord with previous theoretical predictions.22,23 The extremely large value observed for aP in this study indicates that there is a direct mecha nism for getting odd electron density at the nucleus. The orbital containing the unpaired electron must, therefore, be a hybrid orbital containing considerable s character. Since the bond angles in the undamaged molecule were 109°, it is likely that the unpaired electron is localized in an Sp3 hybrid orbital. If this were the case the hyperfine coupling constant for a 3s electron on phosphorus would be approximately 7560 Mc. A calculation of aP using recently calculated 19 H. M. McConnell and]. Strathdee, Mol. Phys. 2, 129 (1959), 20 See, for example, W. V. Smith, P. P. Sorokin, I. L. Gelles, and G. J. Lasher, Phys. Rev. 115, 1546 (1959); R. G. Barnes and W. V. Smith, ibid. 93, 95 (1954). 21 H. M. McConnell, J. Chern. Phys. 28, 1188 (1958). 2' H. M. McConnell and D. B. Chesnut, J. Chern. Phys. 28, 107 (1958). 23 A. D. McLachlan, H. H. Dearman, and R. Lefebvre, J. Chern. Phys. 33, 65 (1960). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 206.196.184.84 On: Wed, 17 Dec 2014 02:22:441792 M. W. HANNA AND L. J. ALTMAN Hartree-Fock functions for phosphorus24 gives aP= 11 000 Me in fair order-of-magnitude agreement. A further point of interest in this study are the large values of the anisotropic spliUings Ad and Bd. Calcula tion of Ad for an Sp3 orbital constructed from 3s and 3p Slater orbitals using a shielding value Z = 4.8 gives a maximum value of Ad= +35 Me, an order of magnitude too low . Neglecting the shielding due to electrons in the same shell only raises Ad by a factor of 2. Either Slater orbitals give a rather poor value of (1/r3 )AV, or the unpaired electron distribution is much more complex than that represented by an Sp3 hybrid orbital. A further possibility is that the odd electron in the 3p. orbital differentially "polarizes" the spins of the paired electrons in the 2p. orbital. This effect would give rise to a larger value of (1/r3)AV. Such a mechanism has been used to explain the large contact hyperfine interactions in transition metal ions.25 As mentioned in the introduction, many previous paramagnetic fragments have had a different hybridiza tion than the undamaged molecule. In the present case, however, the P03 fragment retains approximately the same hybridization as the undamaged molecule. A comparison of the structure of MgHP0 3 with these other compounds reveals a possible reason for this behavior. In the cases where a change in hybridization occurred, the atom on which the unpaired electron was localized was also bonded to another hydrogen atom. After radiation damage it would be possible for this other hydrogen atom to change position in order to give a planar fragment without invoking large changes in the intermolecular forces in the crystal. This would then allow the unpaired electron to reside in a pure p-atomic orbital. In the case of MgHP0 3, the other three atoms are still firmly held in the crystal by hydrogen bonds after damage. For this fragment to rehybridize, these hydrogen bonds would have to be broken or at least drastically distorted from their most stable configuration. The resulting loss of stability would not be compensated by the gain in stability 24 R. E. Watson and A. J. Freeman, Quart. Progr. Rept. No. 41, Solid State and Molecular Theory Group, M.LT. (1961), p. 6. !5 See, for example, J. S. Van Wierengen, Discussions Faraday Soc. 19, 118 (1955). from rehybridization. Weare currently checking this hypothesis by studying irradiated MgH 2P02·6H20. In this case the paramagnetic fragment is expected to contain a second hydrogen atom and should rehybridize. B. Spectroscopic Splitting Factors In the intermediate field strength case the spectro scopic splitting factors must be measured relative to the central frequency given by Eqs. 13 and 14. The results for the P03 fragment are gil = 1.998 and g.l. = 1.999. These values are accurate to ±O.002. c. Satellite Lines When Ho is moved off of the canonical directions i.e., along the z axis and in the xy plane, satellite line~ begin to appear on either side of the main hyperfine lines. The spacing of these satellites corresponds to the proton resonance frequency at the magnetic field of each hyperfine component. Thus, the satellites around the high field line are spaced a little farther apart and are better resolved than those around the low field line [see Fig. 1 (b) J. These satellites have been previously observed in the ESR spectrum of atomic hydrogen,26 and a theoretical treatment of this phenomenon has been given.27 These satellite lines result from a weak magnetic dipole-dipole interaction coupling the electron spin to a neighboring nuclear spin. In the present case the satellites are due to coupling with the protons in the waters of hydration. The theoretical treatment sug gested that under conditions of high sensitivity a second set of satellite lines should be observed corresponding to two neighboring protons concurrently changing state.27 At very high microwave powers this second set of lines appears in the high field component of the P03 spectrum. Figure 3 shows this spectrum in which the two sets of satellites can readily be seen. ACKNOWLEDGMENT The authors wish to thank Professor Harden McConnell for helpful discussions regarding this work and for the use of his ESR spectrometer. 26 H. Zeldes and R. Livingston, Phys. Rev. 96, 1702 (1954). 27 G. T. Trammell, H. Zeldes and R. Livingston Phys. Rev no, 630 (1958). ,. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 206.196.184.84 On: Wed, 17 Dec 2014 02:22:44

1.1735786.pdf

Photoemission in the Photovoltaic Effect in Cadmium Sulfide Crystals Richard Williams and Richard H. Bube Citation: J. Appl. Phys. 31, 968 (1960); doi: 10.1063/1.1735786 View online: http://dx.doi.org/10.1063/1.1735786 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v31/i6 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions968 ACRIVOS, SHAH, AND PETERSEN x-defined by Eq. (7) y-defined by Eq. (7) Yl-a constant defined by Eqs. (22a) and (23) z-polar coordinate, distance perpendicular to the surface of the disk ?)-defined by Eq. (10) ?)!-defined by Eqs. (17) and (18) ?)!-defined by Eqs. (19) and (20) p-density of the fluid r-shear stress q,(?))-defined by Eq. (10) q,!-defined by Eqs. (17) and (18) q,~-defined by Eqs. (20) and (21) w-the angular velocity of the disk JOURNAL OF APPLIED PHYSICS VOLUME 31, NUMBER 6 JUNE, 1960 Photoemission in the Photovoltaic Effect in Cadmium Sulfide Crystals RiCHARD WILLIAMS AND RICHARD H. BUBE Radio Corporation of America, RCA Laboratories Division, Princeton, New Jersey (Received November 5, 1959; revised manuscript received December 17, 1959) A. study has been ~ade of the ,Phot?voltaic effect in Cu-CdS cells and related systems, associated with undlffused metal-semiconductor JunctIOns. The photovoltaic current has been shown to result from the p~otoemissi?n of electrons fr~n: the copper metal into the CdS crystal. Direct evidence is presented for this conclusl~n, and the conditions reqUired tor ~he photoemission process to occur are demonstrated by several expenments. Important factors contnbutmg to the efficiency of Cu-CdS photovoltaic cells of this type are: (a) ~~e optical ?roper~ies of copper, (b) th~ rectifying contact between the metal and CdS, (c) the good conductiVity and high optical transparency which can be achieved in CdS crystals and (d) the favor- able relation between the work function of copper and the electron affinity of CdS. ' INTRODUCTION PHOTOVOLTAIC effects have been observed in a variety of systems in which a metal and a semi conductor are in contact. It has been demonstrated that the effect is produced predominantly by the light which is absorbed near this contact.l The accepted explanation is that light is absorbed by the semiconductor, producing hole-electron pairs, and that both the holes and the electrons are mobile. These carriers are accelerated by the electric field existing in the barrier layer of the semi conductor where it contacts the metal. Holes and elec trons move in opposite directions and give rise to the observed photovoltaic currents. In a number of cases, where the exciting light has energy greater than that of the band gap of the semiconductor, the above explana tion fits the experimental facts very well. In some systems, however, photovoltaic effects are produced with high efficiency by light whose energy is far less than that of the band gap. Two such systems which have been extensively investigated are the Cu Cu20 system and the Cu-CdS system.2 Photovoltaic cells utilizing these junctions may be constructed so that the light must traverse a considerable thickness of the semiconductor before reaching the junctions. This ge ometry is illustrated in Fig. 1. The dotted line is a 1 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, New York, 1948), 2nd ed., p.192. 2 D. C. Reynolds, G. Leies, L. L. Antes, and R. E. Marburger, Phys. Rev. %, 533 (1954); D. C. Reynolds and S. J. Czyzak, Phys. Rev. 96, 1705 (1954). A summary of the results for the Cu-Cu 20 system is given by V. K Zworykin and E. G. Ramberg, in Photoelectricity Oohn Wiley & Sons, Inc., New York, 1949). schematic indication of the semiconductor barrier layer. Under typical conditions this thickness is 1 p. or less.3 Only light of those wavelengths, which are not strongly absorbed, may reach the junction where the effect is produced. Light whose energy is as little as half that of the band gap is effective in both the foregoing systems. These facts raise two major problems in the use of the preceding interpretation of the photovoltaic effect for the description of the results in this part of the spectrum. The first of these may be illustrated by reference to Fig. 1. Only holes and electrons produced in or near the barrier layer are accelerated by the existing internal e~ectric field.to produce a net current with high efficiency. Smce the thIckness of the barrier layer is of the order of a micron, the total thickness of the semiconductor may be more than one thousand times this. For weakly BARRIER LAYER THICKNESS LIGHT I ! ! COPPER CONNECTIONS TO EXTERNAL CIRCUIT FIG. 1. Schematic construction of a typical Cu-CdS photovoltaic cell. S See footnote 1, p. 174. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPHOTOEMISSION IN CADMIUM SULFIDE CRYSTALS 969 absorbed light the absorption in any given layer is approximately proportional to the thickness of the layer. Thus only one-thousandth of the light would be ab sorbed in the barrier and this should set an upper limit of 0.001 for the quantum yield of photovoltaic current. The observed quantum yields may be as much as one hundred times larger. In order to account for the experi mental facts, it is necessary to assume that the absorp tion spectrum of the semiconductor in the barrier is several orders of magnitude greater than that in the bulk. Although there may be some basis for such an assumption in the case of a junction formed by diffusing a metal into a semiconductor, it does not appear appli cable to the experiments described in this paper with plated nondiffused metal layers. The second problem arises from the requirement that both the hole and the electron produced by light absorp tion must be mobile. When the energy of the light is less than that of the semiconductor band gap, then either the electron or the hole produced on absorption must be bound at a level in the forbidden energy gap where its mobility would be very low. This problem can be over come, at least in principle, by the postulation of the existence of an impurity band in which photo-excited holes could have sufficient mobility to produce the photovoltaic effect.2 Such a postulation may be relevant when a semiconductor is used with a high concentration of metallic impurity sufficient to cause impurity band ing, but in the experiments described in this paper it cannot be applied. It is the purpose of this paper to show that the resolu tion of these problems for undiffused plated metallic contacts on transparent semiconducting CdS crystals lies in the interpretation of the effects as resulting from the photoemission of electrons from the metal into the CdS crystal. Several properties of the system combine to make it a favorable one for the observation of this phenomenon. EXPERIMENTAL Most experiments were done with conducting CdS: I crystals grown by L. A. Barton. These had been grown from the vapor phase in an atmosphere containing a few mm partial pressure of iodine. The crystals are n-type semiconductors, with about 3X 1017 electrons cm-8, and with resistivities of 0.1 to 1 ohm cm. They are in the form of thin flat plates about 0.1 mm thick and 4X5 mm2 in area. Their optical properties are very similar to those of pure CdS and they are quite trans parent to wavelengths longer than 5200 A. Spectro graphic analysis of the crystals by H. H. Whitaker showed only the following metallic impurities in parts per million: Cu-3, Si-O.3, and Mg<O.1. Except for about 30 ppm of iodine, therefore, the crystals are relatively free from impurities. Photovoltaic cells were made with the geometry illustrated in Fig. 1. The metal layer, usually copper, was applied to one of the broad faces of the platelike crystal by electroplating. For this purpose crystals were cemented to a sheet of transparent vinylite l2 thick. Figure 2 shows a side view of the mounted crystal. A 0.050-in. diam indium dot was fused to the crystal before mounting by heating briefly to 180°C. A hole about the diameter of the crystal was drilled in the vinylite and the crystal was positioned so that its broad face covered the hole with the indium dot on the face opposite the vinylite. The mounted crystal was placed on a micro manipulator with the crystal on the bottom of the vinylite sheet. A drop of the plating solution was .then placed in the hole so that it covered the exposed portion of the upper surface of the crystal. Contact was made from the negative terminal of a battery to the indium dot and the positive connection was made to the drop of solution through an anode made of the metal being plated. Current densities of the order of 1 ma/cm2 were used to give a proper rate of deposition of the metal on the crystal. To plate crystals of high resistivity, a wire screen anode was used and the crystal was illuminated from above through the anode to give it the necessary conductivity. Silver paste was used to complete the external contacts to the indium dot and the metal layer. Silver and gold were plated from standard cyanide plating solutions obtained from the plating shop in this laboratory. Other metals were plated from solutions made up according to the recommendations· in the Plating and Finishing Guidebook.4 The area covered by the plated metal was typically 0.1 cm2. The nature and appearance of the electroplated cop per layer depended on the plating solution used. Layers deposited from copper cyanide solution were smooth and adherent with the bright metallic lustre character istic of clean copper. Layers plated from acidified copper sulfate solution were darker, less uniform, and less ad herent. Since a number of the experiments to be de scribed required uniform, adherent copper layers, most of the layers were plated from cyanide solution. How ever, somewhat higher photovoltaic yields have been obtained from cells on which the copper was plated from acid solution. Highly efficient photovoltaic cells, consisting of a junction formed by thermally diffusiug copper into CdS crystals, have been made by Reynolds and co-workers.2 In contrast to this method of preparation, the cells de scribed in this paper were prepared without any heat VINYLITE,,\ INDIUM CONTACT' ~HOLE FOR PLATING SOLUTION \CdS CRYSTAL FIG. 2. Schematic drawing of the mounting of crystal for electroplating. 4 Plating and Finishing Guidebook (Metal Industry Publishing Company, New York, 1946), 15th ed. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions970 R. WILLIAMS AND R. H. BUBE -04 0 +~4 Volts applied FIG. 3 (a). Current-voltage curve showing the rectifying proper ties of a Cu-CdS junction. The sign of the voltage is referred to the polarity of the metal side of the junction. treatment, unless otherwise specified. Plated layers of other metals, used in some of the experiments, were applied to crystals in the same way. ',~ To obtain spectral response curves, cells were illumi nated with monocrhomatic radiation from a Bausch and Lomb monochromator equipped with a tungsten light source. These were calibrated to give known relative output in photons/sec over the wavelength range used. Photovoltaic currents were measured with a Leeds and Northrup dc micromicroammeter. Absolute quantum yields were obtained using a calibrated tungsten lamp with known color temperature and controlled geometry, together with cells whose spectral response had been measured independently. RESULTS General Behavior The sign of the photo-emf produced by these cells is always such that the metal layer becomes positive. Thus the direction of electron flow within the cell is from the metal to the crystal. According to diode rectifier theory, the contact between an n type semiconductor and a metal of high work function should be a rectifying one. The current-voltage characteristic curve for a Cu-CdS photovoltaic cell is shown in Fig. 3 (a). Marked rectifica tion is shown and the easy current flow occurs when the metal is positive, which is in agreement with theory. Curvature of the energy bands of the semiconductor near the junction with the metal is responsible for rectification, according to theory. This is illustrated in Fig. 3(b). The electric field resulting from this curvature of the energy bands has the direction necessary to produce the observed sign of the photo-emf. Free elec- METAL -SEMICONDUCTOR INTERFACE -rrrr;..n-7i- __ ~_-_-_-_-_-_-J CON~~~bION -........>...>..l ......... ll.:>~} FILLED BAND FIG. 3(b). Configuration of energy bands at a metal-semiconductor junction. trons would be accelerated away from the metal while free holes would move toward the metal. In either case, the existence of excess free carriers in the barrier region would lead to an emf with the metal becoming positive. As is usual for photovoltaic cells, the open circuit emf increases linearly with light intensity at low intensities and tends toward a fixed value at high intensities. Values of 0.25 to 0.3 v were developed in the light of a focused microscope lamp. The saturation values are probably somewhat higher. Short circuit currents are proportional to light in tensity. Whenever we speak of quantum yield cf> in the following discussion, we mean the number of electrons measured in the external circuit for each quantum of light incident on the crystal. For typical good cells, cf> is about 0.1. Figure 4 shows cf> as a function of the wave length of the exciting light. The response is highest for light whose energy is smaller than that of the band gap and falls sharply for those wavelengths which are strongly absorbed by the crystal. Absorption spectra of the crystals show that there is not enough absorption of .10 .05 SPECTRAL RESPONSE OF Cu -CdS PHOTOVOLTAIC CELL q, IS THE NUMBER OF ELECTRONS FLOWING PER INCIDENT PHOTON 5000 6000 7000 .6000 9000 10000 11000 A{A) FIG. 4. Quantum yield of photo voltaic short circuit current as a function of the wavelength of the exciting lighL light in the bulk of the crystal to account for the photo voltaic current at long wavelengths. It may be inferred from the spectral response that the light producing the effect is absorbed either by the metal layer or by the semiconductor adjacent to the metal, whose optical properties might differ from those of the bulk material. Since no heat treatment or abrasive action was used in the preparation of the copper layers on the crystals, any diffusion of the metal or extensive surface damage is very unlikely. It is possible that a reaction with the surface layers of the semiconductor might occur during the electroplating process in which a layer of material would be formed between the bulk crystal and the metal layer. If this material had strong optical absorption in the range 5000 to 10 000 A, then it might give rise to the observed photovoltaic effect. Visual inspection of the copper layers plated onto crystals makes this seem rather unlikely since they have an appearance indis tinguishable from that of copper metal. Such evidence, however, is suggestive rather than definitive and more direct evidence on this question will be presented. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPHOTOEMISSION IN CADMIUM SULFIDE CRYSTALS 971 IRe suits with Layers of Different Metals on CdS Cells were constructed using CdS crystals with vari ous metals in place of the copper described above. These metals were also applied to the crystals by electroplat ing. To compare the properties of these the following were observed: the short circuit current per unit area of junction produced by room light; the open circuit voltage developed in the light of a focussed microscope illuminator; the current-voltage characteristic of the junction displayed on an oscilloscope. Results are sum marized in Table I. Because of the small currents in volved and other experimental difficulties the spectral response was obtained only for Cu and Au layers. These are plotted together for comparison, as a function of photon energy, in Fig. 5. The points of interest here are that the two curves are distinctly different and that the response of the gold layer begins its steep rise at higher photon energies that that of the copper. This is in agreement with the hypothesis that light absorbed by Metal Cu Fe As Au Cd Ag Ni TABLE 1. Properties of photovoltaic cells made by plating various metals onto CdS crystals. Electron Short circuit work current in Maximum Rectifying function room light open circuit properties of metal, JI. amp/em' emf, v of contact eva 20.00 0.3 good rectifier 4.45 0.08 0.Q2 good rectifier 4.49 0.02 0.5 good rectifier 0.13 0.3 good rectifier 4.89 0.06 very small nonrectifying 4.10 0.05 0.Q2 poor rectifier 4.45 very small very small poor rectifier 4.96 • Electron work functions are taken from a critical compilation made by G. Herrmann and S. Wagener in The Oxide Coated Cathode (Chapman and Hall. Ltd., London, England, 1951), Vol. 2. the metal layer produces the photovoltaic current. Strong optical absorption begins at longer wavelengths in copper than in gold (see Figs. 17 and 18). Thus the relationship between the curves shown in Fig. 5 may depend on the optical properties of the metals involved. The alternative hypothesis that the effect originates in an adjacent layer of modified CdS is not excluded by these considerations. Other metals are less effective than copper and gold as seen in Table I. It is evident that the observed differ ences cannot be explained by differences in the work functions of the metals though one might expect these to be a determining factor. None of the metals tried approaches copper in its usefulness for the construction of photovoltaic cells. Experiments with Semitransparent Metal Layers A definite answer to the question of whether the photovoltaic effects in the long wavelength range are produced by light absorbed in the metal or by light FIG. 5. Photovol taic current vs pho ton energy for a Cu-CdS cell and a Au-CdS cell. " , , I I I I I I I I I /\ ,/ Au-CdS .- absorbed in a layer of the semiconductor adjacent to the metal is provided by experiments with semitrans parent metal layers which will now be described. In the first of these experiments, a layer of copper was plated onto a CdS crystal and the plating was stopped at a point where the layer was thin enough to be semitransparent. The optical transmission spectrum of this layer is shown in Fig. 6 where the ratio of incident to transmitted light intensity is given as a function of wavelength. It is seen that the intensity of incident light is cut down by a factor ranging from 20 to 70 on passing through the layer of metal. Connections were made using the geometry illustrated in Fig. 1. The spectral response was now determined with light in cident on the crystal side of the cell as done in the fore going. It was then determined again on the same cell, but this time with the light incident on the opposite side of the cell, directly on the metal layer (from the bottom in Fig. 1). These two spectral response curves are com pared in Fig. 7. Arbitrary units are employed but the same units are used for each curve and the two curves may be quantitatively compared. When the cell is illuminated from the crystal side the response was simi lar to that in Fig. 4 as expected. When it was illuminated from the metal side the response was quite different. A new strong current appears from wavelengths shorter than that corresponding to the band gap. This is un doubtedly the result of the formation of hole-electron pairs in the barrier layer of the semiconductor by strongly absorbed light; the process discussed in the introduction. It is the response at longer wavelengths which is of principal interest here. At some particular 80r---'---~---'----r---, 60 FIG. 6. Ratio of incident light inten sity 10 to transmitted Jight intensity I for a thin la.yer of copper on a CdS crystal. 20 ~O~OO~~~~0~0~7~0~OO~.~80~O~0~~~OO~~'OOOO ),(A) Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions972 R. WILLIAMS AND R. H. BUBE I Z '" 4 a: a: ::>z uo !::b3 ::>:I: un. !!:IUZ ~~2 00 :I:Z ",- ~~I ~ LEGEND: A-LIGHT INCIDENT THROUGH CRYSTAL 0-LIGHT INCIDENT DIRECTLY ON METAL LAYER SIDE OF CELL '" 0 ~ .L~~~~~ __ ~,-~~~~~~~~~~ a: 3000 4000 5000 6000 7000 8000 9000 10000 11000 X(A) FIG. 7. Comparison of the spectral respu.nse curv~s obtained on illuminating a eu-CdS cell from two different sides. Equal light intensities were used in each case. wavelength, say 7000 A, the current produced when the cell is illuminated from the crystal side is 3.8 times as great as when it is illuminated with the same light in tensity but from the metal side. From Fig. 6 it can be seen that light of wavelength 7000 A is diminished in intensity by a factor of 55 on passing through the metal layer. If the light producing the photovoltaic current were absorbed in a zone of material lying between the metal layer and the bulk crystal, it would have to pass through the metal layer before producing its eff~ct in the case where the light enters from the metal sIde of the cell. Light entering from the crystal side of the cell would not have to pass through any regions of strong absorption to reach this hypothetical zone and would arrive with its intensity undiminished. As a result, the photovoltaic currents produced by equal illumination from the crystal side and from the metal side of the cell, respectively, should differ by a factor of about 55. Alternatively, if light absorbed by the metal produces the effect then this difference should be much smaller. For neither direction of illumination would there be a region of strong light absorption between the light source and the origin of the photovoltaic current. The observed difference, a factor of 3.8, strongly supports the contention that light absorbed by the metal layer, itself, produces the effect. Presumably photo-excited electrons are emitted from the metal into the CdS crystal. A second facet of the same experiment is illus trated by Fig. 8. Here the long wavelength parts of the LEGEND: X -POINTS FROM UPPER CURVE IN FIG.6. ALL VALUES OF CURRENT DIVIDED BY 3.75 0-POINTS FROM LOWER CURVE IN FIG.6 PLOTTED UNCHANGED FIG. 8. Comparison of the curves from Fig. 7 after normalization. curves from Fig. 7 have been replotted after dividing all values of current from the upper curve by 3.75. Both curves now coincide along their entire length and are thus related by a simple factor of proportionality which is independent of wavelength. The physical meaning is that illuminating the cell from the two different direc tions gives rise to two values of current which differ by a proportionality factor and that thi.s factor is the. same for all wavelengths. Referring to FIg. 6, the fractlOn of light transmitted by the metal layer varies by a factor of 3 between the wavelengths of 6000 and 8000 A. Consider first the case where the light is incident from the metal layer side of the cell. Again, if the light needed to pass through the metal layer to be effective then the photo voltaic current should be diminished 3 times as much at 8000 A as at 6000 A due to the light absorption of the metal layer. This effect would be absent when the cell was illuminated from the crystal side. Hence the spectral response curves obtained with the two different .dir~c tions of illumination could not be made to cOlllClde along their entire length by a single proportionality factor as is actually done in Fig. 8. This again can best be explained by assuming that light absorbed in the metal layer is producing the photovoltaic current. In this case variations of the optical density of the metal layer with wavelel1gth have the same effect regardless of the direction from which the exciting light is incident. A second experiment bearing on the same question is the following. A thin layer of copper was plated onto a CdS crystal. The layer was semitransparent, having an optical density of 0.4 at 7000 A. Microscopic examination of the layer showed it to be smooth, continuous, and unbroken. The photovoltaic response was observed with the light coming from the crystal side of the cell. Then the original layer of copper was made thicker by plating more copper on top of it until an optical density of 0.75 was reached. The photovoltaic response was observed again under the same conditions of incident light in tensity and geometry. Finally, more copper was plated on top of the layer until it became opaque and the response was observed for the third time. The same alternatives as before are considered. If the response was caused by light absorbed in a zone between the original metal layer and the bulk crystal, then maki~g the metal layer thicker would not affect the photovoitaic response. On the other hand, if the effect originates in the metal layer, making the layer thicker should enhance the response. A separate experiment with a cell having a metal layer as thin as any used here showed the short circuit current to be directly proportional to light in tensity up to much higher intensities than were used in the measurements. At these intensities, the cells were carrying much higher currents than were observed in the measurements. Therefore, the resistance of the thin metal layer cannot be limiting the performance of the cell. The results are shown in Figs. 9(a) and 9(b). In 9(a) the three spectral response curves are plotted to gether. The photovoltaic current is considerably en- Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPHOTOEMISSION IN CADMIUM SULFIDE CRYSTALS 973 hanced by increasing the thickness of the copper layer, again strongly suggesting that light absorption by the metal produces the effect. There is no quantitative proportionality between the amount of light stopped by the metal and the enhancement of photovoltaic current between curves 1 and 2. This is not an objection to the interpretation of the experiment since the optical den sity of a thin metal film is a complicated combination of reflection and absorption, and there is no simple relation between the optical density and the amount of true absorption by the metal. Figure 9(b) compares more carefully the differences in the three spectral response curves from 9(a). The three curves have been normalized by multiplying two two of them by constant factors such that all coincide at their maxima. It is seen that there are small differences which are easily understood from the fact that light of different wavelengths penetrates different depths into the metal. The point of interest here is that making the layer thicker by adding more of the same metal makes only small changes in the spectral response. ~ :r Q. ~12. '" o U ~ 0: '" Q. 8 ... z '" 0: 0: :::> u ... 4 a 0: u .... 0: o 0 ~~--::-:!-::-:o--'---:::!-:::-::--'--=="'" ili 4000 ,8000 z o ~ Q. ~ 12 '" o U ~ 0: '" Q. 8 .... z W 0: a:: a t: 4 :::l U 0:: U .... 0:: AlA) Cal oo~~~~~~~~-,-~~~ Vi 4000 (b) FIG. 9(a). Response of a Cu-CdS photovoltaic cell as the copper layer is made successively thicker. (1) Optical density of cop per = 0.4. (2) Optical density of copper=0.75. (3) Opaque copper layer. (b) Curves from Fig. 9(a) replotted after normalization to compare their shapes. Curve 3 replotted unchanged. Curves 1 and 2 have been multiplied by constant factors such that all three curves coincide at their maxima. ~ 1 1 1 FIG. 10. Construction of photovoltaic cell with successive layers of two different metals. The final experiment in this vein was the preparation of cells with successive layers of two different metals and observation of the response at each stage, illuminat ing the cell from the crystal side. Some initial experi ments were done by plating semitransparent layers of silver onto CdS, followed by a thicker layer of copper on top of the silver. The short circuit currents produced by exposure to a tungsten lamp of constant intensity were compared for the cell having the silver layer alone and for the finished cell having both layers together. Several such cells were constructed and in each case the response was enhanced by adding the copper layer on top of the silver layer. The factor by which the short circuit current was enhanced ranged from 2 to 4. These cells gave rather small total currents and it was not possible to obtain their spectral response. A more definitive experiment was done in which a semitransparent gold layer (optical density of 0.8 at 7000 A) was first plated onto CdS. This cell gave enough current to allow the spectral response to be recorded with the light incident from the crystal side. Then a thicker layer of copper was plated over the top of the gold. The final cell is illustrated schematically in Fig. 10. After the copper layer had been added, the spectral response of the cell was again determined. Results are in Fig. 11. Again the parts of the curves lying at wave lengths greater than 5200 A are of primary interest. The z ~ o :r Q. .... 2 z w o l3 z 0: w "- f- Z '" ~I a 6000 LEGEND, 1-PHOTOVOLTAIC RESPONSE OF CELL WITH SEMITRANSPARENT Au LAYER 2-RESPONSE OF THE SAME CEl AFTERCu WAS PLATED OVER THE Au lAYER. VALUES OF THE OBSERVED CURRENT WERE All DIVIDED BY 10 TO OBTAIN CURVE 2 BOOO 10000 FIG. 11. Spectral response of cell made by plating copper over a thin layer of gold on CdS. The curves compare the cell having only the gold layer with that having the combined layer with copper over gold. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions974 R. WILLIAMS AND R. H. BUBE spectral response for the gold layer alone is like that shown earlier for a gold layer in Fig. 5. Addition of the copper layer over the top of the gold layer increases the currents by a large factor. (Note that all the values of current from the curve giving the response of the com bined layer have been divided by ten). Also the spectral distribution has been greatly changed and is now that which is characteristic for copper. The only explanation of the experiment seems to be that in the final cell, light passes completely through the gold layer, is absorbed by the copper and produces excited electrons which diffuse back through the gold layer and are emitted into the CdS crystal. Here again the photovoltaic current appears to be caused by photoemission of electrons from the metal into the crystal, but here the electrons are excited in the metal layer furthest from the crystal. The short circuit current per unit light intensity for the cell with combined Cu-Au layer is about i that of the cells made with a Cu layer alone. Experiments with Photoconducting CdS Crystals Some further experiments will now be described to show how the properties of the CdS crystal can affect the behavior of photovoltaic cells. The effect of varia tions in the resistivity was investigated by constructing a photovoltaic cell froni a crystal of high gain photo conducting CdS. A layer of copper was plated on the crystal, which was illuminated during the plating opera tion to increase its conductivity. The geometry of the finished cell was that shown in Fig. 1 and its spectral response was the same as that shown in Fig. 4. It was a -' !!:!3 > ::!E ~ ~2 :l a .... >1 5 .... a: 0.L--J......J'--'--L~-L.-L--'---'-:1:-""---'--''--'~ '1 o 5 10 15.10 CONDUCTIVITYCOHM"CM-'j (a) BOTTOM OF CONDUCTION BAND. / 107 Sl-CM RES I STiVITY " If I Sl-CM RESISTIVITY r ',------------77117'l1i7'l1l7'lJl7'lJl7'l)}7'liimr ----.-ELECTRON FERMI LEVEL \ TOP OF VALENCE BAND METAL , , ..... _----------- CdS (b) FIG. 12(a). Changes in the quantum yield of photovoltaic cur rent of a Cu-CdS cell as the conductivity of the CdS changes. (b) Theoretically expected changes in a metal-CdS barrier region as CdS resistivity changes. illuminated with the light of a tungsten lamp and vari ous known relative intensities were obtained by inter posing calibrated wire screens between the lamp and the cell. The resistance of the cell and the photovoltaic short circuit current were measured for each value of the intensity. It was possible to obtain data over a range of crystal resistivity from 105 to 108 ohm-cm. In Fig. 12 the relative quantum yield of photovoltaic current is given as a function of the conductivity of the CdS crystal. A sharp drop in the quantum yield occurs when the conductivity drops below 10-7 ohm-1 cm-1• This may be caused by changes in the metal-semiconductor barrier brought about by changes in the density of free carriers in the semiconductor. In the low resistivity crystals used earlier, the carrier density is high and the energy of the electron Fermi level lies near that of the bottom of the conduction band, as shown in Fig. 3(b). The difference between the electron Fermi energy EF and the energy Ec of the bottom of the conduction band is given by the equation EF=Ec-kT In(N cln), where n is the density of free carriers and N c is the density of. states in the conduction band and n«N c. As n becomes smaller the Fermi level becomes further removed from the bottom of the conduction band. The Fermi level lies 0.4 v further from the bottom of the conduction band in a crystal having 107 ohm-cm re sistivity than it does in a crystal having 1 ohm-cm re sistivity. If the difference between the work function of copper and the electron affinity of CdS were around 0.4 v then this displacement of the Fermi level with increasing resistivity would ultimately remove the curvature of the bands near the junction which is shown in Fig. 3. The resulting configuration of the bands is in Fig. 12(a). At this stage there would be no strong field to accelerate electrons emitted from the metal and the quantum yield of photovoltaic current would be expected to drop. This interpretation suggests that the height of the metal-semiconductor barrier is around 0.4 v. Experiments with Pure CdS Crystals To observe the effect in a crystal of still higher re sistivity a cell was made by plating copper onto a crystal of very pure insulating CdS whose dark resistance was greater than 1011 ohms. Such crystals are normally rela tively 'free of defects and traps (no spectrographically detectable impurities; trap density about 1013 cm-3),5 and their photoconductivity gain is about unity. Thus in light of reasonable intensity their resistivity is still very high. Figure 13(a) shows the spectral response of this cell which was constructed with the geometry shown in Fig. 1. There is no photovoltaic current at longer wavelengths (for the same reasons given in the previous section), but for wavelengths shorter than that corre- DR. H. Bube and L. A. Barton, R. C. A. Rev. 20, 1959. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPHOTOEMISSION IN CADMIUM SULFIDE CRYSTALS 975 6r----r----.---~----. ~O~O~O--~----~~O~O---3--~6~OOO >.($.) (a) (b) FIG. 13(a). Spectral response of Cu-CdS cell made from in sulating CdS crystal. (b) Configuration of the energy bands of an insulating crystal making contact on opposite faces to two metals having different work functions. sponding to the band gap a response occurs. This is very likely because of the creation of hole-electron pairs by light absorption in the CdS and the subsequent motion of both types of carrier through the crystal. The sign of the photo-emf (0.3 v) is the same as that observed in cells made from low resistance crystals with the copper metal electrode becoming positive under illumination. The assumption of hole motion in pure CdS crystals is sup ported by experimental evidence,6 and in an insulating crystal making contact to two different metals a slope of the energy bands may extend throughout the crystal as shown in Fig. 13(b). Apparently, the motion of holes and electrons through the crystal under these circum stances occurs with considerably greater efficiency than photoemission of electrons into the crystal. The results suggest that a smaller accelerating field suffices to move carriers through the crystal efficiently than is necessary to move electrons into the crystal by the photoemission process. Effect of Heating Layer One experiment was done in order to determine the effect on a Cu-CdS cell of heating to a temperature where a slight diffusion of copper into the CdS occurs. A copper layer was plated onto one of the conducting CdS 6 H. S. Sommers, R. E. Berry, and I. Sochard, Phys. Rev. 101, 987 (1956). crystals described at the beginning of the report The photovoltaic yield and spectral response of this cell were determined. After this the cell was heated to 380°C for 15 sec, a treatment reported to cause slight diffusion of copper into CdS:7 Then the spectral response was d~ termined again. As seen in Fig. 14, the diffusion of copper into the CdS greatly lowers the photovoltaic response and changes its spectral distribution. The re sult indicates that the heat treatment produces a layer of high resistance photo conducting CdS next to the metal and that the photovoltaic effect is then dominated by the properties of this layer. Measurement of the cell resistance showed that there was a large increase in resistance during the heat treatment and the i-v charac teristic indicated the existence of a photoconductirtg layer in the CdS. Experiments Using Other Materials Photovoltaic cells with CdSe crystals in place of CdS have been made using both copper and gold as the barrier electrode. The CdSe crystals contain incorpo rated iodine, show n type conductivity and have a band gap of 1.74 ev.8 Thus, only infrared radiation can pass through the crystal without absorption and the photo voltaic response is confined to this region of the spec trum. Spectral response curves are very similar to those obtained with CdS, rising gradually with decreasing wavelength and cutting off sharply when the rvavelength corresponding to the band gap is reached. Apparently, the fundamental nature of the effect is exactly the same as in the CdS crystals. Attempts to make photovoltaic cells by plating metals onto conducting ZnO crystals9 were not successful. Copper and gold electrodes applied in this way make J' " I' , I r I I \ I ' .... o J ...... ___ 4000 6000 LEGEND: -BEFORE HEAT TREATMENT ---AFTER HEAT TREATMENT .8000 ).,(A) FIG. 14. Change in the properties of a Cu-CdS photovoltaic cell caused by heat treatment. Both curves have the same vertical scale. 1 D. A. Hammond, F. A. Shirland, and R. J. Baughman, WADC Tech. Rept. 57-770. B R. H. Bube, Proc. I.R.E. 43, 1836 (1955). 9 A generous supply of conducting ZnO crystals was provided by the New Jersey Zinc Company. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions976 R. WILLIAMS AND R. H. BUBE LEGEND, CELL WITH COPPER PLATED FROM Cu SO. SOLUTION CELL WITH COPPER PLATED FROM COPPER CYANIDE SOLUTION FIG. 15. Spectral response curves for Cu-CdS photovoltaic cells in which the copper was applied by electroplating from different plating solutions. ohmic contact to the ZnO, and no useful photovoltaic effect can occur at an ohmic contact. It was mentioned earlier that the character of the copper deposit obtained by electroplating onto CdS crystals depends on the plating solution used. The bright adherent layers plated from copper cyanide solu tion are most useful for the type of experiments reported here. Layers plated from copper sulfate solution are darker in appearance and less adherent. The spectral response of a CdS cell with such a copper layer is shown in Fig. 15. Though the general features are the same as those shown in Fig. 4, the relative response in the region from 6000 to 9000 A is enhanced. It is possible that this is simply caused by the obvious difference in the optical properties of the copper deposited in this way. The more fundamental question of what makes the optical prop erties of the copper different is one to which there is no ready answer. DISCUSSION The evidence given above is best interpreted to mea,n that the observed photovoltaic effects are the result of the photo emission of electrons from a metal into an adjacent semiconductor. This picture naturally invites comparison with the phenomenon of photoemission of electrons from metals into vacuum. Recent experiments by Thomas and MayerlO show that the photoelectric effect in potassium is a volume effect. This is in contradiction to the earlier picture of photoemission being essentially a surface effect.!l The experiments of Thomas and Mayer show that light absorbed within the volume of the metal is responsible for the emitted electrons. They were able to demonstrate this by measuring the quantum yield of photoemission at a number of wavelengths for thin layers of metal having various known thicknesses. It was shown that electrons excited within the metal at distances as great as 1000 A from the surface may reach the surface and 10 H. Thomas, Z. Physik 147, 395 (1957); H. Mayer and H. Thomas, Z. Physik 147, 419 (1957). 11 A. L. Hughes and L. A. DuBridge, Photoelectric Phenonema (McGraw-Hill Book Company, Inc., New York, 1932). emerge into the vacuum. The simplest kind of strong light absorption which can occur for visible and ultra violet frequencies in the interior of an alkali metal is a band-to-band transition in which a conduction electron is raised to a state in the next higher unfilled band. The assumption of this kind of transition enabled Mayer and Thomas to account satisfactorily for the wavelength dependence of the photoelectric quantum yield. For the present work the important thing is that in the only thoroughly investigated case the photoelectric effect is a volume effect and that electrons excited at a considerable distance from the surface are able to reach the surface and escape from the metal. Photovoltaic results obtained here will now be com pared with this picture of the photoelectric effect. The most important difference between photoemission of electrons from a metal into an adjacent semiconductor and photoemission into vacuum is, of course, the fact that in the former case the potential barrier which the electron must surmount in order to escape is much smaller. To escape from the metal into vacuum the electron must have excess energy greater than the elec tron work function which is 4.5 ev for copper. To escape from the metal into an adjacent crystal, its excess energy need only be greater than the difference between the work function of the metal and the electron affinity of the crystal. Though the electron affinity of CdS is not accurately known this difference is very likely in the range 0.5 to 1.5 ev. Thus photoemission into the crystal might be stimulated by light having quite low energies. A discussion of this point has been given by Mott and Gurney (footnote 1, pp. 73-74) who were apparently the first to suggest that this kind of photoemission might occur. More recent evidence for the existence of the process has been given by Gilleo.12 Inspection of Fig. 4 3 2 o CIRCLES ON TH EORETICAL CURVE l SHOW EXPERIMENTAL DATA SHIFTED TO COINCIDE WITH THEORETICAL CURVE .I~~ 20 40 60 'X ' hv kT FIG. 16. Fowler plot of the photovoltaic currents near the long wavelength threshold for a Cu-CdS photovoltaic cell. 12 M. A. Gilleo, Phys. Rev. 91, 534 (1953). Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPHOTOEMISSION IN CADMIUM SULFIDE CRYSTALS 977 K .... Ag-x ..................... -"x - x .:;-._ -'x-x -x-x-x_x_ 2oo'~0--~~6~0~00~~--~8~0~00~~--~'0~000 AtAl FIG. 17. Fraction of light at normal incidence which enters metals. as a function of wavelength. Data for all metals except potasslUm taken from R. Wood, Physical Optics, Macmillan and Company, Ltd., London England, 1934), p. 550; data for potas sium from American Institute of Physics Handbook (McGraw-Hill Book Company, Inc., New York, 1957), p. 6-106. shows that the low energy threshold of the photovoltaic effect in the Cu-CdS system occurs for photon energies somewhere around 1 ev. A more quantitative determina tion of the threshold may be provided by a conventional Fowler plot of the photovoltaic currents near the long wavelength limit.l1 For a photoemission process the shape of the curve of quantum yield vs energy is, in the vicinity of the threshold, determined by the distribution of electrons near the Fermi surface. This allows a plot to be made which gives an objective value for the true threshold energy from data taken near the threshold. Such a plot of the data from Fig. 4 is shown in Fig. 16. The indicated threshold value is 1.1 ev. This value is in disagreement with the value of about 0.4 v obtained from the previous discussion of the results with photo conducting crystals. An accurate knowledge of the elec tron affinity of CdS would help to answer the question raised by this experiment. A second consideration is that of the optical properties of the metal involved. Clearly, light which is reflected from the surface can produce no photoeffect. Light which is not reflected enters the metal and is absorbed. The fraction of light at normal incidence which enters the metal is 1 minus the fraction reflected. This is shown as a function of wavelength for four metals in Fig. 17. It is clear that the spectral response curves for photo voltaic cells made with Cu and Au layers are in good agreement with the notion that light must enter into the volume of the metal to produce the effect. The optical absorption coefficient nx is also important here. Once light enters the metal, its intensity diminishes with distance d according to the equation -%nx 1= Ioe---d, ~o nx is the absorption coefficient and Ao is the vacuum wavelength of the light. A high-absorption coefficient leads to light absorption near the surface and the nearer an excited electron is to the surface the better are its chances of getting out. Hence a high value of nx favors a high photoemission efficiency. The absorption con stants for Cu, Ag, and Au are shown as a function of wavelength in Fig. 18, taken from the data of Joos and Klopfer.13 The strong absorption peak lying near 5000 A in the copper spectrum is assigned by these authors as a transition of a 3d electron to a 4s level, 3d104s -+ 3d94s2• It may be noted that the high values of the absorption coefficients of copper and gold between 5000 and 6000 A guarantee that there will be appreciable light absorption in the metal within a few hundred angstroms of the surface. Thus both the reflectivity and absorption data are in agreement with the spectral response data for the photovoltaic cells. The low absorption and high re flectivity of silver are also in agreement with its rather poor performance in the photovoltaic cells. In all these metals there is a weaker absorption extending into the near infrared which presumably arises from transitions in which momentum is conserved through the participa tion of a lattice vibration. Finally, a comparison must be made between the quantum yields obtained in the photovoltaic cells and those obtained for photoemission of electrons into vacuum. From Fig. 4 it is seen that in the photovoltaic cells there may be quantum yields as high as 0.15 elec trons per incident quantum. For photoemission from metals into vacuum the number of electrons per incident light quantum for good metals is likely to be around 0.005. It might appear from this comparison that the efficiency of the process considered here is 30 times as great as that for photoemission into vacuum. However, the comparison based on incident light is not the most instructive one because of the great differences in re flectivity among different metals. (It is given here be cause literature data are so frequently reported in this n~ 3r---'---'---'---~ 2 £hoo , x \ x ., X Ag 'x-x'" x-x-x-x-x-x" 3000 4000 ).(A> 5000 6000 FIG. 18. Optical absorption index for three metals. Data taken from footnote 12. 13 G. Joos and A. Klopfer, Z. Physik 138, 251 (1954). Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions978 R. WILLIAMS AND R. H. BUBE way.) A better quantity for comparison is the number of electrons per absorbed light quantum. This turns out to be 0.3 electrons/quantum absorbed for the Cu-CdS cells. For photoemission from potassium metal into vacuum the quantum yield reaches 0.08 electrons/ quan tum absorbed.10 Thus the quantum yields really differ only by a factor of 3.7 and the high efficiency of the photovoltaic cells is not inconsistent with previous data on the photoelectric effect. It is worthy of note that the quantum yield of 0.15, although somewhat larger than that expected for photo emission into vacuum, is also significantly smaller than that expected for a photovoltaic effect at a semicon ductor-semiconductor junction. Marburger et al.,14 for example, measured a quantum yield of unity on a photo voltaic cell made by diffusing copper into CdS. SUMMARY The general features of the behavior of undiffused metal-cadmium sulfide photovoltaic cells are under standable on the basis of a simple model. It has been demonstrated that photovoltaic currents are produced with high-quantum efficiency by the photoemission of electrons from a metal into the CdS crystal. The princi pal pieces of evidence are the following: 1. Comparison of the spectral response with the cell geometry and the optical properties of the CdS shows that the effect cannot be caused by light absorbed in the bulk CdS. 2. Rectification properties of the Cu-CdS junction show the existence of a typical metal-semiconductor barrier layer. 3. An experiment in which a cell made with a semi transparent layer of copper was illuminated from two different sides shows that the effect is produced by light absorbed in the metal layer. 14 R. E. Marburger, D. C. Reynolds, L. L. Antes, and R. S. Hogan, J. Chern. Phys. 23, 2448 (1955). 4. Experiments on cells constructed with several suc cessive thin layers of metal show that the quantum yield is increased when successive layers of metal are added to an initial layer of the same metal. At the same time only small changes in the spectral response occur. When a layer of copper is added to an initial thin layer of gold there is an increase in the quantum yield and there is also a large change in the spectral response of the cell. When only the gold layer is present the spectral response is that characteristic for gold and when the copper layer is added the spectral response changes to that character istic for copper. Thus the photovoltaic current may be strongly affected by light absorbed in layers of metal which are not immediately adjacent to the CdS crystal. 5. Optical properties of copper and gold agree well with the spectral responses characteristic of photovoltaic cells made with these metals. 6. Comparison with recent work on the photoemis sion of electrons from metals into vacuum shows that the mean free paths of excited electrons in metals are large enough to explain the results obtained here. The results suggest that the following features are important in selecting materials for metal-semiconductor photovoltaic cells of this type. The metal should have low reflectivity and a high absorption coefficient over the wavelength range to be covered. For the semicon ductor the important requirements are that it should be transparent, have high conductivity, make rectifying contact to the metal used and have an appropriate value of electron affinity. Apparently, the chemical properties of the semiconductor are not the dominating ones. ACKNOWLEDGMENTS We wish to thank D. O. North, A. Rose, and W. Spicer for several valuable discussions of this problem, and H. E. MacDonald for assisting in some of the measuremel\lts. Downloaded 31 May 2013 to 155.97.150.99. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.1723365.pdf

Simplified Theory of OneCarrier Currents with FieldDependent Mobilities Murray A. Lampert Citation: Journal of Applied Physics 29, 1082 (1958); doi: 10.1063/1.1723365 View online: http://dx.doi.org/10.1063/1.1723365 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/29/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Onecarrier thermally stimulated currents and spacechargelimited currents in naphthalene crystals J. Appl. Phys. 51, 1619 (1980); 10.1063/1.327766 Fielddependent mobility in liquid hydrocarbons J. Chem. Phys. 64, 1561 (1976); 10.1063/1.432379 Transient Photocurrent for FieldDependent Mobilities J. Appl. Phys. 43, 529 (1972); 10.1063/1.1661151 Theory of OneCarrier, SpaceChargeLimited Currents Including Diffusion and Trapping J. Appl. Phys. 35, 2971 (1964); 10.1063/1.1713140 OneCarrier SpaceChargeLimited Current in Solids J. Appl. Phys. 34, 809 (1963); 10.1063/1.1729542 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:511082 L. H. GERMER and are to be compared with values reported here of lX 10-14 cc/erg and 4Xlo--14 cc/erg. It is natural to wonder why the complicating factors of glow discharge, showering, and activation do not con fuse the results in the Holm and Holm data. The reason is to be found in the large currents, 1.02 to 21.2 amp, at which a steady arc of the inactive type without glow or shower is to be expected at breaking contacts. The data at these large currents on break can, in fact, be taken to supplement and extend the work of the present paper to currents much above the half-ampere range reported here. The current experiments on closure have indeed extended to 100 amp, but the break tests were much more restricted. Many other experimenters have also measured elec trode loss as contacts are pulled apart to break current in an inductive circuit.27-32 Very commonly they have • 7 W. G. Pfann (unpublished). • 8 A. L. Allen, Proc. Inst. Elec. Engrs. (London) 100, Pt. I, 158 (1953). JOURNAL OF APPLIED PHYSICS found net anode loss for low values of circuit inductance changing to net cathode loss at high values.27.28.30.31 In some cases the change has been reported at fairly high values of inductance for metals of high electrical con ductivity and at lower values for low conductivity metals and alloys.SJ.31 These observations are in qualita tive agreement with the work reported here, but in most cases quantitative comparison is impossible. ACKNOWLEDGMENTS This paper has benefited greatly from many discus sions with the author's colleagues of whom the following have made the greatest contributions: Mr. J. L. Smith, Dr. P. Kisliuk and Dr. W. S. Boyle, and Mr. R. H. Gumley. Most of the erosion measurements were carried out by Mr. J. W. Ammons. .. J. Warham, Proc. Inst. Elec. Engrs. (London) 100, Pt. I, 163 (1953) . 30 A. Keil and W. Merl, Z. Metallk. 48, 16 (1957) . 3) W. Merl, Elektrotech. Z. A77, 201 (1956). 32 W. B. Ittner, J. App!. Phys. 27, 382 (1956). VOLUME 29. NUMBER 7 JULY. 1958 Simplified Theory of One-Carrier Currents with Field-Dependent Mobilities MURRAY A. LAMPERT RCA Laboratories, Princeton, New Jersey (Received February 12, 1958) A general method is presented for the calculation of steady-state, one-carrier currents in nonmetallic solids where the mobility is field dependent. The analysis includes the effects of space charge and trapping. The essential mathematical step is the representation of the electric-field intensity as a polynomial in the drift velocity. Detailed applications are made to semiconductor and insulator problems. A new mobility func tion of convenient mathematical form is proposed for the case that the carrier has a constant mean free path. Some very general properties of the current flow are established by geometric analysis. I. INTRODUCTION FIELD dependence of the electron mobility in ger manium has been established by measurements of Ryder1 and more recently, of Gunn,2 Gibson and col laborators,S and Morgan.4 The dependence is com plicated in form and furthermore sets in at relatively low field strengths. In view of these results it would be desirable to extend the analysis of current-flow prob lems so as to include the field dependence of the carrier mobility. The necessity for such an extension has already appeared, for example, in Dacey's· study of one-carrier, space-charge-limited currents in germanium crystals with suitable junctions. In the present study we discuss a general method for the calculation of one-carrier currents where the IE. J. Ryder, Phys. Rev. 90, 766 (1953). Z J. B. Gunn, J. Electronics 2, 87 (1956). a. Arthur, Gibson, and Granville, J. Electronics 2, 145 (1956); b A. F. Gibson and J. W. Granville, J. Electronics 2,259 (1956). • T. N. Morgan, Bull. Am. Phys. Soc., Ser. II, 2, 266 (1957). • G. C. Dacey, Phys. Rev. 90, 759 (1953). mobility is field dependent. The key step in the method is the representation of the electric-field intensity as a polynomial in the drift velocity. We show that by this procedure a substantial class of one-carrier current problems are made analytically tractable. For the case that the polynomial is simply a quadratic it is shown that this leads to a new. mobility function which is appropriate for problems in which the carrier has a mean free path which is independent of its velocity. These results are applied, in some detail, to two simple, prototype problems which have already been studied in the literature, the perfect insulator and the effectively trap-free insulator or semiconductor. The report con cludes with a discussion of the limitations of the simpli fied theory herein presented. In the first appendix some additional problems of current flow are discussed very briefly. In the second appendix a geometric approach to the theory is out lined and the main results presented. Throughout this report a one-dimensional current [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:51ONE -CAR R I ERe U R R E N T S WIT H FIE L D -D E PEN DEN T MOB I LIT I E S 1083 How in a plane geometry is assumed. The units em ployed are mks except where otherwise stated. n. STATEMENT OF THE PROBLEMS A simplified theory of one-carrier,6 space-charge limited currents is based on a simplified current-How Eq. (1), the Poisson equation (2), on Eq. (3') which functionally relates the electric-field intensity and the carrier drift velocity, and on "equation of state" (4) relating the trapped-carrier density to the free-carrier density. J:::::enVa=constantj va=p.o E do --= (n-n)+ (nt-fit) e dx 0= O(Va) or Va= va( 0) nt=nt(n,o) in steady-state equilibrium. (1) (2) (3') (4) These equations are written with regard to the sign conventions, J=-]x, 8=-ox, and Vd=VdX, where x is a unit vector in the x direction; J is the current density, ] = I J j ; 8 is the electric-field intensity, 0= 181; 'lid is the electron drift velocity, Va= IVdj;}L is the electron mobility, defined by the second relation in (1) j e is the magnitude of the electronic charge; ds the static dielectric constant; n,nt are the densities of free and trapped electrons, respectively; they are func tions of position x. The trapped electrons are held in spatially localized states located energywise in the forbidden gap. n,nt are the values of n and nt, re spectively, in the bulk, neutral crystal in thermal and electrical equilibrium (no applied voltage). The theory defined by the Eqs. (1)-(4) is a simplified one in that the diffusive contribution to the current How has been neglected in (1). The physically correct solutions to the Eqs. (1)-(4) are determined by two boundary conditions. Assuming complete space-charge-limitation of the current, in jected via an ohmic contact located at x=O, the appro priate boundary condition is 8=0 at x=O. (5) For the second boundary condition it is convenient to regard the solid as a semiinfinite crystal, extending to x= 00, in which case the it and tit of (2) are taken at x= 00. The se~ond condition then is n(x)-m as x~oo. (6) For a :finite crystal, with the anode contact at x=a, we simply use the solutions for the semiinfinite crystal out to x=a. 'Since this study is concerned throughout with one-carrier currents, we have for the sake of definiteness, taken the carriers to be electrons. AU results obtained are, with appropriate changes of terminology and signs, valid also for hole currents. For further analytical work we find it convenient to express the functional relation between 0 and Vd, (3'), as a polynomial relationship: 0= 'l!a+~(Vd)2+~(Vd)3+ ... +_l_('IId)m+l. (3) }Lo 01}La 022 p.fj Omm p.o Here P.o is the "low-field," field-independent mobility and 01, 02, .. " Om are the coefficients of the expansion, all with the dimensions of electric-field intensity. In all that follows we assume that the Oi are constants, in dependent of the electron density n. For a nondegenerate semiconductor or insulator, under fairly general conditions, the "equation of state" (4), for carriers held in traps of density Nt located at energy Et, can be written nt=JtNli Jt={l+(C,,} Nt-I. (Cn) 2n f (7) Here N =Ne exp{ (Et-Ec)/kT}, where Ee is the energy of the conduction-band minima, k is Boltzmann's constant, T is the lattice temperature in degrees Kelvin, and No is the effective density of states7 in the conduction band. Cn is the probability per unit time per unit density of electrons of capture of an electron from the conduction band into an unoccupied trap. C,,=vO"n(V), where v is the velocity of the electron and 0",,('11) the cross section for its capture by the trap. (Cn) is the average of Cn over the velocity distribution of the electrons, and (cn) denotes the thermal-equilibrium value of {en}. The relation (7) follows from a straightforward treatmentS of the kinetics of trapping under the assump tion that (en), the probability per unit time of ejection of an electron from an occupied trap into the conduction band in the steady state, is unchanged from its thermal equilibrium value {en). For the case that {c,,)"""{c,,}, It reduces to the Fermi-Dirac occupation function referred to the steady-state electron Fermi level,9 and It depends explicitly on n but not on O. In Eq. (7) a statistical weight of the trapping state of 2 has been assumed, i.e., only spin degeneracy exists. If <Cn)~{Cn), then It can depend explicitly on 0 as well, through the factor (c,,)/{c n). m. GENERAL ANALYTICAL PROCEDURE In obtaining the solutions to Eqs. (1-4) it is con venient to define a dimension-less density variable u, distance variable w, and potential variable'll as follows: n u=-; n e'ln2/LoX W=--' v eJ' (8) 7 See, e.g., W. Shockley, Electrons and Holes in SemiconiluctllTs (D. Van Nostrand Compariy, Inc., New York, 1950), p. 240. 8 W. Shockley and W. T. Read, Jr., Phys. Rev. 87, 835 (1952). 9 For a discussion of the role of the steady-state Fermi level in space-charge-limited current flow problems see A. Rose, Phys. Rev. 97, 1538 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:511084 MURRAY A. LAMPERT The potential V in (8) is defined by V = fox 6dx. The polynomial relationship (3) is rewritten as 6= 6J{u+Blu2+B2u3+ .. ·+Bmum+1}, (9) with 6J=J/efip.o, Bl=6J/6l, B2= (6J/ 62)2, ... , Bm = (6J/ 6m)m. It is convenient to define the "equation-of-state" func tion K(u), henceforth referred to as the "kernel," by {(n ) (nt fit)l-l K(u) = ;;-1 + fi -fi f . (10) The kernels for a number of specific current-flow problems are listed in Appendix 1. With these definitions it is now easily seen that that mathematical solution to the problem is given im plicitly through the two quadratures; w= fU {1+2Blu+3B2u2+ ... + (m+l)B mum} ° XK(u)du (11) v= i" {1+Blu+B 2u2+ .. ·+Bmum} o X {1+2Blu+3B2u2+ ... + (m+1)Bmum} XuK(u)du. (12) The lower limits of integration in (11) and (12) are taken equal to zero, since the free-carrier density at the cathode must be infinite, as follows from (1) and (5). The ordinary low-field case, in which the mobility is independent of the field, is obtained from (11) and (12l.. by taking Bl:::=B2=·· . =Bm=O and by taking K(u)=K(u), where K(u) denotes the functional form for the kernel appropriate to "low fields" [that is, no 6 dependence of nt in (4), and (cn) = (cn) in (7)J: "low fields" w= iU K(u)du; v= i" uK(u)du. (13) ° 0 A one-carrier current-flow problem may be said to be "analytically tractable" if the quadratures of (11) and (12) for the field-dependent case, or of (13) for the field-independent, low-field case, may be explicitly carried out in terms of known functions. For the problems examined to date, in the low-field region, successful analytic evaluation of the integrals in (13) has, in ~ach case, proceeded from an expansion of the kernel K(u) in integrable, partial fractions. The major result, and indeed the motivation, of the present formulation is the following theorem, valid for the more general field-dependent case: If the kernel K (u) can be expanded in integrable partial fractions then the current-flow problem is analytically tractable. The proof is very simple and rests on the observation that if J K(u)du can be evaluated through an expansion of K(u) in integrable partial fractions, then so likewise can JuPK(u)du, with p any integer. The integrations in (11) and (12) can therefore be carried out in term-by term fashion. The kernels listed in Appendix I all are of a form so as to yield analytic tractability under quasi-thermal equilibri~m conditions, i.e., for (Cn)=(Cn) in (7) giving K(u)=K(u). We also indicate in Appendix I, for the various problems list~d, the extent to which K(u) may be generalized beyond K(u), i.e., the allowable functional form of (Cn)/(Cn) in (7), without loss of analytic tract ability for the field-dependent case. With analytic evaluation of the integrals in (11) and (12), wand v are obtained as explicit functions of u and the B's; w=!Jo(u)+2B l!Jl(U)+' .. + (m+ I)Bm!Jm(u) (14) v=Ao(u)+3BlAl(U)+2(Bl2+2B2)A2(U)+···. (15) For discussion of the current-voltage characteristic it is convenient to define dimensionless current and voltage parameters, "I and a, respectively: 1 El EoVa "1=-=--; a=va'Y2=--. Wa e2fi2p.oa efia2 (16) In (16), subscript "a" denotes the value of a quantity at the anode, x=a. Va is the applied voltage. Further, the "constants" Bl, B2, ••• are rewritten as Bl=Cl'Y with Cl=efia/ E6l; B2=Cd with C2= (mal E62)2, .... (17) Using (16) and (17), (14), and (15) may be written, at the anode, as 1!'Y=!Jo(u a)+Cl'Y!Jl(ua)+Cd!J2(u a)+,,' (18) a = 'Y2Ao(u a)+ 3CdAl(ua) +2(Cl2+2C2h4A2(Ua)+ .. ·. (19) To obtain current as a function of voltage, "I must be determined as a function of a. Explicit elimination of Ua from the solution, (18) and (19), is generally im possible, even for the low-field case. Therefore it is generally impossible to exhibit 'Y explicitly as a func tion of a, except over certain limited ranges of a. Where analytic elimination of Ua is impossible, an obvious graphical procedure may be employed In this section we have pursued the analytical devel opment of one-carrier current-flow theory. As previously shown,1° geometric methods can also be used to ad vantage in the study of these problems. In Appendix II we outline the geometric approach and present the results obtained from this line of reasoning. 10 M. A. Lampert, Phys. Rev. 103, 1648 (1956). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:51ONE -CAR R I ERe U R R E N T S WIT H FIE L D -D E PEN DEN T MOB I LIT I E S 1085 IV. QUADRATIC CASE The "quadratic case," defined by 8= Vd +~(Vd)2 p'o 81 p'o (20) merits special attention. First, it represents the simplest form of field-dependence of the mobility in the present mathematical framework. Further aspects of interest are brought out by solution of (20) for Vd, which gives for p. = Vd/ 8 : At low fields, (21) approximates to 8«81: P.=P.0(1-:). At high fields, (21) approximates to (22) (23) Regarding the leading terms of (22) and (23), we see that the mobility function (21) has the following properties: (i) At low fields it approaches a constant. (ii) At high fields it has a field-dependence appropri ate to a scattering mechanism which yields a mean-free path A independent of carrier velocityll (e.g., scattering by the acoustic modes of the lattice vibrations). Indeed, the leading term of (23) has already been used in one carrier current calculations, e.g., by Dacey.1i (iii) It smoothly connects the low and high-field regions. (iv) It will not impair the analytic tractability of many one-carrier problems which are tractable in the low-field, constant-mobility case. Therefore we are led to propose the form (21) as a useful mathematical representation of the mobility for constant-mean-free-path problems. The parameter 81 in (20)-(23) is a measure of the field strength at which the transition to substantial field dependence of the mobility takes place. A rough estimate for 81 is that field which, over a mean-free path, imparts an additional kinetic energy to the carrier equal to its thermal energy; &1~3kT /2eA. For scattering mechanisms that yield a constant mean-free path,12 A=4.1 X 10-10 (m*T /m)tp.o with IJ.0 in cm2/volt sec and T in degrees Kelvin. m* is an average effective mass for the carrier. Thus, 81~S.3X106(mT/300m*)lp.o-l. For sufficiently pure n-Ge, the experimental tempera- 11 W. Shockley, Bell System Tech. J. 30, 1006 (1951). 12 See reference 7, pp. 277 and 289. ture dependence18 of the mobility, namely Y-1.66, is close enough to y-t to expect that long-wavelength phonon scattering is the dominant mechanism deter mining the low-field electron mobility. Theoretical analysis14 confirms this. Therefore the preceding relation estimating 81 should be appropriate for this case. Taking m/m*=4 and p.o=3800 cm2/volt sec at 300oK, 81~2.8X108 volts/em at 300oK. Since there is no range of fields over which the measured drift velocity has a simple &t dependence,!" a precise comparison with experiment cannot be made. However, noting that Gunn's low-field data2 is exactly of the form (22), an empirical 81=S.6X10 8 volts/cm at 3000K is thereby determined. This is in good agreement with the pre ceding estimate. For p-Ge at 300oK, the preceding estimate for 81, taking m/m*=4 for the heavy holes and p.o=1800 cm2/ volt sec, gives 81~S.9X1OS volts/em, which is to be compared to an observed break in the Vd vs 8 curve14 at about 1000 volts/em. That our estimated 81 gives such poor agreement in this case is probably due to the fact that lattice scattering of holes is not characterized by a constant mean free path, as evidenced by the experi mental Y-2.8-temperature dependence16 of the hole mobility as well as by theoretical considerationsP Nonetheless the field dependence of the mobility of the holes in Ge at 3000K is empirically14 of the form (23) (leading term), and so (21) is a useful representation of the hole mobility, taking 01= lOS volt/cm. In the following sections we consider applications of the foregoing theory to two specific current-flow problems. V. THE PERFECT INSULATOR The problem of current flow in an insulator con taining neither traps nor free equilibrium-charge is the simplest possible one-carrier, space-charge-limited cur rent problem and the one that has received most attention in the literature. It has been studied by Mott and Gurney18 in its simplest form, by Shockley and Prim19 including the diffusive contribution to the current, and by Dacey" who omitted diffusion but in cluded field dependence of the mobility. The following presentation is a generalization of Dacey's solution and reduces to his, for the quadratic case, in the limit of high 13 F. Morin, Phys. Rev. 93, 62 (1953). 14 W. P. Dumke, Phys. Rev. 101, 531 (1956). This analysis shows that electron mobility is due primarily to the scattering by shears produced by long wavelength longitudinal and trans verse phonons. The magnitude of the mobility at 3000K is thereby predicted to within a factor of two. 15 Figure 3 of reference 3b and Fig. 3 of this paper. Absence of a range of fields over which VdCl' el is very likely related to the early onset, at 4X 1()3 volts/cm at 300oK, of saturation of the drift velocity. (See Sec. VII, ii). 16 F. J. Morin and J. P. Maita, Phys. Rev. 94, 1525 (1954). 17 H. Ehrenreich and A. W. Overhauser, Phys. Rev. 104, 331, 649 (1956). 18 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, New York, 1940), first edition, p.l72. 19 W. Shockley and R. C. Prim, Phys. Rev. 90, 753 (1953). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:511086 MURRAY A. LAMPERT fields. It has the additional property of covering the entire range of fields, from low to high, smoothly. The solution to this problem can be generally useful because it gives the limiting form approached by the solution to any injection problem when finally the injected free charge exceeds the injected trapped charge. The problem is characterized mathematically by Eq. (1), Eq. (2) with ii=iit=nt=O, and Eq. (3). Since there is neither neutralized free charge nor fixed excess charge in this problem we depart from the dimensionless variables u, w, v, and a of (8) and (16) and introduce in their place the dimensionless variables '1/, p, w, and {3 defined by: Ix Vd J p=--; '1/=--=-- E{J.o612 {J.o61 en{J.o61 (24) IV Wa Va w=--; {3=-=-. E{J.o61S Pa a61 The expansion (3) is then rewritten as 8= 81{'1+'12+A27/S+A3'1/4+ ... +Am'lmH} (25) with A2=(6!/62)2, AS=(6!/6s)3, "', A",=(61/6",)"'. Following along similar lines as led to (14), (15), (18), and (19), we readily obtain p= !'12+i'ls+iA2'1/4+tAs'l6+ .. " (26) W=!'IS+h4+HA2+th5+HAs+A2h6+ ... , (27) !+ha+HA 2+tha2+ ... fJ='f/a---------t+ha+iA 2'1a2+ ... (28) To obtain I vs V we must obtain Pa as a function of {3. Generally this can be done only graphically, although over limited ranges of {3 analytic expressions are obtainable. Results are given here for the quadratic case of field dependence, Sec. IV [A2=As='" =Am=O in (25) (28)J. Analytic approximations are given for small and large {3, respectively. (3$O.3: Pa"-'(9/8)fJ2(1-i{3). (29) The first term alone is just Child's law for solids, J = 9E{J.O V N8as, corresponding to the low-field mobility relation, {J. = {J.o, independent of field. 2(5fJ)l[ 15( 3 )1J {3;;;:4: Pa"-'3"3 1-16 5fJ . (30) The first term is just Dacey's6 modification of Child's law: corresponding to the mobility relation, {J. = {J.o( 61/ 8)t. A graphical plot of Pa vs {3 in the intermediate region, 0.3<{3<10, is given in Fig. 1, together with plots of the approximations (29) and (30). As to be expected, the transition from the Child law to the Dacey-Child law occurs in the vicinity of p=1, or Va=a81• VI. EFFECTIVELY TRAP-FREE INSULATOR OR SEMICONDUCTOR (QUADRATIC CASE) In order to study theoretically departures from Ohm's law in crystals with ohmic contacts, one must employ a model which allows for the presence of free carriers in thermal equilibrium. The simplest model for this purpose is that of the effectively trap-free insulator or semiconductor, characterized by ii;060, n,=ii,=O in Eq. (2). The kernel for this problem is K(u)=u(l-u)-l. This problem has previously been studied10 for the case of a low-field, field-independent mobility. The dimensionless variables employed here are those of Sec. III. Here we consider specifically the quad ratic case of field dependence of the mobility: B2=Ba = ... =B",=O and C2=CS='" =Cm=O in (9)-(19). Performance of the integrations in (11) and (12) leads to the equations: a= -tC2-y4ua4-!DyS(2Dy+3)ua3 -h2(20y+ 1) (Dy+ 1)ua2_'Y2(2Dy+ 1) (C'Y+ l)u" 100 10 I O. 0.1 -'Y2(2C'Y+ l)(C,+ 1) In(1-u a) (33) €_i{~p)'Z , V~ 1/ , , j' /: /, >\ v.j' Po '~(iJl)3:2 V (I_~~) I 5 / 9 2 ~'i~ i~/ Q~ ~ -lll(l-iP) I II I I 10 FIG. 1. Theoretical current-voltage characteristic (solid line), in normalized variables, for the perfect insulator, for the "quad ratic case" of field dependence of the mobility: S == ('Od/ p.OJ + (1/ S\) X (Vd/P.O)·. The dashed curves correspond to high-and low-field approximations. p.=Ja/Ep.OSli and {J== V./aSI. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:51ONE -CAR R I ERe U R R E N T S WIT H FIE L D -D E PEN DEN T MOB I LIT IE S 1087 with C=C1=ena/e01. Equations (32) and (33) are the specific forms of Egs. (18) and (19), respectively, for the present problem. It is clearly impossible, through elimination of Ua from Egs. (32) and (33), to exhibit l' explicitly as an analytic function of a. However, since Eq. (32) is only quadratic in 1', there is no algebraic difficulty in elimi nating Ua graphically. The salient features of the re sultant solution for current J (or 1') as a function of applied voltage Va (or a) are exhibited in Fig. 2. These features are conveniently discussed in terms of two characteristic voltages, Vatr and Vacr, defined, re spectively, by (34) In terms of these voltages, C= VatT/Vacranda= Va/Vatr. In Fig. 2, the region to the left of Vatr corresponds to neutrality in the crystal, ua"",1, and the region to the right of V atr corresponds to injection of substantial space charge into the crystal, ua«1. It is seen that for C-::;, 1, Ohm's law is valid over the whole of the neu trality region, whereas for C> 1 there is departure from Ohm's law within this region, the linear J, V relation breaking over into a (V)! dependence in the neighbor hood of Vacr. For C;;::l, Dacey's law (31), giving a Vi dependence, holds throughout the injection region, whereas for C<l, Child's law for solids with its V2 dependence, holds in the voltage range Va tr < Va < Va cr, and Dacey's law (31) for Va> Va cr. L J »1\ 1/ / / VJ / i/ c1: / J V ~l~ c<~ II I ......... II / Vac , I / VIer / v.~c, Va : a / V II / / V ~ / '" V Va / IV v FIG. 2. Possible types of current-voltage characteristics for the effectively trap-free insulator or semiconductor for the "quadratic case" of field dependence of the mobility: s= (Vd/.uO) + (1/S1) X (Va/p.o)2. V4cr=aSl, V4tr=eiia2/., and C= V4tr/V4cr. 30~-1---+--~--+---r--1--~J+-~ 20r-~r-~---+---r---L--~+-1'~~ GUNN'S_I DATA .~ 1~~=+==4===~=+==~=tl~=+~ 8~~r-~---+---r---r-+1---;-~ ~7r--+---r--+-~r--+-+-r--+-~ d 6r-~~~---+---r--~~1---~~ 5r-~r-~---+---r--~7-1---~~ V~~·I+O.047V t -~~~~==l=~-~--t-~ '5- 2 4 5 6 7 8 -6 V=Vd xlO FIG. 3. Replot of Gunn's data2 on the field-dependence of the mobility in 2Ocm. n-Ge at room temperature, and polynomial approximations to the data. VII. LIMITATIONS OF THE THEORY The theory outlined has certain limitations which we now consider. (i) The theory as developed ignores all field effects other than the electrostatic, space-charge effects, Eq. (2) and the field dependence of the mobility, Eg. (3). Therefore such processes as impact ionization of localized states and avalanche breakdown across the band gap are assumed not to occur. Further, in deriving Eg. (7), a possible Stark shifting of the trap levels is ignored, as is also the possibility of field-induced emission of trapped carriers. (ii) In a range of fields where the drift velocity saturates, that is, becomes independent of field, our representation of the field as a polynomial in the drift velocity breaks down. This occurs in germanium, for example, as illustrated by a replot in Fig. 3 of Gunn's data2 for n-Ge at 300°1\.. The vertical segment, occurring between fields of 4X103 and 8XIQ3 volt/cm, is the saturation region. It is seen however that the data at lower fields, below 3000 volts/cm, can be closely ap proximated by a third-order polynomial, but is poorly represented by a linear form. (iii) It has been assumed in our polynomial repre sentation (3) of the field dependence of the mobility that the coefficients 0; are constants which are inde pendent of the free carrier density n. Actually, when the electron density is sufficiently large that the electrons exchange energy amongst themselves at a rate faster than they exchange energy with the lattice, the precise details of the "heating-up" of the electrons under an applied field will depend on the electron density. Such will be the case with many semiconductors, even in the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:511088 MURRAY A. LAMPERT intrinsic range, at room temperature.20 In this case, we may expect an n dependence of the parameters 0;, and average values for the 0; should be used in the calculations. (iv) For problems in which the dynamic equilibrium between free and trapped carriers plays a role, that is, in which Eq. (7) must be used, analytic tractability depends on the detailed functional form of the capture probability ratio (Cn)/(Cn), if tbis ratio departs signifi cantly from unity. Each case must then be individually examined, as is done for several special cases in Appendix I. (v) The neglect of diffusion currents in the simplified theory leads to an incorrect description of the details of current flow in the immediate vicinity of the cathode and anode contacts. Even in thermal equilibrium there will generally be a large diffusion current in such a region which exactly cancels the field-driven current. Under current-flow conditions (for currents substantially smaller than the temperature-limited current), each of these large currents in the contact region is modified only slightly, the measured current being just the differ ence between them. That the simplified theory in correctly describes the fields and carrier densities in the immediate vicinity of the cathode is also evident from the fact that the current-flow Eq. (1) and boundary condition (5) together require that n= 00 at the cathode interface. Therefore, generally speaking, the results calculated from the simplified theory will be quanti tatively useful only under the condition that the widths of the contact regions are not a significant fraction of the total crystal length. If this condition is satisfied then the errors in the calculated current-voltage rela tionship, introduced through the neglect of diffusion, are expectedlO to be correspondingly small. (vi) The details of the theory as presented here are based on the assumption of complete space-charge limi tation of the current, boundary condition (5). There is no difficulty in extending the theory to the case of partial space-charge limitation of the current, 0~0 at x=O. This extension is necessary, for example, in the study of the depletion of majority carriers. ACKNOWLEDGMENTS The author would like to express his particular in debtedness to Dr. H. S. Sommers, Jr., for many stimu lating discussions. He also obtained assistance and encouragement from conversations with Dr. W. J. Merz, Dr. D. O. North, and Dr. L. S. Nergaard. The earlier work of Dr. A. Rose has, at all times, served as a guide for the present studies. 1IO B. V. Paranjape, Proc. Phys. Soc. (London) B70, 628 (1957). APPENDIX I: KERNELS FOR ONE-CARRIER CURRENT-FLOW PROBLEMS Current-Flow Problem A. Effectively Trap Free Solid B. Solid with Fixed, Excess Charge C. Shallow Trap Level D. Deep Trap Level E. Single, Discrete Trap Level K(u) u 1-u u 1+u Ou g(u)-u ug(u) (1-u)g(u)+Lu{g(u)-u} u{g(u)+Su} Rug(u)+ (1-Tu){g(u)+Su} The kernel, K(u), is defined by (10). The trapped carrier density nt, appearing in (to), is given by (7). The total trap density is Nt. Further, we have taken (Cn)/(Cn)=g(u) = 1-Pu-QuL ... , so that nt/n=Rg(u) X{g(u)+SU}-l with R=Nt/n and S=N/2n. N=Nc Xexp{ (E,-Ec)/kT}. Also, 0=N/2N t and it is assumed that 0«1, L=NtN/2n2, and T=l+nt/n. Problem A is discussed, for the quadratic case of field dependence of the mobility, in Sec. VI. Problem B is the "diode" problem of semiconductor physics, previously studied for the case of low-field, field-independent mobility by Shockley and Prim13 and by Dacey5 employing the mobility function J.I.=J.l.o( 01/ 8)1. Here the fixed, excess charge is that of the impurity ions in the "punched-through" region between a pair of back-to-back junctions. Problem B is also the traps-filled-limit problem10 of insulator physics. Here the fixed, excess charge is that required to fill all traps on the insulator which are empty at thermo dynamic equilibrium. In problems A and B, the function g(u) does not appear explicitly in the kernel and therefore the analytic-tractability theorem of Sec. III is valid without qualification. In problems C, D, and E there is an equilibrium between free and trapped charge characterized in part, by the function g(u). This equilibrium is said to be "quasi-thermal" if g(u) = 1, in which case the steady state electron Fermi level alone determines the oc cupancy of the traps, as mentioned in Sec. II. Problem C refers to the case that the trap level is "shallow" energetically, that is, the trap is almost certainly empty, or nt«N t. From the form of the kernel it follows that this problem is analytically tractable if g(u) is not higher than quadratic in u. For quasi thermal equilibrium, the kernel differs from that of problem A only through the multiplicative constant O. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:51ONE -CAR R I ERe U R R E N T S WIT H FIE L 0 -0 E PEN 0 E N T MOB I LIT I E S 1089 In this case, after renormalization of the dimensionless variables, the solution to problem A can be used also for this problem. Problem D refers to the case that the trap level is "deep" energetically, that is, that the trap is almost certainly occupied, or Nt-nt«Nt. This problem is analytically tractable if g(u) is not higher than linear in u. For quasi-thermal equilibrium, the analytical work is simplified through the partial-fraction decom position of the kernel: K(u) = (L+l)-I{ (l-u)-L (1 +LU)-l}. This case is of particular interest for problems involving majority-carrier depletion, which has not been covered in the present report. Problem E has been previously studiedlO for the case of low-field, field-independent mobility and quasi thermal equilibrium. It is of interest because it furnishes the likeliest area for a detailed, quantitative check of the theory, namely through experiments with a solid controllably doped with a suitable impurity. The problem is analytically tractable if g(u) is not higher than linear in u. Problems C and D are just-special cases o(E. How ever these special cases are of particular interest in themselves, and further, represent considerable ana lytical simplifications of the more general case E. Therefore we have listed them separately. APPENDIX II. SOME GENERAL RESULTS BASED ON GEOMETRIC ARGUMENTS It is possible, by geometric reasoning, to obtain some quite general results if the two following assumptions are added to the basic equations of Sec. II: Vd( &) is a nondecreasing21 function of &, (Al) (cn) is a nonincreasing function of Vd. (A2) The importance of (A2) is that it insures that (Cn)/(Cn) in (7) is a nondecreasing function of Vd. By the same token, (A2) is essential for the following develop ments only for those cases where the equilibrium be tween free and trapped carriers is the dynamic one described by (7). From Eqs. (1), (2), and (7), boundary condition (6) and assumptions (Al) and (A2) follows: Theorem 1. In a crystal under injection,22 nand d&/dx are nonincreasing functions of distance from the ca thode, and & is a monotonic-increasing function. This theorem is readily proven by assuming it is untrue and then obtaining a contradiction. The details 21 By specifying that Vd be a non decreasing function of S, instead of simply a monotonic-increasing function of S, we thereby allow Va to saturate over one or more ranges of S, as observed in n-Ge (see reference 2, Fig. 2). 22 In a crystal under conditions of depletion of the majority carrier, the complementary theorem holds: nand de/dx are nondecreasing functions of distance from the cathode, and S is a monotonic-decreasing function. ~------------------------~~~ /-"" ./" ./' / ./' /" /" FIG. 4. Convexity of the electric-field-intensity distribution S and consequent inequalities. The straight-line segment in the plot of S vs x corresponds to a region of saturation of the drift velocity, va=constant, such as observed by Gunn2 for n-Ge. (See also Fig. 3.) The quantities under the geometric figures are the areas of the corresponding figures. of the argument parallel closely those of the "geometric" arguments of Sec. IV of reference 10, and will not be repeated here. From Theorem 1 follows directly: Theorem 2. There cannot exist simultaneously in the same crystal regions of injection and depletion; the entire crystal must be in either one condition or the other. We are further able to obtain a useful inequality: Theorem 3. In a crystal under injection, (A3) Here &0 and &a are the electric-field intensities at the cathode and anode, respectively, and &n is the "ohmic" field intensity, Va being the applied voltage and "a" the cathode-anode spacing. The proof follows from a comparison of three areas as in Fig. 4. This comparison yields directly the inequality, (&a+&0)/2<&n<&a, which is equivalent to (A3). The inequalities (A3) are a generalization of those given in (8) of reference 10, and reduce to them for com plete space-charge limitation, &0=0. There follows readily from theorem 3: Theorem 4. For &0=0, the capacitor relationship, Q = CVa, is valid within a factor of two: CVa<Q<2CV a. (A4) Here Q is the total excess charge in the crystal per unit area, Q= foaqdx, with q the excess charge density (free plus trapped charge), and C is the geometric capacitance per unit area C= e/a. (Al) and (A3) yield directly the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:511090 MURRAY A. LAMPERT inequalities J 1< Jo, (AS) where J=enavd(oa), na being the free electron density at the anode and Jo,=enavd(oO). For both the electrons2 and holes3 in germanium at room temperature the function Vd( 0)/ 0 is a nonincreas ing function of O. In this case, (AS) can be replaced by the inequalities, J 20a Vd(O) 1~-~---~2, for --nonincreasing. (A6) In oa+O< 0 Because of the inequalities (A6), an adequate esti mate of the current often is: J~O=enavd(oO). The form of this relation suggests the usefulness of con sidering separately two ranges of currents, a "neutrality" range over which the free carrier density does not depart substantially from its thermal equilibrium value, say n<na<2n, and a "space-charge-limited range" over which the injected free carriers overwhelm the ther mally generated ones, n> 2n. In these two ranges we have, respectively: n<na<2n: (A7) (A8) with l(Va) =na/(na+nt. a), the ratio of free, excess charge to total excess charge, both taken at the anode. From (A7) it is clear that in the neutrality range, the form of the dependence of J on Va is precisely that of 'lid on V .. /a. Although the ratio l(V .. ) in (A8) is generally a func tion of voltage, it is independent of voltage for the special case of shallow traps in quasi-thermal equilib rium with the free carriers (Appendix I, Case C). In writing (A8) we have used (A4) , taking ~CVa = EVa/a. Further, we have taken Q~ea(na+nt.a), thereby assuming that the excess charge densities at the anode do not differ greatly from the average excess charge densities in the crystal. Although inequalities of the type (A3), relating densities at the anode to average densities, cannot be rigorously established, nevertheless, in practice the errors incurred by the substitution will be small (see, for example, the discussion in Appendix A of reference 10). The voltage V,lr at which the transition takes place from the neutrality range of currents to the space-charge-limited range is given by (A9) Here nt. atr is the excess trapped charge density at the anode when na~2n, that is, under quasi-thermal equilibrium, when the steady-state Fermi level F at the anode has risen from its thermodynamic-equilibrium value F by the amount kT. Because the geometric approach rests on assumptions (Al) and (A2) it is important to explore the extent to which these assumptions are realized in practice. Over the range of fields for which drift-velocity measurements have been made to date, assumption (Al) has always been realized.23 As regards assumption (A2) there is as yet no de tailed quantitative experimental information available on the velocity-dependence of cross sections for the capture of free carriers by localized trapping states in solids. However we can use as a guide the analogous cross sections for the capture of free electrons by atoms and ions in vacuum. For the radiative capture of elec trons by ionized hydrogen atoms un"-''/J2 for E/E,«l, with E the electron kinetic energy and E; the ionization energy for atomic hydrogen.24 Theoretical calculations2~ indicate that this same 1/v2 dependence of the cross section also holds for both the radiative and non radiative (with single phonon emission) capture of free carriers by ionized, hydrogen-like impurity states in solids, under the same condition, E/E,«1. For these important cases assumption (A2) is realized. However for the capture of electrons by neutral hydrogen atoms in vacuum u(E) goes through a maximum at E"'"2/3 ev and for capture by neutral oxygen atoms u(E) goes through a minimum at E"'" 2 ev.24 In both of these cases assumption (A2) is violated over substantial ranges of electron velocity. Summing up, for capture of an electron by a Coulomb attractive center at room temperature, assumption (A2) is very likely correct. For capture by a neutral center, the likelihood is that (A2) is not valid over certain ranges of electron velocity. In any case, from the above remarks it is apparent that the extent of the validity of (A2) for solids is a subject requiring con siderable investiga tion. !3 The theoretical possibility of a negative, differential, bulk resistance at high fields has been pointed out by H. Kroemer, Z. Physik 134, 435 (1953). Such an effect, if present, might well be masked by other high field processes, for example field-or impact induced avalanches, which have not been incorporated into the present work. 24 H. S. W. Massey and E. H. S. Burhop, Electronic and Ionic Impact Phenomena (Oxford University Press, New York, 1952), Chap. VI. 25 M. A. Lampert reported in the final report of Contract II DA 36-039-sc-5548, in 1954 (Signal Corps the U. S. Army) j M. Lax and H. Gummel, Phys. Rev. 97, 1469 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.206.7.165 On: Mon, 24 Nov 2014 19:53:51

1.1722613.pdf

Evidence for Subgrains in MnBi Crystals from Bitter Patterns W. C. Ellis, H. J. Williams, and R. C. Sherwood Citation: Journal of Applied Physics 28, 1215 (1957); doi: 10.1063/1.1722613 View online: http://dx.doi.org/10.1063/1.1722613 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/28/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic properties of single-phase MnBi grown from MnBi49 melt J. Appl. Phys. 115, 17A752 (2014); 10.1063/1.4868204 Formation of MnBi ferromagnetic phases through crystallization of the amorphous phase J. Appl. Phys. 69, 6067 (1991); 10.1063/1.347771 Magnetic Properties of MnBi Single Crystals AIP Conf. Proc. 18, 1222 (1974); 10.1063/1.2947245 Flux Reversal in SingleCrystal MnBi Films J. Appl. Phys. 37, 1486 (1966); 10.1063/1.1708529 Growth of MnBi Crystals and Evidence for Subgrains from Domain Patterns J. Appl. Phys. 29, 534 (1958); 10.1063/1.1723213 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.17.127 On: Tue, 09 Dec 2014 14:56:58LETTERS TO THE EDITOR 1215 into the lower-lying new acceptor states to the extent their relative numbers permit. The thus-emptied donors are then available for trapping conduction electrons in n-type germanium where the initial Fermi level is high. Similarly, the thus-filled acceptors are then available for trapping holes in strongly p-type germanium where the initial Fermi level is low. In weaker p-type material where the initial Fermi level is higher, only the empty level in the lower middle of the gap will be active; it will tend to create some new holes and thus increase the conductivity . . \ccording to this model the "acceptors" described in the first paragraph are really ionized (empty) donors, and the "donors" are reallv ionized (filled) acceptors. The n'ew experimental evidence was developed from studies of the effect of small compressions on the electrical conductivity of germanium specimens of various initial conductivities. The compressions were carried out at 525°C in an inert atmosphere followed by cooling at about 100°C per minute. A standard schedule was always followed. Contamination of specimens was prevented by gold plating7 as demonstrated by the constant conductivity of control specimens which underwent the same thermal cycle. Figure 1 shows the changes in conductivity at room temperature produced by compressions ranging up to 5%. The scatter appears to derive from lack of homogeneity of compressions. The new tinding is the reversal in the behavior of p-type material: the conductivity increases with deformation in low-conductivity germanium and decreases in high-conductivity germanium. In the absence of any electrically active composition changes, these facts can be explained if both interstitials and vacancies are produced during deformation, but not if only one type of defect is produced. Thus, if only vacancies were produced, the number of conducting holes could only increase by ionization of the low-lying vacancy levels. The increase in carriers would then always dominate the decrease in mobility brought about by the new ionized centers,s and the conductivity would therefore increase no matter what the initial conductivity. On the other hand, if only interstitials were produced, they could always act as hole traps and would thus decrease both the number of carriers and the mobility, and the conductivity would decrease no matter what the initial conductivity. If both defects are present, the above model requires the conductivity to behave as is observed. A quantitative treatment of these effects is being worked out on the basis of the model described. It is believed that useful estimates of the numbers of vacancies and interstitials can be obtained separately from conductivity or Hall measurements at two or more temperatures. I C. J. Gallagher, Phys. Rev. 88. 721 (1952). 2 W. C. Ellis and E. S. Greiner, Phys. Rev. 92, 1061 (1953). 'A. G. Tweet, Phys. Rev. 99. 1245 (1955). 4 Greiner, Breidt, Hobstetter, and Ellis, J. Metals (to be published). 'H. M. James and K. Lark-Horovitz, Z. physik. Chern. 48, 107 (1951). 'Cleland, Crawford, and Pigg, Phys. Rev. 99, 1170 (1955). ; C. S. Fuller (private communication). See also R. S. Logan and M. Schwartz, J. Appl. Phys. 26. 1287 (1955). 'M. B. Prince, Ph},s. Rev. 92,681 (1953). Evidence for Subgrains in MnBi Crystals from Bitter Patterns W. C. ELLIS, H. J. WILLIAMS, AND R. C. SHERWOOD Bell Telephone Laboratories, Inc., Murray Hill, New Jersey (Received July 25, 1957) CRYSTALS of MnBi have been made by crystallization from a liquid solution in a temperature gradient. To prepare the crystals electrolytic manganese was continuously dissolved in high purity bismuth at 500°C (solubility -8 wt % Mnl) and crystallized as the compound at 300°C (solubility -1.5 wt % Mnl) in an evacuated Pyrex glass tube about one inch in diameter and ten inches in length. Crystal aggregates grew from the cold end and extended into the tube about four inches in the course C AXIS FIG. 1. Bitt(;'r pattern for an area of ~ ~urfa('e containing the c ~xis of a MnBi crystal. Magnetic field wa~ applif'd nmmal to !-1urfan' examined. of four to 5even days. In one instance, the aggregate was pre dominantly a single crystal; in another, three crystals. The crystals contained a free bismuth phase to the extent of about 18 wt %. A single crystal specimen, about OAXO.iX1.2 cm, cut from an aggregate of about 200 grams had a saturation magnetization (B-H) in the easy direction (parallel to c axis) of 7300 gauss. Allowing for the volume of free bismuth present, a saturation magnetization of approximately 8800 gauss was calculated for pure MnBi at room temperature. Laue reflections from a given plane, although quite sharp, had in some instances a distribution of about 2 deg. This spread indicated that the crystal was composed of subgrains, subcrystal FIG, 2. Pattern for another area of same surface as in Fig. 1. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.17.127 On: Tue, 09 Dec 2014 14:56:581216 LETTERS TO THE EDITOR regions differing by only a small angle in orientation. Bitter patterns2-4 on a surface containing the c axis, shown in Fig. 1, are interpreted as further evidence for the presence of subgrains. An external magnetic field, applied normal to the surface, resulted in the differential collection of colloid depicted. Portions of three subgrains are shown in the figure; the vertical traces are inter sections of subgrain boundaries with the surface of the crystal. The horizontal traces are intersections of domain walls with the surface. The magnetic domains extend along the c axis and across the three subgrains. Each magnetic domain consists of "sub domains" (three are shown for each domain in Fig. 1) because of the slight difference in orientation of the c axis in each subgrain. The c axis is the preferred direction of magnetization in MnBi which has a high uniaxial magnetic anisotropy. If the c axis is tilted up or down with respect to the surface, magnetic poles will be formed on the surface. The applied normal field, by either increasing or de creasing the local fields, causes some subdomains to attract more colloid than do others. This results in the checkerboard pattern which reverses when the applied field is reversed. With no applied field there is no checkerboard pattern, and only the horizontal domain boundaries can be seen extending completely across the figure. These domain boundaries move under the influence of high magnetic fields; however, the vertical traces due to subgrain boundaries do not. The immobility of vertical traces indicates that the associated boundaries are crystallographic. Figure 2 shows sub-boundaries on another portion of this crystal also with a normal applied field. The "spike" pattern at the sub-boundary trace near the center of the section has its origin in reverse domains caused by the presence of magnetic poles at the subboundary. Spike patterns also occur in the proximity of bismuth inclusions where the c axis intersects the inclusion. The curving lines extending in a generally vertical direction are fine cracks in the crystal which developed in the course of the experiments. 1 Seybolt, Hansen, Roberts, and Yurcisin. Trans. Am. Inst. Mining Met. Engrs. 206. 606 (t 956). 'F. Bitter. Phys. Rev. 38.1903 (1931). • W. C. Elmore and L. W. McKeehan. Trans. Am. lnst. Mining Met. Engrs. 120. 236 (1936). 'Williams. Bozarth. and Shockley. Ph)",. Re\". 75. 155 (\949). Addendum: Evaporation of Impurities from Semiconductors [J. App!. Phys. 28. 420 (1957)J KURT LEHOVEC. KURT SCHOENI. AND RAINER ZULEEG Sprague Electric Company. North Adams. Massachusetts IN connection with our above-mentioned paper, reference should have been made to the paper "Heat Treatment of Semi conductors and Contact Rectification" by B. Serin.' In this paper the hypothesis was advanced that heat treatment of impurity semiconductors may generate a depletion of impurities near the surface and thus influences the current voltage relationship and the capacitance of a metallic rectifying contact. The resulting impurity distribution is derived under assumptions identical with those leading to our Eq. (5). 1 B. Serin. Phys. Rev. 69. 357 (1946). Erratum: Electrical Conductivity of Fused Quartz D. App!. Phys. 28. 795 (1957)J JULIUS COHEN Physics Laboratory. Sylvania Electric Products. Inc .• Bayside, New York IN Fig. 3, I(d) should be equal to 1.1XlO-4 amp. Estimate of the Time Constant of Secondary Emission * A. VAN DER ZIEL Electrical Engineering Department, University of Minnesota, Minneapolis. Minnesota (Received July 31, 1957) IT is the aim of this note to show that energy considerations allow a simple estimate of the time constant 7' of secondary emission. To do so, the lattice electrons are divided into two groups: the unexcited or "normal" electrons and the "hot" electrons that have been excited by the primaries; part of the latter can escape and give rise to the observed secondary emission. The time constant 7' of secondary emission can now be defined as the time necessary to build up a steady-state distribution of "hot" electrons in the surface layer; since one "hole" is created for each hot electron, there is a corresponding steady-state distri bution of the holes, too. Let Jp be the primary electron current density, J.=oJp the secondary electron current density, where 0 is the secondary emission factor, and Epo the energy of the primary electrons. If N is the equilibrium number of hot electrons per cm2 of surface area and if E, and Eh are the average energies of the electrons and the holes, taken with respect to the bottom of the conduction band, then the total energy stored per cm2 surface area is The primary electrons 'deliver a power per cm' P=J "Epo=J.Ep%. (1) (2) If it is assumed that the primary electrons are 100%)ffective in the production of hot electrons, the value of 7' is (3) The problem is thus solved if the quantities N / J. and (E.+E h) can be calculated. This is not difficult, since it is known that the velocity distribution of the escaping secondaries is nearly Max wellian with a large equivalent temperature T.(kT.le~2-3 vJ. The hot electrons should therefore also have a Maxwellian distri bution with an equivalent temperature T.. Since the energy distribution of the secondaries depends very little upon the primary energy, it may be assumed that T, is independent of the primary energy and independent of the position in the lattice. Because of the interaction with the other electrons and with the lattice, the velocity distribution of the hot electrons should be isotropic in space. It is thus possible to calculate E. and to express J. and N in terms of the surface density no of the hot electrons. In metals one can only talk about "hot" electrons when their energy is above the Fermi level E[; in semiconductors and insulators their minimum energy is zero. Both cases can be considered simultaneously by defining a hot electron as an electron with a speed v~to with Vo= (2eEolm)t; one then has Eo=E/ for metals and Eo=O for semiconductors and insulators. Let n(x) be the density of the hot electrons at a depth x below the surface. If (vx,vy,v.) are their velocity components, their velocity distribution is dnx = Cn(x) (2trkT./m)-J exp(!mv2/kT,)dv xdvydv., (4) where V= (vl+vy2+vz2)! and the normalization factor C is defined such that fdnx=n(x) when the integration is carried out over all hot electrons. Let no and dno be the values of n(x) and dnx at the surface (x=O). If x is the electron affinity of the material then only those electrons at the surface can escape for which v.> (2ex/m)!. We thus have J.= fvxdno=eCno(kT./2rrm)! exp( -ex./kT.), (5) where the integration is carried out over all escaping electrons. C-1=2rr!q exp( -q2)+1-erf(q), (6) E.=C(kT /e){rr-'(2tf+3q) exp( -q2)+Kl-erf(q)J}, (7) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.17.127 On: Tue, 09 Dec 2014 14:56:58

1.1731676.pdf

Capacitive Energy and the Ionization of Aromatic Hydrocarbons Felix T. Smith Citation: The Journal of Chemical Physics 34, 793 (1961); doi: 10.1063/1.1731676 View online: http://dx.doi.org/10.1063/1.1731676 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/34/3?ver=pdfcov Published by the AIP Publishing Advertisement: This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20THE JOURNAL OF CHEMICAL PHYSICS VOLUME 34, NUMBER J MARCH, 1961 Capacitive Energy and the Ionization of Aromatic Hydrocarbons* FELIX T. SMITH Stanford Research Institute, Menlo Park, California (Received April 21, 1960) Large polynuclear aromatic molecules ionize like fragments of graphite, and their ionization potentials approach the sum of the graphite work function Vgr and the electrostatic work of charging a conductor the Slze and shape of the molecule. The capacitive work for a single charge is u=e2/2C, where C is the molecular capacitance. In the limit of large size, the work to remove n electrons is Wn=nvgr+n2u. This limiting law implies a relation among successive ionization potentials and the electron affinity, which is well satisfied by anthracene and phenanthrene, and moderately well by naphthalene and even benzene. For larger mole cules, C, and therefore u, can be estimated classically by assuming the conducting region is bounded by the carbon skeleton, and computed by analog measurements on metallic models; results are tabulated for a series of aromatics. The relation b~tween the limiting law and more detailed quantal formulations is dis cussed, and the magnitude and trend of some expected deviations are explored with the help of a simple free electron picture. INTRODUCTION THAT many properties of large polynuclear aro matics must approach those of graphite in the limit of large size is common knowledge. Among these properties are the ionization potential and the electron affinity, which are expected to approach the work function of graphite. Although the work function itself depends upon the inevitably quantal behavior of the 1r-electron gas, it may be looked upon as a macroscopic property of the material, measured by experiment, and as such, it gives a classical limit for the ionization potential and electron affinity of the large aromatics. So much is well known. It seems not to have been noticed that an improved classical limit can be found by considering the electrical properties of finite par ticles of graphite. My purpose in this paper is: to develop this new classical limit; to show that it leads (a) to a relationship between successive ionization potentials, and (b) to an approximate, classical calcu lation of the ionization potentials; to demonstrate that both (a) and (b) are in reasonable agreement with experiment; to provide predictions for other un measured molecules for comparison with future meas urements and with more detailed calculations; to esti mate roughly the points at which this classical limit must diverge from the correct microscopic quantal behavior; to show how this classical limit must ulti mately merge with a more detailed quantum me chanical formulation of the problem; and to point out that quantal calculations should be tested by seeing if they extrapolate properly to this classical limit. I. CAPACITIVE ENERGY. THE CLASSICAL LIMIT AND ITS CONSEQUENCES This approach originates from recent work on the ionization of solid conducting particles, where the potentials for successive levels of ionization differ from * Supported by U. S. Air Force, Air Research and'Develop ment Command. the work function of the solid in bulk by the classical electrostatic work required to charge up the particle. If the particle is a conductor of capacitance C, its cou lombic capacitive energy when it is charged to a level of n electronic charges is (ne)2/2C; to this must be added n times the work function v, which is the energy needed to raise an electron from the Fermi level in the lattice to a level where it could escape from the semi infinite solid, for which the coulombic charging energy vanishes. The total work to remove n electrons is then Wn=nv+ (ne)2/2C, (1) and the nth ionization potential, corresponding to the process A+n-4A+n+e, is1.2 CPn=v+ (n-!)e2/C. (2) If n<O, electron affinities are obtained. It is often assumed that the particles are spherical, in which case C=41rEor, where Eo=8.854·1Q-12 farad/meter is the dielectric constant of space. Another simple shape, which might be approached by graphite particles, is the thin circular disk, for which C=8Eor. It is generally accepted that the ionization potential and electron affinity of the polynuclear aromatics approach a common limit which is close to the work function of bulk graphite. This seems justified by the fact that the layers in bulk graphite are so far apart and so weakly bound (about 1.5 kcal or 0.065 ev per atom interlayer binding energy3) ; there is also evidence 1 F. T. Smith, J. Chern. Phys. 28, 746 (1958) and Proceedings of The Third Conference on Carbon (Pergamon Press, London, 1959), p. 419; see also T. M. Sugden and B. A. Thrush, Nature 168, 703 (1951). 2 H. Einbinder, J. Chern. Phys. 26, 948 (1957), interested pri marily in the average behavior of large highly charged particles, makes an approximation equivalent to neglecting a term e2/2C in Eq. (2); in this he is followed without comment by A. A. Arshinov and A. K. Musin, Doklady Akad. Nauk S.S.S.R. 118, 461 (1958), and Yu. S. Sayasov, ibid. 122, 848 (1958). 3 H. Inokuchi, S. Shiba, T. Handa, and H. Akamatsu, Bull. Chern. Soc. Japan 25, 299 (1952). 793 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20794 FELIX T. SMITH TABLE I. The relation between successive ionization potentials.a Estimated Experimental values Effective work triple Molecule Benzene Naphthalene Anthracene Phenanthrene Graphite WI (A->A+) 9.38 8.26 7.55 8.03 a All values in electron volts. W. (A->A++) 26.4 22.7 21.1 23.1 b Extrapolated value (reference 9). function W_I=-AI (A->A-) V21 Vl,_l (+0.54)b 5.55 4.42 -0.65 5.15 4.46 -1.19 4.55 4.37 -0.69 4.50 4.36 4.39 4.39 from the dielectric susceptibility that the interlayer effect on the conduction electrons is very small.' We may thus assume that large enough particles of a single graphitic layer will have successive ionization potentials governed by Eqs. (1) and (2) with'll approximated very closely by the work function of graphite, 'IIgr=4.39 ev.o This, then, is the limiting behavior that must be approached by the ionization potentials of large fused ring-aromatic molecules. Two consequences can be derived from Eqs. (1) and (2) which have predictive value and can be compared with experiment. These will now be examined in turn. A. Relationship between Successive Ionization Potentials Besides the work function, which is known, Eqs. (1) and (2) contain only one additional parameter, the capacitance C of the molecule or particle. H this is unknown, these equations nevertheless predict a relationship between successive ionization potentials for a single molecule. It will be informative to see how closely this limiting relationship is obeyed by the molecules for which measurements are available. Since the capacitance occurs inconveniently in the denominator, let us introduce the specific capacitive potential (or simply specific potential) by the equation u=e2/2C; with this, Eq. (1) becomes (3) (4) Eliminating u between equations of this form for two different levels of ionization, nand m, we can predict successive ionization potentials (or electron affinities) from the measured value of anyone of them. Equiva lently, from measured values W" and W m, we can • R. R. Haering and P. R. Wallace, Proceedings of the Third Conference on Carbon (Pergamon Press, London, 1959), p. 183. 6 A. Braun and G. Busch, Helv. Phys. Acta 20, 33 (1947). Effective specific potential ionization potential 'V2.-1 UIO U20 U_I,O 1~ Ul,_l 112._1 (limits) Wa 4.0 4.99 4.40 4.96 3.8 4.96 4.58 55.5:1::2.7 4.2 3.87 3.48 3.81 3.1 3.80 3.57 46.2:1::1.8 4.3 3.16 3.08 3.18 3.0 3.18 3.12 41.4±0.5 4.3 3.64 3.58 3.67 3.5 3.67 3.62 45.8±0.4 4.39 compute Vnm=[mn/(m-n)][(W .. /n2) -(W",/m2)]=vgr, (5) and test the predicted relation by seeing how close the values Vnm come to each other and to Vgr. On the other hand, eliminating v, we can also compute Unm= (n-m)-l[(W,,/n) -(W ... /m)], (6) and (7) and see, for a given molecule, how close these come to a common value u. Electron affinities can be used in the same comparison, since the electron affinity is A".=-W ......... (8) A special case of Eq. (5), derived long since on other grounds,S is the rule that the electron affinity and the first ionization potential tend to be equidistant from the work function of graphite, A1+W1=2'111._1:::::::2vgr. The recent experiments of Wacks and Dibeler7 make it possible for the first time to compare single and double ionization potentials for a series of polynuclear aromatics.s Their values of WI and W2 are used for such a comparison in Table I. In addition, I have used what seem to be the only available experimental data on electron affinities for these molecules, those of Blackedge and Hush, cited by Hush and Pople9 but apparently not yet published elsewhere; they seem to give a striking confirmation of the rule that Al and WI are balanced about the graphite work function. It is worthy of remark that '1121, depending on W2, also converges rapidly to the graphite value, the deviation being only 3% for anthracene and 2% for phenanth- 6 R. S. Mulliken, J. chim. phys. 46,497 (1949). 7 M. E. Wacks and V. H. Dibeler, J. Chern. Phys. 31, 1557 (1959). 8 For benzene, the double ionization potential was observed earlier by A. Hustrulid, P. Kusch and J. T. Tate, Phys. Rev. 54, 1037 (1938). 9 N. S. Hush and J. A. Pople, Trans. Faraday Soc. 51, 600 (1955). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20IONIZATION OF AROMATIC HYDROCARBONS 795 TABLE II. Ionization potentials and the classical specific potential. Classical specific Experimental Effective potential Corrected classical specific ionization specific potential potential (from metallic model) potential (Eq.39) Molecule Reference WI (A----A+) UIO=WI-V g, uo=t?/2C. Benzene Naphthalene Anthracene Phenanthrene Tetracene 1,2-Benzanthracene 3,4-Benzphenanthrene Pyrene Chrysene Perylene Coronene • 3.05 calculated for circular disk. 7 9.38 7 8.26 7 7.55 7 8.03 12,11 7.71,7.00 11 7.60 12 8.40 11 7.55 11 7.8 11 7.15 11 7.6 rene. One may confidently predict that the agreement will be even better for larger molecules, and not much worse for triple and higher ionization. Predicted values of W3, based on Eq. (7) and the spread of values of Uno, are given in Table I. B. Classical Estimates of Molecular-Capacitance If the molecule is large enough, it should be possible to estimate its capacitance, and thus the parameter u, classically. It seems reasonable to assume that the molecule behaves as a conducting plate, bounded by the peripheral carbon skeleton of the aromatic system. How small a particle can be treated by this model? The molecule coronene, C24H12, is sufficiently symmetric to be treated as a circular conducting disk bounded by the outer carbon atoms, for which r=3.70A; from this C=8Eor=2.62·1O-8 pp,j, u=e2/2C=3.0S ev, and W1=vgr+u=7.44 ev. Unfortunately the ionization potential of coronene has not been measured directly, but Matsen has derived a value of 7.S0 ev from a correlation of ionization potentials and electronic spectra10 while Briegleb and Czekalla have deduced 7.6 ev from charge-transfer spectra of complexes.l1 This agreement is sufficiently encouraging to prompt similar comparisons for other molecules, even though computation is not so easy for less symmetrical con figurations. However, numerical computation can be evaded by taking advantage of the fact that capaci tances scale just as a linear dimension. I have, there fore, resorted to measuring the capacitances of flat metallic models composed of hexagons three inches 10 F. A. Matsen, J. Chern. Phys. 24, 602 (1956). 11 G. Briegleb and J. Czekalla, Z. Elektrochem. 63, 6 (1959). 4.99 3.9 3.2 3.6 3.3,2.6 3.2 4.0 3.15 3.4 2.75 3.2 8.24 5.89 4.43 4.63 3.81 4.04 3.98 4.22 3.81 3.67 3.34-6.26 4.73 3.71 3.95 3.00 on an edge; details of the method and necessary cor rections are given in the Appendix. A comparison with available experimental values7•11,12 is given in Table II. The computed and measured values of u appear to be converging satisfactorily for the larger of the still small molecules for which they are known; beyond phenanthrene, a considerable error no doubt resides in much of the experimental data for WI. In general, Table II shows that the simple classical models predict values for u that come within about 30% of the experi mental ones; in terms of the total ionization potential WI the error is reduced to about lS%. Thus, the simple classical limit gives an excellent first approximation to these ionization potentials. In the expectation that further experimental data will improve the quality of the agreement, and perhaps reveal systematic effects worth more detailed study, results of the analog com putation with metallic models are given for an extensive series of molecules in Table III. II. MOLECULE AND SOLID THE MICROSCOPIC PICTURE A. Molecular Capacitive Energy How does the macroscopic, classical treatment of the last section tie in with a microscopic, quantum mechanical description of the molecular system? Consider the molecule as an array of N more or less graphitic carbon atoms, surrounded by other atoms such as hydrogen that need not be closely specified. The total energy of the molecule, or of any of its ions, in the lowest state, can be divided uniquely into kinetic and potential energies, plus magnetic and spin 12 D. P. Stevenson (unpublished). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20796 FELIX T. SMITH TABLE III. Specific potential computed classically.- Molec.ule b Uo Molecule b uo Molecule b Uo • 8.24 ~ 2.95 .... < 2.66 -5.89 ~ 2.95 ~ 3.41 --4:43 ~ 2.63 Y 2. T3 -3.81 V 3.98 I 2.48 "\ • • • • III 3.30 '\....J 3.33 ~ I 2.72 \ 4.63 \...-J 2.98 >-< 3.18 ~ 4.04 ~ 3.40 Z 3.67 ~ 3.32 ~ 2.96 l 3.22 .... \ 2.98 ~ 2.61 L 3.21 \ 3.26 ~ 2.60 L 3.01 ~ 2.96 V 3.27 .+. 2.% ~ 3.81 \..r 3.31 ~ 2.97 ~ 3.28 -0 3.53 x: 3.00 ~ 2.99 -( 3.85 V 2.46 \ ... \ 2.63 -( 3.41 L0y ~ 2.98 ~ 3.41 ---< 2.98 \1 4.22 uo ~ e I2Co' Co from measurement on metallic model. Units are electron valls. b A dot or the intersection of two lines represent a benzene hexagon, a line represents the junction of two hexagons through a common edge. MOleculeb ~ --+ + .-lSI .--LsI ~ ~ JI ~ ~~ :a' ~ 4 txl><I ~ <txt> uo Molecule b uo 3.68 ~ 2.51 3.22 ~ 2.53 3.23 ~ 2.60 3.56 IStSl 3.48 I 3.17 ....1S!r I 2,79 3.23 psi 2.64 2.56 « 3.53 3.25 ... ~ 3.10 2.56 i . f? · 2.80 2.86 I IA 3.34 I 2.95 /Sl5lSl 3.01 3.23 ® 3.34 I 3.23 Xl 3.00 I 2.79 ®r 2.84 2.98 @ 2.73 3.05 «K'V 2.28 I ylJJA 2.54 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20IONIZATION OF AROMATIC HYDROCARBONS 797 terms that can be neglected here: Eo= To+Uo+··· E+n= T+n+U+ n+···. (9) Sin~e we need not actually compute them, we may envIsage T and U as average values obtained by ap plying the appropriate quantal operator to the com plete, accurate, wave function for the ground state of the molecule or ion. The potential energy U +n includes the mutual repulsion of all the nuclei as well as the electron interactions, so it includes all the energy of the electrostatic field. The nfold ionization potential is given by W+n=E+n-Eo = (T+n-To)+(U+n-U O)+· .• , (10) and the potential energy contribution (U +n -Uo) includes the nuclear repulsion as well as their attrac tion for the electrons. Let us now compare this with the description of a large flat graphitic sheet, in which the total energy per atom is Es= T.+U.+···. (11) The work function, the energy required to remove an electron from the semi-infinite two-dimensional solid, can be formally defined by taking the limit of Eq. (10) as the number of atoms in the particle goes to infinity (in such a way that the size of the particle grows as N4 in 2 perpendicular directions) : v= lim W+1(N) = Tv+U v+···. (12) N_ro It is useful to consider the difference function defined by (13) This includes all effects due to the edge of the molecule (including the nongraphitic border atoms), and also the energy of compressing the excess positive charge into the region of the molecule. The effects around the edge (such as (]' bonds to the bordering H atoms) will in large part cancel out when we subtract 00 from O+n to construct the quantity (14) Let us now look at the electrostatic energy contribu tion to r n,O. It is convenient to use units such that 47rEo= e= 1. If the total electron density at any point r, when the molecule is in the +n ionized state, is P+n(r) and the nuclei are fixed at the positions r i (their zero-point vibrations are unimportant in this context), the potential energy of the system is U -~ qiqj ~ jP+n(r)d3r +n-L.... ~-L....qi . "-<·2r·· . 1 r-r'l t,J7"'"""'i. tJ t t +(1_~1_) ({P+n(r)P+n(r')d3rd3r' M-n JJ 21 r-r' 1 +l+n, (15) where M is the total number of electrons in the neutral molecule, and 1 +n is a correlation term which com pensates for the use of the product of densities P+n(r) P+n(r') in the double integral instead of the correct correlated double density per, r'). We can define the quantity which obeys the condition (17) nS+n(r) is the distribution of electron holes, by com parison with the neutral molecule. Subtracting Uo from U +n, we find =njs+n(r)[ ~ 1 r~iri 1-( 1-M~JjPlo~~:,r;}3r n ({po(r) po(r')d3raar' M(M-n)}} 21 r-r' I + 2(1-_-.!_~_)lyS+n (r) S+n (r') d3rd3r' ( _ ) n M 21' 1 + l+n 10• ~-n r-r (18) In the limit of large N the first term in (18) becomes the potential energy contribution to the work function, Uv, the second vanishes as M-!, and the third gives the classical capacitive energy. The correlation term (l+n-10) may also contribute to Uv and to the double integral that becomes the capacitive energy. However, its contribution to the latter integral is tantamount to a correlation of the holes represented by the distribution S+n(r), and must become insignificant as the density of holes diminishes with the increase in N. For any ionized system, small or large, a capacitive energy can be defined by the third term in (18), that is, by inserting the distribution of holes nS+n(r) in the classical integral for the electrostatic energy of the sys tem. This will make it possible to compare the results of quantum mechanical calculations with the simple classical limit, n2u=n2(e2j2C). Obviously, the distri bution of holes S+n will tend to become independent of n and will closely approach the classical distribution of charge, in order to minimize the total energy of the system. B. The Ionization Potential as a Power Series The character of the limiting law can be seen more clearly if one expands W +n in negative powers of the average molecular radius R. In the limit, u is inversely proportional to R, and it is convenient to introduce a quantity that depends only nn the shape S of the This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20798 FELIX T. SMITH mol("cule or particle and can be computed classically: gs= lim (Ru) = lim (Re2/2C). (19) R-+oo R-+co Since the work function v is defined by the limit (12), it may be necessary to include in the effective work function (as it appears in Eq. (5), for instance) an R-l term involving another, possibly shape dependent, parameter as. Such a term represents, among other things, the second term on the right of Eq. (18). With these terms, the expansion is W+n(R, S) =n(v+asR-l)+n2R-lgs +R-2hs(n)+ .. ·. (20) motion of the electrons parallel to the plane of the molecule as an isolated problem. (In the perpendicular direction, all 'II' orbitals have the same structure, so the electrons behave as two-dimensional conduction elec trons only.) The energy levels in a two-dimensional box of area A are where U is measured from the bottom of the box, -Eo. The highest filled level is the Fermi level nF, with energy U F, and the total number of filled levels, with U ~ U F, is roughly 'II'nF2. Each level can ac commodate 2 electrons, so the total number of 'II' elec trons is related to nF and U F by (25) The form of the terms of order through R-l is such as to satisfy Eq. (12) and the requirements of classical electrostatics when R is large but finite. and Using this expansion Eqs. (5)-(7) become (26) vnm= v+asR-l+R-2(m-n)-1[(m/n) hs(n) In graphite, N,,=N and -(n/m)hs(m)]-··, (21) A =agrN = (i)v3"re2N, (27) Unm= R-lgS+ R-2(n-m)-1[n-1hs(n) -m-1hs(m)]- . " where agr is the area per C atom in a graphite layer and (22) ro= 1.42A is the C-C distance, so (28) +R-lg S+R-2n-2hs(n) .. ·. (23) and we can make the identification As we have seen, the difference (v-vgr) is probably negligible (<<0.1); and it appears from the close convergence of the values of Uno for each of the mole cules in Table I that I as I is probably very small compared to gs, so that the entire first term in Eq. (23) can usually be neglected. In practice, one would like to use the first terms of Eq. (20) not merely as a limiting law, but to estimate ionization potentials of real, not very large, molecules. For this purpose it is desirable to have some idea, even if very rough, of the principal sources of deviation from the ideal limiting classical model. The next sub sections are devoted to this question. C. Kinetic Energy It has so far been assumed implicitly that the kinetic energy contribution to the ionization potential W +n is assignable to the work function v. In order to verify this and assess possible corrections, it is useful to assume that the kinetic energy of the 'II' electrons that participate in the ionization behaves much like the kinetic energy of particles in a box. This permits us to use the simple considerations of the free electron theory of metals. If one treats the molecule as a fragment of graphite -containing N aromatic carbon atoms and N 11' conduc tion electrons, and neglects edge effects and the periodicity of the lattice, the separation of variables in the dynamical problem allows one to examine the (29) In the aromatic ions, the number of 'II' electrons, N ... =N-n, (30) is not very large, and the Fermi level shifts as electrons are added or removed. The work of removing an elec tron from the (+n-l) ion is then Wn,n-l= Eo-UF,n-l= Eo-e/1-[(n-l)/NJ} =vgr+(n-l)eN-l. (31) Including the capacitive term, the nfold ionization potential becomes The kinetic energy thus contributes a correction varying as N-l or R-2 to both the effective work function and the effective specific potential. The magnitude of this correction is not reliably estimated by this simple argument (a more sophisticated metallic model would introduce the band structure and a corrected effective mass for the electron, changing the magnitude of e), but taken at face value it would amount to about 0.3 ev for anthracene and phenanthrene. In this simple model, the pairing of the electrons in an orbital has been neglected insofar as it would introduce a correction of similar magnitude, alternating as n is even or odd. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20IONIZATION OF AROMATIC HYDROCARBONS 799 Another effect of the kinetic energy term is revealed by this picture of the electron as a free particle in a box. Conditions at the edge of the molecule may give the electron an effective area for free motion different from that corresponding to N graphitic unit cells. Introducing the area A+oA in place of A = Nagr in Eq. (26), one finds +n2( u+ 2~ A :OA). (33) As an edge effect, the term ilA/(A+oA) varies as R:-1, and diminishes slowly with size. However, it is expected to be very small, since any pronounced redistribution of electrons to or from the edge will lead to a charge separation opposed by the electrostatic forces, which tend to maintain charge neutrality in each unit cell in the neutral molecule and the classical capacitive distribution in the ion. This discussion suggests a general observation. It is often assumed, on the basis of Koopman's theorem, that ionization potentials and electronic excitation levels are linearly related. However, electronically excited mole cules are still electrically neutral, and probably main tain charge neutrality very closely in each unit cell. Thus, the levels of electronic excitation can be thought of as involving mainly changes in the kinetic energy of the electrons; this would make the success of the free electron model in its various forms understandable. If this is the case, the electronic spectrum should tend to vary as R:-2 with increase in molecular size, while we have seen that the ionization potential depends mainly on the capacitive energy which varies as R:-l, D. Polarization In the customary phenomenological discussion of the work function, it is pointed out that the electron, once it has been removed to a finite distance from the solid, exerts a polarizing influence on the conducting surface. This interaction is included in the work function, but its magnitude will depend on the size and shape of the particle, and it is worthwhile to estimate how it changes. For an ideal semi-infinite plane conductor, the inter action can be treated as being due to an image force which varies as -ell/161rEo.r, where x is the distance from electron to surface. This corresponds to an effec tive potential energy -ell/161rEoX. When x is as small as atomic dimensions, the surface no longer appears as an ideal plane conductor; this is usually treated by assuming a cutoff in this image potential at some' distance Xo. The work function then includes a term of the form -vi=e2/167rEoXo. When the molecule to be ionized is not very large, the image force is reduced, and an estimate of the magnitude of the resulting change in the effective image potential can be obtained in the case of a spherical particle of radius R, for which the image force is Fi,R= -(e2/167rEo) (x-2-(x+2R)-2). (34) This can be derived from the effective image potential This will have a cutoff at the same distance Xo, and the work function then has a term -Vi,R(XO) =e2/161r1;oXo-e2/161rEo(xo+2R). (36) The effective work function for a sphere is therefore reduced below what it is for the semi-infinite solid: AVi=ell/167reo(xo+2R) = (ell/8C) (1+xo/2R)-l, (37) It is not quite clear how the computation should be made for a nonspherical molecule, since the cutoff distance Xo will then have to depend on the path the electron is supposed to follow, However, we can ignore Xo in (37) and assume the correction in general to be represented roughly by e2/SC=u/4. Adding this to the principal terms in the effective work function from Eq. (33), we have v=vgr+[(EOA/ A) -(u/4) J-(E/2N). (38) The terms in parentheses are grouped together because they fall off as R-l. The experimental data in Table I suggest that the polarization effect has been con siderably exaggerated by this argument, and that if it exists it is much smaller than u/4. E. The Classical Capacitance The classical capacitances used in Tables II and III were obtained from metallic models which assumed the capacitive charge distribution to reside inside the carbon skeleton of the molecule. As Eq. (IS) shows, the capacitive energy properly is computed from a dis tribution of holes in an electron cloud that may extend somewhat beyond the framework of the nuclei. Thus the capacitive energies were overestimated, and the error is worst for the smallest molecules; this is borne out by Table II. To correct for this error, the classical estimate Ucl can be multiplied by (1+or/R)-l. Including the term from Eq, (33), the effective specific potential is U=ucl(1+or/ R)-1+Ej2N =ucl+[(e/2N) -Ucl(or/ R) ] +ucI(or/R)2.... (39) In the limit, all the correction terms vanish faster than Ucl itself. The requirement that the electron distribution lead to approximate charge neutrality in the region of each nucleus in the neutral molecule suggests an alternative simple model for the electrical behavior. Let us suppose that the molecule is simply cut from a sheet of graphite in such a way as to bisect all the exterior C-C bonds, leaving each nucleus surrounded by a triangular unit This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20800 FELIX T. SMITH cell throughout which the electrons are free to move. Metallic models constructed for this configuration would give another classical estimate for the capacitive energy. Benzene and coronene are then embedded in hexagonal figures, for which the capacitive energies are uci'=4.76 ev (benzene), uci'=2.38 ev (coronene). Alternatively, one may use Eq. (39), assuming or= 0.71A, half the C-C distance, and taking R as the geometric mean of the longest and shortest diameters of the molecular skeleton. The values of uct" in Table II were obtained in this way, with the addition of the term ej2N (which varies from 0.76 ev for benzene to 0.19 ev for coronene). It is interesting to compare the quantal distribution of holes S+n(r) with the classical distribution in an ideal conductor. If the conductor is a circular disk, the classical distribution is well known,13 Near the center of the disk, the charge density is half of what it is in a uniform distribution. Quantally, the infinity at the edge will be reduced to a high peak near the edge, and a gradual tail will remain beyond the classical edge. F. Discussion In a properly rigorous treatment, the argument of the last three subsections would have been based on Eq. (18) and its analog for the kinetic energy. However, such a line of argument could not have been pursued very far without introducing drastic approximations, and it seemed more fruitful at the beginning to use very simple intuitive arguments, realizing that they should be superseded later and may be wrong at some points. The modern structure of molecular orbital theory has been quite successful in predicting some single and double ionization potentials,9.14 and it is clear that a bridge should eventually be built between that ap proach and the one of this paper. Among the points that have been neglected in this paper are spin dependent effects, which are of some importance in the smaller aromatics because the spin multiplicity changes from level to level of ionization. Also, the change in nuclear equilibrium positions has been ignored, on the double grounds that it rapidly becomes small for the larger molecules, and that we are usually confronted experimentally with "vertical" ionization potentials, in which the nuclei have not had time to move toward a new equilibrium position. It hardly need be said that many of the considera tions in this paper apply to other large systems of con jugated double bonds as well as to the fused-ring aromatics. It will be interesting, for instance, when the 13 J. H. Jeans, Mathematical Theory of Electricity and Mag netism (Cambridge University Press, 1925), 5th ed., p. 249. 14 R. M. Hedges and F. A. Matsen, J. Chern. Phys. 28, 950 (1958) . potentials for single and double ionization of other such systems become known, to compute the effective work function and specific potential from them, and see how the former compares with the work function of graphite. It may also be possible to estimate their molecular capacitances by classical metallic models judiciously designed. It has also been suggested to me recently by Professor D. F. Hornig that the same principles may apply to the ionization of large aliphatic systems, where the electron may be removed from the aggregate of (J electrons of the molecular skeleton; in this case, the effective work function will be quite different from that for 7r electrons. APPENDIX An Analog Estimate of Molecular Capacitances Formulas are known for the capacitance of con ducting objects of various simple shapes including circular and elliptic disks, and they can be used for a classical estimate of the capacitance of symmetrical molecules like coronene. For less regular shapes nu merical methods would be tedious, and it is possible instead to make use of the fact that capacitances scale in proportion to a linear dimension for objects of similar shape and can be measured readily for models of reasonable size. We have therefore been able to use metallic models composed of hexagonal disks to simulate the conducting region within the C-C skeleton of polynuclear aromatic molecules. Chemical syntheses can be performed rapidly with a soldering iron, and capacitances measured with a Q-meter do the duty of appearance potentials in a mass spectrometer. The models were made of hexagons, 3 in. on an edge, cut from brass sheet (the models of benzene, naph thalene, and anthracene were bigger, 5.635 in. on an edge). A wire running in the plane of the model and from an extremity of it connected it to the Q-meter. The wire was 73 in. long, 0.01264 in. in diameter. Measurements were made in the open, away from conductors and dielectrics; the model lay on a polyfoam support, 3 ft from the nearest wood and 6 ft from the ground. The frequency used was about 50 kc; capaci tance was measured by comparison with a known impedance. The capacitance of the wire alone was Cw= 12.1 /L/Lf, and that of the model plus wire, Cmw• Since the capacitance of the model Cm was of about the same magnitude as that of the wire, and the charge on each influenced the other, the capacitance of the model could not be computed simply by subtraction, Cm~ Cmw-Cw, but an approximate correction was used which will now be derived. Consider a sphere S of radius rs and a wire W of length hw running normal to it, both charged to a potential V'. In this condition, the total charge on the sphere is Qs', and the charge on an element dx of the wire is qw'(x)dx, where x is measured from the center of the sphere. Compare this with the situation where the wire has been removed, the sphere is again charged This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20IONIZATION OF AROMATIC HYDROCARBONS 801 to the potential Va" = V' with a charge Q.", and the potential along the path of the wire is V w" (x) = V'r./x, while qw" (x) =0 everywhere on this path. The dis tributions of potential and charge in these two situa tions are connected by Green's reciprocation theorem, Lq/'V/= Lq/V/', (At) i i so jrs+hw Q."V' Q.'V'+ V'rs x-1q'w(x)dx. r. (A2) Since When the model is not a sphere, a further approxi mation is required. The potential which is used in the integrand in Eq. (A2) should now contain higher terms of a multipole expansion, but these fall off rapidly with distance and can be ignored, and the leading term in the expansion is taken to be the potential which would be created by a sphere with the same capacitance as that of the model, Cm. Since C.=47rEors= Cm, and the wire begins at a distance rm from the electrical center of the model, we now have instead of (A7) Cm= Cmw-Cw+ (CmCw/41rEohw) In (1 +hw/r m). (AS) (A3) This was used to compute the capacitance of the model Cm from the measured Crow and Cw in the form one finds Cm=[1-(Cw/47r€ohw) In(1+hw/r m)]-l(C mw-Cw). j,.+hw Qa"=Qsw'- (1-r./x)qw'(x)dx. (A4) (A9) r. Far from the sphere the charge distribution on the wire approaches constancy, qw'(x)-+qwO, and if end effects are neglected this constant value is related to the capacitance of the isolated wire by (AS) If one substitutes qwO for qw'(x) in the integral in Eq. (A4) , the error that is introduced at small values of x will be diminished by the factor (1-rsl x). With this approximation, we have The capacitance of the sphere is Cs=Q." IV' and that of sphere plus wire is Csw=Qsw'IV', so For a symmetrical molecule, the electrical center is at the center of symmetry, and rm was measured from there to the point of attachment of the wire. Where there was no center of symmetry, the effective center of mass was estimated by eye, and rm measured from there. Measurements on a sphere, a circle, and an elliptic disk, for which the capacitance could be computed exactly, gave agreement within ±3%. Results for the molecular models should be reliable to better than ±S%. Reduced to molecular dimensions, these results are tabulated as the quantity Uo= e2/2Co in Table III. ACKNOWLEDGMENTS I am grateful to Dr. Robert L. Tanner for help with the capacitance measurements and advice about the correction required, and to John A. Briski for making the models and the measurements. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 93.180.53.211 On: Mon, 28 Oct 2013 21:51:20

1.1734118.pdf

Electron Spin Resonance Studies of Irradiated Glasses Containing Boron Sook Lee and P. J. Bray Citation: The Journal of Chemical Physics 39, 2863 (1963); doi: 10.1063/1.1734118 View online: http://dx.doi.org/10.1063/1.1734118 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/39/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An electronspinresonance study of γirradiated barium galliosilicate glasses J. Chem. Phys. 96, 4852 (1992); 10.1063/1.462775 Electron Spin Resonance Studies of GammaIrradiated Alkali Titanate Glasses J. Chem. Phys. 53, 716 (1970); 10.1063/1.1674049 Electron Spin Resonance Studies of GammaIrradiated Glasses Containing Lead J. Chem. Phys. 49, 1298 (1968); 10.1063/1.1670223 Electron Spin Resonance Study of Trapped Electrons in GammaIrradiated Triethylamine Glasses J. Chem. Phys. 47, 256 (1967); 10.1063/1.1711855 Electron Spin Resonance Study of Irradiated Anhydrous Maltose J. Chem. Phys. 42, 2388 (1965); 10.1063/1.1696305 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03THE JOURNAL OF CHEMICAL PHYSICS VOLUME 39, NUMBER 11 1 DECEMBER 1963 Electron Spin Resonance Studies of Irradiated Glasses Containing Boron* SOOK LEEt AND P. J. BRAY Department of Physics, Brown University, Providence, Rhode Island (Received 12 July 1963) Electron spin resonance (ESR) spectra have been observed for various types of alkali borate and boro silicate glasses which were exposed to -y-ray or thermal neutron irradiations. The irradiated alkali borate glasses (LhO-B 203, Na20-B203, K20-B203, Rb20-B20a, and Cs20-B20a) containing less than 25 mole % alkali oxide exhibit spectra consisting of five peaks in the resonance absorption curve at an operating micro wave frequency in the vicinity of 9.2 kMc/sec. The same five-line resonance has also been observed in the irradiated borosilicate glasses (Corning 7070, 7740, and 7720). Investigations of the irradiated glasses by means of a K-band ESR spectrometer operating at approximately 23.3 kMc/sec have yielded spectra which are partially resolved into six lines. Alkali borate glasses isotopically enriched with BlO display a 10-line resonance at the X-band frequency when irradiated with l' rays or neutrons. It is concluded that the five and six-line resonances arise from an axially symmetric hyperfine interaction of a hole with a BlI nucleus, while the 10-line structure arises from such a hyper fine interaction with a BlO nucleus. The atom containing this BlI or BIO nucleus is believed to be in a B04 tetrahedral configuration in the glass structure. I. INTRODUCTION ELECTRON spin resonance (ESR) studies of defect centers in irradiated glasses containing boron were first reported by Yasaitis and Smaller.! They observed an ESR spectrum consisting of four lines in irradiated borosilicate glasses at an operating frequency of 375 Mc/sec. This four-line structure was attributed to a hy-perfine interaction with a Bll nucleus. No detailed experimental results and analysis of the structure have been reported. The authors of the present paper have previously reported2 that irradiated alkali borosilicate glasses simi lar to the glasses investigated by Yasaitis and Smaller display ESR spectra which are partially resolved into five absorption lines at X-band frequencies. The same five-line structure has also been observed by the au thors in irradiated alkali borate glasses containing less than approximately 25 mole % alkali oxide.2,3 This report is concerned with detailed analyses of the five line structure observed in the irradiated alkali borate and borosilicate glasses. It will be shown that the five-line resonance can be explained satisfactorily as arising from an axially symmetric hyperfine interaction with the Bll nucleus. The atom containing this nucleus is believed to be the boron atom which is at the center of a B04 tetrahedral configuration in the glass struc ture. An appropriate spin Hamiltonian has been found for the center giving rise to the five-line structure. II. EXPERIMENTAL The apparatus used in this investigation was a Varian Model V-4S00 spectrometer operating at a microwave * Research supported by the U.S. Atomic Energy Commission under Contract AT (30-1)-2024. t Based on work performed by Sook Lee in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Brown University. 1 E. L. Yasaitis and B. Smaller, Phys. Rev. 92, 1068 (1953). 2 S. Lee and P. J. Bray, Bull. Am. Phys. Soc. 6, 246 (1961). 3 S. Lee and P. J. Bray, Bull. Am. Phys. Soc. 7, 306 (1962). frequency near 9.2 kMc/sec with 100 kc/sec field modu lation. The spectrometer provides a recorder tracing of the first derivative of the resonance absorption curve. Measurements at K-band microwave frequency were made elsewhere.4 Unless specified in the figures, the ESR data presented in this paper were obtained with the X-band ESR spectrometer at room temperature; measurement at liquid-nitrogen temperature yielded the same data obtained at room temperature. Samples of alkali borate glasses (Li20-B203, Na20- B203, K20-B203, Rb20-B20a, and Cs20-B20a) were pre pared in this laboratory using the methods reported previously.5,6 More details of these methods can be found in Ref. 7. Borosilicate glasses-Corning 7070, 7740 (Pyrex), and 7720 (Nonex)-were obtained com mercially. The samples were irradiated with C060'Y rays (lXlOLIX108 R), or in the reactor at the Brookhaven National Laboratory for thermal neutron doses rang ing from 1017 to 1019 n/cm2• All of the irradiations were carried out at ambient temperatures near room tem perature. The 'Y-or neutron-irradiated alkali borate glasses containing less than approximately 25 mole % alkali oxide exhibit ESR spectra which are partially resolved into five major absorption lines. The first-derivative curve of the spectra is displayed in Fig. 1 (a). The derivative peaks denoted by P2, Pa, P4, and P6 are approximately equally spaced with an average spacing of 14.0±0.5 G, and the g value at the center of these four peaks was measured to be 2.0024±0.OO06. The separation between the derivative peaks Pl and P6 is 61.0±0.5 G. (The basis for the choice of derivative peaks to be labeled by PI, P2, •• ·P6 in Fig. lea) becomes clear later in this paper.) In addition to the 4 MIT Lincoln Laboratory, Lexington, Massachusetts. 6 A. H. Silver and P. J. Bray, J. Chern. Phys. 29, 984 (1958). 6 P. J. Bray and J. G. O'Keefe, Phys. Chern. Glasses 4, 37 (1963) . 7 S. Lee, Ph.D. thesis, Brown University, 1963. 2863 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:032864 S. LEE AND P. J. BRAY (a) (b) Ff I -oj. ~ 61 GAUSS--1 Ff 1 1 I! 14 GAUSS II--f V I ~ g=20035 FIG. 1. The ESR spectrum obtained for a 'Y-ray irradiated (1X108 R) sodium borate glass of molar composition 15% Na20- 85% B,O. (a.) before and (b) after thermal treatment at 250°C for 10 min. 10 derivative peaks of the five-line resonance, the ob served spectrum appears to display another broad de rivative peak on the low-magnetic-field side. This addi tional broad peak will be denoted by PB• Since the additional derivative peak is very broad, the exact location of this peak in the magnetic field cannot be determined; however, it is located in the vicinity of g=2.045. Figure 1 (b) shows the ESR spectrum obtained after heat treatment of the samples at 250°C for approxi mately 10 min. The five-line structure has been re moved completely; the resultant spectrum has no resolved structure and its line shape indicates an anisotropic resonance. [It should be noted that the intensity gain of the spectrometer for the structureless, anisotropic resonance shown in Fig. 1 (b) was approxi mately five times larger than that for Fig. 1(a).] The g value of this latter resonance, measured at the point where the derivative curve crosses the horizontal base line, is 2.0095±0.OO08, and the linewidth (taken to be the separation between the positive and negative extremum points in the derivative curve) is approxi mately 14 G. The thermal annealing experiments were also performed at 100°, 150°, and 200°C for a period up to one hour; however, these heat treatments were not sufficient to remove completely the five-line struc ture. It was difficult to investigate in detail the response of the broad derivative peak PB to thermal treatment of the samples, since the location of this peak (g value) is approximately the same as that of the peak PI of the structureless, anisotropic resonance. However, it appeared that the broad peak PE was removed in the thermal annealing experiments in which the five-line structure was destroyed completely. (The derivative peak PB has been observed in all of the irradiated glasses investigated at X-band frequency in this ex periment, and its g value and response to heat treat ment were found7 to be nearly the same as described above.) Detailed investigations of the five-line resonance were made as a function of the type and concentration of alkali oxide. These investigations were made on the samples which received a ')I-ray irradiation of 1 X 108 R. The locations of the derivative peaks (in the mag netic field) of the five-line spectrum shown in Fig. 1(a) are independent of the kind of alkali oxide incorporated in the alkali borate glasses. It was also noted that the intensity of the five-line resonance tends to increase as the alkali oxide concentration increases. That is, the sample containing about five mole % alkali oxide exhibits spectra consisting of both the five-line and the structureless resonance which are superimposed on one another (Fig. 2); however, when the alkali oxide concentration becomes higher than approximately 10 mole %, the five-line resonance becomes so intense that the structureless resonance can hardly be recog nized. In connection with the above experiment, irradiated "pure" B20a glasses have also been investigated. The B203 glass which received a /,-ray dose of 1 X 108 R also displays the spectrum exhibiting the five-line structure observed in the irradiated alkali borate glasses [Fig. 1 (a)]. However, the intensity of the five-line spectrum in the B203 glass is very roughly 20 times less than that of the five-line resonance in the alkali borate glasses containing 15 mole % alkali oxide. When the B203 glass received heat treatment at 250°C for about 10 min, the five-line resonance was destroyed com pletely; the resultant spectrum displayed the struc tureless, anisotropic resonance observed in the irradi ated alkali borate glasses after similar heat treatment [Fig. 1 (b)]. Figure 3(a) displays a typical ESR trace obtained from the /'-or neutron-irradiated borosilicate glasses. The spectrum consists of a five-line structure simi lar to that observed in the alkali borate glasses, and in addition contains a"sharp, intense line which is superimposed on one of the five lines. (No such additional intense line appeared in the spectra ob served in the borosilicate glasses which were irradi ated with 40-kV x rays for up to 200 h, or which received a /,-ray dose less than 5X 106 R.) The posi- If-61 GAUSS ---lI FIG. 2. The ESR spectrum observed for a 'Y-ray irradiated (1 X 108 R) sodium borate glass of composition 5% Na,O-95% B~3. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03PAR A MAG NET I C RES 0 NAN CEO FIR R A D I ATE D G LAS S E S 2865 tions of the derivative peaks of the five-line structure are the same as those of the five-line structure obtained in the irradiated alkali borate glasses. Thermal annealing at 250°C for about 15 min was sufficient to remove the five-line resonance observed in the irradiated borosilicate glasses, leaving the sharp, intense line and an anisotropic resonance without re solved structure [Fig. 3(b)]. [The intensity gain of the spectrometer for Fig. 3(b) is ten times larger than that for Fig. 3(a).J The g value, linewidth, and line shape of the latter resonance are the same as those of the structureless resonance observed in the alkali borate glasses after heat treatment [Fig. 1 (b)]. (a) (b) H I ~ 61 GAUSS -iI FIG. 3. The five-line spectrum observed for a Corning 7740 (Pyrex) borosilicate glass (a) before and (b) after thermal treatment at 250°C for 15 min. A microwave power-saturation experiment was per formed on the five-line resonance observed in both the irradiated alkali borate and borosilicate glasses. All the peaks in the derivative curve of the five-line resonance have the same response to the increase of the microwave power level. Two other observations also suggest a common origin of the five lines; the growth rate with irradiation dose is the same for each of the peaks in the derivative curve; all of the peaks decrease at the same rate when the spectrum is held at 250°C. The irradiated alkali borate glasses were also in vestigated by means of a K-band ESR spectrometer operating at a microwave frequency in the vicinity of 23.3 kMc/sec. This investigation showed that the ir radiated glasses which display the five-line structure at X-band frequency exhibit a spectrum consisting of six major absorption lines at K-band. The first-deriva tive curve of this spectrum is shown in Fig. 4. The FIG. 4. The six-line ESR spectrum obtained at a K-band frequency of 23.3 kMc/sec for a sodium bo rate glass of .composition ---- 15% Na20-85% B20,. This -J:4 sample has received a ')'-ray irradiation of 1 X 108 R. ~ 83 GAUSS -of If Il I I derivative peaks denoted by P3, P4, Po, and Ps are nearly equally spaced with an average spacing of 14± 1 G, and the g value measured at the center of these four peaks is 2.0026±0.0009. The separation between the derivative peaks PI and P2 is 17±1 G. In order to investigate the role played by the boron nucleus in the observed five-line structure, alkali bo rate glasses were isotopically enriched with Blo. This was done by using boric acidS in which about 96% of the boron nuclei are BIO. Investigations of these glasses irradiated with "Y rays or neutrons have yielded an entirely new type of spectrum. The spectrum obtained at liquid-nitrogen temperature with the X-band ESR spectrometer consists of 10 absorption lines with a resolution poorer than that for the five-line structure observed in the normal (nonenriched) glasses. Figure 5 shows the first-derivative curve of this lO-line reso nance. (The magnetic-field-scanning rate used in ob taining this spectrum was 2.5 times slower than that used for Figs. 1, 2, and 3.) The seven derivative peaks from P4 to PIO are nearly equally spaced with an average spacing of 4.8±0.s G; the peak P7, which is at the center of these seven peaks, has a g value of 2.0027±0.0008. The separation of approximately 5.5 G between the peaks PI and P2 is nearly the same as that between peaks P2 and P3• In addition to the derivative peaks of the lO-line resonance, the broad derivative peak PB was also observed at the low magnetic-field side of the spectrum. Investigation of the BID-enriched sample at room temperature yielded the same type of resonance, ex cept that the resolution of the spectrum became so poor that it was difficult to recognize all of the 10 lines. After heat treatment of the samples at 250°C for 10 min, the lO-line structure disappeared com pletely, and there remained a structureless, anisotropic ~ 4S GAUSS -----iI 'is 'r ~ ~ I I I I ;"2.045 FIG. 5. The 10-line resonance obtained for a ')'-ray irradiated (1 X 10' R) sodium borate glass enriched in the B'0 isotope. The composition of this sample is 15% Na20-85% B203• 8 The boric acid enriched with BIO was prepared by the Oak Ridge National Laboratory. Oak Ridge. Tennessee. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:032866 S. LEE AND P. J. BRAY resonance. The line shape, g value, and linewidth of the latter resonance are the same as those of the struc tureless resonance observed in the normal alkali bo rate glasses after the same heat treatment. For the purpose of investigating effects of water on the observed five-line spectra, some alkali borate glasses were deuterated.7 The ESR spectra obtained from the deuterated samples after a 'Y-ray irradiation were found to be the same as those obtained from the normal (nondeuterated) samples (Fig. 1). III. ANALYSES OF THE HYPERFINE STRUCTURE The following facts suggest that the five lines ob served in the irradiated glasses may arise from only one type of paramagnetic center: (1) All of the peaks in the derivative curve of the five-line resonance have the same response in the thermal-annealing and micro wave-power-saturation experiments: (2) The growth rate for each of the derivative peaks is the same with respect to the irradiation dose. It is then reasonable to assume that the five partially resolved lines in the resonance spectra arise from a hyperfine interaction in the irradiated glasses. If a hyperfine interaction takes place with the nu cleus of an alkali ion in the glass, the resultant hyper fine structure should be dependent on the kind of alkali oxide incorporated. The basis for this is the fact that the spins and magnetic moments associated with the nuclei of the alkali ions arc not the same. Experi mentally, the five-line structure is independent of the type of alkali oxide incorporated into the alkali borate glass. In addition, the samples enriched with the BIO isotope did not show the five-line structure, although these samples were made from the same batch of alkali oxide chemical used in the normal samples. It can be seen, then, that the alkali ions or impurities present in the alkali oxide are not responsible for the five-line structure. It appears unlikely that impurities which may be introduced into the glasses along with boron oxide are the primary source for the five-line resonance. If such impurities give rise to the observed five-line structure, the resonance intensity of this structure would prob ably decrease as the alkali oxide content increases and the boron oxide content correspondingly decreases. However, the experimental results show that the in tensity of the five-line resonance tends to increase rather than decrease as the alkali oxide content in creases. If hydrogens, which are believed to be present in the glasses, are responsible for a part or all of the five-line resonance, the deuterated samples would dis play ESR spectra which differ from those observed in the normal glasses. Experimentally, the deuterated alkali borate glasses exhibit the same type of five-line resonance observed in the normal samples. It appears quite unlikely that the five-line resonance has any direct connection with the presence .of water in the irradiated alkali borate glasses. The arguments presented in the preceding para graphs lead one to consider the following possibilities: The nucleus which is responsible for the observed five line hyperfine structure is either Bll with a spin 1 of t Bill with a spin 1=3, or 017 with 1=!. The natural abundances of BIl, BlO, and 017 arc 81.17, 18.83, and 3.7X10~2 at. %, respectively. However, the possibility that 017 is responsible for the spectrum can be ruled out by the fact that the alkali borate glasses enriched with the BIO isotope exhibit a spectrum entirely differ ent from the five-line resonance. If the 017 is responsi ble for the five-line resonance, the same structure should be observed regardless of the type of boron nucleus. It is then concluded that the five-line reso nance observed in the normal glasses arises from a hyperfine interaction with the WI nuclei, while the lO-line resonance observed in the glasses enriched with the BIO isotope arises from a hyper fine interaction with the BlO nuclei. The structure of alkali borate glasses has been stud ied extensively by means of nuclear magnetic resonance (NMR)5.6 and other techniques.9 It is believed that an alkali borate glass is composed of triangular BOa and tetrahedral B04 units connected by the sharing of oxygen atoms. (The alkali ions reside near B04 units in the boron~oxygen network as positively charged ions.6•g) The fraction of boron atoms contained in B04 tetrahedral units in the alkali borate glasses has been shown to increase as the alkali oxide content increases up to approximately 40 mole %.6 The experimental result that the (ESR) intensity of the five-line struc ture tends to increase as the alkali oxide concentration increases suggests that the Bll nucleus responsible for the five-line structure is the one which is at the center of a B04 tetrahedral configuration in the glass struc ture. As for the five-line structure observed in the irradi ated borosilicate glasses, the following facts should be noted: (1) irradiated alkali silicate glasses containing no boron atoms do not exhibit any resolved hyperfine structure/'lO (2) NMR investigations have shown that the borosilicate glasses also contain some B04 tetra hedral units,5 and (3) the five-line structure observed in the borosilicate glasses is the same as that produced in the alkali borate glasses. These facts suggest that the five-line structure observed in the borosilicate glasses also arises from a hyperfine interaction with the Bll nucleus which is at the center of a B04 tetra hedral configuration. The remainder of this section is devoted to detailed studies of the boron center which gives rise to the observed five-line hyperfine structure. In these studies 9]. Biscoe and B. E. Warren, ]. Am. Ceram. Soc. 21, 287 (1938) . 10 ]. S. van Wieringcn and ;\. Kals, Philips Res. Rcpt. 12, 432 (1957) . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03PAR A MAG N" E TIC RES 0 NAN CEO FIR R A D I ATE D G 1. ASS E S 2867 the possibility of a quadrupole interaction in the boron center will be neglected in view of the following con siderations. (1) According to the NMR studies," the electric-field gradient at the site of the nucleus of a four-coordinated boron atom in the alkali borate glasses is very small. Consequently, the quadrupole interac tion would be very weak, although the Bll and BIO nuclei have a substantial quadrupole moment. (2) The ESR spectra observed for the irradiated glasses in this experiment show no definite indication of quadrupolar effects. Thus it appears to be reasonable to neglect the quadrupole interaction in order to simplify the analysis of the observed hyperfine structures. The di rect interaction between the nuclear spin and the ex ternal magnetic field can also be neglected in this case. A. Influence of Crystalline Fields If the crystalline electric field experienced by the boron center were spherically symmetric, the resultant hyperfine structure would consist of four equally spaced lines of equal intensity.ll It is therefore highly improb able that the boron center responsible for the observed five-line structure experiences a spherically symmetric crystalline field. If the crystalline-field symmetry is axial, one can write A =K., B=K",=K y, (1) where (gx, gy, g.) and (Ke., Ky, K.) are the principal values of the g and hyperfine tensor, respectively, and the symmetry axis is taken to be the z direction. The first-order hyperfine resonance condition in single crys tals is given by12 Il = (1/ g(3) (hvo-KM) = (hvo/{3) (g1l2 cos28+g.L2 sin28)-i -(M/{3) (A2g112 cos28+B2g.L2 sin28)! X (g1l2 cos28+g.L2 sin20)-r, (2) where K is the hyperfine splitting constant, M is the magnetic quantum number of the nuclear spin I, and 8 is the angle between the symmetry axis and the direction of the applied magnetic field. In this equa tion it is assumed that the principal axes are the same for both the g and hyperfine tensors. The signs of the components, gil and g.L, of the g tensor are always taken to be positive, while the components A and B of the hyperfine tensor may have either sign. How ever, the absolute sign of A and B cannot be deter mined from the experimental ESR data when the hyperfine resonance condition is given by Eq. (2). Only I A I and I B I can be found experimentally. In the case of a glassy or powdered substance, 0 II See, for example, D. J. E. Ingram, Free Radicals as Studied by Electron Spin Resonance (Butterworths Scientific Publica tions, Ltd., London, 1958). 12 B. Bleaney, Phil. Mag. 42,441 (1951). ~.--- A H( I) H , ,iL OJ' i ~ ~ , f ~ ~ I + + , ~ 1§ ~ ~ :::: :::: ~ ~ ~ AHo H; ~ FIG. 6. The theoretical hYFerfine structure in glass for the case of I=i!, I:II>I:~, and 1 A I> B 1>0. takes random values, and the hyperfine structure will be the envelope of the properly weighted spectra ob tained in single crystals at all possible orientations. According to the theoretical calculations,13.14 the hyper fine structure in glass consists of two sets of 21+1 equally spaced peaks, as illustrated in Fig. 6 for the case oU =!, gil> g.l., and I A I > I B I >0. (The "shoul ders" or "wings" at HI1(M) are also called peaks.) The locations of the peaks in the magnetic field are given by IlII(M) = (/Zru/gll(3) (M I A l/gll(3), (0=0); (3a) H.L(M) = (hvo/g.L{3)-(M iB l/g.L(3), (o=i1r). (3b) The spacing of one set of 21+1 equally spaced peaks is given by I A I / gll{3, and that of the other set of 21+1 equally spaced peaks by I B l/g.L{3. If one denotes Ilil 0 and H.L 0 as the centers of the 21+1 equally spaced peaks of IlI1(M) and H.L(M), respectively, the separation between III I 0 and H.L 0 , llHo is given by AHo ° 0 hvo( 1 1 ) 4.l =IlJ.. -HII =----. (3 g.L gil (4) This expression indicates that llIlo is linearly propor tional to the operating frequency Vo. On the other hand, the spacing between the 21+1 peaks, I A I / gll(3 and I B I / g.L(3, is independent of vo. Furthermore, if the spacing [Il.L(M=-I)-H11(M +I)J between the peaks at Il.L(M=-I) and H11(M=+I) is denoted by !:J.Il(I) , the following relation holds: I( I A I /glli3) + !:J.Ho +1 ( I B I /gJ./J) =llH(I). (5) It should be pointed out that in certain cases one peak may overlap another peak, so that fewer than the total of 2 (21 + 1) peaks are observed in the hyper fine structure in glasses. A typical example of this 13 R. H. Sands, Phys. Rev. 99, 1222 (1955). 14 R. Neiman and D. KivelsolJ, J. Chern. Phys. 35,156 (1961). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:032868 S. LEE AND P. J. BRAY "'x 1 I-X ..,x I , f ~ <:: ~ ~ , , (a) " H Hs l HU .:;r:. HO .:;r:. -i ~.,.. J: .. § ~ & FIG. 7. The hyperfine structure in glass characterized by Eq. (6) for the case of I = ~, gil> g J.. The solid curve is the theoretical line shape obtained by neglecting the fact that the resonance components are broadened by spin-spin and spin-lattice interac tions and other mechanisms. The dashed curve represents the case for which such broadening effects are included. The structure displayed in (b) is an approximate derivative curve of the dashed curve in (a). occurs if the following relations holds: I A l/gW3= I B l/g.l.)3=AHo. (6) Under this condition, a hyperfine interaction with a nuclear spin of J =! produces five peaks in the reso nance absorption curve, as illustrated in Fig. 7. In this figure the solid curve represents the theoretical line shape obtained by neglecting the fact that the resonance components are broadened by spin-spin and spin-lattice interactions, while the dashed curve repre sents roughly the case for which such broadening effects are included. It is therefore possible that the observed five-line resonance arises from an axially symmetric hyperfine interaction with a Bll nucleus. If the symmetry of the crystalline field were lower than axial, the theoretical hyperfine structure in glass would be more complicated. According to the theo retical calculations for this case,I5 in the absence of any overlap, the hyperfine structure would consist of a total of 3 (2I + 1) peaks in the resonance absorption curve. Even if some peaks overlap, it is likely that a hyperfine interaction with a nuclear spin of J =! would produce more than five peaks in the resonance absorp tion spectrum, when the symmetry of the crystalline field departs substantially from axial. B. Spin Hamiltonian for Boron Center In order to analyze the observed five-line structure in terms of a hyperfine interaction of an unpaired spin, S=!, with a BIl nucleus, it is assumed that the boron center which is responsible for the five-line struc- )5 H. Sternlicht, J. Chern. Phys. 33, 1128 (1960). ture experiences a crystalline field whose symmetry is axial, with gll>gl.. Now if one considers the hyperfine structure in Fig. 7 (a), for which the broadening effects are included (the dashed curve), it is probable that the first derivative of the absorption curve will exhibit extrema at or near the values of magnetic field labeled HI through Hs in Fig. 7 (a) [see Fig. 7 (b) J. Considering the hyperfine structure shown in Fig. 7, it is suggested that the peak PI in the derivative curve of the observed five-line resonance [Fig. 1 (a) J arises from the extremum in the first derivative at HII (M = +t), while the peaks P2, Pa, P4, and Pa arise from the extrema in the first derivative at Hl.(M=+t), Hl.(M=+!), Hl.(M=-!), and l!J.(M=-!), re spectively. However, it is not necessary that the con ditions given in Eq, (6) hold strictly true for the observed five-line resonance. For the hyperfine struc ture characterized by Eq. (6) (Fig. 7), the peak (for example) at HII(M =+!) is exactly coincident with that at Hl.(M=+t). If these two peaks do not actu ally coincide, but are separated by an amount which is small compared to ! A ! /gll{3, I B l/gJ.{3, or AHo, the resultant hyperfine structure that can be observed experimentally would still consist of five peaks in the absorption curve, due to the line-broadening effects. If the tentative identification of the peaks in the first-derivative curve of the five-line resonance is ac cepted, the values of I B I and gl. can be determined. That is, the value of I B I /gl.{3 is given by the separa tion between the four equally spaced peaks, P2, p., P4, and Po in Fig. l(a), which was measured to be 14.0±O.5 Gj gJ. is given by the g value at the center of these four peaks. Thus, one obtains I B 1 /gl.{3=14.0±O.5 G, gl.=2.0024±O.0006, I B I = 1.31±O.05X 10-3 cm-I, (7) The locations of HII(M=+!), HII(M=-!), and HII(M = -t) and, consequently, the values of I A l/gII{3 and gIl could not be found from the observed five-line structure. However, since the peaks Pl and Po were assumed to correspond to the positions of HII (M = +!) and Hl.(M = -!), respectively, the separation between the peaks PI and Ps is simply the AH(I) of Eq. (5). I::..H(J) was found to be 61.0±O.5 G. One can there fore write the following relation using Eq. (5): !( I A I /gll(3) +I::..Ho+21 =61 (in gauss). (8) The _ third term in this equation comes from the fact that I B I /gl.{3=14 G, and consequently HI B I /gl.(3) = 21 G. Here, I::..Ho is unknown from the five-line structure because gil is unknown. If the assumptions and analysis of the preceding paragraphs are correct, one should anticipate that the five-line structure would be altered in the case of ESR investigations at a K-band microwave frequency, since I::..Ho and 1::..1I(I) depend on the operating frequency. Indeed, the samples which display the five-line struc- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03PAR A MAG NET I eRE SON A NeE 0 FIR R A D I ATE D G LAS S E S 2869 ture at 9.2 kMc/sec exhibit a six-line structure at 23.3 kMc/sec. The separation between Pl and P6 in the derivative curve of the six-line resonance (Fig. 4) is measured to be 83±1 G, which would be the value of f:J.H(I) for the case of the K-band measurement. Furthermore, the K-band microwave frequency is about 2.5 times the X-band frequency. Thus the expression correspond ing to Eq. (8) is !( I A l/gll.B)+2.5f:J.Ho+21=83 (in gauss). (9) Solving Eqs. (8) and (9) for I A I /gll/3 and f:J.Ho, one obtains I A I /gll/3=16.9±1.0 G, which yield gil =2.0121±0.OO09, f:J.Ho=14.6±1.0 G, (10) I A I = 1.59±0.09X 1o-a cm-I• (11) The theoretical hyperfine structure constructed for the case of the K-band microwave frequency by using the values of Eqs. (7) and (11) is shown in Fig. 8. The solid lines refer to the case in which the broaden ing effects are neglected. If one takes into account the broadening effects, the resultant hyperfine structure would consist of six peaks in the absorption curve, as indicated by the dashed curve in Fig. 8(b). Figure 8(c) G' GAUSS II-14 G-II (a) (b) (c) FIG. 8. The hyperfine structure in glass constructed from the spin Hamiltonian in Eq. (12) for the case of K-band frequency. The summation of all of the hyperfine components in (a) yields the structure shown in (b). An approximate derivative curve of the dashed line in (b) is displayed in (c). G=GAUSS It-14 G-!I It-16.9 G -iI (a) H (b) H P, (c) I .. :~~"" .,~""""··'I. H ) /\ FIG. 9. The hyperfine structure in glass constructed from the spin Hamiltonian of Eq. (12) for the case of X-band frequency. The structure displayed in (b) is the result of the summation of all of the hyperfine lines in (a). A rough derivative curve of the dashed curve in (b) is shown in (c). is an approximate derivative curve of the dashed line in Fig. S(b). The positions of the peaks denoted by pI in Fig. S(c) are in good agreement with those of the derivative peaks for the six-line structure observed experimentally (Fig. 4). The predicted hyperfine struc ture for the case of the X-band frequency (Fig. 9) also fits well with the observed five-line structure in Fig. lea). It appears that the boron center which gives rise to the five-line hyperfine structure observed at X-band frequency in the irradiated glasses can be satisfactorily described by the following spin Hamiltonian: JC=/3[gIIHzSz+gJ.(HxSx+HIISII) ] +AlzSz+B(IxSx+IIISII)' (12) where S=t, I =t, gil =2.0121, gJ.=2.0024, I A I = 1.59X1o-a cm-I (16.9 G), and I B I =1.31X1o-a cm-I (14G). If the preceding analysis is correct, the ESR spec trum observed for the samples enriched with the BIO isotope should also be describable in terms of the quoted spin Hamiltonian. The value of /-t/I (where /-t is the magnetic moment of the nucleus) for the BIO nucleus is approximately one-third of that for Bll. Therefore, the hyperfine spacing for the case of a hy perfine interaction with a BIO nucleus should be one third of that obtained for the hyperfine interaction with a Bll nucleus, assuming that the wavefunction of the unpaired spin is unaltered in replacing Bll by BIO. In this case the parameters for the spin Hamil- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:032870 S. LEE AND P. J. BRAY Ca) (b) (c) H ... H H I FIG. 10. The hyperfine structure in glass characterized by the parameters in Eq. (13) at X-band frequency. The summation of all of the hyperfine components in (a) yields the structure ex hibited in (b). The structure shown in (c) represents an approx imate derivative curve of the dashed curve in (b). tonian in Eq. (12) should be s=!, 1=3, gJ. = 2.0024, I A I ~0.53XlO-3 cm-! (5.6 G), I B I ~0.44X1O-3 cm-1 (4.7 G). (13) Figure 10 shows the hyperfine structure predicted from the spin Hamiltonian using these values for the case of X-band frequency. This hyperfine structure involves 10 peaks in the absorption curve, as indi cated by the dashed curve in Fig. lO(b). Experimen tally, a lO-line structure was observed in the samples enriched with the BlO isotope (Fig. 5). The positions of the observed derivative peaks in the magnetic field are found to be in good agreement with those predicted in Fig. lO(c). The quoted spin Hamiltonian should also account for the four-line structure observed in the irradiated boro silicate glasses investigated by Yasaitis and Smaller! at an operating frequency of 375 Mc/sec. If the oper ating frequency becomes 375 Me/sec, I1Ho of Eq. (4) becomes extremely small (less than 1 G). Figure 11 shows the hyperfine structure predicted for this fre quency from the quoted spin Hamiltonian. If one takes account of the broadening effects, the hyperfine struc ture is expected to consist of only four peaks in the resonance absorption curve, which is quite consistent with the four-line structure observed by Yasaitis and Smaller. IV. DISCUSSION It was assumed in the preceding analysis that the first derivatives at H11(M) and HJ.(M) in the theoret ical hyperfine line shape reach a positive and negative extremum, respectively. This assumption may be true only in a first approximation. Strictly speaking, the first derivatives at H11(M) and HJ.(M) may not be extrema, thereby introducing errors into the values obtained for the various parameters in the spin Hamil tonian of the boron center investigated here. However, these errors can hardly be found or estimated in the present work. The main reasons for this are: (1) the theoretical hyperfine line shape in glass, for which the width of the resonance in single crystals is taken into account, has not been calculated; and (2) the line shape or width of the resonance in single crystals for the boron center is not known, since the boron center observed in this experiment has not been identified in any single crystals. For the case of a single resonance (without hyperfine interaction) with an axially symmetric g tensor, the resonance line shape in glasses has been calculated taking account of the line shape and width of the resonance in single crystals.16•17 From results of this calculation, one can predict not only the location but also the intensity of each peak in the resonance ab sorption curve. However, since such a calculation has not been made for the case of hyperfine interactions, it is not possible to predict the ESR intensity of each peak with respect to other peaks in the observed hyper fine structures. Furthermore, the true intensity of each peak in the observed hyperfine structure could not be measured or estimated experimentally, since the ob served hyperfine structure was superimposed on a structureless, anisotropic resonance, as the thermal annealing experiments showed. (The intensity of the r 14 G + 14 G+ 14 G -1 ) I 0,,-2.0121 ~f2.0024 r-16.9 G+ 16.9 G + 16.9 G -1 G· GAUSS FIG. 11. The hyperfine structure in glass constructed from the spin Hamiltonian of Eq. (12) for the case of an operating fre quency of 375 Mc/sec. 16 J. W. Searl, R. C. Smith, and S. J. Wyard, Proc. Phys. Soc. (London) 78, 1174 (1961). 17 J. A. Ibers and J. D. Swalen, Phys. Rev. 127, 1914 (1962). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03PAR;\ l\l A G NET I eRE SON A NeE 0 FIR R ,\ D 1;\ TED G L.\ SSE S 2871 lattcr rcsonancc could not be obtained accurately be cause the thermal annealing necessary to destroy the hypcrfine structure seemed to affect the intensity of the structureless resonance.) It should also be noted that the lO-line hyperfine structure (Fig. 5) arising from a hyperfine interaction with BIO should be present underneath the spectra exhibiting the five-line struc ture, sincc the naturql abundance of the BlO (18.83 at. %) is not very small compared to that of the Bll (81.17 at. %). The presence of the lO-line resonance presumably affects not only the shape but also the resolution of the observed five-line structurc. Recently, Keiman and Kivelson14 havc pointed out that, for certain intcrrelationships of the paramctcrs in thc spin Hamiltonian, H of Eq. (2) is not a mono tonicfunction of 0 or cosO in the range of 0=0 to 0=1r/2. In this situation thcre appear one or more additional peaks in the theoretical hyperfine structure in glass which normally consists of two sets of 21+1 peaks. It was assumed that such has not been the case in the hyperfine structure observed in irradiated glasses in this experiment. The result that the preceding analysis of the boron center is consistent with all of the hyper fine structures observed in the irradiated glasses is an indication that this assumption is justified. In view of the fact that the principal values of the g tensor of the boron center are larger than the free electron g value, it is indicated that the unpaired spin which interacts with the Bll or BIO nuclcus in aBO, unit is a hole rather than an electron. The production of a hole in an oxygen tetrahedral unit by irradiation has already been discussed for the case of the aluminum center observed in irradiated quartz single crystals.18,19 For the case of the aluminum center, it was suggested that the hole is mainly associated with an oxygen atom and spends only a small fraction of its time on the aluminum nucleus at the center of an AI04 tetrahedral unit, resulting in a small hyperfine interaction with the aluminum nucleus. Such a conclusion could not be drawn directly from the experimental data obtained for the boron center in this investigation. However, there are two possibilities to be considered: that the hole is trapped on (1) an oxygen, or (2) the boron atom in the B04 unit. In the following paragraphs, discussion will be presented, which suggests that the hole is probably trapped on one of the oxygen atoms and only weakly associated with the boron atom of the BO, unit. In the alkali borate glasses containing less than 30 mole % alkali oxide, most of the oxygen atoms are shared by two boron atoms. In the event that the hole is produced on an oxygen atom which is bonded to two four-(or three-) coordinated borons, it is probable 18 M. C. M. O'Brien, Proc. Roy. Soc. (London) A231, 404 (1955) . 19 J. H. E. Griffiths, J. Owen, and 1. M. Ward, Report of the Conference on Defects in Crystalline Solids (The Physical Society, London, 1955). that the hole will interact approximately equally with two boron nuclei. This suggestion is made on the as sumption that the eharacteristics of the two R·O bonrl ings are identical, and the hole associated with the oxygen ion "sees" two identical boron electronic con figurations. This situation corresponds to a hyperfine interaction with the total nuclear spin of 1 =3. As the analysis of the preceding section showed, there is no indication of such a hyperfine interaction in the ob served spectra. The hole may be trapped on an oxygen atom which is shared by one four-and one three-coordinated boron atom. In this case the B-O bonding between the oxy gen and the three-coordinated boron atom is not the samc as that between the oxygen and the four-coordi nated boron, and the hole will "see" two different boron electronic configurations. The preceding analysis of the five-line structure has indicated that the hole interacts predominantly with only one boron nucleus. It was also suggested that the boron nucleus interact ing with the hole is the one which is at the center of a tetrahedral BO, configuration. If this suggestion is cor rect, then the hole giving rise to the observed resonance spectrum is trapped on an oxygen shared by one three coordinated and one four-coordinated boron atom. It is, in fact, probable that the hole will be attraded more to the four-than to the three-coordinated boron, since the former atom has a higher electron density than the latter; that is, the four-coordinated boron has obtained an extra electron in order to form a fourth bond. In the alkali borate glass containing low alkali oxide concentration (less than approximately 30 mole %), the majority of oxygen atoms are shared by two boron atoms. However, some oxygen atoms may be bonded only to one boron atom; such singly bonded oxygen atoms9 are known to exist in various glasses. Thus it is also possible that the hole interacting with the boron nucleus is produced on a singly bonded oxygen atom in the B04 tetrahedral unit in the glass network. The second possibility is that whereby the hole is trapped on the boron atom in the BO, unit. If this is the case, one may predict the strength of the hyperfine interaction with the Bll nucleus in the following manner. When the hole is mainly associated with the boron atom, the wavefunction of the hole can be considered to consist mainly of a tetrahedral bond ing orbital of the form C (4-!) C I 2s )+v'J I 2p», where I 2s) and I 2p) are, respectively, the 2s and 2p or bitals of the boron atom that are used in the B-O covalent bondings. The hyperfine interaction of an s electron is charac terized by the interaction energy20 JCs=i1rg{3CJ.l/I) \1{;(0) I 2S·1 = <XsCS,,,I,,,+SyIII+ S.I.) , (14) 2Q E. Fermi, Z. Physik 60,320 (1930). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:032R72 S. LEE A N () P. J. B R ,\ Y where I 1/;(0) 12 is the density of the s electron wave function at the nucleus, and as is defined to be (15) On the other hand, the interaction between the nucleus and an electron in a single p orbita121 is JCp= -!g{3(/-L/1) (r-3 ) (lxSxlx+lySv1y+1zSzlz) , (16) where 1z=-%, lx=ly=t for the orbital pz. If one de notes ap= !g{3(J!./ 1) (r-3), Eq. (16) can be written as (17) JCp=ap( -tSx1x-! Syly+iSzI.). (18) Then the hyperfine interaction of the orbita14- 1(12s)+ VJ I 2pz» is characterized by with Kx=KlI=B=t(a.-a p), Kz=A =t(a s+2ap). (19) (20a) (20b) Here, a. and ap have the same sign; for boron they are positive. In order to estimate the isotropic interaction part, as, one needs to know the values of 11/;(0) 12. So far as the authors know, the value of 11/;(0) 12 for the 2s electron of the boron atom has not been found or cal culated theoretically. It is therefore not possible at this time to estimate the value of as. However, for the boron center, the experimental value of 1 A I is larger than that of I B I, and ( I A I -I B I) is small com pared to I A I or I B I. This indicates that as must be larger than ap for the hyperfine interaction of the boron center, as can readily be seen from Eq. (20). It can also be concluded from Eq. (20) that the sign of both A and B is positive for the boron center, since a. and ap are positive and a.>a p• To evaluate the anisotropic part of the hyperfine interaction, ap, it is necessary to know the value of (r-3) for the boron atom. This can be obtained from the well-known expression22 aJ= -({32gr/h) [2L(L+l) / J(J +1) ] (r-3)5'J (21) and the experimental value of at=366.2 Me/sec ob tained for the 2 Pi ground state of the boron atom by Wesse1.23 Neglecting the relativity correction factor, 5' J, one finds (22) for the ground state of the boron atom. Using this 21 B. Segall, G. W. Ludwig, H. H. Woodbury, and P. D. John son, Phys. Rev. 128, 76 (1962). 22 L. Davis, Jr., B. T. Feld, C. W. Zabel, and J. R. Zacharias, Phys. Rev. 76, 1076 (1949). 23 Gunter Wessel, Phys. Rev. 92, 1581 (1953). value in Eq. (17), one obtains (ip=5.69XI0-3 cm-I• (23) That is, if the hole of the boron center is trapped en tirely on the boron atom, the value of (ip must be that of Eq. (23). From Eq. (20), (ip can be expressed as (ip=t(A-B) (24) The upper limit of the value of (A -B) found for the boron center in this experiment is 0.42 X 10-3 cm-I• This indicates that if the hyperfine splitting constants found for the boron center are correct, the value of (il' for this center must be less than 0.56X to-3 em-I, which is approximately an order of magnitude less than the value required in Eq. (23). It can be seen, then, that the experimental data obtained for the boron center suggest that the hole interacting with Bll or BIO nucleus is only weakly associated with the boron atom. It is interesting to note that irradiated glassy B203 also exhibits, in addition to the structureless, aniso tropic resonance, the same five-line resonance observed in the alkali borate glasses. If the interpretation that the five-line structure arises from a hyperfine interac tion with the BII nucleus in a B04 tetrahedral unit is correct, it is indicated that the B203 glass contains some B04 units. However, the possible number of B04 units in this glass is considered to be very small, if one notes the following facts. In the alkali borate glasses containing less than approximately 30 mole % alkali oxide, the fraction of four-coordinated boron atoms is given by x/(1-x), where x is the mole frac tion of the alkali oxide incorporated in the glasses.6 For x=0:15, x/(1-x) is about 0.18; that is, in the alkali borate glasses containing 15 mole % alkali oxide, 18 % of the borons are four-coordinated. The intensity of the five-line resonance in the irradiated B203 glass is roughly 20 times smaller than that of the five-line resonance observed in the irradiated alkali borate glasses containing 15 mole % alkali oxide. Therefore, the fraction of four-coordinated boron atoms in the B20a glass may be estimated as only 1% or less. In connection with the present experiment, it should be pointed out that Nakai24 has also made ESR studies of irradiated alkali borate glasses. The latter author has observed a fIVe-line resonance which is the same as that observed in this experiment. In some instances, however, he observed only four lines; such has not been found to be the case in any of the irradiated alkali borate glasses (containing less than 25 mole % alkali oxide) investigated in the present work. Nakai has suggested that only four lines in the five-line reso nance arise from a hyperfine interaction with a Ell 24 K. Nakai, Nippon Kagaku Zasshi 82. 1629 (1961) [Ahstract 138, Phys. Chern; Glasses 4. 13A (1963)]. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03PAR A MAG NET I eRE SON A NeE 0 FIR R A D I ATE D G LAS S E S 2873 nucleus, considering an isotropic crystalline electric field. In view of the analyses of the five-line structure made in the present work, this interpretation appears to be unjustified. Very recently, Karapetyan and Yudin25 also inves tigated sodium borate glasses irradiated with l' rays and observed the same five-line resonance studied in the present work. Their analysis of the five-line struc ture differs from the analyses by both Nakai and the present paper. Final comments will be concerned with the reso nances other than the five-line structure observed for the irradiated glasses in this experiment. The line shape, g value, and linewidth of the structureless, anisotropic resonance observed in both the irradiated alkali borate and borosilicate glasses are nearly the same as those of the hole resonance observed in ir radiated Si02 glass by Weeks.26 Details of a study of this anisotropic resonance will be reported later in conjunction with ESR investigations of irradiated alkali borate glasses containing high alkali oxide concentra tions. As for the sharp, intense line produced only in the irradiated borosilicate glasses, the line shape, line width, and g value are the same within the experi mental error as those found for the trapped-electron resonance produced in irradiated Si02 glass.26 The origin of the broad derivative peak, denoted by PB, in the observed spectra is not certain at the pres ent time. Since the resonance giving rise to this de rivative peak is partially masked by the five-line structure, difficulties are encountered in finding details of the resonance. Experimental results obtained7 so far for this broad peak indicate that the paramagnetic center giving rise to the latter peak is not associated 25 G. O. Karapetyan and D. M. Yudin, Fiz. Tverd. Tela 4, 2647 (1962) [English transl.: Soviet Phys.-Solid State 4, 1943 (1963)]. .e R. A. Weeks, J. Appl. Phys. 27, 1376 (1956). with boron, alkali ions, or protons. The authors believe that the paramagnetic center is associated with oxygen in the glass structure. V. CONCLUSIONS Irradiated alkali borate and borosilicate glasses have exhibited ESR spectra consisting of a five-line struc ture at X-band. This hyperfine structure is independ ent of the kinds of alkali oxide incorporated in the glasses. It has been shown that the five-line structure arises from an axially symmetric hyperfine interaction of a hole with a Bll nucleus. The atom containing this nucleus is believed to be the boron atom which is at the center of a B04 tetrahedral configuration in the glass structure. The hole interacting with this nucleus is produced in the B04 unit by irradiation and is prob ably trapped on an oxygen atom that is shared by the four-coordinated boron and a three-coordinated boron. A spin Hamiltonian has been found for the boron center responsible for the five-line structure. This spin Hamiltonian has been shown to describe satisfactorily the other hyperfine structures observed in the irradi ated alkali borate and borosilicate glasses. The authors have recently extended their ESR work to irradiated alkali borate glasses with higher alkali oxide concentration and alkali borate crystalline com pounds. The results of these investigations will be re ported later. ACKNOWLEDGMENTS The authors wish to thank Dr. W. H. From of the MIT Lincoln Laboratory, who made available to the authors the K-band ESR measurements made in his laboratory. They also wish to express their apprecia tion to Dr. H. O. Hooper, Mr. Y. M. Kim, and Mr. D. Griscom of this laboratory for their assistance with the experiment. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.97.178.73 On: Sat, 22 Nov 2014 19:43:03

1.1702530.pdf

LowFrequency Negative Resistance in Thin Anodic Oxide Films T. W. Hickmott Citation: J. Appl. Phys. 33, 2669 (1962); doi: 10.1063/1.1702530 View online: http://dx.doi.org/10.1063/1.1702530 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v33/i9 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 9 SEPTEMBER 1962 Low-Frequency Negative Resistance in Thin Anodic Oxide Films T. W. HICKMOTT General Electric Research Laboratory, Schenectady, New York (Received February 5, 1962) Negative resistance and large current densities have been ob served in the direct-current-voltage characteristics of five metal oxide-metal sandwiches prepared from evaporated metal films. The systems studied and their voltages for maximum current are: AI-SiO-Au, 3.1 V; A1-Al.Oa-Au, 2.9 V; Ta-Ta20.-Au, 2.2 V; Zr-Zr02-Au, 2.1 V; and Ti-Ti02-Au, 1.7 V. For aluminum oxide, which has been most extensively studied, the voltage for maximum current is independent of film thickness for films between 150 and 1000 A thick; the phenomenon is not field dependent. Peak to-valley ratios of 30: 1 and current densities of 10 A/cm2 are typical. Maximum current densities at peak voltage are 25 A/cm2; I. INTRODUCTION THIN insulating films of controlled thickness can be formed on a number of metals, including aluminum, tantalum, titanium, and zirconium by anodization of the metals in a suitable electrolyte. Anodic oxide films formed in this way have been widely used in electrolytic capacitors, and many studies of the formation and properties of oxide films have been made.1,2,2& However, electrical conduction in thick oxide films is still poorly understood. Recently, electron tunneling through very thin «50 A) oxide films has proved to be a powerful tool in the study of super conductivity.3,4 Tunneling has also been proposed as the basis of a family of solid state devices.6 Anodic oxide films between 150 and 1000 A thick, too thin to be good capacitor dielectrics and too thick for electrons to tunnel through, have received little study. Some of the electrical properties of anodic oxide films in this inter mediate thickness range is described in the present work. II. EXPERIMENTAL PROCEDURE Aluminum oxide films have been investigated most extensively because of their ease of preparation and their excellent dielectric properties. Steps in the prepara tion of the aluminum oxide sandwiches are indicated schematically in Fig. l.4 A 2-mm strip of aluminum, 2000-4000 A thick, is evaporated onto a carefully cleaned glass microscope slide, the aluminum film is anodized in a suitable electrolyte to form an insulating oxide, and a counterelectrode of gold, 100-1000 A thick, is then evaporated through a mask onto the slide. Capacitor grade aluminum, 99.99% pure, has been 1 T. P. Hoar, Modern Aspects of Electrochemistry (Academic Press Inc., New York, 1959), Vol. 2, p. 262. 2 D. Vermilyea, Advances in Electrochemistry [Interscience Publishers, Inc., New York (to be published)]. 2. L. Young, Anodic Oxide Films (Academic Press Inc., New York, 1961). a I. Giaever and K. Megerle, Phys. Rev. 122, 1101 (1961). 4 J. C. Fisher and I. Giaever, J. AppJ. Phys. 32, 172 (1961). • C. A. Mead, J. Appl. Phys. 32, 646 (1961). minimum current densities are 0.01 A/cm2. Switching time from peak current to valley current is <0.5 !,sec but negative resistance is not found for 6O-cycle voltages. Establishment of the de char acteristics and dependence on temperature and atmosphere are described. Electron emission from aluminum oxide sandwiches can occur at 2.5 V. Space-charge-limited currents in the insulator provide a possible mechanism for the current-voltage curves and large currents below the voltage for maximum current through the oxide films. The mechanism responsible for negative resistance is uncertain. used. Evaporations are carried out in a bell jar at 1 X 10-4 Torr or less. Hydrogen ions incorporated into aluminum oxide films prepared by anodization in aqueous solutions may adversely affect the resistivity of the film and increase its leakage current.6 To minimize hydrogen ions in the oxide, a molten salt bath of NH4HS04 and KHS0 4 in equimolar portions was used as the electrolyte in form ing the anodic oxide films that have been most exten sively studied in this work.7 The eutectic melts at 150°C and has sufficiently low viscosity above 180°C for EVAPORATE ALUMINUM ANODIZE BATTERY FIG. 1. Preparation of metal-anodic oxide-metal sandwiches. Circuit for measuring electrical characteristics. 6 W. E. Tragert (private communication). 7 S. Tajima, M. Soda, T. Mori, and N. Acta 1, 205 (1959). Baba, Electrochim. 2669 Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2670 T. \V. HIe K MOT T 14 '2 FIG. 2. Tracing of X-V recorder plot of the establishment of con ductivity in a 350-A. aluminum oxide film. Au= +, AI=-. anodization to occur readily. In prolonged use, or above about 275°C, the melt decomposes by the evolution of H20 to form S207~. Deterioration of the melt is retarded by using a stainless steel cathode through which oxygen is bubbled.8 The aluminum strips are anodized at a constant current density of 1 to 2 mA/cm2. Film forma tion proceeds rapidly and capacitance of the films is inversely proportional to the final anodizing voltage. Oxide films formed in molten bisulfate eutectic are dense and resistant to chemical attack.7 X-ray fluores cence analysis indicates about 5% sulfur is incorporated in the oxide film; x-ray diffraction shows it is in the form of KAb(S04k Electron diffraction shows the film to be amorphous and no hydroxyl ion was detected by infrared reflection from the oxide film.8 After anodiza tion the aluminum strips frequently have many small blisters. Fortunately, the presence of these blisters does not seriously affect the electrical properties of the oxide films. Although most of the experimental results have been obtained on oxide films containing a significant amount of sulfate impurity, the electrical properties of aluminum oxide films formed in a solution of boric acid and borax in water are essentially identical. Gold counterelectrodes are formed by evaporation through a mask with alternating !-and 1-mm slits to give total contact areas of 1 and 2 mm2. Contact to electrodes is made mechanically rather than soldering to either gold or aluminum. Advantages of the crossed film configuration are the well-defined contact areas and elimination of mechanical damage to the thin oxide films in making contact. Film thicknesses in this report have been derived from capacitance measurements using a value of 8 for the dielectric constant of AhOa formed by anodization.9 . A conventional potentiometer circuit, shown in Fig. 1, was used to measure direct-current-voltage 8 W. Lambrechtse and W. E. Tragert (private communication). • W. Ch. van Geel and B. J. J. Schelen Philips Research Rept 12, 240 (1957). ' . characteristics of the oxide sandwiches. Film currents below 10-4 A were measured with a Keithley model 415 micromicroammeter. Leakage currents in the voltage divider network of the potentiometer limited minimum currents to about 10-12 A. Above 10-5 A, current-voltage curves were plotted directly with a Moseley X-Y recorder. Potentials across the oxide films were deter mined by either two-probe or four-probe measurements; potential values reported that are affected by the use of only two probes will be indicated. All measurements have been made with the oxide sandwiches in an evacuated bell jar because of the effects of atmosphere in modifying conductivity and electrical properties. III. EXPERIMENTAL RESULTS de Electrical Conductivity and Negative Resistance After anodization, the aluminum oxide films are very good insulators. Resistivities of freshly anodized films are between 1014 and 1018 Q-cm. An exponential relation between current and voltage is usually found which could easily be mistaken for electron tunneling between the metal films; however, oxide thicknesses of over 100 A preclude this.1O As the voltage across a film is gradually increased, a critical voltage is reached at which a sharp increase in current through the oxide occurs, as shown in Fig. 1. In Fig. 2 the current increase occurs at 4.1 V. On lower ing the voltage across the 350-A film, a pronounced negative resistance region occurs. On raising the voltage to successively higher values, the current-voltage characteristic continues to show a negative resistance region for both increasing and decreasing voltages. Currents through the oxide film are noisy and highly erratic during the first time that a voltage range is covered. However, on successive tracings of a character istic the currents are much less erratic. The voltage for the onset of the sharp increase in film conductivity varies from film to film but usually occurs at higher voltages for thicker films. Breakdown of the oxide film, in the sense of a sharp increase of current, has produced an irreversible and permanent change in the aluminum oxide film; the original high resistivity and exponential dependence of current on voltage are not recovered. However, break down that causes increased oxide conductivity and negative resistance can be distinguished from dielectric breakdown which destroys the insulating film. The term "forming" of an oxide film will be used to mean estab lishment of a negative resistance region in the current voltage characteristic by a nondestructive breakdown of the film. During forming the maximum current through the oxide film increases as the voltage applied to the film is increased. Figure 3 shows the current voltage characteristic that developed after forming the 10 R. Holm, J. Appl. Phys. 22, 569 (1951). Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsNEG A T I V ERE SIS TAN eEl NTH I NAN 0 DIe 0 X IDE F I L M S 2671 film of Fig. 2 by applying 10 V to it. Maximum current through the 2-mm2 film is 125 rnA. The negative resist ance region was not traced out by the recorder because of circuit instabilities. Figures 2 and 3 illustrate the general features of forming of conductivity in oxide films; a sharp and irreversible rise in film conductivity at some voltage which increases with film thickness, the appearance of a negative resistance region above 2.9 V, and the increase of maximum film current by applying higher voltages to the oxide film. Figure 4(a) shows a typical current-voltage character istic for a completely formed aluminum-aluminum oxide-gold sandwich prepared by anodization in a molten bisulfate mixture. Four distinct current regions can be distinguished: (1) Below 2.9 V, film current increases rapidly with increasing voltage. [A two-probe potential measure ment has shifted the voltage for maximum current to 3.4 V in Fig. 4(a)]. Maximum current densities of 25 AI cm2 have been observed; typical peak current densities are 5 to 10 AI cm2 and well developed charac teristics with peak currents as low as 0.01 A/cm2 have been found. (2) Above 2.9 V, the current decrease is exponential in voltage. In Fig. 4(a), as in a number of other figures, a dotted line shows that the complete current-voltage characteristic was not traced out because of circuit instability in the negative resistance region. However, in many instances this portion of the curve can be traced by a recorder if the impedance of the film is greater than the impedance of the voltage source. (3) A shallow minimum is reached between 8 and 10 V, followed by a small increase in current. Peak-to valley ratios are typically 30: 1 and may be as high as 300: 1. (4) Beyond 9 V, current through the film in Fig. 4(a) decreases again. This is the region of dielectric break down in which the gold film will melt back in small globules, and the oxide film is destroyed. Holding the 140 FIG. 3. Tracing of x-v recorder plot of the fully developed direct-current-voltage characteristic of a 350-A aluminum oxide film showing negative resistance for increasing and decreasing voltage. Au=+, Al=-. 80 70 60 OXlI£ THICKIIESS -300 A 20 (a) 10 10 60 (b) FIG. 4. Dependence of direct-current-voltage characteristics of Al-AbOa-Au sandwich on aluminum oxide thickness. Al = +, Au=-. voltage above that required for dielectric breakdown destroys the oxide and reduces the peak current at 2.9 V, since the effective area of the film is reduced. Dielectric breakdown occurs at higher voltages in thicker films; in Fig. 4(b), about 15 V were required for dielectric breakdown. Negative resistance is observed with either aluminum or gold positive, with no qualitative difference in characteristics. Varying polarity for forming the film may change the characteristic between 10 and 12 V slightly; with aluminum positive, the current rises above the minimum at 8 V, with gold positive, the minimum is reached when dielectric breakdown of the Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2672 T. W. HICKMOTT ... z ... a: a: :> <.> ~ ;;: INCREASI NG VOLTAGE DECREASING VOLTAGE \ VACUUM \ \ \ \ \ \ \ \ \ \ \ \. FIG. 5. Direct-current-voltage characteristics of 450-A aluminum oxide film when placed in atmospheric pressure of dry nitrogen and dry helium. AI= +, Au=-. film occurs. The voltage for maximum current V M is voltage dependent, not field dependent, since the location of the voltage for maximum current is independent of film thickness as shown in Fig. 4. Varying oxide film thickness between 1.5 X 10-6 cm and 9.5 X 10-6 cm produces no change in the voltage for maximum current. However, V M will change if metals other than gold are used as the evaporated electrode. Some hysteresis occurs for increasing and decreasing voltage with the maximum current usually smaller for decreasing voltage. Switching time from peak current to valley current can be measured by inserting a small resistance in series with the negative resistance element to define a suitable load line. Switching times of <! p,sec have been measured for a film with an RC time constant estimated at about t p,sec. Switching times are ex tremely fast and are limited by the time constant inherent in the capacitative nature of the oxide film sandwich. Effects of Atmosphere on dc Conductivity Development of negative resistance characteristics such as those in Figs. 2-4 have been accomplished with the aluminum oxide sandwich in vacuum. Attempts to form a negative resistance region with the films in air have been unsuccessful. Instead, currents through the film are small and extremely erratic for voltages less than the voltage at which dielectric breakdown occurs. Dielectric breakdown of the films occurs at the same voltage in either air or vacuum and the destroyed films look the same after dielectric breakdown in either medium. Microscopic examination of dielectric break down in air shows that it occurs over the whole film and is not confined to structural defects or blisters in the oxide. Some preferential dielectric breakdown does take place at edges of the metal films. Once the electrical characteristics of the oxide sand wiches are established, increasing pressure in the bell jar to atmospheric pressure does not change qualitative features of the current-voltage curves but markedly reduces the peak current and the valley current and may cause the negative resistance to disappear com pletely. Polarity effects become particularly noticeable at atmospheric pressure. Figure 5 shows current-voltage curves for a film in vacuum and in atmospheric pressure of two unreactive gases, dry helium and dry nitrogen. With the aluminum film positive, both helium and nitrogen reduce the peak current but helium has a greater effect than nitrogen. The peak-to-valley ratio in nitrogen is three hundred, the highest observed for any oxide sandwiches, and the reproducibility between successive curves is good. With aluminum negative, the effect of atmospheric pressure is to reduce the peak current to nearly equal values (about 10 mA) in both helium and nitrogen, and to give a rather noisy charac teristic. Figure 5 was obtained by successive cycling between vacuum, helium, vacuum, nitrogen and INCREASING VOLTAGE DECREASING VOLTAGE ~ i j ;;; ~ 20 a ~ ;;: 4.0 VOLTS FIG. 6. Temperature dependence of the direct-current-voltage characteristic of a 4so-A aluminum oxide film. Note constancy of voltage for maximum current and disappearance of negative resistance at 195 OK. Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsNEG A T I V ERE SIS TAN eEl NTH I NAN 0 DIe 0 X IDE F I L M S 2673 0.10 -0.10 FIG. 7. Derivative of direct-current-voltage characteristic of a 4so-A aluminum oxide film at different temperatures. Note con stancy of maximum, zero, and minimum slope. vacuum. In each cycle, the high peak current in vacuum was constant within 10% indicating that an oxide film which has had a negative resistance characteristic formed in vacuum is not permanently changed by operation at atmospheric pressure of unreactive gases. Temperature Dependence of dc Conductivity An evacuated cryostat was used to measure the temperature dependence of the conductivity of alumi num oxide sandwiches between 1800 and 300°K. A copper-Constantan thermocouple attached to the glass slide behind the aluminum film measured temperature. Power dissipated by the oxide elements was sufficient to increase the glass temperature by as much as lOoK at 2S00K when more than 70 mA of current was carried. Temperatures in the film are not accurately known when large powers are dissipated. Figure 6 shows the change of current-voltage charac teristics of a 4.5 X 10-6 cm film as the temperature is lowered. The shape of the curves, particularly the location of the peak current, remains unchanged. Between 300° and 206°K, the peak current decreased from 130 to 35 mA, while between 206° and 195°K the peak current decreased from 35 to 2 mAo At 195°K, negative resistance was observed when the voltage was t:ncreased to 10 V; the negative resistance characteristic vanished for decreasing voltage leaving only a smoothly varying, small film current. Further voltage cycling at temperatures below 195°K produced no dc negative resistance region and very low currents « 1 rnA for 11 V applied). Increasing temperature above 195°K resulted in the re-establishment of the negative resist ance region, and when the film was raised to room temperature the maximum current was as large as previously. In Fig. 6 the apparent shift of voltage for maximum current to 3.6 V at 3000K arises because two probe potential measurements were made. Figure 7, which plots the derivative of the current-voltage curves, shows that the maximum and minimum values of dIldV, as well as the point of zero slope, are unchanged as a function of temperature. A "memory" effect, the maintenance of a room temperature characteristic at a temperature, 188°K, where no negative resistance would normally occur is shown in Fig. 8. The current-voltage characteristic was established at 3000K, and the film was then cooled. At 188°K, increasing voltage gave a large peak current nearly equal to the maximum current at 300oK, and a pronounced negative resistance; decreasing voltage showed no negative resistance region and much reduced currents. Cycling of the voltage to 10 V and back at 188°K retraced the decreasing voltage curve in Fig. 8 and raising the film to 3000K restored the original characteristic. Thus, reducing film temperature decreases the film conductivity reversibility while the negative resistance region remains unchanged until it disappears com pletely over a narrow temperature range of only one or two degrees. Raising the temperature allows higher film conductivity and negative resistance to re-establish, ,w 90 80 T'188'K 70 ~ ~60 \ ~ , , ::E , -50 , ~ , I g; , ., 40 \ :I I lA: , , 30 , I I I 20 I 1.0 , I I I I , 6 VOlTS 8 10 FIG. 8. Persistence of room temperature direct-current-voltage characteristil; of a 4so-A aluminum oxide film at low temperature. Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2674 T. W. HICKMOTT and raising the film temperature to 3000K restores the initial current-voltage characteristic. ac Conductivity of Aluminum Oxide Films Negative resistance observed when slowly varying dc voltage is applied to aluminum oxide sandwiches is no longer found when 60-cycle voltage from a low imped ance source is applied to the oxide film. Instead, the locus of the maximum ac voltage traces out the dc negative resistance characteristic. Figures 9 and 10 show the changes in current-voltage curves as the applied 60-cycle voltage across an oxide film is gradually increased. The film had an initial low conductivity. For low voltages, the current is ohmic. As the voltage increases the conductivity, measured by the slope at zero voltage, increases steeply when more than 1.8 V is applied to the film until a maximum conductivity is found with 3.3 V (6.6 V peak-to-peak) across the film. Further increasing the ac voltage decreases the conductivity of the oxide film until at 10 V the film conductivity has returned to its original low value. If the voltage is turned off, the low conductivity that is characteristic of 10 V across the film will remain and the cycle can be repeated. For all voltages, current voltage traces are symmetrical. In Fig. 11 film conduc tivity is plotted against maximum applied potential for the traces in Figs. 9 and 10. Peak values of conductivity occur at voltages corresponding to the maximum current for slow dc voltage changes. If the dc negative resistance were traced for 60-cycle voltages, there would be a peak I mA/em I volt I em 20 mAl em Ivolt lem 5 mAl em I volt I em 50mA/em Ivolt/em FIG. 9. 60-cycle current-voltage characteristic of an aluminum oxide film with initial low conductivity as the voltage across the film is increased. Abscissa, voltage; ordinate, current. 20 mA/em 2volt I em 10 mA/em 2 volt I em 20mAlem 2 volt I em 10 mAiem 2volt/em FIG. 10. 60-cycle current-voltage characteristic of an aluminum oxide film with further increase in voltage. Abscissa, voltage; ordinate, current. in the current at about 3 V and a region of negative slope above 3 V. Instead, "hash" appears at this voltage, as if the film were trying to follow a negative resistance, but the slope of the current-voltage curve is positive at all voltages across the film. Film conductivity can be established and negative resistance formed, as described previously, by slowly increasing the dc potential applied to the film, by increasing an alternating potential across the film or, in addition, by applying 100-,usec pulses of suitable magnitude to the unformed film. With each method of formation of the film, the same dc negative resistance characteristics are found. Once film conductivity has been established, ohmic current-voltage curves are found for voltages across the film of less than O.S V. The film conductivity under small signal conditions can be varied over wide limits, and is extremely stable if only small ac or dc voltages are applied to the oxide film. Photoconductivity of the film causes an increase of conductivity, measured with small applied voltage, 40 ~ 35 :E 30 ~ 25 l;1 ~ 20 ~ 15 3 10 ;;: MAXIMU~ CURRENT, 175 mA %~~~---74----~6----'~O MAXIMUM 60 'V VOLTAGE FIG. 11. Change of conductivity of aluminum oxide film by increase of 60- cycle voltage across film. Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsNEGATIVE RESISTANCE IN THIN ANODIC OXIDE FILMS 2675 when a microscope light shines on the film. The response is to the blue light transmitted by the thin gold film. Thus, there is an anomaly here in that switching from maximum to minimum current through the negative resistance region for slowly varying dc voltages occurs in less than 0.5 J.lsec, but 60-cycle voltages will not trace out a negative resistance. Conduction processes with widely varying time constants must be involved in aluminum oxide sandwiches. Pulsed Voltage Measurements Conductivity of aluminum oxide sandwiches can be varied over wide limits by varying the voltage across the film as shown for 6O-cycle potentials in Fig. 11. If conductivity is measured with a small voltage, where current-voltage relations are linear, changes of conduc tivity reflect changes in the number of mobile charge carriers in the oxide. Pulsed voltages of I-J.lsec duration can remove or inject carriers as well as 60-cycle voltages can. Figures 12 and 13 show changes in oxide film con ductivity that can be made by single voltage pulses. To obtain such curves, film conductivity was established at some desired value Uo, and measured by a small (0.3 V) signal. With the ac measuring signal removed, a single lO-J.lsec pulse of arbitrary amplitude was applied to the oxide film from a Tektronix type 163 pulse generator. Current-time and voltage-time traces for the single pulse were measured on a Tektronix type 555 dual-beam oscilloscope to determine the voltage across the film just as the voltage pulse was turned off. Film conductivity u was again measured with the small ac 0.2 01 -0.1 -0.2 -o.~ 11" u;--0.4 -0.5 -0.6 -0.7 -0.8 -0.9 " • • 5.0 " " FIG. 12. Change of conductivity of aluminum oxide film by lO-JLsec pulses of varying voltage. 1.2 "."~B' 10"1l1-cmt' " • 10 P SIC PlJLSES 1.0 • '100 P sec PULSES 0.8 0.6 0.4 .Ju:. 0.2 fl. 5.0 ·0.2 " -0.4 •• • " -0.6 A -0.8 -1.0 FIG. 13. Change of conductivity of aluminum oxide fihn by lO-l'sec and 100-l'sec pulses of varying voltage. signal, the conductivity was re-established to its initial value Uo, and the cycle was repeated. Figure 12 shows ;lu/uo for different values of the voltage pulse. For films of low initial conductivity, the number of charge carriers in the oxide could be either increased or de creased. For films with higher initial conductivity, single voltage pulses below 3 V left the conductivity unchanged while pulses above 3 V removed charge carriers from the oxide and reduced the conductivity. For still lower initial film conductivities, as in Fig. 13 even larger increases in ;lu / Uo by lO-J.lsec pulses were found. Identical results are found for pulses as short as 1 J.lsec. In Fig. 13 it can be seen that pulses of longer duration produce greater changes in film conductivity although voltages that increase or decrease conductivity are the same. The extreme example of this effect is that slowly varying dc voltages which produce negative resistance in the current-voltage curves will also produce the maximum film conductivity. FIG. 14. Dependence of voltage for maximum current through metal oxide-metal sandwiches that show negative re sistance on the square root of the dielectric constant of the oxide. Au = +, base metal I 1-.-1 4 6 8 10 (DIELECTRIC CONSTANTI'" Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2676 T. W. HICKMOTT VACUUM 'a .....,.~-h-.,-.,.,....;tf ~ lUNNELING CURRENT J .. liP G6.83.1()1(";") ';'inlfU FIG. 15. Schematic of metal-oxide-metaI sandwich as a cold cathode. Multifilm Structures Limited experiments have been made on sandwiches with three metal films separated by aluminum oxide layers.5 In several multifilm structures, a negative resistance characteristic and high conductivity could be established between the base electrode and the top electrode without having any detectable conductivity between base and middle electrode, or between middle and top electrode. For example, O.S V between base and top electrode produced a current of 43 rnA between these two electrodes. The same voltage between middle and top electrodes or between middle and base elec trodes caused less than 0.01 rnA to pass between these electrodes. If a voltage high enough to develop con ductivity and form a negative resistance characteristic were applied to the middle electrode, current could then be drawn between any pair of electrodes. Such large currents between base electrode and top electrode with no current drawn to the middle electrode can occur if establishment of film conductivity and negative resistance of the oxide film is a surface phenom enon. The base electrode and top electrode were evapor ated in such a way that there was no direct path between them except through the middle electrode. The middle electrode was formed as an aluminum film a few hundred angstroms thick and then anodized to form an aluminum oxide film that was 200 A thick. This oxide film provided insulation between middle electrode and the top electrode. Evaporated films of this thickness usually have a discontinuous structure with many small holes in them.!! Anodizing such an initially discon tinuous film would produce even more tiny holes in the middle aluminum electrode. Through these holes, current could pass between base electrode and top electrode during establishment of film conductivity. The lack of current to the middle electrode as well as its 11 G. A. Bassett, J. W. Menter, and D. W. Pashley, Structure and Properties of Thin Films (John Wiley & Sons, Inc., New York 1959), p. 3. ' inability to affect current flow between base and top electrode strongly imply that the barrier to conduction is at the oxide-metal interface, and that forming of conducting oxide films occurs at the same interface. Negative Resistance in Other Oxides Negative resistance for slowly varying dc voltages has so far been found in five oxide systems, though properties of other systems have not been studied in the detail that aluminum oxide has. For tantalum, titanium, and zirconium, preparation of the metal oxide sand wiches was the same as indicated in Fig. 1, except that the base metal film was prepared by sputtering in argon. The oxide film for these metals was formed by anodizing in a saturated solution of ammonium borate in ethylene glycol with 4% of water added to increase solution conductivity. Silicon monoxide films were prepared by evaporation onto an aluminum strip.l2 Gold counter electrodes were evaporated through the same mask as was used with the aluminum oxide films. Table I summarizes the location of the peak voltage with base metal negative and gold positive for all the oxide systems in which negative resistance occurs, and Fig. 14 shows the correlation that exists between voltage for peak current and the square root of the dielectric constant of the anodized oxide film. Values of K used are; SiO=5.2-6,13 AI203=8,9 Ta206=25,14 Zr02= 28/5 and Ti02=40.!6 For tantalum, titanium, and zirconium it is difficult to obtain a negative resist ance region. These films tend to break down to a very high conductivity state rather than exhibiting negative resistance. Table I also lists the position of the minimum in the current-voltage curves. There is some correlation between this voltage and the band gap of anodic oxide films, where the gap is known. Thus, high conductivity and negative resistance regions in the current-voltage curves seem to be general properties of metal-insulator systems if proper conditions for their establishment can be determined. TABLE I. Voltage for the onset of negative resistance in metal-oxide-metal sandwiches. Insulator AI-SiO-Au AI-AJ.03-Au Ta-Ta20.-Au Zr-Zr02-Au Ti-Ti02-Au Voltage for maximum Voltage for minimum current current 3.1 2.9 2.2 2.1 1.7 8-9 7.5-8.5 5-6 4-5 3.5-4 12 R. L. Wilson of the General Engineering Laboratory, Schenectady, New York prepared the evaporated SiO films. 13 P. Molenda, General Engineering Laboratory (private communication). 14 D. A. Vermilyea, J. Electrochem. Soc. 102, 655 (1955). 15 L. Young, Trans. Faraday Soc. 55, 632 (1959). 16 F. Huber and J. Bloxsom, Trans. IRE CP·8, 80 (1961). Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsNEGATIVE RESISTANCE IN THIN ANODIC OXIDE FILMS 2677 Electron Emission into Vacuum-Possible Cold Cathode It has recently been proposed that electron tunneling through oxide sandwich structures similar to those described here could be used as a cold cathode.5 Figure 15 indicates schematically the basic concept. cPmin is the barrier to the emission of electrons from metal A into the insulator. Electrons from metal A tunnel through the insulating oxide film under the influence of an elec tric field of the order of 107 V jcm. In metal B, the tunnel electrons are "hot" electrons which are not in equilib rium with other electrons in metal B. If the potential drop between metals A and B is greater than the work function of metal B, cPB, some of the tunnel electrons could pass through a thin layer of metal B without being attenuated significantly, surmount the barrier to electron emission from B, and appear in the vacuum. Current between A and B would be exponential in voltage and insensitive to temperature changes if tunneling from metal to metal occurs. Since the work functions of metals are generally larger than 4 V, there are two contradictory requirements for such emission into vacuum; the oxide must be thin enough (d<50 A) for electrons to tunnel through it and it must be thick enough to support between 5 and 10 V without being destroyed by dielectric breakdown, in order to obtain significant amounts of electron emission into the vacuum. In electron tunneling between metals, the oxide film is simply a barrier or spacer between electrodes. Unfortunately the dielectric breakdown strength of 60 i 50 ~ 40 ::: ~ 30 ... z w '" '" 20 " u '3 ;;: 10 OXIDE THICKNESS~ 520A \ V'I I I I • I I JL..L...L~_L L _1 __ _ 6 6 4 2 0 VOLTS FIG. 16. Electron emission from an aluminum-aluminum oxide gold sandwich during establishment of film conductivity. 14 12 ~ ~ 100 ::E 3 80 ~ ... z '" <r " '3 40 ..: I I I \ OXIDE THICKNESS ~ 520 A \ I I I I I \ , \ \ \ \ \ I \ I I 1 FIG. 17. Electron emission from an aluminum-aluminum oxide gold sandwich after direct-current-voltage characteristic is fully established. most insulators ",5 X 106 V jcm, is less than the fields of 107 V j em needed for electrons to tunnel from metal to metaP7.18 One approach to this dilemma is to reduce the work function of metal B by adsorbing cesium or some other alkali metal. A second approach is to go to thicker films where problems of dielectric breakdown are minimized and try to get sufficient electrons into metal B by mechanisms other than tunneling from metal A through the insulating oxide. In such a case the oxide ceases to be just a spacer and its electrical properties become of prime importance. Electron emission into vacuum from oxide sandwiches occurs both during forming of the film characteristics and after a negative resistance has been established. Electrons appearing in the vacuum are not due to tunneling between metals. In Figs. 16 and 17, the current between metal electrodes and the current emitted into vacuum are both displayed for the same voltage across the oxide film. Figure 16 shows that electron emission during forming of a 5 X 10-6 cm film can occur at applied voltages as low as 2.5 V, substan tially below the work function of gold, 4.7 V.19 Once a negative resistance characteristic is established, elec tron emission is closely tied to whatever mechanism is responsible for switching from maximum to minimum current, as shown in Fig. 17. Two probe measurement of film potential causes the current maximum to appear at 6 V instead of 2.9 V. No electron emission could be J7 S. Whitehead, Dielectric Breakdown in Sulids (Clarendon Press, Oxford, England, 1951). 18 P. D. Lomer, Proc. Phys. Soc. (London) B63, 818 (1950). 19 J. C. Riviere, Proc. Phys. Soc. (London) B70, 676 (1957). Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2678 T. \V. HIe K MOT T BEFORE CONTACT METAL VACUUM + AfTER CONTACT FIG. 18. Schematic diagram of the band structure of a metal, an insulator, and a metal-insulator contact. detected above noise until a sharp burst occurred at the same time as the oxide film sandwich switched from maximum to minimum conductivity. Electron emission into the vacuum shows a second maximum at voltages just below those which will produce dielectric break down of the oxide films. Maximum electron emission which has been observed is 2 X 10-6 A with 13 V across the film and a current between metal films of ",1XI0-3 A. DISCUSSION Band Structure of an Insulator and the Metal-Insulator Interface The essentials of a simple model for the band struc ture of an insulator, developed by Mott and Gurney,20 are indicated schematically in Fig. 18. The character istics of an ideal insulator are: The valence band is completely occupied, the conduction band is un occupied, and the forbidden energy gap Eg is large enough that thermal excitation of carriers across the band gap is negligible. The electron affinity X is the energy required to take an electron from the lowest state of the conduction band into the vacuum. No impurity levels have been included in Fig. 18 though they will probably be present in an amorphous anodic oxide film. The simple model of Fig. 18 was developed for an ideal single crystal; its application to highly amorphous and impure substances such as anodic oxide films is dubious at best. However, it is useful in defining some of the energy relationships in an insulating film and will serve for approximate calculations. We shall assume that electrons and not holes are the charge carriers in thin insulating oxide films. Partial justification is provided by Fig. 16 which shows that signficant electron emission into vacuum can occur from oxide sandwiches. 20 N. F. Matt and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, London, 1948), 2nd edition. A metal-insulator contact is also indicated schemati cally in Fig. 18. The work function of the metal cf> is the energy required to raise electrons from the Fermi level of the metal into the vacuum. On contact with an insulator, a barrier to the emission of electrons into the insulator is established. The minimum value of this barrier is cf>min =cf>-X, as shown in Fig. 18; however, the surface barrier may also be larger than this. Space-Charge-Limited Currents in Solids An unusual feature of the currents through thin oxide films after conductivity has been established is the magnitude of the current for voltages less than 2.9 V. Current densities as high as 25 A/cm2 can be passed through films 3X 10-6 em thick without destroying or changing the oxide film. Mott and Gurney20 first showed that large currents could be drawn through very thin insulating films if space charge in the insulator rather than injection of charge into the conduction band of the insulator con trolled the current. More general treatments21-27 that include diffusion of charge carriers, and traps in the insulator, have extended the simple analysis. Space charge-limited currents have been observed in thin single crystals of CdS,28.29 ZnS/o and GaAs,31 and their possible use in solid state electronics has been dis cussed.32.33 However, de space-charge-limited currents in anodic films or in thin amorphous oxide films have not been reported. Two requirements must be satisfied if space-charge-limited currents of appreciable magni tude are to be observed22; at least one electrode must make ohmic contact to the insulator and the insulator must be relatively free from trapping defects. If these requirements are met, the space-charge-limited current IS (1) where V is the applied voltage, d is the thickness of the insulator, p. is the drift mobility of charge carriers, and K is the dielectric constant of the insulator. Equation (1) is Child's law for a solid-state space-charge-limited diode. Shallow traps reduce the effective mobility of the charge carriers by an amount 8, the fraction of total charge injected into the insulator which is free. The space-charge-limited current retains the same voltage 21 A. Rose, RCA Rev. 12, 362 (1951). 22 A. Rose, Phys. Rev. 97, 1538 (1955). 23 M. A. Lampert, Phys. Rev. 103, 1648 (1956). 24 S. M. Skinner, J. App!. Phys. 26, 498 (1955). 25 G. H. Suits, J. App!. Phys. 28, 454 (1957). 2. G. T. Wright, Solid State Electronics 2, 165 (1961). 27 F. Stockmann, Halbleiterprobleme (Friedrich Vieweg & Sohn, Braunschweig, Germany, 1961), Vol. 6, p. 279. 28 R. W. Smith and A. Rose, Phys. Rev. 97, 1531 (1955). '" G. T. Wright, Nature 182, 1296 (1958). 30 W. Ruppel, Helv. Phys. Acta 31, 311 (1958). 31 J. W. Allen and R. J. Cherry, Nature 189, 297 (1961). 32 W. Ruppel and R. W. Smith, RCA Rev. 20, 702 (1959). 33 G. T. Wright, J. Brit. lnst. Radio Engrs. 20, 337 (1960). Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions~ EGA T I V ERE SIS TAN eEl NTH I NAN 0 DIe 0 X IDE F I L M S 2679 110 0 T:300'K .. T. 253'1< D Tc tit 't( 100 0 T· Z06'K 90 80 iii 70 ~ FIG. 19. Space-.. <C charge-limited cur-:3 60 rents in a 450-A ii aluminum oxide film .... at different tempera- ~ 50 tures. 3 40 ii: 12 dependence, and (2) where JLO is the mobility of free carriers. For anodically formed aluminum oxide, K=8, but values of the mobility JL are not known. A value of 10 cm2/V-sec will be used for an order of magnitude calculation. For d=5X1O-6 cm and V=l V, I~6X104 A/cm2, for a trap-free insulator, an extremely large current. In Fig. 19, the current through a 4.5X1o-6 cm film before negative resistance occurs is plotted against V2 for different temperatures. From the excellent fit and from the magnitude of the currents which can be drawn through thin anodic oxide films, we conclude that currents below the negative resistance region of the current-voltage curves are space-charge-limited. The curves of Fig. 6 that are plotted in Fig. 19 were meas ured under slowly varying dc voltages that maximized currents through the films. If the voltage were increased across films of lower initial conductivity, a steep rise in current would occur above 2 V until the maximum film conductivity was obtained. For films of maximum conductivity, Child's law for solids (10:: P) is obeyed at temperatures at which negative resistance is observed in aluminum oxide sandwiches. An ohmic contact between electrode and insulator is required to observe space-charge-limited currentsP With an ohmic contact, the metal serves as a reservoir of electrons in thermal and electrical equilibrium with the insulator; the value of "'min in Fig. 18 should be low enough that thermionic emission into the insulator is significant. The number of electrons/ cm3 emitted into the insulator from the metal may be approximated by No~2.5XlO19 exp("'min/kT). TABLE IT. Thermal emission of electrons into an insulator, T=300DK. -====-.------ cpmin (eV) 0.1 0.2 0.3 0.4 0.5 No (electrons/cm3) 5X1017 lX1016 2X1014 5Xl012 lXl011 IMt (A/cm2) 8Xl0· 2X104 3X1Q2 8 0.2 -------------- :------cc=== In Table II, No at 3000K is calculated for different values of "'min. Saturation in a solid-state diode is not predicted by Eq. (1), but will occur when all injected electrons are drawn to the anode. In Table II, Isat is calculated from Isot= ]\;' oJLeF, assuming that F= 106 V /cm, and JL is 10 cm2/V-sec. Saturation currents for "'min=0.3 V are in reasonable agreement with maximum currents observed through anodized films. In calculating Isat, it is assumed that JL is independent of field and that no electrons are produced in the insulator by impact ionization. Current in the space-charge-limited current region of a solid-state diode will be relatively insensitive to temperature so long as No is larger than the number of electrons being drawn to the anode. Figure 6 shows this to be the case for aluminum oxide sandwiches. The role of traps22,23 in the oxide film in determining the experimental current-voltage characteristics is un certain. The relative absence of traps is a requirement for observing space-charge-limited currents in an insulator. Anodic oxide films generally have an amor phous structure that gives no identifiable electron or x-ray diffraction pattern. The density of traps in such structurally irregular films may be very high. In addi tion, anodic films formed in molten bisulfate have about 15% of sulfate which could provide trapping sites. In spite of the possible high density of traps, fair agree ment between experiment and a theory for trap-free insulators is found for oxide films that have maximum conductivity. Some mechanism for the effective neutralization of trapping sites in the oxide appears to minimize trapping of electrons in the conduction band. Electrons injected by an ohmic contact into an insulat ing film may neutralize many of the electron traps in the insulator provided No is greater than the number of traps. Such neutralization of traps in the oxide film would make possible the observation of space-charge limited currents in thin films of highly amorphous and impure substances. The sensitive dependence of current On film thickness predicted by Eq. (1), I 0:: ljd3, is not confirmed quantita tively. However, thick films (>600 A) generally have much lower maximum currents than can be developed in thin films ( <300 A) made under the same conditions. Establishment of Oxide Conductivity Space-charge-limited currents in thin oxide films re quire a low barrier to the emission of electrons from the metal into the insulator. Simple band theory of a metal insulator contact predicts that the barrier should be '" -x, the difference between the work function of the Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2680 T. W. HICKMOTT metal and the electron affinity of the insulator. For clean AI,cJ>=4.2 VI9while oxidized aluminum has been reported (0 have cJ>=5.4 V.:J4 Values of the electron affinity of oxides are poorly known. For BaO and SrO, x"",0.5 V,a6 while Mgoa6 has an electron affinity of """ 1 V; the value for Ab03 has not been determined. Figure 16 shows that electron emission from AI-AI20a Au sandwiches can occur when only 2.5 V are applied across the film. If emission were from electrons that have passed through the oxide, into the gold anode, and then out into the vacuum, as in Fig. 15, no emission should be detected below 4.7 V, the work function of gold. However, since very thin evaporated gold films «100 A) have many small holes,!1 electron emission can occur at 2.5 V, two volts less than the work function of gold, if the electrons escape into vacuum directly from the conduction band of the oxide through pinholes in the gold film.37 For this to happen, the maximum value of the electron affinity of aluminum oxide is 2.5 V; electron emission can occur if X is less than 2.5 V and the electrons lose significant amounts of energy in traversing the oxide. If 2.5 V is the maximum value of x, cJ> -X is greater than 1. 7 Vat the aluminum-aluminum oxide interface, and greater than 2.2 V at the gold aluminum oxide interface, assuming that cJ> has the same value as the work function in vacuum. Resisti vities of 1014 to 1017 !2-cm at 300 oK, observed for oxide sandwiches after anodization, are in agreement with such high barriers at the metal-insulator interface. On this model, establishment of conductivity and negative resistance in the oxide film by the application of a high field to the film in vacuum, results in the reduction of cJ>min to a value between 0.2 and 0.5 V. The barrier height is reduced by at least 1.2 V and possibly by as much as 3.5 V during forming. Reduction of the barrier will occur if a positive dipole is produced at the metal-oxide interface with the positive end of the dipole in the oxide. One possible source of a positive dipole is field ionization of neutral donors, homogene ously distributed throughout the bulk of the oxide film, to produce immobile positive traps. However, reason able numbers of donors, 1016 to 1018/cm3, will only reduce the barrier a few hundredths of a volt if they are fully ionized. To reduce the barrier height by more than 1.2 V, positively charged centers in the oxide need to be located close to the metal-insulator interface rather than distributed uniformly throughout the insulator. The reduction in the work function at a surface due to positive charges at a distance d from the metal surface is ~ V = 41rued, where e is the electronic charge and u is the number of positive charges per unit area. If d""" 10-7 34 A. F. Ioffe, Physics of Semiconductors (Academic Press Inc., New York, 1960), p. 229. 35 D. A. Wright, Proc. Phys. Soc. (London) 60, 13 (1948). 36 J. R. Stevenson and E. B. Hensley, J. App!. Phys. 32, 166 (1961). 37 R. N. Hall, Solid State Electronics 3, 320 (1961). cm, u""" 1013/cm2 positive charges will reduce the barrier by 1 V. Thus, reasonable numbers of positively charged states at the metal-oxide interface will lower the barrier to electron emission from the metal into the insulator, while positive centers homogeneously distributed throughout the thickness of the oxide film will not. From this point of view, establishment of film conductivity may result from the field ionization of initially neutral states to form immobile positive states at the metal insulator interface. Since the field-free oxide film will be electrically neutral, the number of positive centers at the surface and in the bulk of the oxide will equ~l the number of injected electrons. Surface states in the insulator, of the type discussed by Bardeen38 for semi conductors, may be the source of the positive centers. Leakage Currents in Anodized Films Anodic films with thicknesses between 150 and 1000 A generally have very high resistivities and expo nential current-voltage characteristics before the irrever sible establishment of conductivity. Experimental values of the currents for small applied voltages, 10-10 to 10-7 A/cm2, while small, are much larger than should be observed if electrons are thermally excited from the metal into the insulator, and very much larger than if valence electrons were thermally excited across the band gap. Tunneling of electrons from one metal film to the other would produce an exponential current voltage characteristic 4,5 but tunneling currents are insignificant for insulating films greater than 50 A thick when the value of cJ>min is greater than 1 V. FrenkeP9 showed that the lowering, by high fields, of the barrier to thermal emission of an electron from an impurity into the conduction band gave a conductivity U=Uo exp[(e3F/K)t/kT]' Vermilyea40 applied the theory to measurements of conductivity in anodized Zr02 films41 with good fit to the data, but he has recently criticized the validity of the agreement with experi ment.2 Tunneling of electrons from donor impurity levels in the oxide42 is an alternate process that provides semiquantitative agreement with experimental current voltage curves for freshly anodized oxide films. For oxide films that are greater than 50 A thick, great care will be needed to distinguish between expo nential current-voltage characteristics due to tunneling of electrons from one metal electrode to the other, due to tunneling combined with thermionic emission, or due to field ionization of impurities in the oxide. The mere existence of an exponential current-voltage characteristic in insulating films is not sufficient 38 J. Bardeen, Phys, Rev. 71, 717 (1947). 39 J. Frenkel, Phys. Rev, 54, 647 (1938), 40 D. A, Vermilyea, Acta Met. 2, 346 (1954). 41 A. Charlesby, Acta Met. 1, 348 (1953). 42 W. Franz, Handbuch der Physik (Springer-Verlag, Berlin, 1956), Va!. 17, p. 155. Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsNEG A T I V ERE SIS TAN eEl NTH I NAN 0 DIe 0 X IDE F I L M S 2681 evidence that electron tunneling from one metal to the other is the dominant conduction process.5 Dielectric Breakdown The establishment of space-charge-limited currents in thin insulating films presents certain difficulties in determining their dielectric breakdown strength. The breakdown strength is usually determined experi mentallyl7,43 as a value of the electric field at which a rapid and destructive increase of current through the insulator occurs. In establishing the conductivity of oxide films by applying a voltage, as in Fig. 2, a rapid and irreversible current increase occurs at fields of around 106 V / cm, a rather small breakdown strength. The current increase looks like dielectric breakdown. However, the current increases are not destructive and the simultaneous establishment of a negative resistance region shows unequivocally that dielectric breakdown has not occurred. Instead, destructive breakdown of the films occurs at higher voltages, as in Fig. 4, and manifests itself in melting back of the gold counter electrode in small islands for dc voltages. Final break down of the aluminum oxide film causes a decrease in current through the film due to loss of contact with the gold electrode. Thus, it is somewhat uncertain what the definition of the dielectric strength of these films should be. Negative Resistance in Oxide Films Negative resistance similar to the characteristic for anodized aluminum has been found in five metal-oxide metal systems.# It should be of general occurrence in metal-insula tor-metal sandwiches provided certain limiting conditions are satisfied: (1) The insulator should be thin enough that fields needed to establish conductivity are less than the dielectric breakdown strength of the insulator. Maxi mum insulator thickness will be around 1000 A. (2) The insulator should be thick enough that dielec tric breakdown of the insulator does not occur at voltages below the voltage for peak current. This sets a minimum thickness around 80 A. (3) The band gap of the insulator should be larger than the voltage for peak current. Otherwise, currents due to band-to-band tunneling and other avalanche processes may swamp space-charge-limited currents and a negative resistance region. (4) Low dielectric constant insulators form a charac teristic most easily. For high dielectric constant materials, K> 25, breakdown to a high conductivity state may obscure the negative resistance region. 4. R. Stratton, Progress in Dielectrics (John Wiley & Sons, Inc., New York, 1961), Vol. 3, p. 233. 44 G. S. Kreinina, L. N. Selivanov, and T. I. Schumskaia Ra?iotekh~ika i elektronika 5,1338 (1960), have reported negativ~ reslstance In AbO. films. A few insulators that might show negative resistance, if thin films without pinholes can be obtained, are MgO, Nb206, CaO, SrO, BaO, ZnS, and most of the alkali halides. Although predictions of the occurrence of negative resistance in metal-insula tor-metal sandwiches can be made, the mechanism responsible for the effect is essentially unknown. Experimental observations on aluminum oxide may be summarized: (1) Location of the voltage for peak current is independent of film thickness. It is a voltage phenomenon not a field phenomenon unless the field is across a surface barrier whose thickness is independent of film thickness but dependent on dielectric constant. (2) The shape of the current-voltage curve is inde pendent of temperature although the peak current decreases with decreasing temperature. Thermal effects do not appear to determine the negative resistance since a seventy-fold decrease in power dissipated in the oxide film leaves the shape of the current-voltage characteristic unchanged, as shown in Fig. 6. (3) Electron emission into the vacuum seems to be closely coupled with onset of dc negative resistance. (4) Current decrease is exponential in voltage through the negative resistance region. (5) Switching from peak to valley occurs in less than 0.5 J.lsec and current through the film can be increased or decreased in J.lsec. On the other hand, a 60-cycle voltage does not trace out the negative resistance. Processes with widely varying time constants may be involved in both conductivity and in the observation of negative resistance. (6) Qualitatively, current-voltage curves in the negative resistance region are independent of polarity or of the electrode metal. This eliminates the possibility that a p-n junction in the oxide is responsible as in tunnel diodes. The precise region of the voltage for maxi mum current depends on the metal electrode. (7) Negative resistance regions are of general occurrence in metal-insulator sandwiches. Some possible mechanisms can be suggested. Negative resistance in aluminum oxide above 2.9 V may be due to an increase in tPmin. Tunneling of electrons from the metal into positive states located in the insulator at a distance of "'" 10-7 cm from the metal would neutralize them and increase tPmin. An increase of only 0.1 V in tPmin will reduce the number of thermally injected elec trons by nearly a factor of 100. Such tunneling would be extremely rapid and could be voltage dependent if the voltage drop in the oxide were primarily across a surface layer independent of total film thickness. An alternative source of a decrease in oxide conductivity is voltage-sensitive traps, traps that become effective when the free charge carriers reach a certain voltage. Further experiments are necessary to determine whether either of these possible mechanisms is correct or whether Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions2682 T. W. HICKMOTT some other phenomenon such as a decrease in carrier mobility is responsible for the occurrence of negative resistance in metal-insulator-metal sandwiches. The possibility that singular sites in anodic oxide films are the primary factors in determining current-voltage characteristics must also be explored.2 ACKNOWLEDGMENTS Discussions with many people at the General Electric Research Laboratory have assisted in this work. In particular, L. Apker, I. Giaever, W. A. Harrison, R. H. Pry, J. J. Tiemann, and D. A. Vermilyea have offered advice and encouragement. JOURNAL OF APPLIED PHYSICS VOLUME 33. NUMBER 9 SEPTEMBER 1962 Interdiffusion of Gases in a Low Permeability Graphite at Uniform Pressure R. B. EVANS, III, G. M. WATSON, AND J. TRUITT Oak Ridge National Laboratory,* Oak Ridge, Tennessee (Received February 26, 1962) An experimental investigation of the interdiffusion behavior of gases in a low permeability graphite was performed by sweeping the opposite faces of a graphite septum with helium and argon at uniform pressure and measuring the diffusive flux of both gases. The objectives were to ascertain the diffusion mechanism, to verify the applicable equations and associated theories, and to determine the parameters required to use these equations. At all experimental pressures, contributions of both normal and Knudsen diffusion effects were detectable via the pressure dependence of the diffusion fluxes. It was found that a previously proposed I. INTRODUCTION A. Applied Processes THE experimental results described in this paper are not limited to any particular application; however, it may be pertinent to consider the applied problem which stimulated our initial interest in the transport behavior of gases in graphite. The problem concerns the maintenance of low concentrations of radioactive contaminants in the coolant gas streams of nuclear reactors. A partial solution involves attempts to segregate gases which surround the fuel elements (the contaminant source) from those serving as a heat transfer fluid. A simplified diagram of such a system is shown in Fig. 1. The fuel element system' is composed FIG. 1. Possible graphite canned-fuel element. -----* Operated for the U. S. Atomic Energy Commission by Union Carbide Corporation. 1 G. E. Lockett and R. A. U. Huddle, Nuclear Power 5, 112 (1960). dusty-gas model formed an excellent basis for correlating the results. The dusty-gas model yields flux equations which predict the diffusion behavior over a wide range of pressures for particular gas concentrations at the boundaries. Only two experimentally determined parameters (characteristic of the gases and graphite) are required. These are: an effective normal-diffusion coefficient obtained through interdiffusion experiments and a Knudsen coefficient obtained through single-gas (permeability) experiments. The procedures used to evaluate these parameters in terms of the experimental data are described in detail. of two graphite components: a low-permeability graphite sleeve and a uranium-fueled graphite body. During operation, the pressure in the annulus between the components is to be adjusted such that a forced flow of coolant gas enters the annulus through the sleeve. This should backflush the gaseous fission products which tend to diffuse out through the pores of the sleeve, while reducing the concentration driving force upon which the diffusion depends. Perhaps, for convenience and by coincidence (rather than by intent), a majority of the studies of the steady state interdiffusion behavior of gases within porous media have been carried out in the laboratory [under uniform or nonuniform total-pressure conditions (see Fig. 2)] with an experimental setup2 which is nearly identical to the scheme outlined in Fig. 1. For an experi ment, the fueled body is replaced by a source of a second gas and the operating conditions such as tem perature, pressures, and gas concentrations are not as drastic as those visualized for a nuclear reactor. Another interesting application involves the develop ment of methods to bring about a separation within a flowing gas mixture. Separations are accomplished under steady-state conditions by allowing a mixture to flow through screens or barriers composed of either a porous medium3 or a condensable third gas com- 2 The experimental apparatus described in Fig. 2 was first employed by E. Wicke and R. Kallenbach, Kolloid-Z. 97, 135 (1941 ). 3 Typical examples of barrier separation processes are de scribed by P. C. Carman, Flow of Gases Through Porous Media (Academic Press Inc., New York, 1956), p. 139. Downloaded 06 May 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.1732141.pdf

Thermal Dissociation Rate of Hydrogen W. C. Gardiner and G. B. Kistiakowsky Citation: J. Chem. Phys. 35, 1765 (1961); doi: 10.1063/1.1732141 View online: http://dx.doi.org/10.1063/1.1732141 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v35/i5 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 11 Apr 2013 to 152.14.136.96. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsMATRIX ELEMENTS FOR ASYMMETRIC ROTORS 1765 where use has been made of the fact that A,L, and 1/, are pure imaginary operators so that M'",=M." and el"I'!'''=0 unless p,~p,', p,~p,", p,'~p," ( -lV, P=parity of the permutation of p,'p,p," from Z, X, Y LeI"!'!''' A,"A., !'" = LAv" 1" eV",.1 (A.8) IJ.Jj" JIll and EM JI'IIleVIlJlyl = (p") S .. I TIN'" (l)=} (v") s (A.9) The other terms in TIN(l) arise from the decomposition of M .. , into trace, skew tensor (axial vector), and sym~ metric second rank traceless tensor. The term [S]I[I]m+[I]I[S](l} is treated in the same way, except that S and I commute. It yields three terms, one is absorbed in the dipole-dipole interaction, another in the Fermi interaction leaving only [T5N(l) XS]l,1 as a new form. THE JOURNAL OF CHEMICAL PHYSICS VOLUME 35, NUMBER 5 NOVEMBER, 1961 Thermal Dissociation Rate of Hydrogen W. C. GARDINER, JR.,* AND G. B. KISTIAKOWSKY Gibbs Chemical Laboratory, Harvard University, Cambridge 38, Massachusetts (Received February 3, 1961) The thermal dissociation rate of hydrogen in xenon-hydrogen mixtures has been studied with shock-wave techniques over the temperature range 30CJ0-4500oK. The observed density profiles were consistent with the rate constant expressions: H2+Xe=Xe+2H H2+H2=Hz+2H Hz+H=3H k= 1.8X1017T"i exp( -V/ RT)cm3 mole-1 secl k=1.8X1Q20T-i exp( -V/RT)cm3 mole-1 sec-l k= 1.2X101sT-i exp( -V/RT)cm3 rnole-1 sec-l. THE dissociation of hydrogen molecules into atoms is of interest as the simplest of all chemical reac tions. Extensive studies of the reverse reaction have been carried out at low temperature, but a direct study of the dissociation reaction itself at shock-wave tem peratures has not been amenable to the usual shock wave techniques. On the one hand, it is difficult to heat hydrogen strongly with shock waves due to its high sound speed. On the other hand, the progress of the dissociation reaction is difficult to observe by previ ously used techniques. The electronic transitions available for absorption measurements are all at very short wavelengths, and an interferometric technique would be insensitive due to the low refractive index. The use of soft x-ray absorption for density measure ments in detonation waves has been shown to combine reliability and high time resolution.l Its application to the measurement of the hydrogen dissociation rate solved both the sound-speed problem and the analytical problem, since the use of xenon as diluent gas and • Present address: Department of Chemistry, The University of Texas, Austin 12, Texas. 1 G. B. Kistiakowsky and P. H. Kydd, J. Chern. Phys. 25, 824 (1956); J. P. Chesick and G. B. Kistiakowsky, ibid. 28,956 (1958). x-ray absorber decreased the sound speed of the experi mental gas to a point where strong incident shocks could be obtained with moderate driver-gas pressures. We have assumed the dissociation to proceed by three bimolecular mechanisms: H2+Xe-+Xe+2 H, H2+H2--tH2+2 H, H2+H--t3H. (1) (2) (3) Recent theoretical studies2 have shown that such a simple scheme certainly does not represent the correct mechanism for the dissociation of diatomic molecules. It is clear that the rate-determining step is the gradual collisional activation to successive vibrational levels until the dissociation limit is approached, rather than a direct transition from the ground or first vibrational level to the continuum. The details of the collisional activation process, however, are still so uncertain that there is no immediate prospect of making reliable rate 2 E. V. Stupochenko and A. T. Osipov, J. Phys. Chern. U.S.S.R. 32, 1673 (1958); E. M. Montroll and K. E. Shuler, Advances in Chern. Phys. I, 361 (1959); E. E. Nikitin and N. D. Sokolov, J. Chern. Phys. 31, 1371 (1959); and others. Downloaded 11 Apr 2013 to 152.14.136.96. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1766 W. C. GARDINER, JR., AND G. B. KISTIAKOWSKY TABLE I. Shock conditions before dissociation. Shock Shock Shock Run Mixture velocity pressure temperature (m/sec) (atm) (deg K) 12 4:1 1346 0.6081 4350 13 4:1 1358 0.5926 4420 14 4: 1 1313 0.6197 4155 15 4: 1 1438 0.5190 4910 18 4:1 1405 0.5334 4700 19 4: 1 1413 0.5485 4750 20 4:1 1510 0.4887 5360 21 1:1 1890 0.3629 4670 22 1: 1 1583 0.5286 3430 23 1:1 1735 0.4830 4015 24 1:1 1747 0.4827 4065 25 4:1 1364 0.5945 4455 26 1:1 1518 0.5062 3190 calculations using this model. For the purposes of data analysis there would be no advantage in doing so, since dissociation-rate data are not capable of distinguishing between "one-shot" and ladder climbing models.3 We shall therefore use the classical collision theory expres- sion for the dissociation-rate constants k=CT(!-s) exp(-DIRT), (4) where D=dissociation energy, C=a constant, and s=! the number of classical degrees of freedom of internal energy which can contribute to the dissocia tion energy, if itlis assumed that relative translational energy only along the line of centers can contribute. Ideally, an experimental study of the dissociation rate, should provide values for both sand C. In prac tice, however, it is difficult to extend measurements over a sufficiently wide temperature range to obtain an independent measurement of s. In this investigation we have assumed values of s and'determined the con stants C in (4) for reactions (1)-(3). S~paration of the"rate constants for reactions (1) and (2) would be accomplished in an ideal case by observ ing the initial dissociation rates in mixtures containing almost 100% xenon and almost 100% hydrogen, re spectively. With these two rate constants determined, the hydrogen-atom rate constant could be found by observing the later stages of the dissociation reaction. The limitations of our apparatus, however, forced a retreat to considerably less ideal conditions. At high xenon concentrations the density changes due to dissociation become small. At high hydrogen concen trations the sound speed of the gas increases to a point where exceedingly high driver-gas pressures are re quired to produce shocks strong enough to dissociate a E. V. Stupochenko and A. T. Osipov, reference 2. appreciable fractions of the hydrogen. Preliminary experiments indicated that the former problem would become serious when the xenon content exceeded 80%, and the latter problem would become serious when the hydrogen content exceeded 50%. The compositions used in the experiments reported here were xenon 82.97%, hydrogen 17.03%, xenon 52.33%, and hydro gen 47.67%. It should be pointed out that since the increase in hydrogen-atom concentration through the shock is accompanied by a very substantial decrease in tem perature, any misassignment of temperature depend ence to the rate constants (i.e., any error in s or D) will result in an incorrect assessment of the effect of the growing hydrogen-atom concentration on the over-all rate. This inseparability of temperature and hydrogen atom effects is the major source of uncertainty in the present experimental method. The rate constant for dissociation upon collision with hydrogen atoms which is derived from the results of these experiments is therefore meaningful only to the extent to which the assumed temperature dependence of the rate constants is correct. This point will be discussed again later. EXPERIMENTAL The apparatus described previouslyl was rebuilt to permit shock-wave studies under clean vacuum condi tions. The shock tube was made from 3-in. i.d. Rockrite steel tubing and had a 10-ft driver section and an l1-ft expansion section. It could be pumped to a vacuum of less than 10-5 mm Hg and had an outgassing rate of 2-3 M/hr. It was cleaned thoroughly with various solvents before experiments were started. The two sections of the tube were separated by Kodapak cellulose acetate diaphragms. Before each experiment all diaphragm fragments from the previous experiment were carefully removed from the tube. Shock velocities were measured with a series of five ionization gauges equally spaced upstream and down stream of the x-ray slits. The signals were displayed on a raster sweep oscilloscope. Attenuation of the wave was found to be less than 0.2% in 50 cm. A Machlett AEG 50-A x-ray tube with a copper anode was operated at 28 kv, 21 ma emission current to supply the x-ray power. Sixty-cycle ripple in the x-ray output intensity, a source of data scatter in the previous apparatus, was reduced to less than 5% by using a dc filament supply to the x-ray tube and strongly filtered high voltage. A redesigned slit system allowed the entire solid angle of radiation emitted by the tube to be utilized for absorption measurements. The x-ray beam was defined by lo5-mm slits through the wall of the shock tube. The curved O.OIO-in. beryllium windows were recessed 0.012 in. back from the inside wall of the shock tube. This indentation was the only irregularity presented to the gas flow. The intensity of the x-ray beam emerging from the shock tube was measured by a detector consisting of a plastic scintillator block (Plastic Scintillator B, Pilot Downloaded 11 Apr 2013 to 152.14.136.96. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsDISSOCIATION RATE OF HYDROGEN 1767 Chemical Company) machined into a truncated rec tangular pyramid and a DuMont 6364 photomultiplier tube. To minimize fluorescence loss the sides and top of the pyramid were coated with a thin layer of alu minum. The photomultiplier tube signal was converted to low impedance by a cathode follower and displayed on a Tektronix 531 oscilloscope. A single sweep of the oscilloscope was triggered by the arrival of the shock at a platinum resistance gauge upstream from the slits and was photographed. A circuit for supplying accu rately measured simulated photomultiplier currents to the cathode follower allowed calibration lines to be traced on the same photograph. The effective extinction coefficient of xenon for the 28-kv x rays was determined by static calibration. Ex periments with shock waves in pure xenon confirmed that density measurements correct to about 3-4% could be made in single experiments, and that there were no systematic errors. A shutter prevented the x-ray beam from passing through the shock tube until about 5 msec before the arrival of the shock wave. Synchronization of the shutter and the shock was accomplished by allowing a signal from the opening shutter to trigger a thyratron driven, solenoid operated, diaphragm slitting device. Experimental mixtures were prepared in a glass vacuum system of conventional design. Hydrogen was prepared from uranium hydride.4 Xenon was purified by gettering over titanium at 900°C. Mass spectro metric analysis of the gas mixtures indicated that the impurity level was below 0.005%. The compositions of the mixtures were determined by careful manometric measurements during their preparation. RESULTS Thirteen experiments were analyzed in this study. Observations were made in incident shocks only. The conditions in these shock waves before the onset of dissociation are summarized in Table I. Shock densities and temperatures were determined from the measured shock velocities by interpolation on graphs constructed using the Rankine-Hugoniot equations and standard enthalpy tables.s It was assumed that the vibrational relaxation of the hydrogen was complete before dis sociation was appreciable.6 Sample oscilloscope records are shown in Fig. 1. Density values were calculated from photomultiplier currents read from the records at 5-or 10-,usec inter vals. Reference point for the density measurements was the final density at dissociation equilibrium. During the dissociation reaction all the hydro dynamic and thermodynamic parameters change by large amounts. Under these circumstances data reduc tion becomes a complicated affair. The arguments in- 4 F. H. Sped ding et al. , Nucleonics 4, 4 (1949). • S~lected Values of Chemical Thermodynamic Properties (NatIOnal Bureau of Standards Washington D. C. 1952) Ser. III. "" 6 M. Salkoff and E. Bauer, J. Chern. Phys. 29, 26 (1958). FIG. !. Sample ~scilloscope records. Density increases up wards. 1< ull sweep = ;,00 /Lsec. volved are described in Byron's work on oxygen.7 The adva~tages of a digital computer for analyzing the expenmental data in a situation like this are obvious. We were fortunate in receiving an offer from the United Aircraft Corporation Research Laboratories to use their computation facilities and a program written by R. W. Patch for integrating the hydrodynamic and kinetic equations describing a shock wave in a mixture of dissociating diatomic gas and inert gas. This program accepts an assumed set of rate constants and the condi tions in a shock before dissociation commences as input and computes the reaction profile through the wave until dissociation equilibrium is attained. The data reduction procedure used in this work was therefore a series of comparisons between computed and experi mental density profiles, revising the input-rate con stants each time until agreement between computed and experimental density profiles was obtained. Experimental values for s in Eq. (4) could not be obtained in this work. The pre-exponential temperature dependence of the rate constants was taken to be the same as that obtained by Byron in his \york on the dis sociation rate of oxygen.7 For the atom-molecule reac tions, we assume s = 1 and for the molecule-molecule reaction s=2, values which are in accord with the classical collision-theory analysis.8 Of the several studies of the gas phase recomBination rate of hydrogen atoms the most reliable appears to be that of Farkas and Sachsse.9 These workers studied the room-temperature recombination rate in the presence of argon as inert gas by a photochemical steady-state method. They found that the rate con stants for argon, hydrogen molecules, and hydrogen atoms were the same within the accuracy of their mea~urements. If it is assumed that the third body effiCiency of xenon for recombining hydrogen atoms is 7 S. R. Byron, J. Chern. Phys. 30, 1380 (1959). 8 R. H. Fowler and E. A. Guggenheim Statistical Tlzermo- 1r.amics (Cambridge University Press, Ne\~ York, 1952), Chap. 9 L. Farkas and H. Sachsse Z. physik. Chern. (Leipzig) B27, 111 (1934). ' Downloaded 11 Apr 2013 to 152.14.136.96. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1768 W. C. GARDINER, JR., AND G. B. KISTIAKOWSKY 14 RUN 20 25 --- ,..",.--;;: 1,4 g. ....... RUN 23~: ~O. RUN 24 RUN 25 " ....... !4~ ...... . ...... ..I..-_....I.....L. ..... 10 17 ...... "'--..1..-..1..-_....1.. 14 '{}"-_ .... _. ...:1"0" _----10....... .-2 n../ b· 1.8.IBO Y RUN 23 c· 2 .(}{)---""" .. {J'~--~:-;- / b-18 RUN 23 C' .2,20 12 L..;..J.......L....L...L....L 21 -~~ .... ~;:,~---rI , .... r/ 1/ • -2 1,6 .-2 b • LO,IOO , b • 18 C • 2 ,e .. 2,20 RUN 18 RUN 18 RUN 18 '41-..I..-_....I.. ....... ...L 14 ..... ..I..-..&-....I.....&..-I. 14 ..... ..I..-..&--'-...&..""""" I--~Of·8C4 FIG, 2. Comparison of calculated and experimental density profiles. Abscissa = laboratory time, Ordinate = density, g/ccX 1()5 .••• = equilibrium density. --=computed using final rate constant expressions. -----=Computed using the rate-constant expressions kx.=aXI017r-t exp(-D/RT) , kBl=bX1019 :r-t exp( -D/RT), kH=cXI017r-t exp( -D/RT). the same as the third body efficiency of argon, then the recombination-rate constants of Farkas and Sachsse can be converted to the bimolecular dissociation rate constants kXe=2XI0l7T-!exp(-DIRT) ems mole-I seci kH2=1.8XI02or-! exp( -DIRT) cm3 mole-l seci kH=2XlO17T-l exp( -DIRT) ems mole-I seci. D=4.4769 ev. These expressions for the rate constants were taken as a starting point for the first trial integrations. Density profiles were computed for runs 18, 19, 22, and 23. Additional integrations were carried out with increased and decreased values for each of the rate constants separately to determine the magnitude of the effect that such variations would have on the density profile. A check was made to ensure that reducing the size of the integration increment used had no effect on the results. The density profiles computed for runs 18, 19, and 23 were in reasonable agreement with the experimental results, whereas there was serious disagreement in run 22. The lack of agreement in 22 is discussed below, It appeared that the agreement between calculation and experiment could be improved by reducing the xenon rate constant and increasing the hydrogen-atom rate constant. The former was reduced by a factor of 0.9 and the latter was increased by factors of 6 and 10 for the second set of trial integrations. These were carried out for runs 14, 20, and 21. Comparison of the results indicated that more satisfactory agreement was ob tained with the sixfold increase in hydrogen-atom efficiency. The serious disagreement in run 20 is dis cussed below. At this point all of the density records were compared with one another and with the computed density pro files. It was decided that no significant improvement in the agreement between computed and experimental density profiles would be obtained by further adjust ment of the rate constants. Density profiles were then computed for all 13 experiments using kxe=1.8XI0I7T-!exp(-DIRT) ems mole-I seci kH2=1.8X1020r-J exp( -DIRT) ems mole-I seci kH=1.2XlOlsT-! exp( -DIRT) ems mole-l secl • These profiles and the corresponding experimental data are shown in Fig. 2. The solid lines are computed profiles corresponding to the final rate constants. The other lines correspond to rate constants altered as indicated. DISCUSSION The classical collision theory expression for the dis sociation rate constant isS k=pu2(87rRTlp.)i Sl-I(DIRT)8 exp( -DIRT), (5) Downloaded 11 Apr 2013 to 152.14.136.96. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsDISSOCIATION RATE OF HYDROGEN 1769 where Il=reduced mass of colliding pair, u=collision diameter, p = 1 for collisions between unlike molecules and! for collisions between like molecules. Equation (5) allows the experimental rate constants to be con verted into effective collision diameters for the three collision partners. The collision diameters are thus found to be for xenon 1.2 A, for hydrogen molecules 0.31 A, and for hydrogen atoms 2.6 A. The exceptionally high value for hydrogen atoms is not in agreement with the low-temperature recombination results of Steiner and Wicke.10 The form of data reduction used does not lend itself readily to a determination of the error limits within which the rate constants have been measured. An estimate of this can best be made by comparing the agreement between experimental and computed density profiles with the variations in computed density profile effected by altering input-rate constants. The rate constant for xenon is of course the most closely deter mined and is unlikely to be incorrect by as much as a factor of 2. The rate constants for hydrogen atoms and molecules are probably correct to a factor of 4, if the assumed temperature dependence of the rate constants is correct. To the extent that the classical collision theory is an adequate description of the dissociation process it would seem unreasonable to ascribe such a relatively large collision diameter to the hydrogen atom as follows from the above expressions for the rate constants. The most uncertain factor in the interpretation of our results lies in the temperature dependence of the rate constants. During the course of the dissociation the temperature changes by as much as 14000 • This means that any misassignment of temperature de pendence will lead to a misinterpretation of the relative contribution of the hydrogen atom reaction to the over-all dissociation process, since changing this was the means used to adjust the later portion of the density profiles to agree with the experimental data. There are two indications that the temperature de pendence that we have assigned to the three rate constants may not be correct. First, there is a uniform tendency for the experimental points to lie below the computed density profiles at the start of the dis sociation.H This means that the rate of dissociation at 10 W. Steiner and F. W. Wicke, Z. physik. Chern. Bodenstein Festband, 817 (1931). 11 In the coolest shocks in the 1: 1 mixture (runs 22 and 26) the observed density after 5 pSec is in fact less than the calculated initial density before dissociation starts. If this is a real effect, then it is also an interesting one. An obvious interpretation would be incomplete vibrational equilibration. Theoretical cal culations indicate, however, that vibrational relaxation in hydro gen is too fast under these conditions to be observed on the time scale of our experiments.6 An independent experimental measure ment of the hydrogen vibration relaxation time would be required to check this point. A complication due to incomplete rotational equilibration would seem still more unlikely, even though rota tional relaxation in hydrogen and translational energy exchange between xenon and hydrogen are inefficient processes. As a result of the extreme difference in mass between hydrogen and xenon the two gases are thermally insulated from one another in passing through the shock front. Immediately behind the wave the the higher temperatures prevailing at the start of the dissociation may in fact be lower than it should be compared to the rate later on. Moreover, there is sig nificant disagreement between computed and experi mental density profiles for the hottest and coolest shocks (runs 15, 20, 22, and 26). The sense of the dis agreement is such as to indicate that the computed rates are too slow at low temperatures and too fast at high temperatures. The agreement with experiment could be improved by reducing the temperature dependence of the rate constants.l2 For a start it would be quite reasonable to increase the value of s. Camac and Vaughn 13 have recently completed a detailed study of the dissociation of oxygen by argon over the temperature range 3300°- 7SoooK and find s=1.S±0.2. We would not feel justi fied in recomputing all of our results with s = 1.5 rather than 1.0, since there is no strong analogy between collisions where the masses of the collision partners are about equal and collisions where the mass of one partner is 65 times the mass of the other. Qualitatively, however, we see that such a change would reduce the increase in the xenon rate between 30000 and 50000 by about 30%. This in itself would allow the disagreement at the highest and lowest temperatures to be reduced considerably. Furthermore, it is by no means impossible that the activation energy for the dissociation might be less than the spectroscopic dissociation energy. The usual requirement that the activation energy of an endo thermic reaction must be at least equal to the endo thermicity of the reaction presupposes that the reacting molecules are in thermal equilibrium with the heat bath which defines the temperature of the system. This condition, however, is almost certainly not fulfilled in the situation at hand, since it has been shown that the dissociation causes an extensive depopulation of the highest vibrationallevels.2 It does not seem unreason able to suggest that the process of depopulating the high vibrational levels may be so much faster than the vibrational excitation process, since the former can readily occur by thermal collisions, that all molecules which reach the highest discrete level can just as well be counted as part of the continuum. This is supported xenon temperature is therefore much higher than the hydrogen temperature. Translational relaxation to a common tempera ture, however, would be expected to be complete in a few hundred collisions at most. Rotational equilibration in hydrogen requires about 300 molecular collisions at room temperature [W. Griffith, J. Appl. Phys. 21, 1319 (1950) J. Under the conditions of these experiments 1 J.tsec laboratory time corresponds to about 20 thousand molecular collisions, so that processes on the time scale of hundreds of collisions are not observable. It might be possible to observe a combined translational-rotational relaxation in low pressure shock waves in xenon-hydrogen mixtures. 12 In the limited temperature range studied here it would be quite difficult experimentally to determine which temperature dependent term, i.e., activation energy or s, should be adjusted. These two terms are separable from one another, however, if a wider temperature range is covered. 13 M. Camac and A. Vaughn, Avco-Everett Research Labora tory Research Rept. 84, AFBMD-TR-60-22 (1959); J. Chern. Phys. 34, 448, 460 (1961). Downloaded 11 Apr 2013 to 152.14.136.96. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions1770 W. C. GARDINER, JR., AND G. B. KISTIAKOWSKY by Osipov and Stupochenko's conclusion that transi tions from the highest vibrational levels to the con tinuum are much more probable than transitions between adjacent discrete levels near the dissociation limit.2 The net result of such a situation would be a slight reduction in the activation energy for the dissocia tion process as a whole. This, in turn, would easily alter the temperature dependence of the rate constants enough to allow the highest and lowest temperature ex periments to be described by the same set of rate constants. We hesitate, however, to ascribe the disagreement between experimental and computer results to such an effect without further investigation. The strongest argument for this is the excellent agreement with ex periment obtained in the oxygen-dissociation study of Camac and Vaughn using an expression of the form of Eq. (4) and the spectroscopic dissociation energy. The upper vibrational levels are more closely spaced in oxygen than in hydrogen, which would mean that any depopulation effect should show itself more strongly in the former. Unless some new evidence becomes avail able, then, we would be reluctant to propose an activa- THE JOURNAL OF CHEMICAL PHYSICS tion energy for the hydrogen dissociation reaction other than the spectroscopic dissociation energy. A large share of the disagreement observed between computed and experimental density profiles for the highest and lowest temperature shocks can be ac counted for as indicating that higher values of s in Eq. (1) are more appropriate to the hydrogen-dissocia tion process. For the present we interpret the remaining disagreement to indicate the presence of an unap preciated experimental difficulty. It does not seem likely that there could be a complication in this tem perature range due to incomplete vibrational relaxation of the hydrogen4 or ionization or electronic excitation of the xenon.14 It is hoped that an extension of the temperature and composition range studied in the ex periments reported here may resolve the difficulty. ACKNOWLEDGMENTS We are indebted to R. W. Patch of the United Air craft Corporation Research Laboratories for his generous assistance with the computations and to P. H. Kydd for helpful discussions. 14 W. Roth and P. Gloersen, J. Chern. Phys. 29, 820 (1958). VOLUME 35, NUMBER 5 NOVEMBER,1961 Proton Magnetic Resonance Study of Ferroelectric Potassium Ferrocyanide Trihydrate R. BLINC,* M. BRENMAN, t AND J. S. WAUGHt Department of Chemistry, Laboratory of Chemical and Solid State Physics, and Research Laboratory of Electronics,§ Massachusetts Institute of Technology, Cambridge, Massachusetts (Received April 13, 1961) Proton magnetic resonance and vibrational spectra of potassium ferrocyanide trihydrate have been studied between 77° and 300oK. From the observed two sets of O-H stretching frequencies it has been possible to suggest the probable positions of four previously undetermined water molecules in the unit cell. Changes in the vibrational spectra and in the proton resonance second moment in the neighborhood of the Curie point indicate that the onset of ferroelectric behavior is associated with a dynamical orientational ordering of the hydrogen-bonded water molecules. Apparently there is also a contribution to the spontane ous polarization and internal field from displacements and polarizability of the K+ and Fe (CN) ,-4 ions. INTRODUCTION FERROELECTRIC phenomena in hydrogen bonded crystals have been extensively studied in recent years, and every year several new ferroelec tries are discovered. However, our understanding of the basic interactions and the nature of ferroelectricity is not increasing so rapidly, and no comprehensive theory yet exists. In order to improve this situation it * Alfred P. Sloan Postdoctoral Fellow in the School for Ad vanced Study, on leave from the Institut J. Stefan, Ljubljana, Yugoslavia. t Buenos Aires University Postdoctoral Fellow, on leave from the National Microbiological Institute, Buenos Aires, Argentina. t Alfred P. Sloan Research Fellow. § This work was supported in part by the U. S. Army (Signal Corps), the U. S. Navy (Office of Naval Research), and the U. S. Air Force (Office of Scientific Research, Air Research and Development Command). seems highly desirable to clarify the role of hydrogen atoms in the ferroelectric transition, i.e., to get precise knowledge of the distribution and dynamics of protons in a number of ferroelectric crystals. Solid-state proton magnetic resonance studies pro vide an ideal approach to this problem in many ways, particularly if the interproton distances are of the same order of magnitude as the distances over which important protonic motions occur. In contrast to the situation in ferroelectrics of the KH2P04 type, this condition is obviously fulfilled in the object of the present investigation, potassium ferrocyanide tri hydrate, K4Fe (CN) 6· 3H20 (hereinafter abbreviated KFCT), which has recently been found to show ferro electric behavior below -24.S°C. ' 1 S. Waku, H. Hirubayashi, H. Toyoda, and H. Iwusaki, J. Phys. Soc. Japan 14, 973 (1959). Downloaded 11 Apr 2013 to 152.14.136.96. 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1.1702716.pdf

GalvanoThermomagnetic Phenomena. IV. Application to Anisotropic Adiabatic Nernst Generators T. C. Harman, J. M. Honig, and B. M. Tarmy Citation: Journal of Applied Physics 34, 2215 (1963); doi: 10.1063/1.1702716 View online: http://dx.doi.org/10.1063/1.1702716 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theory of GalvanoThermomagnetic Energy Conversion Devices. V. Devices Constructed from Anisotropic Materials J. Appl. Phys. 34, 2225 (1963); 10.1063/1.1702717 Erratum: OPERATING CHARACTERISTICS OF TRANSVERSE (NERNST) ANISOTROPIC GALVANOTHERMOMAGNETIC GENERATORS Appl. Phys. Lett. 2, 44 (1963); 10.1063/1.1753765 Theory of GalvanoThermomagnetic Energy Conversion Devices. III. Generators Constructed from Anisotropic Materials J. Appl. Phys. 34, 189 (1963); 10.1063/1.1729064 Theory of Galvano-Thermomagnetic Energy Conversion Devices. I. Generators J. Appl. Phys. 33, 3178 (1962); 10.1063/1.1931132 OPERATING CHARACTERISTICS OF TRANSVERSE (NERNST) ANISOTROPIC GALVANO THERMOMAGNETIC GENERATORS Appl. Phys. Lett. 1, 31 (1962); 10.1063/1.1753692 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:39FA 11 L T I :\ G A:"-J DEL E C T R 0 L U MIN ESC E N C E I:"-J Z n S PH 0 S P H 0 R S 2215 3-1ayer system, is at 120°. The calculated functions shown are for the indicated Jagodzinski parameters lX and (:3. The misfit was not essentially improved by changes from these values and has been attributed to heterogeneity.13 Series 1 granules showed sharper maxima for both cubic and hexagonal phases than did those in Series 2. In both Figs. 6 and 7 there is indication of the superposition of scattering from coherent and extensively disordered material. The observed scatter ing can be better fitted by the addition of two Jagod zinski functions, each with its own set of lX, (:3 param eters, but this approach was not pursued any further. The diffractometer patterns obscure much, or most, of the disorder in the background; the line broadening of whatever hexagonal material is left contains only a fraction of the total disorder information. The bright EL powders were of this type; their granules show very low-order birefringent colors which could easily be mistaken for isotropy. The close connection between Cu entry and faulting suggests that Cu enters by means of, and perhaps resides in, defects due to faulting. This view has been summarized and particularly emphasized recently.6 Where faulting is much less, or nonexistent, the EL activity may be associated with other defects,21 with due reservations concerning the accuracy of the structural identifications. A by-product of faulting may be the significant surface roughening reported above which could assist in releasing light from the highly refractive (n= 2.37) crystal. ACKNOWLEDGMENTS We wish to express appreciation to F. Palilla of this Laboratory for his help with problems in the phosphor technology. 21 A. Wachtel, ]. Electrochem. Soc. 107, 682 (1960). JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 8 AUGUST 1963 Galvano-Thermomagnetic Phenomena. IV. Application to Anisotropic Adiabatic N ernst Generators T. C. HARMAN, ]. M. HONIG, AND B. M. TARMY* Lincoln Laboratory,t Massachusetts Institute of Technology, Lexington 73, Massachusetts (Received 24 October 1962; in final form 14 February 1963) Tensor components have been derived for the matrix relating entropy flux and gradient of electrochemical potential to gradient of temperature and electrical current. The entries have been specified in terms of six basic galvano-thermomagnetic transport coefficients; the Casimir~Onsager reciprocity conditions have been utilized to reduce the number of independent coefficients. These relationships have also been employed to extend the Bridgman-Heurlinger identities to anisotropic materials. Using these equations along with the phenomenological relations, the characteristics for galvano-thermomagnetic Nernst generators operating under adiabatic conditions have been formulated. The results are compared with the equations which are valid for the isothermal case. INTRODUCTION WE present here a rather detailed derivation of the tensor for the representation of galvano-thermo magnetic (GTM) phenomena exhibited by anisotropic materials under the combined influence of temperature gradients, electric and magnetic fields. This is followed by a derivation of the Reurlinger-Bridgman relations, on the basis of which the operating characteristics of transverse Nernst generators, constructed from aniso tropic materials, and working under adiabatic condi tions, are investigated. In so doing, we extend to the present situation the methods employed in earlier publications.! In applying the theory of paper III, we are interested * Summer staff member. Permanent address: University of Pennsylvania, Philadelphia, Pennsylvania. t Operated with support from the U. S. Army, U. S. Navy, and U. S. Air Force. IT. C. Harman and J. M. Honig,]. App\. Phys. 33, 3178, 3188 (962); 34, 189 (1963), hereafter referred to as I, II, and III. in the tensor which connects the x, y, z components of the entropy flux J s and of the Fermi level gradient VY;=:Vf/e with those of the temperature gradient vT and the current density J. Rere, f is the electrochemical potential (Fermi level relative to an electron at rest at infinity) of the electrons. A schematic representation of the desired relations is given by lr !~A = r~ '~~( H,) ~~ ! TI" (If') ~ T I [~~ ~jl 'l"y; C9n,(H.) I PAA' (Hz) fA' l : J (A,A'=X,Y,z). (1) As is seen by inspection of (1), the matrix has been partitioned into four blocks, each of which contains nine entries. We distinguish between the two blocks -KIT and p on diagonal locations (DB) and those not on the diagonal (ODB). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:392216 HARMAN, HONIG, Al\D TARMY The phenomenological equations (1) have been written in partially inverted form, as originally suggested by Mazur and Prigogine2 and by Callen? The advantage of this formulation is that all entries in the tensor (1) can be given a simple physical interpretation, as is detailed later. For the present, we note that the entries in each block of the partitioned tensor are of the general form [Gxx(Hz) Gxy(Hz) Gxz(Hz)] Gyx(Hz) Gyy(Hz) Gyz(Hz) Gzx(Hz) Gzy(Hz) Gzz(Hz) [s xx(H z2)+ H zT .Ix(H z2) = Syx(H})+H,T"x(H z2) Szx(H})+HzTzx(H}) ... ] ... , (2) where G= -KjT, IIIT, (p, or p depending upon the block under consideration. As is mathematically per missible, Gu' has been split into contributions Su' and HzTxx'(}.., }..'=x, y, z) that are, respectively, even and odd with respect to reversal of magnetic field direction. One should note that T u' by itself is an even function of the magnetic field. Without loss of generality, it is assumed throughout that this field lies along the z axis. The entries in (1) and (2) are not independent of one another. One can prove by the simple method of paper III or by more elegant procedures4 that for the entries in the DB, G= -KjT and p, Gu' (Hz) =Gx'x( -Hz), (G= -KIT, p) (}.., }..' = x, y, z). (3) The relations between entries in the ODB are summa rized bya,6 (Px'x(Hz)='+IIn,(-Hz)IT (}.., }..'=x, y, z). (4) It must be emphasized that, in general, there exist no relations between entries within each of the ODB; as shown in (4), the Casimir~Onsager reciprocal conditions (CORe) apply to the entire matrix (1), and hence relate entries in different ODE. Only if the crystal exhibits a sufficiently high degree of symmetry will the more stringent conditions (3) also apply to the case where G=(P or IIIT.7 DIAGONAL ENTRIES At this point, we digress to consider some proper ties of the diagonal elements in each block. When Gu= -KuIT or pxx, condition (3) for }..=}..' reads Gu(Hz) =Sn(H})+ HzTn(H}) =Gn( -Hz) = Sn(Hz2) -HzTn(Hz2). If these conditions are to hold for arbi-trary Hz, one must require that Tn=O for }..=x, y, z. In other words, only the symmetric parts survive in the diagonal entries of the DB. To examine the general situation, let us carry out a rotation of the sample by an angle 0 about the z axis. The transformation matrix for this rotation is given by' [R(O)]= [~~~!O ~~:: ~]. o 0 1 (5) The entries in (2) before and after the rotation are re lated by a similarity transformation. Thus, if we first write the tensor (1) as a sum of even and odd matrices, then the latter entries before and after the rotation are connected as follows: [T xx T xy T xz] [T']= [R][T][R] = [R] Tyx Tyy T yz [R], Tzx Tzy Tzz (6) where the tilde symbol designates the transpose of [R(O)]. When (5) is substituted in (6) and the indicated matrix calculations are carried out, one obtains the following entries of interest: T xx' = cos20T xx+sin20T yy+cosO sinO(T Xy+ T yx), (7a) T yv' = sin20T xx+cos20T yy-cosO sinO(T Xy+ T yx), (7b) T xv' = cos20T xy-sin20T yx+cosO sinO(T yy-T xx), (7c) T YX' = cos20T yx-sin20T Xy+cosO sinO(T yy-T xx). (7d) Let us suppose that prior to rotation, the diagonal entries Txx=Tyy=O; Eq. (3), as applied to the odd entries, reads T Xy(Hz2)= -Tyx(Hz2). Hence, it emerges from (7a) and (7b) that T xx' = T vv' =0 regardless ofthe value of O. This is the situation encountered in the diagonal blocks. In general, the CORC are of no relevance for entries within a given ODB; under these conditions, there is no relation between T xy and Tyx; T xx' and T yy' will represent nonzero entries even if T xx and T yy vanish. Hence, all entries in ODB will, in general, contain both even and odd terms. Only for sufficiently great crystal symmetry, where T>.0Hz2) = TX'>.(Hz2) within a given ODB, are the odd entries missing from diagonal positions of the ODE. THE GTM TENSOR IN GENERAL FORM With the above remarks as a guide, we now return to our main discussion. The following partial matrices can be constructed, as explained below: [ Pxx Pxy+HzlSt xy pxz+HzlStxz] [PH,(Hz)]= Pxy-HzlSt xy pyy pyz+HzlSt yz , Pxz-HzlSt xz pyz-HzlSt yz pz. (8) 2 P. Mazur and I. Prigogine, J. Phys. Radium 12, 616 (1951). 3 H. B. Callen, Phys. Rev. 85,16 (1952); H. B. Callen, Thermodynamics (John Wiley & Sons, Inc., New York, 1960), Chaps. 16,17, 4 H. G. B. Casimir, Rev. Mod. Phys. 17, 343 (1945); R. Fieschi, Nuovo Cimento Suppl. 1, 1 (1955). 5 Consistency wi~h the results of ~II may be est.ablished as shown in the following example: According to III, £ol4(H ,) = -£41 (-H ,). Th1s seems to contradiCt Eq. (4) of th1s paper, but one must note that in matrix (4) of paper III, the entries G", read: Gn(II,) = -£ou(H z) and Gl4(H ,) = -£o41( -Hz), respectively, and these entries are consistent with Eq. (4) cited here. 6 R. Fieschi, S. R. de Groot, and P. Mazur, Physica 20, 67 (1954). 7 C. Herring, T. H. Geballe, and J. E. Kunzler, Phys. Rev. Ill, 36 (1958). 8 J. F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, 1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:39GAL V A :\T 0 -THE R !VI 0 MAG :\T E T [C P H E :\T 0 :'vI E N A. [V 2217 (9) (10) where we have used the definitions pxx= V' xif;1 I X)e} pyx: V' yif;1 I:)e IY= Iz= V' xT= V' yT= V'zT=O HziJtyx= V' yif;1 I )0 (lla) (llb) (llc) (lld) (lle) ( 11£) (llg) (llh) ( lli) ( llj) Kxx= -T I sxlV' .cT)e} Kyx= -T jsylV' xT)c Ix=Iy=jz= V' yT= V'zT=O Hz'Jrryx= V' yT IV' xT)o jx=]y= jZ= j sy= V'zT=O with analogous ones applying to the remaining sub scripts. It is noted that odd diagonal terms are missing from (8) for reasons detailed earlier. The entry Pxy+HziJt xy arises as follows: For anisotropic materials, J and vif; do not necessarily coincide in direction; Pxy thus correlates V' xif; with Iy under conditions where V'xT=Ix=Iz=O. This, however, does not fully take account of the ex perimental situation. For, in the presence of a magnetic field, an additional contribution to V' xif; arises from the Hall effect, as represented by V' xif;=HziJtxyjY when V'xT=jx=jz=O. For this reason, we have introduced the subscripts e and 0 in (llb) and (11c); the two contributions can easily be distinguished experimentally by reversal of the magnetic field direction. We see that when the entry PXy+HziJtXY is used in (1) under the indicated restrictive conditions, we obtain the relation V' xif;1 ]Y= (V' xif;1 jY)e+ (V' xif;! ]Y)o= PXy+ HziJtxy which conforms to the definitions (llb) and (11c). The first and second index refer to the components of the de pendent and independent variable, respectively. The entry Pxy-HziJt xy now follows directly from the CORC. Similar arguments yield the remaining terms in (8). The entries in (9) are slightly more complicated. Again, only even terms appear in the diagonal positions, and those in the off-diagonal slots are associated with the intrinsic anisotropy of the sample under considera tion. For example, we find from (9) and (1) that Kxy= -(T j s"/V'yT)e when V' xT= V'zT=]X=O, and this is fully in accord with the definition (11e). In con structing the odd entries, we are guided by the require ment that they must reduce to the form HzK/im[ in the isotropic case.! The subscripts are dictated by the following consideration: According to Eqs. (1) and (9), T j sx= -KxxV' xT -Kxy V' yT+HzKxx;;rrxyV' yT when V'zT =]X=O. If we now set j sx=O, we obtain V' xTIV' yT = (V' xT IV' yT) c + (V' xT IV' yT)o = -(K xyl K xx) + H Z;;rrXY' Here the ratio KxylKxx is even; hence we can identify (V' xT IV' yT)o with H Z;;rrXY' which, under the restrictive conditions enumerated above, conforms to (11f). Finally, as concerns (10), we note the presence of terms of the type H;;J(xx, and we again call attention to the fact that there is in general no relation between Cxx' and CA'A within this matrix. Entries in the lIlT block are constructed in accordance with requirement (4). If the crystal exhibits a sufficiently high degree of sym metry, however, the off-diagonal entries in (10) are re lated as shown in (3). A special case of this is repre sented by an isotropic material, where (Pxx' = O(A~ t..') and where -:nXX,=+:nA'A=:n(A~t..'). In view of our intended applications, we henceforth restrict ourselves to the two-dimensional form of Eq. (1). As will be shown in a subsequent publication,9 it is permissible to use the two-dimensional analog of Eq. (1) providing one assumes (a) 1'= V'zT= 0 and (b) that the transport coefficients in (1) are independent of position (and hence, of T). One can use (8)-(10) to write out the resultant matrix in final two-dimensional form. However, the subsequent work is materially simplified by deferring the total use of the CORC to a later stage of the derivation. Accordingly, in setting up Eg. (12) below, we employ Eqs. (8) and (9) only as a general guide to establish the form of the entries in the DB of Eq. (12). However, the two-dimensional analog 9 T. C. Harman, ]. M. Honig, and B. M. Tarney, J. App!. Phys. 34,2225 (1963), Part V, following paper. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:392218 HARMA~, HONIG, A~D TARMY of (10) is the lower ODB in (12), and the upper right ODB is found by use of the CORC [Eq. (4)]. The fundamental tensor of interest now takes the form (12a). For easy identification of entries, we have also set up the companion matrix (12b), where the entries are identifIed as F i/ or F ,r according as the antisymmetric portions in (12a) are preceded by a positive or negative sign. ( -Kxy(I)+Hz'JfLxyKxx(l)IT -Kuy(l)IT l~fj (P Xl/([)+ H z:rLx/l) (p yy w+ H z:rLyy(l) or In the above, we have introduced the superscript (I) explicitly to indicate that the entries refer to isothermal conditions of measurement. Summarizing, we note that all ODB entries in (12b) conform to the pattern (13a) Splitting the general entry F into even and odd contri butions Fi},(Hz)=Fij,(e)(H.2)+H zF'j'(O)(H z2), we fur ther note that s= -KIT, p, (P, For isotropic materials Special note should be taken of the relationship (l3b) (13c) (13d) (13e) (13f) Kxx(l)ffi!xy= -Ky)l)ffi!yx. (13g) This completes the fundamental considerations required for our later development. THE HEURLINGER-BRIDGMAN RELATIONS Before proceeding with the device analysis, we need to investigate interrelations pertaining to the general set of galvano-thermomagnetic transport coeffi cients (GTMTC). These form the analogs of the Reurlinger-Bridgman relationslO that apply to isotropic materials. Relations between the isothermal and adia batic Seebeck and Nernst coefficients may be estab lished on the following basis: Let us compute the ratio 10 R. G. Chambers, Proc. Phys. Soc. (London) A65, 903 (1952); H. B. Callen, Thermodynamics (John Wiley & Sons, Inc., New York, 1960), Chap. 17. (l2a) (12b) 'V yif;I'il xT under the conditions (a) J"= jy= j sy= O. On imposing these constraints, the second and fourth rows in (12b) become 'V yif;=F41+'il xT+F42+'V yT, (14a) O=F2!+'il xT+F22'VyT. (14b) After eliminating 'il yT from (14), we find 'V yif;I'V xT= F41+-F 21+F 42+ IF 22=(P YXCA)+ Hz:rLYXCA) (for P=jy=j sy=O), (lSa) where the right-hand side follows by definition [see Eqs. (l1h), (ltj)] from the ratio on the left, since the quantities of interest were determined subject to the condition T j sY=O. Let us now reverse the magnetic field direction and the F ij superscripts in (1Sa), keeping in mind Eqs. (13); this converts Eq. (lSa) to F 41--F 21-F 42-IF 22=(PyxCA) -lIz:rLyx(A) (ISb) on reversal of the magnetic field. Adding or subtracting (1Sa) and (ISb) and replacing the F ij± with the entries from Eq. (l2a), we obtain the desired relationships (P yx(A) = (P yx(l) -(P yyCl)Kyx(l) I Kyy(I) + lIz2:rLyy(l)'JfLy x, (16a) :rLy x(.4) =:rLy x(I) -:rLyy (l)Kyx(I) I Kyy(I) +(Pyy(l)'JfL yx. (16b) Using precisely the same approach in connection with the ratio 'il xif;/'V yT we find (P Xy(A) = (P XyU) -(P xx(l)Kxy(l) IKxxU) + Hi:TLx/f)ffi!XY, (17a) :rLXy(.4) = :rLXYU) -:rL xxWKx/l) / Kxx(l) +(P xxCl)ffi!XY' (17b) Next, we determine the quantity 'il x1f'=F31+'il xT + F 32+'il yT which is obtained from the third row of Eq. (12b) under the constraint (a). Again, Eq. (14b) applies [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:39GALYA:\lO-THERMOMAGNETIC PHENOME:\lA. IV 2219 here so that on elimination of v!JT, one can write v zf/v xT=F:l1+-- F:l2+F21+/Fn = Cp xxCI)+ H zm:xx(·1) (lSa) which is the analog of (lSa). Reversal of the magnetic field direction yields the companion relation F31--F32-F21-IF22=CP xx(·4)- H/J'{xx(A). (18b) On adding and subtracting, this leads to the results Cp xx(A) = Cp xx(1) -Cp xy (l)Ky xU) 1 Kyy(I) + H /m:Xy(l):m: yx, (19a) ;)I xx (A) = ;)I xx ([) -;)I xy (1)Ky x(I) 1 Kyy (l) +CP xy (l):m:y x. (19b) Similarly, one may proceed to determine the ratio 11 yif;/Il yT as Cp yy(A) = Cp yy (I) -Cp y x(l)K y xU) 1 K xxU) + H }:m:xy;)Iyx(l), (20a) ;)Iyy (A) = m:yv (l) -;)Iy x(l)Ky x(1) 1 K x/I) +CP y x(l):m:XY' (20b) Let us now impose condition (a) on the first row of (12b); we then find TJsx=F1lllxT+F12+llyT. Elimi nating 'ilyT through (14b) and applying (13), one can solve for -TJ sxl'il xT =Kxx(A) = -TF 11[1- F 12+F21+ IFI1F22J =K (I){l-K (l)2IK (I)K (1)-H 2CYn' CYn'} (21a) xx yx Xl; yy z ulLxyJILyx , where the F ij have been replaced as indicated on the right. Using precisely similar techniques, one obtains an interrelation involving Kyy(A), namely, Kyy(A) = Kyy(I) {1-Kyx(I)2/ KxxCI)KyyCI) _Hz2:m:Xy:m:yx}. (21b) From (21) it now follows that (22) Still another set of interrelations is obtained bv im posing the boundary condition (b) 11 ,] = J y = J ,; y = O. The second and third rows of (12b) then read 0=F22'ilyT+F32-J~, (23a) 'ilxf=F32+llyT+FaaJx (23b) and when 11 yT is eliminated, one can solve for 11 xf/ J"= F 33(1-F 32+F 32-1 F 22F 33) = Pxx(A). (24) The quantity on the right follows as a definition from the ratio on the left, subject to conditions (b). After substituting for the F ij±, one obtains Pxx(A) = Pu(l){ 1 + T[CP Xy(I)2- Hz2;)Ix1J(I)2JI Kyy(1)pxP)}. (2Sa) The companion relation to (2Sa) reads pyyCA) = pyyU){ 1 + T[cpyx(I)2-Hz2;)Iyx(l)2JI Kxx(1)pyy(I)}. (2Sb) We can obtain other relationships by imposing con dition (b) on the fourth row of Eq. (12b), which is thereby reduced to the form 'il1l>./;=F42+'ilyT+F43+JI. On eliminating v yT via (23a), one can solve for the ratio where the central quantity is the definition appropriate to the ratio indicated on the left under condition (b). Magnetic field reversal brings (26a) into the form pyxCA)-Hzffi.yx(A) =F43--F32+F42-IF22. (26b) Equations (26) may be solved separately for the adia batic resistivity and Rall coefficient by appropriate additions and subtractions. On subsequently replacing the F;j± with the entries in (12a), one finds py x(·t) = py x(1)+CP Xy(l)cp yy (l) T / Kyy (I) -H };)IXY (l);)Iyy(l)T 1 Kyy(l), (27a) ffi.y x (A) = ffi.y x (ll +CP xy (ll;)Iyy (l) T / Ky/I) -;)IXy(l)CPyy(l)TIK yy(/). (27b) The companion relations to (27) are obtained by solving for the ratio 'ilxflJY under the boundary condition (c) 'il yT= J.r= J sX= O. This procedure leads to the results Pxy (.4) = Pxy (l)+cp y x (l)CP xx (I) T 1 K xx (I) - H z2;)Iyx(1);)Ixx(l)T 1 Kx/I), (2Sa) ffi. xy (A) = ffi. x/I) +CP y x(I);)I x x (I) T 1 K xx(I) -'J'Lyx(I)(J>xY)TIKxx(I). (28b) Other Reurlinger relations may be found using similar methods, but only those listed above are of relevance in the further development. EFFICIENCY OF TRANSVERSE ADIABATIC GTM GENERATOR CONSTRUCTED FROM ANISOTROPIC MATERIALS With the preliminaries out of the way, we proceed with an analysis of the characteristics of GTM genera tors, constructed from anisotropic materials and oper ating under adiabatic conditions. We follow closely the procedures outlined in papers I and III. The reader should refer to Fig. 1 for a schematic diagram of the apparatus geometry and conventions, which follow those of the earlier papers. Under operating conditions considered here, Jx= 0; the phenomenological equations of interest may then be rewritten as (29) where the matrix entries are given by Eq. (12a). Under steady-state conditions, V·J=O; however, since Jx=O, the above condition specializes to the result V' xJx = 11 yJY = O. Again, under steady-state conditions, V· J g =0, where J e is the total energy flux vector, given by3 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:392220 HARMA:\', HO~IG, t\l\'D TARMY (w) (w) Rq FIG. 1. Schematic diagram for apparatus geometry. J s= TJ s-if;J. Under conditions contemplated here, the law of conservation of energy reads V' x(T] sX)+ V' y(T] sY-if;]v) = O. (30) As explained earlier, the conservation equation does not involve z components when experimental conditions have been presupposed such that ]z=V'zT=O. On substituting from Eq. (29) into (30) and simpli fying, one obtains the following differential equation for the temperature in a device arm TFn V' ,,2T+ T(F12++F21+)V' xV' yT+ TF22V' y2T + ]Y(F42--F 42+) V' yT+ (F41--F41+)]yV' xT -F44(Jv)2=0. (31) We now introduce the definitions a=.F44(JY)2, b=. (F42--F42+)]Y, C=.FllT, d=.T(F 12++F21+), j=]Y(F 41--F 41+), g=1'f22 (32) in connection with which one should note that the product TF ij in c, d, and g is independent of T,as may be verified by inspection of the matrix entries in (12a). We then attempt to solve Eq. (31) using a Taylor's series expansion about a point (xo,yo): T(x,y) = T(xo,yo)+ (x-xo)Tx'(xo,yo)+ (y-yo)T u'(xo,yo) +H (x-xo}2T xx"(xo,yo) + 2 (x-xo) (y-yo)T x/' (xo,yo) + (y-yo)2T yy" (xo,Yo)}, (33) where the primed quantities indicate partial derivatives with respect to the independent variable occurring as a subscript. To determine the constants in (33), we re quire boundary conditions. One of these is obtained by introduction of the trial solution and (33) in (31). This yields the following relationship, valid at x= XI) and y=yo: gT yy" (xo,yo)+cT xx" (xo,yo)+dT Xy" (xo,yo) + jT x'(xo,Yo)+bT,/ (xo,Yo) = a. (34) Another boundary condition may be formulated on the basis that, under adiabatic operation, the power trans port remains constant across any junction between the lead wires (w) and the device arm (d). This means that at any junction point, the condition Ay(w)] sY(w) = AyCd)] cy(d) = Ay(W) (T] sew) -if;w] wY) = AyCd) (T] SCd) -if;d]aV) must hold. On substituting from Eq. (12b), this condition may be reformulated as Ay(W) (F21+T'V xT+F22TV'yT+F42-T ]Y-if;]u)(w) = A y (d) (F 21 +TV' xT + F 22TV' yT + F 42-T ]Y_if;]y)(d). (35) We now introduce the following simplifications: First, we neglect the junction resistance between the wire and device arm. This means that if;(w) _if;(d) = 0, or, alter nately, that the two terms containing if; cancel each other in Eq. (35). Further, we can safely assume that V' xTCw) = 0 for the wire, thereby eliminating the first term on the left of Eq. (35). Next, we presuppose an experimental arrangement in which F22cw)Aycw)V'yT(w) «F22Cd) AyCd)V' yTCd); in this event, the heat flow through the wire by conduction is much smaller than that by conduction through the device arm. Finally, we assume that the device is constructed from semiconducting materials, and that the leads are metallic and outside the magnetic field. Then the term involving F42-on the left of (35) is much smaller than that on the right. We are thus left with the simple relation (36) where (37) and where the properties in (37) refer to those of the TI ----------------------------- T2 (0,0) (O,Ly) FIG. 2. Geometry pertaining to one de vice arm. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:39GALVANO··THERMOM:\G~ETTC PHE:\Ol\1El\A. IV 2221 device arm. Strict ly speaking, this relation applies only at the position of the junction between the lead and the device. In practice, we see that under the assumed con ditions, no reference is made to the wire in the final equation; for this reason, Eq. (36) can be assumed to hold at any position along the lateral edges of the device. On introducing Eq. (33) into (36) and carrying out the evaluation at x= Xo and y= Yo, we find that Ty' (xo,yo) =aT x' (xo,Yo)+iJT(xo,Yo). (38) The remaining requirements are found by reference to Fig. 2 where we have indicated the temperatures pre vailing at each corner of the device arm. In order not to come into conflict with the assumption of adiabatic operating conditions, we have replaced the actual bar, whose ends are in contact with the reservoir, with a somewhat shorter bar, as indicated by the dotted lines. Initially, all algebraic manipulations are done on the T x' (0,0) assumption that the four temperatures TI, "', T4 differ; later, we allow T2 ---. TI and T3 ---. T4• We study first the situation in the upper left-hand corner, where Xo= yo= 0. On applying the boundary con ditions T(O,O)= TI, T(O,L y) = T2, T(Lx,L y) = T3, T(Lx,O) = T4 in Eq. (33), we obtain the relationships TI = T(O,O), (39a) T2= T(O,O)+ LyT/ (0,0)+!L y2T,,/' (0,0), (39b) T3= T(O,O)+ LxT x' (0,0)+ LyT u' (0,0)+!L x2T xx" (0,0) + LxLyT Xy" (O,O)+tLyT y,," (0,0), (39c) T4= T(O,O)+ LxT x'(0,0)+!Lx2T xx" (0,0). (39d) The six equations (39), (38), and (34) with Xo= Yo= 0, may now be solved for the various T' (0,0) and T" (0,0). After straightforward though lengthy algebraic manipu lations, one obtains the following results of interest (40) Lxj Lx b Lx g 1-----a-+--a- 2 c 2 C Ly C where 2D-==(T l-T2)+(T3-T4). Proceeding similarly with the upper right-hand corner, where xo=O and Yo=L y, one finds that T3-T2 1 d Lx g Lx g Lx b Lx a --+- -D+--(TI-T2)+--iJT2+-iJ-T 2--- Lx LyeLl e Lv C 2 C 2 e (41) Lx j Lx b Lx g 1-----a---a- On the basis of Eqs. (40) and (41), one can now com pute the average temperature gradient near the plane X= ° according to the relation (\1 xT)o=![T x' (0,0)+ T x' (O,Ly)]. (42) The resulting expressions are extremely complex due to the difference in signs of the denominators of (40) and (41). To keep the subsequent theory manageable, we have therefore resorted to an expansion of these de nominators. Under the assumption that an experimental arrangement is achieved whereby Lx«Ly and jy as well as D is small, so that 13, jle, and ble are small, we obtain the following result: (43) where only the leading terms have been retained. Here, (~xT)=[(T4-Tl)+ (Ta-T2)]/2. Aside from (43) which 2 e 2 C Ly C is needed for our subsequent development, we also require the following average: (\1 yT)o=![\1 yT(O,O) + \1 yT(O,Ly) J; according to (36), this quantity is given by THE ENERGY FLUX AND ELECTRIC CURRENT Let us now extend the region under consideration so that the entire device arm is being covered. This means that we are now concerned with the plane x=o im mediately adjacent to the heat reservoir at temperature To. In these circumstances, we can remove the averaging symbols from (43) and (44) and replace (TI+T2)/2 with To. Equations (43) and (44) thus modified may then be used in the following expression for the rate of energy flux past the plane x= 0: ex~o= TvA xj sx=A xTo[FuO(\1 xT)o +F12+O(\1yT)o+F 41-jY]. (45) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:392222 HARMA\f, HONIG, A:\[) TARMY One is thereby led to the relations tx~o= + A xTo{ (F 1l0+aF 12+0) (\7 xT)u+{3F l2+OTO+ F41-.fY} (46) where the superscripts 0 on appropriate Fij signify that the temperature in the corresponding matrix entries is to be equated with To. The F ij combinations encountered in (46) can be eliminated through the use of Eqs. (lSb) and (21a). We then obtain (47) It is expedient to replace Kxx(A) with the relation on the right-hand side of Eq. (21a). In general, the Righi-Leduc coefficients and the quantity Ki/x(I) may be considered small, so that their product with fl2e, abl2e, or ale may be neglected compared to other leading terms. In effect, we thereby replace Kxx(A) with Kxx(l) in products containing the quantity e; Kx'/!) and c cancel, and Eq. (47) may be rewritten as (48) Returning to Eq. (IS), we see that the quantity in curly braces is given by -2lJz'J'l1l./AJ- ToF42-2H z;myx' Since we are only interested in the zero-order approximation, we simplify matters further by neglecting the second term relative to the first. With this cascading of approximations, we finally arrive at the relationship where the superscript (v) has been introduced to remind us that we have so far dealt with only one device arm. The total energy flux, taking account of all device arms is given by where As in the earlier publications, we determine the cur rent by evaluation of the line integral §'il yif;dy= 0 around the transverse circuit; as shown, this quantity vanishes. For the various device arms, the integrand is specified by the fourth row of Eq. (12a); reference to Fig. 1 shows that the integrand is to be evaluated at x= Lx12. We neglect all junction and lead wire resist ances; since J x = 0, we then find (SO) (51) Here, pq and Lq are the resistivity and length of the load resistor; the integration covers the y dimension of each arm. We determine 'il xThxl 2 as \7 xT) Ld2= H ![T :x' (0,0)+ T x' (O,Ly) ] +![Tx'(Lx,O)+Tx'(Lx,L y)]}. (53) The first term on the right may be evaluated from (40) and (41); the leading terms are (TcTl)ILx and (Ts-T2)1 L!J), respectively. Following the steps taken in deriving Eqs. (40) and (41), one can obtain corre sponding expressions for use in the second term on the right of Eq. (53); it turns out that the leading terms are exactly the same as those just cited. After substituting these results in (53), and allowing T2 -+ T1, Ts -+ T4, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:39GALVA:'.JO-THERMOMAGNETIC PHENOMEXA. IV 2223 one obtains (\7 xT)-=\7 xTk/2,,=,b. xT/Lx, (54) which result may then be used in the first term of Eq. (52). To determine the second term, we avail ourselves of Eq. (36). Strictly speaking, this relationship applies only at the boundaries between lead wires and device arms. However, to the extent to which the approximations introduced 'so far are applicable, one may use an averaging procedure by utilizing Eq. (36) in the form (\7"T)=a(\7 xT)+i3T and then introducing Eq. (54). On carrying out the indicated operation, the result and Eq. (54) can be employed to rewrite the integrand in the central expression of (52) as b.xT (b.xT_) Vyt/;=F41+--+F42+ a--+{31' +F4dy Lx Lx b.xT = --(cp yx(A)+ H.:n.yx(A»+P!ly(B) J", (55) Lx where use was made of Eq. (15a) and of the matrix entries in (l2a); furthermore, we have introduced the definitions R"y(B)=R yy(1){1+'YL(l)T}, (56) 'YL (1) -= [('13yy(I})2- (H.9cyy(l»2J/ K yy(1) Ryy(l}, (57) where and 'YL (1) being a generalized figure of merit for the iso thennal, longitudinal mode of operation of the generator. We now introduce Eqs. (55)-(57) into (52), carry out the indicated integration, and solve the result for I y= -b.xT('13 !Jx(,1)+Hz91yx(·4»/ (Ryy(B)+ Rq); (b.xT<O) (58) in which we have used defining relations analogous to (51). OPERATING CHARACTERISTICS We are finally at the point where the efficiency of the device of Fig. 1 can be computed according to the relation 'Y/=I y2Rq/ 8x=0, into which we substitute from Eqs. (50) and (58). Consistent with our aim of keeping the zero-order theory free from unnecessary encum brance, we follow the method used in an earlier paper by an ad hoc replacement of Ryy(I} with Ryy(B) in Eq. (SO). Such a substitution is not necessary but it does simplify considerably the subsequent mathematical operations. This is not likely to lead to serious errors in the final results, since the replacement is made in the smallest term of (SO). Let us then use the revised equation (SO) and Eq. (58) in the efficiency relation; on simplifying, the final result reads (b.xT<O), (59) lIT(A) = ('13y,,<.1)+H,91 y.,(A»2/K "",(A) RyyCB). (60) (61) Finally, we note that the third term in the denominator of (59) may be rewritten as '13y",(A) -TLH.91y",(A) /To T('13yxCA) -H .91y",(A»-~b.xT('13yP)+ Hz91yx(A» m (A)+Hffi (A) +,yx z;,l'lyx To ('13 yx(A)+ H .91yx(A» which defines the quantity QA as shown on the right. Equation (59) may now be written as b.xT{ Oy(A) } 'Y/= -To (1+0y(A»)2I lIT(A)To+b.xT/2To-QA(1+0/A»/To (b.xT<O) (62) (63) which is identical in form to Eq. (23) of paper III. It follows that the various statements made in paper III con cerning the characteristics of generators operating under isothermal conditions also apply here. Except for the replacement of Ryu(I) by RlIy(B), the symbol I appearing either as a sub-or a superscript in paper III can now be replaced by the sub-or superscript A. DISCUSSION In view of the formal identity between the results obtained here and those cited in III, only a summary of the principal conclusions need be presented. The operating efficiency obtained on optimizing the ratio (60) is given as b.xT{ 1-0y*(A) } 'Y/*-- (b.xT<O) (64) To Tr A (1 +Oy *(A»/To-b.",T(1-o y *(A»/2To [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:392224 HARMAX, HO)JIG, Al\D TARIVIY FIG. 3. Plot of V vs '1J1' according to Eq. (64), for To=310oK, TL=300oK. in which and (65) (66) It is seen that the optimal device efficiency is governed not only by the figure of merit [Eq. (61)J but also by the anisotropy parameter r A defined above. In general, the smaller r A, the smaller is the value of 'Y T(Al required for efficient operation of the generator. The method of paper III may also be used to show that Carnot effi ciency can only be achieved in the range -1:::; r A:::; 1 ; this presents no problem since r A can always be adjusted by proper choice of magnitude and direction of Hz. Further, since we cannot exceed Carnot efficiency, the quantity in curly braces in Eq. (64) cannot exceed unity. One can show that this leads to the requirement 1'H/"Ryx(Al2/K xx(Al Ryy(B):::; 1, (67) Carnot efficiency being achieved only if the inequality is replaced with an equality. It is worth noting that the limits on 'YT(Al1' are given by -rYJ <'YT(Al1'<1. We now consider briefly the limitations of the present treatment. The principal restriction is the requirement that !!..T/1'«I, which must be imposed for the following reasons: (1) The entire treatment is based on a linear set of macroscopic equations, as given by Eqs. (1). The linear formulation presupposes only minute deviations from equilibrium3; in principle, any theory purporting to deal with large departures from equilibrium (such as obtain when !!..T","1') must be based on a nonlinear theory. However, it is found in the zero-field, thermo electric power generation case that the predictions derived from the linear theory check experimental re sults to within 10% even when !!..T"-'1'. It remains to be established whether this happy cancellation of errors also occurs in the present case. (2) To obtain analyti cally manageable results, we have assumed that the transport coefficients are independent of position and hence, of T. For AT comparable to T, this is a very poor approximation. Any attempts to extend the theory by elimination of this assumption would require machine calculations. In this connection, one should note that it is no longer permissible to "decouple" the z-direction effects from those in the x-y plane if the transport coefficients are allowed to depend on spatial coordi nates. (3) In the derivations the simplifying assumption that all P are constant has been used throughout. It can be shown that this is a good approximation only when AT/1'«1. In line with the above, we have therefore approxi mated the temperature dependence on position by the Taylor's series expansion (33) ; inclusion of higher-order terms or attempts to obtain exact solutions are unwar ranted in the present approximation scheme. For AxT/To«l, the second term in the denominator of Eq. (64) is small, relative to the first. Therefore, a good approximation to the quantity (-1'}*To/ AxT)= V IS V"'" (l-oy *(Al)/r A (1 +Oy *(A»), (68) which is independent of the Carnot efficiency A xT /To and depends only on the material parameters involved in the quantities 3T(A) and r A as well as on 1'. Therefore, the extent to which Carnot efficiency is approached at a given l' is essentially determined by the properties of the material of construction. In Fig. 3 is shown the dependence of Von 'Y1' for a set of rA values; here we have set To=31OoK, TL=300oK. This curve is analogous to Figs. 3 and 4 of Ref. 1, paper III. It is seen that the smaller the r A value the smaller is the value of 'Y1' required to reach Carnot efficiency. For rA=O, 'YA vanishes likewise; Eq. (64) is now inde terminate since 0 y *(A) = 1. Application of I'Hopital's rule shows that (69) Thus, to remain below the Carnot efficiency limit (VrA-+O = 1 in this case), it is sufficient that Eq. (61), mUltiplied by 1'( < To), be less than 4. In view of the requirement (67), the approach to Carnot efficiency for r A = 0 leads to the condition that both 'J,3yx(A)1'/K xx(AlRyy(A) and Hz2"Ryx(A)1'/Kxx(A)Ryy(A) be unity. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.105.215.146 On: Tue, 23 Dec 2014 15:13:39

1.1728756.pdf

Current Voltage Characteristics of Germanium Tunnel Diodes Marshall I. Nathan Citation: Journal of Applied Physics 33, 1460 (1962); doi: 10.1063/1.1728756 View online: http://dx.doi.org/10.1063/1.1728756 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Schottky barrier height extraction from forward current-voltage characteristics of non-ideal diodes with high series resistance Appl. Phys. Lett. 102, 042110 (2013); 10.1063/1.4789989 Current-voltage characteristics of metal-oxide-semiconductor devices containing Ge or Si nanocrystals in thin gate oxides J. Appl. Phys. 106, 064505 (2009); 10.1063/1.3190520 Current–voltage characteristics of Schottky barriers with barrier heights larger than the semiconductor band gap: The case of Ni Ge ∕ n - ( 001 ) Ge contact J. Appl. Phys. 97, 113706 (2005); 10.1063/1.1923162 Nonlinearities in the current–voltage characteristics of neutron transmutation doped germanium at millikelvin temperatures J. Appl. Phys. 82, 3341 (1997); 10.1063/1.365644 Exciton-induced tunneling effect on the current-voltage characteristics of resonant tunneling diodes J. Appl. Phys. 81, 6221 (1997); 10.1063/1.364409 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:551460 S. AMELINCKX AND P. DELAVIGNETTE evident that the relative magnitudes of the stacking fault energies are correctly reflected in the widths of the stacking fault ribbons. Contrast effects fluther show that lines (a) and (c) and lines (b) and (d) have the same Burgers vectors, while (ef) behaves differently. The directions of the Burgers vectors as determined by diffraction are also in agreement with this model. Depending on the direction and sense of the X1Xa glide vectors, one can obtain ribbons with the high energy stacking fault at the right (at the left) or in the center [Fig. 3 (i), (ii) ] ; both cases are observed (Figs. 4 and 5). There are two kinds of fourfold ribbons, as shown in Fig. 3(iii) and (iv). The first is more or less sym-metrical: no high energy stacking fault occurs (Fig. 6). For the second kind one ribbon contains a high energy stacking fault and the two bordering partials combine; the fourfold ribbon looks threefold. [Fig. 3 (iv)]. The dislocations in Bils are similar in most respects to those in CrCla, while in cadmium iodide they consist of two partials of the Shockley type. ACKNOWLEDGMENTS We wish to thank Mr. J. Goens, director of the C.E.N., for permission to publish this paper. We also wish to thank Mr. J. Nicasy for skillful preparation of specimens and Mr. H. Beyens for careful photographic work. JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 4 APRIL, 1962 Current Voltage Characteristics of Germanium Tunnel Diodes* MARSHALL I. NATHAN International Business Machines Corporation, Thomas J. Watson Research Center, Yorktown Heights, New York (Received August 7, 1961) The current-voltage characteristics of germanium tunnel diodes with phonon assisted current predominant have been measured for reverse biases at 297°K. The tunneling exponent is deduced from the data and found to be in agreement with that obtained from high pressure experiments. The value of the pre-expo nential factor obtained from both experiments is much larger than predicted by theory. Calculations of the tunneling current which take into account nonparabolic effects of the energy band structure and the electric field variation with position in the junction are presented and compared with experiment. The results of current-voltage measurements on germanium diodes with non-phonon current predominant are also pre sented and discussed. THE properties of tunnel diodes have been investi gated by a number of workers. Esakil has shown that simple tunneling theory accounts for many of the electrical properties. Esaki and Miyahara2 and Hall and co-workers3 have demonstrated that phonons can par ticipate in the tunneling process in diodes constructed in a large number of materials. The pressure dependence of the tunneling current has been examined,4 and its general aspects have been found to be in agreement with theory; however, there are quantitative dis crepancies. Chynoweth and co-workers5 have studied the current-voltage characteristics of backward diodes * A preliminary account of this work was presented at the March, 1961 APS Meeting, Bull. Am. Phys. Soc. 6, 106 (1961). 1 L. Esaki, Phys. Rev. 109,603 (1958). 2 L. Esaki and Y. Miyahara, Solid State Electronics 1,13 (1960). 3 N. Holonyak, I. A. Lesk, R. N. Hall, J. J. Tiemann, and H. Ehrenreich, Phys. Rev. Letters 3, 167 (1959); R. N. Hall, Proceedings International Conference on Semiconductors, Prague, 1960 (Publishing House of the Czechoslovak Academy of Sciences, Prague, 1961),p. 193. 4 S. L. Miller, M. 1. Nathan, and A. C. Smith, Phys. Rev. Letters 4, 60 (1960); M. 1. Nathan and W. Paul, Proceedings International Conference on Semiconductors, Prague, 1960 (Pub lishing House of the Czeckoslovak Academy of Sciences, Prague, 1961), p. 209. Ii A. G. Chynoweth, W. L. Feldman, C. A. Lee, R. A. Logan, G. L. Pearson, and P. Aigrain, Phys. Rev. 118, 425 (1960); A. G. Chynoweth and R. A. Logan, Phys. Rev. 118, 1470 (1960). in several materials. These diodes, which have lower impurity densities than tunnel diodes, exhibit tunneling current at low reverse voltages (p side negative) but have at most a small amount of tunneling in the forward directJon and no negative resistance region. they found general agreement with theory. The widespread interest in tunneling phenomena would seem to make a detailed comparison of simple tunneling theory with tunnel diode current-voltage characteristics worthwhile. It might seem at first sight that it would be best to make initial comparisons on backward diodes. The relatively low impurity density will minimize effects due to the discrete nature of the charge density and due to rapid changes in potential, which have not been treated theoretically. However, the theory6.7 is most apt to be applicable for small applied biases. The tun neling will be between the lowest conduction band minima and the highest valence band maxima, and the dependence of the junction field on bias, as we shall see, should be simple. It is important to be certain that the tunneling current predominates at small bias. The ob- 6 P. J. Price and J. M. Radcliffe, IBM J. Research and Develop. 3,364 (1959). 7 E. O. Kane, J. App!. Phys. 32, 83 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:55CUR R E N T VOL TAG E C H A RAe T E R 1ST I C S 0 F GeT U NNE L D I 0 DES 1461 servation of negative resistance in the current-voltage curve ensures the predominance of tunneling current. Therefore we have chosen to study tunnel diodes. We have examined germanium diodes, mainly those which exhibit predominantly phonon assisted tunneling current at 4.2°K. We have restricted the investigation of the characteristic to reverse voltages, so that compli cations due to distortion of the energy bands near the edges are not expected to be important. A simple expression for the tunneling current density J as a function of applied voltage V which can be ob tained from the results of Price and Radcliffe6 and of Kane7 is, for reverse voltage, J=AVexp[ -XCV)]. (1) We shall take the pre-exponential factor A to be a constant. This assumption, which is an important one for what follows, is not strictly valid, A is expected to depend on both V and X. The exact variation depends on the par ticular tunneling process involved. However, we shall show in the discussion section that these dependences can be accounted for and do not affect the conclusions of this paper. Therefore, in order to keep the analysis simple at the outset, we shall take A constant. The exponent X (V) is given by fX' lX' a XCV) = 2 adx= 2 -dE, Xl Xl eF (2) where Xl and Xz are the classical turning points, a is the imaginary part of the x component of the wave vector at the energy cp; of the tunneling electron, F is the elec tric field, e is the electronic charge, and E is the energy of the conduction band edge Ecex) or the valence band FIG. 1. The potential profile for a reverse biased junction. </>,. and </>p are the quasi-Fermi energies for electrons and holes, respectively. F FIG. 2. The field as a function of distance x in the space-charge region at OaK. x" and Xp are the edges of the depleted region. The field has two linear portions between x" and Xp. edge E.(x) relative to rpi. The situation is illustrated in Fig. 1, which is the conventional diagram of a reverse biased p-n junction. CPi is any energy between the quasi Fermi energies8 CPn for electrons on the n side and CPP for holes on the p side. Equations (1) and (2) are derived with the assumption that X»l, that the potential does not change much in a distance 1/a, and that X is approximately independent of the energy cp;. The wave vector a which goes into Eq. (2) is the smaller of the wave vectors ac and a. of the conduction and the valence bands. For "spherical parabolic" bands Ec-cp;=ft2acz/2me (3) cP;-E. =ft2a.2j2m., where me and m. are the effective masses in the conduc tion and valence bands. If the field is taken to be con· stant and if the bands are assumed to be spherical and parabolic, Eq. (2) can be integrated with the result X(V)=H2m*)!ENheF, (4) where (1/m*)= (1/mc)+ (l/m.) and Eg=Ee-E •. For an actual junction the field is not constant. If thej unction is abrupt the field has two linear portions in the spaceccharge region as shown in Fig. 2. (We are neglecting the discrete nature of the ionic charge.) At OOK the maximum field can easily be shown to be Fmax=[81r(E g+ir,,+irp-eV)N o/KJi, (5) where rn=CPn-Ec; rp=E.-cpp; (1/No) = (1/Nd) + (1/ N a) ; N d is the net impurity density on the n side; N a is the net impurity density on the p side; and K is the dielectric constant. For the constant field approxi mation we might take the field to be a constant fraction of Fm"". If then we expand Eq. (5) for rn, rp, and eV small compared with Eg and substitute in Eq. (4), we find for the current density in Eq. (1) J=AV exp[ -h(O) (l+eV /2EgH (6) This equation does not apply at low temperatures (......,4 OK) to phonon assisted tunneling current. Since an the phonons which induce current are "frozen out", only tunneling with phonon emission is possible and the S w. Shockly, Bell System Tech. J. 28,435 (1949). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:551462 MARSHALL I. NATHAN usual structure is observed in the current-voltage curve.2.3 A term like Eq. (6) with an onset voltage equal to the phonon energy hw and with Eo replaced by Eg+hw is expected for each participating phonon. At higher temperatures (3000K), however, both emission and absorption of phonons occur in the tunneling process. Moreover, the onset voltage for emission dis appears since kT""'hw' It seems reasonable to see if a law of the form of Eq. (6) then holds. The reasons for making the attempt to fit the current voltage curves of germanium diodes with phonon as sisted current predominant are twofold: First, the diodes were readily available in a range of current den sities for which the series resistance is negligible for a fairly large voltage range. Second, high pressure experi ments on them gave results,4 which could be interpreted in terms of simple theory. EXPERIMENTAL METHOD The diodes studied were made by alloying nominally 1% gallium in indium dots into antimony-doped ger manium substrates. The alloying process was carried out on a heater strip. The heater strip temperature was raised until the dot started to alloy. Then the power was either turned off immediately or it was held constant for about one second and then reduced over a period of a few seconds. No dependence of the diode characteristics on the heat cycle was found. It was not possible in most cases to determine accurately the impurity concentra tion in the substrates, since they were most times small single crystals from a polycrystalline ingot. In order to determine the current density of the diodes it was necessary to measure their area. This was accomplished by etching the diodes until they were in the shape of well defined pedestals of several thousandths of an inch in diameter and then measuring 200 J!! 100 ~ i 50 II: ... ~ .. "'I> 20 0.02 0.04 0.06 008 0.10 0.12 0.14 -v (VOLTS) FIG. 3. In (J IV) vs -V for phonon-type diodes. T=297°K. ~'r-r-------~------------------~ 10' 0;., 10' . Ii g ~ 10· 16 18 20 22 24 26 28 FIG. 4. In JO' vs -d In (J /V)/dV for phonon-type diodes. T=297°K. the projected diameter in several directions through a microscope. The average was taken and the cross section was assumed to be circular. It is felt that this method could be in error by as much as ± 50%. An attempt was made to eliminate those diodes which had nonuniform current density. Both the peak current and the area were measured between successive etches. If the peak current was not a linear function of the area, the diode was discarded. This technique will not elimi nate diodes with microscopic inhomogeneities. EXPERIMENTAL RESULTS Figure 3 shows In (J I V) plotted as a function of reverse voltage at room temperature for two typical diodes. The linear dependence suggested by Eq. (6) is found until about -0.11 v in one case and -0.07 v in the other. The increase in slope above these voltages is undoubtedly at least in part due to the onset of tunnel ing from the valence band to the (000) conduction band minimum.9 The (000) minimum is approximately 0.14 ev above the Fermi level on the n side of the junction. Since the energy distribution of carriers is smeared out by a few kT around the Fermi energy, this direct tun neling would be expected to begin at approximately -0.1 v at room temperature. The exact value of the onset voltage will depend on the impurity densities on the two sides of the junction. (The p-side density can be involved since the top of the valence band can provide a lower limit on the onset voltage.) According to Eq. (6) the slopes of In (J IV) vs V plots in Fig. 3 should be given by dIn (JIV)ldV=-eA/2E g• (7) • J. V. Morgan and E. O. Kane, Phys. Rev. Letters 3, 466 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:55CUR R E N T VOL TAG E C H A RAe T E R 1ST I C S 0 F GeT U NNE L D I 0 DES 1463 From Eqs. (7) and (6) we see that Jo'= (dJ/dV)v=o is expected to vary exponentially with dIn (J /V)/ dV for different diodes. In Jo'=ln A+(2Eg/e)(d In (J/V)/dV). (8) ] 0' is determined by extrapolation of Fig. 3 type plots to V = O. In Fig. 4 In] 0' is plotted against the slope obtained from plots like Fig. 3 for several diodes. The systematic relationship is evident. All the diodes except one are on the line labeled "1= 1.25 within experimental error. Unfortunately, the exceptional diode was broken before it had been checked for uniformity of current density over its area in the manner previously men tioned. The line labeled "I = 2 has the slope 2 E / e pre dicted by Eq. (8). It is seen that a better fit to the data is obtained for the line labeled "1= 1.25, which has a slope 1.25 Eg/ e. This suggests that Eq. (6) is replaced by In (J/V)=lnA-X(l+eVhE g), (9) where "1= 1.25. To account for Eq. (9) it is sufficient for the field F to be changing faster with bias than the square root law given by Eq. (5). The derivation of Eq. (5) assumes that the junction is abrupt, that is that the impurity density on both sides is constant and that t~e transition from n-to p-type doping occurs in a distance s~all compared to the width of the space charge regIOn. If, however, the junction is graded so that th:re. is a gradual transition from n to p type, the field Will III fact vary faster with bias. For instance lin.ear gradin? would make F proportional to (Eg-e V)i; thiS would give a slope of 1.5 Eg/ e in Eq. (9) in much cl?ser ~greem~nt ~ith experiment. The space-charge Widths III the JunctIOns investigated are in the range of 100 to 200 A. Over such distances it is difficult to exclude the possibility of grading caused by diffusion of imp~rities. This is regarded as unlikely, however, since varylllg the alloying time for the junctions did not seem to affect markedly the peak current densities. In the derivation of Eq. (4) it is assumed that the ~eld i.n the junction is constant. Actually for an abrupt JunctIon the field has two linear portions as shown in Fig. 2. The tunneling occurs between the classical turning points, which at zero bias coincide with the ~dges. of .the space-charge region, XnO and XpO. If the JunctIOn IS reverse biased, only the maximum field in t~e junction Fmax is proportional to (Eg-eV)!, pro Vided r n, r p«Eg. The field at all other points in the space-charge region (Xn <x <xp) will increase with in creasing reverse bias by the same absolute amount as F ~ax. Hence, the relative change in the average field Will be greater than (Eg-eV)!. This will account for the fact that the coefficient of e V in Eq. (9) is larger than the 1/ (2Eg) predicted by the simple theory. A c~lculation of the dependence of the exponent on bias With a varia~le junction field is given in the Appendix. The calculatIOn shows that in agreement with experi ment In(J IV) or the exponent varies faster with bias than the constant field approximation suggests. How ever, the calculated exponent does not vary linearly with bias. A curve like that in Fig. 5 results. This is to be compared directly with the In(] IV) vs V plots ob tained experimentally. It is seen that the experimental curves are much more nearly linear than the calculated ones. A calculation of the exponent which takes account of the nonparabolic nature of the valence band and the variable junction field is also in the Appendix. This does not materially change the conclusions. The ]/V vs V data show that the pre-exponential factor for current has a linear dependence on V from -0.01 to approximately -0.1 v. For a linear depend ence Kane's7 calculation requires that This condition is obviously not satisfied for the voltage range of the data. This might also account for the fact that "I~2 is found experimentally in Eq. (9). Similar experiments have been carried out on diodes made by alloying 1% gallium in indium dots into arsenic doped germanium substrates. These diodes exhibit evidence of structure at 4°K caused by phonon participation in the tunneling process, but the major part of the tunneling current shows no evidence of phonon participation. Typical curves of] /V plotted on a logarithmic scale as a function of reverse bias are shown in Fig. 6. The same kind of linear dependence as for the "phonon diodes" shown in Fig. 3 is found. The 0.43,------------------, 0.4~ 0.47 0.02 0.04 0.06 0.08 0.10 0.12 0.14 - V (VOLTS) FIG. 5. I/; = Xo(tre'Noh'/m*K)! vs -V. The curves are computed from Eq. (.'\2) and the straight line is for the constant field approximation. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:551464 MARSHALL I. NATHAN 100r---------------"""7'! 50 ~ " 10 5 0: .... 0; ~ "'\> 5 IL--~-~-~--L---L-~-~O~ o 0.02 004 0.01 001 0.10 0.12 -Y (VOLTS) FIG. 6. In (J IV) vs -V for nonphonon diodes. T= 297°K. results of In Jo' vs dOn J /V)/dV for several nonphonon diodes is shown in Fig. 7. The line drawn has the slope of 2Eg/ e predicted by Eq. (8). The systematic trend is evident but comparison with Fig. 4 shows that the scatter in the points is somewhat greater than for the phonon diodes. This scatter is outside the experimental error. A similar correlation of tunneling current density for different diodes can be made with the results of high pressure experiments. For some kinds of diodes it has been found4 that the major effect of high pressure is to cause A to vary linearly with pressure, P, i.e., A=A(P=O)(I+aP), (10) where a is a constant independent of A. Equations (1) and (10) imply that for f!xed small applied bias J will vary exponentially with pressure. It has been found experimentally4 that the voltage V p at which J is equal to its maximum value J p for forward bias is independent of pressure in germanIum. For phonon assisted tunnel ing the exponential variation of J p with pressure is shown in Fig. 8 curve (a). It has also been found4 that din J / dP is independent of bias for small bias. If we differentiate Eq. (1) with respect to P and eliminate A between the equation obtained and Eq. (1), we have the following relationship between the conductances at zero voltage J 0' in different diodes and their pressure coefficients provided we neglect the small pressure de pendence of A : In Jo'=lnA+(l/a)d In Jp/dP. (11) Equation (11) shows that for different diodes In Jo' should vary linearly with din J 1'/ dP with a slope of 1/ a. Figure 9 shows the data on phonon-type ger manium diodes of Miller, Nathan, and Smith4 together with some more recent data.l° For parabolic energy bands and constant field we see from Eq. (4) that A a: Eo", (12) where n is usually between 1 and 2. Its actual value is determined by the manner in which m* and F vary with Eg. Equation (12) implies that a=n(d In Eo/dP). (13) (d In Eg/dP) is known from experiments on pure mate rial to be 0.OnXlO- 4 (cm2/kg). The line in Fig. 9 is drawn for n= 1.1. The results of high pressure experiments on the "nonphonon" diodes are complicated. J p does not show a simple exponential dependence on pressure as found for the phonon diodes. A typical example is shown by curve (b) of Fig. 8. A change in slope in the pressure dependence of J p occurs at approximately 8000 kg/ cm2 over a range of J p from 10-1 to 103 amp/ cm2• As yet there is no satisfactory explanation of this. With In J p plotted as a function of its low pressure coefficient (P<8000 kg/cm2), on the basis of early data a relation of the form of Eq. (11) was found4 with n=2. However, more recent measurements carried out at Harvard UniversitylO have shown this to:be incorrect. The data are shown in Fig. 10 where In J p is plotted as a function of d In J p/ dP for different diodes. I(is necessary to plot J p rather than J 0' on the ordinate of Fig. 10 since, for the higher current density diodes, series resistance pre vented an accurate determination of Jo'. The results I~~-'---------~ I 10 10' I~ _ dl.tn J/Y) (YOLTS)' dY FIG. 7. In Jot vs -dIn (J/V)/dV for non phonon diodes. T=297°K. 10 M. I. Nathan and W. Paul (to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:55CUR R E N T VOL TAG E C H A RAe T E R 1ST I C S 0 F GeT {) NNE L D I 0 DES 1465 for three kinds of diodes are plotted in Fig. 10: (1) The circles represent diodes made by alloying 1% gallium in indium dots onto arsenic-doped substrates; (2) the squares represent diodes made by alloying 1% gallium in tin dots onto arsenic-doped substrates; and (3) the triangles represent ones made by alloying 1% gallium in tin dots onto phosphorus doped substrates. All three types of diodes have the same shape of pressure de pendence as pictured in Fig. 8 curve (b), and all of them have nonphonon tunneling current predominant at 4°K. The scatter in the data in Fig. 10 is large when all the points are considered. However, if just units of type (1) are considered, the data are fairly consistent with a line of almost the same slope as for the phonon diodes discussed earlier. This suggests that the pre-exponential factor A for the nonphonon current in the three types of diodes is different. I'0lk--~--------------~ 0.5 .,1 .,;;: Q2 0.1 e A INCREASING P 6 t. DECREASING P P \kl/em2) 20000 30000 FIG. 8. In Jp vs P. Curve (a) is for a germanium diode with phonon tunneling current predominant. Curve (b) is for a diode with nonphonon current predominant. DISCUSSION The data of Fig. 4 verify the linear relation between Info' and dIn (J/V)/dV predicted by Eq. (9) for phonon assisted current. However, the best fit to the data of Fig. 4 is a slope of 1.25 Eo/ e, rather than 2Eo/ e as predicted by Eq. (8). We can determine an exponent AV= -1.25 (Eo/e)d In (J/V)/dV, (14) where the subscript V refers to the fact that Av is deduced from the f vs V data. A can also be deduced from the high pressure data Ap= -(d In fp/dP)/a. (15) For those phonon diodes on which both high pressure data and f vs V data exist, Av=Ap to within 15%. The pre-exponential factor for phonon current Aph can also be determined from the experimental data. However, before doing this it is necessary to consider the dependence of Aph on A. At T=OoK and for para-10' r--------.-----------, '0 ~ 10' Ln SLOPE a 11,9.10. !!.. em 10°L-_J-_~ __ L__~_---'~~ 0.4 0.6 0.8 1.0 1.2 1.4 1.6 _ d.in Jp (164 em') dP kl FIG. 9. In Jo' vs -d In J p/dP for germanium diodes with phonon current predominant. bolic energy bands, Aph can be written in the form6,7,1l-13 ne2(mcXmcymczmVxmVymvz)tEotM2V Aph=-------------- 12y'J1l'~h6mTX!Aj 10' .....-.---------------, 10' o ./.n SLOPE -12.4 xlO· 4 em 10~ L--_-'-_--'- __ "'--_-'-_---''--_-'--_-' 0.4 0.6 0.8 1.0 d.in Jp --dP-1.2 ~ -4 em') 10 -k, 1.4 I •• 1.8 (16) FIG. 10. In Jp vs -d In Jp/dP for nonphonon diodes. 0 repre sents diodes made by alloying 1% gallium in indium dots into arsenic doped substrates; 0, 1% gallium in tin dots into arsenic doped substrates; L., 1 % gallium in tin dots into phosphorus sub strates. The line has almost the same slope as in Fig. 4. 11 L. V. Keldysh, J. Exptl. Theoret. Phys. U.S.S.R. 34, 962 (1958) [translation: Soviet Phys.-JETP 34(7), 665 (1958)]. 12 Eq. (38) of reference 6, which gives the value of A contains an arithmetic error and is too large by a factor 16r. ' 13 The numerical value of the ratio of direct to phonon current quoted on p. 88 of reference 7 is in error. E. O. Kane (private communication). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:551466 MARSHALL I. NATHAN where M is the phonon matrix element for scattering from one extremum to the other at the point of sta tionary phase; V is the volume; n is the number of valleys in the conduction band favorably oriented for tunneling; mex, mey, mez, m.x, m.y, mvz are the com ponents of the conduction band and valence band mass tensors; and mrx is the reduced mass in the direction of the field. We see that Aph is proportional to A-i. This means that dIn (J / V)dV will contain a term din Aph/dV. Moreover the ordinate in Fig. 4 should strictly be multiplied by At to obtain a linear relation. However, since the range of the experimental din (J/V)/dV only from 17.4 to 25.5 (47%) this makes little difference to the form of Eq. (9), but it does affect somewhat the actual values of Aph and 'I' deduced from experiment. Taking this variation into account we find from the data of Fig. 4 that '1'=1.13 and Aph=7·106 X (Q cm2)-1 for A= 15. Similar corrections must be made to the high pressure data in Fig. 9. We assume that dAph/dP=O since the pressure dependence due to A will tend to be cancelled out by the pressure dependence of Eo and the masses. Then from Fig. 9 we find Aph=5·10 6(Qcm2)-1 for A= 15. The values of Aph deduced from the two experi ments are in much better agreement than the experi mental errors would indicate. The theoretical value of Aph can be computed from Eq. (16). We take n=3; mex=mey=0.08 mo, mez=1.58 mo; mvx=mvy=mvz=0.04 mo; mo=free electron mass. Eo=0.65 ev; M2V=4.3·1O-49 erg2 cm3, the value used by Kane.7 It is necessary to correct Aph to the finite temperature of the experiment (297°K), since the phonon occupation numbers are finite at this tempera ture. There is also a correction due to neglect of the finite phonon energy in the exponent. We have esti mated these corrections to cause a factor of three in crease in Aph. Then from Eq. (16) we find Aph=4·103 X (Q cm2)-1 for A= 15 at 297°K. The experimental values are 103 times the theoretical one. The question might be raised as to whether the dis crepancy might be due to an incorrect estimate of A. E E 1.1 Ibl FIG. 11. (a) E vs ",2. The solid curve is the (000) band for the two band approximation. The dotted line represents the (111) conduction band. (b) The approximate E vs ",2 curves used in computation of the integrals which take into account variable field. (See Appendix.) There are two pieces of evidence against this. First, from measurement of the Hall effect it was possible to estimate the density of impurity atoms in one substrate to be 3.5.1018 cm-3• Diodes made on this substrate had values of A v between 17 and 19. If we take N a = 2 . 1019 cm-3, me=0.08 mo and mv=0.04 mo, we find from Eq. (4) that the average field F=0.63 F max' F= 2/3F max is expected on the basis of the calculations in the Ap pendix. Thus the values of A found from experiment are reasonable. Second, diodes have been made14 on vapor deposited antimony doped substrates which have Jo'=4·103(Q cm2)-1. Observation of the current-voltage curve at 4.2°K leads to the estimate that the tunneling current is approximately 3/4 phonon assisted, which gives J 0' = 3 .103 for phonon current. If the calculated Aph is correct this corresponds to A = 3 for these diodes, which is unreasonably smalL We have assumed in making all calculations of A thus far that the energy bands are spherical and para bolic. In germanium the constant energy surfaces of the lowest conduction band minima are prolate ellipsoids along the (111) directions. The valence band maximum, which is at the center of the zone is doubly degenerate with warped energy surfaces. It is difficult to do any thing about the fact that the bands are degenerate and non-spherical but we can examine the effect on A and A of nonparabolicity. If the tunneling is from the light hole valence band, Kane7 has pointed out the major nonparabolic effects will be due to the valence band. He has calculated the E(a) curve for the light hole band taking into account three interacting bands. It can be shown,15 however, that the three band approximation changes the tunneling probability only very slightly from the two band approximation,16 We shall use the latter. If mv is small compared to the free electron mass, as it is in germanium we have for the light hole valence band E(I-E/ Eo) =h2a.z/2mv. (17) Eo is the energy gap between the (000) valence and con duction band extrema, and mv is taken to be equal to (000) conduction band mass. The zero of energy is taken at the valence band maximum. For E <Eo/2, Eq. (17) represents the valence band, and for E>Eo/2 it represents the (000) conduction band. E(a) for the (111) conduction band is (18) if nonparabolic effects are neglected. It is easy to show that this is justified for our purposes. The energy E. of the point of stationary phase, where the integration in Eq. (2) changes from the valence band to the con duction band is given by the condition ac=av. Hence, by Eqs. (17) and (18) 14 J. C. Marinace (private communication). 16 W. P. Dumke (private communcation). 16 E. O. Kane, J. Phys. Chern. Solids 12, 181 (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:55CUR R E N T VOL TAG E C H A RAe T E R 1ST I C S 0 F GeT U NNE L D I 0 DES 1467 4.Z·K 10-1 ~ ~ i .! ~ I -2 10 2.10' L--.JJ._--!-:---:l::---:l:-.......l--=---!:---:=---::! o .04.06.08.1 J2 .14 .16 .18 - V lVOLTS) FIG. 12. Current density -J vs reverse voltage -V for a phonon-type diode at 4.ZOK. E8= {(mc+mv)_[(mc+mv)2_ EumcJl}Eo. (19) 2mv 2mv Eom. If we take Eo=0.8 ev, Eg=0.65 ev, mv=0.04 mo, mc=0.08 mo we find E.=0.57 ev. The tunneling wave function belongs to the (000) system of Eq. (17) almost all the way across the energy gap. Nonparabolic effects due to the (111) conduction band will become important when Eg-E is some reasonable fraction of the vertical energy gap to the nearest interacting band, which in germanium is probably 2.1 evP Since this is large com pared to Ee-E. we are justified in using Eq. (18). The E(a) curves are illustrated schematically in Fig. 11 (a). The solid curve belongs to the (000) system and the dotted line to the (111) conduction band. To evaluate A the path of integration is OSG. For the con stant field approximation we find from Eq. (2) m.t {[ (2E. ) 7rJ A=--EoJ sin-1 --1 +- hF2! Eo 2 + 16(me)!(Eg~E8)!}. (20) 3 mv Ro In the Appendix, A for this band structure is evaluated approximately allowing for a variable field. The form 17 J. C. Phillips, J. Phys. Chern. Solids 12, 208 (1960). of the voltage dependence shown in Fig. 5 is found to be unaffected by inclusion of both the variable field and nonparabolic effects. A value of a in Eq. (10) may be calculated by differ entiating Eq. (20) and using the experimentally deter mined values of dEg/dP and dEo/dP.18 If we assume Frx.Egt, m.rx.Eo, me is constant we find a=0.235. From the simple theory given by Eq. (4) for parabolic bands, we find a=O.077 if we take the masses independent of pressure. The experimental value de duced from the data of Fig. 9 is a=0.09 where we have corrected for the A dependence of Aph. It is seen that the parabolic approximation gives much closer agreement with experiment than the two band theory. The reason for this is not understood, but it may be related to the fact that we have not taken account of the degenerate valence band structure. Using the methods outlined in references 6 and 7 we have calculated the effect of the two band approxima tion on Aph. The result is to increase Aph by only a factor of 1.7 over its value obtained by the parabolic approximation. For reverse biases in germanium direct tunneling from the valence band to the conduction band minimum at k=O is observed9 at low temperatures with an onset voltage given by -(Eo-Eg-r n)/ e. The ratio of Aph to the pre-exponential factor Ad for direct tunneling can be estimated roughly from experi ment in the following-manner. Figure 12 shows current density vs reverse voltage for a phonon type diode at 4.2°K. The inflection at about -V =0.135 v we assume to be the onset of direct tunneling. We take the approxi mate onset voltage for the phonon current to be -0.03 v, since three of the phonons involved start to give current near this voltage.3 From comparison of the currents at equal reverse voltages above these onset voltages we estimate the ratio of direct current to phonon current to be 100. This ratio is expected to be independent of current only for voltages above the onset large compared to E.J./e""0.02 v/ so that the roughness of the estimate should be appreciated. This ratio is approximately the same for the diodes having current densities in the range shown in Fig. 4. The value of the exponent Ad for the direct tunneling process is (21) for the two band approximation. If we compare this with Eq. (20) for A for phonon assisted tunneling and we allow for the difference of field due to the comparison of currents at different voltages we find that the ex ponents for the processes are within 5% of each other. Therefore the ratio of currents is expected to be equal to the ratio of the pre-exponential factors. This leads to an experimental value of Ad= 2 '108(n cm2)-1 for A= 15 at 4°K (we have corrected ApI> by a factor of 3 for the 18 A summary of high pressure results is given in W. Paul, J. Phys. Chern. Solids 8,196 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:551468 MARSHALL I. NATHAN TABLE.1. The experimental and theoretical values of the pre exponential factors Ad, Aph, A for germanium diodes in units of (n cm2)-I. Phonon Nonphonon Direct assisted assisted Process WK) 297 oK 297°K Experiment 2X108 6X106 107 Theory 107 7X103 temperature change). From references 6 and 7 the theo retical value of Ad can be put into the following form for the two band approximation and constant field: (22) For Eo=O.9 ev and m.=O.04 mo this gives Ad=1·107 X (12 cm2)-1 for X = 15. In view of the roughness of the method of determining the experimental value this is regarded as being within experimental error. For non-phonon current by procedures similar to those outlined previously, an experimental value of A = 107 (12 cm2)-I for X = 15 can be estimated from the ~ig~ pressu~e data of .Fig. 10 for diodes made by alloying mdmm-gallIum dots mto arsenic doped substrates. (We have taken Jo'/Jp=50 V-I for X=15 and assumed A is proportional to X-7/2.) The J vs V data are not re garded as good enough to estimate a value of A. SUMMARY We have seen that from J vs V data we have been able to deduce the exponent for germanium tunnel diodes which exhibit predominantly phonon assisted current. The value obtained is in agreement with that found from high pressure measurements provided that we accept a dependence of the field on voltage stronger than the simple theory predicts. Calculations which take into account the linear variation of the field through the junction give the necessary stronger de pendence; however, the calculated curve for the ex ponent vs voltage is not as straight as the one deduced from experiment. The very good straight lines obtained experimentally are not completely understood. The theoretical and experimental values of the pre exponential factors are given in Table I for X= 15. For A d the discrepancy is not regarded as significant in view of the roughness of the experimental estimate. The dis crepancy for phonon induced current on the other hand is significant and suggests that the theory is inadequate. The theoretical valueI9 of Aph takes account of non parabolic effects of the valence band. J .vs V measurements have also been made on ger mamum diodes which exhibit predominantly non phonon current. The data can be analyzed similarly to 19 A similar calculation for A ph has been made by P. J. Price, IBM Research Note NW-2. He obtains a value a factor of two larger. The difference is mainly due to the use of a different value of the the phonon matrix element. that for the phonon type diodes. However, the correla tion of current density with the exponent deduced from the In (J / V) vs V data is not as good as for the phonon case. ACKNOWLEDGMENTS I should like to thank Dr. P. j. Price, at whose sug gestion this work was undertaken, for many very helpful discussions. His encouragement throughout its course was invaluable. My colleagues at the IBM Research ~enter have helped me in many ways. I am particularly mdebted to Dr. W. P. Dumke, Dr. L. Esaki, Dr. R. R. Haering, Dr. R. W. Keyes, and Dr. P. B. Miller for helpful discussions and to Mr. J. Reinhold for experi mental assistance. The high pressure experiments were carried out at Harvard University in collaboration with Professor W. Paul with equipment supported by the Office of Naval Research. I want to acknowledge his permission to discuss the results prior to publication. APPENDIX It is a straightforward matter to see what quantita ~ive ~ifference the variation of the field through the JunctIOn makes to the exponent X. It is necessary to inte_grate Eq. (2). First Ec(x) and E.(x) must be cal culated. Their variation through the junction is de termined by the field, i.e., where Ec(a) is the conduction band edge energy and X= a is a point on the n side of the junction. At T= OOK, ~her~ degenerate statistics apply throughout the JunctIOn, the field can be calculated as a function of Ec(x) throughout. In the charge depleted region (x" <x<x~), the field is linear in x. The integration of Eq. (A1) IS elementary. Ec(x) is a quadratic function rath~r th~n the linear func~ion in the constant field ap ~roxI.matlOn. For zero applIed bias the path of integra tIon IS at the Fermi energy, EF=CPn=CPp, XI=X"O and X2~XpO. (We are neglecting the phonon energy.) For fimte reverse bias the current is given by a sum over all equi-ener~y paths with occupied states on the p side ar:d unocc~pled states on the n side, i.e., those paths with energies between the quasi-Fermi energies CPn and CPP in Fig. 1. Rather than sum over all paths it will be sufficient for our purposes to consider just a few paths such as those at CPn, CPP' CPo, the path of minimum distance between classical turning points. To see explicitly what effect the variation uf field has on the exponent we shall consider a very simple case-a completely symmetrical junction with "spherical para bolic" bands, mc=m., Na=Nd, t,,=tp=t. We shall ca~c~late Xo. the exponent for the path at energy CPo of mlllimum distance through the space charge region. We [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:55CUR R E N T VOL TAG E C H A RAe T E R 1ST I C S 0 F GeT U NNE L D I 0 DES 1469 find AO=( m*K )!{[Eg~(Eg-eV+1.2.\)!J 7re2Noh2 (A2) The function 1/I=Aoh(7reWo/m*K)i is plotted in Fig. 5 as a function of -V for Eg=0.65 ev and two values of .1,0.01 ev and 0.025 ev. The straight line in Fig. 11 has the slope of t predicted by the constant field approxi mation from Eqs. (3), (4), and (5). It is seen that the slope of the curves is greater than t, in agreement with our experimental results. The calculated plot in Fig. 5 should be compared directly with the experimental curves of In (J / V) vs V in Fig. 3. The marked curva ture evident in the calculated curve is not present in the experimental one. It is possible that the onset of tun neling to (000) minimum at about 0.1 v masks the curvature at the high voltage end of the curve. Never theless, the experimental curve still appears much straighter than the theoretical one. The explanation of this may be related to the fact that the experiment was carried out at room temperature while the theory is for T=OoK. We have carried out the integration of Eq. (2) for other paths-q,,, and r/>p in Fig. 1. We have also allowed for an asymmetrical junction me"" m. and N a"" N d. None of this makes any difference to our conclusions: (1) A varies faster with V than predicted by theconstant field approximation. (2) The experimental curve of A vs V is much more nearly linear than the theoretical curve. It should be pointed out that the exponent given in Eq. (A2) has to a first approximation, for eV, r«Eg, a linear dependence on Eg in agreement with simple theory. Allowance for asymmetry does not affect this conclusion. Hence the interpretation of the results of the high pressure experiments is not affected by the variable field. Nonparabolic effects of the valence band and variable field can be taken into account. The E(a) curves given by Eqs. (17) and (18) must be used in Eq. (2) with the variable field. The integration can be performed in terms of elliptic integrals. Rather than do this we have approximated the E(a) curves as shown in Fig. 11 (b). The path of integration is OS'. For the (000) band E=h2av2/2mv Eo-E=h2av2/2mv For the (111) band E<Eo/2 E>Eo/2. The integral can then be easily evaluated. The result for eV, .In, ,Ip«Eg is (A3) where !J.E= Eo-Eg. The path of integration is at the energy r/>p in Fig. 1. If we differentiate Eq. (A3) with respect to voltage we find as before that the derivative is greater than that obtained from simple theory and that there is more curvature in the A vs V plot than the simple theory predicts. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.120.242.61 On: Sat, 22 Nov 2014 22:36:55

1.1729057.pdf

Sputtering Experiments with 1 to 5keV Ar+ Ions A. L. Southern, William R. Willis, and Mark T. Robinson Citation: Journal of Applied Physics 34, 153 (1963); doi: 10.1063/1.1729057 View online: http://dx.doi.org/10.1063/1.1729057 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Angular distribution of Si atoms sputtered by keV Ar+ ions J. Appl. Phys. 51, 2884 (1980); 10.1063/1.327957 Sputtering Experiments with 1–5keV Ar+ Ions. Displacement of Ejection Pattern Spots J. Appl. Phys. 40, 4982 (1969); 10.1063/1.1657329 Sputtering Experiments with 1 to 5keV Ar+ Ions. III. Monocrystal Targets of the Hexagonal Metals Mg, Zn, Zr, and Cd J. Appl. Phys. 39, 3463 (1968); 10.1063/1.1656798 Sputtering Experiments with 1 to 5keV Ar+ Ions. II. Monocrystalline Targets of Al, Cu, and Au J. Appl. Phys. 38, 2969 (1967); 10.1063/1.1710034 Sputtering Yields of Single Crystals Bombarded by 1 to 10keV Ar+ Ions J. Appl. Phys. 34, 3267 (1963); 10.1063/1.1729175 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 1 JANUARY 1963 Sputtering Experiments with 1-to 5-keV Ar+ Ions A. L. SOUTHERN, WILLIAM R. WILLIS,· AND MARK T. ROBINSON Solid State Division, Oak Ridge National Laboratoryt, Oak Ridge, Tennessee (Received 13 July 1962) Sput~ering yields have been determined with 1-to 5-keV Ar+ ions normally incident upon targets of type 304 stamless steel, three different types of polycrystalline Cu, a wide variety of Cu monocrystals and mono crystals of Si and Ge. Ejection patterns have been recorded from these targets and from a mo~ocrystal of InS.b .. The.sputtering yi:ld of.polycryst~lline Cu depends on the source of the metal, apparently due to vanatlOns m preferred onentatlOn. The yIeld from Cu monocrystals is strongly dependent on orientation the effect becoming more pronounced as the energy is increased. A simple model is presented which account~ for this behavior in terms of the variation with direction of the initial mean free path of the incident ion. The ejection patterns leave little doubt that focusing collision chains are primarily responsible for the transport of momentum to the surfaces of close-packed metals. INTRODUCTION THE erosion of metal targets during ion bombard- ment has been widely studied in recent years, both for technological reasons and because of its rela tionship to fundamental radiation-damage processes. The experiments reported in this communication were begun in the hope that a quantitative understanding of the mechanism of sputtering might be more readily obtained than could the corresponding understanding of more conventional radiation damage processes and with a view towards aiding in the interpretation of the latter. Perhaps the most striking aspect of sputtering is the fact, first observed by Wehner,! that the bulk of the material ejected from metal monocrystals leaves the surface in a limited number of principal crystal lographic directions, a fact that is generally taken as the primary evidence for the reality of the correlated collision sequences ("focusing collisions") postulated by Silsbee.2 Some of the experiments reported here were intended to settle certain doubts remaining in our minds about the correspondence between the experi mental phenomenon and Silsbee's theory. The quanti tative aspect of sputtering is generally reported in terms of the yield, that is, the average number of target atoms ejected per incident ion. The remaining experi ments were performed to obtain information on the dependence of the yield on crystallographic variables and on the energy of the bombarding ions in the 1 to 5 keY range, primarily for eu targets. EXPERIMENTAL PROCEDURE A block diagram of the apparatus used to determine sputtering yields is shown in Fig. 1. The ion source was of the type designed by Moak, Reese, and Good.3 It * Professor of Ph~si~s,. West Virginia Wesleyan College, Buckhannon, West VlrglnIa. Consultant to ORNL Solid State Division. , t Oak ~idge N ational Labora~ory is operated by Union Carbide (.orporatlOn for the U. S. AtomIc Energy Commission. IG. K. Wehner, J. App!. Phys. 26,1056 (1955)' Phys. Rev. 102,690 (1956). ' 2 R. H. Silsbee, J. App!. Phys. 28, 1246 (1957); G. Leibfried, J. App!. Phys. 31, 117 (1960). 3 C. D. Moak, H. Reese, Jr., and W. M. Good Nucleonics 9 No.3, 18 (September 1951). " was mounted in the tank coil of a self-exciting oscillator operated at 38 Mc. The rf power was smoothly variable from 0 to 150 W. Argon ions formed in the source were accelerated and were then focused onto a defining iris by a zero-energy electrostatic lens. The collimated beam fell at normal incidence onto the surface of the target which was an integral part of a Faraday cup provided with a guard ring to suppress the current of secondary elec trons generated at the target. The apparatus used to record ejection patterns was essentially similar, except that the target holder shown in Fig. 2 replaced the Faraday cup. Without the beam, the pressure in the system was reduced to about 2X 10-7 Torr by an oil diffusion pump fitted with a liquid nitrogen cooled baffle. When the beam was turned on, the pressure in the target chamber reached 2 to 7 X 10-6 Torr, the increase being due to argon gas. Since current densities of SO to 200,uA/cm2 were employed, the rate of arrival of background gas atoms at the target surface (not induding Ar) was about t to 1/20 the rate of arrival of Ar+ ions. No evidence of a pressure effect on sputtering yields4 was found in this work, over the total pressure range from 1 to lOX 10-6 Torr. This result suggests that contamina tion of our target surfaces does not play a significant role in determining the observed yields. FIG. 1. Block diagram of the apparatus for ___ ~~ determining sputtering yields. '0. C. Yonts and D. E. Harrison, Jr., J. App!.l'hys 31 1583 (1960). . , 153 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29154 SOUTHERN, WILLIS, AND ROBINSON ION BEAM 1 0 1 2 """""""" -- 1""""1 INCHES STAINLESS STEEL BASE PLATE TARGET __ STAINLESS STEEL TARGET CLAMP BRASS FLANGE FIG. 2. The target holder used for recording ejection patterns during ion bombardment. The energy distribution of the ion beam was studied by inserting a counter-field lens5 between the electron suppressor and the Faraday cup. Results of analyses made with }-and i-in. defining irises are shown in Fig. 3. In yield determinations above 2 keY, the smaller iris was used, while below 2 ke V the larger was employed in order to make use of the greater current available. Ninety-five percent of the beam current had energy within 100 eV of the desired value: The fraction of energetic neutral atoms in the beam was estimated by measuring the sputtering yield of Cu at 2.5 keY, first with the normal beam and then with the charged particles in the beam deflected by the counter-field lens and an external permanent magnet.. The results suggest that less than about 1.4% of the beam par ticles are uncharged. Since the Ar-Ar+ charge exchange cross section probably does not vary greatly in the 1 to 5 keY energy range, this result is believed to be representative of our beam over the entire range of measurements reported here. The fraction of multiply charged particles is unknown, but the experience of others6 with similar Ar+ ion sources suggests that there will be less than about 6% Ar++ ions produced from the source. The fraction of ArB in the beam arriving at the target will be reduced by action of the focusing lens and by the effects of space charge, both of which will tend to throw Ar++ to the outside of the beam where it will be removed by the defining iris. Sputtering yields were determined by measuring the weight loss of the target during bombardment and the total charge collected in the Faraday cup. Weight losses, which ranged from 0.2 to 1.2 mg, were deter mined on an Ainsworth microchemical balance. The current collected by the Faraday cup was recorded on a Minneapolis-Honeywell strip chart recorder equipped with a transmitting slide wire which operated a Brown continuous integrator. The total collected charge could 5 G. Forst, Z. angew. Phys. 10, 546 (1958). 6 P. H. Ste!son, Oak Ridge National Laboratory (private [.olllmllnj(:ation). be determined to within about ±2% and the weight loss to within about ± 1 to ±4%, depending on its magnitude. The reported yields are believed to be ac curate to within ±S percent. All yield experiments were carried out at room temperature with normally incident ions. Ejection patterns were recorded by allowing the material ejected from the target to deposit onto Pyrex glass plates 3t in. in diameter and l6 in. thick. In some experiments with Cu, a thin (",,25 A) layer of Pd was evaporated onto the plate before recording the ejection pattern in an attempt to suppress any surface migra tion of the depositing atoms, but no clear-cut result was obtained. The ejection patterns, particularly those of Cu, do not keep especially well,7 although some success has been had in preserving them by spraying with a plastic film after removal from the apparatus. The patterns were photographed and in several cases densitometer measurements were made. All ejection pat tern experiments were carried out at room temperature. The argon gas used in the ion source was assayed by mass spectrometry to contain 99.97% Ar. The target materials used, their sources, and their preparative treatment were as follows: (1) Type 304 stainless steel. The targets were cut from a single sheet of steel, cleaned in dilute nitric acid, degreased in perchloroethylene, and hydrogen fired. They were then placed in a vacuum dessicator, evacuated, and stored under 1 atm of air. For this alloy, an effective "atomic weight" of 55.13 was de duced from its composition, for use in converting weight losses to the molar basis. (2) Polycrystalline Cu. Material from three different sources was used. Targets designated ASR were cut 100 "\' 80 60 1025v 40 £ 20 I- ~ 0 co (a) co a 100 ,. <f w CD 80 f- 60 1025V 40 f--- 20 o o (b) r--, '\ 1985V 2900V 3760 V ACCELERA~'ON VOLTAGE Va-in. DE,INING IRIS ~ ~ "\ ~ 1985 V 2900 V \ 3760V ACCELERATION VOLTAGE 1/4~in. OE"INING IRIS COUNTER FIELD LENS VOLTAGE (kV) (a) (b) 4 FIG. 3. The energy distribution of the ion beam. (a) With t-in. defining iris. (b) With i-in. defining iris. ---- 7 R. S. Nelson and M. W. Thompson, Proc. Roy. Soc. (Lohdon) A259,458 (1961). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29SPUTTERING EXPERIMENTS WITH 1-TO 5-keV Ar+ IONS 1SS 7 -+ 1 o I "" / / / / / ---?7 ~ OFHC Cu ~ ~ -~-+==-V f- 4 /-~- DUTCHCu' V V --,I-~--- 7-~~ STAIJ.ESS rEEL -,- II I i 2 3 4 5 A'+ ION ENERGY (keV) FIG. 4. The sputtering yields of type 304 stainless steel and ?f polycrystalline Cu as functions of the normally incident Ar+ IOn energy. from rolled sheets of 99.999% Cu purchased from the American Smelting and Refining Company. Those designated OFHC were cut from a sheet of OFHC grade Cu obtained from O. C. Yonts of this Laboratory; the sheet was from the same stock as that used by Yonts, Normand, and Harrison8 in their sputtering experiments. The third set of targets, designated Dutch, was cut from a sheet of Cu obtained from Professor J. Kistemaker, Laboratorium voor Massascheiding, Amsterdam, which was from the stock used in the experiments of Rol, Fluit, and Kistemaker.9 All of these targets were cleaned by etching in SO% nitric acid, followed by rinses in 5% nitric acid and in dis tilled water. (3) M onocrystalline Cu. Single crystals grown from 99.999% Cu were cut to provide targets of the desired orientation. Some of these targets were prepared in this Laboratory and some were purchased from the Virginia Institute for Scientific Research, Richmond. The targets were polished on an acid polishing ma chine,1O cleaned in a mixture of equal parts of nitric, phosphoric, and glacial acetic acids, and, finally, electropolished. (4) Monocrystalline Si, Ge, and InSb. The targets were cut from high-purity single crystals, obtained from E. Sander and J. W. Cleland of this Laboratory, and were etched in CP-4 before use. The orientations of the various monocrystalline tar gets were determined by the Laue x-ray back-reflection technique, the indices of the surface normal being evalu ated to within ±2 degrees, except as noted below. The '0. C. Yonts, C. E. Normand, and D. E. Harrison, Jr., J. Appl. Phys. 31, 447 (1960). 9 P. K: Rol, J. M. Fluit, and J. Kistemaker, Proceedings Third I nternatwnal Conference on Ionization Phenomena in Gases Venice 1957 (Societa Italiana di Fisica, Milan, 1957), p. 871' Physica 26' 1000 (1960); Cf. also references 18, 19, and 23.' , 10 F. W. Young, Jr., and T. R. Wilson, Rev. Sci. lnstr. 32 559 (1961). ' targets, the Laue photographs, and the glass plates used to record ejection patterns were all marked in a corresponding way so that direct comparisons of the x-ray and sputtering patterns could be made reliably. EXPERIMENTAL RESULTS Polycrystalline Materials The sputtering yields obtained from type 304 sta.in less steel and from the three sorts of poly crystalline Cu are shown as functions of the Ar+ ion energy in Fig. 4. The data obtained for ASR Cu are in excellent agreement with the values reported by Bader, Witte born, and Snousell and by Yonts, Normand, and Harrison4 (who did not work below 5 keV); those ob tained for OFHC Cu are consistent with the results of Rol, Fluit, and Kistemak er9 (who did not work below 5 keV); and those obtained for the Dutch Cu agree well with the results of Keywell.12 In attempting to find a reason for the variation of the yields from the various types of Cu, an ejection pattern was recorded from each one. Most surprisingly, the ASR Cu showed clear evidence of spots of the type observed from mono crystals (see below), whereas the OFHC Cu showed only a continuous deposit of circular symmetry, as expected from polycrystalline targets. The Dutch Cu showed a continuous deposit, but of approximately elliptical symmetry, not unlike the pattern from a (011) monocrystal. The ASR pattern is shown in Fig. 9(d). It appears probable from these results that differ ences in preferred orientation of the individual grains are responsible for the yield differences among the three sources of Cu. In view of the very large differ ences in yields observed from various Cu monocrystals (see below), it is not at all surprising that polycrystal line materials from different sources and in the hands .9 3 " E o o :l 2 w >' (!l z a:: w ~ 1 ::0 ll. (f) Si 2 3 4 5 Ar+ ION ENERGY (keVl FIG. 5. The sputtering yields of monocrystalline Si and Ge as functions of the normally incident Ar+ ion energy. ---- 11 M. Bader, F. C. Witteborn, and T. W. Snouse, Nationa, Aeronautics and Space Agency Report NASA-TR-R-I05 (1961). 12 F. KeywelI, Phys. Rev. 97,1611 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29156 SOUTHERN, WILLIS, AND ROBINSON FIG. 6. Orientations of Cu monocrystals used in Ar+ sputtering experiments. The crystals marked with open circles were ex amined only at 5 keY. The other crystals were studied at several ion energies. of different investigators should give rather disparate yield data. Almen and Bruce13 have found similar variations in the sputtering of Ag from different sources by 45-keV Kr+ ions. Monocrystalline Semiconductors The yields obtained from (001), (011), and (Ill) Ge and from (Ill) Si monocrystals are shown as functions of the Ar+ ion energy in Fig. 5. No yield measurements were made on InSb. There do not appear to be signifi cant differences in the yields of the three Ge mono- o , 9 - I 6 :l '" 5 -;:: ~ <001) I I (011) l_..I,.-+--+-+-+ Ar i'" ION ENERGY (keVl FIG. 7. The sputtering yields of some Cu monocrystals under normally incident Ar+ ion bombardment, as functions of the ion energy. The arrows indicate points obtained on freshly electro polished (111) surfaces. 13 O. Almen and G. Bruce, Nuclear rnstr. Methods 11, 257 (1961). crystals except possibly at the higher energies where the order of yields seems to be (001»(111)? (OIl). Ejection patterns from Si and Ge showed only con tinuous deposits. No evidence of a "body-centered" pattern, as reported by Anderson and Wehner,14 was seen. In the energy range from 1.0 to 1. 7 ke V, a (111) InSb monocrystal produced an ejection pattern con sisting of three light spots superimposed on a continu ous background. The best definition of the spots was obtained at 1.5 keY. Unfortunately, the loss of con trast in the photographic process prevents a useful copy of this remarkable pattern from being included in this communication. The three light spots, indicative of less than average probability of sputtering, were located at the (011) poles of the pattern. This result provides a rather compelling argument for the reality of the Silsbee correlated collision sequences2 and seems at the same time to exclude "tunnel focusing"!' as the primary source of ejection pattern spots. In space groups F43m (InSb) and Fd3m (Si,Ge), there are rather large channels running in (011) directions, bordered by what may be regarded as either zig-zag lines (made up of (Ill) segments) of atoms or as closely spaced double lines parallel to (011). As an atom in one of the double lines moves parallel to (011), it does not first strike the next member of its line, as would be necessary to obtain Silsbee focusing, but, instead, suffers a glancing collision with an inter vening member of the other line of the doublet, thus causing both atoms to move away at substantial angles from (011). The consequence is that ejection of atoms parallel to (011) is inhibited in F43m by pre cisely the analog of the process causing preferential ejection parallel to (011) in space group Fm3m. Further more, if tunneling were primarily responsible for ejec- 8 1-----1 OR'~NTAJ'ON 1 1 1 1 1 rl I 0(0,1,6) I I o <0,1,11) ~ (1,1,14) I f----.. (1,6,26) I ! • (1,3,15) I ---.- ~--- j!...--: .....- ~O",6) V I ~ I ~ (0,1,11) c-~-\P V I I 3 2 o 3 4 5 6 Ar+ ION ENERGY (keVi FIG, 8. The sputtering yields of some Cu monocrystals under normally incident Ar+ ion bombardment, as functions of the ion energy. 14 G. S. Anderson and G. K. Wehner, J. App!. Phys. 31, 2305 (1960). 16 M. T. Robinson, D. K. Holmes, and O. S. Oen, Bull. Am. Phys. Soc. 7, 171 (1962); C. Lehmann and G. Leibfried (to be published) . [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29SPUTTERING EXPERIMENTS WITH 1 TO 5-keV Ar+ IONS 157 (a) (b) (e) (d) FIG 9 Ejection patterns obtained from some Cu monocrystals and from ASR polyerystalline Cu under normally incident A~+ion bombardment. (a) {001) at 4 keY, (b) {111) at 2.5 keY, (e) {122) at 4 keY, (d) ASR polycrystalline at 4 keY. tion pattern spots, one would expect that Ge and Si would show tendencies toward spot formation and that InSb would give dark spots at the (011) poles. No attempt was made to study the chemical composition of the deposits from InSb. Monocrystalline eu The orientations of the Cu monocrystals used in our sputtering experiments are shown on the stereographic projection of Fig. 6. The observed sputtering yields for twelve of these crystals are shown in Figs. 7 and 8 as functions of the Ar+ ion energy. In Fig. 13 the sputtering yields at 5 keY of ten (Okl) monocrystals are shown as a function of the angle between the surface normal and (001) (note that the incident beam was always perpendicular to the target surface!). Figure 9 shows the ejection patterns obtained from monocrys tals of (001), (111), and (122) orientation, as well as the pattern obtained from an ASR polycrystalline target. Ejection patterns were also recorded from the fourteen other crystals shown in Fig. 6. In agreement with previous observations,1,7,14,16,17 the ejection patterns always consist of a limited number of intense spots accompanied by varying amounts of low intensity background. Comparison of the ejection pat terns with Laue back-reflection photographs made from the same surfaces of the same crystals shows all of the intense spots to be accounted for by the (011), (001), and (111) directions. The (011) spots seem to be the consequences of Silsbee focusing collision chains,! while the (Om) and (111) spots result from the very similar "assisted" focusing chains discussed by Nelson and Thompson.7 No spots corresponding to (112) or (013), both of which are more closely packed lines than (111), have been observed, as was to be expected since assisted focusing is not possible in these directions because of the relatively unsymmetrical disposition of 16 M. Koedam, Thesis, Utrecht (1960); M. Koedam ~nd A. Hoogendoorn, Physica 26, 351 (1960); M. Koedam, Physlca 25, 742 (1959). 17 V. E. Yurasova, N. V. Pleshivtsev, and 1. V. Orfanov, Zhur. Eksp. i Teoret. Fiz. 37,966 (1959); Soviet Phys.-JETP 37,689 (1960). parallel neighboring lines about a (112) or (013) line. The patterns did not appear to change significantly over the energy range from 1 to 5 keY. The (001) pattern, Fig. 9(a), shows four (011) and one (001) spots, accompanied by considerable haze. Crystals cut a few degrees from (001) yield patterns in which these five spots move in the directions and to the extent expected from crystallographic considerations. This result greatly increases our confidence that the spots are really the consequences of focusing collision chains. Furthermore, as a spot moves away from the surface normal, its intensity decreases rapidly, pre sumably because of the increased length of the collision chains leading to the surface. At least near (001), the orientation of a crystal can be determined about as accurately from sputtering ejection patterns as from Laue back-reflection of x rays. The intensity of the (001) (central) spot in Fig. 9(a) is at least as great as that of one of the (011) spots. Note that the ellipticity of the (011) spots on this and other patterns results from the geometry of the apparatus used to record the patterns. The (111) pattern, Fig. 9(b), shows the expected group of three intense (011) spots. The central, (111), spot does not appear in this particular pattern because of its small size compared to that of the hole in the plate, but in other patterns, taken at small angles of beam incidence ('"'-'2 degrees from the surface normal), this spot was clearly evident. Midway between the (011) spots, there are streaks, running from (111) in the general direction of (114) and (001), which give the over-all pattern a hexagonal appearance. The outer bulge of each of these streaks may be interpreted as due to (001) focusing events. The appearance of "streaks" seems to be largely illusory: if the outer bulges towards (001) and the (011) spots are ignored, the background seems to be quite uniform. The (122) pattern, Fig. 9(c), shows six spots. Near the center is a very intense (011), two others appearing in the lower right quadrant of the pattern. Two faint (Om) spots and one (111) are also present. A point of some interest is that relatively less background appears in this pattern than in the (001) and (111) projections. This [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29158 SOUTHERN, WILLIS, AND ROBINSON r;THEOR , SURFACE G G G FIG. to. Sketches to illustrate deviations of ejection directions from the directions of close-packed lines (see text for discussion). is also true of other patterns obtained at large angles from the principal directions. Patterns from a (011) crystal (not shown) consisted of an intense central (011) spot, two much weaker (001) spots, and four faint streaks leading from the center of the pattern towards the positions of the peripheral (011) poles, which are at an angle of 60 degrees from the surface normal. After the yield from this crystal proved to be lower than expected theo retically (see below), the ejection pattern and Laue photograph were re-examined. The surface normal was found to be about 3 degrees from (011); the indices of the normal were approximately (1,16,17). The noncentral ejection pattern spots seem always to occur somewhat closer to the surface normal (by '" 2 to '" 5 degrees) than is expected from crystallo graphic considerations. Since location of the center of a spot is rather subjective, no attempt was made to study these angular deviations in detail. The angular discrepancies are easily understood as due to the in fluence of neighboring rows on the motion of the last member of a sequence of correlated collisions. Figure lO(a) shows a correlated collision sequence along the row AIA2 . . . which makes an angle cf>theor with the surface normal; the last atom in the row Al is ejected. Note that this atom experiences a force from atom BI which is unbalanced by an atom in the row C1C2 ... and which causes atom At to be ejected at an angle from the surface normal cf>obs<cf>theGr. In an unpublished calculation based on the momentum approximation, C. Lehmann of this Laboratory has obtained a value of about 3 degrees for cf>ti.£Or-cf>obs for (011) chains at a {oo1} surface assuming a Born-Mayer interaction potential between the atoms of the crystal. It should be pointed out that any distortion of the lattice near the surface which increases the interatomic separation normal to the surface (or decreases that within the surface) will also cause the surface atoms to be ejected closer to the surface normal than is expected from bulk crystallography. Such distortions might arise either from radiation damage to the crystal or from the in fluence of trapped argon atoms on the structure. If such distortions of the crystal are carried to an extreme, so that surface atoms are displaced from their normal lattice sites, the direction of ejection may be quite different from that of the associated correlated collision sequence, as is shown in Fig. lOeb), where the atom BI of Fig. toea) has been removed and Al has "relaxed" into the vacancy. A similar effect would be shown, if, instead of there being a vacancy in the last complete atomic layer, an "extra" atom were placed above the last full layer, but not always in a normal lattice site. During sputtering, on account of the high rate of removal of material, the last atomic layer is surely incomplete and, even if all momentum reaches the surface via correlated collision sequences, some ejection in quite general directions must occur. This presumably could account for the background observed in our patterns. The point is that the presence of con tinuous background in these ejection patterns does not definitely indicate the existence of a mechanism for sputtering which does not involve collision chains. The dependence of the sputtering yield on the orien tation of the target crystals, shown in Figs. 7, 8, and 13, is very marked. Although there are resemblances to the experiments of Rol, Fluit, and Kistemak erl8 with 15-to 20-keV Ar+ on Cu and to those of Almen and Bruce13 with 45-keV Kr+ on Cu, there are also marked differences, which we attribute partly to the fact that the present experiments were always carried out at normal incidence, while the others were per formed at varying angles of incidence on a small number of principally oriented monocrystals, and partly to the differences in energy in the various ex periments. Some of the detailed difference may, of course, be due to uncertainties in the orientations of our crystals. It must also be noted that the pretreat ment of the targets influences the yields observed from Cu monocrystals. The (111) crystal points in Fig. 7 marked with arrows were obtained on freshly electro polished surfaces, whereas the other points were ob tained on surfaces which had been previously irradi ated, but which were not repolished. While no de tailed study of this effect has been made, it is clear from the figure that significant changes in the yield can occur after a previous irradiation has been per mitted to modify the crystal surface. A partial inter pretation of this behavior can be given in terms of the model which we shall now discuss. 18 P. K. Rol, J. M. Fluit, and J. Kistemaker, Progress in Astro nautics and Rocketry (Academic Press Inc., New York, 1961), Vol. 5, p. 203; P. K. Rol, Thesis, Amsterdam (1960). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29s PUT T E R IN G EX PER 1M E N T S WIT H 1-TO 5 - k e V A r + ION S 159 A MODEL OF THE SPUTTERING YIELD' Prescntly available theories of sputtering19-21 are not sufficiently developed to account for the dependence of the yield on crystallographic variables. Harrison22 has had some success in fitting his theory to some of our data on eu monocrystals, insofar as the energy dependence is concerned, but the orientation depend ence of the resulting "fitting parameter" remains un explained. It is possible, however, to obtain a reasonable account of the crystallographic orientation dependence of our eu data by using the theory of Rol, Fluit, and Kistemaker19 in conjunction with a "transparency" model similar to that of Fluit.23 It should be empha sized that the present treatment of this model is quite different from that of Fluit. The basic assumptions of the theory are that the sputtering yield is determined with sufficient accuracy by the first collision of the incident ion with an atom of the target and that the collision may be imagined to take place between hard spheres whose size de termines the total (microscopic) cross section for scattering of the incident ions by the target atoms.24 The yield is regarded as proportional to the stopping power of the target for incident ions at their initial energy, i.e., as proportional to the average energy transferred to a target atom in a collision and as in versely proportional to the mean free path of the in cident ions to their first collisions. Thus, the sputtering yield for ions of energy E, normally incident upon a target surface in a direction specified by the unit vector u, may be written as S(E,u) = Cf.T(E,u)E/A(E,u), (1) where a is a proportionality constant, X(E,u) IS the mean free path of the incident ion, and r(E,u) = 2T(E,u)/T m(E), (2) where T(E,u) is the average energy transferred to a target atom, and T m(E) is the maximum energy that can be transferred in a single collision. No mass factors 19 P. K. Rol, J. M. Fluit, and J. Kistemaker, Physica 26 1009 (19~0) i Proce.edings of t~e In~ernational Symposium on Elect;omag netzc Separatwn of Radwactzve Isotopes, Vienna, 1960 (Springer Verlag, Vienna, 1961), p. 207. 20 D. E. Harrison, Jr., Phys. Rev. 102, 1473 (1956) i J. Chem. Phys. 32, 1336 (1960) i Proceedings of the International Colloquium on Ion Bombardment, Paris, 1961 (Centre National de la Recherche Scientifique, Paris, to be published). 21 R. S. Pease, Rend. Scuola Intern. Fis. "Enrico Fermi" 13 158 (1959). ' 22 D.~. H~rrison, Jr., U. S. Naval Postgraduate School (private commumcatlOn) . 23 J. M. Fluit, Proceedings of the International Colloquium on Bombardment, Paris, 1961 (Centre National de la Recherche Scientifique, Paris, to be published). 24 For discussions of the validity of the hard sphere approXI mation, see M. T. Robinson, D. K. Holmes, and O. S. Oen, Proceedings of the International Colloquium on Ion Bombardment Paris, 1961 (Centre National de la Recherche Scientifique Paris' t? be publi~hed) i and D. K. Holmes, Proceedings of the I~terna~ twn.al Atomtc Energy Agency Symposium on Radiation Damage in Sohds and Reactor Materials, Venice, 1962 (to be published). appear in Eq. (1) since they have been absorbed into the constant a. The reason for specifying a directional dependence in Eq. (2) will appear presently. The target is considered to be an array of sphcres, arranged on a crystalline lattice. The radius of each sphere R defines the total cross section for those ion atom interactions which will, eventually, lead to sput tering. For ions which are incident upon the surface of the crystal in an (hkl) direction, it is necessary to consider only the "elementary crystal" defined by the elementary translations thkl, th' k' I', th" k" I" where, for cubic lattices, (h'k'l') and (h"k"l") are perpendicular to (hkl) and to each other, since the unit so defined con tains all but the translational symmetry of the lattice. If the surface of this elementary crystal is uniformly irradiated with ions, a fraction Phkl of them will make collisions within the element, while the balance will pass through it without making any collisions at all. It is from this property of the model that the term "transparency" arises. The average distance from the surface which the colliding ions move in reaching their collision points Xhkl is their contribution to the first mean free path of all the ions. It is assumed that those ions which do not make collisions in the elementary crystal have a mean free path AO measured from the surface of the crystal, which is independent of their direction. Thus, in terms of this model, the mean free path for all ions may be written as The energy transferred in a collision may be defined in terms of the impact parameter b, the perpendicular distance from the path of an ion to the center of the atom with which it collides. For hard sphere interactions, (4) where ( ) denotes the average value. If the spheres comprising the elementary crystal are sufficiently small that they do not overlap when projected onto an {hkl} plane, then W/R2) = 1/2 and r(E,u)=l, but for large enough spheres this will not be true because of shadow ing. It is assumed that shadowing may be ignored for those ions which make collisions beyond the elementary crystal. Then, r(E,u)==Thkl(E) = 1+Phkl[1-2(b2/R2)hkl]. (5) Finally, introducing Eqs. (3) and (5) into (1), S hkZ(E) =o{1 +phkl(1- 2(b2j R2)hkl) JE/ [PhkIXhkl+ (1-PI.kI)Ao]. (6) If the sizes of the spheres of which the lattice is composed are sufficiently small that the contents of an elementary unit do not overlap when projected onto an {hkl} plane, the several quantities introduced above may be written in terms of the indices. For fcc crystals [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29160 SOUTHERN, WILLIS, AND ROBINSON of lattice constant an, thk1/ aO= alQ/2, (7a) P)'kl= 47r(R/ ao)2tkhZ/ ao= 2al7r(R/ ao)2Q, (7b) Xhkl/ au= (aIQ2-az)/4Q, (7c) where Q= W+k2+12)1/2 and, if h+k+l is even if lz+k+l is odd, { 1 if lz+k, h+l, or k+l is odd 02= 2 if h+k, h+t, and k+l are even. Unfortunately, except for the lowest index directions. even for quite small spheres, these simple formulas will not suffice [except, of course, for (7a)J. An IBM 7090 program has been written to permit the easy evaluation of the various quantities for arbitrary sphere sizes and indices in the fcc lattice. The elementary crystal is defined by the three mutually perpendicular axes (hkt), (O,l, -k), and (k2+l2, -lzk, -hi), where it is assumed that O~h~k~l and that each set of indices is coprime. The machine automatically orders the in cident beam indices if necessary and removes any common factors to make the sets coprime. Storage limitations in the current version of the code limit the program to those for which i.e., to those elementary crystals containing fewer than 1000 atoms. The starting coordinates of the incident PROJECTION OF THE fcc LATTICE ONTO THE (122) PLANE. SPHERE RADIUS: 0.147 LATTICE CONSTANTS. 24661012141618 NUMBER OF (122) INTERPLANAR SPACINGS BELOW SURfACE (a) (b) FIG. 11. (a) Projection of the fcc lattice onto the (122) plane. The radius of the atomic spheres is O.147ao. The shaded atoms are in the first four (122) layers, the darkest being nearest the surface. (b) The collision probability for (122) as a function of penetration. O~~ __ -J ______ ~ __ -L __ -L __ ~J-- o 2 4 6 8 10 12 14 16 IMPACT PARAMETER (lunil=0.01414 00) FIG. 12. The impact parameter distributions for (122) and (0,1,11) monocrystals for spheres of radius 0.21 ao. ion are selected uniformly over the rectangle defined by the unit translations th'k'l' and t""k"l" by the use of random numbers and the ion is followed until it makes a collision with a lattice atom or until it passes com pletely through the elementary unit. Its collision point and impact parameter are scored and the run is re peated as often as desired. Generally, 10 000 histories constitute a single "experiment," giving results for Xhkl and (b2/R2)hkl which are accurate to within about ±1 percent. Such a run requires from about 15 sec for a principal direction to a few minutes for a high-index direction such as (049). Figure 11 shows the projection of the fcc lattice onto the (122) plane and the proba bility of a collision occurring at various depths beneath the surface for ions incident in a (122) direction. For small spheres (R/ ao<0.118 for this direction), the collision probability is independent of depth, but as the size of the spheres increases, the probability gradually approaches the exponential behavior expected for ran dom orientations. Similar behavior is observed in all directions, although for principal directions, shadowing effects set in only for rather large spheres (R/ ao> 0.306, 0.250, 0.204 for (011), (001), and (111), respectively). Study of the projection in Fig. 11 shows also the effects of shadowing on the impact parameter distribu tion. The shaded spheres, which are the ones nearest to the surface, prevent collisions of large impact pa rameter (i.e., of small energy transfer) with atoms lying deeper within the crystal. In the upper plot of Fig. 12 is shown the distribution of impact parameters for a (122) direction for a sphere radius of 0.2Iao. The dashed line shows the "no shadowing" histogram. The preference for small impact parameter shown in this figure is not universal. For other directions the reverse [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29S P II T T E R IN G EX PER 1M E N T S WIT H 1-TO 5 -k e V A r + ION S 161 TABLE 1. Comparison of observed and calculated sputtering yields for 5.0-keV Ar+ normally incident monocrystalline Cu. Scale Sob, Sob,-Seale (hkl) Pkkl Xhkl/ ao (b2/ R2)hkl atoms/ion atoms/ion atoms/ion 011- 0.397 0.177 0.500 3.41 2.60 -0.81 001 0.555 0.250 0.500 4.36 4.20 -0.16 013 0.754 0.656 0.467 5.57 6.30 +0.73 111 0.947 0.577 0.494 9.44 9.35 -0.09 123 0.682 0.661 0.497 4.65 4.90 +0.25 012 0.803 0.753 0.500 5.50 5.65 +0.15 015 0.813 0.808 0.516 5.27 5.20 -0.07 122 1.000 1.066 0.451 6.98 6.85 -0.13 113 0.998 0.944 0.469 7.59 7.53 -0.06 • As noted in the text, the normal to this crystal actually was (1,16,17). may be true, as is shown in the lower plot of Fig. 12 for a (0,1,11) direction. As a consequence of the effects shown in Fig. 12, it is necessary to retain a directional dependence in the average transferred energy as was done in writing Eq. (6). In principle, the isotropic contribution to the mean free path Ao ought to be related to the hard sphere radius so that Eq. (6) contains only two parameters. For fcc crystals, one should have Ao/ ao= l/47r(R/ ao)2. (8) Furthermore, the radius should be calculable from the interatomic potential between the incident ion and a target atom, e.g., from the distance of closest approach of the two particles in a head-on (b = 0) collision. If the Bohr exponentially screened Coulomb potential is used as the basis of such a hard sphere approximation, the radii calculated for Ar+-Cu interactions in the 1-to 5- keY energy region are in the range of 0.08 ao to 0.11 ao, although, on the basis of studies of the ranges of ener getic atoms in solids,25 values some 30% higher are probably more reasonable. Equation (6) was fitted to our experimental data on the sputtering yield of monocrystalline Cu under normally incident Ar+ ion bombardment. The pro cedure was to evaluate Phkl, Xhkl, and (b2/ R2)hkl for a particular hard sphere radius using the computer pro gram. The values of a and AO were then obtained from the data by the method of least squares. This procedure was repeated for each of several values of the radius. The quantity (9) where N the number of observations in the data set, was plotted against the radius and the "best" value of R was selected from the minimum in this curve. It was not felt worthwhile to locate the "best" value of the radius with better precision than about ±O.OOS ao. The calculated and observed yields at 5 keY are com- 25 D. K. Holmes and G. Leibfried, J. App!. Phys. 31, 1046 (1960); V. A. J. van Lint, R. A. Schmitt, and C. S. Suffrerlini, Phys. Rev. 121, 1457 (1961). pared in Table I, which also lists the values of Phkl, Xhkl, and (b2/R2)hkl used. The parameters derived at each of four bombarding ion energies are listed in Table II, along with a test of Eq. (8) and the quantity O'ext2• Finally, in Fig. 13, the theoretical sputtering yields for (Okl) Cu monocrystals are plotted against the angle between the surface normal and (001), using the S.O-keV parameters of Table II, and the resulting curve is compared with our experimental data. The results of the data fitting are very gratifying in that the calculated and observed yields are always in quite good agreement. The calculated yield for (011) crystals is consistently higher than the observed value and the calculated yield for (013) is rather too low, but otherwise the calculations are always within the experi mental uncertainties of the observed values. Perhaps the most remarkable feature of Fig. 13 is the fact that the two deep minima do not occur precisely at (001) and (011), but are displaced by about 2 degrees. (Note that the exact location of these minima is in some J r: ~ 5r---~~--~~--~~----~r---~ '" E o <; o -' w >= 4 ~ 3r-----T------r-----+----~--_b~ cr w >- !; 2 0.. en O~ __ --~-----L----~------L- __ ~ o 10 20 30 40 50 ANGLE FROM SURFACE NORMAL TO <001> (deg) FIG. 13. The sputtering yields of (Okl) Cu m()nocrystals under normally incident 5-keV Ar+ ion bombardment, plotted against the angle between the surface normal and (001). The line repre sents the theoretical yield, the points the experimental. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29162 SOUTHERN, WILLIS, AND ROBINSON TABLE II. Parameters derived from Cu-Ar+ sputtering data. Uext2 (atoms2/ion2) a/ao (atoms/ion keY) Ao/ao (ao/411'Ao) 112 R/ao RBobr/ao 2.0 0.094 2.25 ±O.OS 2.51 ±0.20 0.17S±0.OO7 0.21O±0.OO5 0.102 doubt; the computer program does not permit calcula tions within about 2.5 degrees of the principal direc tions, except, of course, exactly at their positions.) An error of orientation of as little as about 0.5 degrees in our (011) crystal could account for the entire dis crepancy between calculation and observation. Further more, the great scatter observed in data obtained from this crystal (d. Fig. 7) can be accounted for by small random variations in the angle of incidence of the beam because of the great sensitivity to direction of the yield in the vicinity of (011). The same sensitivity, which seems to occur near all three principal directions, presumably is responsible for the scatter observed in data from the other crystals also. As noted above, the discrepancy between calculated and observed yields led us to re-examine the Laue pattern from the "(011)" crystal, with the result that the surface normal was found actually to lie about 3 degrees from (011), its indices being expressible as (1,16,17). Similar re-exami nation of the (001), (013), and (111) crystal patterns showed no observable error in orientation (i.e., less than about 0.5 degrees). The parameters derived from the fitting of Eq. (6) to our experimental data are shown in Table II. The parameter 0:, expected to be independent of energy, decreases with increasing energy, and in fact is pro portional to E-1I2. Put another way, the factor o:E in the numerator of Eq. (6) should be replaced by a factor 0:' E112, suggesting that the "driving force" of sputtering is not the transferred energy, but the trans ferred momentum. The point should not, however, be much emphasized, since it likely results from over simplifications in the theory. The value obtained for R, on the other hand, is apparently independent of energy, instead of decreasing with increasing E as was anticipated. Furthermore, Eq. (8) is not satisfied, the value of (ao/%Ao)112 being 15 to 20% smaller than that of (R/ao). Finally, both radii (R/ao) and (ao/411'Ao)112, are substantially larger than expected from the hard sphere approximation to the Bohr potential, as is shown by the values of RBohr/ ao entered in the table. The origin of the behavior of 0: may, of course, be con nected with those of Rand Ao and represent merely the inadequacy of the theory. The large values ob tained for R are similar in magnitude to the value (R/ao)=0.19, deduced by Fluit2S from experiments 18 on the 20-keV Ar+ ion bombardment of monocrystalline Ar+ ion energy (keV) 3.0 4.0 0.132 1.85 ±0.07 2.S2 ±0.23 0.168±0.OO7 0.21O±0.OO5 0.091 0.147 1.56 ±0.05 3.06 ±0.25 0.161±0.OO7 0.21O±0.OO5 0.084 5.0 0.190 1.35 ±0.05 3.18 ±0.25 0.158±0.006 0.21O±0.OO5 0.080 Cu. We suggest that tbese large radii result, at least in part, from one or the other of two seriously com plicating features which must be taken into account before the transparency model of monocrystal sputter ing can be said to have been adequately tested. First, as Almen and Bruce13 have emphasized, the fate of the stopped incident beam atoms is important. If one assumes that sputtering is the only mechanism for removal of the trapped material (Le., if diffusion is ignored), it is easily seen that the average steady state concentration of beam material in the irradiated region of the target is (1 +S)-l atomic fraction. Thus, in our Cu monocrystal experiments, the sputtered material originates not in pure Cu, but in an "alloy" containing from 10 to 30 at. % Ar, depending on the orientation. There would seem to be little question that considerable distortion of the crystal could be conse quent upon this large impurity content, or, at the least, that the possibilities of interaction of the incident ions with their trapped predecessors (interstitially located?) would have to be included in the formulation of the transparency modeL Second, the question of radiation damage is impor tant. Following Kinchin and Pease,26 assuming a hard sphere scattering law and ignoring the mass difference between Ar and Cu (an Ar atom can transfer up to 94.8% of its energy to a Cu atom in a single collision), and taking 25 eV as the displacement threshold in Cu, it can be calculated that, on the average, a 5-keV Ar+ ion will initiate a cascade ultimately involving about 100 displaced Cu atoms as it slows down to rest. Using 60 A as the range27 of 5-ke V Ar+ in Cu and as suming that the displacements are produced uniformly along its track, then the total displacement rate from iJLA Ar+/cm2 is about iX1021 displaced Cu atoms/cm s sec. At 100 JLA/cm2, the mean life against displacement of a Cu atom in the disturbed region of the crystal is only about 1 sec. The very existence of the ejection patterns shows clearly that the annealing of this dam age must be extremely rapid. It may be noted that a current density of 100 JLA Ar+/cm2 produces a damage rate equivalent to a fission neutron flux density of'" l()2o neutrons/ cm2 sec, higher by more than five orders of 26 G. H. Kinchin and R. S. Pease, Repts. Progr. Phys. 18, 1 (1955). 27 Unpublished calculation. The method is discussed in refer ences 15 and 24. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29s PUT T E R I N G E X PER I MEN T S WIT H 1 - T 0 5 - k e V A r + ION S 163 magnitude than is available in any presently operating nuclear reactor. The effects of trapped gas atoms and of radiation damage can be included, at least to some extent, in the transparency model, and work along this line is pro gressing. Further tests of the model must await this development as well as experimental studies of the sputtering of monocrystals of other than fcc crystals, a topic which we are also investigating. ACKNOWLEDGMENTS The authors are greatly indebted to D. R. Burrow bridge, a cooperative program student from the Virginia Polytechnic Institute, for his assistance in performing many of the experiments. They express their appreciation to T. Giles for his continuous help in constructing the electronic apparatus, to R. M. Wallace for his skilled photographic work, to F. A. Sherrill for making the many Laue photographs, and to L. D. Hulett and F. W. Young for guidance in the treatment of Cu targets. They wish to thank J. W. Cleland, L. D. Hulett, E. Sonder, O. C. Yonts, and F. W. Young of this Laboratory, and Professor J. Kiste maker of the Laboratorium voor Massascheiding, Amsterdam, for providing many of the samples studied. They are also grateful to D. K. Holmes, C. Lehmann, G. Leibfried, and O. S. Oen of this Laboratory, and to D. E. Harrison of the U. S. Naval Postgraduate School, Monterey, California, for many stimulating and fruitful conversations. JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 1 JANUARY 1963 Photoconductivity in CdSe EDWARD H. STUPP* International Business Machines Corporation, Thomas 1. Watson Research Center, Yorktown Heights, New York (Received 27 June 1962) The photoconductivity in cadmium selenide single crystals is investigated to determine if the Klasens picture properly describes the electronic phenomena in this material. It is found that the simple two-state model is inadequate and at least three recombination centers, and possibly four, are needed for consistency. It is furthermore found that for the model to be acceptable, it must be assumed that activation occurs at impurity levels below that available in the purest material. Some of the limitations of using sintered photo conductors in a study of this type are also described. INTRODUCTION A MODEL of electronic processes introduced to describe photoconductivity in insulating materials must explain the occurrence of (a) luminescence, (b) superlinearity of photoconductivity and lumines cence, (c) activation, (d) infrared quenching, and (e) thermal quenching. It has been shown by Rosel and Klasens2 that a single recombination center could not account for these phenomena. Rosel was able to obtain a qualitative description of all of these by using distributions of two types of states in the forbidden gap. Klasens2 was able to describe semi-quantitatively the complete steady-state behavior of insulating photo conductors with two discrete recombination centers by using an approximation due to Duboc.3 In this paper4 we investigate the ability of the Klasens model to describe some of the above phenomena in cadmium selenide photoconductors, with specific empha- * Present address: Philips Laboratories, Irvington-on-Hudson, New York. 1 A. Rose, Phys. Rev. 97, 322 (1955). 2 H. A. Klasens, J. Phys. Chern. Solids 7, 175 (1958). 3 C. A. Duboc, Brit. J. Appl. Phys. Suppl. 4, S107 (1954). 4 An account of some aspects of the present work has previously been reported: E. H. Stupp, Bull. Am. Phys. Soc. 7, 173 (1962). sis on the temperature effects. A brief discussion of this model is given and the nature of the experimental results to be expected on the basis of it are described. The data are compared to the theory in the discussion section and a possible interpretation is offered. It is shown that the experimental results can be consistent with the model and other published data if assumptions are made about the relative lifetimes that are associated with the different activators and the number of centers present. It is also necessary to reconsider the nature of the acti vation process to admit the possibility that the addition of activator impurities to the photo conductor only serves to compensate for shallow impurities. Finally, some effects arising only in sintered photo conductors are presented to indicate the extent to which this type of material may be used for a study of this type. EXPERIMENTAL DETAILS The CdSe used was either obtained from the General Electric Company or was made by reacting the elements in a sealed quartz capsule. The reacted material was preferred since the foreign impurity level, as determined spectroscopically, could be kept below 1 ppm while the purchased CdSe had 10-20 ppm group I impurities [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 24.210.71.168 On: Mon, 21 Apr 2014 16:32:29

1.1702409.pdf

Dislocation Planes in Semiconductors H. F. Mataré Citation: Journal of Applied Physics 30, 581 (1959); doi: 10.1063/1.1702409 View online: http://dx.doi.org/10.1063/1.1702409 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Scattering of carriers by charged dislocations in semiconductors J. Appl. Phys. 113, 163705 (2013); 10.1063/1.4803121 Reconstruction defects on partial dislocations in semiconductors Appl. Phys. Lett. 79, 3630 (2001); 10.1063/1.1421623 Dislocation glide in {110} planes in semiconductors with diamond or zincblende structure Appl. Phys. Lett. 62, 2206 (1993); 10.1063/1.109443 Generation of misfit dislocations in semiconductors J. Appl. Phys. 62, 4413 (1987); 10.1063/1.339078 Dislocation energies and hardness of semiconductors Appl. Phys. Lett. 46, 54 (1985); 10.1063/1.95849 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14J 0 URN A L 0 F .\ P P LIE 0 P H V SIC S VOLUME 30. NUMBER 4 APRIL. 1959 Dislocation Planes in Semiconductors R F. MATARE Research Labamtaries, Sylvania Electric Products, Inc., Bayside, New York* (Received September 3, 1958) Dislocations and mainly dislocation planes, as the most important nonchemical imperfections in semi conductor crystals, are discusse:i from the point of view of their influence on carrier transport. After a short review of the general propertie3 of grain boundary planes of medium angle of misfit, the electrical effects are discussed, including the barrier behavior, the band structure, and lifetime anisotropy. Devices tructures based on these properties of dislocation planes are described among which the dislocation field effect tran sistor shows promising features as the first temperature independent transistor in the range 3000K to 2°K. 1. INTRODUCTION THE electrical properties of dislocations randomly distributed over large crystal volumes (plas tically deformed crystals) have been studied for some time. Following the concept of the supposed electrical behavior of a single edge dislocation, which is believed to be the major form of an electrically active imper fection, the influence on the transport of charge carriers can be found. Some conclusions drawn from the picture of arrays of dislocations (lineage) and their influence on crystal properties could first not be verified experimentally (see e.g., the paper by Read!). In a simple model, an edge dislocation site is assumed to be a free bond with acceptor character whose energy level is situated below the Fermi level in N-type material. Free electrons may combine with the free acceptor level and form double dangling bonds. This causes the buildup of a space charge region around the electrically active zone which repels other conduction electrons. If lineage is formed with an overlap of dangling bond levels a space charge cylinder will form with high conductivity along the pipe. In order to study the characteristics of such anom alous structure in crystals other than by microscopic (optical) methods it is necessary to make electrical measurements with direct contact to these regions. This can be achieved by growing dislocation planes or grain boundaries such that potential measurements, those of barrier-layer characteristics, capacity measure ments and conductivity measurements can easily be performed. The following survey covers the field of the electrical behavior of dislocation layers as it appears today, and extends into the field of electronic devices that make use of the remarkable properties of dislocation planes. 2. GROWTH AND PROPERTIES OF DISLOCATION PLANES Although some original work on grain boundary layers was performed on accidentally grown grain boundaries of uncontrolled angle of misfit,2 some meas urements of resistivity and Hall coefficient of gold doped bicrystals were done on medium-angle grain boundaries grown in bicrystals from the melt.3 Further work on the behavior of grain boundaries showed the desirability of defining clearly the range of misfit and symmetry of these planets.4 Figure 1 gives a schematic view of a bicrystal ar rangement. There are three degrees of freedom for the seeds (e, r, t). The angle of inclination e defines the number of dangling bonds. The angles of rotation and twist must be kept near zero to avoid screw-type dis locations. The dislocation plane has two degrees of freedom. The symmetrical case is that in which the angle of orientation cp= 90° (Fig. 1). Figure 2 shows the grain boundary plane cross sectioned and the energy amounts combined with the local stress and the isolated dangling bond. The distance D= a/ (2 sine) (a= lattice [100] G.B.plane lG.B.plane FIG. 1. Bicrystal-seed arrangement and grain boundary orien tation in semiperspective representation. ---- * Now TE KA DE-Semiconductor Laboratory, Numberg, 2 Taylor, Odell, and Fan, Phys. Rev. 88, 867-875 (1952). Germany. 3 A. G. Tweet, Phys. Rev. 99, 1182-1189 (1955). 1 W. T. Read, (a) Phil. Mag., 45, 6, 775-796 (1954); (b) 45, 4 H. F. Matare and H. A. R. Wegener Z. Physik 148 631-645 1119-1128 (1954); (c) 46, 111-131 (1955). (1957). " 581 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14582 H. F. MATARE FIG. 2. Schematic picture of cross section through a grain boundary plane in the simple cubic case with energv expressions (Shockley-Read). < constant) is dependent on the angle of misfit e. In this context it is important to define three ranges of misfit: (1) low-angle boundaries, H< 10 ; (2) medium-angle boundaries, 10< (-:) < 25°; (3) high-angle boundaries, (>-»25°. The first case leads to lineage problems where no clear overlap of the wave function of the dangling bond levels occurs laterally from pipe to pipe. The Bohr radius for Ge, for example, is (2.1) where K = dielectric constant, m* = effective mass, h= Planck's constant/27r, and e= electron charge. For K=16 and m=m* this gives rB~8.5 A which is larger than a lattice constant, (a~5 A). With K higher than 16 inside the distorted region and m*«m, due to band gap changes by a local pressure com ponent, one has to assume rather large radii. Thus for medium-angle grain boundaries, overlap can safely be assumed. Large-angle boundaries are excluded here in order to avoid diffusion problems and a region in which t he energy E can be no longer be expressed in terms of FIG. 3. Preferentially etched (IOO)-surface of a grown bicrystal. (1500X.) the Read-Shockley model. 5 Figure 3 shows a micro scopic view of a grain boundary on a (100) surface of a Ge crystal. The oil immersion microscope is focused on a structural etch pit. Conforming to the above, one of the most outstanding properties of these medium angle dislocation planes is their conductivity. It has been shown6•i that the conductivity of these planes is practically independent of the doping of the original material from which the bicrystals are grown. Gold doping is in no way essential to bring out their conductivity mechanism. Contact to these layers can be made in germanium by P-type alloy zones either jet deposited or evaporated. The leakage current to the bulk can be maintained at a level small compared to the cross current through the sheet. This method is very efficient, especially at low temperature, since the leakage current is given by the expression,8 (Dnnp DpPn) i,,=e---+-- [l-exp(-eV/kT)], Ln Lp (2.2) with Dn; Dp=diffusion constants for electrons and holes; np; pn = minority carrier densities; Ln; Lp= dif fusion lengths for electrons and holes, respectively; V=voltage applied; and e/kT=usual exponent in Boltzman factor. Formula (2) shows that eet. par. the decrease of temperature brings the leakage current rapidly to its ultimate saturation value « 10-8 A). At liquid nitrogen temperatures this is already below the current through the dislocation sheet. At 4.2°K the leakage junction current may be neglected completely, since the sheet resistivity stays practically constant throughout the entire temperature scale with a linear I-V characteristic of slope 10 p,A/v for about 1-mm sheet width. Figure 4 shows the temperature dependence of the resistivity (in ohm cm) of a Ge monocrystal as com pared to the grain boundary sheet. The thickness of the sheet is assumed to be 100 A. This is understood to be the innermost part of the disturbed layer with its dangling bonds. Potential measurements, capacity measurements and etching with successive microscopic studies revealed a maximum thickness of 10-4 cm (104 A) which is the actual space charge width. The inversion layer region formed by holes from the valence band at either boundary side is imbedded into the space charge region ('" 10-6 cm= 102 A) and may be considered the actual high conductivity zone. As Fig. 4 • W. T. Read and W. Shockley, Phys. Rev. 78, 275 (1950). • Weinreich, Matare, and Reed, "Electrical and photoelectrical properties of grain boundary layers," paper presented at the International Conference on Solid State, Brussels, June 1958. Mata[(~, Reed, and Weinreich, Bull. Am. Phys. Soc. Ser. II 3, 14 (1958). 7 H. F. Matare, "Anisotropy of carrier transport in semicon ductor bicrystals," paper presented at the International Con ference on Solid State, Brussels, June (1958). 8 E. Spenke, Electronische Halbleiter (Springer-Verlag, Berlin, 1954), p. 99. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14DISLOCATION PLANES 1"1 SEMICONDUCTORS 583 shows, the grain boundary conductivity departs clearly from the conductivity of the monocrystal below lOOK. Its conductivity, however, can be shown to be constant also above lOoK. If, as mentioned before, rectifying contacts are applied (Fig. 5), the grain boundary layer shows clearly an almost constant re sistivity while a Ge monocrystal shows the well-known minimum at lower temperatures and a strong increase in resistivity at very low temperatures. The resistivity minimum being the onset of predominant impurity scattering at low temperature. The medium-angle dislocation plane therefore may be considered as having a form of degenerate behavior with a quite different conductivity mechanism from an extrinsic or intrinsic semiconductor. The overlap of the wave functions and the resulting change in statistics of the filling of available free bonds7 causes a very low activation energy over the entire temperature scale from 3000K to 2°K. This behavior, as well as the building up of a barrier layer at either side of the plane, makes the bicrystal interface interesting from a device standpoint. The minor importance of impurity segregation at the boundary can be demonstrated by the fact that per pendicularly to the individual edge pipes the dilation and compression regions D and L overlap to a certain extent, diminishing the Cottrell-atmospheres. For angles of misfit 8> 25° this is apparently no longer the case. Measurements of penetration in diffusion experi ments show that grain boundary diffusion depends on the angle of misfit and is important for 8> 25°. Work in this laboratory has shown also that the properties of grain boundary layers are in no way correlated to impurity density and type of the crystal in general. 6 e2Xl02 u ~ :r £ <I.. 2xIO' 2.0 12 G.B.15 T¢PERATURE DEPENDENCE -X-BULK ri--, -<>-GRAIN (l00ftil00) BOUNDARY FIG. 4. Temperature dependence of the resistivity for a Ge mono crystal (1 ohm em) and a bicrystal interface. 10 , 9 8 7 6 5 4 <>...3 E U2 ~ ~ Ge(5n~' ;: I ' t:~ / ~ l , w 4 *i 5 \ tr 3 •........•.•... /' P(FOR 100 II LAYER) ;/..,_",,-_ (IOO~~O~) --..L..... __ ._ T .... lo'L----"-------,--'-:-"..,..,---100 200· K FIG. 5. Temperature dependence of resistivity over wide tempera ture range for Ge monocrystal (5 ohm em) and bicrystal layer. Ge bicrystals with a resistivity of 36 ohm cm down to 1 ohm cm, doped with gallium, antimony or copper, show a constant sheet resistivity of p=0.007±5.1O-3 ohm cm (for 100 A) without correlation to the bulk impurity type or amount. Figure 6 gives a schematic view of the probable energy band situation along the grain boundary plane perpendicularly to the pipes. For germanium with a negative pressure band gap coefficient the forbidden gap widens at the compression zones. Since partial overlap sets up a stabilizing grain boundary energy, the total plane is a zone of a higher gap than the mono crystal. Thus the original gap E is not widened to the full amount El = E+ 2t::.E bu t to an amoun t Eg """ E+ t::.E =E+2t::.El. For silicon new gap pressure coefficients are available (the figure of -30 given by Shockley9 is not correct) FIG. 6. Probable band scheme at grain boundary interface. Dilation (D) and compression regions (e) correlated to band gap changes. 9 W. Shockley, Electrons and Holes in Semiconductors (D. Van Nostrand Company, Inc., Princeton, 1950), p. 334. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14584 H. F. MATARE 2.88 2. SINGLE POINT PROBING 3.5-G.B. -10 2.38 INTEGRA TEO i RESISTIVITY 2.18 IN(.n.Cm) 1.98 ...... ___ ._._ ~m) .31~--~~~~~~~ __ --~ __ - 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 FIG. 7. Plot of integrated resistivity taken over bicrystal surface using a one-point microprobe equipment. which show a slightly positive value.lO One can calculate the energy of an edge dislocation El in dynes, and express from this the total energy in dyne/cm over the width of the layer (100 A) which is acting on the adjacent lattice points at the monocrystal grain bound ary plane interface. This gives a specific strain per lattice site!::.E= E/ / a (a= lattice constant) on the adjacent lattice, or using the stress-strain relation with Poisson's constant 11=0.4, one can deduce from this the unit compression ratio at the adjacent lattice, !::.P E//a[dynes/cm2] -=0.4 , P ,u[dynes/cm2] with J.L the elastic constant for Ge. With the gap pressure constant EgI= -5 (ev) this gives a band gap change of !::.E"-'0.15 ev for Germanium. We have to take into account uncertainty in the determination of El since the grain boundary width is not exactly known. Also, the stress-strain relation might no longer be governed by Hooke's law. The gap change described would explain certain features of the grain boundary layers such as high injec tion efficiency and the lifetime anisotropy. These phenomena will be described in what follows. 3. GRAIN BOUNDARY BARRIERS The early work on grain boundary layers began with a search for an explanation of the high blocking action at the interfacey,I2 Carrier injection into bicrystal 10 L. J. Neuringer, Bull. Am. Phys. Soc. Ser. 2 2, 134 (1957). See also W. Paul, "The effect of pressure on the properties of Ge and Si," International Conference on Semiconductors, Rochester, New York (1958). 11 G. Pearson, Phys. Rev. 76, 459 (1949). 12 Taylor, Odell, and Fan, Phys. Rev. 88, 867-875 (1952). interfaces and modulation effects were studiedI3 and measurements were extended to bicrystals produced by defined growth with double seeds.14 This technique gave a means for studying the de pendence of the electrical behavior of the misfit and to grow very precisely4 larger dislocation planes to which contacts (formed points or alloyed) could be applied.I5 The photoelectrical behavior of these interfaces was also studied and showed interesting properties.I6 Here we can only briefly describe the main features of these blocking layers . . If the bicrystal is polarized across the N-P-N interface, one of the junctions works in the forward direction while the other one is biased in reverse and shows a high blocking voltage. A single-probe mapping shows the integrated resistivity as in Fig. 7. The voltage is taken between the probe and one end of the bicrystal and lR p*=qj-dX o x (3.1) (q=cross section, l=length, R-resistance) is plotted against x. The steep region of the potential drop occurs in a distance of less than 1/10 mm. Actually the inversion region is much smaller and could be measured more precisely optically. As mentioned already, the upper limit for the space charge region is 10-4 cm or 104 A which is a width measured photoelectrically with a fine beam. One must assume that the actual grain boundary width lies within a factor of between 10 to 100 below this value. The I-V characteristic Fig. 8, of a grain boundary layer, measured in both directions, is rarely symmetric because of slight differences at the surfaces. In dc operation the blocking voltages may reach values as high as 100 v and more. In pulsed operation these voltages are much higher (up to 1500 v with leakage currents below 1 rnA have been measured). \ 1(1'0) C I "o'\"'''~T ) J(j.L0) I I --01 A I-- I I I I I I I III 1 .... / I II_--r 0 I ---1-)1.---- I I I •• ~ I _+--1 I -50 (' : I I I -100 I _VOLTS FIG. 8. 1-V, characteristic of Ge bicrystal. 13 H. F. Matare, Z. Naturforsch. 9a, 7-8 (1954). "H. F. Matare, Phys. Rev. 98,1179 (1955). ,. H. F. Matare, Z. Naturforsch. lOa, 640 (1955). 16 H. F. Matare, Z. Physik 145, 206-234 (1956). 150 100 50 0 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14DISLOCATION PLANES IN SEMICONDUCTORS 585 The rather sharp increase in current near the zero point (Form A, Fig. 8) is a peculiarity of the grain boundary. It shows that empty states are being filled as the voltage increases and that this buildup is com plete for voltages of the order of 1 v. This can be deduced from the simplified model of the grain boundary as shown later in this paper. Assume a grain boundary barrier of a height e V D to which an external voltage V. is applied such that VI=VD+V e (Fig. 9). Assume furthermore that the carrier densities ni and n2 on either boundary side are equal. The fields at both boundary sides are then EI = -grad VI; E2= -gradV2; K V<[n(1)~n(2)~nJ =--(E12- E22). 87ren (3.2) (3.3) Equation (3.3) follows from the potential/field ex pression for equidistributed charges in a medium with dielectric constant K.2,15 Now we need a relation between the charge density q in boundary states and the fields at both sides. This is the line integral along the grain-boundary plane (q is actual charge density in grain boundary states). In equilibrium we have V10= -V20=ct>/e with ct>=bar rier height and ct>1=ct>2=ct>. This gives, from (3.4), ElO= -E20= -27rqo/K, qo= equilibrium charge in boundary states. If we now introduce E12 from (3.4), E12= E2L 87rqE2/K + (47rq/K)2, into Eq. (3.3), we get E2=27rq/K- (en/q)V e, E1= -27rq/K- (en/q)V e• (3.5) (3.6a) (3.6b) These two equations give a relation between the field strength at both bicrystal sides and the polarizing voltage in terms of the surface charge density in dangling bonds. Introducing now the field expression in terms of the equilibrium number of charges in boundary states, (3.5) into (3.6) gives -27rqo/K=27rq/K- Ve(en/q) , II 1;;(, , 1 I , I I XTp N FIG. 9. Band scheme of grain boundary. Ef)=fixed donors, P=dangling bond layer (,,-,100A wide), 8=fixed acceptors, + -= free holes, electrons, N d -N a = space charge region = .1.XT = 10-4 cm. (See text.) with the critical voltage relation 27r ct> Vc=-q02=4- Ken e for the case E1=E2=O from (3.6) using (3.3) and (3.4). This is an important ratio of the actual number of surface (boundary) states as caused by an applied voltage Ve and the equilibrium number qo without an external voltage applied. Actually q/ qo is equal to the ratio of the filling factor of boundary states, f, and the factor fo, without voltage applied (Vc=O). fo is the statistical value for thermal equilibrium, 1 ~ ,(3.8) 1 +exp[(Eo-Ep)/kT] where Eo=dislocation level and Ep=Fermi level, if Fermi-Dirac statistics applies. This is the case for the condition foEo<kT (3.9) with Eo= foe2/k·c, where e=electron charge, K=di electric constant, k= Boltzman constant, and c= spacing between dangling bonds. We can express fo in terms of the dislocation spacing c and the spacing between added electrons, ao, at equilibrium, fo=c/ao. (3.10) Thus if q/ qo changes due to a voltage Ve applied, f/ fo changes, or f is a new distribution fUnction, depending on Ve q(Ve) f(Ve) f(Ve) -~=--=--·ao. qo fo C (3.11) or q/qo= ![1+ (1 +eVe/ct»i] The implication of this for the distribution law has (3.7) been considered. 7 Also~the change in number of electrons [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14586 H. F. MATARE in boundary states in the case of voltage applied has been calculated.l6 Starting from a Shockley-type barrier layer equation one may express the electron density in terms of the equilibrium density nco and the voltage VIand V D(V 2) at the barrier sides in equilibrium. Introducing relation (3.7) and differentiating leads to16 : iJn. 2neo 1 -=-.-~---- iJVe qo 1+[1+ (eVeI</»]i x{ V2!C~2-k~);;: exp( _:~2) -Vl{2~1- k~);;>xp( _:~l) 1 e/</> + ----- l+[1+(eV.!<p)]! 2[1+(eV e!<p)]t x[ Vliexp( _ e;)_ V2texP( _ e;) ]}, (3.12) where V2= V D=voltage on one side of the barrier, V1=voltage on other side of the barrier, V.=external voltage V.= V1-V2, and </>=barrier height in ev. Analysing this change for normal conditions16 leads to the conclusion that ne increases rather rapidly for small voltages Ve applied. One may express the voltage VIand V 2 (V D in Fig. 9) directly by inserting (3.6) into (3.3) and finds under consideration of (3.7) an expression for the voltages of the form (see references 2, 15, 16) 7rqo2 enK 1 VI(2)=--a±tV.+-_V.L. 8enK 27rqo2 a (3.13) (The negative sign is valid for V 2) with the abbreviation { [ (eVe)!] eVe} a= 2 1+ 1+-; +-; (all other notations as in the foregoing). FIG. 10. Efficient injector electrode using gap increase oE in grain boundary region. Function (3.13) can be developed into a series for a= (eVe/</» :::;1; a= (x+l)2 and x2= l+a. With the abbreviation C=7rqo2/enK one gets16 In this range the quadratic member is predominant. For a> 1 or a»l we get [ C 1 (</»2] V. V1(2)" ...... a -+--±-. 8 2C e 2 (3.15) Since </>"'" 1 ev, V. is less than 1 v for case (3.14) or a quadratic increase of the barrier height with V. is bnly valid for small polarization. From there on the linear increase as in (3.15) is predominant. The current voltage relation for a grain boundary barrier may be approximated by assuming a uniform charge density in the boundary plane. For </>>>kT this leads to the Schottky-type exhaustion layer with variable thick nessP It is, however, still important to know, as pointed out here, how strong a dependence on the number of states arises from the external field applied to the barrier. This may be the explanation for the extremely high blocking voltages of over 1500 v measured on some Ge boundaries in pulsed operation. 4. BAND STRUCTURE AND LIFETIME ANISOTROPY As discussed in Sec. 2, the grain boundary in Ge is to be considered as a region with large band gap. It is known that a contact region between two semicon ductors of different band gaps exerts forces on the charge carriers (Fig. 10). Thus electrons have a drift component to the left, holes to the right. An external voltage Ve applied in the forward direction at such a P-N interface therefore causes enhanced hole injection into the N-type material. (The widening oE of the gap causes an effect similar to a forward voltage applied.) Therefore such a zone could be used as an efficient injector (Fig. 10). Other consequences of this inhomogeneity are strong photoelectric effects and anisotropy with respect to minority carrier lifetime. Figure 11 shows a set of measurements of the voltage at a collector probe with respect to the base plating versus the distance of a light spot (usual ..1x-lifetime method7). It turns out that these curves show a st-rong change in slope or even a minimum and maximum when the light spot crosses the grain boundary.ls With the 17 R. Stratton, Pmc. Phys. Soc. (London) B64, 513 (1956). 18 H. F. Matare and B. Reed, Z. Naturforsch. 10, 876 (1956). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14DISLOCATION PLANES IN SEMICONDUCTORS 587 collector very near to the boundary the effect is less strong, also with increased collector bias the strong anisotropy decreases leaving only a change in slope behind (Fig. 12). Ordinarily, for monocrystals the results are expressed in terms of a single lifetime parameter 1', which is assumed to be a homogeneous property of the crystal. The results attained for bicrystals do not lend them selves to such easy interpretation. The usual diffusion equation for minority carriers has to be modified to take into account the sink behavior of the boundary field for X~Xl with Xl = boundary location. The influ ence of the boundary field depends on the diffusion length. After the light line or light spot has entered the 6 10 GB 25 ~ .,. -0 55 mils from G.B. b,-b 80 .. O-c 110 .. FIG. 11. Collector voltage vs light spot position (Ax-lifetime method) on bicrystal for different positions of the collector probe with respect to the boundary. boundary region there is an increased electron-hole pair production due to the hook effect and the internal stress field (band gap change). The diffusion equation with a sink influence at Xl may be written div gradP=~(l--=-), (4.1) Dr Xl with p=added hole density, Dr= (diffusion length)2 =Lp2, 1'= lifetime, D=diffusion constant, and xl=sink abscissa. In cylindrical coordinates and with the light line at r= 0 this is equivalent to 11GB 20 Bias effect 4 'Vbios' 0.00 VOLTS .. -0.10 X· -0.20 T' -0.40 Q. -1.42 X(miIS), .I-'---.,-.--r-;h-....---.---,-·--i,--... 20 40" 60 BO 100 120 140 FIG. 12. Same as 11 only for fixed position of collector probe but different collector base potentials . Since, by definition, r=Lp'x, this becomes IFp +~ dp -P(l--=-) =0, (4.3) dx?-x dx Xl or Solution to this must be found for three different ranges, 1. XI»X. This is the normal case with the solution alnV alnp (4.5) --=--=-------, ar ar Lp iHo(I)(ix) H 0 (1) and HI (I) = Hankel functions of first degree and zero and first order. For the light point source we find M= a InV = a lnp = _ [~+~], (4.6) ar ar Lp r where M = experimental slope on natural log paper. For the case Xl = X a logarithmic solution follows from (4.4), d Inpjdr= Inr. For the normally small value of r this means a small or at least decreased slope, as is found. However, the full detail of the complicated a InV jar curve cannot be brought out by a sink alone. As mentioned before, an increase in injection efficiency or a higher number of holes reaching the collector is the main reason for the increase in collector voltage some times as high as the maximum voltage when the light falls directly on the collector spot. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14588 H. F. MATARE +20 + 15 +10 +5 jPHOTO VOLTAGE (mV) I / / " ~' rDISPLACEMENT OF Ge CRYSTAL mils. /+ O'+---~~~--~~~--~--,~~'--ao ";-'_'- 20 40 60 ///+ -5 / -10 ~! -15-1---------=------ __ FIG. 13. Photoelectric scanning of bicrystal. Photo voltage vs position of light beam. 5. GRAIN BOUNDARY DEVICES (a) Photoelectric Devices At the end of our survey of the properties of bicrystal interfaces we discuss a few device applications starting with the grain boundary photocell. The most important property of these layers is their high photoelectric sensitivity, combined with a strongly localizing action due to the fineness of the P-type layer. As described at the beginning, the innermost highly conductive region may have a width of only 100 A while the resulting space charge region is as wide as 10-4 cm or 104 A. When a light beam falls on the sample, say to the left of the grain boundary (Fig. 13), the barrier layer I I I ELECTR0"4 I I I FIG. 14. Grain boundary transistor and band scheme. as a hole sink separates the charges and a current through the sample results. This current is highly increased for voltages applied to the bicrystal because in this case the injected holes lower the barrier and an increased photocurrent through the sample results. When the light spot is located to the right, the charge separation works in the opposite direction and an opposite current results. If the light spot is placed so that the grain boundary lies exactly in the middle of the illuminated area, no photovoltage develops. The finer the P-type region the steeper the zero passage of the photovoltage across the sample.19 This device has the unique property of following in its response to any optics since the grain boundary layer width cannot be matched by any focusing system for ordinary light. Zero passage from + max to -max of the photo voltage of less than 10-4 cm have been measured. An example of such a sensitivity curve is shown in Fig. 13. Photocells with these properties can be used for homing purposes where utmost precision of localization is required. (b) Grain Boundary Transistors The P-type grain boundary layer could be considered as an ideally thin base layer if it were not for the high recombination probability in this zone. There is, how ever, the possibility of using such a device in an emitter-to-ground connection such that the signal is fed in between the grain boundary layer and one side (ground) while the load is in series with the cross polarization of the sample. The probable band scheme , 2 POWER DIFFERENTIAL AW'/AWz' AI,' V,/Ah.Vz· f (V"A l2 ) --.1, (mA) .3 I , , , I .:h .4 _ V,IVOLTS) 4 6 14 FIG. 15. Power differential of grain boundary transistor in emitter to ground connection as function of cross current (voltage) with sheet current as parameter. 19 Weinreich, Matare, and Reed, Enlarged Abstract 58, Elec trochemical Society Meeting, Washington, D. C., May (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14DISLOCATION PLANES IN SEMICONDUCTORS 589 of such an arrangement is shown in Fig. 14. The extra levels in the forbidden gap are indicated in the grain boundary zone. Here the bands curve up. An external polarization, as indicated, creates a drift field for holes and electrons. The injection of holes into the grain boundary layer modulates the barrier height-the quasi Fermi level-and in this way power amplification is achieved. In Fig. 15 the power differential for different currents II (voltages VI) across the bicrystal bar is plotted with tlI2, the grain boundary modulation current, as parameter. Saturation values (""30 db) are reached for modu lation (sheet) currents of 20-30 j.l.A. Due to the high capacity of the sheet with respect to the bulk the frequency response is rather limited. Figure 16 describes schematically another possi bility, namely, to build an N-P-N-P structure using a bicrystal. For bistable devices this scheme might have merit because of the high sensitivity of the grain boundary layer for injected minority carriers and its fast recovery. ( c ) Grain Boundary Field Effect Transistors A class of devices, in which the logical use of grain boundaries can be made, are the field effect transistors. The temperature-independent sheet current is sensitive to field changes at both sides. Since the sheet has already a natural width of less than an inversion layer (10-4 cm) this device is relatively easy to build and represents a unique transistor amplifying throughout a temperature range from below liquid helium tem perature up to room temperature. The sensitivity of (Tl G.B. ! I I -+l1~ I I I ELECTRO~ I I I I I ENERGY I I I £~eV I /////'i / / EF ..LI_r+--+- eVa E ' F FIG. 16. N--P-N-P unit using a grain boundary layer 11 and band scheme. 1200 I, V, 4 -29 -58 1100 ~BULK F R 1000 SHE! 900 I2 V2 800 7 o~~~~ 2 4 6 8 10 12 14 16 18 V2 VOLTS FIG. 17. Characteristic chart for grain boundary field effect transistor. Sheet conductance at 4.2°K; 150 em N-type Ge sample #100. the sheet current to field changes is expressed by Eq. (3.12) which gives the change in electron density at boundary states with the voltage Vc applied. We have seen that saturation for this process is reached at rather low voltages. Therefore one expects the current modulation to level off for rather low field voltages. Figure 17 shows this to be the case. The cross current is plotted vs the cross voltage with the field voltage as parameter. [Corresponding to Ve in (3.12).J Maximum changes occur for higher cross voltages but for low values of the field potential VI. While the contacts to the sheet have to be made by careful alloying to the sheet in order to avoid leakage in the blocking direction of these junctions with respect to the bulk, one may use ohmic or nonohmic contacts for the field electrode since currents here are minimized « 10-8 A) due to blocking layers at both sides and the contact geometry. Transconductance values of several 100 J.i.mho are easily reached with devices a few mm long and 1 mm wide. An increase by a factor of 10 in transconductance can easily be achieved by proper geometry. The frequency response of these devices is similar to the one of other known field effect structures. ACKNOWLEDGMENTS The author wishes to thank his colleagues at the Sylvania Research Laboratories for helpful comments. Especially E. M. Conwell, A. Many, H. Vasileff, O. Weinreich, and B. Reed who contributed directly or in discussions. Thanks are due to P. H. Keck and G. D. O'Neill for comments and help with the editing. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.83.63.20 On: Wed, 26 Nov 2014 18:41:14

1.3058129.pdf

Horizons in physics: Cryogenics John G. Daunt Citation: Physics Today 15, 4, 26 (1962); doi: 10.1063/1.3058129 View online: http://dx.doi.org/10.1063/1.3058129 View Table of Contents: http://physicstoday.scitation.org/toc/pto/15/4 Published by the American Institute of PhysicsHORIZONS IN PHYSICS: CRYOGENICS By John G. Daunt The following paper was presented as part of a symposium entitled "New Knowledge in Physics: A National Resource", which was held on September 28, 1961, during the fourth annual meeting of the Corporate Associates of the American Institute of Physics at the Ardcn House, Harriman, N. Y. THE present activity in low-temperature physics research, together with the current rapid appli- cation of the results of this research to com- mercial and military uses, is on a far grander and, we may hope, on an equally significant scale as that occurring in that great period at the turn of the cen- tury which saw the birth of cryogenics. In the space of little over a decade at that time, the first major achievements in the production of low temperatures were realized, namely the liquefaction of air, hydrogen, and helium; new laboratories dedicated for the first time to low-temperature research were established and through their work the frontiers of our physical knowl- edge were handsomely enlarged, especially concerning the early foundations of quantum theory. As if this were not enough, this early period saw the beginning of cryogenic engineering on a substantial scale in the almost immediate application of air liquefaction to the separation and production of liquid oxygen and liquid nitrogen. This application of cryogenics, as we are now well aware, was destined to grow into a major industry, separating and supplying not only oxygen and nitrogen but also other gases and products. It is not my purpose to dwell on the many significant contribu- tions which this industry makes to our complex society. I only wish to note how immediate were the uses it made, and still makes, of frontier knowledge in re- search. The present period is outstanding also in its broad coverage of every facet of cryogenic endeavor. Several hundred research laboratories have facilities for work to temperatures down to 1°K, thanks to the com- mercial production of a reliable helium liquefier. Many J. G. Daunt was born in Ireland and earned his PhD in physics at Oxford. He is professor of physics at Ohio State University.have facilities for reaching to temperatures more than a hundred times lower. The output of research work, both theoretical and experimental, is considerable. It is some of the highlights of this work which I wish to dis- cuss today, in the hope that this may give some insight into present, and more importantly future, objectives and the means whereby these objectives may be reached. BEFORE coming to these matters, I wish to men- tion, in passing, items which concern the recent application of this research to engineering uses at much lower temperatures than heretofore and the stimulat- ing feed-back these applications have on the tech- nology of research. First, concerning the production and handling of low temperatures, liquid hydrogen and liquid helium, so long available only in small quan- tities in advanced laboratories, are now commercially produced and distributed, the production of liquid hydrogen for rocket fuel being on a tonnage basis. Liquid hydrogen is being handled on a scale similar to that already long familiar with liquid oxygen. Trailer trucks for 5000 gallons are not uncommon. However, in making the step from liquid oxygen to liquid hydro- gen, storage and insulation difficulties increase greatly, corresponding to the almost order-of-magnitude differ- ence in their latent heat of evaporation per unit volume of liquid. This step, and the more difficult step to liquid-helium handling, has provoked many new de- velopments and reappraisals in cryogenic-engineering techniques. A most notable development here has been in multilayer insulations (the so-called "superinsula- tions"), which are replacing the conventional "Dewar systems in many applications. The new engineering at liquid-hydrogen and liquid- helium temperatures has highlighted our previous 26 PHYSICS TODAY27 scarcity of information on the general properties of materials, and especially on the mechanical properties of metallic alloys and of nonmetallic materials (insu- lators, seals and gaskets, adhesives, embediment and encapsulating components). The past few years have witnessed the establishment of many laboratories dedi- cated to gathering the required information and to component and material testing. Such laboratories and test installations are engaged now in large-scale opera- tions, especially stimulated by the needs of the missile industry. A whole new race of cryogenic engineers is being evolved, and this in turn is generating its own problems in education. A relatively recent development has been the re- introduction of closed-cycle refrigerating systems, using a reversed Stirling cycle and a variant of this which I call the Taconis cycle. Such systems, used singly or in cascade, can provide refrigeration reliably and con- tinuously at temperature levels down to helium tem- peratures and can be made in sizes from miniature to very large to suit the individual heat load. They have great advantages over the conventional cryogenic liquid baths and will be of increasing future impor- tance for providing no-loss liquid storage systems and for maintaining the desired temperatures for cryogenic devices, for example, infrared detectors, masers, super- conducting computers, gyroscopes, dc transformers, high-field magnets, and a host of others which have become or are just about to become current engineer- ing realities. This refrigeration development is so cru- cial for so many future applications that it deserves much more emphasis than it has already been accorded. A recent industrial operation at liquid-hydrogen temperatures has been the construction, in several European countries, of distillation plants for the iso- topic separation of deuterium from hydrogen. These plants produce several tons of heavy water per year for moderator material for nuclear reactors. Low- temperature techniques for isotopic separation are, of course, not new on a small scale; for example, not only the hydrogen isotopes but also the neon and the helium isotopes have been separated cryogenically. The helium separation techniques were interesting in that the preliminary enrichment of the He3 in He4 from about one part per million to about one percent was done not by distillation but by superfluid osmosis. Operations at liquid-hydrogen temperatures, and below, now concern large-scale "space" simulators— large tanks providing an environment similar to that encountered by satellites or space vehicles. Cold walls simulate the thermal radiative environment. Surfaces cooled to 20°K or below act as vacuum pumps by condensing the residual gas on them and maintain the desired high vacua. This simple application of low temperatures now goes by the name of "cryopumping". It would be difficult and tedious to enumerate all the present applications of the new cryogenics—the extension of work to liquid-hydrogen temperatures and below—and any enumeration would rapidly becomeincomplete. However, my brief list will serve, I hope, to show that the business of making and maintaining very low temperatures is no longer a laboratory gim- mick and consequently it is timely for the serious promotion of cryogenic devices and processes fathered by low-temperature research. Interwoven with the business of making and main- taining very low temperatures is an active and grow- ing industry in instrumentation. Many companies now market an increasing diversity of specialized instru- ments and equipment for low-temperature work (ther- mometers of many varieties, depth gauges, level con- trols, pressure transducers and controls, dewars, tanks, valves, etc.) benefitting both the research laboratories as well as the industry they serve. FUNDAMENTAL research at very low tempera- tures falls into two general categories: the first, in which the low temperatures are employed only as an extension of observation over a wider temperature range, and the second, in which the phenomena of interest are characteristic of the low-temperature en- vironment itself. The first category cannot be dis- cussed here, nor indeed would such discussion be justi- fied. It represents, however, a large field and has encouraged many research institutes to regard low temperatures as just another research tool and to aban- don any specialized cryogenic groups. There are, how- ever, many laboratories which still believe in low- temperature physics as a viable area to itself and, in my opinion, they have amply justified their belief. The most significant phenomena associated uniquely with very-low-temperature physics include the proper- ties of the heliums, He4 and Hea, especially in their liquid phases, superconductivity, and certain aspects of magnetism and metal physics. Liquid He4 has long been known to transform to a superrluid phase below about 2.2°K, and the properties of this peculiar phase have been of great interest, theoretically and experimentally, for many years. In the past several years more interest has been paid to the rarer isotope, He3, which in pure form has recently been available as a man-made product. Both He3 and He4 remain liquid to the absolute zero of tempera- ture, but it was anticipated at first that liquid He3, being a fermion system, would not show the compli- cation of a superrluid phase. Experimental results to date confirm this anticipation. However, a general theo- retical picture of superfluidity both in boson (He4) and fermion systems has evolved, based on the suc- cessful theoretical treatment of superconductivity in metals by Bardeen, Cooper, and Schrieffer, which now postulates the occurrence of superfluidity in liquid He3 at temperatures of a few tenths of a millidegree Kelvin. This prediction provides one of the more inter- esting problems in low-temperature physics facing the experimentalist today, requiring him to push his meas- urements on liquid He3 to yet lower temperatures. Al- ready, measurements have been made to 0.008°K, but April 196228 this is not far enough. The problem is not trivial, since it pertains to current concepts of the many-body prob- lem, concepts which are applicable not only to the macroscopic superfluid systems occurring at low tem- peratures, but also to nuclear matter. In passing I must note that temperatures of a few microdegrees Kelvin have been obtained, first by Kurd and co-workers, by adiabatic demagnetization of copper nuclei starting from about 0.01 °K. However, there is still a big gulf between cooling a nuclear-spin system itself to these unbelievably low temperatures and using the technique to cool a calorimeter filled, for example, with He3. Technical problems of thermal con- tact, thermal isolation, and relaxation times remain, and they art- formidable. Solid He1 and He:i are valuable "model" solids. No other solid is so compressible. By pressurization, both can be readily changed in volume by over 30 percent. Measurement of the properties of these solids at widely different possible densities, therefore, allows the theory of insulating solids to be thoroughly ex- plored. He3, moreover, is magnetic, with nucleus of spin J. A big field lies open here for observation of possible ferromagnetic to antiferromagnetic transitions under changing density and its consequential signifi- cance to our understanding of magnetism. Nuclear ferromagnetism or antiferromagnetism in solid He3, however, is another very-low-temperature problem (T < 0.01 °K), so that again these problems emphasize the need for exploration well below the now commer- cially familiar temperature range of liquid He1 (1°K to 4.2°K). Magnetic-cooling techniques of the most advanced kind are required. Unlike what I shall discuss about superconductivity, there appears no obvious or immediate application stemming from research on He3 and He4. It appears to be a fine example of research for knowledge only. I would be happy, however, to be proved wrong in this. Superconductivity in metals, one of the most startling macroscopic phenomena occurring at low temperatures, has recently received and continues to receive much attention. For a long time it was con- sidered to be almost worked out experimentally, until in 1957 the successful theory of Bardeen, Cooper, and Schrieffer stimulated further investigations. Today, the study of this effect is intense, both of the fundamental properties and of its applications in devices. Work continues on pure superconducting metals, especially in thin films, and one of the more interesting develop- ments relates to the quantization of magnetic flux trapped in superconducting rings. The properties of superconducting alloys also are being re-examined and the question of the applicability of existing theory to them, and also to isotopes of transition elements, is a matter of recent concern, and of increasing importance. Development of studies of superconducting thin films has led to a new technique of measurement of impor- tant parameters, such as the energy gap, which sup- plement those based on thermal, infrared absorptionand other techniques. The next several years will COT tainly see a profound consolidation of our knowledge of the phenomenon. In the low-temperature field, the phenomenon on superconductivity is generating more devices and use-1 ful instruments than any other. The superconducting switch, or "cryotron" as it is now called, was first used more than two decades ago in conjunction with superconducting galvanometers. Recent interest in it stems from its development into a very-high-speed, bistable element suitable for computer applications in memories and in logical circuits. One recent type con- i sists of a thin film of superconducting tin as the "gate". crossed by and insulated from a thin, evaporated j ribbon of lead, acting as "control", current through the control serving to make the adjacent part of the gate go "normal" and hence exhibit resistance. Other ty use the phenomenon of persistent currents in su conducting rings. These devices, and related ones st ming from them, hold great promise. Cheaper, morl compact computers are in the offing and their develi ment proceeds. There are many other possible applications of sup conductivity which are now being investigated developed: superconducting, almost-frictionless, be ings for gyroscopes and motors; superconducting tran formers (both dc and acj; superconducting rectifie tunnel diodes, amplifiers, oscillators, etc. One of more interesting and potentially significant is the high- field electromagnet. It has been recently found that many high-transition-temperature alloys and com- pounds, as for example Nb-Zr, Nb-Ti, V-Ti, Nb3Sn, when drawn into fine wire, not only remaifl superconducting in very high external fields but also can carry high current densities, thus lending them- selves to the construction of high-field magnets. Solenoids yielding fields of over 60 kilogauss havj already been constructed, and research observations! so far indicate that, with proper choice of materials, fields of 200 kilogauss or even higher are feasible. Fields of this magnitude are not impossible to obtain^ by more conventional means, but with the super- conducting magnet no Joule heating is developed; the power requirements are negligible and the bottleneck of heat transfer is nonexistent. The future here is a] rosy one, not only for research magnets, but also fon application to masers, microwave devices, accelerators, and plasma physics, to name but a few. We are begin- ning here a new era in the technology of physics. I regret there is no time left to discuss other active areas of low-temperature physics research; for exam- ple, the many fascinating techniques for study of energy bands and the Fermi surface in metals, the magnetic properties of solids with application to masers, developments in ferromagnetism and ferro- magnetic superconductors, nuclear orientation and t study of hyperfine structure in solids, thermoelectricity. and observations on frozen free radicals. I know that this cryogenic area of physics will continue to be fertile and repay the effort expended on it. PHYSICS TODAY

1.1728874.pdf

Electrical Conduction and the Photovoltaic Effect in Semiconductors with PositionDependent Band Gaps P. R. Emtage Citation: Journal of Applied Physics 33, 1950 (1962); doi: 10.1063/1.1728874 View online: http://dx.doi.org/10.1063/1.1728874 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Semiclassical electrical current expressions for materials with position-dependent band structure in the near equilibrium approximation J. Appl. Phys. 81, 6800 (1997); 10.1063/1.365224 Erratum: ‘‘Analytical solution for minoritycarrier transport in heavily doped semiconductors with positiondependent band structures’’ [J. Appl. Phys. 6 8, 1710 (1990)] J. Appl. Phys. 69, 4153 (1991); 10.1063/1.348979 Analytical solution for minoritycarrier transport in heavily doped semiconductors with position dependent band structures J. Appl. Phys. 68, 1710 (1990); 10.1063/1.346626 Carrier injection in semiconductors with positiondependent band structure: Electronbeaminduced current at heterojunctions J. Appl. Phys. 64, 2505 (1988); 10.1063/1.341633 The effect of positiondependent dielectric constant on the electric field and charge density in a pn junction J. Appl. Phys. 52, 6783 (1981); 10.1063/1.328632 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 6 JUNE 1962 Electrical Conduction and the Photovoltaic Effect in Semiconductors with Position-Dependent Band Gaps* P. R. EMTAGE Electronics Laboratory, General Electric Company, Syracuse, New York (Received November 6, 1961) The semiphenomenological transport equations that are commonly used in the theory of semiconductors have been rewritten in variables appropriate to the discussion of an illuminated isothermal semiconductor in which the band gap is a linear function of one of the position coordinates. It has been found necessary to neglect those terms in the transport equations that arise from variations in mobility and effective mass. The first section considers electrical conduction in a uniformly doped specimen under weak illumination; deviations from Ohm's law were found to be small. In the second section, the potential distribution in a highly illuminated sample of such material, both with and without a junction present, is found. The major contribution to the total potential is the same as that found earlier by Tauc. These results are then used to determine the best geometry for a solar-energy converter in which such materials are used, and to calculate the efficiency of this converter as a function of illumination. The greatest efficiency possible was found to be 43%. I. INTRODUCTION THERE exists now sufficient theoretical evidence to show that it may be possible to treat the alloyed transition region between two similar semiconductors as a region of continuously varying band gap, mobility, effective mass, and density of states, provided the alloying is sufficiently gradual. That is, we may at each point define an effective mass, a band gap, etc., having the same values as those which would be found in an infinitely extended homogeneous crystal of the same composition as has the inhomogeneous crystal at that point. The first treatment of this problem was that due to Bardeen and Shockley,! whose discussion was confined to a description of inhomogeneities arising from a varying elastic strain in an otherwise homogeneous crystal, and who showed that in this case the concept of a variable band gap was legitimate. The basis of their work has been criticized by Hunter and Nabarro,2 who find a weaker dependence of the mobilitjes on the rate of change of strain than was found by Bardeen and Shockley. This work has been extended by Kroemer,3 who considers the general case of an arbitrary change in the shape of the local crystal potential without reference to the origin of such variation. In view of these results, it will be supposed that it is valid to describe an alloy of variable composition as being at each point equivalent to an infinite homogene ous crystal of the proper composition, the electron dy namics being governed by the same quantities as would be found in the infinite. crystal. The subsequent argu ment of this paper is based on the usual semiphenome nological type of transport theory that is commonly used in semiconductor theory, extended to cover the case where the band gap and other quantities are func tions of position. The first part of this paper develops * This work was supported by Air Research and Development Command, U. S. Air Force. 1 J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950). 2 S. C. Hunter and F. R. N. Nabarro, Proc. Roy. Soc. (London) A220, 542 (1953). 3 H. Kroemer, Arch. Elek. Ubertragung 8, 499 (1954). the appropriate transport equations, in the absence of a thermal gradient or magnetic field, and uses them to discuss the deviation of the electrical conductivity of such a system from Ohm's law. It has been suggested to the author4 that if a variable band-gap system can be formed, it may be of use in the construction of an efficient photovoltaic converter. For in a simple junction converter, one of the major losses arises from the dissipation of excess energy when a high energy photon is absorbed, while if a continuous range of band gaps is present, each photon will be absorbed at, or close to the point at which its energy is equal to the band gap. Furthermore, in the simple converter the band gap must be quite high if any substantial voltage is to be extracted from it, so that all low-energy photons will be lost, while with the variable band-gap material it may be possible to make the least band gap small, thereby eliminating this loss. The use of such material may therefore form a practicable substitute for the cascaded cell. 5 The last section of this paper is devoted to an investigation of this possibility. II. CONDUCTION PROPERTIES UNDER LOW ILLUMINATION In this section the isothermal transport equations pertinent to an illuminated semiconductor in which the band gap varies in a distance that is comparable to the diffusion and Debye lengths will be derived, and used to discuss the conduction properties when the illumination is low. The doping will be taken as constant, since variations in doping will tend to obscure any peculi arities arising from the variation in band gap. Distri bu tion Suppose that at a point x the conduction and valence band edges have energies E1(x) and E2(x), measured from the zero of electrostatic potential, the electrostatic 4 J. F. Elliott (private communication), General Electric Com pany, Syracuse, New York. S See for example, W. R. Cherry, Proc. 14th Annual Power Sources Conference, May 1960. 1950 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21ELECTRICAL CONDUCTION AND THE PHOTOVOLTAIC EFFECT 1951 potential at x being cf>(x). The band gap is defined as EB(x)=E1(x)-E2(x), see Fig. 1. If a current is flowing, or the system is illuminated, the hole and electron distributions must be charac terized by the two different pseudoelectrochemical po tentials!t' and !2' introduced by Shockley; in thermo dynamic equilibrium these potentials will be equal. The total energy of an electron in the conduction band is H]=-ecf>+E1+€1, where €l is its kinetic energy. The Fermi distribution for the electrons in the conduction band is then 11 = 1/[1+exp(H 1-!1')/kT] =1/[1+exp(€1-!1)jkT], (1a) where !I=!t'+ecf>-Et, the difference between the electrochemical potential and the conduction band edge. The energy of an electron in the valence band is H2= -ecf>+E2-€2, where €2 is the kinetic energy. The hole-distribution function is therefore 12= 1-1/[1+exp(H 2-!2')/kTJ = 1/[1 +exp( €2-!2)/kT], (1b) where !2=E2-ecf>-!2'. It should be remarked here, that El and E2 are properties of the material alone, since they are measured from the zero of potential energy, and do not depend on the doping or electrostatic potential. In semiconductor physics, the electron and hole distributions are generally far from degeneracy, i.e., -!I, -!2»kT, so that the electron distribution, for example, becomes exp(!l-€l)/kT. If the bands are spherical, then the density of states is Pie €i) = 41r(2mi/h2)!€it, i= 1, 2, where mi is the effective mass. Multiplying by the distribution function and integrating over € from 0 to a;; , the hole and electron densities are found to be (2) where Transport Equations The partial currents due to the electrons and the holes have been written by Shockley6 in the form II = nel-'l(at//aX), 12 = Pel-'2 (a!2'/iJX) , (4) where II and 12 are the electron and hole currents, 1-'1 and 1-'2 their mobilities. It is readily verified that these forms hold even when the band gap and effective masses vary, either from the Boltzmann equation or from thermodynamic arguments. In a steady state the net current I = I 1+ 12 is a con stant, independent of position, since the rate of change of charge at any point must be zero. The equation of 6 W. Shockley, Bell System Tech. J. 28, 435 (1949). E o -e '" "2 FIG. 1. Energy level diagrams and notation. continuity is then (alt! ax) = -(aId ax) =e(R-g), where R is the recombination rate, g the rate of pair production by both phonons and external illumination. It will be assumed that the recombination rate has the simplest form possible (R cr. np). But in thermodynamic equilibrium, with no current or illumination, R= go, go being the equilibrium rate of pair production, and the product np=n?, where ni is the intrinsic density, defined by (5) The equation of continuity therefore takes the form (all/iJx)=ego[(np/n;2)- (g/go)]. (6) One further equation is needed in conjunction with the above for the system to be defined completely, this being Poisson's equation relating the rate of change of the electrical displacement to the charge. Using Poisson's equation: aD/ax = 41rp= 47re(N+p-n) ; N is the total background charge in the ionized donors and acceptors (N = N D+ -N A-) and D is the electrical displacement. Since D=Kacf>/ax, where K is the dielec tric constant, this becomes a2cf> a InK acf> 41re -+--=--(N+p-n). (7) ax2 ax ax K These equations are now sufficient, but the form in which they have been written in not suitable for further computation. It will be found convenient to introduce new dimensionless variables ~ and 'Y/ defined by e~= (n/p)l and e~=np/n,2. (8a) In terms of the potentials used before, these are ~=[ecf>+Hrt'+r2')-Eo]/kT, 'Y/= (rt'-.\2')/kT, (8b) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:211952 P. R. EMTAGE where Eo=HEl+E2)-!kT InA 1/ A2. In terms of these variables, the current and potential equations above are / = /1+ / 2= constant, a/I = ego(e~-~), ax go a2tP a InK atP 47re -+---=-[ -N+2nie hsinhH ax2 ax ax K (9) (10) (11) (12) (13) These constitute a set of seven equations for the unknowns p, n, ~, 'I), /1, /2, and cp, so that, provided ap propriate boundary conditions are given, the problem is completely determined. The final transport equations will be written in terms of ~ and 'I) alone. By adding the pair of equations (10) the gradient of the potential can be expressed in terms of the total current acp / kT[a~ a'l)] 1 aEo -=--+- -+Ht 1-t2)-+--, ax 17 e ax ax e ax (14) \\here the following abbreviations for local quantities have been introduced: 171 = nej.ll = n;ej.lleHh, 172= pej.l2= n;ej.lle-H!\ 17=171+172, t1=U1/U, t2=u2iU. (15) tl and t2 will be used extensively in the ensuing calcula tions. It should be borne in mind that t1+t2= 1, and that they are functions of ~ alone. For an n-type semiconductor, t2 is nearly zero, tl nearly one, and vice versa if the material is p type. On substituting (14) into (10), it is found that kT 171172 OTJ /l=td+---. e 17 ax (16) Now by substituting (16) into (12) and (14) into (13), the set of equations (9)-(13) can be reduced to a pair of linked, nonlinear, second-order differential equations relating ~ and TJ. The results of the substitution are a InK{ / kT[a~ aTJ] 1 aEO} 47re2 +----+--+Ht 1-t2)-+-- =--(-N+2n;ehsinh~), ax 17 e ax ax e ax KkT (17) and This pair of expressions constitutes the final form for the transport equations, and it is clear that they cannot be solved in their entirety. Since we are only interested in those peculiarities of the system that arise from the variation in band gap the following simplifying as sumptions are made immediately: (1) The dielectric constant will be assumed constant; this assumption is fairly good, as K is a very weak function of the band gap, (2) the mobilities will be assumed constant; this is not a good assumption, but is necessary, as the terms involving the mobility variations are among the most (18) complex of those present, (3) the effective masses are constant, again a poor assumption. The complete set of assumptions necessary will be given later. Before proceeding to find the effect of weak illumi nation, the forms of ~ and TJ in equilibrium will be determined. Thermodynamic Equilibrium If / =0 and g= go it can be seen by inspection that TJ=O is a solution of (18). This is clearly the solution [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21E LEe T RIC ALe 0 N Due T ION AND THE P HOT 0 VOL T A ICE F FEe T 1953 required, for then r/ = r2' so that the electrochemical tron concentrations are potentials of the two systems become equal. Equation (17) then reduces to no=N[1+(~y/(1-,82LN2)J po=nNN. (23) where ~o is the equilibrium value of ~. It was remarked in the introduction that N would be set constant, so that effects arising from the variation of band gap would not be obscured by those arising from variations in the doping. If any reasonably simple de pendence of ~o on the band gap is to be found, iJ2 Eo/ iJx2 must be made zero, and ni must depend on the band gap alone. Since Eo contains the ratio of the effective masses, and ni contains their product, the effective masses must be supposed independent of position. The remaining condition is that the band edges be assumed linear functions of position. Under these conditions, solutions to (19) can be found (1) when the doping is zero, ~0=0, and (2) when the doping is everywhere larger than the intrinsic density. In this latter case, the solution is found by observing that for n-type material, to is large and positive j further, the intrinsic density can be written in the form ni=niOe-!~x, where niO is the intrinsic density at the origin, and ,8= (l/kT) (aEBjax). (20) That the length 1/,8 characterizing the rate of change of the band gap is a constant is ensured by the condition that the band edges are linear functions of position. It is now found that being the first two terms in a series in (ni/N)2. Here LN is defined by (22) and will be referred to as the Debye length in the doped crystal. It is the length LN that governs the rate of change of internal fields arising from small perturbations in the doped crystal, when no current is flowing. With a doping of 1016, K = 10 and T= 300 OK, LN= 7X 10-6 cm and is therefore much smaller than any of the other lengths that will be considered with reference to the system. The above expression for ~o is only valid if LN«1/,8. It is unlikely that the alloying can be carried out over a distance less than 10 I-' or that the description in term of local states would be valid for much smaller lengths. If the band gap changes by 1 ev in this distance, then at room temperature 1/,B=2.SX10- o cm. This condition will be complied with in all systems that will be con sidered. With this form for h, the equilibrium hole and elec-For a p-type material, /;0 changes sign and the hole and electron concentrations are switched. Since the calcula tion with the p-type material is exactly the same as with the n type, only the latter will be given. Approximations Before listing the approximations made, one more that will be necessary is discussed. On the right-hand side of (18) there is a term which may be reduced, in equilibrium, to the form k;~/::+::)= ~2' (24) where L is the diffusion length. The quantity go may be written in the form (function of lattice parameters)Xe-EBlkT. Since the materials to be alloyed must have a similar structure, the function of lattice parameters will vary only slowly, and will be taken as constant. If the doping is large, compared with the intrinsic density, then no will be nearly constant, and therefore, so will the diffusion length. The approximations which have been made are now listed. (a) The effective masses ml, m2 are constant; (b) The mobilities 1-'1, 1-'2 are constant; (c) The dielectric constant K is constant; (d) The recombination factor can be written in the form go=constXe-EBlkTj (e) The doping N is constant, the ratio (n;jNY being everywhere of the first order of small quantities. (f) The band edges EI and E2 are linear functions of position. For the purposes of this section, it will also be as sumed that the current and illumination are so small, that changes in I; and TJ resulting from them are less than unity. It is not possible to calculate the dependence of the current on the applied potential and illumination in any closed form, so that instead the potential will be expressed as a power series in the current and illumina tion, the series being carried only as far as quadratic terms in I and g. This procedure then gives an estimate of the current necessary before the conduction properties deviate from linearity. We now write g=go+g', where g' is the rate of pair production due to the incident pho tons, and 1;= I;o+h+h' . " TJ=TJI+TJ2 where h, TJI are of the first and h, TJ2 are of the second order in I and g'. Only first-order terms in (n;j N)2 will be included. Equation (17) will then reduce to a set of equations in h+!TJI, /;2+!TJ2, etc., and (18) to a set of equations in T}I, TJ2' . '. For reasons which will appear ~+!TJ will be needed to only first-order terms in I, g', and (n;jN)2, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:211954 P. R. EMTAGE while 71 will be needed as far as quadratic terms in I and g' but only zero order in (nil N)2, if the conduction properties are to be found as far as quadratic terms in I and g', and linear terms in (niIN)2. Under these condi tions, Eq. (17) reduces to a2 1 -(h+h)--(h+h) ax2 LN2 where On regarding the right-hand side as a set of source terms, it appears that h+!71 is of the order of (n;/N)2, so that the first term on the right can be ignored. Equation (18) then reduces to and a2711 a7l1 1 1 g' Ie --{3---rJl= ----(3--, ax2 ax L2 L2 go kT(Jo (26) The only one of these to which an explicit solution will be needed is (26); this solution can readily be found, and is 1 g' Ie f(x) = ----(3-. L2 go kT(Jo Potential Considering the perturbations of ~ and 71 from equilib rium as being small, the expression (14) for the gradient of the potential may be rewritten in the form act> =~ aEo+ kT a~o+ kT(ah+~ aTf)_~ ax e ax e AX e ax 2 ax (Jo (29) The potential tending to produce a current in the positive direction along the sample is -f (act>jax)dx. The first and second terms in the above are a difference in contact potentials and must be subtracted in order to find the observed potential. The boundary conditions will be so chosen that the integral of the third term will vanish, and the fourth term is a simple ohmic resist ance; the terms of interest are therefore the fifth, sixth and seventh. The fifth term is multiplied by I, so that ~l+h need be found to only first order in I and g' in order to calculate quadratic terms in the current. The sixth and last terms are multiplied by (n;/N)2, so that 7J need to be found to only zero order in this parameter. The approximations made in deriving (25)-(27) are thereby justified. This calculation is concerned only with the intrinsic conduction properties of the variable band-gap semi-conductor, it being desirable to exclude effects arising from contact potentials at the ends, or any other term depending on how the sample is terminated. It can be seen that ~l and 7J would both be zero if it were not for the presence of the inhomogeneous terms on the right of Eqs. (25)-(27); it is the presence of these source terms that gives rise to any peculiarities the system may ex hibit. In order to eliminate any specific method of terminating the material it is supposed that the sample extends between -!l and +!l only. Between these limits hand 71 obey Eqs. (25)-(27); outside these limits hand 71 obey the homogeneous equations, with the source terms on the right made zero. The boundary conditions are therefore that hand 71----t 07 as x ----t ± 00. Using these boundary conditions, the total potential along the sample may be reduced to a set of integrals over source terms, the result being that the potential V tending to produce a current in the positive direction 7 Many of the terms in the potential are of the form i: e-pxTJdx or i: e-Px(~,+h)dx. When x is large and negative, TJ -> exp[~+ (b+~r} as can be seen from (28), therefore, this term does not diverge. Also, ~,+h -> e-XiLN so that this term does not diverge provided LN< 1/{J. This last condition has already been introduced in the discussion of thermodynamic equilibrium. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21E LEe T RIC ALe 0 N Due T ION AND THE PH 0 TO VOL T A ICE F FEe T 1955 takes the form where V a= Jl.2(niO)2 ~ AE, Jl.l N goo el po=l/NeJl.l, 1 (niO)2 G { Jl.2[ {3V ]} pa=---1+-1--[(1+4/{32V)-!-!{3L] , uo N goo Jl.1 I Where G is the total number of incident photons, AE the change in band gap along the length I, niO and goo are the values of ni and go at the origin, and nm is the intrinsic density at the point -!l where the band gap is least. In calculating the above, it was supposed that g' was con stant between -!l and +!l. Terms of the order e-I1ElkT have been ignored in comparison with unity, so that it appears that P2 does not vanish when AE=O. Here Va is the voltage produced by illumination, Po the decrease in resistivity due to the current-carrier pairs formed by illumination, PO+P1 is a simple resistivity, P2 a second order term arising from the change in band gap. In a real system the mobilities and effective masses will not be constant, so that the values to be used in Eqs. (30) must be specified. The contributions to the integrals giving po, Pa, and Va come uniformly from the whole sample, for which reason average values of all quantities should be used in numerical calculation. The main contributions to PI and P2 come from near x= -!l, so that the mobilities used in evaluating them should be those found in the narrow band-gap material. (30) III. HIGH ILLUMINATIONS It appears from the previous section that a potential is developed in a variable band-gap semiconductor under illumination, even when the doping is uniform. This result has been reached by previous investigators.8 It is therefore possible that such materials may be used in the construction of photovoltaic cells. As is well known, such cells cannot be efficient unless the rate of pair production due to incident illumination is much greater than that due to lattice excitations. For the moment the discussion of the high illumination case will still be confined to the case of a uniformly doped sample. Suppose that the semiconductor is illuminated with an intense uniform spectrum, the number of photons in a given energy interval being constant. The rate of pair production per unit volume g' will then be constant, the criterion for high illumination being g'»go at all points. Then, dropping the terms due to variations of mobility, Eqs. (17) and (18) may be written as a2~ a21/ a~ a1/ Ie [ a~ 1 a1/ 1 ani] 47re2 N 87re2niO -+Ht 1-/2)-+2/ 1t2--+- (t1-t2)-+--+-- +--=--eH>rPx) sinh~, ax2 ax2 ax ax kTu ax 2 ax ni ax KkT KkT (31) and a21/ 1(a1/)2 a1/ a~ a1/ 2Ie a~ e(Jl.le~+Jl.2e-~) -+---!(3-- (t1-t2)--+--= [gooeH>rPx)-g'e-H>rPx)]. ax2 2 ax ax ax ax kTu ax Jl.IJl.2kTniO (32) It is first necessary to find the forms of ~ and 1/ in the bulk of the illuminated region, neglecting end effects. On supposing that Eqs. (31) and (32) are valid for all x, then by inspection the required solution is N (gOO)l 1/a={3x+lng'/goo, sinh~a=-- --; , 2niO g (33) where goo, niO are the values of go and ni at the origin. It should be noted that ~a is a constant, and that both ~a and 1/a are independent of I, so that the material is strictly ohmic within the boundaries of illumination. The subscript G will be used to distinguish quantities 8 Jan Tauc, Revs. Modern Phys. 29, 308 (1957). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:211956 P. R. EMTAGE . I ~I' .. ---.. ' , , . ~ , ...... 1 I /' -"7::"--1'" I , , ./ \. r .... -__ '2 ~. ..... --... FIG. 2. Sketch of po tentials in limited region of intense illumination. pertinent to the illuminated region, far from its boundaries. and where Here LG and LNG stand for the diffusion and Debye lengths in the illuminated region, and are smaller than Land LN, respectively. LNG will therefore be small com pared with 1/{3, and calculation with various systems shows that LG will remain large compared with 1/ {3 unless the illumination is far greater than any that will be considered. The solutions to the pair of Eqs. (35) must be of the form fJ1 =Aex/\ ~1 = Bex/X. Substituting these forms into (35) and taking LNa«1/{3«L G, approximate expres sions for these quantities are 1/}"= 1/ LNG-t{3{ 1-(t1O-t2G) tanh~G} with A/ B={3LNG(t1O-t2G), 1/}..= -1/ LNG-!{3{l- (t1O-12O) tanh~G}l with A/B= -(3LNG(t1O-t2G), . 1/}"=2/{3LG2{1+(tw-t2G) tanh~G} with A/B= -2 coth~G, 1/}"= -!{3{1+ (tW-t2G) tanh~G} with A/B= -2 coth~G' (1+1{32LN2(t1O-t2G)}. (37a) (37b) (37c) (37d) Figure 2 shows the forms of the Fermi levels and of ~ and fJ arising from a limited region of intense illumi nation; if there is no current, ~ and fJ revert to the values corresponding to thermodynamic equilibrium outside the region of illumination. For this reason f (au ax)dx does not contribute to the potential between the ends of the material, so that it will be convenient to use a pseudopotential tj/ defined by acjJ' 1 kT afJ -=---(11-/2)-' ax 2 e ax (34) The change in cjJ' between the ends of the material gives the observed potential. Consider the illuminated region, and suppose that near its edge ~ and fJ change to ~G+~l and fJG+fJ1 where band fJ1 are small. Then on neglecting second-order terms, Eqs. (31) and (32) become, in the absence of a current, (35) (36) These are of two main physical types: In (37a) and (37b) the lengths}.. are close to ±LNG, and the change in fJ associated with a given change in ~ is small. The change in partial currents is therefore small [see Eq. (16)J, so that these solutions correspond to a change in electrostatic potential such as is found at a junction, in the absence of a current. Solutions (37c) and (37d) correspond to diffusion processes, the lengths }.. being close to the forward and backward diffusion lengths [d., Eq. (28) J with large changes in ~, fJ, and the partial currents occurring simultaneously. Edge and Junction Potentials Consider one sharp illumination boundary only, all the region to the left being illuminated, and all to the right being dark. Near the boundary, rapid changes in ~ FIG. 3. Sketch of ~ and 7/ at illumination boundary. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21E LEe T RIC ALe 0 N Due T ION AND THE PH 0 T 0 VOL T A ICE F FEe T 1957 and 1/ occur simultaneously, as shown in Fig. 3. There are also large changes in the partial currents, which are zero on the right and equal and opposite on the left, where (a1/1 ax) = (3. The changes in potentials to the left of the boundary will therefore be mainly those asso ciated with diffusion processes, and will be predomi nantly of the types (37c) and (37d). Since it is not possible to solve the equations govern ing ~ and 1/ accurately when large changes occur, the following method will be used in estimating the po tential change across the boundary: choose some point Xo such that Hxo) is so large that t1(x) is very close to unitywhenx>xo. Then f~ (t1-t2) (a'Y/lax)dx= -1/(xo), since 'Y/ ---t 0 as x ---t 00. To the left of Xo the integral is evaluated using the forms of ~1 and 1/1 calculated in (37). Since the result so found will be shown to be only weakly dependent on Xo, Xo can be replaced by the value of x at the boundary. On the left, assume 'Y/1=Aex/\ b=Bex/A; then fXO a'Y/1 A Ioh(XO) (t1-t2)-dx=-- db(/J.1e~G+h-/J.2e-~G-h)1 -00 ax B 0 (/J.1e~G+h+/J.2e-~G-h) Therefore e r -dx= -!kT 'Y/o(xo)--lntw , acp' [ A ] J boundary aX B (38a) if the material is n type. If the material is p type then e r -dx=!kT 'Y/o(xo)+-lnI2G, (38b) acp' [ A ] J boundary aX B where AlB is given by (37c). Similar forms may be ob tained if the illumination is on the right of the boundary, with AlB given by (37d). If there is a junction at some point in the illuminated material, then ~ changes sign in the neighborhood of the junction. Assume that ~G is small, having the value ~GI on the left of the junction, ~Gr on its right. The potential change across the junction may be found by using the forms (37) for small changes in ~ and 'Y/, so as to fit ~, 1/, and their derivatives at the junction. On evaluating the integrals, it is found that acp' e r -dxrovkTaOI-~Or), (39) J junction aX when I ~Ol-~Grl is small. The existence of this term depends on the fact that {3 is not zero, but if the condition 1/{3«LG is satisfied, then {3 disappears from the result, to a first approximation. Total Potential The remaining contribution to f (acp' I ax)dx is the region between the boundaries of illumination. If the illumination extends between Xl and X2, this bulk con tribution is fX 2 acp' 1 kT fX 2 -dx=-- (tw-t2G){3dx Xl ax 2 e Xl 1 liE =--(lw-t2G)' (40) 2 e Uniform doping. If the material is n type, then on adding the bulk term (40) and the two boundary terms, the total potential developed under illumination is elicp= -t2GliE+"ikT{32LN2(tw-t2G) coth~G lntw. Here the first term is the same as that found earlier by Tauc, while the second is an end correction, being the difference in the ratios of A to B for forward and back ward diffusion. This end-correction is the difference be tween two large and nearly equal terms. The two terms are nearly equal because of the assumption that the mobilities are constant, but even if this is not so, the magnitude of the correction still cannot exceed kT coth~G lntw, which will be fairly small. If the material is p type, the potential developed is elicp=twliE+(correction). It will be seen that for a substantial potential to be developed, the illumination must be so intense that the partial conductivity of the minority carriers becomes appreciable; the tendency of very strong illumination is to make the ratio of the concentration of the minority carriers to that of the majority carriers nearly equal to unity. The largest value of tw for a p-type material under intense illumi nation is /J.11 (/J.l+/J.2), so that to achieve a high potential from a variable band-gap p-type material, one requires /J.1> /J.2· This condition is satisfied for most semicon ductors, so that it will generally be desirable to use acceptor doping in the variable band-gap material. Junction. Suppose that the illumination extends be tween Xl and X3, there being a junction at a point X2 between Xl and X3, the material being n type on the left of the junction, p type on the right. Quantities on the left and on the right of the junction are distinguished by the subscripts land r. For convenience, it is supposed that the dopings on the left and on the right are equal and opposite, so that ~GI = -~Gr= ~G, ~G being positive. Then on adding the various contributions, the total potential is found to be e!1cp= kT'Y/o(X2) -!1E12t2GI+ !1E23tWr +2kT coth~G IntlGlt2Gr+2kT~G, (41) where liEij is the change in band gap between Xi and Xj. The first term on the right is the usual value for the potential developed across a junction; the second and third terms are the major parts of the potential de- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:211958 P. R. EMTAGE veloped in the variable band-gap material on the left and on the right, and have the same form as is to be expected from (40); the remaining two terms are end and junction corrections. It should be remarked that using Eqs. (33), the junction term can be written in the formkn]G(x2)=E B(x2)-2kTln[(2A/N) sinhtG], where A = (A1A2)!, Al and A2 being defined in (3). IV. PHOTOVOLTAIC CELLS Clearly, one need not consider the uniformly doped semiconductor alone as being a possible energy con verter, for the greatest voltage that can be obtained from it is less than the change in band gap, while if a junction is incorporated, a voltage of the order of the greatest band gap can be attained. Further, as has been shown, the uniform material is ohmic under high illumi nation, while one might hope that with a junction present the current-voltage relationship would be nearer to the square box form that is ideal in a generator. As can be seen from (41), the graded material to the left of the junction produces a potential opposite in sign to that produced in the junction and on the right, so that it appears best to have the junction as far to the left as is possible. Nevertheless, it might be possible to have the junction on the right provided t2G! could be kept small, were it not that the diffusion length to the right is of the order of 1/13, so that none of the pairs formed on the left will be separated by the junction. Such an arrangement, therefore, could not be better than a single junction in wide band-gap materia1.9 For these reasons, the only geometry that will be considered is that with a sharp junction at the narrow band-gap end of the graded material, it being considered that n-type material with a constant and small band gap extends to the left of the junction, the variable band-gap material being p type. There is no need to consider a diffuse junction, for if the graded material extends to the illuminated surface there is ipso facto a strong field at the illuminated surface tending to move the current carriers towards the junction. Current-Voltage Relationships Only the geometry mentioned above will be con sidered, it being assumed that the junction is so narrow that recombination within it is negligible. The starting point of the discussion is the realization that even under open circuit conditions two large partial currents of holes and electrons flow into the junction from the right. When some current I is flowing, the partial cur rents in question are I1r and I2r, with I =I1r+I2r. On the left the currents will change to Ill=I and 121=0. There is therefore a change in the electron current from I1r to I occurring on the left of the junction alone, since recombination within the junction is negligible. 9 Much the same conclusion has been reached by WolflO for this geometry, although he did not consider the contribution to the potential due to the graded material. 10 M. Wolf, Proc. Inst. Radio Engrs. 48, 1246 (1960). Let the electron current at some point on the left be Ill; then if there is no illumination on the left, Eq. (12) gives Since this is a diffusion process, we may write '1]'" Aexl L on the left, then But Ilr=Ilor+tlGrI, where I10r is the value of IJr on open circuit. The current therefore takes the form I=IoI[l-eA-Ao], where and eAo= 1 + I lorl egO!L. The potential developed across the junction is pro portional to A ; on using (37) the potential on the left is given by HkTle)[tanh~G+1]A. In general, if <P is the total potential across the junction, it is found that A = e<pt! fkT, where f is a factor of the order of unity. Then if <PO! is the open-circuit potential existing across the junction I = I 0t{1-exp[e(<Pl-<POI)/ fkT]}. (43) It has already been shown that the graded material on the right is substantially ohmic; then if <POr is the open circuit potential developed across it, and <Pr the po tential when a current I is flowing, the ohmic property implies (44) where Ior= 2110r is the short-circuit current for the right. The total potential across the cell when a current I is flowing is therefore (45) where <Pr and <PI are found from (43) and (44) when the current I is given. But since 101, lor, <POI, and <POr may be found from formulas already given, these relationships are sufficient to determine the connection between I and <P for any particular case. Since it is not possible to write a general form for the I -<P curve, a particular case has been chosen to help clarify the physical situation. 101 and lOr can be expected to be of the same order of magnitude, so they have both been set equal to 10; it has also been supposed that <POl=<POr=! v. The resulting set of curves for <PI, <Pr, and <P against 1/10 is shown in Fig. 4. The essence of the above treatment for estimating the current-voltage relationships is that the junction and the graded material have been considered as separate, the junction being so close to the graded material that it is affected by the proximity. This appears in Fig. 4, where the net potential <P is controlled almost entirely [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21E LEe T RIC ALe 0 N Due T ION AND THE PH 0 T 0 VOL T A ICE F FEe T 1959 by the graded material when the current is low, by the junction when the current is high. Physically, it is the injection of holes into the junction that is responsible for this separation into regions. As long as the hole current remains high, the junction is swamped by this current and the potential across it alters very little. When the hole current becomes sufficiently small, the junction characteristics take over and the total current rapidly reaches the junction saturation value. If the number of photons incident on the cell is G, then the short-circuit current lois 10= (1-r)eG, (46) where r is the fractional recombination. No accurate estimate of r can be given, since to do so would require a precise knowledge of ~ and T/ everywhere. However, it is possible to calculate r when the illumination is low, and hence to estimate it for strong illumination. By using (28), T/ may be found when g'«go; the excess rate of recombination within the illuminated region is then L~~l gOT/dx. On assuming L»1/{3 we find 1( 1)2 kT r=tl/L2{3=-Z L I:J.E· (47) This result is plausible, since D{3 is the diffusion length to the left within the graded material. It will be assumed that under illumination the recombination is given by replacing L with LG in (47). Efficiency If the potential across the cell is cp when the current is I, the output power is W =Icp. If W is plotted against cp, it reaches a maximum at some point between cp=O and Cp=CPo, where CPo is the open circuit potential. It is 01 02 03 o. o. o. ~a 0.1 ---------, , , , : J I , , J I , J , ! , , , I I , I : I I ,/ --Particukr case (in lextl -------Simple p-n junction FIG. 4. Current-voltage relationships for particular photovoltaic cell discussed in text, showing contributions of <PI and <Pr to <p. The current-voltage relationship for a simple junction giving 1 v on open circuit is shown for comparison. ~ 5i ~ 0.2 IIJ 0.1 S lOS 10ZS J03S Intensity FIG. 5. Efficiency vs illumination (S=3X 1017 photons/cm2/sec) for GaAs-InAs combination. common practice to write the value of Wat this maxi mum in terms of CPo and 10, 10 being the short-circuit current Wmax=aloc/Jo, (48) where a is a number between 0 and 1. In the case of an ohmic generator, a=0.25; for a p-n junction, ~0.85, it being this high value of a that gives the junction-type cell its high efficiency. For the case shown in Fig. 4, a is found graphically to be 0.45. For the case under investigation, a must be found graphically for each arrangement of band gaps and each intensity of illumination. If the mean energy of the incident photons is written as e(cp) , the incident energy is Ge(cp); the greatest efficiency is therefore t=a(1-r)cpo/(cp). (49) The system will be considered to be illuminated with a uniform spectrum extending between the greatest and least band gaps present only, so that e(cp) = HEmax + Emin). The efficiencies are calculated using the light proper to the system rather than sunlight, since it is these values of the efficiency that are pertinent if a system is designed for use under sunlight. Numerical values of the efficiencies attainable have been calculated using a GaAs-InAs alloy (e(cp)=0.89 ev). These compounds have identical structures, and will alloy at all compositionsY Some relevant quantities arel2: EB (ev) ni InAs 0.4 1.6 X 1015 GaAs 1.3 7 107 J!.dJ!.2 110 15 K 13 14 mdm 0.03 0.078 mdm 0.4 0.65. The minority carrier lifetime is about 10-7 sec when the doping is 1017, so that gO=3X1020 for InAs, and the hole diffusion length is 8 jJ.. It has been considered that the shortest distance over which the alloying can plausibly be carried out is about 10 jJ., so 1 has been set equal to this amount. 11 B. A. Smith and ]. C. Wooley, Proc. Phys. Soc. (London) 72, 214 (1958). 12 N. B. Hannay, Semiconductors (Reinhold Publishing Corpora tion, New York, 1959), pp. 406-415, 451. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:211960 P. R. EMTAGE TABLE 1. Potentials and efficiency in GaAs-InAs cell under various illuminations; S=3X 1017 photons/cm2/sec. Intensity S lOS 30S lQ2S 3X1Q2S l()3S 3Xl()3S l()4S 105S <por 0.017 0.10 0.25 0.50 0.65 0.80 0.90 0.93 0.96 <Po/ 0.045 0.14 0.22 0.36 0.39 0.40 0.40 0.40 0.40 <Pc 0 -0.02 -0.03 -0.037 -0.045 -0.06 -0.116 <Po 0.062 0.22 0.44 0.82 1.00 1.14 1.18 1.19 1.20 r 0.04 0.04 0.04 0.04 0.045 0.055 0.06 0.10 0.18 a 0.47 0.45 0.43 0.39 0.38 0.36 0.35 0.34 0.34 E 0.031 0.106 0.203 0.344 0.403 0.431 0.435 0.407 0.374 • The last two terms in the end-correction are inaccurate; if (41) is used, they become so large that the observed potential decreases. <Po has therefore been estimated by using the criterion that the open-circuit potential does not decrease when the illumination increases. The number of photons incident on a square centi meter of surface per second in sunlight is 3X 1017; if these are absorbed in a distance of 10 j.L, the pair forma tion density g' is 3X1()2°. The intensities used in the calculation of efficiency have been expressed as mUltiples of the intensity of sunlight, S=3X1017• The values of CPOr are calculated using the average of t2G for InAs and GaAs, Table I gives the relevant quantities and effi ciencies, CPc being the end and junction corrections, the temperature being 300°K. The resulting curve for efficiency vs illumination is shown in Fig. 5, It will be seen that the efficiency reaches a maximum of 43% when the illumination has an intensity of about 2000 times the intensity of sun light, but is still above 40% when the illumination is as low as 300 times sunlight. The necessity for high intensities of illumination with the GaAs-lnAs combination arises from their very low minority carrier lifetime, which in turn is probably related to the fact that the optical transition is direct. In materials such as silicon, where the transition is in direct, a lifetime as high as 10-3 to 10-4 sec is found, so that the minority carrier concentration can be made high under much weaker illumination. The high values of CPOr under strong illumination arise from the high mobility ratio, The efficiency is only high so long as the light proper to the system is used: thus if amplified sunlight (e(cp) = 1.8 ev) were used on the combination under consideration, the maximum efficiency would be about 20%, If such a system is to be designed for use under sunlight, the range of band gaps present must span the regions of greatest intensity of sunlight. From the above considerations one may find the re quirements for the materials necessary to construct an efficient solar cell of this type: A pair of alloyable materials must be found to fit the following specifica tions; (a) the lesser band gap must be near 0.75 ev, the greater near 2.5 ev; (b) the minority carrier lifetime must be of the order of 10-3 to 10-4 sec when the doping is between 1015 and 1016; and (c) the mobility ratio must be greater than 10. If these requirements can be met, an efficiency of 35% can be achieved under direct sunlight. ACKNOWLEDGMENTS I wish to thank Dr. J. F, Elliott, who has suggested this problem to me, and has supplied me throughout with such information as I needed; Dr, N, Schwartz, for whose continued encouragement and suggestions I am most grateful; and Dr. B. Segall, whose excellent dis cussion of the uniformly doped case has suggested many useful methods. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Sun, 30 Nov 2014 11:39:21

1.1742161.pdf

Effect of Fermi Resonance on the Centrifugal Stretching Constants in CO2 C. P. Courtoy and G. Herzberg Citation: The Journal of Chemical Physics 23, 975 (1955); doi: 10.1063/1.1742161 View online: http://dx.doi.org/10.1063/1.1742161 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/23/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Fermi resonance distortion of the Ru–CO stretching mode of CO adsorbed on Ru(001) J. Chem. Phys. 108, 5035 (1998); 10.1063/1.475910 Finite Nuclear Mass Effects on the Centrifugal Stretching Constant in H35Cl J. Chem. Phys. 45, 2433 (1966); 10.1063/1.1727958 Computation of Asymmetric Rotator Constants from Energy Moments. III. FirstOrder Centrifugal Stretching Effects J. Chem. Phys. 31, 1227 (1959); 10.1063/1.1730573 Errata: Influence of Fermi Resonance of the Centrifugal Stretching Constant of a Linear Molecule J. Chem. Phys. 25, 800 (1956); 10.1063/1.1743085 Influence of Fermi Resonance on the Centrifugal Stretching Constant of a Linear Molecule J. Chem. Phys. 24, 44 (1956); 10.1063/1.1700851 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 17:15:49LETTERS TO THE EDITOR 975 \ \ o \ ' \ 0 \ 2.3 0 \ " \ 0 '\ " '\ '\ '\ .... 0 '\ '\ \ 0 \ \ " 0 \ \ 0 \ \ " \ .... 0 .... ~A " 0 N'O "' ",0 2,1 "- 2, 0.8 e 5 4 3 -. '0 ~ , '0 '0 , \ '0 \ 0 ,0 , '\ , \0\ , \ ~ '" '\, ~ 'o\, , 0, \'0 " ",\ 'o~ "" 0.9 1.0 FIG. 1. Plot of the sums of coexistent densities (ordinates on left) and coexistent volumes (upper curves, ordinates on right) against reduced temperature (abscissa) for van der Waal's type equation. The limiting slopes are indicated by dotted lines (Table I). Open circles and full curves represent calculated data •. ~ecause of the proximity of the cu!'ves and limiting slopes for the densIties, the curves have not been drawn m. so that it is possible to calculate x for any value of Z and from the inverse transformations, the corresponding values of Ll1 and Ll2. It is interesting to note that the form of the equations is such that the sets of coexistent densities are simply related to temperature by a scale factor 8"+1. The behavior of the coexistent densities and volumes is shown in Fig. 1 for different values of the exponent n=O, i, and 1. All three equations indicate reasonably good agreement with the law of rectilinear densities, Ll,+Ll2-2=k(1-8), the best fit over the widest range being given by the Berthellot equation (n= 1). None of them, however, shows good agreement with the law of rectilinear volumes, 4>, +4>2 -2 = k' (1-0). However, in the immediate vicinity of the critical, (Le., in the range 8=0.998 to 1) both laws are obeyed to within ±0.01 percent by all three values of n. Table I gives the limiting slopes (ko and ko') for this region. Guggenheiml has proposed an empirical formula for t.he difference of the coexistent densities, (Llz-Lll) =7/2(1-8)1, whJle both Gug genheim and Fowl(~r2 and Landau and Lifschitz3 have predicted on theoretical grounds a parabolic relation, (Ll2-A,) =a(1-0)i. In fact it has been generally supposed that van der Waals' type equations will have a parabolic coexistence plot in the immediate neighborhood of the critical. However,.if 10g~Ll2-Ll:) is I:l~t~ed against log(1-(J), then it is found that In the ImmedIate VICinity of the critical the slope of the curve is 1/1.68 for n =! or 1 and 1/1.84 for n =0. This seems to indicate some serious difficulty with TABl.E I. Values of limiting slopes for rectilinear plots. n=O n=t n=l k. 0.80 1.19 1.58 k.' 7.25 10.9 14.4 the expansion methods which have been used in the afore mentioned derivations. In conclusion it should be noted that the equation with n= i gives a reasonably good representation of the properties of non associated liquids. At 8=0.625 which corresponds to the normal boiling point of most liquids, the Trouton constant is 9.4 (ob served 10) and the reduced liquid density is 2.47 (observed 2.68). This is a marked improvement over both van der Waals' and Berthellot's equations. We are pleased to acknowledge our indebtedness to Mr. David Pearson of this department for his assistance with the computa tions and to the Office of Naval Research for their support (Contract No. N6-onr-23811). 'E. A. Guggenheim, J. Chern. Phys. 13,253 (1945). t R. Fowler and E. A. Guggenheim, Statistical Thermodynamics (Cam. bridge University Press, Cambridge, England, 1949), Section 729. • L. Landau and E. Lifschitz, Statistical Physics (Oxford University Press, Oxford, England, 1938), Section 50. Effect of Fermi Resonance on the Centrifugal Stretching Constants in CO 2 C. P. COURTOY'" AND G. HERZBERG Dit'is;on of Physics, N aUona! Research Council, Ottawa, Canada (Received Marcl1 2, 1955) USUALLY it has been assumed that the centrifugal stretching constants D are essentially the same for all vibrational levels of a polyatomic molecule in its electronic ground state. Since D itself is very small and therefore not measurable with high accuracy, any small changes of D with vibration would normally be expected to be beyond the accuracy of most measurements. We were therefore greatly surprised when we found large differ ences in the D-values of various Fermi polyads of vibrational levels in CO •. The CO2 bands Vt+"3, 2"2+V3; 2"'+"3, ".+2".+"3, 4"2+V3; 3Vl+va, 2"1+2"2+"3, 11,+4"2+113, 6"2+"3; "2+2"3; 3"3 and most of the corresponding "hot" bands were measured with a high resolution infrared spectrometer with a PbS receiver using up to 68 traversals' through a 1-m absorption cell. The D-values in Table I were obtained for the levels indicated. The relative accu racy of these numbers is much better than the absolute accuracy. The V-values for the TI states (upper and lower states of hot bands, l= 1) are less accurate than those of the 2: states (l=O). Within the accuracy of the determination, the average D-values of the polyads are the same as those of the lowest vibrational level and of the single unperturbed states (monads). However, within a polyad the individual D-values differ greatly, in the case of the tetrad by as much as a factor three. v, .,1 v. 0 (JO 0 1 (JO 1 0 2· 1 2 (JO 1 1 2' 1 0 4' 1 3 (JO 1 2 2' 1 I 4· I 0 6" 1 0 (JO 3 0 11 0 0 l' 2 0 l' 3 1 11 1 0 31 1 2 I' 1 1 3' 1 0 51 1 2 3' 1 I 51 1 TABLE I. :l;i :l;i :l;i :l;i l] :l;i l] l] l] :l: l!: II II II II II II II II II II D(10- S cm-') 13.7 11., 16.1 9 .• 13.7 18 .• 7 •• 10 .• 16 .• 21.. 13 .• 14 .• 12 12 12 .• \ ii:'} 14.8 16 13., 16 •• Mean for polyad 13 .• 13 .• 14.0 14 .• This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 17:15:49976 LETTERS TO THE EDITOR The D-values have been determined by means of the combina tion sums R(J)+P(J) which, when plotted against J(J+1), would give straight lines if D' -D" were zero. To illustrate the effect of the variation of D within a polyad, Fig. 1 shows R(J) + P (J) + 2 (B" -B')J (J + 1) for the four bands of the tetrad 3PI+P3. If the D'-values were all the same as in the ground state, four horizontal straight lines should be obtained. The strong R (J)+P (J)+2 (e'!...e') J(J+I) =2I1o+2e'-2(D'-D")J2(J+1)2 FIG. 1. Combination sums for the Fermi tetrad 3.1 +., of CO,. deviation from linearity actually found is a measure of the devia tion of D' from Dooo. The symmetrical way in which D' varies for the four components of the tetrad is striking. If the D-values of the ~ states are plotted against the order of the polyad, the diagram Fig. 2 is obtained. The variation for the II states is similar but less completely observed. The D-values of the II states given in Table I are averages of the D-values of the two l-doubling components. Actually there is a systematic difference in the D-values of the two l-doubling components which increases rather rapidly with V2. However, this difference may equally well be considered as due to a quartic term in the formula for the l-type doubling. ----__ 0 ..""...",.",." ..."..".."" ___ 0 0-- ....-_--tJ---- .",...,.,."..."., 0:::::- 0----... .................. ---0 ......... -------- .................. 0 -0 ____ -_ .... _ -0 __ ...... --20 l' !j 'l Q 0' lOr- O~~I ________ ~ ______ ~ ________ ~ ________ -J 3 4 FIG. 2. The D-values of CO, in Fermi polyads (2: states). It is remarkable that while the B-values in a Fermi polyad approach one another more closely than they would without Fermi resonance, the D-values are pushed apart from one another quite strongly. Following the preliminary report of this work at the Columbus meeting last June, Amat, Goldsmith, and Nielsen have discussed the theory of this effect. We understand that they will report about it shortly. The detailed analyses of the bands on which the above discussion is based will be submitted for publication shortly. A similar study for Cl302 is in progress and has already yielded similar differences in the D-values. * National Research Laboratories Postdoctorate Fellow. Ionization Potential of Benzyl Radical IKUZO TANAKA* AND CHIMAO KOMATSU Division of Pure Ch,mistry, National Research Council, Ottawa, Canada, and Laboratory of Physical Chemistry, Tokyo Institute of Technology, Tokyo, Japan (Received March 8, 1955) BENZYL radical has C2v symmetry (see Fig. 1). The electronic structure has already been investigated by Tanakal using simple LCAO approximation. The authors calculated the energy levels and wave functions by the SCF method. The SCF determinant was constructed for benzyl cation, a six ,..-electron system, using 2jnr Slater functions for neutral carbon atoms (Z=3.18). According to Parr and Mulliken,2 a solution of SCF determinant is given to be Ei = 1.+"1:. (2Jii-Kii)' (1) i It was assumed here that the lowest three MO's of the benzyl cation obtained can be applied to benzyl radical, a seven ,.. electron system as approximation. Then, using (1) the energy E4 was represented as (2) and it can be used for benzyl radical as approximation. Also cf>4 can be used for it. Moreover, as the radical is a neutral molecule, use of Z=3.18 gives a better approximation to E4 and cf>4 than lower levels and MO's. The energy E4 and the corresponding 4>4 obtained from SCF calculation are E4=E2p+3.30 ev, (3) 4>4 =O.04OXI -O.506X4 -O.679x7+0.414(x2+Xs) +O.OO2(X3+XS). (4) The ionization potential of benzyl radical becomes immediately from (3), Recently Lossing and his co-workers3 obtained by electron impact, These two values are in good agreement. Lossing4 also obtained 7.61±O.05, 7.65±O.05, and 7.46 ±O.05 ev for the ionization potentials of ortho-, meta-, and para-methyl radicals respectively. Using the aforementioned orbital, and making simple pertur bation calculations, the authors reached the following results. According to second-order perturbation theory, Wk" IHkl'12 WkO-W1o (5) where Wk" is second-order perturbation energy, Wko is above E4 the energy of cf>4 orbital, and W10 is the energy of the methyi radical in its ground state. Now Kkl'= JXSH'4>4 dT (6) where xs is 2p,.. Slater function of the new-coming eighth carbol1 atom. Neglecting all {3=fxsH'XidT but for the nearest neighbors it becomes ' (7) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.114.34.22 On: Tue, 25 Nov 2014 17:15:49

1.1717071.pdf

Instrumentation for Medium Energy GammaRay Scattering Measurements E. L. Garwin and A. S. Penfold Citation: Review of Scientific Instruments 31, 853 (1960); doi: 10.1063/1.1717071 View online: http://dx.doi.org/10.1063/1.1717071 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/31/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Fermi Gammaray Burst Monitor Instrument AIP Conf. Proc. 1133, 34 (2009); 10.1063/1.3155916 Neutron and Gammaray Measurements AIP Conf. Proc. 988, 249 (2008); 10.1063/1.2905077 Ruler: An instrument to measure gammaray burster distances AIP Conf. Proc. 307, 665 (1994); 10.1063/1.45824 Instrumentation for gammaray spectroscopy Phys. Today 19, 86 (1966); 10.1063/1.3048395 Scattering and Absorption of GammaRays J. Appl. Phys. 22, 350 (1951); 10.1063/1.1699954 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48E-H GRADIENT SPECTROMETER 853 however, the balance condition can be approximately satis fied for such beams. For example, with a beam width of 0.025 em in a field whose gradient is 2.SE v/cm2, the maxi mum unbalanced force suffered by beam atoms is about 5% of the force due to either E or H alone. The average unbalance is about 2.5%. Furthermore, the balance condi tion may be achieved for very broad beams by choosing a subs tate and field such that f.Leff is proportional to Hover a wide range of H (e.g., the F= 1, mF=O state of hydrogen in the field region 0 to 150 gauss). An apparatus of this kind has been built and is presently being used to perform precision measurements of the po larizabilities of the alkalis. Figure 1 shows a beam profile taken with the balance condition being satisfied for the -jJ.o state of K39, for a magnetic field of 290 gauss and an electric field of about 81 500 v/cm. Beam profiles with no fields, and with an H field of 290 gauss (no E field), are shown for comparison. It is seen that the -f.Lo state is indeed quite narrow, while the +f.Lo state is broadened by both E and H fields. By adding a second E-H gradient spectrometer, flopout THE REVIEW OF SCIENTIFIC INSTRUMENTS magnetic resonance experiments may be performed. Figure 2 shows a sketch of such an arrangement. The usual collimator slit can now be replaced by a number of optically aligned slits to produce very accurate alignment. The resolution may thus be significantly improved. By using narrow slits and a suitable geometry, states with f.Leff of the order of nuclear magnetons can be separated by fulfilling Eq. (1). Thus the method is applicable to atoms (or molecules) in ISO (or 12:0) ground states. When J~O, it is possible to separate nuclear magnetic substates by working in the low field region where different mI states have effective moments which differ by fractions of a Bohr magneton.2 A given nuclear magnetic subs tate can then be made to pass through the gradient spectrom eter without deflection or broadening, with all the original velocity distribution present. Such an application would be useful in producing high intensity polarized ion beams, which are of considerable importance for nuclear physics scattering experiments. 2 Since "'eff is then somewhat field-dependent it is important to choose th e so-called focusing state. See S. Millman, 1. 1. Rabi, and ]. R. Zacharias, Phys. Rev. 53, 384 (1938). VOLt;ME 31. NUMBER 8 AUGUST. 1960 Instrumentation for Medium Energy Gamma-Ray Scattering Measurements* E. L. GARWlNt AND A. S. PENFOLDt The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois (Received May 6, 1960) A system for the energy analysis of scattered gamma rays of up to 70-Mev energy is described. A 5-in. diam, 4-in. long NaI(TI) crystal was the energy sensitive element. The net amplification between the ten-stage photo multiplier output and the pulse-height analyzer input was 1.8. The pulse-height-to-channel transfer characteristic of the system departs from linearity by less than 0.05 channel between the 50th and 16th channel and by one channel between the 16th and sixth. It is demonstrated that the light-to-voltage transfer characteristic of the photomultiplier is linear up to at least 30-v output levels. Provision for direct determination of the effect of super posed pulses (pileup) is incorporated into the system. Careful shaping of the pulses reduced pileup to a minimum, commensurate with obtaining good resolution from the NaI(TJ) crystal. Some typical pulse-height spectra are given, and the novel features of the circuits are described. I. INTRODUCTION RECENTLY, there has been considerable interest in the measurement of nuclear gamma-ray scattering in the energy range from 10 to 70 Mev. The measurements which have been made at the University of Chicago 100- Mev betatron1,2 have employed pulse-height analysis in conjunction with as-in. diam, 4-in. thick NaI (TI) scin tillation crystal. The equipment, whose novel features are * Research supported by a joint program of the Office of Naval Research and the U. S. Atomic Energy Commission. t Now at the Department of Physics, University of Illinois, Urbana, Illinois. t Now at Litton Industries, Beverly Hills, California. 1 E. L. Garwin, Phys. Rev. 114, 143 (1959). 2 A. S. Penfold and E. L. Garwin, Phys. Rev. 116, 120 (1959). described here, was designed to minimize pileup of small pulses and to ensure good resolution for gamma rays of high energy (10 Mev and above). Small pileup was par ticularly necessary in these experiments because the ratio of the number of gamma rays in a 1-Mev energy interval at 3 Mev to the number in the same energy interval at 20 Mev was many thousand under some conditions of irradiation. In addition, there were many scattered low energy electrons present. Data were recorded on a 50-channel analyzer of the pulse-height-to-digital conversion type. The analyzer was capable of handling pulses with rise times of 50 mf.Lsec and had a voltage sensitivity of 1 v/channel. The net This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48854 E. L. GAR\VIN AND A. S. PENFOLD Anti-Coine. Signal Key All Cobles RG-1I4/U Int. Disc.-Integral Discriminator, Threshold Set at Channel 51 edge in Anal er amplification employed between the ten-stage photomul tiplier tube and the analyzer input was 1.8. In addition, a single tube feedback amplifier with a gain of 13 was incorporated to allow calibration of the photomultiplier gain by means of a Na22 radioactive source. The electronic equipment employed as low a gain as possible to minimize effects of overload, baseline shift, and electronic instability so often observed in experiments with short duty cycle accelerators. Highly fedback cathode followers3 and amplifiers were employed throughout. Du mont type 6364 photomultiplier tubes (5 in. diam, ten stage) were used at high gain to deliver, directly, stable pulses of magnitude up to 30 v. The high photomultiplier gain was attained without exceeding the manufacturer's voltage ratings. Experiments with a light pulser demon strated that there were no nonlinear effects in the photo multiplier when it was producing the 30-v output signals. The pulses at the photomultiplier anode had a full rise time of 0.2 j.Lsec (0 to 100%). These pulses were subse quently processed by a pulse shaping circuit which rendered them bipolar and of total length 0.5 j.Lsec. The pulses were presented to a 0.2-j.Lsec linear gate opened by a fast trigger branch of the circuit only for pulses: exceeding a set threshold, arriving during the time of the x-ray burst, and not associated with a pulse in the anti-coincidence counter. The equipment has provision for direct determination of the effect of pileup on the pulse-height spectrum and for measurement of counting losses in the pulse-height analyzer. II. GENERAL ARRANGEMENT A block diagram of the electronic system is shown in Fig. 1. The current pulses from the Dumont type 6364 photomultiplier were integrated with 0.1S-j.Lsec time con- a E. L. C. White, U. S. Patent No. 2,358,428. f· J<'JG. 1. The block dia gram for the energy analy sis system. stant at the grid of cathode follower 1. The latter drove cathode. followers 2a and 2b through 3 ft of 18S-ohm RG-114/u cable. These powerful cathode followers then drove the pulses through 30 ft of cable to the remaining circuitry located outside the beam area. The differentia tors in the fast sides delivered prompt pulses of height proportional to the ultimate height of the pulses passed by the slow side. The anti-coincidence circuit produced an output pulse (in the absence of a charged particle pulse, and for photon pulses exceeding the threshold set by the attenuators in the fast branches) which was transmitted through the betatron beam gate and used to trigger the linear gate of 0.2-j.Lsec duration.4 This trigger pulse could only pass through the betatron beam gate during a time encompassing the betatron beam pulse. Thus, the cosmic ray background was reduced by the duty cycle of the betatron beam. The pulses in the slow side of the equipment were first separately clipped with delay lines of 0.1-j.Lsec length, and added. This sum pulse was then rendered bipolar by a shorted 0.2-j.Lsec length of delay line and presented to the signal input of the linear gate. All pulses which passed through the linear gate were presented to the SO-channel pulse-height analyzer for analysis. The threshold above which pulses were analyzed was, therefore, determined entirely by the setting of the step attenuators in the fast branches of the equipment. If the pulse exceeded channel SO in the pulse-height analyzer, it did not appear in the memory or memory totalizer, but instead triggered the integral discriminator whose threshold was set at the upper edge of channel SO in the analyzer, and so was counted in the overbound scaler. The pulse-height analyzer circuitry was so arranged 4 E. L. Garwin and A. S. Penfold, Rev. Sci. Instr. 28, 116 (1957). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48GAMMA-RAY SCATTERING 855 that only one pulse per betatron burst could be analyzed. This restriction did not apply to pulses of heights corre sponding to channels larger than channel SO, however, which had only a few microseconds dead time per pulse. In order to correct for counting losses, this procedure was followed: The difference of counts in the gate scaler and the overbound scaler was formed. This difference was then divided by the count in the memory totalizer to yield the correction factor by which the spectrum in the pulse height analyzer was multiplied. The resultant spectrum was then quantitatively correct, as long as no pulses were lost in the pulse-height analyzer due to thresholds set in its internal circuitry. To ensure that, the low channel cutoff of the spectrum was always set above channel three by means of the attenuators in the fast sides of the circuit. The voltage-to-channel transfer characteristic of the system departs from linearity by less than 0.05 channels between the 50th and 16th channel, and by one channel between the 16th and the sixth. The differential linearity for the electronic system is shown in Fig. 2. This curve was obtained with a motorized sliding pulser. The fine structure (one channel high and the next low) was due to irregularities in the motion of the motor-driven poten tiometer slider which stuck momentarily in one chAnnel and then traversed the next at higher than normal speed in order to maintain the constant average velocity set by the synchronous motor drive. This mechanism explains the two channel correlations of Fig. 2. The output of cathode follower 1 was transmitted to the paralleled inputs of cathode followers 2a and 2b, whose outputs in turn supplied parallel branches of the succeed ing circuitry. These parallel branches were designed to accept and add pulses from two separate photomultipliers for the study, for instance, of gamma-ray cascades. When used as shown in Fig. 1, the gain of the adder circuit was reduced by a factor of two. The parallel arrangement was most useful for measuring the effect of small pulse pileup present at any given betatron intensity level. Pileup tests were performed by introducing, at one of the cathode follower inputs, standard flat-topped pulses from a pulser triggered at the peak intensity of the betatron photon pulse. The linear gate was opened only by these standard pulses. The photomultiplier was connected to the other cathode follower input. Since the pulses from the two branches were added before being presented to the linear gate, the observed pulse-height spectrum yielded a direct measure of pileup at the given beam intensity. When the x-ray intensity was reduced to zero, the standard pulses were all recorded in one channel in the analyzer. As the x-ray intensity was increased, the pileup produced by small pulses from the NaI(Tl) crystal caused the recorded pulses to spread into more than one channel. The effect of pileup was always a symmetric broadening with no 200 ..J ... Z z 150 .. :r: <.J II: ~ 100 '" ... z ::> 0 u 50 0 10 20 30 40 50 CHANNEL NUMBER FIG. 2. The differential linearity of the electronic system. The two channel correlations are due to imperfections in the pulser used for the test. shift of the peak position, because the pulses were rendered bipolar at the input to the linear gate. The broadening was always kept less than plus or minus one channel in experiments. III. THE PHOTOMULTIPLIER AND ASSOCIATED CIRCUITS Cathode follower 1 was in the same box which held the photomultiplier socket. The schematic diagram for this assembly is shown in Fig. 3. The divider chain for the photomultiplier was chosen so that the lens between the photocathode and first dynode would remain saturated over a large range of supply voltages. The focus electrode was connected to the first dynode, as this gave very nearly the optimum resolution for all the tubes tested. The last four dynodes were supplied from a separate high current, -420-v source, so that saturation effects in these dynodes were minimized and made independent of multiplier gain (at a fixed output voltage). The last dynode was grounded to eliminate one large bypass capacitor and the anode was direct-coupled to the cathode follower grid to eliminate the dc shifts associated with caflacitor coupling. The ca pacitors in the divider chain were so chosen that (to first approximation) the photomultiplier gain remained con stant if the often repeated pulse currents alter (sag) the voltages across the capacitors. This was made possible by forcing the redistribution of the fixed power supply volt ages so that the decrease of voltage across some stages was compensated by an increase across others, in such a way as to maintain constant gain. The time constant of the RC combination was on the order of a millisecond, allowing ample time to recover in the 16-msec interpulse interval of the betatron. The stray capacity at the anode of the photomultiplier, coupled with the impedance at the cathode follower grid, gave an integration time constant of 0.15 J-Lsec for the signal. The cathode follower was fed-back; with an output impedance of 20 ohms for negative pulses. The thermionic diode-catch in the grid limited the phototube output to This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48856 E. L. GA R WIN AND A. s. PENFOLD +300'1 6 AU 8 REGULATED -H.V • • 001 01 47K 37.51< 2.7K 2w 68K tCK5829 04 .001 OUTPUT 05 33K .001 .25 06 .001 •• 01 07 -420'1 .008 08 .10 .001 .1 820 .01 -50'1 09 .01 FIG, 3. The photomultiplier socket, cathode follower 1 assembly. 30 V to prevent severe overloading of succeeding circuits due to cosmic-ray pulses which were as big as 70 v. This diode is a subminiature tube and was wired directly into the circuit to reduce stray capacity. A thermionic diode was necessary because the impedance variation with tem perature of typical germanium diodes is large enough to have a I or 2% effect on the magnitude of the signal for the changes in ambient temperature encountered. Silicon diodes were too slow for this application (at the time of construction), as a fast recovery diode was required. To determine the linearity of the light-to-current trans- 100 1000 12.00 1400 PHOTOTUBE HIGH VOLTAGE FIG. 4. Results of a linearity test carried out with the light pulser and a typical photomultiplier tube. The parallel nature of the curves, and their spacing, indicate lack of saturation at the 24-v level. fer characteristic of the photomultiplier and associated circuitry, a light pulser was constructed from a type 5823. gas triode run as a relaxation oscillator. The light attendant to the discharge in the tube illuminated the photomultiplier, while the electrical breakdown pulse furnished a convenient coincident trigger signal. Many photomultipliers were tested with the light pulser, and a typical set of data is presented in Fig. 4. In Fig. 4, we have plotted the logarithm of the reciprocal of the incident light intensity (varied by the insertion of Kodak neutral density Wratten filters) versus the photomultiplier high voltage. This procedure was carried out for various output levels from the photomultiplier. The similarity of the curves indicated that any saturation effects were inde pendent of output voltage level, while the relative spac ings of the curves indicated that the filters attenuated by the amount obtained by calibration against a con tinuously variable density wedge. Because the curves of Fig. 4 are quite closely parallel for photomultiplier output voltages from 2.4 to 24 v, we concluded that there were no serious saturation effects present at those output levels. Indeed, some tubes were found to have no saturation effects at 50-v output. Saturation effects in the photo multiplier used in the experiment were not apparent at 30-v output levels. IV. CATHODE FOLLOWER 2 AND THE GAIN-OF-13 AMPLIFIER Figure 5 shows the circuit diagram of one of the power ful cathode followers employed to transmit pulses through 30 ft of RG-114/u cable (185-ohm impedance). This circuit was quite similar to that of cathode follower 1, This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48GAMMA-RAY SCATTERI"\G 857 6BC5 470 100M ( 10K IW' 220K 150K 0.1 15K 6BQ7A 6Cl6 1800 IW. 100 100 0.1 330K lOOK +300v. REGULATED FILAMENT +----._TRANSFORMER 0.1 CENTER TAP lOOK FiG. 5. The circuit for cathode follower two and the gain-of-13 amplifier. The plate supply voltage is regulated. with" the exception that the tubes in this circuit were capable of higher current operation. The 6BCS tube served as a fedback amplifier of gain 13. It is essentially a grounded-grid amplifier with feed back applied to the grid. This allowed the photomultiplier gain to be standardized against the 'Y rays from natural radioactive sources, Na22 in our case. The unfedback gain of this low input impedance amplifier was 60, which allowed a comfortably large feedback ratio. V. THE PULSE-SHAPING CIRCUITS All the pulse-shaping circuits were constructed on one chassis. The description given will be for one branch only. The differentia tor employed in the fast side was simply a 6AU8 tube connected as a fedback cathode follower3 (similarly to Fig. 3), with a 33-,uj.Lf output condenser which fed the junction of a shorted 6 ft length of RG-114/u cable and a length of similar cable leading to the impedance matched attenuator and thence through two Hewlett Packard wideband amplifiers to the anti-coincidence cir cuit. The anti-coincidence circuit was a fourfold coinci dence onefold anti-coincidence of the type described by Garwin.5 One of the fourfold inputs was switched out of circuit; the other was short-circuited and the associated switch used to start and stop data taking. Directly from the input to the differentiator, a 7S0-ohm resistor (to match the l80-ohm line to the 950-ohm delay cable) led to the slow pulse-shaping circuit shown in Fig. 6. The pulse shaping was effected by the carefully terminated RG-65/u delay line, as will be discussed. The shaped pulse is transmitted to the adder circuit by the 6AU8 cathode follower, to be added to the pulse from the other branch of the slow side of the circuit. The adder is a 6AH6 tube connected as in a "see-saw"6 5 Richard L. Garwin, Rev. Sci. Instr. 24, 618 (1953). 6 B. Chance, V. Hughes, E. F. MacNichol, D. Sayre, and F. C. Williams, W/71'eJnrms (~fcGraw-Hi1l Book Company, Inc., New York, 1949), p. 28. circuit. In the circuit of Fig. 6 the negative feedback applied through the l5-k resistor to the 6AH6 grid from the double 6CL6 cathode follower caused the grid to act as a ground point. The ratio of output from the cathode follower to the sum of the applied inputs was, therefore, the ratio of the feedback resistor value to the input resistor value, here a factor of three. The output polarity was positive, while the input was negative. This is a simple, fast rise time (30 mj.Lsec), highly fedback circuit, which has proved quite reliable. The small capacitor across the l5-k feedback resistor compensates the feed back for the capacity at the grid of the 6AH6. The variable capacitors across the 5-k resistors are adjusted until the output pulse shape is identical to the input pulse shape. The powerful cathode follower is required to drive the pulses into the 1900-ohm impedance formed by the 950- ohm shorted delay cable and the 950-ohm series terminat ing resistor. The output from the adder circuit was delayed by 0.2 j.Lsec in the 58 in. length of RG-65/u cable to center it in the 0.2 j.Lsec gate interval of the succeeding linear gate. This was required by the delays in the various amplifiers and trigger circuits in the fast sides. At the end of this 58-in. delay cable (at the input to the gate circuit) was a 58 in. length of shorted RG-65/u cable, which rendered the input pulse to the gate bipolar in nature. When the gate opened, it accepted only the positive part of the bipolar pulse; however, the bipolar nature of the pulses prevented baseline shifts due to input circuit condensers sagging while the gate was closed. The major pulse shaping was performed, as mentioned previously, by the 0.1 j.Lsec length of delay cable at the input to the cathode follower preceding the adder. This delay cable was not short-circuited, but rather terminated by a carefully chosen parallel combination of resistance and capacitance. The time constant at the photomultiplier anode was 0.15 j.Lsec, while the decay time of light from This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48858 E. L. GAR'vVIN .-\;\JD A. S. PENFOLD "A" Input Negotiv~ 750 r·OI To Differentiator "e" Input Negative r 750p.p. To Dlflerentiator 150k .01 L 29" I ... RG 65/U 200 As Above 6AU 8 2.2k 6 Ati.6 56k 1.5 -7 p.p. 58" RG65/U 960 Gel6 Gel6 220k Ik +300 A + e Out (Positive) 58"RG~ ..JlL 6" RG 114/U () To Linear Got. FIG. 6. The slow side shaping circuitry and adder. Only one branch of the shaping circuit is shown in detail, the other is indicated by the box. NaI(TI) was 0.25 J.Lsec. These parameters gave rise to a pulse which had the shape of the curve marked "input pulse" in Fig. 7. The remaining curves of Fig. 7 show the calculated response of a O.l-J.Lsec long shaping cable of characteristic impedance 950 ohms (RG-65/u), when pre sented with the input pulse shape indicated and termi nated by the parallel combinations of resistance and ca pacitance noted. The resistance values are in ohms and the capacitance values in micromicrofarads. It is obvious that one can do very much better than merely to short circuit the line. 2 ? '" :> ,.. ~ I :0 ~ ~ :I: '" W :I: t; Of--~~- ~ :::> Cl. L 'R'120 C'1240 .~~' ~~~~-<----.J o 0.2 0.4 0.6 0.8 1.0 TIME IN f'sec FIG. 7. The calculated response of a 950-ohm pUlse-shaping cable (of 0.1 !,sec length) to the input pulse indicated, when the cable is terminated by the parallel RC combinations indicated. Values of R are in ohms; values of C in Jl.jJ.f. Oscilloscope observation of the actual pulses indicated agreement with Fig. 7, and gave the following optimum values for the termination parameters: R= 200 ohms; C= 750 J.LJ.Lf. These values differ from the calculated ones somewhat, not only because of small deviations in the actual pulse from the assumed shape, but also because of the nonideal characteristics of the delay line which has significant attenuation. The amplitude of the shaped pulse was 30% of the amplitude which would have been attained for a I-J.Lsec long pulse, providing the resistance at the grid of cathode follower 1 were infinite, and the capacity the same as in our case. Since the major contribu tor to the lack of crystal resolution at high energies was not the smallness of the amount of light collected, this shaping scheme reduced pileup significantly without no ticeably increasing the width of the resolution function. VI. THE LINEAR GATE The linear gate circuit faithfully reproduced pulses with SO-mJ.Lsec rise times, had an open period of 0.2 J.Lsec, accepted positive input pulses, and provided negative out put pulses. It has been described in detail.4 VII. THE INVERTER Because the pulse-height analyzer required positive in put pulses, it was necessary to invert the output signal from the gate circuit. The circuit of the inverter is shown in Fig. 8. It was a single input "see-saw" circuit6 with a This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:486AH6 39K 47K 1.5-7p.p. SAU8 IN38A +300v REGULATED 220K lOOK 85<> .01 qf 10p.f . 150 50V .1 r-f 22K .14S•SK 2p.p. 10K .01 1% .01 loon 0 +OUT FIG. 8. The inverter circuit following the linear gate. The cathode follower is modified to deliver large positive output pulses. gain of two, similar to the circuit of Fig. 6. Here, however, it was necessary to have large positive output pulses and the 6AU8 was connected in a somewhat different manner. The triode section of the 6AU8 was used as a constant current load for the pentode section which became the cathode follower. The screen grid was capacitively coupled to the output, so that during the pulse the screen-cathode voltage was constant, thus increasing the maximum current available from the pentode. The diode from the positive supply to the screen grid assured fast recharging of the cathode-screen capacitor after the pulse had passed. Be cause the output impedance of this cathode follower was about 90 ohms, the series terminating resistor for the open circuited i8S-ohm cable leading to the pulse-height ana lyzer was made 100 ohms. The diode and capacitor at the inverter input stretched the pulse with a O.S-j.lsec decay time in order to reduce the requirements on the pulse height analyzer and thus assured improved long term reliability. VIII. THE BETATRON BEAM GATE The betatron beam gate allowed pulses from the anti coincidence circuit output to pass through to trigger the linear gate only during the time encompassing the x-ray pulse from the accelerator. The (negative) ejection current pulse from the betatron cut off a high gain triode in whose plate was a parallel RC combination which caused the plate voltage to rise with a time constant of 400 j.lsec. The plate was capacitively coupled to a Schmidt dis criminator, whose trigger threshold determined the open ing time of the betatron gate with respect to the beginning of the ejection pulse. The output of this discriminator triggered a phantastron which controlled the open time, variable from 25 to 500 j.lsec. The screen gate pulse of the phantastron actuated one side of a diode "and" gate, while the output from the coincidence circuit was con nected to the other side. The output from this gate led to a trigger circuit and thence to a pair of conventional cathode followers which fed the trigger input of the linear gate and the gate scaler. The circuit was conventional and does not require a figure. IX. THE INTEGRAL DISCRIMINATOR The integral discriminator shown in Fig. 1 was a diode with variable bias feeding a fixed threshold trigger circuit. The diode bias was set so that the discriminator fired at the level of the top of channel 50 in the pulse-height analyzer. X. THE PULSER The electronic gain of the system was monitored by means of a mercury switch pulser whose output was care fully shaped to match the output pulse from the photo multiplier. The repetition rate of the pulser was deter mined by a S823 gas triode in a relaxation oscillator circuit, which fed the 6J6 mercury reed driver at a frequency different from 60 cps. In this way, 60-cps pickup in the circuits would have appeared as a broadening of the channel edges in the pulse-height analyzer. The pulser , was arranged to deliver either positive or negative output pulses in order that it could also be used to test the pulse height analyzer directly. The electronic gain of the system was found to be stable to better than 1% over periods of several months. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48860 E. L. GARWIN A~D A. S. PEXFOLD a ...J W • Z lOll Z • ."a « • :::c • i U i .! i 0:: W Q. 0 C/) I- Z :::> 0 U 10 20 30 40 50 CHANNEL NUMBER FIG. 9. A typical pulse-height spectrum for the IS. I-Mev level in 02• The logarithm of the number of counts per channel is plotted versus channel number. The solid line indicates the extrapolation used to find the resolution function of the NaI(Tl) crystal at this energy. XI. PHOTOMULTIPLIER GAIN In order to monitor the gain of the photomultiplier, a spectrum of a radioactive source (Na22) was taken by connecting cathode follower 1 to the gain-of-13 amplifier shown in Fig. 2. The position of the peak (11% wide at half-height) was determined immediately before and after each experimental run, and the average taken to indicate the gain during the run. The largest variation of Na22 peak position over the 2-hr period of a run was observed to be 2%. Prior to determining a Na22 spectrum, the betatron was always operated for several minutes at the intensity to be used during the run, with the scattering target ("I-ray source) in place. This procedure was necessary because an increase in gain of the photomultiplier with high counting rate was observed. This amounted to 5% gain change at the highest counting rates employed in the experiment. After the betatron was shut off, the gain de cayed to its original value with a 1O-min half-life, allowing ...J W Z Z lI! « :I: U • •• • •••••• 1..Ii • a:: W Q. C/) l- Z :::> ---0 o U '. 00. 10 20 30 40 50 CHANNEL NUMBER FIG. 10. A typical spectrum from the radioactive source Na22, taken with the gain-of-13 amplifier. The logarithm of the number of counts per channel is plotted versus channel number. ample time to make the O.S-min Ka22 peak positioning run without significant change in gain. XII. TYPICAL PULSE-HEIGHT SPECTRA A typical pulse-height spectrum for the 1S.1-l\lev CI2 level is shown in Fig. 9. The logarithm of the number of counts per channel (corrected for counting rate losses) is plotted versus channel number. This was the result of a 2-hr irradiation of a t-g/cm2 scatterer in a 42-"Mev brems strahlung beam, with the counter at 135° to the beam direction. The solid line in the figure shows the extra polation used to find the resolution function of the NaI(Tl) crystal at this energy. There is evidence2 that this resolu tion function is a universal function of energy from 15 to 60 Mev. The sharp edge in the spectrum near channel 20 is due to photons associated with the scattering target. Figure 10 gives a typical Na22 spectrum for comparison with Fig. 9. Note that the positions of the Na22 peak (taken with the amplifier) and of the 1S.1-Mev peak nearly coincide. This was a considerable convemence m scattering experiments carried out on C12. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 146.189.194.69 On: Sat, 20 Dec 2014 16:13:48

1.1777022.pdf

Band Structure of the Intermetallic Semiconductors from Pressure Experiments William Paul Citation: J. Appl. Phys. 32, 2082 (1961); doi: 10.1063/1.1777022 View online: http://dx.doi.org/10.1063/1.1777022 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v32/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32, KG. 10 OCTOBER, 1961 Band Structure of the Intermetallic Semiconductors from Pressure Experiments* WILLIAM PAUL Division oj Engineering and Applied Physics, IIarvard Uni!'ersity, Cambridge, ,~f assac/msetis Three types of conduction band extrema in the (000), (100), and (111) directions in k space seem to determine many of the properties of the group 4 and group 3-5 semiconductors. Early experimental work on the pressure coefficients of the energy separations of these extrema from the valence band maximum energy, carried out on Ge (111), (000), (100), Si (100), and InSb (000), suggested that the three pressure coefficients might be independent of the specific element or compound in the group 4 and group 3-5 series. This work is discussed in detail, and the theoretical basis is briefly considered. All of the completed pressure measurements on these compounds are critically reviewed, and the correlation of unique pressure coefficients with specific band edges examined. It is demonstrated that pressure experiments can be planned to show up details of the band structure unavailable for study at atmospheric pressure. Particular attention is paid to GaP, and a new model for excess absorption occurring in n-type samples of this compound and in Si, GaAs, and AISb is suggested. The application of similar techniques to PbS, PbSe, and PbTe is discussed, and results of electrical and optical measurements of energy gap and electron and hole mobilities presented. I. INTRODUCTION THE earliest experiments on the effect of pressure on the electrical resistivity of Ge1a demonstrated that the minimum energy gap between conduction and valence bands increased linearly at low pressures; later work2 was interpreted to mean that at higher pressures this linear change decreased and possibly reversed sign, due to the increasing importance of some new set of states having a different pressure coefficient from those important at low pressures. In Si, pressure decreased the minimum energy gap.3a After theoretical and experimental work had identified the conduction and valence band "edges" in Ge and Si, investigation of Ge-Si alloys, interpreted by Herman, plausibly identified this new set of Ge states as a (100) conduction band set, possessing properties similar to the (100) set forming the conduction band extrema in Si.4 Quantitative studies showed that the energy gap between these (100) extrema and the (000) valence band extremum decreased at about the same rate in Ge * This work was supported by the Office of Naval Research. 1 (a) P. W. Bridgman, Proc. Am. Acad. Arts Sci. 79, 129 (1951); P. H. Miller and J. H. Taylor, Phys. Rev. 76, 179 (1949); J. H. Taylor, Phys. Rev. 80, 919 (1950); H. H. Hall, J. Bardeen, and G. L. Pearson, Phys. Rev. 84, 129 (1951). (b) D. M. Warschauer, W. Paul, and H. Brooks, Phys. Rev. 98, 1193 (1955); H. Y. Fan, M. L. Shepherd, and W. G. Spitzer, Photoconductivity Conference at Atlantic City, edited by R. G. Breckenridge, B. R. Russell, and E. E. Hahn (John Wiley & Sons, Inc., New York, 1956); W. Paul and D. M. Warschauer, J. Phys. Chem. Solids 5, 89 (1958); A. Michels, J. van Eck, S. Machlup, and C. A. ten Seldam, J. Phys. Chem. Solids 10, 12 (1959). 2 W. Paul, Phys. Rev. 90, 336 (1953); W. Paul and H. Brooks, Phys. Rev. 94, 1128 (1954). 3 (a) W. Paul and G. L. Pearson, Phys. Rev. 98, 1755 (1955), (b) M. I. Nathan and W. Paul, Bull. Am. Phys. Soc. 2, 134 (1957); W. Paul and D. M. Warschauer, J. Phys. Chern. Solids 5, 102 (1958); H. Y. Fan, M. L. Shepherd, and W. G. Spitzer, Photocanductivity Canference at Atlantic City, edited by R. G. Breckenridge, B. R. Russell, and E. E. Hahn (John Wiley & Sons, Inc., New York, 1956); L. J. Neuringer, Phys. Rev. 113, 1495 (1959); T. E. Slykhouse and H. G. Drickamer, J. Phys. Chern. Solids 7, 210 (1958). 4 F. Herman, M. Glicksman, and R. H. Parmenter, Progress in Semiconductors, edited by A. F. Gibson, P. Aigrain, and R. E. Burgess (John Wiley & Sons, Inc., New York, 1957), Vol. 2. and Si.5 This observation, emphasized by the un expected difference in sign of the effect for the (111) and (100) band edges6 in Ge and Si, naturally led to some speculation regarding the uniqueness of the association of pressure coefficient and type of band edge in the group 4 semiconductors. Subsequent experiments that qualitatively supported this associa tion broadened the extent of the speculation to include the group 3-5 and group 2-6 compounds and led to a cautious use of the correlation that does exist in the investigation of these materials.7 It is the purpose of this paper to examine the sources of this speculation, the extent of its success, and its prospects for continued usefulness. II. BASIS OF SPECULATIVE ASSOCIATION OF BAND EDGE PRESSURE COEFFICIENTS The basis for any association of a unique pressure coefficient with the energy gap between the common valence band maximum and a particular type of conduction band minimum is primarily experimental. Since lattice dilatation would seem to be a more fundamental parameter than pressure, we shall on occasion quote experimental results in terms of either pressure or dilatation, or both. Theoretical arguments are naturally based on dilatation, while greater corre lation exists among the pressure coefficients. The lattice compressibilities are different enough so that the one correlation does not imply the other. 5 M. r. Nathan, thesis, Harvard University (1958); Report HPI (1958); M. I. Nathan, W. Paul, and H. Brooks, Phys. Rev. (to be published); see references 2 and 3. 6 We shall frequently refer to a coefficient pertaining to the energy separation between the valence band and a band extremum X as the coefficient of the extremum X. However, this does not imply that we have any information regarding the absolute value or sign of the coefficient for the extremum X relative to the energy of an electron at infinity. This has particularly to be kept in mind when discussing "different signs of effects for different band edges." In some cases, notably germanium, the actual deformation potential for a particular band edge can be found. 7 A. L. Edwards, T. E. Slykhouse, and H. G. Drickamer, J. Phys. Chern. Solids 11,140 (1959); see reference 35. 2082 Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsBAND STRUCTCRE FROi\l PRESSURE EXPERIME:-.JTS 2083 TABLE I. Pro erties of group 4 and group 3-S compounds. Compou~ds contain!ng boron, ni~rogen, thall!um, and bismuth are not included Cond!ction band minima are labelled "speculative" ("spec.") If the type IS a systematIc extrapolat!O~ or based ~n a press,;!re coefficie~t, and are unlabelled if the type is considered assured through measurement of c:fclotron resonan~e, optIcal abs?rpt!On~ e~ectIve mass, etc. Temperatures of energy gaps are mixed; the room temperature gaps more eaSIly. allow compansons where hIgher mlmma are present Little attempt has been made to obtain the very latest values of the parameters III columns 1-3. On the other hand, columns 4--5 rep~esent our best present assessment of the pressure coefficients. Numbers in parenthesis are references. Lattice constant Energy gap Conduction (dE./dP)T (dE./d In V)T (ev) Compound (A) (25°C) (ev) band minima (ev /kg cm-2) C 3.567 5.3 (3000K) ~I(spec.) <10--6(31) Si 5.43 1.21 (OOK) III 0.66(3000K) LI Ge 5.66 0.803 (3000K) r2' 0.85 (3000K) III + 1.SX 10-6(3) 5XlO-6(1)(2) 12X 10-6(23) (25) (27) o to -2X 10-6(19)(20)(21) +1.5 -3.8 -9 o to +1.5 Sn 6.489 0.08(OOK) LI(spec.) 5:><10-6(31) AlP 5.47 3.1 (3000K) Il, (spec.) AlAs 5.66 2.16(3000K) III (spec.) -1.6X 10--6(35) AISb 6.10 1.6 (3000K) III (spec.) GaP 5.47 2.2 (3000K) D.I(spec.) { -1.7XlO-6(38) -1.8X 10--6(39) 2.6(3000K) r\(spec.) GaAs 5.66 1.53 (OOK) r\ -7 -9 {9.4X 10--6(7) 12X10--6(45) 1.89 (OOK) III (spec.) negative GaSb 6.10 0.81 (OOK) rl {16X 10--6(49) 12X 10-6(35) -9 -6.75 L\ ~5X 10-6(47) -2.8 Il\(spec.) negative (35) InP 5.9 1.34(OOK) r\ 4.6X 10--6(35) -6.15 InAs 6.07 0.36(3000K) III (spec.) -lOX 10--6(35) +7.45 -3.3 rl lnSb 6.49 0.2357 (OOK) 1\ Systematic Trends Since the band structures of the 2-6 compounds are less well determined, we confine our attention, for the moment, to the group 4 and group 3-5 materials. They are listed in Table I, along with selected data about them. We note that the minimum energy gap decreases as the average atomic number increases. The minimum gap is greater in a compound than in its isoelectronic group 4 element. The valence band structure is similar in all of the compounds, and will not further concern us. On the other hand, the lowest identified states in the conduction bands are of three types8: (1) at the (000) position in the Brillouin zone (f2' or rl), (2) along the (100) directions (~,), (3) along the (111) directions (LI). There appears to be some systematic trend of the relative energies of these three minima with average atomic number. Thus the ~, states are lowest in Si, are probably lowest in GaP, and perhaps also for diamond. The L, states are . lowest in Ge, where the atomic number is higher, and they appear to be close to the extreme position in GaSb and in gray Sn. The r, minimum is lowest in InSb and tends to be low for compounds of high average atomic number. The variation is not entirely systematic, but if we knew 8 The states r2' and r2.' in the diamond lattice become r, and r,., respectively, in the zinc blende. The III minima may shift to the Brillouin zone edge point X I, but for ease in writing, we shall refer to ~l states only. {5.SX 10--6 (S8) 8.5X 10--6(49) 4.8X 10--6(35) f15.5XlO-S(29) \14.2X 10-6(28) -5.1 -2.9 -6.7 -6.1 how to account correctly for electronic energy changes due to changes in ionicity (for want of a better term), it might well become so. Alloy Studies Where such examination is possible, alloys of group members show intermediate properties. However, strikingly nonlinear effects occur when the changeover in properties involves a change in conduction band extrema. An early example of this arose from the study of the optical energy gap in Si-Ge alloys.9 From 0 to about 15% Si the lowest conduction band minimum is of the Ll type, while from 15-100% Si, the ~, states form the extrema. Measurements of magnetoresistance9 have confirmed this interpretation, and have also shown that the mass ratio (and quite probably the masses) in the LI and ~l minima in the alloys are very close to their values in the pure substances. Although changes in lattice constant accompany changes in alloy com position, this is not the prime cause of the change in energy gap; thus, for example, the energy gap changes by about 0.15 ev between 0 and 10% Si content, whereas the gap change that would result from the change in lattice constant by this composition is only 0.05 ev. For the (100) minima, a decrease in the lattice constant through alloying increases the appropriate 9 See reference 4. Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2084 WILLIAM PAUL energy gap, whereas a decrease caused by pressure decreases it. Extrapolation from its variation in energy in Si-rich alloys is clearly useful in fixing the energy of the Al minima in pure Ge. A thorough study of the effects of alloying on all possible alloy systems of the group might establish a systematic behavior of the different minima with lattice constant and ionicity, and in fact, this seems a logical consequence of any systematic variation among the compounds them selves.IO This systematic behavior, however, does not have to extend as far as a relative constancy of energy shift with dilatation for anyone type of extremum, independent of substance studied, which is the question we are addressing ourselves to here. Many of the properties of the r2' or rl, AI, and LI extrema are similar in the different materials. The similarity in properties that depend only on the symmetry of the states is trivial. Less obvious are similarities in effective mass, yet the Al states seem to maintain the same mass ratio (and thus probably mass) between 100 and 15% Si in Si-Ge alloys, and analysis of pressure results in Ge involving higher minima, that are most probably of a Al type, requires an effective mass very similar to that of Si. The masses in all r2' or rl states are small, and the matrix element between the light hole states and the r2' or rl states appears to be constant for all members of the group,!l so that, neglecting spin orbit interaction, the mass vanes almost directly with the energy gap at k=O. Bond Theory Various authors have discussed the systematic variations of energy gap, carrier mobility, melting point, and hardness with lattice constant, average atomic number, and degree of ionicity.12 The basis for these discussions has been the connection between bond strengths, bond lengths, and atomic constitution. Extrapolations from such systematic variations have been rather successful, especially in predicting new semiconductors, but the theory used does not give the details of band structure and band interaction, and fails to explain the changes in effective mass and carrier mobility associated with changes in the extrema of the conduction band. To choose one example, the violent effect of pressure on the conductivity of extrinsic GaAs,l3 which depends probably on a change of band extrema, would be inexplicable in this "bond" theory. However, it is not our purpose to examine critically the explanations offered for this systematic behavior, but simply to recognize that systematic variations do exist, 10 Solubility differences might thwart this study in specific cases. 11 See references 43 and 53. 12 See, for example, H. Welker and H. Weiss, Advances in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1956), Vol. 3; O. Folberth and H. Welker, J. Phys. Chern. Solids 8, 14 (1959). 13 W. E. Howard and W. Paul (to be published); see reference 43. and that the explanation of some of them may be related to the pressure coefficient correlations that are the subject of this paper. We can examine the predictions of a bond theory for the pressure coefficient of an energy gap. Decrease in lattice constant implies decreased bond length and so increased bond strength. Since the energy gap is the difference in energy between a bonding and an anti bonding configuration, this implies an increased energy gap; the increase in energy gap in the Sn, Ge, Si, C sequence is consistent with this argument. In practice, a small decrease of the lattice constant produced by pressure may increase or decrease the energy gap, as is indicated by the opposite effects in Ge and Si. However, in both of these cases, the dielectric constant decreases with decreases in lattice constant,14 from which we infer that there is a general tendency for the separation between conduction and valence bands to increase. Thus the predictions of the bond theory probably correlate with the behavior of the average separation of conduction and valence bands, but do not describe well the behavior (induding pressure behavior) of the extrema of the bands, on which many of the critical properties depend. Band Theory Apart from some work of Parmenter on dilatational and alloying effects in Ge and Si,16 the author knows of no attempt, from a band theory viewpoint, to estimate the perturbation of band structure caused by changes in lattice constant. It would appear, nevertheless, that satisfaction of the known experimental behavior of different band edges with dilatation is not an unfair test for any theory pretending to compute the band structure ab initio. It has been argued that the valence and conduction band electrons in our group of com pounds move in an almost-free electron potential caused by the cancellation of the attractive periodic potential of the cores by a repulsive potential; this repulsive potential is introduced to simulate the result of orthogonalizing the valence electron wave function to the core wave function.16 To the extent that this reduces the importance of the details of the charge structure of the core and yields a band structure only slightly perturbed from that of the empty lattice, we should expect similar band structures in all of the compounds. In terms of the absolute energies, the fluctuations in energy of the r2', rl, AI, and LI minima are relatively small. The differences between the empty lattice energies and those in the actual lattice depend on the symmetry of the state considered. In general, one expects that states of s character will be raised in 14 M. Cardona, W. Paul, and H. Brooks, Solid State Physics in Electronics and Telecommunications, edited by M. Desirant and J. L. Michiels (Academic Press, Inc., New York, 1960), Vol. 1, p.206. 16 R. H. Parmenter, Phys. Rev. 99, 1759 (1955). 16 J. C. Phillips and L. Kleinman, Phys. Rev. 116, 287 (1959). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsBAND STRUCTURE FROM PRESSURE EXPERIMENTS 20S5 energy relative to those of p character, due to the action of the pseudorepulsive potential. Presumably, the larger interatomic separation of the compounds of higher atomic number can be regarded as part of a scaling operation involving the mean radius of the host ion. It is not clear what the effect of a change in lattice constant on the energies of the different states will be, but it is very probable that it will be different for states of different symmetries and it is not im plausible that it will be about the same for states of the same symmetry, especially if the constitutions of the cores are nearly the same. While this is encouraging, it is not apparent, even on a qualitative basis, what the relative changes will be. We might expect to obtain some clues by observing the changes in energy of the different states as we change from the empty lattice to the real one,17 as this is qualitatively similar to the effect of pressure in increasing the ratio of the size of host ion to unit cell (we regard the host ion core as relatively incompressible). Then we find that in Ge, the behavior of the r2', L1, and ~l state energies with respect to the r 25' state is not well reproduced; for example, the ~l state energy increases rapidly over the r26', opposite to the effect of decreasing the lattice constant. Similarities in the dilatational (or pressure) coeffi cients for any particular set of minima may depend partly on their occupying similar positions in the over all band structure. This is assured by the nature of the experiments performed, which usually examine minima fairly close to the absolute maximum of the valence band. The minima might have a different pressure coefficient if that could be measured when the minima were far from being the lowest conduction band states. The latter type of measurement has not been carried out but, in certain specific cases, it seems that it is feasible (see Sec. IV). These qualitative arguments are not, of course, the starting point of the present article, and are little more than an attempt to rationalize the experimental situation. Our conclusion is that there is no clear-cut reason why wide correlations of behavior under pressure of similar minima in the compounds should exist, even though experimentally it appears that the different types of minima can be identified by their pressure coefficient. We examine next the experimental situation. Experimental Basis Experiments carried out by Bridgmanl established that the conductivity of n-type Ge decreased rapidly at pressures above 12 000 kg/ cm2 and went through a minimum near 50 000 kg/cm2• Later work by Paul and Brooks2 on pure and impure material showed that this effect was probably due to the growing importance of a second set of minima in the conduction band. The investigation of Ge-Si alloys,4 interpreted by Herman, 17 F. FIerman, Revs, Modern Phys. 30, 102 (1958), suggested that it was very plausible that the new states needed were a ~1 set, which also formed the conduction band edge in Si. Smith's experimentsl8 on the drift mobility of electrons at pressures up to 30 000 kg/ cm2 established that the observed conduc tivity decrease was a mobility effect, not a carrier density variation. The analysis of Paul and Brooks and a more complete study by Nathanl9 concluded, inter alia, that the pressure coefficient, relative to a fixed valence band maximum, of the ~l states was between a and -2X10-a ev/kg cm-2• More recent work on magnetoconductance of n-type Ge at pressures up to 20 000 kg/ cm2 by Howard20 has shown this pressure coefficient to be -1.5 X 10-6 ev /kg cm-2• Optical absorption experiments of Slykhouse and Drickamer21 to pressures of 100 000 kg/ cm2 have shown that the energy gap of Ge passes through a maximum with pressure near SO 000 kg/ cm2 and that at higher pressures the decrease7 is at -1.2X10-6 ev/kg cm-2• To the author's knowledge, the symmetry of the higher states in Ge to which these pressure phenomena are attributed has never been directly established. The Harvard laboratory has been unable so far to reach pressures above 20 000 kg/ cm2 in nonmagnetic pressure vessels so that magnetoresistance measurements could be performed; probably the most feasible experiment would be a study of elastoresistance under pressure, following the methods described by R. W. Keyes and his collaborators.22 However, this is very likely only a question of experimental tidiness as the Si-Ge alloy work makes it almost certain that the additional states are of the Al type. The energy of these states, found from extrapolation to 0% Si content, is roughly 0.22 ev. Nathan, Paul, and Brooksl9 find from the pressure data O.lS±O.03 ev, Howard20 0.21±0.03 ev, and Slykhouse and Drickamer21 0.2 ev. The first measurements of the pressure coefficient of the r26/-~1 gap in Si, by Paul and Pearson,3& gave a coefficient of -1.5 X 10-6 ev /kg cm-2• Subsequent optical measurements by Paul and Warschauer3b gave -1.3XlO-6 ev/kg cm-2, and by Fan, Shepherd, and Spitzerb +5X10-6 ev/kg cm-2. Later Neuringer23 found results in agreement with Paul and Warschauer, and Slykhouse and Drickamer,2l in experiments to 140000 kg/cm2, determined a coefficient of -2X10-6 ev/kg cm-2. Quite different measurements by Nathan and Paul,3h on the change in ionization energy of gold impurity in Si, determined the gap change at low 18 A. C. Smith, thesis, FIarvard University (1958); Report HP2 (1958); Bull. Am. Phys. Soc. 3, 14 (1958). 19 FI. Brooks and W. Paul, Bull. Am. Phys. Soc. ], 48 (1956); Also see reference 5. 20 W. E. Howard, thesis, Harvard University (1961); Report HP7 (1961). 21 T. E. Slykhouse and H. G. Drickamer, J. Phys. Chern. Solids 7, 210 (1958). 22 R. W. Keyes, Advances in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1960), Vol. 11. 23 L. J. Neurin~er, Phys. Rev. 113, 1495 (1959), Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2086 \\7 ILL I AMP A U L pressures as -1.5 X 10-6 ev /kg cm-2• It, therefore, seems to be fairly well established that the Si gap decreases with pressure, and we shall assume the rate to be -1.S X 10-6 ev /kg cm-2. Thus we see that the behavior of the r25'-L1 gap and the r25'-~1 gap in Ge is qualitatively different and that there is quite close quantitative agreement between the coefficient for the r25'-~1 gap in Ge and the corresponding gap in Si. The agreement persists if we convert the pressure coefficients into dilatational coefficients by computing El = dEy/ d In V, as we see by examining Table 1. Second Experimental Basis The similar behavior of the ~1 states in Ge and Si under pressure form the first basis for our speCUlative scheme of pressure coefficients. The second is afforded by a comparison of the pressure coefficients of the r2' minimum in Ge and the r1 minimum in InSb. The pressure coefficient of the r26'-r2' energy separation in Ge was first observed in results of Fan, Shepherd, and Spitzer.1b They found an increased pressure coefficient (over that appropriate for the r26'-L1 separation) for that part of the absorption edge attributed to direct transitions. Later, oscillatory magnetoabsorption experi ments24 showed that the experiments of Fan et at. were carried out at energies just below the edge. Paul and Warschauer25 deduced a coefficient for the r26'-r2' gap by assuming that the r2' state was the only intermediate state in the indirect transition absorption into the Ll extrema, and using the expressions of Bardeen, Blatt, and Ha1l26 in analyzing the change of shape in their absorption edges with pressure. Neuringer23 also reported a coefficient for this edge, also at energies slightly below that corresponding to the onset of direct transitions. Still later, Cardona and Paup7 measured the direct transition absorption in very thin films well into the appropriate energy region. Their coefficient for the r25'-r2' energy separation, measured at different absorption levels, was between 1.2 and 1.3 X 10-5 ev/kg cm-2 which is in substantial agreement with all of the earlier determinations despite their different shortcomings. Although the reasons for this agreement are not established, two can be considered: (1) that the absorption at energies just less than the direct gap is due to direct optical-phonon-aided transitions, and thus has the same pressure coefficient as the direct gap, (2) that the spectral resolution of the early experiments was low, so that at a nominal energy setting less than 24 S. Zwerdling, B. Lax, K. J. Button, and 1. M. Roth, J. Phys. Chern. Solids 9, 320 (1959). 25 W. Paul and D. M. Warschauer, ]. Phys. Chern. Solids 5, 89 (1958). 26 J. Bardeen, F. J. Blatt, and 1. H. Hall, Photoconductivity Conference at Atlantic City, edited by R. G. Breckenridge, B. R. Russell, and E. E. Hahn (John Wiley & Sons, Inc., New York 1956). ' 27 M. Cardona and W. Paul, J. Phys. Chern. Solids 17, 138 (1960). the direct gap, some direct gap absorption spectrum was included. Long28 and Keyes29 measured the pressure coefficient of the resistivity of intrinsic InSb and deduced there from the coefficient for the energy gap. Keyes' value, determined to the higher pressures, was 1.SS X 10-5 ev /kg cm-2; Long's result was 1.42 X 10-5 ev /kg cm-2• Thus these two sets of results for the 1'2' and 1'1 states in Ge and InSb give coefficients two and a half times larger than that for the Ll states in Ge, and an order of magnitude larger, and with the opposite sign, than the coefficient for the ~1 states. The agreement for the dilatational coetlicients is however inferior (see Table I). It therefore appears that the coefficients are grouped-so far-into three sets; the first, appropriate to the r2' or 1\ state, of order approximately 2.S in arbitrary units; the second, for the Ll states,la,lb,2 of order 1; and the third of order -0.4. We shall not consider further the exactness of the agreement within anyone set. On the one hand, the pressure coefficients have experimental errors almost sufficient to allow exact agreement. On the other, we do not need exact agree ment for further progress, and so shall be content to keep a watching brief on the matter. III. SPECULATIONS AND RESULTS \Ve now speculate that the pressure coefficients of the energies of the r2' or rl, LI and ~I states with respect to the r 25' (or r 15) valence band maximum in all of the group 4 and group 3-S semiconductors fall into three groups and that the coefficients are close to the values 12.S X 10-6 ev /kg cm-2, S X 10-6 ev /kg cm-2, and - 2 X 10-6 ev /kg cm-2, i,e" in the ratio 2.S: 1: -0.4. For ease of expression we shall refer to the r25'-1'2', r25'-LI, and r26'-~I, energy gaps and pressure coeffi cients as r2', LI, and ~I [or less precisely as (000), (111), and (100)J gaps and coefficients. In this section we shall briefly review the knowledge or speCUlation regarding the conduction band minima in the com pounds, and the related pressure experiments. Diamond The conduction band extrema in diamond have not been clearly established. Systematic extrapolation from the Si band structure suggests either the ~l conduction band state or some other energy extrema not en countered in the group 4 and group 3-5 compounds of larger atomic number. Optical absorption data of Clark30 may be tentatively interpreted as indicating minima of ~I type. Regarding the pressure coefficient, there is one rather inconclusive piece of evidence. Champion and Prior31 report that they found no shift 28 D, Long, Phys, Rev. 99, 388 (1955), 29 R. W. Keyes, Phys. Rev. 99, 490 (1955), 30 C. D. Clark, J. Phys. Chern, Solids 8, 481 (1955), 31 F. C, Champion and J, R. Prior, Nature 182, 1079 (1958), Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsBAND STRUCTURE FROM PRESSURE EXPERIMENTS 2087 in the absorption edge to pressures (nonhydrostatic) of about 10000 kg/cm2, although their spectrometer was capable of detecting a shift of 1 A, or a coefficient of about 2 X 10-7 ev/kg cm-z. They also report a theoretical expectation of 10-8 ev/kg cm-2 without quoting a source for their theory. Whatever the theory, and whatever the nature of the experimental stresses applied by Champion and Prior were, it would appear that the pressure coefficient is small. We note that this agrees with our systematic extrapolation to AI-type minima. Silicon Only the Al states in the conduction band have been positively identified, and their pressure coefficient is one of the bases for our discussion. Gray Tin The band structure of gray tin is being actively investigated at atmospheric and elevated pressures. Magnetoresistance32 and pressure33 measurements both indicate complexity in the band structure, in which Ll minima are probably involved. Paul has reported34 a result of Groves on the variation of intrinsic resistivity with pressure to 2000 kg/cm2 at -40°Cj without mobility corrections, the energy gap is deduced to change at 5 X 10-6 ev /kg cm-2, precisely the coefficient for the LI gap in Ge. Measurements at lower tempera tures, however, give a temperature-dependent pressure coefficient of electron mobility which is possibly the result of a complex conduction and valence band structure. It is the author's view that Groves' coefficient will prove to be that of an LI gap, and that it will therefore agree very well with the LI coefficient in Ge. The low electron mobility in n-type Sn and the magneto resistance results support this contention. It is empha sized, however, that even if electrons in the LI states make the major contribution to the conductivity at -40°C, these are not necessarily the only low-lying conduction band states and they may not even be the lowest minima at this temperature or lower temperatures. Aluminum Phosphide, Arsenide, Antimonide Data pertinent to these relatively neglected sub stances are shown in Table 1. For technical reasons the conduction band extremum has not been investig~ted either by cyclotron resonance, magnetoresistance, or e1astoconductance, all of which could potentially identify the lowest states. Systematic extrapolation from Si might suggest that the ~l states are lowest in ali three. 32 A. W. Ewald (private communication). 33 S. H. Groves and W. Paul (unpublished measurements). .14 Prague International Conference on Semiconductors (1960). Optical absorption measurements under quasi hydrostatic pressure conditions to 50000 kg/cm2 have recently been carried out by Edwards and Drickamer31i on AlSb; they deduce a coefficient of -1.6Xlo-6 ev/kg cm-2 from the shift of the frequency with pressure for a fixed (low) absorption coefficient. If the Al states are indeed lowest, this coefficient would correlate very well with our chosen one. No pressure experiments on AlP or AlAs have been reported. Gallium Phosphide The gallium compounds have provided a rich harvest for pressure studies. Data for GaP are listed in Table I. The lowest conduction band minimum has not been identified by any of the conventional methods although systematic extrapolation from Si suggests one of Al type. Spitzer et a[.36 have reported two optical absorp tion edges, the lower at 2.2 ev and apparently an indirect transition, the higher nearer 2.55 ev, and probably a direct transition. Extrapolation from absorption studies on GaAs-GaP alloys37 suggests that the latter transition may involve a rl conduction band minimum. Edwards, Slykhouse, and Drickamer38 have measured the absorption spectrum to pressures of 50 000 kg/ cmz and have deduced from the slope of curves of hv versus P at constant absorption coefficient that (1) the lowest minima have a pressure coefficient of -1.7XlO-s ev /kg cm-2 (2) there is probably a higher minimum which produces direct transition absorption, and whose energy gap has a positive pressure coefficient. Thus, on the surface, these two studies qualitatively agree. However, the direct transitions in Spitzer's study supposedly occur above 2.55 ev and at absorption coefficients greater than 1000 em-I, while Edwards et al. estimate that the higher minimum is 0.1 ev above the lowest set, and that it contributes heavily to the absorption at atmospheric pressure for energies above 2.4 ev and absorption coefficients greater than 100 em-I. Severe discrepancies exist therefore in the interpretation of the edge. Our later discussion suggests that neither interpretation is well-established. Zallen39 has recently remeasured the GaP absorption edge over a wide energy range. At atmospheric pressure, his results agree within experimental error with those of Spitzer et al. At high pressures, he finds that the shape of the low-ertergy edge changes in such a way that graphs of hv versus P at constant absorption might be interpreted to give either sign of pressure 35 A. L. Edwards and H. G. Drickamer, Phys. Rev. 122 1149 (1961). ' 36 W. G. Spitzer, M. Gershenzon, C. J. Frosch. and D. F. Gibbs, J. Phys. Chern. Solids 11, 339 (1959). . ~7 H. Welker .and H. Weiss, Advances in Solid State Physics edited by F. Seltz and D. Turnbull (Academic Press, Inc., New York, 1956), Vol. 3. 38 A. L. Edwards, T. E. Slykhouse, and H. G. Drickamer, J. Phys. Chern. Solids 11, 140 (1959) . 39 R. Zallen and W. Paul (unpublished measurements). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2088 \V Ii L L I AMP A U L coefficient. This is simply explained on the basis of a few reasonable assumptions. If we ignore phonon energy terms, and assume that only one intermediate state is involved (which may not be strictly accurate), we find that the absorption coefficient for parity allowed indirect transitions is (1) where a is the absorption coefficient, A a combination of constants, Eg(P) the indirect gap, and tlE(P) the energy separation of the lowest state from the mter mediate state. Then, at fixed a, and close to P=O, Eg(P) = Eg(O), (aEg ) = (~(hP») -Chp-Eg(O)J(a IntlE) . ap T ap a ap l' (2) We can determine (a(hp)/ap)a and Jtp for various ct, and then plot the first quantity against the second. The resultant line has slope (a IntlEjaPh and intercept at (a(hp)/ap)a=O of Eg-(aEg ) / (~ntlE) . ap T ap T From (2), it is clear that if (aEg/aph is negative, and (a IntlE/aPh positive, then the sign of (a (hp)/ap)a can be either positive or negative. Carrying out the above procedure for measurements on a pure sample of GaP and assuming Eg(O) = 2.2 ev, Zallen finds (aEg/aph=-1.8X1O-6 ev/kg cm-2 and (1/ tlE) (atlE/ap) = l.4X 10-5 kg-l cm2• This pressure coefficient agrees very well with that of Edwards et at., although derived by a different pro cedure, and is very close to that for the gap in Ge and Si. If the intermediate state is indeed a 1\ state, and we assign it the appropriate pressure coefficient, we shall require tlE(O) """ 1 ev. This is quantitatively different from Spitzer's deduction; however, his experimental curves seem to us to be consistent with a second minimum at.a somewhat higher energy, and our assumption of a single intermediate state of large pressure coefficient naturally leads to a high value of tlE(O). Spitzer et at. also report an absorption peak near 4 JI., due to absorption in excess of the normal free carrier absorption. Similar excess absorption has been observed in Si,40 GaAs,41 and AISb.42 The explanation proposed by Spitzer et at. is that the absorption is due to transi tions between the two conduction bands that give the 40 W. G. Spitzer and H. Y. Fan, Phys. Rev. 108, 268 (1957). 41 W. G. Spitzer and J. M. Whelan, Phys. Rev. 114,59 (1959). 42 R. F. Blunt, H. P. R. Frederikse, J. H. Becker, and W. R. Hosler, Phys. Rev. 96, 578 (1954). W. J. Turner and W. E. Reese, Phys. Rev. 117, 1003 (1960). z 20- o iJj Ul ~ Ul Z <! cr: I- ~ 10 w u cr: W Q. 0.4 0.6 0.8 hv (ELECTRON VOLTS) FIG. 1. Transmission curves at two pressur~s for aGe-GaP sandwich, illustrating the small pressur~ coeflicle!lt of. the trans mission near 4 p.. The shift of the germamum edge III thiS measure ment is in agreement with that found in the work of Paul a~d Warschauer (reference 25). The magnitude of the Ge edg~ shift (not the energy gap change) is ~9XlO-6 ev/kg cm-'; I~ .the transmission minimum in GaP were due to a 4.1 -; r1 tranSitIOn, its shift would be about 14X 10--6 ev /kg cm-2 (Zallen and Paul, unpublished data). two parts of the main absorption edge. This mechani.sm is favored over excitation or ionization of deep lymg impurities. .. To digress for a moment, at least partlal resolutlOn of the source of the absorption should be afforded by photoconductivity measurements.. ~hotoionizati?n should give a distinct effect, photoexcltatlOn none, whIle free carrier interband transitions could yield slight photoconductance or photoresistance, dependi~g on .the mobilities in the two bands and the relaxatIOn tIme between bands. Measurement of the pressure coefficient of the two parts of the main absorption edge and of the excess absorption should unambiguously confirm whe~her interconduction band transitions cause the absorptIOn. Preliminary measurements by Zallen39 have not yet clearly established a coefficient for the higher minimum. Measurements of the transmission in the excess absorp tion region at pressures of 90 and 4250 kg/cm2 are shown in Fig. 1. The shift of the "bump" with pressure is certainly at a rate smaller than 10-6 ev /kg cm-2• If the minima involved were tll and rl, and if our postulates regarding the rl state are correct, the rate might be expected to be 1.4X1Q-5 ev/kg cm-2• It seems very unlikely, from Zallen's measurement, that tll ---+ rl transitions cause the absorption. We shall return to this problem in Sec. IV. Gallium Arsenide GaAs has been intensively studied recently by Ehrenreich,43 who has concluded that the lowest conduction band minimum is at the (000) position; presumably it is of the rl symmetry. Pressure measure- 43 H. Ehrenreich, Phys. Rev. 120, 1951 (1960). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsBAND STRUCTURE FROM PRESSURE EXPERIMENTS 2089 ments by Howardl3 and Sagar,44 Hall coefficient data, and data on GaAs-GaP alloys are analyzed to demon strate the presence of a second set of minima, of the Lli variety, roughly 0.35 ev above the r1 set. Free carrier absorption supposedly involving interconduction-band transitions suggest an additional set of minima only 0.25 cv above the lowest set,41 but as noted by Ehrenreich, the threshold for absorption into the Lli minima may be reduced by 0.10 ev in highly doped samples, due to filling of the lowest states of the rl minimum. Optical absorption measurements at high pressures by Edwards, Slykhouse, and Drickamer7 have been interpreted to give a coefficient for the rl gap of 9.4XIQ-6 ev/kg cm-~2. Sagar44 and Howardl3 have measured the extrinsic n-type conductivity as a function of pressure at room temperature. Howard's striking curve for this variation is shown in Fig. 2; the rapid resistivity increase, caused presumably by change-over of carriers from the r1 minimum to a higher set, continues beyond 30 000 kg/cm2• Ehrenreich assumed the coefficient of 9.4XIQ-6 ev/kg cm-2 for his analysis and deduced a coefficient of -1.4XIQ-6 ev /kg cm-2 for the higher set of minima. This coefficient would be in fine agreement with the value we postulated for the Lll minima. However, again there are disturbing features about the experimental situation. Unpublished measurements of Paul and Warschauer45 gave a coefficient for the rl gap of ",12XIQ-6 ev/kg cm-2• The difference between this value and that of Edwards et al. is significant for Ehren reich's interpretation, and might radically alter his conclusions regarding identification of the higher 32 28 24 16 12 8 4 GoAs n~type o LlA p,.13J1.em, R'230 em'leaul, R'P1800 emzlv·see A LLB p •. 27fl.·em °O~~~4~~~8--~~12~~1~6~~2~0~~24~L-2~a~ P(IO'K9/em") FIG. 2. Resistivity vs pressure for n-type GaAs, at room tempera ture (W. E. Howard and W. Paul, to be published). 44 A. Sagar, Westinghouse Research Rept. 6-40602-3~R1 (1959). 4. W. Paul and D. M. Warschauer (unpublished measurements). 4.0r----,---r--,---,-----r--,---" / 3.0 2.0 ~/ • n~type GoSb n_IO,s/em' Rcr -2600 em" lv-sec on-type Go Sb n ~ 3 x lOiS/em' Rcr -2280 cm2/v_ sec • p-type Ga Sb I.O,l-<>-------------------~-- o 4 8 12 16 20 24 P(IO' r.9 lem2) 28 FIG. 3. Resistivity vs pressure for n-type GaSb, at room tempera ture (W. E. Howard and W. Paul, to be published). minima.43 Either coefficient, however, lies close to the value for the same minimum in Ge. Furthermore, Edwards et at. determine a maximum gap near 60000 kg/cm2 whereas Ehrenreich's analysis predicts that the r1 and ,11 minima are equal in energy near 33000 kg/cm2• Beyond the maximum gap, Edwards et al. determine that the energy gap decreases at a rate of -8.7XIQ-6 ev/kg cm-2• More recent quasi-hydrostatic measurements46 to higher pressures seem to reach a plateau of resistivity a factor of 600 greater than that at atmospheric pressure. To our mind, the situation in GaAs is not adequately explained. The coefficient for the r1 minimum (which ever of the two determinations is nearer correct) agrees well enough with our postulates. However, in order to allow interpretation of the other data, this measurement should be repeated. We do not believe all of the higher energy minima have been accounted for, and suspect the positions of the r15 and the Ll minima should also be considered. Gallium Antimonide Sagar has reported47 measurements of resistivity, Hall effect, and piezoresistivity as a function of tem perature and pressure. His results confirm previous deductions that the lowest conduction band minimum is in the (000) position, and presumably of r1 symmetry. He finds that he can fit all his results reasonably well by assuming a second set of minima along the (t 11 ) 46 W. E. Howard (unpublished measurements). 47 A. Sagar, Phys. Rev. 117, 93 (1960); R. W. Keyes and M. Pollak,_Phys. Rev. 118, 1001 (1960). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2090 WILLIAM PAUL axes of the Ll type and by postulating that the de formation potentials for the r1 and Ll states are similar to those for germanium. Sagar's model has been con firmed in part by Keyes and Pollak.47 Since Dr. Sagar's paper in this volume analyzes the present situation regarding GaSb in detail, we need not go into it here. Special attention should be drawn, however, to the unpublished measurement of Howard,48 reproduced in Fig. 3, which shows that the resistivity increases sharply at pressures near 25000 kg/cm2. We suppose that a third set of minima is beginning to contribute; these states have not been positively identified, nor their pressure coefficient estimated from the electrical measurements. Edwards and Drickamer35 have reported measure ments of the change in the optical absorption spectrum with pressure. For pressures up to 18000 kg/cm2, they find a shift of the frequency for fixed low absorption coefficient of 12XIo--6 ev/kg Cn12, which confirms their previous measurement,38 and qualitatively agrees with earlier data of Taylor49 on the optical absorption coefficient, which gave 15.7 X 10-6 ev /kg cm-2. Thus the pressure coefficient for the r1 minimum is in the range of values we postulated. Edwards and Drickamer also report a change in slope of their curve of frequency versus pressure (at fixed absorption coefficient) which they attribute to a change in the extrema from the r1 type to extrema in the (111) direction, approximately 0.09 ev~,higher in energy at atmospheric pressure, and with a pressure coefficient of 7.3X1O-6 ev/kg cm-2• At 45000 kg/cm2 the pressure coefficient passes through a maximum, and then decreases, indicating the existence of a third set of minima. These observations are generally consistent with the experiments of Sagar and of Howard. We shall not, however, use their coefficient quoted for the (111) minima as we feel sure that it is hard to obtain from the published data, and moreover, in cases where indirect optical transitions are involved, shape changes of the absorption edge tend to give misleading coeffi cients in (hIJ)a versus P plots. The shape change we expect in this case will depend on (E/JlJO_ EPl)-2 where E/JOO and E}l1 are the energies of the (000) and (111) minima. This quantity is increasing rapidly with pressure above 18000 kg/cm2, which will tend to increase pressure coefficients estimated from iso absorption curves. Similarly high values50 (in our opinion) of the pressure coefficient of the Ll states in Ge were obtained by Slykhouse and Drickamer,21 Neuringer,23 and Fan, Shepherd, and Spitzer,3b by neglecting the possibilities of shape changes. Paul and Warschauer25 found that inclusion of such effects reduced apparent coefficients of 8XIo--6 ev/kg cm-2 " W. E. Howard and W. Paul (unpublished measurements). 49 J, H. Taylor, Bull. Am. Phys. Soc. 3, 121 (1958). 60 It should be remarked that this statement may be debated by all of these workers. The work by Paul and Warschauer is the only one where the necessity for this sort of correction is asserted. by about half, bringing the gap coefficient close to that determined from intrinsic resistivity. It would appear that similar corrections would bring the coefficient for the (111) states in GaSb close to our postulated one for the Ll type of extrema. Although no coefficient is given for the third set of states, important above 45000 kg/cm2, we note that the coefficient is negative, which is the distinguishing feature of the .11 type of extrema. Indium Phosphide Measurements of optical absorption51 and magneto resistance 52 show that the lowest states in this com pound lie at (000), and the effective mass is low, consistent with r1 states.63 The pressure dependence of the optical energy gap has been measured recently by Edwards and Drickamer,35 who find a coefficient of 4.6XIO--6 ev/kg cm-2. This coefficient is close to that we found for minima of the Ll type in Ge and GaSb and is therefore not in agreement with our postulates. The optica151 and galvanomagnetic52 measurements have also been interpreted to suggest the existence of higher minima. The interpretation of Edwards' measure ments gives a maximum gap near 40000 kg/cm2, and a decrease at a rate54 of -1OXIo-- 6 ev/kg cm-2 at high pressures. The latter decrease is far greater than we postulated for the .11 states; it implies that the higher minima are 0.7 ev54 higher in energy than the r1 states at atmospheric pressure. The pressure dependence of the n-type resistivity at room temperature has been measured by Sagar56 and by Howard,56 whose results qualitatively agree. Howard's measurements show that the resistivity of his purest sample increases linearly to 30 000 kg/ cm2 by 35%; this agrees roughly with the effects of changes in the elastic constants and in the effective mass of the r1 minimum. The separation of the minima implied by the optical work is consistent with the fact that there is no evidence of any effect of higher minima on the electrical resistivity a.t high pressures, but is inconsistent with the indications of the presence of higher minima shown by the galvanomagnetic and optical experiments at atmospheric pressure. Indium Arsenide Experiments on infrared cyclotron resonance 53.57 indicate that the conduction band structure of this compound has an extremum at (000), probably of the r1 type. 53 The pressure coefficient of the electron mobility and the energy gap were first measured, to 61 R. Newman, Phys. Rev. 111, 1518 (1958). 62 M. Glicksman, J. Phys. Chern. Solids 8, 511 (1959). 63 H. Ehrenreich, J. Phys. Chem. Solids 12, 97 (1959). 64 Our figures. 65 A. Sagar (private communication). See this volume. 56 W. E. Howard and W. Paul (unpublished measurements). 61 B. Lax, Revs. Modern Phys. 30, 122 (1958). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsBAND STRUCTURE FROM PRESSURE EXPERIMENTS 2091 2000 kg/cm2, by Taylor,58 who deduced a gap coefficient of 5.S X 10-6 ev/kg cm-2• Later optical measurements49 by the same worker were interpreted to give an optical gap coefficient of 8.5 X 1(}-6 ev /kg cm2• Recently published optical measurements by Edwards and Drickamer35 to SO 000 kg/ cm2 are interpreted to give an optical gap coefficient of 4.8X10-6 ev/kg cm-2 to 20000 kg/cm2 with a change-over to a coefficient of 3.2 X 1(}-6 ev /kg cm-2 bet ween 20 000 kg/ cm2 and SO 000 kg/cm2. Taylor's variation58 of electron mobility to 2000 kg/cm2 at room temperature has been confirmed by DeMeis,59 who has also extended the measurement to 30 000 kg/cm2, without any spectacular change of the coefficient. His results are thus qualitatively consistent with those of Edwards and Drickamer. We note that all of the coefficients quoted are considerably lower than the one we have postulated for r1 type minima. Indium Antimonide The pressure coefficients for lnSb have been discussed already. These measurements have been extended to 30000 kg/cm2 in the Harvard laboratory, where it has been confirmed that there is no evidence for phenomena attributable to minima higher than the lowest (r1) one at atmospheric pressure. IV. SUMMARY AND PROSPECT We must now examine the agreement between our speculative pressure (or dilatational) coefficients and experiment. In doing so we should weigh most heavily data on identified conduction band minima, by proved methods on good samples, and from several laboratories. Minima of A 1 Type From the last two columns of Table I, we see that the correlation of the pressure coefficients for .11 minima is good. All of the pressure coefficients are negative, and, except for the single measurement at very high pressures on InP, in rather good agreement. The six measurements (on C, AISb, GaP, GaAs, GaSb, InP) confirming our basis (Ge, Si) are all on minima labelled "speculative," but it seems hardly possible that all could be in error. The discrepancy for the .11 minima in InP found by Edwards and Drickamer is not, in our view, serious. The optical experiments to extremely high pressures of Drickamer and his collaborators are very informative, but need very careful interpretation. The technique is such that the fine details of the structure of absorption edges, and their variation at pressures below 20 000 kg/cm2 may not be resolved.36 Nevertheless, the experiments are superb in delineating the gross behavior of the extrema of the band structure, and giving guide lines for more detailed experiments. The interpretative 58 J. H. Taylor, Phys. Rev. 100, 1593 (1958). 69 M. DeMeis (unpublished Harvard measurements). method usually used is to determine the change in frequency for fixed absorptioll coefficient. We empha ~ize, as we did under Sec. III in discussing GaP and (iaSb, that this may lead to quite erroneous results in the case of indirect transition absorption. For InP, if the pertinent intermediate state for transitions into the .11 state is the r1 state, the shape change will be like that in GaP, leading to either sign of pressure coefficient for the .11 states. The negative pressure coefficient, when measured, will be too small. That of Edwards and Drickamer is already an order of magnitude too large. Changes of deformation potential and compressibility with volume have not been considered here, yet they clearly cannot be omitted. Such changes make it surprising that we get any correlation at all with Drickamer's coefficient at pressures above SO 000 kg/ cm2• If we agree to restrict our examination of correlations to pressures below 30 000 kg/ cm2, a reasonable (but still possibly high) limit for constancy of compressibilities and deformation potentials, the agreement among the .11 coefficients is excellent. Minima of L 1 Type The correlation in Sn and GaSb is adequate. Further identification of the minima in gray 8n is required, and confirmation of the coefficient in GaSb. Minima of r 2' or r 1 Type In GaAs and GaSb, the correlation is adequate, while in In As and InP it is certainly inadequate. However, on examining both the electrical and optical experi ments,35.49,58 it seems to us that at least two effects have not been considered, both of which alter the observed coefficient. (1) The transitions in InAs and InP are presumably from r16 to r1 and are allowed. Thus the absorption coefficient can be shown to be, under certain limiting conditions on the oscillator strengths, a=------- nhv where 11 contains constants only, and n is the refractive index. If we assume Ey~hv, the band edge shape depends on m*t/n, m will increase with pressure, n will decrease, and the slope of the edge will become steeper. As a result, if the energy gap is increasing, isoabsorption plots will give a spuriously low coefficient. (2) In semiconductors of low electron effective mass, n-type samples can be degenerate at room temperature. The increase of mass with pressure tends to reduce the degeneracy by lowering the Fermi level in the conduc tion band, and decreasing the photon energy required for the transition. This also reduces the apparent pressure coefficient of the energy gap. Crude estimates of the corrections necessary because of (1) and (2), Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2092 WILLIAM PAUL which depend on the actual conditions of (he experi ment, indicate that they are considerable. The samples used by Edwards and Drickamer are not too well characterized, so that we cannot estimate what corrections are required. However, for InAs, they report a decrease in the slope of the absorption curve by 50000 kg/cm2 by a factor of about 6. We do not see any explanation for this, but it seems quite apparent that isoabsorption plots cannot give coefficients of the energy gap in such circumstances. Summary In summary, we believe a large degree of correlation exists. It is not evident whether or not there is a systematic slow trend in the coefficients for anyone minimum with starting energy gap or ionicity. The correlation of the dilatational coefficients is, if any thing, poorer than that for the pressure coefficients. We have no explanation to offer for this. Obviously, experiments to check and confirm the coefficients of Table I should be carried out. It would also be advisable if other energy separations could be measured under pressure. Chief among these are the "vertical" energy gaps supposedly investigated in the reflection spectra.60 From such measurements we might expect further correlations, and also we might gain information on the rigidity of the whole valence band structure under dilatation. We have said very little about the 2-6 or 1-7 com pounds, although these have been investigated in tensively by Drickamer and his co-workers, and by others. We believe it is too early to say, either from an experimental or theoretical standpoint, whether much correlation is to be expected. Space limitations forbid our considering the present evidence at any length. However, it is certainly valid to remark that we should, on the basis of our experience with the group 4 and group 3-5 compounds, look for internal correlations among the 2-6 compounds themselves, or indeed among any group of compounds that are similarly derived. Experience shows that different band edges very probably have different deformation potentials so that the pressure experiments have great value in sorting out effects in complex structures. The experiments on Ge, Sn, GaP, GaAs, GaSb, and InP demonstrate this fact, which is independent of the presence of correlated pressure coefficients, although correlation improves the technique. A few illustrative examples of investigations that are based on a knowledge of the pressure coefficients can be given. Already published are experiments on hot electrons6l and tunnel diodes.62 Near infrared free carrier absorp- 60 H. R. Philipp and E. A. Taft, Phys. Rev. 113, 1002 (1959); J. C. Phillips, J. Phys. Chern. Solids 12, 208 (1960). 61 S. H. Koenig, M. I. Nathan, W. Paul, and A. C. Smith, Phys. Rev. 118, 1217 (1960). 62 M. 1. Nathan and W. Paul, Prague International Semi conductor Conference (1960). tion involving interband transitions is a suitable example for our present purpose. Thus we have already demonstrated that the "extra" absorption in GaP36 is probably not caused by a ~l to rl transition. We intend to extend this examination to Si,40 GaAs,4l and AISb.42 Our investigation is based on the supposition63 that the excess absorption observed by Spitzer and Fan in Si is caused by vertical ~l ---> ~2' conduction band transitions. In the intermetallics, the degeneracy at X 1 splits into Xl and X 3. The minima mayor may not be on the zone boundary; the splitting has been calculated to be '" 1.4 ev for BN64 but may be smaller. Whether or not the minima are on the boundary, the transition from ~l to ~l (or Xl to X3) is allowed, whereas it is disallowed in Si. We speculate that the absorption in GaP and AlSb is caused by such transitions, and will give a small pressure coefficient, which can be tested. It is perhaps significant that no absorption of this type has been seen in Ge, GaSb, InP, and InAs; these materials do not have ~l states lowest, and have no close minima vertically above the Ll and rl states. A disturbing feature is the observation of excess absorp tion in GaAs. It is not inconceivable, however, that this could be caused by rl ---> rl6 transitions. Again, we can suggest that the pressure measurement may be definitive. As a second example of the use of pressure measure ments in illuminating semiconductor properties, we wish to report our recent measurements on the series PbS, PbSe, and PbTe. We have been concerned in examining several aspects of their behavior: (a) any similarity in the conduction or valence band structures and scattering mechanisms, as shown by similarity in ¢ tJ n-PbS-E2 ot 196°1( 4.0 o tJ n-PbS-E2 at 296°1( 9.3 '" Ii n-PbSe H at 196°1( .48 A d n-PbSe H at 296°1( 1.2 o d p-PbSe G at 196°1( 4.1 0.6 o d p-PbSe F at 296°1( 13.7 v <I p-PbTe -C2 at 196°1( 3.5 v <I p-PbTe-C2 at 296°1( .16 o 2 :3 4 5 6 7 8 POO' Kg/em') FIG. 4. Resistivity vs pressure, at 196°K and 296°K, of extrinsic samples of PbS, PbSe, and PbTe (L. Finegold, M. DeMeis, and W. Paul, unpublished data). 63 W. Paul (to be pUblished). 64 L. Kleinman and J. C. Phillips, Phys. Rev. 117,460 (1960). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsBAND STRUCTURE FROM PRESSURE EXPERIMENTS 2093 00 1.9 1.8 1.7 1.6 Log,O _ 1.5 IA .. 1.3 1.1 To 2.9 2.8 • ..,v:;- 2960 K .. " ... , . • 0 <f 316°K~ Iyi 0 OD6-Ja3 306°K~ .y • -£2 ~6r:fO 296°K .0 a • o'b . 270 4 6 8 10 12 14 16 18 20 22 24 P(IO' Kg/em2) FIG. 5. Resistivity vs pressure for a near intrinsic (Ja3) and impure n-type sample (E2) of PbS, at 296°K (L. Finegold, M. De Meis, and W. Paul, unpublished data). the pressure coefficients of the extrinsic conductivity; (b) any nonlinearity with pressure of the coefficients for the extrinsic mobility which may be evidence of complexity in the band structure; (c) any dependence of the pressure coefficients of the extrinsic mobility on temperature or impurity concentration, which gives us clues concerning the presence of different scattering mechanisms, particularly impurity scattering, and of multiple extrema having different pressure coefficients; Cd) the pressure coefficient of the energy gaps, which allows us to separate the explicit and implicit (volume) contributions to the temperature coefficient of the gap; (e) the changes with pressure of the position and shape of the optical absorption edge, which gives similar information to (c) and (d). Part of this program has been completed, the rest continues. Thus, we are unable to distinguish much qualitative difference in the pressure coefficients of the extrinsic conductivity of n-type PbS, n-and p-type PbSe and p-type PbTe shown in Fig. 4, although the magni tudes of the effects are different. We find nonlinearity in all of these compounds, but neither the magnitude nor the nonlinearity is sufficient in itself to establish a complex band structure. Thus, the size of the pressure coefficient of p-type PbTe at low pressures can be ex plained on the basis of a simple band structure, if we assume that the mobility is (roughly) quadratically de pendent on the effective mass, and the effective mass changes are given by the pressure coefficient of the energy gap we find for PbS. The nonlinearity is similarly accounted for. This evidence is not conclusive, though, since we can think of combinations of circumstances that would obscure complexity in the band structure. We are at present investigating these combinations (see below). From Fig. 4 we see that there is no change in the pressure coefficient of extrinsic n-type PbS and n-type PbSe between 196° and 296°K. This is consistent with the absence of all scattering except lattice scattering, and suggests that, if the band structure is complex, the pressure coefficients of its parts must be similar. Other wise, we should require that the additional extrema do not contribute much to the conductivity. A difference in the pressure coefficients of p-type PbSe and p-type PbTe at 196° and 296°K is found; for PbTe this is consistent with the model of a two-band valence band that has been suggested previously.60 The result for PbSe suggests that its valence band structure is similar. We find measurable differences in the pressure coeffi cients for samples of n-type PbS of different impurity concentration at 196°K, but have done insufficient measurements to draw firm conclusions from these. We have determined the pressure coefficient of the energy gap of PbS from both measurements on photo conductivity and absorption spectrum under pressure, and on the intrinsic resistivity. The optical measure ments carried out by Prakash66 gave a coefficient of 8 X 10-6 ev /kg cm-2, and some evidence (which has to be confirmed) of a change of shape of the absorption edge with pressure. Figure 5 shows a determination of the variation of the intrinsic resistivity with pressure at several temperatures.67 The variation of the resistivity at low pressures is due primarily to mobility effects, but the rapid decrease of gap makes the sample more intrinsic at high pressures. If the curves for the different temperatures are corrected for the variation of the mobility of the electrons, determined at 23°C on an impure sample, and if (in lieu of such data for p-type PbS, our material being insufficiently impure) the correction for hole mobility is taken to be the same as that for the electrons, then a gap coefficient of 7 X 10-6 ev /kg cm-2 is obtained. The optical and electrical determinations are thus in fair agreement. One im portant consequence is that the deduced volume contribution to the temperature coefficient of the energy gap is only + 2.2 X 10-4 ev rK, assuming a volume thermal expansion coefficient of 6X 10-'rK and a compressibility of 1.9X 10-12 cm2/ dyne. This is to be com pared with a total temperature coefficient of +4X 10-4 ev rK, and implies a positive explicit effect of tempera ture on the energy gap. This seems to us to be incon sistent with any of the present theories for this effect of the electron-phonon interaction. ACKNOWLEDGMENTS I am happy to acknowledge stimulating di!!!cussions on the subject of this paper with Professor Harvey 6. J. R. Burke, Jr., B. B. Houston, Jr., and R. S. Allgaier, Bull. Am. Phys. Soc. 6,136 (1961); R. S. Allgaier, this volume. 65 V. Prakash and W. Paul (unpublished measurements). 67 L. Finegold, M. DeMeis, and W. Paul (unpublished measurements ). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2094 WILLIAM PAUL Brooks, Dr. H. Ehrenreich, Dr. W. E. Howard, and Dr. G. Peterson. The measurements on GaP were carried out by Mr. R. Zallen and on the lead salts by Dr. L. Finegold and Mr. M. DeMeis. All of us are grateful to Mr. J. Inglis and Mr. A. Manning for necessary machine work and to Mr. D. Macleod for fashioning the samples used in the optical and electrical investigations. The samples of GaP measured in the new data reported were generously given us by the Monsanto Chemical Company and by Dr. W. G. Spitzer. For the PbS samples, we are indebted to Professor R. V. Jones of Aberdeen University and Dr. W. D. Lawson of the Radar Research Establishment; for the PbSe samples, to Dr. W. D. Lawson and Dr. A. C. Prior of R. R. E. and Dr. A. Strauss of Lincoln Laboratory, and for the p-type PbTe samples, to Dr. W. W. Scanlon of Naval Ordnance Laboratory. JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32, NO. 10 OCTOBER. 1961 Energy Band Structure of Gallium Antimonide* W. M. BECKER, A. K. RAMDAS, AND H. Y. FAN Purdue University, Lafayette, Indiana Resistivity, Hall coefficient, and magnetoresistance were studied for n-and p-type GaSb. The infrared absorption edge was investigated using relatively pure p-type, degenerate n-type, and compensated samples. Infrared absorption of carriers and the effect of carriers on the reflectivity were studied. The magneto resistance as a function of Hall coefficient for n-type samples at 4.2°K gave clear evidence for a second energy minimum lying above the edge of the conduction band; the energy separation is equal to the Fermi energy for a Hall coefficient of 5 cm3/coulomb. The shift of absorption edge in n-type samples showed that the conduction band has a single valley at the edge, with a density of-state mass mdl =0.052 m. By combining the results on the edge shift, magnetoresistance, and Hall coefficient, it was possible to deduce: the density-of-states mass ratio mdjmdl = 17.3, the mobility ratio ~2/~1=0.06, and the energy separation 1l=0.08 ev between the two sets of valleys at 4.2°K. Anisotropy of magneto- I. INTRODUCTION INFORMA TION on the band structure of GaSb has been obtained from various investigations. Roberts and Quarrington1 found that the intrinsic infrared ab sorption edge extrapolated to 0.704 ev at 2900K and 0.798 ev at 4.2°K and had a temperature coefficient of -2.9X1O-4 ev;oC in the range 100o-290°K. The shape of the absorption edge led the authors to suggest that either the minimum of the conduction band or the maximum of the valence band is not at k=O. Ramdas and Fan2 attributed the absorption at high levels to direct transitions but found a temperature dependent absorption tail indicative of indirect transitions. They reported also effective mass values obtained from in frared reflectivity measurements: me= 0.04 m and mh=0.23 m. From studies of the resistivity and Hall coefficient in the intrinsic and extrinsic temperature * Work supported by Signal Corps contract. 1 V. Roberts and J. E. Quarrington, J. Electronics 1, 152 (1955-56). 2 A. K. Ramdas and H. Y. Fan, Bull. Am. Phys. Soc. 3, 121 (1958). The value of hole effective mass reported was in error and should have been mh=0.23 m. The experimental data used are shown in Fig. 8. resistance, observed at 300oK, showed that the higher valleys are situated along (111) directions. The infrared reflectivity of n-type samples can be used to deduce the anisotropy of the higher valleys; tentative estimates were obtained. Infrared reflectivity gave an estimate of 0.23 m for the effective mass of holes. The variation of Hall coefficient and transverse magnetoresistance with magnetic field and the infrared absorption spectrum of holes showed the presence of two types of holes. Appreciable anisotropy of magneto resistance was observed in a p-type sample, indicating that the heavy hole band is not isotropic; this was confirmed by the infrared absorption spectrum of holes. The results on the absorp tion edge in various samples seemed to indicate that the maximum of the valence band is not at k=O. However, it appears likely that transitions from impurity states near the valence band produced ahsorption beyond the threshold of direct transitions. ranges, Leifer and Dunlap3 deduced EG(T=0)=0.80 ev, me=0.20 m and mh=0.39 m. Zwerdling et aZ.4 ob served magneto-optical oscillations in the intrinsic infrared absorption which indicated that the absorption at high levels corresponded to direct transitions. By attributing the oscillations to Landau levels in the con duction band, an electron effective mass m.= 0.047 m was obtained. Sagar5 studied the temperature and pressure dependences of the Hall coefficient of n-type samples. The results were explained by postulating a second band with a minimum above the minimum of the conduction band. The second band was assumed to have minima along <111) directions by analogy with germanium, and piezoresistance effect was observed which supports the suggestion that the band has many valleys. Assuming the valleys to have the mass parame ters as in germanium, Sagar estimated a density-of states ratio of 40 and an energy separation of 0.074 ev at room temperature between the two conduction bands. The two-band model has since been used by other authors to interpret measurements on resistivity 3 H. N. Leifer and W. C. Dunlap, Jr., Phys. Rev. 95, 51 (1954). 4 S. Zwerdling, B. Lax, K. Button, and L. M. Roth, J. Phys. Chem. Solids 9, 320 (1959). 5 A. Sagar, Phys. Rev. 117, 93 (1960). Downloaded 31 Aug 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1735738.pdf

On the Neutron Bombardment Reduction of Transistor Current Gain J. W. Easley and J. A. Dooley Citation: Journal of Applied Physics 31, 1024 (1960); doi: 10.1063/1.1735738 View online: http://dx.doi.org/10.1063/1.1735738 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Defect-driven gain bistability in neutron damaged, silicon bipolar transistors Appl. Phys. Lett. 90, 172105 (2007); 10.1063/1.2731516 Thermally switched superconducting weak-link transistor with current gain Appl. Phys. Lett. 76, 2295 (2000); 10.1063/1.126325 Current gain reduction by 0.45 eV oxygen related level in graded bandgap AlGaAs baseemitter of heterojunction bipolar transistors Appl. Phys. Lett. 59, 3407 (1991); 10.1063/1.105690 Resonant tunneling hotelectron transistor with current gain of 5 Appl. Phys. Lett. 49, 1779 (1986); 10.1063/1.97242 Current gain of the bipolar transistor J. Appl. Phys. 51, 5030 (1980); 10.1063/1.328384 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.42.202.150 On: Sun, 23 Nov 2014 18:03:49JOURNAL OF APPLIED PHYSICS VOLUME 31. NUMBER 6 JUNE. 1960 On the Neutron Bombardment Reduction of Transistor Current Gain* J. W. EASLEY Bell Telephone Laboratories, Inc., Whippany, New Jersey AND J. A. DOOLEY Wright Air Development Center, Wright-Patterson Air Force Base, Ohio (Received January 5, 1960) Detailed measurements of the fast-neutron and gamma-ray bombardment behavior of germanium-alloy transistor current-gain have been obtained concurrent with exposure. These data indicate that previously reported analyses, which lead to a linear dependence of common-base current-gain on fast neutron exposure, yield a good approximation for the npn device, but are not of general validity for the pnp germanium transis tor. The extent of departure from the linear approximation depends on the width and conductivity of the base-region and can be appreciable in many cases of practical interest. For the pnp germanium transistor it is necessary to take additional account of both changes during bom bardment of the minority-carrier recombination rate at bombardment introduced and initially present recombination centers and changes in the width of the collector junction depletion layer. Observed bom bardment curves are in good agreement with an analysis which includes these effects. From initial slopes of the current-gain bombardment curves, values of the product of fast-neutron-exposure times minority carrier-lifetime at bombardment introduced recombination centers are 9.7X107 for 2.7 ohm-em p-type and 14.2X107, 6.0X107, and 1.3 X 107, for 3.6 ohm-em, 1.2 ohm-em, and 0.2 ohm-em n-type germanium, re spectively, in units of sec-neutrons/cm2• INTRODUCTION IT has been reported1•2 on the basis of analysis and experiment that the common-base current gain of the transistor decreases linearly with fast neutron ex posure. Although this is a good approximation for silicon and npn germanium transistors/ our experimental data show that it is not of general validity for the pnp ger manium transistor. The extent of departure from the linear approximation depends on the width and conduc tivity of the base-region. This departure can be appre ciable in many cases of practical interest. vides a good fit to the observed bombardment curves for the current gain of both pnp and npn devices. For the pnp germanium case it is necessary to con sider two effects which result from any appreciable bombardment induced change in the net donor concen tration of the n-type4 base region. The first is the de crease in recombination rate at bombardment intro duced or initially present recombination centers as the Fermi-level moves toward the lower half of the band gap; and the second is the change of effective base width, for a given applied collector voltage, as the col lector depletion layer widens. Consideration has been taken of these two effects in an analysis appropriate to the germanium-alloy struc ture. The alloy-transistor affords a considerable sim plification in both analysis and experiment and although the following arguments will be quantitatively most exact for this structure the qualitative results apply to other structures. The application of this analysis pro- * This work was supported by the Air Research and Develop- ment Command, United States Air Force. 1 J. J. Loferski, J. Appl. Phys. 29, 35 (1958). ANALYSIS The case of a uniform conductivity base-region alloy transistor will be considered herein for simplicity. The symbols employed are defined in the Appendix. It can be shown that the injection efficiency and the collector multiplication factor are relatively insensitive to bombardment arid the argument is therefore un changed by considering 'Y=O:*= 1. Therefore, the current gain5,6 can be written as: W2 SA8W W2 0:(<1»=1-------, 2DTo DA. 2DTb (1) where the effective base width, W = W(<I», is not a con stant but is a function of the bombardment exposure. The quantity Tb is the minority carrier lifetime of bom bardment introduced recombination centers and de pends not only on the density of these centers but also depends, as does TO, on the position of the Fermi-Ievel,1 which is modified with bombardment. In the subse quent analysis it is assumed that the surface recom bination velocity is not altered by bombardment. The validity of this assumption is discussed in a later section. A very simple expression results, if, following Webster,6 it is assumed that surface recombination occurs pre dominantly in an annular region about the emitter of 2 G. C. Messenger and J. P. Spratt, Proc. LR.E. 46, 1957 (1958). 3 J. W. Easley, IRE-WESCON Conv. Rec. 3, 149 (1958). 5 W. Shockley, Bell System Tech. J. 28, 435. (1949). • J. W. Cleland, J. H. Crawford, and J. C. Pigg, Phys. Rev. 6 W. M. Webster, Proc. I.R.E. 42, 914 (1954). 98, 1742 (1955). 7 W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952). 1024 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.42.202.150 On: Sun, 23 Nov 2014 18:03:49REDUCTION OF TRANSISTOR CURRENT GAIN 1025 width W. Then the quantity (i-a) can be expressed as (i-a) (<I»= [(l-ao)+ W02](W)2, 2DTb Wo (2) where the zero subscripts denote initial unirradiated values and Wand Tb are the exposure dependent variables, since dD/d<I><.<dTb/d<I>. The value of effective base width is given by (3) where d is the geometric width of the base-region and X m (<J» is the width of the collector depletion layer. For constant applied voltage, the depletion layer width varies during a bombardment because the net donor acceptor concentration N1(if» depends on exposure. This variation can be expressed and determined experi mentally in ~erms of the collector capacity since" Xm(if»=4.3X103(V /N1)!= 1.44[Ac/C c(<I»] cm, (4) where V is the collector voltage in volts, Ac the area of the collector junction in cm2, C(if» the collector ca pacity in J.l.J.I.f, and the numerical coefficients are appro priate to germanium. In expression (2) the change in recombination rate at initially present centers has been neglected because the relative large values of TO in ger manium alloy transistors permit the neglect of these changes in the second term of expression (1). In transis tors exhibiting initial low values of TO, such that at exposures of interest TO and Tb are of the same order of magnitude, this approximation may not be valid. Em ploying a first order approximation for this change in rate of recombination at bombardment introduced re combination centers, the quantity Tb can be written as Tb= (a+bif»/<I>. (5) Relations (2), (4), and (5), in conjunction with values for the constants a and b and the empirical relationships of Cleland et al.4 for N1(<I», provide a description of the current gain bombardment curve applicable to either the pnp or npn device. For cases in which bif>«a and W(if»~Wo, Eqs. (1) and (2) reduce to the previously reported form W02 a(<I»=ao---4> 2Da (6) by which a decreases linearly with exposure. These con ditions are satisfied to good approximation for p-type base-regions but not in general for those of n-type caused by the greater relative change in net carrier concentration for equal exposure and generally larger values of b for the latter material. EXPERIMENTAL PROCEDURE Bombardment curves of current gain have been ob tained from measurements of (i-a) concurrent with irradiation. A fission plate source at the Brookhaven National Laboratory reactor was employed to provide a mixed flux of neutrons and gamma rays. A C060 source was employed separately to approximately duplicate the gamma dose-rate encountered in the mixed spec trum and thereby determine the effect of the gamma-ray component. The effect of the gamma component on current gain behavior was determined to be negligible. Various lots of transistor samples were prepared at the Allentown branch of Bell Telephone Laboratories with 'as uniform characteristics as possible except for a variation in base-width within each lot. For npn devices, a single lot with base material of 2.7 ohm-cm average resistivity was employed. For pnp devices, three lots with base material of 3.6, 1.2, and 0.2 ohm-cm average resistivity were employed. These values of initial base resistivity were determined from collector and emitter junction capacity and area measurements. Values of initial base widths employed in all calculations were determined indirectly from frequency cutoff measure ments for each unit. For these determinations, values of the diffusion constant D as reported by PrinceS were employed. Direct measurements from subsequent cross sections of approximately 10% of the devices yield values of base width which are on the average 15% less than those obtained indirectly from electrical measurement. Values of (i-a) were obtained from small signal measurements and recorded on a Leeds and Northrup recorder with a duty cycle of 8% per device, ten of which were employed per bombardment. The calibra tion of the complete measurement channel was similarly monitored. The resultant bombardment curves and the calibration record, which each comprise more than a hundred data points, are, therefore, essentially continu ous. The maximum experimental error in the measured value of (1-a) is estimated as approximately 3% so that the measurement uncertainty in a is of the order of a few tenths of a percent. The emitter bias for all measurements was approximately 0.3 amp/ cm2 so that any correction to the low current level approximation, implicit in Eq. (1), should be small. The temperature of the transistors was monitored throughout the bombardments by a thermocouple soldered to the case of one of the ten devices measured during a bombardment. The case of these devices was in good thermal contact with the collector. The recorded temperature was constant to within a few" tenths of a degree centigrade throughout the bombard~ents. The values of fast neutron exposure employed herein were determined from the measured change in conduc tivity of approximately 2 ohm-em n-type Ge exposed with the transistors in each bombardment. The ob served conductivity change was converted to a value of "fast neutron flux," employing the data of Cleland et al.4 Values of fast neutron flux for neutrons of energy 8 M. B. Prince, Phys. Rev. 92, 681 (1953). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.42.202.150 On: Sun, 23 Nov 2014 18:03:491026 J. W. EASLEY AND J. A. DOOLEY FIG. 1. Representative bombardment curves for five npn ger manium-alloy-transistors of 2.7 ohm-cm resistivity base-material. The values of frequency cutoff are 7.2, 5.8, 5.0, 2.4, and 1.4 Mc for units 7,9, 11,3, and 2, respectively. The solid lines each consist of a smooth curve drawn through the experimental points. greater than 0.1 Mev obtained by threshold-reaction activation techniques indicate that the fast neutron exposure determined and specified in this manner is approximately 60% of that determined from the con ductivity alteration method. The values obtained from conductivity alteration are employed herein as a con venience in correlating data with that of the ORNL group.4,9 DATA AND DISCUSSION Representative bombardment curves of current gain vs fast neutron exposure for the 2.7 ohm-cm p-type base transistors are shown in Fig. 1. The curves shown have been selected to include units over the range of base width available in the sample lot. The approximations 4.10-15 NPN Ge-ALLOY 2.10-15 PB~ 2.7 Jl.-CM , N ::Ii <.> I 1.10-15 z 0 '\ .', • •• ),. a: ... . :::> ~ ... z Gle 4.10-16 <1<1 '~~ ... '\ . " 2.10-16 '\ '\ '\ 2 4 6 8 10 20 fa. MEGACYCLES PER SECOND FIG. 2. Values of the initial slope of the bombardment curves versus frequency cutoff for 2.7 ohm-cm base-material npn transistors. 9 O. L. Curtis, Jr., J. W. Cleland, and J. H. Crawford, Jr., J. Appl. Phys. 29,1722 (1958). of Eq. (6) are valid and an approximately linear a-<I> relationship is observed. The small departure from linearity is attributed to annealing of damage during the bombardment. Since the frequency cutoff of the current gain5 is inversely proportional to W2, Eq. (6) predicts a slope of the a vs <I> curve inversely propor tional to frequency cutoff for any given base material, injection level and junction temperature. Values of the observed initial slope vs frequency cutoff for the 27 samples measured of this lot are shown in Fig. 2. The dotted line is that determined via Eq. (6) employing the average value of the constant a= Tb<l> computed from the measured data. Values of a determined from the initial slopes of the bombardment curves and cor responding to the pre-bombardment base-material re sistivity of the four sample lots are tabulated in Table 1. The departures of individual points from the average is most likely on account of random errors in the fre quency cutoff measurements and departures in device structure from the model on which Eq. (1) is based. Any variations in base resistivity among samples of a lot also result in a displacement of a point perpendicular to the average line and contributes to the spread. TABLE 1. Values of the product Tb'P=a corresponding to various prebombardment values of germanium base-material resistivity as determined through Eq. (6). Resistivity a Base ohm-cm sec cm--2 p-type 2.7 9.7±1.3X107 n-type 3.6 14.2±2.1X107 1.2 6.0±0.9X 107 0.2 1.3±0.3XI0 7 Representative bombardment curves of current gain vs fast neutron exposure for the 1.2 ohm-cm n-type base transistors are shown in Fig. 3. As in Fig. 1, the curves have been selected to include units over the range of base width available in the sample lot. A distinct cur vature is observed for these n-type base region units in contrast to the approximately linear curves for the p-type base units. Values of WND of Eq. (2) cover a similar range for the two cases. For the exposure dura tions employed, the a vs <I> curves for the units of smaller W 0 exhibit a minimum, gradual increase and subsequent sudden rapid decrease. The point of initiation of this rapid decrease corresponds to "punch-through" or con tact of the collector depletion layer with the emitter. The approximations of Eq. (6), as discussed previously, are not valid for pnp units and Eq. (2) must be em ployed. The initial or approximately zero exposure value of the slope is to good approximation given by lim -"'---(.1a) W02 <1'-+0 .1<1> -2Da' (7) and consequently the relationship between initial slope, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.42.202.150 On: Sun, 23 Nov 2014 18:03:49REDUCTION OF TRANSISTOR CURRENT GAIN 1027 frequency cutoff and the constant a is equivalent to that for the npn devices discussed previously. Values of the observed initial slope vs frequency cutoff for the 38 samples measured of the 1.2 ohm-em resistivity n-type base material are shown in Fig. 4. The dashed line is determined as in Fig. 2 and the consideration of dispersion of points is equivalent. Bombardment curves obtained from ten samples each for equivalent devices of 3.6 and 0.2 ohm-em n-type base material are quali tatively similar to those shown in Fig. 3 except for those differences expected from the different initial resistivi ties and the associated differences in recombination rates at bombardment introduced recombination centers. In the devices of 3.6 ohm-em base material the slopes of the a vs <I> curves are substantially less and punch-through occurs at an earlier point in the exposure history. In the devices of 0.2 ohm-cm base material the 0.98 0.96 0.94 0.92 C:l 0.90 0.88 0.86 0.84 0.82 ~ ______ .... _,E-5 ""... ...... . ~---="':..:"'::-:..-- 0-9 • . ..,." ... -. 0-1 0.80 '----'-_-'-_..L-..-J-..L---'-_....L..._J.-..-J_---'----' o O.~ 1.0 1.5 2.0 2.~ 3.0 3.5 4.0 4.5 5.0 q. (FAST NEUTRONS/ CM2 x 10-14) FIG. 3. Representative bombardment curves for five pnp ger manium-alloy-transistors of 1.2 ohm-cm resistivity base-material. The values of frequency cutoff are 9.8, 6.0, 4.7, 2.7, and 2.3 Mc for units E-5, D-9, D-l, C-3, and C-4, respectively. The solid lines are those calculated from Eqs. (1) and (5) as outlined in the text. slopes of the bombardment curves are substantially greater. For values of base width available in these samples, a is drastically reduced (i.e., to the order of 0.5) by exposure in which the term b<I> is small com pared with a [see Eq. (5)J and consequently only a small amount of curvature is observed. For this initial resistivity, punch-through is not observed for exposures of the order tlf magnitude obtained. From Eqs. (1), (4), and (5), values of the constants a and b of Eq. (5) required to fit each of the 38 measured bombardment curves of the 1.2 ohm-em base material devices have been determined. The calculations are not sensitive to the relative values of TO and S. An estimated value of TO equal to 10 .usee was employed and the cor responding value of S obtained from the preirradiation values of (l-a). The average value of the constant a employed to fit the curves was 6.0±0.9X 107 sec neutrons/ cm2 and was determined from the initial slope N 2 <.) 'j z 0 II: .... J '" Z ale <1<1 4,10-15 2,10"'15 ~, " 1,10-1$ 4,10-16 2>10-16 1,10-16 I PNP Ge -ALLOY PB=I.2n.-CM '. ~~ . .... . . . I~· ~" I .... ~, " , " , 2 4 6 8 10 20 40 fa, MEGACYCLES PER SECOND FIG. 4. Values of the initial slope of the bombardment curves versus frequency cutoff for 1.2 ohm-cm base-material pnp transistors. in the v~cinity of zero bombardment, as previously mentioned. The average value of b required to fit the curves was 2.5±0.4X 10-7 sec. The fit obtained from these constants begins to depart from the observed curves in the latter portion of the exposure, as shown in Fig. 3. This arises in part from a less rapid decrease in Tb with <I> than is given by the first-order approximation of Eq. (5). A portion of this departure is also possibly caused by the effect of departures from uniform plan arity of the collector and emitter interfaces. These values of curve fitting parameters can be com pared with values obtained independently from other data. Values of Tb<P products obtained from initial slopes of the bombardment curves for three sample lots and from other sources9•10 are plotted versus equilibrium electron concentrations in Fig. 5. The value of a for any 10 9 [J [J 0 A o DATA OF THIS PAPER A CURTIS ET AL [] WALTERS 10"5 ELECTRONS/CM3 A C 0 FIG. 5. Values of the product 1'b<fl = a versus equilibrium electron concentration for n-type germanium from the data of Curtis el al.,9 Walters,lO and that reported in this paper. 10 A. E. Walters, private communication. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.42.202.150 On: Sun, 23 Nov 2014 18:03:491028 J. W. EASLEY AND J. A. DOOLEY net donor-acceptor concentration is equal to the cor responding Tbell product. Values of d(Tbell)/dNr can be obtained from these data and employed with the values of dN r/ dell reported by Cleland et al.4 to yield a value of b. The value of b obtained in this manner is equal to approximately 1 X 10--7 sec. This value corresponds to a partially re duced donor concentration at an exposure value such that the term b<P has become appreciable with respect to a. The average value of a required to fit the curves is in good agreement with the data of Curtis et al. and Walters. Although the corresponding agreement in b values is not as good, it is considered reasonable in view of the approximations involved. The justification of the assumption that surface re combination velocity is a constant, implicit in the pre ceding analysis, is based on correlation between the results of analysis and observed behavior, and on the post-irradiation re-etch of devices. The npn bombard ment curves and relation (1) require that any variation in surface recombination velocity be either small or varying approximately linearly with bombardment. The latter appears unlikely as the bombardment alteration of other surface sensitive parameters which have been measured is generally observed to saturate early in a bombardment. The bombardment behavior of current gain has been observed to not be sensitive to variations in initial surface conditions and ambients. Post-irradia tion re-etch and subsequent measurement of current gain indicates that the bombardment alteration of the surface recombination velocity is negligible after any appreciable exposure. CONCLUSIONS The bombardment curves of current gain versus in tegrated fast neutron flux for the npn germanium tran sistor are represented to good approximation by relation (6), which indicates a constant rate of decrease of current gain with fast neutron exposure. This linear relationship, however, is not of general validity for the germanium pnp device. For the pnp germanium transis tor, it is necessary to include consideration of both changes during bombardment of the minority-carrier recombination rate at bombardment introduced and initially present recombination centers and changes in the width of the collector junction depletion layer. A good approximation to the major portion of the ob served current gain-exposure curves can be obtained through expressions (1), (4), and (5). The constants required to fit the curves are in reasonable agreement with independently determined macroscopic electrical properties of the semiconductor material. ACKNOWLEDGMENTS The authors wish to express their appreciation for the assistance of W. P. Knox and G. Kiriokos in the prepa ration of experimental apparatus and Mrs. A. K. Knoop for assistance in the considerable amount of data reduc tion and numerical calculation. APPENDIX Symbols Employed in Text a Common-base short circuit current gain 'Y Injection efficiency a* Collector mUltiplication factor W Effective base width of transistor, i.e., distance in base from interfaces of emitter and collector de pletion layers d Width of base as measured between discontinuities in net donor-acceptor concentrations at emitter and collector X m Width of collector depletion layer TO Initial minority carrier lifetime in base region Tb Minority carrier lifetime in base region associated with bombardment introduced recombination centers D Diffusion constant for minority carriers in the base S Surface recombination velocity A 8 Effective surface recombination area A e Emitter junction area A c Collector junction area ell Fast neutron exposure, i.e., integrated fast neutron flux N I Net donor-acceptor concentration in base Cc Collector capacity a Constant defined by Eq. (5) and equal to the product Tbell for any given resistivity b Constant defined by Eq. (5) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.42.202.150 On: Sun, 23 Nov 2014 18:03:49

1.1743110.pdf

GasPhase Oxidation of Ammonia by Nitrogen Dioxide Willis A. Rosser Jr. and Henry Wise Citation: The Journal of Chemical Physics 25, 1078 (1956); doi: 10.1063/1.1743110 View online: http://dx.doi.org/10.1063/1.1743110 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/25/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ammonia decomposition on Ru(001) using gas-phase atomic hydrogen J. Vac. Sci. Technol. A 16, 984 (1998); 10.1116/1.581282 Gasphase reactivity of sulphur ions with ammonia J. Chem. Phys. 99, 9664 (1993); 10.1063/1.465499 Gasphase electron paramagnetic resonance absorption in nitric oxide. II. The effects of nitrogen15 and oxygen17 and 18 substitution J. Chem. Phys. 64, 3097 (1976); 10.1063/1.432644 Gasphase solvation of the ammonium ion in ammonia J. Chem. Phys. 62, 4576 (1975); 10.1063/1.430331 Kinetics of the gasphase reaction between ozone and nitrogen dioxide J. Chem. Phys. 60, 4628 (1974); 10.1063/1.1680953 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.26.31.3 On: Mon, 22 Dec 2014 09:37:181078 LETTERS TO THE EDITOR Fermi Interaction Terms in C1?0216 and N 214016t HARALD H. NIELSEN AND K. NARAHARI RAO Department of Physics. The Ohio State University. Columbus. Ohio (Received September 7. 1956) RESULTS reported earlier from this laboratoryl.2 on some of the carbon dioxide bands at 15 jJ. enable us to evaluate a few other constants pertaining to the Fermi diad 10°0 and 02°0 of the C120216 molecule. The object of these evaluations is first to examine how the Fermi interaction term of C1202I6 calculated from these data compares with that derived by Dennison3 and secondly, to see how the CO2 interaction term differs from that in the nitrous oxide molecule. The relevant theoretical formulas are ~O [B,ooo-B o2oo]pe,tu,bed = (-0'1+20'2) (~o2+2K2)t approx, (1) where (2) ~ being the separation between the perturbed levels 10°0 and 02°0, ~o, the separation for no perturbation and K the Fermi interaction term which is (Wv'1) in Herzberg's notation. The remaining sym bols are as described by Herzberg.' Since (0',+20',) is not affected by Fermi resonance, the B values for the levels 10°0 and 02°0 reported earlier2 give (0'1+20'2)= -0.21 5XlO-S cm-1. Moreover, the measurement 1 of v, of CO2 gave rise to (B01'o(d) -Boooo) = 0.410 X lO-s cm-I= -0'2PR. This value of 0'2PR combined with qOl0 = 0.65 X lO-s cm-1 quoted by Herzberg and Herzberg' gives 0',= -0.73,XlO- s cm-1 since 0'2= -(qolo/2)+0'2 PR. The values of (0'1+20'2) and 0'2 thus derived lead to (-"1+20'2) = -2.72,X 10-s cm-I. Equations (1) and (2) can now be solved for ~o and K be cause [Blooo-Bo2oo]p",tu,bed=-0.27,XlO-s cm-I and ~=102.76 cm-1 are measured quantities. From these calculations we arrive at ~o= 10., cm-I and K = 72.3 cm-I. Similar processing of the data2 on (VI +VS) and (2v,'+vs) of 020216 gives ~o' = 16.7 cm-I and K = 71.1 cm-I. These values for K agree very well with that derived by Dennison. s Rao and Nielsen6 have recently remeasured and discussed the VI and 2V2' bands of nitrous oxide by following the same procedure as outlined above and deduced that ~o= 100±3 cm-1 and K =42±3 cm-I for the N214016 molecule. In other words, in nitrous oxide, although the separation of the unperturbed levels of the Fermi diad 10°0 and 02°0 is nearly ten times the separation between similar levels in the carbon dioxide molecule, the Fermi interaction term in N20 is comparatively quite appreciable. t Supported in part by the Office of Ordnance Research. U. S. Army. through contracts with the Ohio State University Research Foundation. 'Rossmann. Narahari Rao. and Nielsen. J. Chern. Phys. 24. 103 (1956). 'Rossmann. France. Narahari Rao. and Nielsen. J. Chern. Phys. 24. 1007 (1956). 3 D. M. Dennison. Revs. Modern Phys. 12. 175 (1940). • G. Herzberg. Infrared and Raman Spectra of Polyatomic Molecules (D. Van Nostrand Company. Inc .• New York. 1945). 'G. Herzberg and L. Herzberg. J. Opt. Soc. Am. 43.1037 (1953). 'K. Narahari Rao and H. H. l\ieIsen. Can. J. Phys. (to be published). Gas-Phase Oxidation of Ammonia by Nitrogen Dioxide* WILLIS A. ROSSER. JR.t AND HE"RY WISEt Jet Propulsion Laboratory. California Institute of Technology. Pasadena 3, California (Received August 30. 1956) THE reaction between N02 and NHs is conveniently di vided into two temperature regions. At temperatures below the melting point of NH,NOs, NO, reacts with NHs to form NH,NOs.'·2 At higher temperatures (>6000K) the course of the reaction changes and neither NH,NOs nor its principal decomposi tion produce (N,O) are observed. The present experimental measurements in the temperature range 600 to 8000K were carried out in a static system identical to that described in reference 3. The reactants NHs and N02 were introduced separately into the reaction vessel. Initial concentra tions of (N02) were varied from 10-8 to 10-7 mole/cc, and initial ammonia concentrations from 10-6 to 10-5 mole/cc. The progress of the reaction was followed by continuous measurement of the optical density of the nitrogen dioxide in the reaction vessel.s After a time sufficient for mixing of reactants, the rate of disap pearance of N02 was found to follow the rate expression -d(N0 2)/dt= k1(N02) (NHs). (1) As shown in Fig. 1, the temperature variation of the specific rate '0 \ 0 • 1'\ '\ \ ,\ . ~~ : . ~ 10 .... '\ '\ r'l9" \ '. , 1\ Po-v :\ \ \ 1.20 .... 1.40 1.50 1.60 1.10 liT (OK. 10') FIG. 1. Specific rate constant as a function of temperature. constant kl corresponds to the Arrhenius expression kl = lO'2.7e-27500/RT cc/mole sec. 'AO (2) As the reaction proceeded in a Vycor reaction vessel, the ap parent value of kl decreased. However, in the presence of a surface coating of KCI or of additional surface (Pyrex beads in the uncoated Vycor vessel) this time variation of kl disappeared. Although under these conditions Eq. (1) was obeyed throughout the reaction, the rate constants were lower than those obtained from initial rates in an uncoated vessel. Similarly, with the addi tion of NO to the reaction mixture the rate of disappearance of N02 was reduced and kl was independent of time. At 6400K an initial concentration ratio (NO)/(N0 2)=8 resulted in a fourfold reduction in rate. The reaction products were found to include nitrogen, nitric oxide, and water. Analysis for N2 and NO based on fractional distillation of the product mixture followed by catalytic com bustion indicated that approximately equal molar concentrations of NO and N, were formed: (NO formed) / (N02 consumed) = 0.53, (N2 formed) / (N02 consumed) = 0.45. Therefore, within the experi mental error, the sum of the concentrations of NO and N2 is equal to the concentration of N02 reacted. The ratio (N2 formed)/(NO. consumed) increased with the addition of NO to the reaction mix- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.26.31.3 On: Mon, 22 Dec 2014 09:37:18LETTERS TO THE EDITOR 1079 ture from 0.45 for (NO/N0 2)i=0 to 1.6 for (NO/N02)i= 15 and appeared to approach the value 2 corresponding to the limiting stoichiometry (3) The rate expression, Eq. (1), suggests a bimolecular reaction between NH3 and N02 as the initial step. However, the decrease in the rate constant with time and its dependence on surface imply complexity in the mechanism. In addition, the inhibition by NO suggests that the over-all reaction includes some chain character. These observations may be interpreted in terms of an initiating reaction (4) followed by a chain reaction involving NH2• The amino radicals may also be destroyed by wall reactions or by reaction with NO, a reaction product (2HN02 = NO+ N02+ H20). The accumulation of NO during the course of reaction causes a reduction in the concentration of NH2 radicals essential to the propagation of the chain. Consequently, the rate constant kl decreases with time. The reaction4 between NO and NH2 produces N2 according to the stoichiometry (5) which is in accord with the observation that the total number of moles of NO and N2 produced in the over-all reaction equal the number of moles of N02 reacted. * This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory under Contract No. DA-04-495-0RD 18, sponsored by the Department of the Army, Ordnance Corps. t Present address: Department of Chemical Physics, Stanford Research Institute, Menlo Park, California. 1 D. M. Yost and H. Russell, Systematic Inorganic Chemistry (Prentice- Hall, Inc .. New York, 1946), p. 17. 2 R. N. Pease and R. Falk, J. Am. Chern. Soc. 76, 4766 (1954). I W. Rosser and H. Wise, J. Chern. Phys. 24, 493 (1956). • C. H. Bamford, Trans. Faraday Soc. 35, 568 (1939). Catalytic Effects of Ti02 and Mn02 on the Phase Transformation of Gothite S. C. CHAKRABORTY* AND A. Roy Indian Association for the cultivation of Science, Calcutta 32, India (Received July 31, 1956) IN recent years a number of workers studied the effect of the impurities on the phase transformation of different minerals. It was found in most cases that the transition temperature for the same mineral varies with geological origin. Extensive physical and chemical studies of the minerals showed that the amount of impurities present played an important role in the phase trans formation of the minerals during thermal treatment. Dasgupta1 studied the complete phase transformation of the two minerals gothite and limonite during thermal treatment. Posnjack and Merwin2 and Dasgupta 3 have shown from their x-ray analysis that gothite and limonite have identical crystal structure. Dasgupta3 also observed that though these two minerals have identical structure and the same chemical formula, the transition tempera ture for the two minerals were different; limonite and gothite changes into Fe20, at 210°C and 250°C, respectively. Chemical analysis showed presence of Si02 in limonite as impurities, whereas in gothite no trace of Si02 was found. Dasgupta3 attributed this cause for different transition temperatures for the two minerals to the presence of impurities in the minerals. To verify this explana tion, Dasgupta4 also studied the effect of Si02 on the phase transformation of gothite and found that with the gradual increase of the amount of Si02 added to gothite, the transition temperature fell down to 210°C. From mineralogical data, it is seen that the other impurities which can be found in limonite and gothite are Ti02, MnO, etc. The present investigation was carried out to see the effect present of Ti02 and Mn02 on the transition temperature of gothite and whether the transition temperature also depends on the nature of impurities added. Known weight of Mn02 and Ti02 was added separately to a known weight of powdered gothite. The resulting mixture was then heated at a particular temperature for at least 24 hours. Different samples having different percentages of Ti02 and Mn02 (Ti02 from 0.397% to 5.2% and Mn02 from 0.67% to 1.74%) in gothite were then heated to various temperatures. X-ray powder diffraction photographs of all samples having different thermal history were taken in cylindrical cameras using Fe-K" radiation from a sealed tube, run at 30 kv 20 rnA. X-ray diffraction study reveals that the gradual addition of Mn02 and Ti02 causes a lowering of the transition temperature of gothite from 250°C; the decrease being regularly up to a certain range of the percentages of the two agents added. In the case of Ti02 the transition temperature of gothite comes down to 210°C, (Ti02 -1.1 % by weight) which cannot be changed with further addition of Ti02. But in the case of Mn02, transition starts at 190°C. The microphotometric records (Fig. 1) of the powder diffraction pattern will clearly show the change of gothite in Fe203. In the case of Ti02 and Si02 (Dasgupta)3 the transition tempera ture of gothite came down to 210°C, but in the case of Mn02 the transition temperature was 190°C. The cause for this different ---lcr----------------------fo~---- ------------------5rj-----------------------t6'--- -------------.-----;;,-------------- idJ '0' BRAGG ANGLE IN DEGREES FIG. 1. Micrqphotometric records of the powder diffraction patterns. (a) Gothite and TiO, (1.1 %) before heating. (b) Gothite and TiO. (1.1 %) heated up to 210°C. (c) Gothite and MnO, (1.009%) before heating. (d) Gothite and MnO, (1.009%) heated up to 190°C. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 138.26.31.3 On: Mon, 22 Dec 2014 09:37:18

1.1731899.pdf

Effect of Chemisorbed Hydrogen on the Magnetization of Nickel R. E. Dietz and P. W. Selwood Citation: The Journal of Chemical Physics 35, 270 (1961); doi: 10.1063/1.1731899 View online: http://dx.doi.org/10.1063/1.1731899 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/35/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of nickel grid parameters on production of negative hydrogen ions Phys. Plasmas 21, 063514 (2014); 10.1063/1.4886149 Interaction of magnetic transition metal dimers with spin-polarized hydrogenated graphene J. Chem. Phys. 138, 124709 (2013); 10.1063/1.4795500 Effect of magnetic field on the electrodeposition of nickel AIP Conf. Proc. 1461, 279 (2012); 10.1063/1.4736904 Magnetic shape memory effect in thin foils Appl. Phys. Lett. 93, 022503 (2008); 10.1063/1.2957675 Effect of Chemisorbed Hydrogen on the Magnetization of Nickel at Low Temperatures J. Appl. Phys. 30, S101 (1959); 10.1063/1.2185841 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02270 J. N. BRADLEY AND G. B. KISTIAKOWSKY carbon with an odd number of carbon atoms as a final product. No products with masses in the range 12-16 have been observed, but since argon in large excess was used as a diluent, Ca products with masses in the range 36-44 would have been difficult to detect. In particular it is possible to construct mechanisms involving 2C2H2+OH or 0 which produce CO+CaH4(allene). The mass spectrum of the latter has its parent peak at mass 40 and successively smaller peaks at masses 39 to 36. The individual steps in these mechanisms are exothermic and obey the spin conservation rule render ing them plausible. That some such Ca intermediate is formed is suggested also by the observation that the concentration of carbon monoxide continues to increase after that of acetylene has reached a stationary value. The chief result of these experiments has been to demonstrate the presence of hydrocarbon intermediates in both the pyrolysis and oxidation of acetylene. In the pyrolysis there are three intermediates which are formed from acetylene by rapid bimolecular reactions and which reach equilibrium concentrations. After a time which correlates with the measured induction time THE JOURNAL OF CHEMICAL PHYSICS for carbon precipitation their concentrations drop sharply to a new steady state value. There is some evidence that the carbon is formed directly from the polymers. In the oxidation reaction there is an induc tion period after which there is a rapid branching chain reaction consuming acetylene and oxygen and producing diacetylene. By combining this with earlier results a plausible mechanism for the chain may be written. There are also reactions which produce carbon monoxide at a rate comparable to that of the chain reaction. It is felt that these reactions must involve a second hydro carbon intermediate which was unobservable in the present experiments. ACKNOWLEDGMENTS This work was supported by a grant from the Na tional Science Foundation. One of the authors (J. N. B.) is indebted to the Commonwealth Fund of New York for the award of a Fellowship. The authors wish to express their appreciation to Dr. P. H. Kydd for valuable assistance in the interpretation of the data and the preparation of the manuscript. VOLUME 35, NUMBER 1 JULY, 1961 Effect of Chemisorbed Hydrogen on the Magnetization of Nickel R. E. DmTz* AND P. W. SELWOOD Department of Chemistry, Northwestern University, Evanston, Illinois (Received October 20, 1960) The effect of chemisorbed hydrogen on the saturation magnetization of nickel was investigated on fine particles of nickel. For certain preparations the saturation moment of the nickel was within 1 or 2% of that of massive nickel; this is considered evidence that the surfaces of the nickel in these samples were sub stantially free from chemisorbed impurities, and that the electronic state of the nickel was identical to that of massive nickel. For these preparations, hydrogen decreases the saturation moment of the nickel by about 0.7 Bohr magneton per average atom of hydrogen adsorbed. This effect appears independent of temperature up to 3000K (the highest temperature investigated) and of surface coverage over nearly the entire range. 1. INTRODUCTION THE effect of chemisorbed hydrogen on the mag netization of fine nickel particles has been investi gated by Selwood and co-workers1,2, and by Broeder, van Reijen, and Korswagen.a Measurements at high magnetic fields and at temperatures as low as 200K indicated that the magnetization of nickel suffers a decrease, but measurements at still lower temperatures were necessary for a quantitative determination of the * Present address: Bell Telephone Laboratories, Murray Hill, New Jersey. I P. W. Selwood, S. Adler, and T. R. Phillips, J. Am. Chern. Soc. 76, 2281 (1954). 2 P. W. Selwood, S. Adler, and T. R. Phillips, J. Am. Chern. Soc. 77, 1462 (1955), and later papers. 3 J. J. Broeder, L. L. van Reijen, and A. R. Korswagen, J. chim. Phys. 54, 37 (1957). effect on the saturation. Measurements of the effect of hydrogen were also carried out in alternating mag netic fields of low intensity; at 3000K a decrease in the magnetization was observed while at 200K an increase was observed in some samples. The present work extends these measurements to 4.2°K, and to dc magnetic fields of 10 000 oe, conditions which enable the saturation magnetization of nickel particles in certain nickel-silica catalysts to be meas ured precisely; this investigation occupies the first four sections of this paper. In the fifth and remaining sec tions are discussed the magnetic properties of these systems measured under less strenuous conditions of temperature and magnetic field intensity; these non saturation investigations not only further characterize the hydrogen-nickel surface interaction, but also This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02C HEM ISO R BED H Y D R 0 G E NAN D THE MAG NET I Z A T ION 0 F N I eKE L 271 establish a basis for comparing the present saturation experiments to the results of those investigations per formed under less rigorous conditions, which have been previously reported in the literature. 2. EXPERIMENTAL TECHNIQUE Procedure In a typical experiment the following sequence of steps was carried out. A pellet containing fine particles of nickel was reduced for about 12 hr in flowing hydro gen at 350°C. To remove the residual hydrogen, the sample was evacuated at 350°C for several hours until a pressure of 10-6 mm Hg was attained. The sample was then cooled, and magnetization measurements as a function of magnetic field intensity were made at temperatures ranging from 4.2° to 300°K. The sample was warmed to room temperature, and a measured volume of hydrogen was admitted. The magnetization measurements were then repeated. In all cases, about 0.1 ml of helium gas was introduced to induce thermal equilibrium. Since the maximum changes in magnetization for these systems are about 10% or less, the saturation must be measurable to ±0.1 % to provide a precision in the relative change in saturation magnetization of 1 %. Since actual measurement of the saturation magnetiza tion is impossible, the magnetization must be extra polated to infinite field intensity. This improves pre cision by almost another order of magnitude under the most favorable circumstances. Thus, magnetization measurements and the corresponding measurements of the magnetic field intensity are sufficiently precise if made with an average error of 1 % or less over a linear portion of the magnetization curve comprising about 10 experimental points. Construction of the Apparatus The apparatus developed for the magnetic measure ments, shown in Fig. 1, was adapted from a method of Weiss and Forrer.4 Magnetization was measured by moving the sample from one set of sensing coils to another set having the same number of turns, but wound in opposition; the current induced in the coils by the change in flux was integrated by a Leeds and Northrup model 2290 ballistic galvanometer used as a fluxmeter. The bucking sets of coils balanced out currents induced by transient changes in the applied magnetic field. Thermal emf's were balanced out by superimposing a small reverse emf. The sample, consisting of a single cylindrical pellet, 0.63 cm in diameter and about the same length, fitted snugly in the bottom of a Pyrex tube, the upper end of which was attached to a glass-encased iron slug. A movable solenoid then raised or lowered the assembly in the vacuum envelope. The sample tube was fitted 4 P. Weiss and R. Forrer, Ann. Phys. 5, 153 (1926). WIRE LOOP FOR SUSPENDING I-t--- SAMPLE CARRIER TUBE DURING REDUCTION AND OUTGASSING TO GAS HANDLING r.----'---i-T.-iiii-i----- H2 EXHAUST DURING REDUCTION (j .. ---- TO FLUXMETER CONTROLS ~-- FLANGE PLATE FITS AGAINST DEWAR O-RING SAMPLE HOISTED TO HERE L---++-+-l--fi'--- DURING REDUCTION AND OUTGASSING 1.,,----- SENSING COIL 300 TURNS ~~-------------SAMPLE r-------MAGNET POLE FIG. 1. Apparatus designed for in situ measurements of mag netization after various surface treatments. with a hook at its upper end which permitted the sample to be suspended about 15 cm above the sensing coils. This was necessary during reduction and evacua tion at elevated temperatures to avoid overheating the sensing coils by the electrical resistance sleeve furnace used to obtain these temperatures. A homogeneous magnetic field of intensities to 10 000 oe was provided by a 4-in. electromagnet of the Weiss design. The field in the region of the sensing coils was uniform within the sensitivity of a Rawson model 720 rotating coil gaussmeter used to monitor the field while the magnetization was being measured. The lower portion of the vacuum envelope around the sample tube was cooled by immersion in a double Dewar containing a refrigerant. Boiling helium was used for the saturation experiments. Measurements could also be made at elevated temperature by using a small sleeve type resistance furnace. Analysis of the Method The question now arises as to whether the current integrated by the fluxmeter actually was a measure of This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02272 R. E. DIETZ AND P. W. SELWOOD the magnetization of the sample. Unfortunately two complicating effects render a complete analysis of the method impracticable, but these can be treated semi empirically. They are (a) demagnetizing fields and inhomogeneous magnetization of the samples, and (b) the effects of polarization of the pole pieces of the magnet by the induced field of the sample. If these effects are ignored, elementary electromagnetic theory shows that the integrated current induced in the sensing coils is proportional to the specific magnetiza tion of the sample. Because the samples never contained more than 10 volume percent of nickel, effects of demagnetizing fields were small. Corrections were ap plied by subtracting the demagnetizing field from the applied field to obtain the internal field; since the de magnetizing field approaches a constant value propor tional to the magnetization at high fields, it affects only slightly the slope of a M vs l/H plot near saturation, and has no influence on the saturation intercept. When corrections for demagnetizing fields were necessary at low magnetic fields, the demagnetizing factor for the samples was approximated by that of the sphere. Although the contributions to the apparent, meas ured magnetizatiQn from the polarization of the pole pieces by the sample (commonly known as the "image effect") amounted to several percent of the observed magnetization, it was possible to show· that corrections for the effect were unnecessary for the relative meas urement techniques employed in this investigation. Calibration of the Apparatus The apparatus was calibrated to obtain the saturation of the various samples relative to pure, polycrystalline nickel, for which the saturation megnetization is pre cisely known.6 Very pure, powdered nickel, obtained from the International Nickel Company, was mixed with graphite and silica gel and was pressed into pellets of the same shapes, sizes, and density of nickel as those employed in experiment. To insure complete reduction of the nickel, the pellets were heated in hydrogen for 12 hr at 350°C prior to measurement. The saturation per gram of nickel, of the various cali bration samples, showed a standard deviation of 0.6%. The extrapolations to infinite field were made from M vs 1/H2 plots7• The approaches to saturation on these plots were quite linear above 2000 oe. Preparation of Samples Since a large surface to volume ratio is necessary to observe appreciable changes in the saturation mag netization due to absorbed molecules, the selection of samples was restricted to those containing nickel par ticles of diameters less than 100 A. Such systems are most stable and reproducible when the nickel particles 6 R. E. Dietz, doctoral dissertation, Northwestern University, 1960. 6 H. Danon, Compt. rend. 246, 73 (1958). 7 C. P. Bean and J. S. Jacobs, J. AppJ. Phys. 31, 1228 (1960). are suspended in a matrix of silica, and several different preparations of this type were used. These samples were generally prepared either by impregnating an existing silica matrix, such as kesselguhr or Davison gel, with a concentrated solution of nickel nitrate, drying, com pressing into pellets, and finally reducing in situ in a stream of flowing hydrogen at 350°C for about 12 hr; or by coprecipitating nickel hydroxide and silica by mixing boiling solutions of nickel nitrate and basic sodium silicate according to the method described by van Eijk van Voorthuysen and Franzen.s The impregnation-type samples were commercial nickel-silica hydrogenation catalysts prepared by the Universal Oil Products Company. For some saturation experiments, these catalysts were sintered at elevated temperatures in hydrogen. Graphite was used as a binder and lubricant during the pellet forming process for all but the coprecipitate samples. All samples were evacuated after reduction with a mercury diffusion pump for four hours at 350°C. This lowered the pressure, during pumping, to 10-6 mm Hg. Electrolytic tank hydrogen, used both for reducing the nickel samples and subsequent chemisorption, was purified by catalytically combining any free oxygen with the hydrogen in a "Deoxo" unit. The resulting water was then frozen out of the gas stream in a silica gel trap at 2oooK. The helium used to attain thermal equilibrium was purified by passage over finely divided, hydrogen-reduced copper at 600°C, and then through a dry-iced silica gel trap. A mercury micro gas buret was used to measure aliquots of hydrogen to the sample, and the pressure was measured by a McLeod gauge. Each sample was quantitatively analyzed for nickel by solution in hydrofluoric acid and electrodeposition of the nickel on weighed platinum electrodes. The samples were also examined spectrochemically for impurities. Calcium and sodium present to 1 % were the chief impurities in the impregnation-type samples. These were believed present in the preexisting silica matrix. Other impurities, such as the transition metals, con stituted less than 0.3% by weight of the sample. The coprecipitate samples contained less than 0.1% im purities. 3. SATURATION MAGNETIZATION OF FINE NICKEL PARTICLES Since the saturation magnetization of massive poly crystalline nickel has been thoroughly investigated, a comparison of the sa~uration of the experimental systems of fine particles and the massive nickel calibra tion samples was first attempted. Any differences found should arise chiefly from the differences in specific surface area. The polycrystalline nickel had a negligible surface area; the particles used for ad sorption studies, on the other hand, had approximately 8 J. J. B. Van Eijk van Voorthuysen and P. Franzen, Rec. trav. chim. 70, 793 (1951). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02C HEM ISO R BED H Y D R 0 G E NAN D THE MAG NET I Z A T ION 0 F N I eKE L 273 20 to 30% of their atoms on the surface. Thus any factors which significantly alter the saturation proper ties of these surface atoms would produce large differ ences in the relative saturation. We now must consider the problem of how to deter mine the saturation from the magnetization data: Should extrapolations to infinite field intensity be made from a plot of M vs l/B, or M vs 1/B2, or per haps some other function of B? The answer must lie in the equation of state for the magnetization of the particular sample, and it is obvious that different types of samples probably follow different laws of approach. The question of the proper inverse power of B for extrapolation to saturation is a critical one, since serious errors may result in the determination of saturation from improper extrapolation technique.7 It is not surprising that the samples containing the smallest particles are also the most difficult to saturate. These particles approach molecular dimensions and are quite disordered by thermal energy, even at liquid helium temperatures (in the absence of a magnetic field). In Sec. 5 we shall show that such particles follow a l/B law of approach to saturation if measurements are carried out under certain conditions. For some samples, however, the particles are so small that our maximum field intensities were inadequate to bring all of the particles into the l/B law region. Figures 2 and 3 depict the approach to saturation for two types of samples, measured at different temperatures, as a function of l/B. Inspection of these curves shows that for disperse systems of small particles the curves are concave upward, and linear for larger particles. Al though these samples contain approximately the same volume fraction of nickel as the calibration samples of polydomain nickel, they show no indication of the 1/B2 approach law. This behavior is due, apparently, to the large effect of thermal disordering of the particle moments which obscures the effects of the magneto- ~ z 4.2°K Q 12 .... D U D w \ D it , D t..I 10 , 0 \ a: \ !oJ \ .... 8 \AA W 77°K ~ , A A 0 A Z \ A A .c \ A A ~ 8 OqOo A -< 296°K \!) " 000 \ 0 ° 4 \ 0 7 FIG. 2. Approach to saturation ma!!;netization of a sample of coprecipitated nickel in silica at 4.2°, 77°, and 296°K. The dashed line is used to estimate the mean particle volume by Eq. (14). 11~~ __ ~ __ -L __ ~ __ ~ __ ~ __ ~ __ ~~ o o.~ 1.0 1.~ 2.0 2.~ 3.0 3.~ 4.0 4.~ l/H oe-I Xl04 FIG. 3. Approach to saturation magnetization of a sample of sintered UOP catalyst with and without chemisorbed hydrogen, at 4.2°K. crystalline anisotropy. If the l/B approach law con verts to a 1/B2 law at very high magnetic fields, then the saturation as extrapolated from the portion of the curve going by a l/B law will be higher than the true saturation. However, if the difference between the magnetization represented by the last experimental point and the saturation magnetization is already a small fraction of the saturation magnetization, as shown in Fig. 3, then the error induced in performing the l/B extrapolation will be at least as small, and probably smaller. At any rate, the ratio of particular interest is the saturation after adsorption of hydrogen to the saturation before adsorption. Since the law of approach does not seem to change greatly after adsorp tion, the error here will also be very small. The measurements showed that the saturation of the samples prepared by the impregnation process was very close to that of massive nickel. If these samples are sintered slightly at SOO°C in helium, the approaches to saturation become long linear curves for M vs l/B plots, affording accurate extrapolations to infinite field. Five experiments with the sintered UOP nickel-silica catalyst produced an average saturation of 9S.4±1.0% that of massive nickel. The preparations made by the coprecipitation procedure showed consistently high saturations, about 130% that of massive nickel. These samples consisted of extremely fine particles, and the approach to saturation was quite steep. Consequently the extrapolations are somewhat uncertain for these samples, but we do not feel that this uncertainty can account entirely for the observed discrepancy with the saturation of massive nickel. There are several reasons why the saturation mag netization of these systems might be less than that observed for massive nickel. A common apprehension regarding adsorption on powders concerns the difficulty of obtaining and characterizing "clean" surfaces under the moderately high vacuum conditions of this in vestigation. Such criticism may not apply to this investigation, however, for two reasons: (a) The surface area of nickel in each pellet is of order of 10 m2, while the external area of the pellet is about one square centimeter. Thus the rate at which the internal surface area of the pellet is covered is limited by the rate at This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02274 R. E. DIETZ AND P. W. SELWOOD 15~--~--~----.---'----r---'r---' ~ u ~ z a ~ w -' ... w o cr 12 w I-w ::!! a z 11 ~ -' ° ° ~ 10~ __ ~ __ ~ __ ~~~~~~~~ __ ~ o 1.5 2.0 3.5 I/H oe-1Xlo4 FIG. 4. Approach to saturation magnetization of a sample of coprecipitated nickel in silica with and without chemisorbed hydrogen, at 4.2°K. which the gas molecules impinge on the external surface of the pellet. The rate at which the internal surface of the pellet is covered with gas, in this case at 10-6 mm Hg, is comparable to the rate of coverage for a plane surface at 10-11 mm Hg. (b) Since the individual particles are insulated from one another by the silica matrix, the gas molecules crossing the surface of the pellet must penetrate inward by a gaseous diffusion mechanism. As the interstices between the particles are of the same order of size as the particles (less than 100 A), the diffusion mechanism is of the Knudsen type and is, therefore, relatively slow. The factors discussed above determine that provided the surface were initially clean, it would remain rela tively clean for the duration of an experiment. The point that the surfaces were initially clean remains to be established. To this effect, we may use the data of Roberts and Sykes9 who measured adsorption isotherms for hydrogen on coarse nickel powders at low pressures and to about 430°C. They found that very little hydro gen is adsorbed at 305°C and pressures less than 10-3 mm Hg, relative to the amount adsorbed at higher pressure and lower temperatures. From the isotherms published by Roberts and Sykes, we estimate that the equilibrium adsorption at 305°C and 10-3 mm Hg represents no more than 4% of the coverage at -183°C and 0.1 mm Hg. Another reason for considering that the surfaces of the nickel as prepared under the conditions described above are essentially clean is that the satura tion moment of the nickel particles in the sintered samples coincides with that of massive nickel within 1 or 2%, depending on the run. Since the admission of hydrogen to atmospheric pressure results in a decrease in the saturation of about 10%, we can safely say that at the onset of an experiment, the surface is 80-90% free of hydrogen atoms. The saturation is also reduced by 30-40% when the sample is exposed to air; although such oxidation probably proceeds to a depth of several 9 M. W. Roberts and K. W. Sykes, Trans. Faraday Soc. 54, 548 (1958). monolayers, it nevertheless serves to support the view that reduction is virtually complete. Although there has been a reportIo that supported nickel suffers an appreciable decrease in magnetization because of electronic interaction between the support (generally alumina or alumina doped with semi conducting oxides) and the metallic phase, no such be havior was observed for the silica supports. The saturation data also suggest that samples showing good agreement with the saturation of massive nickel are in a similar electronic state with respect to ferromagnetic ordering and number of unpaired spins. Considerable conjecture and evidence have been pre sented in the past few years that the exchange energy is a function of the particle size (or in the case of films, the film thickness) and that below certain critical dimensions this energy rather abruptly becomes very small compared to thermal energy, and a transition from a ferromagnetic, ordered state to a paramagnetic, disordered state ensues. However, other investigations, chiefly measurements of the saturation magnetization of fine ferromagnetic particles precipitated from copper alloys or from mercury amalgams, have demon strated that the saturation magnetization at 3()()OK and below is not significantly different from that of the bulk substance.ll Neugebauer12 has also shown that the spontaneous magnetization of thin nickel films prepared under ultrahigh vacuum conditions exhibit a tempera ture dependent spontaneous magnetization similar to that of massive nickel. In conclusion, these saturation results demonstrate that, with the exception of the coprecipitated sample~, the adsorption occurs on a nickel substrate that IS electronically similar to massive nickel, insofar as the saturation moment is sensitive to these electronic properties. Since we may, therefore, use the saturation magneton number for massive nickel to describe the saturation of these experimental systems, we are justified in relating the fractional change in saturation of the samples to a change in the saturation magneton number. For the experiments on the sintered uor samples, there appears to be good evidence that the surface is substantially free from impurities. 4. EFFECT OF CHEMISORBED HYDROGEN ON SATURATION MAGNETIZATION Because of the complexity of the nonsaturation magnetization of these systems, an accurate determina tion of the saturation magnetization is possible only at very low temperatures. This necessitates adsorption of hydrogen at ambient temperature, followed by cooling with liquid helium to the temperature of measurement. The unadsorbed hydrogen remaining in the free space 10 G. M. Schwab, J. Block, and D. Schultze, Z. angew. Chern. 71, 101 (1959). ... . 11 C. P. Bean, in Structure and Properttes of Thln Ftlms, edIted by C. A. Neugebauer et al. (John Wiley & Sons, New York, 1959), p.331. 12 C. A. Neugebauer, Phys. Rev. 116, 1441 (1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02C HEM ISO R BED H Y D R 0 G E NAN D THE MAG NET I Z A T ION 0 F N I C K E L 275 was maintained at such low pressures that any sub sequent adsorption on the sample during cooling would represent a small fraction of the total amount of gas initially adsorbed. Figures 3 and 4 represent typical experiments of this type, performed on two different nickel substrates. It is apparent that some difficulty was encountered in making precise extrapolations to saturation mag netization for samples containing very fine particles. We note that these results are in fair agreement with Moore and Selwood13 on sintered samples, but that they do not confirm that the fractional change in mag netization diminishes as saturation is approached for unsintered samples. The new results also differ from the work of Lee, Sabatka, and Selwood14 who observed that the low-field magnetization of similar nickel prepa ration increased with chemisorption of hydrogen. The reasons for our differences with Moore and Selwood are doubtless based on the increased precision of the present work. The observations of Lee et at. are now understood and will be discussed in the next section. We must express the relative change in saturation t:.M./M. in terms of the fundamental electronic inter action. The appropriate unit for this purpose is the Bohr magneton (3. Since saturation magnetization at absolute zero corresponds to the state with maximum multiplicity for the particle and for the assembly of particles, we can write t:.Mo/Mo = t:.(NNi'uNi) /NNi'uNi, (1) where NNi is the total number of nickel atoms, and ,UNi is the saturation magnetic moment of a metallic nickel atom defined by (2) V. being the volume of the sample. 'uNi is well known to be about 0.60 (3; NNi may be obtained by analysis of the sample. t:.(NNi,UNi) must be related to the effect of the chemisorbed hydrogen. We assume an average attri tion ~ of the magnetic moment of the nickel for each hydrogen atom chemisorbed so that (3) where NH represents the total number of hydrogen atoms chemisorbed. By combining Eqs. (1) and (3), (4) The parameter ~ is not a completely experimental quantity since it depends on an assumed value of 'uNi; however, if we restrict our consideration to samples which show good agreement with the saturation magnetization of massive nickel, then we may consider the direct calculation of ~ justified. 13 L. E. Moore and P. W. Selwood, J. Am. Chern. Soc. 78, 697 (1956) . 14 E. L. Lee, J. A. Sabatka, and P. W. Selwood, J. Am. Chern. Soc. 79, 5391 (1957). To express ~ more graphically, we may relate it to the number of electron spins cancelled by pairing in the d band. In fact, we occasionally identify ~ as "the number of spins cancelled per hydrogen atom adsorbed." To validate this interpretation, however, we must consider the sources of magnetic moment ,UNi. If we define ,U={3(L+2S) = g{3(S), (5) where Land S are the orbital and spin angular mo mentum respectively, g is the "spectroscopic splitting factor." We can then write g= 2+~, where ~ is a measure of orbital contribution to the magnetic moment. Bagguley,16 using electron spin resonance on fine particles of nickel gives g=2.22. Standley and Reich16 obtained g=2.19, and Meyer17 obtained g=2.193. Argyres and KitteP8 have pointed out that the number of effective electron spins n. contributing to the satura tion magnetization is just ,UNi(2/g); thus n.=0.54. If the change in magnetic moment arises solely from a cancelling of electron spins, then ~ calculated by Eq. (4) represents the actual number of spins can celed. If the change in magnetic moment arises from a simultaneous quenching of the orbital magnetic mo ment with the spin moment so as to keep g constant, then the number of electron spin moments canceled per hydrogen atom adsorbed will be ~X (0.54/0.60) =0.90~. Values for ~ have been determined for the various preparations of nickel previously described. For the unsintered samples (this does not include the co precipitated preparations which exhibit high saturation moments), ~=0.56±0.09. For the sintered UOP samples, ~ = o. 71±0.04. Because of the uncertainty in the extrapolations for preparations containing very small particles, we do not believe that definite con clusions can be reached from these data alone as to whether ~ varies with particle size,.and we shall there fore use the value 0.7 in the following discussions to characterize the interaction of hydrogen and nickel surface. The fact that only a fractional number of spin moments are canceled when a hydrogen atom chemi sorbs on nickel may be rationalized as follows: One can consider that the protons create a set of levels, either continuous or discrete, which the electrons may fill with the restriction that only states below the Fermi energy may be occupied; the remainder of the electrons will occupy holes in the d and s bands of the nickel. A second possibility pertains to the similarity between ~ and 'uNi, suggesting that for each hydrogen atom chemisorbed the moment of a nickel atom is lost to the crystal. One may argue that this atom no longer ,. D. M. S. Bagguley, Proc. Roy. Soc. (London) A228, 549 (1955) . 16 K. J. Standley and K. H. Reich, Proc. Phys. Soc. (London) B68,713 (1955). 17 A. J. P. Meyer, Compt. rend. 246, 1517 (1953). 18 P. Argyres and C. Kittel, Acta Met. 1, 241 (1953). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02276 R. E. DIETZ AND P. W. SELWOOD 1.0'.---..---..---,--,--,--,--,-----, FIG. 5. The Langevin function and its approximations (x=p.H/kT). contributes to the moment of the particle, and that the hydrogen-nickel interaction does not affect the moment of neighboring atoms. The first mechanism has an analogy in the magnetic properties of the transition metal alloys, such as the copper-nickel series. Recent experiments19,20 have shown that the saturation magnetization of copper-nickel alloys decreases linearly with added copper, reaching zero at 53 at. % copper, corresponding to an E for this system of about 0.46 for each copper atom added. (This is just the number of 4s electrons a metallic copper atom possesses in excess of metallic nickel atom). It is generally considered that the copper atoms in Cu-Ni retain all ten of their d electrons, the remaining electron being divided between the sand d bands of the alloy. A fraction of this electron must occupy the 4s state of the copper atom as dictated by the Fermi level, and the remainder may then occupy states in the d band of the alloy, thus accounting for the fractional value of E for Cu-Ni.21,22 This case may be analogous to that for hydrogen chemisorbed on a nickel surface, and if so, the value of E=0.7 suggests that the electron of the chemisorbed hydrogen atom may likewise have an appreciable density about the hydrogen nucleus. 5. NONSATURATION PROPERTIES Several questions, important for this investigation, cannot be answered by data derived from our satura tion measurements. These concern the dependence of the spin-pairing effect of chemisorbed hydrogen on tem perature and of surface coverage over a wide range of coverages. The effect of temperature cannot be ob tained from saturation measurements because satura tion can only be approached at low temperatures with 19 S. A. Ahern, M. J. Martin, and W. Sucksmith, Proc. Roy. Soc. (London) A248, 145 (1958). 20 A. J. P. Meyer and C. Wolff, Compt. rend· 246, 576 (1958). 21 Lomer and W. Marshall, Phil. Mag. 3, 185 (1958). 22 Absorbed hydrogen has also been observed to d~crease the susceptibility of palladium metal to zero at a H:Pd ratio of about 0.7. Although. is generally taken to be unity for this system [see J. Wucher, Ann. phys. 7, 317 (1952)], there is no reliable, inde pendent measurement of the moment of palladium to substantiate this assumption. the highest available magnetic fields. The effect of surface coverage is difficult because of lack of sensi tivity in the method, as described. But both factors influence the nonsaturation magnetization strongly, and if we can relate the nonsaturation magnetization to the spontaneous magnetization of the particles, we can separate out the effects. Fortunately, a theoretical model exists which has been very successful in describing the magnetization curve of fine-particle ferromagnetic systems. Bean and Livingston23 review the magnetic properties of fine single domain particles. Bulk ferromagnetic materials consist of regions, called "domains," in which the magnetization is es sentially constant and arises spontaneously because of the ferromagnetic exchange energy. The upper limit for the volume of a single domain particle depends on the shape of the particle, but an approximate figure corresponding to a sphere is 150 A in radius. Although a polydomain particle may magnetize by both domain boundary motion and then rotation of the atomic moments, the single domain particle is constrained to magnetize solely by rotation processes. Particles not too eccentric in shape probably magnetize by a coherent rotation of all of the atomic moments in the particle. If such rotation is uniformly facile over all directions, then the particle may be thought of as a paramagnetic atom of very large quantum number, and provided the particles do not interact with one another, their mag netization obeys the classical Langevin equation of state: MIM.= coth (p.HlkT) -(kTlp.H), (6) where M is the magnetization under the conditions of field intensity H and temperature T, p. is the magnetic moment of the particles, and M. is the saturation mag netization corresponding to perfect alignment of the particle moments with the field. Since p.=np.Ni, where n is the number of atoms in a particle, approximately 102 to 105, and P.Ni is the moment per ferromagnetic nickel atom (=0.6/3) the particle acts as an atom with a very large atomic moment: hence the appellation, "super paramagnetism." This equation is derived by applying the Boltzmann distribution law to the assembly of isotropic, noninteracting dipoles, and therefore requires that the assembly be in thermal equilibrium. Real particles never exhibit such isotropic behavior, however, and magnetization in certain directions al ways requires more energy than in others. The sources of this anisotropy may lie in the dependence of the magnetostatic energy on the shape of the particle, in differences in energy for magnetizing along the different crystal lattice directions, or from magnetizing energy differences resulting from applications of stress. If the anisotropy is of cubic or randomly oriented uniaxial symmetry, the above equation can be shown to hold at 23 C. P. Bean, J. D. Livingston, J. Appl. Phys. 30, 120S (1959). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02C HEM ISO R BED H Y DR 0 G E NAN D THE MAG NET I Z A T ION 0 F N I C K E L 277 very low fields where the approximation MIM.=p,HI3kT (7) is valid, and also for very high fields where the approxi mation MIM.=l-(kTlp,H) (8) is valid. Figure 5 shows Eqs. (6)-(8) plotted vs x=p,HlkT, and l/x. However, if the anisotropy energy is very great, the particle moments may not be in thermal equilibrium, and the Langevin function would not apply. Since the total anisotropy energy is pro· portional to the volume of the particle, larger particle:> will be found often to deviate considerably from super paramagnetic behavior, while very small particles should obey the Langevin function more closely. Neel24 has shown that the approach to equilibrium for an assembly of initially aligned uniaxial particles proceeds as a decay in the remanent magnetization, characterized by a relaxation time T, where l/T=jo exp( -KvlkT), (9) 10 is a frequency factor of the order of 109 secI, K is the anisotropy energy per unit volume, and v is the volume of the particle. Thus there will be some critical particle volume for which T will become very long with respect to experiment, while smaller particles will approach equilibrium almost instantaneously. If particles larger than the critical volume are present, appreciable remanence will be measurable, and this effect is in itself indicative of nonsuperparamagnetic behavior. Throughout the above discussion we have assumed that Ji. was independent of the variables of state T and H. This is not strictly true, for the spontaneous mag netization of nickel is a function of both variables, but to a negligible extent for the conditions of this in vestigation. Consideration of Eq. (6), and the approximations, shows the magnetization to be a function only of the combined state variable HIT. Thus magnetization 0.8 '" 0.6 :E -:E 0.4 0.2 FIG. 6. Test of HIT superposition for a sample of sintered UOP catalyst. 24 L. Neel, Compt. rend. 228,664 (1949) j see also W. F. Brown, Jr., J. App!. Phys. 30, 130S (1959). ::I 6 I I I I I r d II ""\ 0 ° II . ~ oil" 5f- Oil". ~ • z ° II • ..". 0 II i= 4"-'? ". • 0 0 .. - ~ II • "-OIl. BEFORE ADMISSION OF Hz .... 3-0 .. ° 3000K II: <1>. 11 77°K w dll 'WITH Hz I-2f--.... - :E III .300OK 0 £; + '77°K ~ 1,· - ~ oP I I 0 Z 4 6 8 10 12 14 18 18 HolT FIG. 7. Test of HIT superposition with and without chemi sorbed hydrogen for a sample of coprecipitated nickel on silica. Small corrections were made to correct H for demagnetizing fields as described in the text. measurements for a variety of field intensities and temperatures should fall on the same curve when M is plotted versus HIT. Such behavior has been observed by Bean and Jacobs25 with fine particles of iron sus pended in mercury, Becker26 with cobalt particles pre cipitated from a homogeneous copper-cobalt (copper rich) alloy, and others. The data of Heukelom et al.,27 who were the first to apply the Neel theory to catalyst systems, may also be made to show HIT superposition for nickel-silica catalysts. We have also made such a study as an adjunct to the chemisorption experiments. Figures 6 and 7 describe tests of HIT superposition for two types of samples. Of our preparations the coprecipitated samples alone ex hibit good HIT superposition below 3000K and can, therefore, be classified as superparamagnetic in that region. The behavior of the other samples is apparently complicated by effects of anisotropy, particle inter action, or perhaps other phenomena to cause varying degrees of deviation from the HIT superposition criterion. Since a consequence of the short relaxation time criterion for superparamagnetic behavior requires that no remanence should be observed, the appearance of remanent magnetization is also a quick indication of nonsuperparamagnetic behavior. The coprecipitated samples, for instance, fail to exhibit measurable re manence at 77°K and above. They do possess re manence at 4.2°K amounting to about 0.10 M •. Meas urements of the remanent magnetization are sum marized in Table I for a number of samples. In such preparations, the particles of metal always have a distribution of sizes. The expression for the magnetization must, therefore, be modified to account 25 C. P. Bean and I. S. Jacobs, J. App!. Phys. 27, 1448 (1956). 26 J. J. Becker, Trans. Am. Inst. Mining, Met. Petrol. Engrs. 209,59 (1957). 27 W. Heukelom, J. J. Broeder, and L. L. van Reijen, J. chim. phys. 51, 474 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02278 R. E. DIETZ AND P. W. SELWOOD TABLE I. Remanent magnetization as a function of temperature. Relative remanent magnetization M r/ M. Sample type 4.2°K nOK 3000K Coprecipitate UOP Sintered UOP 0.10 0.18 0.26 0.00 0.02 0.07 for such a distribution, as28 M= LniJ.l.i£(J.l.iH/kT) , 0.00 0.00 0.00 (10) where ni represents the number of particles having a volume Vi and £ is the Langevin function. By sub stituting J.l.i= vJs, we obtain for the low-field approxi mation M = (J.2H/3kT) Lnivi2 (11) i from which an average particle volume (V2)Av/V may be derived: (V2)AV/V= (3kTll.H) (MIM.). (12) The high-field approximation (note, this equation pre dicts a IIH law for the approach to saturation, as is experimentally observed for our samples) M =1 Ln 'v .(1-~) 8 i " vJ.H (13) yields an expression for the mean volume ii: kT[ 1 ] ii=I.H l-(M/M.) . (14) Because of the deviation due to anisotropy from superparamagnetic behavior at the high field intensities and low temperatures needed to effect saturation, the use of Eq. (7) is generally prohibited, and the low-field volume average must be relied upon. This average (v2 ) Aviv, of course, is always larger than the mean volume v by a factor characteristic of the volume dis tribution. As shown in Fig. 2, use of the 77°K data for a coprecipitated sample, which exhibits H/T superposi tion, would lead to a very low value for the saturation and would not reflect the true slope of the approach to saturation at that temperature. If we suppose that the 4.2°K data reflect true saturation, then the dashed line represents the maximum possible slope. Measurement at higher fields would certainly reveal an even steeper slope for the actual magnetization curve. The dashed line, however, when used to calculate the high field average particle volume gives v=4.SXlO-21 cms, cor responding to a spherical particle radius of 10 A. This value, although an upper limit for v, is considerably smaller than the value (V2)AV/V calculated from the 28 J. w. Cahn, Trans. Am. Inst. Mining, Met. Petrol. Engrs. 209,1309 (1957). initial slope of the magnetization curve for the same sample: (v2)Av/v=63XIO--21 cms, corresponding to a spherical particle radius of 2S A. The low-field average volume, therefore, must be used with the understand ing that ii may be appreciably smaller. If we use the value 10 A to calculate the surface area per gram of nickel, we obtain a value of 31 m2. The volume of hydrogen adsorbed at complete coverage for similar samples has been estimated by Dr. P. G. Fox in this Laboratory to be about 24 cc/g of nickel. This figure may be related to the surface area by mUltiplying the number of hydrogen atoms adsorbed by 6.4 A2, the area of a single adsorption site.8 Thus the adsorption meas urements indicate an approximate surface area of 43 m2/g of nickel.29 This figure is in agreement with the magnetic measurement in the sense that the figure 10 A represents an upper limit for v and therefore a lower limit for the surface area. Average particle volumes were also determined for two other preparations using the low field calculations. The corresponding sphere radius for the unsintered UOP preparation was 42 A, and for the sintered UOP 64 A. But as these two preparations did not exhibit HIT superposition at 77° and 300oK, these determina tions must be regarded as uncertain. 6. EFFECT OF HYDROGEN ON LOW FIELD MAGNETIZATION The simplest description for the nonsaturation mag netization is from a viewpoint of superparamagnetism: that is, we consider the effects on systems of particles which show good HIT superposition and zero re manence. The effects of adsorption on nonsuperpara magnetic systems are then approached by considering how these systems deviate from ideal superparamag netic behavior. In Sec. S we showed that the spontaneous mag netization of samples which exhibit H/T superposition, such as the coprecipitated samples, does not change ap preciably over the range of temperature measure ments. A similar experiment after hydrogen was ad sorbed shows the magnetization curve depressed below that of the same sample with a "clean" surface as expected. Figure 7 shows magnetization vs HIT for the clean sample and also with hydrogen adsorbed. Although this sample did not show quite as good superposition as other samples of the same preparation, H/T superposition after chemisorption is as good, or better than prior to chemisorption. To ensure that the same amount of gas was adsorbed at both 77° and 3000K, the hydrogen was admitted to the sample at 3000K until a pressure of several mm Hg had been attained. The vacuum pumps were then con nected, and the free space evacuated to an equilibrium 29 Measurements on similar preparations by adsorption of carbon monoxide have produced nickel surfaces between 31 and 75 m2/g (see reference 2). The total BET surface area on these preparations is about 200 m2/g, but a large fraction of this is of course, due to the silica support. ' This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02C HEM ISO R BED H Y D R 0 G E NAN D THE MAG NET I Z A T ION 0 F N I C K E L 279 pressure of about 0.01 mm Hg. The magnetization was then measured as a function of field intensity at 300°, at 77°, and finally again at 3000K. There was no hysteresis between the two sets of measurements at 300°K. Although the HIT superposition of this particular sample was poorer than desired, it probably permits the following conclusions. Since HIT superposition implies constancy of the spontaneous magnetization with temperature, we must conclude that the degree of spin pairing by the chemisorbed hydrogen does not change appreciably over the range between 77° and 3000K. Thus if two different kinds of chemisorbed species are present in equilibrium at some temperature T then either (a) the equilibrium constant does not vary appreciably over the range 77 < T<300, or (b) the two (or more) species have the same spin-pairing effect. In order for (a) to be correct, AH for conversion of one type of adsorbed species into another would have to be very small. However, if the conversion is from an atomically adsorbed state to a molecular species as often postulated to account for changes in heats of adsorption and surface potentials with coverage, AH would be rather large, making (a) implausible; also (b) would be equally implausible for such a system. On the other hand, if the distribution of bond types is governed by some temperature invariant surface heterogeneity, as the nature of the lattice site, then the distribution of bond types need not necessarily be an equilibrium distribution characteristic of any particular tempera ture at very high coverages. A better defined consequence of HIT superposition of the sample with chemisorbed hydrogen is that the spontaneous magnetization has decreased by approxi mately the amount expected from the spin-pairing mechanism only. These results provide no evidence that the exchange interaction is altered to produce large decreases in the Curie temperature of the particles, as was previously suspected. But the possibility that this may occur at higher temperatures is not yet ex cluded. Because of the unsuitability of saturation measure ments for investigating the effect of adsorbed gases over wide ranges of coverage at room temperatures and above, Selwood30 developed a low frequency ac induc tion apparatus capable of high sensitivity for low field magnetization measurements at elevated tempera tures and pressures. Many studies have been performed with this apparatus, most of the results being reported as magnetization-volume isotherms similar to that in Fig. 8. The relative magnetizations so measured have been considered proportional to the relative saturation magnetization. If this is true, then the linearity of the magnetization-volume isotherm is good evidence for the independence of the number of electrons paired per gas atom chemisorbed over a wide range of coverage. Also, ao P. W. Selwood, J. Am. Chern. Soc. 78, 3893 (1956). 0.8!-__ -!I:-__ -=-'1:--__ ~I:-_ _7. o !) 10 1!) 20 CC H,./G NL FIG. 8. Three successive magnetization-volume isotherms for hydrogen adsorbed on a sample of unsintered UOP nickel-silica catalyst (after Lee, Sabatka, and Selwood). The ordinate is the ratio of magnetization after some degree of coverage to the initial magnetization. The measurements were performed in a low field (about 100 rms oe) ac induction apparatus at 300oK, and are described in detail in reference 14. the slope of the isotherm is proportional to the number of spins paired per atom adsorbed E and if this slope could be related to a similar experiment for hydrogen, then the number of d electrons involved in the chemi sorption bond with any other molecule could be determined. As discussed in Sec. 5, the magnetization of an assembly of fine nickel particles is evidently not always linearly related to their saturation magnetization; thus changes in nonsaturation magnetization may not be proportional to the change in saturation. Therefore, we must determine under what conditions these changes are proportional to the corresponding changes in saturation, to validate analysis of the data described in the previous paragraph. Since we would not expect linear relations between the low field magnetization and saturation magnetiza tion in samples which exhibit remanence or other non equilibrium behavior, the following discussion will be limited to superparamagnetic systems. This is not a severe practical limitation since the coprecipitated prep arations show good superparamagnetic behavior, and in addition have large surface areas. Moreover, other preparations such as nickel-impregnated Davison Gel also exhibit ideal superparamagnetism, and possess superior adsorption properties. at In constructing a theory for the change in mag netization resulting from the adsorption of a given volume of gas, considerable mathematical simplifica tion is attained by limiting consideration to the region jJ.HlkT<0.5, where we may use thtl low field approxi mation of the Langevin function with a maximum error 31 G. C. A. Schuit and L. L. van Reijen, Advances in Catalysis 10,242 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02280 R. E. DIETZ AND P. W. SELWOOD TABLE II. Values of a and a/o for certain arbitrary particle size distributions. Particle size distribution a a/o Delta function 2.000 2.000 Infinitely wide rectangle 1.600 2.400 Maxwellian distribution of volumes 3.734 2.100 Maxwellian distribution of radii 1.471 2.500 of 2%. For nickel particles of 100 A in diameter, this region includes fields up to 100 oe at 300°K. Both the unsintered UOP and the coprecipitated samples meet this requirement at 300°K. By allowing for an arbitrary change in the moment of a particle, tJ.J.l.i characterized only by the size of the particle, the low field magnetization of such an as sembly of superparamagnetic particles is M' = (H/3kT) Lni(J.l.i- tJ.J.l.i) 2. (15) i The relative change in the magnetization tJ.M/M IS then (tJ.M/M) = Lni(tJ.J.l.i)L2LniJ.l.itJ.J.l.i. (16) LniJ.I.i2 Notice that Eq. (16) is independent of Hand T. If we assume spherical particles, (17) where r i is the particle radius, fI is the number of molecules adsorbed per unit area of surface at complete coverage, and e is the fraction of the surface covered. We notice that (7]fJe) A/I. Vo=tJ.M./M., (18) where A is the total surface area of the particles, and Vo is the total vQlume of ferromagnetic metal. Upon solving Eq. (18) for flee and substituting into Eq. (17), and converting the radius to the sphere volume, we obtain tJ.J.l.i= (tJ.M./Ms) (I.Vo/ A) 411'(3v;/47r) 2/3. (19) It is convenient to transform ni into a continuous volume distribution function26: n,--"7f(v)dv/v, (20) where {O f(v)dv= Yo. (21) Expressing A in terms of the continuous distribution function: A = {O 41rT2[f( v)dv/v ] = 47r(3/47r ) I{O [f(v)dv/v1]. (22) Substitution of Eq. (22) in Eq. (16) gives I1M/M = -a (tJ.M 8/M.) +o(LlMs/M.) 2, (23) where a and a are parameters related to the particle size distribution: {"VY(V)dV {"f(V)dV a=2-------- 1aoo v-y( v) dv 1aoo vf( v) dv 1aoo vY( v) dv 1aoo v-Y( v) dv a/o=2--------- From the symmetry of the ratio of integrals, we expect that a/a and a both are approximately equal to the number 2. Calculations applying these functions to arbitrary volume distributions, summarized in Table II, indicate this expectation to be substantially correct. Furthermore, experimental values for a are: For un sintered UOP, a""2.5; for coprecipitate, a",,3.3, in ap proximate agreement with the theoretical values in Table II. Provided a/a is of the order of 2 or larger, the second term of Eq. (23) may be neglected, and tJ.M /M is proportional to tJ.M./M •. By Eq. (18), tJ.Ms/M. is linear with e provided e is independent of e. Since linear magnetization-gas volume isotherms are obtained experimentally, as shown in Fig. 8, we conclude that e is indepe~dent of surface coverage from e=o to that coverage corresponding to a hydrogen pressure over the nickel of nearly an atmosphere. So far we have not discussed in detail the behavior of particles which do not exhibit superparamagnetic behavior. It is now obvious that the anomalous effects observed by Lee et al.,14 namely, that the effect of chemisorbed hydrogen increased the magnetization of a nickel-silica catalyst when measured at 200K but decreased the magnetization at 3000K probably arose WITHOUT Hz C 4.Z0K '" 71"K ° 303°K WITH Hz • 77°K • 30,OK FIG. 9. Approach to saturation of a sample of sintered UOP nickel-silica catalyst, with and without chemisorbed hydrogen. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02C HEM ISO R BED H Y D R 0 G E NAN D THE MAG NET I Z A T ION 0 F N I C K E L 281 because the nickel particles were not behaving super paramagnetically at such a low temperature. Taking a cue from Leak's observation82 that larger particles appear to give the increases in magnetization at higher temperatures than smaller particles, we determined the magnetization curves of a sample of sintered UOP, shown in Fig. 9. Adsorption of hydrogen decreases the magnetization at 300oK, as Lee observed. However, the corresponding data at 77°K show a decrease at high fields, but an increase at low fields. One observes the same sort of effect by merely lowering the temperature of the sample, as evidenced by the curve at 4.2°K. Thus the increase in magnetization at 77°K observed at low fields after adsorbing hydrogen does not result from an increase in the spontaneous moment, but rather from an alteration in the process by which the particle moments rotate to align themselves with the applied magnetic field. It seems possible that the magnetic anisotropy is decreased by the hydrogen, thereby decreasing the relaxation time for the particles; however, as we observed earlier, the remanence of a sample at low temperatures did not appear to be altered appreciably by adsorbed hydrogen. In any event the effects pre viously thought to be evidence for a change in bond type with, perhaps, formation of a negative hydride ion are seen to be a consequence of deviations from super paramagnetic behavior. 7. CONCLUSIONS The effect of chemisorbed hydrogen on the saturation magnetization of nickel was investigated on fine particles of nickel. For certain preparations the satura tion moment of the nickel was within 1 or 2% of that of 32 R. J. Leak and P. W. Selwood, J. Phys. Chern. 64, 1114 (1960) . massive nickel; this is considered evidence that the surfaces of the nickel in these samples were substantially free from chemisorbed impurities, and that the elec tronic state of the nickel was identical to that of massive nickel. For these preparations, hydrogen de creases the saturation moment of the nickel by about 0.7 Bohr magneton per average atom of hydrogen adsorbed. The decrease in the saturation magnetiza tion was attributed, as previously suspected,2 to spin moment cancellation of the nickel 3d electrons by the hydrogen electrons. A comparison was made with the magnetic properties of homogeneous alloys. If valid, this comparison indicates that the interacting electrons have an appreciable density around the hydrogen nucleus. Investigations of the effect of hydrogen on the magnetization of these systems under conditions far from superparamagnetic saturation indicate that: (1) The effect of chemisorbed hydrogen on the saturation magnetization of nickel appears independent of tem perature up to 3000K (the highest temperature in vestigated) and the surface coverage over nearly the entire range; (2) the effect of hydrogen on the satura tion magnetization can be simply related to the effect on the low field magnetization of superparamagnetic nickel particle systems; and (3) anomalous effects observed by other investigators probably arose because of deviations from superparamagnetic behavior. ACKNOWLEDGMENTS The authors would like to express their gratitude to Professor J. A. Marcus for his advice and assistance in supplying liquid helium, and to C. R. Abeledo for in formative discussions. This work was performed under contract with the Office of Naval Research. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.18.123.11 On: Sun, 21 Dec 2014 16:56:02

1.1728927.pdf

Some Properties of Dirty Contacts on Semiconductors and Resistivity Measurements by a TwoTerminal Method George G. Harman and Theodore Higier Citation: Journal of Applied Physics 33, 2198 (1962); doi: 10.1063/1.1728927 View online: http://dx.doi.org/10.1063/1.1728927 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extraction of the tunnel magnetocapacitance with two-terminal measurements J. Appl. Phys. 107, 093904 (2010); 10.1063/1.3407509 A Method Of TwoTerminal Excess Noise Measurement With A Reduction Of Measurement System And Contact Noise AIP Conf. Proc. 780, 677 (2005); 10.1063/1.2036842 Transverse magnetoresistance: A novel twoterminal method for measuring the carrier density and mobility of a semiconductor layer Appl. Phys. Lett. 64, 3015 (1994); 10.1063/1.111389 Twoterminal bias induced dual wavelength semiconductor light emitter Appl. Phys. Lett. 63, 3367 (1993); 10.1063/1.110147 TwoTerminal Asymmetrical and Symmetrical Silicon Negative Resistance Switches J. Appl. Phys. 30, 1819 (1959); 10.1063/1.1735062 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:47JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 7 JULY 1962 Some Properties of Dirty Contacts on Semiconductors and Resistivity Measurements by a Two-Terminal Methodt GEORGE G. HARMAN AND THEODORE HrGIER Electron Devices Section, NaJional Bureau oj Standards, Washington 25, D. C. (Received December 8, 1961) . The surface and bulk properties of semiconductors have been studied by a two-terminal method using duty contacts. These contacts are defined as ones that are easily applied and removed and that are separated from the bulk by surface states, oxides, adhered gasses, and chemical films. The method essentially involves measuring the resistance-voltage (R-V) characteristics from the millivolt range up to about 100 V, and from these data the sample resistivity can be obtained directly. The effect of work damaging or etching the semi conductor surface can be readily evaluated. By observation of the shape of the R-V curves it is possible to separ~te the bulk from .the surface effects, ca~cul~te the surface barrier height and thickness from tunneling equahons, and determme whether the barner IS a surface film or due to metal-semiconductor contact potential difference. An application of the theory of electric field tunneling of Holm to the data of silicon carbide gives values of about 2.4 eV for the barrier height and 13 A for the barrier width of the film on the surface. ~ufficient information is included on electrode techniques and pitfalls so that the experimentalist can readily make ilie measurements. Efforts were concentrated on silicon carbide and silicon but the techniques are applicable to all types of semiconductors. ' I. INTRODUCTION THE properties of metal-semiconductor contacts have been studied by many investigatorsl-4 and reviewed by Henisch.5 In most studies the contacts were applied in such a manner that they produced either definitely ohmic or rectifying conditions. Some in vestigators, however, studied the case in which an artificial barrier layer existed between the electrode and the semiconductor.6,7 But in general most workers are interested in either the surface or the bulk properties and go to great lengths to exclude the effects of the undesired portion. Surface measurements are frequently made using the field effect method, aptly described by Statz.s Four-terminal techniques are usually considered essential for bulk measurements, such as resistivity.9 However, the present work is a study of both the semi conductor bulk and surface properties, using only two dirty contacts applied to opposite sample faces. Dirty contacts are defined, for this paper, as ones that are easily applied and removed, and that are separated from intimate contact with the bulk by barriers consist ing of surface states, atmospheric contaminants, worked semiconductor surfaces, oxide layers, chemical films, etc. The electronic representation of these are shown later in Fig. 6. In general, these contacts are applied at t This work was supported by Air Force Research Division Hqts. Det. 2, L. G. Hanscom Field, Bedford, Massachusetts. A more detailed report on this subject was issued on 5/15/62 AFCRL-62-190. ' 1 J. Bardeen, Phys. Rev. 71,717 (1947). 2 R. F. Schwarz and J. F. Walsh, Proc. IRE 41, 1715 (1953). • E. H. Borneman, R. F. Schwarz, and J. J. Stickler, J. AppJ. Phys. 26, 1021 (1955). • E. C. Wurst, Jr., and E. H. Borneman J. App!. Phvs. 28 235 (1957). ,- , 'H. K. Henisch, Rectifying Semicondttctor Contacts (Oxford University Press, New York, 1957). 6 C. C. Dilworth, Proc. Phys. Soc. (London) 60, 315 (1948). 7 P. A. Hartig and R. N. Noyce, J. App!. Phys. 27, 843 (1956). 8 H. Statz, G. deMars, L. Davis, Jr., and A. Adams, Jr., Phvs. Rev. 106,455 (1956). • • G. H. Rudenherg, Semiconductor Products 2,28 (1959). room temperature and include such materials as liquid gallium alloys, air-drying silver paint, graphite, and simple pressure contacts of soft metals such as indium and lead. This definition is made to specifically exclude diffused and alloyed contacts and in most cases to exclude the type of contacts used on thin film tunnel devices. These latter ones may be dirty in the sense of having oxide layers interposed, but the techniques and conditions of application are completely controlled and very sophisticated compared with those employed in the present work. The major problem to be overcome in measuring bulk resistivity using dirty contacts is the resulting contact resistance which may be much larger than the bulk resistance of the sample. The 4-point probe is most commonly used to avoid these barrier effects. Many of the available new wide-gap semiconductor single crystals (e.g., {1-SiC, BP, and AlB12) are simply too small (approximately 1 mm on a side) to satisfactorily attach leads for 4-terminal measurements from room temperature to perhaps as high as 1200°C. In addition easily applied ohmic contacts may not have been de~ veloped. Thus, a 2-terminal method based upon the characteristics of readily available electrodes is both desirable and necessary. One method of doing this is to measure with a high frequency ac bridge,lO,l! in order to reduce the impedance of the barrier by capacitive shunt ing. For this method one must use a special crystal holder. Also barrier relaxation effects may be encount ered that can confuse interpretation of the data. Therefore, it is desirable in many instances to have an alternate 2-terminal method of measuring resistivity that can be adapted to existing crystal holders, cryo stats, ovens, etc. Easily applied and removable ohmic contacts are not available for most semiconductors, but in the study of 10 M. Pollak and T. H. Geballe, Phys. Rev. 122, 1742 (1961). II W. Keller, Z. Angew. Physik 11, 346 (1959). 2198 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:47DIRTY CONTACTS ON SEMICONDUCTORS 2199 dirty contacts it was found possible to usc their voltagc characteristics to obtain accurate bulk measurements. From the variation of surface resistance with voltage it is possible to distinguish whether a barrier is a surface film or an electrode-semiconductor exhaustion region. For the case of 'surface films,information is obtained about the barrier height and width. Variation of the surface resistance with different etching treatments, impurity diffusion, and the work function of the applied electrodes are apparent. Thus far, measurements by this method have been successfully made on SiC, BP, GaAs, AlBa, silicon and germanium single-crystal samples in wide resistivity ranges and of both carrier types. The limitations of the method have been determined, and, bearing these in mind, the procedure is applicable for measurements on any semiconductor. It is possible to adapt it to the production testing of small semiconductor slices using a single-point probe and a large conducting plate. Details of this latter technique will be described in a later publication. . IL EXPERIMENTAL METHOD A. The R-V Curve The present semiconductor-electrode study is bascd upon the resistance vs voltage (R-V) characteristics of the contact resistance of non-ohmic area electrodes. These characteristics have been used for many years in studies of switch and relay contacts and of thin film tunneling by Holm.I2,l3 Other workers14.15 have used similar curves to experimentally evaluate different electrical contacts on ferrites and ceramic semicon ductors to determine the electrode material best suited for their particular application. In general, however, modem workers in the field of semiconductors use the standard I-V characteristics, and such data are fre quently not as revealing as the R-V method. In the present work a convenient electrode is chosen, and by applying the correct voltage its characteristics are forced to conform to the desired use. The experimental procedure was to apply the appro priate dirty contacts to opposite faces of the sample. Then a known voltage was applied and the current was measured. The resistance was calculated from these data. The dc measurements were made in the low voltage region ($1 V) where currents < 1 jlA are usually encountered. At higher voltages it is generally necessary to use a low-duty cycle «1%) pulse supply to avoid JOUle heating. However, the pulse width must be long enough to overcome any transient effects. These can arise from stray capacity but at times appear to be associated with the nature of the barrier. Conductive 1! R. Holm, F.kclric Contacts (H. Gebers, Stockholm, 1946). 13 R. Holm, J. Appl. Phys. 22, 569 (1951). uS. S. Flaschenand L. G. Van Uitert, J. App!. Phys. 27, 190 (1956). U H. A. Sauer and S. S. Flaschen, Am. Ceram. Soc. Bull. 39, 3,04 (1960). A-25 t -... _-...... , 'flO',· \ o \ I \ c:: )::'10' f:> i= SQIO' (f) W 0::: C-600 ·C f-IOZ~_ --.. -- \ z \ W 0::: <! - &10' <! IO°,--:;,--'-U"",-;--'-'-~,;;--<-.u..u~~..w.u.j-:;-'- 10·' 10.' 10° 10' 10' APPLIED VOLTAGE FIG. 1. R-V characteristics of an unetched 560 O-cm n-type SiC sample using liquid tin-gallium electrodes. samples ($1 Q-cm) normally do not exhibit transient effects so that one may use duty cycles,,-,O.Ol%. For such samples, pulse widths of 20 to 50 jlsec are usually satisfactory. But widths of up to 1000 jlsec are desirable for semiconductors in the > 100 Q-cm range. The pulser used in the early part of this investigation was a Rutherford model B-2, which had a peak output of 90 V at 200 rnA. Later a system was constructed using power transistors, which would supply 180 V at 6 amp with up to 10 amp at lower voltages. At all times voltage and current were measured on an oscilloscope, the latter across a one-ohm resistor, which is placed in series with the sample. In all measurements except those with qualifications, a value of 100 V was found adequate to reduce the barrier resistance to approxi mately ohmic.I6 Figure I gives the R-V characteristics of an unetched high resistivity (560 Q-cm at 25°C) n-type SiC sample at 25°, 400°, and 600°C using liquid tin-gallium elec trodes. Voltage in the dotted curve is in the reverse direction from the solid curve. The curves show sig nificant differences in the forward and reverse character istics in the 0.1-1.0-V region, which appear frequently in these studies. This is partly due to different surface conditions of the opposite electroded surfaces and their effect upon tunneling, to be described later. These curves demonstrate how the resistance of the metal semiconductor surface barrier is reduced by voltage. On the extreme left, region I, in the low voltage portion, the surface barrier dominates the measurement, its resistance area product being obtained by multiplying the apparent resistivity in region I by the sample length. On the right, region III, in the high voltage portion the 16 A few measurements were made on polycrystalline silicon samples, with unsatisfactory results. It is quite probable that a much higher voltage than was available is needed to overcome the multitudinous grain boundaries and obtain a bulk value. However, a sample containing only a few grain boundaries should give accurate results. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:472200 G. G. HARMAN AND T. HIGIER E u CIO" ~ 1-10· :> f= C:!?IO' (j) W 0:: 10' I Z ~IOI ct B-elched 0... 10°L,--,--,-,-.L.U-uL..----,--,-",-,-,..c.ul;;--'----''--'-'...J..U'-';--'---'--'...w..,-,:,: « 10-' 10-' 10° 10' 10' APPLIED VOLTAGE FIG. 2. R-V characteristics of a 3 O-cm SiC sample ~t ~oom temperature showing the effect of etching. Dashed hnes mdlcate change to pulser. barrier is negligible, and this is defined as the ohmic or bulk properties region. The middle transition region (II) shows the reduction in barrier resistance by the tunnel ing process. Figure 2 presents similar characteristics for a 3 r2-cm p-type SiC crystal. For curve A the crystal was cl.eaned in HF to remove oxides and other surface contammants (this produces no etching of the SiC), and tin-gallium electrodes were applied. A barrier exists, but its re sistance is only about two orders of magnitude higher than the bulk. There is a considerable spread between the forward and reverse current in the 0.1-to 1-V region indicating different semiconductor surface tun neling conditions under each electrode. The electrodes were removed, and the sample was electrolytically etched in an alcohol-HF solutionl7 to a depth of '" 100 IJ.. This great depth was chosen to assure that all damaged surface was totally removed. The electrodes were applied and the result is given in curve B. The barrier resistance is almost 5 orders of magnitude higher than that of the bulk. In addition, the variations between forward and reverse directions are negligible, indicating that the semiconductor surface barriers are nearly identical. The resistance differences between curves A and B in the 100-V region are probably caused by slightly different areas being wet by the liquid metal electrodes after etching, since the sample ends to be electroded were uneven. At any rate, the values are within 20%, which is considered the accuracy of the measurement, as discussed in Sec. lIB. Thus the R-V method offers a simple means of studyin~ the effect of an etchant or the depth of w~rk damage on SiC. One should also be able to determme the depth of a diffusion or alloying process by measuring the low voltage R-V effect after successive shallow etches. The voltage used in Figs. 1 and 2 was extended down into the millivolt range to demonstrate the measuring 17 H. C. Chang, Research on Silicon Carbide Transistors, Con tract AF33(616)5592 (1958), ASTIA No. AD214, 955. circuit impedance required (0 make resistivity, thermo electric Hall and other type measurements. These curves 'readil; demonstrate why there is difficulty in obtaining good 4-point probe resistivity measurements on high resistivity (> 10 n-cm) SiC at room temperature. The potential probes may have only a'few millivolts on them and thus the barrier resistance is at its maximum. Valu~s of > 109 r2 have been measured from such probes. The low voltage data is also useful in calculating the surface barrier height and width from electric field tunneling theory as shown in Sec. IVB. . It should be pointed out that the R-V effect IS a com pletely reversible process and is in no way related to capacitor-discharge forming of small-point contacts. Area contacts are used in this procedure and thus no local heating occurs as could happen with a point con tact. The reversibility of the characteristics has been demonstrated by repetitive oscilloscope traces. How ever if electrodes are removed and then re-applied, ther~ may be resistance differences of about a factor of 2 in the low voltage region. This is negligible compared to the differences due to changes in surface treatment as shown in Fig. 2. Experiments similar to those of Figs. 1 and 2 were run on etched and unetched, 145 r2-cm p-type silicon samples and the results are shown in Fig. 3. There is a resistance factor of 17 between the work-damaged (sandblasted) surface, curve B, and the well-etc~ed surface curve A. Curve A also presents the pnme situati~n to be avoided, namely avalanche injection. Minority carrier injection by an avalanche breakdown process occurred and was sufficient to reduce ~he bulk resistance by a factor of 3.8 to 85 V. The great dIfference between the shapes of curve'3A and those of Figs. 1 and 2 is immediately apparent; the curve does not level out at high voltages. If a curve is not relatively flat by 75 to 100 V, then the chosen contact or surface treat ment is not satisfactory for bulk-resistivity measure ments. There remain two choices. The first is to use the same electrode but work damage the surface, as was done in curve B. It produces ohmic results, within 20%, either at low or high voltages. The reason for the resistance increase in the middle range is not presently understood, but it occurs consistently on other p-type silicon samples given the same surface treatment and electrodes. The second alternative is to choose a differ ent electrode, such as was done in curve C. Here it is shown that graphite rubbed onto etched p-type silicon surfaces produced ohmic resultS.17a In all cases, the dotted portion on the right side of the curve represents the change from dc to low-duty cycle pulse measure ments. The pulsed current experiences a higher barrier than dc, indicating barrier relaxation effects, but this becomes less significant at higher voltages. 17. Note added in the proof. Rubbed-on graphite will not s~ick to a highly polished, mirror-like silicon surf3;c~, but a collOld~1 graphite suspension may be substituted provldmg the sample IS desiccated before making the measurement (see reference 24). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:47DIRTY CONTACTS ON SEMICONDUCTORS 2201 B. Resistivity Measurements Using the characteristics presented in Figs. 1-3, a simple measurement of the semiconductor bulk resis tivity can be made with only two electrodes. One merely records the voltage and current in the ohmic or bulk region III of the curve and uses the following well-known formula: Pb= (VA/IL) Q-cm, (1) where Pb is the .bulk resistivity, V the voltage, I is the current, A the cross sectional area, and L is the length of the sample. If high voltage is needed as in Figs. 1 and 2, it is applied by low-duty cycle pulses to avoid heating. Certain precautions must be observed with respect to avalanche injection [Fig. 3(A)] and the maximum allowable field strength. These will be dis cussed in following sections. As mentioned in the Introduction, the resistivities of many different semiconductors have been measured by this method. These were within 20% and usually 10% of results obtained by a 4-terminal method. Normally the results are on the high side because, even when 100 V is applied, 5 or so volts must appear across the barrier. Therefore, unless the contact is definitely ohmic [Fig. 3(C)] or injecting [Fig. 3(A)], the measured resistivity will be at least 5 to 10% higher than the true bulk value. This inaccuracy varies with the type of sample and the condition of the surface. Since pulses are used, the measurement is generally made with an oscilloscope, which may introduce a 5% error in reading. Other errors may be introduced in the measure ment of sample geometry and in the improper applica tion of electrodes to the intended area. These latter errors are also present in other 4-terminal measurements and may be minimized. In some cases, reversing the polarity of the applied voltage results in 5 to 10% lower resistance in one direction. This results from the two barriers having different heights or widths, so the lower value should be adopted for the more accurate results. III. ELECTRODES A. Electrode Techniques and Temperature Limitations The same type of electrode material was used for all except one of the curves presented in this paper. It was semiliquid tin-galliuml4 which was applied by dipping a piece of tin into liquid tin-gallium eutectic and then rubbing it on the semiconductor. In addition to being easily applied, this material has the advantage that after measurements it can be easily removed by dipping in dilute HCI and wiping or shaking off the remainder. This alloy is quite useful in the high temperature range (in both air or inert atmospheres) since both tin and gallium have high boiling points. However, diffu sion of the constituents into the semiconductor may become a serious problem long before the boiling point A 10-' 10° 10' 10' APPLIED VOLTS FIG. 3. R-V characteristics for a 145 Q-cm p-type silicon sample with various surface treatments and electrodes, measured at room temperature. (A) Etched surface with Sn-Ga electrodes; (B) sandblasted surface with Sn-Ga electrodes; and (C) etched sur face with rubbed graphite electrodes. The arrow on the resistance axis indicates 145 Q-cm. Dashed lines indicate pulser. is reached. Typically one may wish to make an experi mental measurement of the type shown in Fig. 1 from room temperature to perhaps 1000°C. Based on a very conservative diffusion coefficient of 5 X 10-14 cm2/sec and a maximum of 1 h of testing time near the peak temperature, diffusion should not become a problem with tin-gallium electrodes up to ",750°C for germa nium, l000°C for silicon,18 and 1400°C for SiC.19 Verification tests have been made on SiC up to 850°C with no noticeable diffusion. But when silicon, with tin-gallium electrodes, was heated to only 300°C, there was a definite change in the contact characteristics that indicated some diffusion had occurred. The contact became relatively ohmic on p-type samples resembling Fig. 3(B) and could be used thereafter as an ohmic contact. However, it produced a barrier with avalanche effects on n-type, similar to Fig. 3(A) and just the opposite of the results at 25°C. These changed condi tions remained when the temperature was lowered. The electrodes were removed with HCI (which does not etch Si) and then re-applied. For most samples the initial contact conditions were again observed, which indicates that no true diffusion occurred but rather a purely surface interaction. Nevertheless, this surface effect would ruin a measurement. In an effort to reduce the barrier effects the surface of an n-type silicon sample was sandblasted before electrodes were applied. Subsequent high-temperature tests showed that although avalanche injection was eliminated, a barrier was still created which resulted in apparent resistivities of from 25 to 50% higher than the bulk value. Therefore, the tin-gallium electrode cannot be safely used on n-type silicon at temperatures sig nificantly higher than room temperature. Graphite can 18 Diffusion data for both silicon and germanium were taken from F. J. Biondi, Transistor Technology III (D. Van Nostrand Co~pa!1Y' Inc., ~rinc~ton, New Jersey, .1958), Chap. 3 . • 1 ~hls value IS esumated from published data on aluminum dIffuSIOn, J. R. O'Connor and J. Smiltens, Silicon Carbide (Perga mon Press, New York, 1960), Chap. V-3. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:472202 G. G. HARMAN AND T. HIGIER be used to very high temperatures on p-type Si in an inert atmosphere, and as mentioned previously, Sn-Ga produces no noticeable diffusion effect on SiC to at least 850°C. The III-V semiconductors should be amenable to indium-gallium electrodes. But choice of electrodes must be investigated with preliminary tests for each particular semiconductor before it can be relied upon for accurate measurements. There are two additional precautions that should be observed when using liquid gallium (alloys) at high temperatures. The first is that it readily alloys with most metals. Therefore, in addition to the semicon ductor, gallium should touch only such materials as tungsten, graphite, glass, or ceramic20 portions of a crystal holder. The second is that occasionally on the first temperature cycle the Sn-Ga will cease to wet one or more small portions of the semiconductor surface. The extent of this will depend upon the particular semiconductor and the condition of its surface. This retraction may not be visible upon examination, but it can produce apparent changes in resistivity. The simplest solution is to heat the sample to the peak desired temperature on a quick preliminary run. The contact will then be stable for actual measurements, and the data may be normalized to the initial value for resistivity measurements. Another alternative is to re-apply the electrode after the preliminary run by re-rubbing. Then it will generally be stable. Tin-gallium can only be used for electrodes down to '"V lO°C. Below this it solidifies, and, because of its large expansion coefficient, usually separates from the semiconductor. Similar liquid mixtures of indium mercury14 and indium-thallium-mercury21 can be used to about -40°C. Rubbed-on graphite, or indium either pressed against the surface or soldered to it, can be used for measurements down to at least liquid nitrogen temperature. In general, one may use any electrode that covers the surface area uniformly, is convenient to apply and reasonably matches the semiconductors' expansion coefficient in the desired temperature range, or is in the form of a thin film so that the expansion differences are less significant. B. Physical Considerations Some consideration must be given to the effect of the electrode material on the observed contact resistance to the particular semiconductor. By careful etching fol lowed by jet electroplating of contacts, Wurst22 has 20 For a discussion of gallium corrosion (alloying) properties see c. A. Hampel, Rare lvI etals Handbook (Reinhold Publishing Corporation, New York, 1961), Chap. 10. 21 V. J. King, 'Rev. Sci. Instr. 32, 1407 (1961). 22 See reference 4. They used pressure contacts for their alkali metal results. These could fall within the present definition of dirty contacts, except that these measurements may have been made in vacuum. No information was given, but it was obvious that the work was not performed in an oxygen or water vapor rich atmosphere. shown that there is a reasonable correlation among the nature of the contact, the work function of the metal, and the semiconductor carrier type. In general, he finds that metals with low work functions ( <4 eV) are ohmic when used as contacts on n-type silicon and injecting on p-type silicon. For metals with work functions >4 eV the converse is true. We have also found this to be the case in our work using dirty contacts, where the work function of the pure metal and the semiconductor are of importance irrespective of the surface condition. Tin-gallium and indium-gallium electrodes which have low work functions23 show very slight barriers (essentially ohmic) on etched n-type silicon and quite high ones with avalanche injection in the lOO-V region on p type. Pressure indium and tin contacts produced similar results. However, rubbed graphite (see Fig. 3) or a water-based paste of it,24 pressure-gold contacts, and air-drying silver paint,25 all of which have work functions >4 eV, gave exactly the opposite results. The water-graphite mixture introduces surface states which lead to an ohmic contact at low voltages « 1 V) and non-ohmic behavior at higher voltages. C. Field Strength Considerations There is a field strength precaution that must be observed in using the 2-terminal method. At very high field strengths (E) the carrier velocity decreases, from a linear function of E to an Et relationship, as was shown by Ryder26 for silicon and germanium. His values of maximum fields for a linear velocity relationship are reproduced in Table I for convenience. In addition, we have added data for SiC based on a drift-mobility extrapolation. These SiC data are approximate but should be within a factor of two. When using the present method one should not exceed these values if good resistivity measurements are to be made. If, however, a higher field strength is necessary, then a correction for Material Ge n type Ge p type Si n type Si p type SiC n type SiC p type TABLE I. Critical field (V /cm at 25°C) 900 1400 2500 7500 35000 100 000 23 H. B. Michaelson, J. App!. Phys. 21, 536 (1950). 21 For instance, Dag dispersion No. 226, obtainable from A~heson C:olloids Company, P?rt Huron, Michigan. When dry, thiS matenal produces an ohmiC contact on p-type germanium. For a discussion of some properties of this type of contact on p-type silicon, see G. G. Harman, T. Higier, and O. L. Meyer, J. App!. Phys. 33, 2206 (1962), following paper. 26 Of the several silver paints tested, Degussa #200 produced the most consistent results. It is obtainable· from Materials for Electronics, Inc., Jamaica, New York. In general, such paints required that ~150 V be applied before they would yield ohmic results on p. type silicon. 2. E. J. Ryder, Phys. Rev. 90, 766 (11).'i3). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:47DIRTY CO:'llTACTS O'\' SEMICO~DUCTORS 2203 ~Time FIG. 4. 145 [I-cm p-type silicon sample. Sn-Ga electrodes, 25°C. (a) Pre-breakdown current pulse, 26 V applied, 0.5 rnA/major div vertical and 200 J'sec/div horizontal; (b) first stable break down current pulse, 35 V applied, 1 mA/div vertical and 200 J'sec/div horizontal; (c) voltage pulse on top (75 Vat Y intercept) and current pulse (0.17 A at intercept) decaying rapidly below, and (d) same sample with graphite electrode showing linear volt age-current relationship. Upper pulse is voltage; lower is current. the E! mobility relationship could be applied and the field increased by at least a factor of five. From Table I, it is apparent that there should be no critical field problems when working with very small crystals of SiC and other semiconductors that have relatively low drift mobility. IV. DISCUSSION There are several mechanisms capable of producing current flow across a semiconductor surface barrier, but only two are of significance in the present work: (1) The tunnel effect6,12,13; this dominates the situation in Figs. 1 and 2 and any others where the barrier width is <50 A; (2) avalanche breakdown27,28; this is ac companied by noise pulses in the pre-breakdown region and has been observed in several cases [Fig. 3(A)]. A. Avalanche Breakdown Avalanche breakdown and its resulting minority car rier injection have been definitely observed on J!-and p-type etched silicon samples, depending upon the work function of the applied electrode, Where this breakdown occurred it resulted in significantly lower apparent resistivity values than measU'Ied by a 4-terminal method, as was shown in the injection region part of Fig. 3(A). It is always evident that the current-voltage relationship varies during the pulse duration. Figure 4(a) gives the current pulse just before breakdown. Except for an initial capacitive spike, the current wave shape is identical to the voltage wave. Figure 4(b) shows the current shape just after the breakdown had become stable enough to photograph. The current in this picture is almost four times that in 4(a), while the voltage was only raised 35%. Figure 4(c) gives the 27 P. A. Wolff, Phys. Rev. 95, 1415 (1954). 28 Y. Yamashita, Profiress in Semiconductors (John Wiley & Sons, Inc., New York, ]<)()O), Vo!' 4, p. oS. voltage and current curves for the stabilized breakdown (~75 V). The top curve is voltage and the lower is cur rent. Note the non-ohmic I-l~ relationship as the cur rent drops away from the voltage. This cannot be at tributed to heating since the current changes in the wrong direction. Figure 4(d) is present for compari son. It represents the usual current and voltage pulse shape obtained when the barrier is overcome by the tunneling process. The voltage and current maintain the same relationship as a function of time. A valanche injection can be eliminated from silicon and most other materials, regardless of the electrode, by sandblasting or other methods of work damaging the surface before electrodes are applied (Fig. 3). This has the effect of enormously increasing surface recombi nation, thus preventing minority carrier injection into the bulk, and relatively ohmic results are obtained by applying voltage pulses, as was shown in Fig. 3(B). This offers one explanation of why neither n-nor p-type silicon carbide, nor other semiconductors that have short minority carrier lifetimes and low mobilities (diffusion lengths of only a micron or so) , have not shown avalanche injection and the resulting lowered bulk resistance as in the discussed case of silicon.z9 Wolff27 has given the threshold for electron-hole pair production (and thus the initiation of avalanche break down) as 2.3 eV for silicon. This is just over twice the band gap. Assuming the same general relationship to hold for SiC, it is necessary for the electrons to be ac celerated to ~6 eV before avalanche breakdown could occur. The attainment of such high energies is less prob able than 2 eV, so avalanche injection should be less significant in large-gap semiconductors. The occurrence of avalanche breakdown should also be inversely related to the critical field discussed in Sec. lIIC. B. Tunneling Holm13 and Dilworth6 have given equations for the tunnel effect through an insulating film in the case where both contact members consist of the same semicon ducting materials. In the present work their theory is applied to the somewhat different situation of Fig. 6(A) in which electrodes are applied to two faces of one semiconductor which is covered by a uniform surface film. Holm gives the equation (including V in the numera tor30) for the surface resistance area product qs in units of \1-cm2 as follows: 1014Vexp[0.683(S/V)(4>~- (4)-V)~)] ---------------\1- cmz, 2.5nT~[1-exp( -eV/kT)] (2) 29 One very high purity n-type SiC sample (",10000 [I-cm) out of the 13 that were tested did show evidence of an avalanche effect. The R-V curve did not decrease in regions II or III; thus it was obvious that the data could not be used for resistivity measurements. '" R. Holm (private communication). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:472204 G. G. HARMAN AND T. HIGIER -EXPERIMENTAL .-THEORETICAL 10' ~IO~'2~----IO~'I~-----IO~o------~------~I02 VOLTAGE FIG. 5. Curve A is the data of Fig. 1 (A) (560 U-cm, n-type SIC) replotted in units of U-cm2 accompanied by its theoretical curve. Curve B is similar data for the 3 U-cm p-type SiC crystal of Fig. 2(B). S is the surface film thickness in angstroms, cp is the barrier height in volts, V is the total voltage across the barrier, n is the density of majority carriers per cmS in the bulk of the semiconductor, T is the absolute tem perature, e is the charge of the electron, and k is Boltz mann's constant. In order to adapt this equation to the situation of Fig. 6(A), it is necessary to extend Eq. (1) as follows: q81+q 82+PBL= VI/IA Q-cm2, (3) where q8t and q82 are the resistance area product of the two semiconductor surface films, and VI is the total voltage applied to the electrodes. This relates the bulk and surface resistance, in units of Q-cm2, to the experi mentally measured parameters. The data of Figs. 1 (A) and 2(B) are re-plotted in Fig. 5 in units of Q-cm2 and accompanied by the theoretical curves calculated with Eqs. (2) and (3). For the best fit with curve SA it was necessary to use the entire Eq. (3) to include the resistance of two barriers, but curve 5B was calculated on the basis of only one barrier. Thus it is apparent that the barrier on the p-type sample, biased in the forward direction, is significantly lower than the one in the reverse direction. The main features of the R vs V I curves of Fig. 5 can be discussed by considering the three separate regions as described in Sec. IIA. For region I, at low voltages, Holm gives a simple expression for the exponential in the numerator of Eq. (2), exp[1.025S(cp-YV)t]. (4) This exponential can be considered constant when V <O.2cfi. All of the voltage drop is across the surface barrier in this case. The factor V /[l-exp( -eV/kT)], (5) accounts for the rectification of the semiconductor and Holm has shown that it leads to an increase i~ resistivity with voltage when V"'kT. The curve, Fig. 1(B), taken at T=400°C shows a maximum at 0.5 V, which agrees with the value predicted31 (see Appendix). In the reverse direction of this curve the resistivity decreases when V>O.l V. It is probable that there is a built-in voltage V 0, and this rectifying effect could be accounted for by changing the entire factor to the form of V+Vo l-exp[ -e(V o+ V)/kT]' (6) Dilworth6 uses an expression of this general type. A built-in voltage Vo of this type will lead to a more rapid fall-off of the resistance vs voltage in region II. In the second region V>O.2cfi. The dominant resistance is again qs, and the main variation in resistivity vs voltage is due to electric-field-induced tunneling. A further discussion of the application of these equa tions was recently given by the present authors.S! This work considered experimental. curves of the type shown in Fig. 1 (B) in greater detail. By using expression 4 we can obtain an estimate of cp from the value of the voltage at which the resistance starts to decrease rapidly in Fig. 5. The values obtained by this method are cp= 2.5±.5 eV for the n-type sample and cp=2.3±.5 .eV for the p-type sample. The barrier thickness S can be determined from the slope of the log-log plot and is S= 11 A+2 A for the n-type sample and S= 16 A+2 A for the p-type one. Sand cp are esti mated by determining the variations of Sand cp which change the exponential by a factor of 10. The agreement of the two curves is good at the low voltage end. How ever, the theoretical curve gives a somewhat low value at intermediate voltages. The agreement could be im proved by taking into account the fact that the two films may have slightly different Sand cp and by assum ing a small amount of field-induced band-bending, as described by Spenke.32 Although only a few measure ments have been made on GaAs, this material was the only one measured that had a decrease sharp enough to produce a good fit throughout the entire region II. The value S of surface thickness which is determined by these measurements will be related to the true surface thickness by the relation S' =KS, where S' is the true surface thickness, and K is a relative dielectric coefficient. A discussion of this is included in Henisch5 (Chap. 7). c. General From the preceding it is apparent that there are two different types of barriers of significance in this study • 31 T. Higier and G. G. ~arman, Paper presented at the Interna tional Research SymposIUm on Electric Contact Phenomena Orono, Maine, November 1961 (to be published in the Conferenc~ Proceedings). 32 E. Spenke, Electronic Semiconductors (McGraw-Hill Book Company, Inc., New York, 1958), Chap. 10. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:47DIRTY CONTACTS ON SEMICONDUCTORS 2205 B .FIG. 6. Equilibrium metal-semiconductor energy level diagram for n-type semiconductors. (A) is for approximately equal work functions, but with an artificial layer-type barrier. q, is the energy difference of the barrier above the Fermi level of the metal. S is the barrier width; (B) is for a metal with a work function greater than the semiconductor and results in an additional barrier due to bending of the bands. of dirty contacts. One results in avalanche breakdown and the other in tunneling. The most important consideration for silicon is the contact potential difference between the electrode and the specimen, the ever-present layer of Si02 ( ",30-40 A· and of low barrier height) being of little consequence. For an ohmic contact (indium or tin-gallium on n-type, or graphite on p-type) there is no bending of the bands. Adhered· oxygen, water vapor, or other surface con taminants produce only slow states which do not hinder dc measurements and are swamped by high voltage pulses as shown in Fig. 3, curves Band C. For this situation in the band picture of Fig. 6(A) is representa tive. The thin barrier due to the artificial layer is penetrated by tunneling. When the electrode work func tion is such that avalanche breakdown and injection occur (less than 4 eVan p type and greater than 4 eV on n-type Si), then another model is applicable. In this case, there is significant bending of the bands, and an exhaustion layer is created as shown in Fig. 6(B). When voltage is applied, the height and width of the reverse-biased barrier will increase, and its character istics will totally dominate current flow through the sample. The width (> 1000 A on a typical sample) of this region is so much greater than that of the Si02 that it dominates the characteristics. Tunneling or field emission is improbable through such a wide region so, as reverse voltage is applied to the barrier, avalanche breakdown occurs. There is, however, always the pos sibility that tunneling can occur for the case shown in Fig. 6(B), providing that the composite barrier width is <50 A. This might occur if the semiconductor is degenerate or near degenerate. If the composite barrier is > 50 A, then there is the possibility of field emission or Schottky emission. These will be obvious by the relatively high voltage required to enter region II and the impossibility of fitting the resulting data to Eq. (3). Nevertheless good two-terminal resistivity measure ments can be achieved for either of these cases, since they do not result in minority carrier injection as did avalanche breakdown in Fig. 3(A). The nature of the surface layer and/or surface states on SiC is quite different from those of silicon. No dirty contact could be found that produced ohmic results similar to those obtained by rubbing graphite on p-type silicon [Fig. 3(C)]. There is always a significant barrier on both n-and p-type SiC. Since the Si02 layer on the surface was removed by HF and this layer re builds very slowly, if at all, at room temperature, there must be a very dominant atmospheric type surface state present. It must be of sufficient width or height to cause tunnel-effect limiting of the current. Dillon et al.33 have shown that oxygen is adsorbed on the SiC surface with a sticking coefficient of approximately O.Ol. Its effect is sufficient to reduce the photoelectric yield and increase the work function by 0.61 eV above the argon-bombarded high vacuum-treated surface. Such a surface state, with perhaps a contribution from water vapor or other ambient gases, could well explain the present observations. ACKNOWLEDGMENT The authors wish to acknowledge many valuable discussions with Owen L. Meyer of the Diamond Ordnance Fuze Laboratories. APPENDIX The barrier thickness S and height cp can be calculated from a given experimental R-V curve by considering the variation of the main terms in Eq. (2). The case when only one barrier contributes to the surface resistance is given first. Two equations are required to determine Sand cp. The experimental ratio at two different voltages V 1 and V 2 are equated to the theoretical ratios as deter mined from Eqs. (2). To obtain the first equation, two values of voltage V 1 and V 2 can be chosen in the range 0.1> V 2': 0.3cp. The theoretical ratio will then be given by q,theory V 1exp[1.025S(cp- V 1/2)IJ q,theory V 2exp[l.025S(cp- V z/2)tJ (A1) This ratio is equated to the experimental resistance ratio between V 1 and V 2. A second equation is obtained by taking a point in region II. A value V 3 can be taken which lies between 2 and 4 V.6 One limit on the value V3 is that the experi- 33 J. A. Dillon, Jr. R. E. Schlier, and H. E. Farnsworth, J. App\. Phys. 30, 675 (1959). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:472206 G. G. HARMAN AND T. HIGIER mental resistance should be greater than 10 times the bulk resistance. Now the theoretical ratio will be given by q,theory (A2) q,theory This ratio is now set equal to the experimental ratio of resistances at V 2 and V 3. When V 3 is chosen so that V3=3V, a reasonable approximation is to take (<p-Va) = O. By repeating this calculation for a few points an estimate of the error in Sand cf> can be obtained. When both barriers contribute to the surface re sistance, in general the values of S]<p]! and S2<P21 must JOURNAL OF APPLIED PHYSICS be equal within 2 or 3%. In this case, the resistance of the two barriers will be equal, and the value of voltage used in formula 2 will be just t the experimental volt age. In general, when SI differs from S2 and <PI differs from t/!2, four experimental points will be needed to determine these parameters. One other value can be calculated from Eq. (2) when Sand <P are known. This is the voltage V at which Eq. (2) has a maximum. The main variation in Eq. (2) when V is greater than 0.1 v is due to 6.q."" Vexp[1.025S(t/!V/2)iJ. (A3) The maximum of this expression is determined by put ting its derivative equal to zero, which gives 1.025SV(t/!- V /2)-b4. (A4) VOLUME 33. NUMBER 7 JULV 1962 Some Electrical Properties of the Porous Graphite Contact on p-Type Silicon * GEORGE G. HARMAN AND THEODORE HIGIER National Bureau oj Standards, Washington, D. C. AND OWEN L. MEYER Diamond Ordnance Fuze Laboratories, Wasltington, D. C. (Received January 22, 1962) Some unusual properties of the porous graphite contact on p-type silicon are described. Current through the sample reaches a saturation level that is inversely proportional to the amount of adhered water vapor. Other gases such as ammonia, H2S, and HCl modify the shape or amplitude of the saturation current. There is a hysteresis loop in the low voltage region which is similar in appearance to that of a ferroelectric. The general method of measurement can be applied to studying the semiconductor surface as well as the contact phenomena. The possible applications include such devices as current regulators, humidity detectors, and surface-barrier radiation detectors. An electronic band model, which includes a trap-dominated inversion layer, is presented to explain the phenomena. This model also integrates various conflicting theories of metal-semiconductor contacts. IN the course of studying the surface properties of semiconductors, using "dirty contacts," it was found (after a study of work function dependence) that gra phite rubbed onto the surface of p-type silicon produced an essentially ohmic contact.! For ease of application, a water-based paste of graphite was applied to etched,2 p-type silicon. The characteristics were essentially ohmic when the graphite dried. However, this contact was not ohmic when it was in high humidity. Instead the current reached a saturation value that was stable to about 100 V, as shown in Fig. 1 (a). Stable charac teristics with values intermediate between the saturated * Part of this work was sponsored by the Air Force Cambridge Research Center, Bedford, Massachusetts. I G. G. Harman and T. Higier, J. Appl. Phys. 33, 2198 (1962), preceding paper. 2 All samples described in this letter were etched in 90% HNOs + 10% HF for one minute. One sample was etched in CP4 and appeared to have a lower breakdown than the others, but this etching effect was not investigated further. humidity and dry curves of Fig. 1 (a) were achieved by controlling the sample humidity. • It should be pointed out that the experimental curves (Fig. 1) were obtained with two identical contacts back-to-back, the reverse electrode (determined by applied-voltage polarity) controlling the current flow. A single contact was studied by alloying aluminum, for a coiwentional ohmic contact, as the counter electrode. When reverse-biased (positive on graphite), the graphite contact had the same characteristics as in Fig. 1, but it showed injection under forward bias. At low voltage there is a hysteresis loop [Fig. 1(b)J which changes shape and magnitude with the bulk resistivity, the humidity, and the rate of change in applied voltage with respect to time (dv/ dt). The barrier capacity was measured in the hysteresis region and showed an increase with reverse bias, in contrast to tht. usual decrease with depletion layer widening. This [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 141.210.2.78 On: Tue, 25 Nov 2014 18:21:47

1.1735850.pdf

Contribution of Anode Emission to Space Charge in Thermionic Power Converters Anthony F. Dugan Citation: Journal of Applied Physics 31, 1397 (1960); doi: 10.1063/1.1735850 View online: http://dx.doi.org/10.1063/1.1735850 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Cascaded space solar power system with high temperature Cs-Ba thermionic converter and AMTEC AIP Conf. Proc. 420, 1565 (1998); 10.1063/1.54785 Cascaded Thermionic Converters Applied to Space Nuclear Power Systems AIP Conf. Proc. 301, 819 (1994); 10.1063/1.2950275 Fast neutron thermionicconverters for highpower space nuclear power systems AIP Conf. Proc. 217, 657 (1991); 10.1063/1.40121 Extended SpaceCharge Theory in LowPressure Thermionic Converters J. Appl. Phys. 33, 2485 (1962); 10.1063/1.1729001 Electron SpaceChargeLimited Operation of Cesium Thermionic Converters J. Appl. Phys. 33, 1445 (1962); 10.1063/1.1728752 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Wed, 24 Dec 2014 04:55:43ELECTRODES IN A LOW-PRESSURE PLASMA 1397 in the low ion energy region when a light ion bombards a material of much heavier atoms because the accommo dation coefficients here are expected to become sub stantially less than unity. ACKNOWLEDGMENT The skillful and patient work of B. Richelman, R. Moseson, and our glass experts, Emil Benz and Arthur Haut, is gratefully acknowledged. JOURNAL OF APPLIED PHYSICS VOLUME 31. NUMBER 8 AUGUST, 1960 Contribution of Anode Emission to Space Charge in Thermionic Power Converters ANTHONY F. DUGAN Lockheed Missiles and Space Division, Sunnyvale, California (Received December 16, 1959; and in final form April 6, 1960) The space charge theory of Langmuir has been extended to include the effects of anode emission on the performance of a vacuum thermionic power converter. The basic equation is similar to Langmuir's ~-7J equation, but it involves two additional parameters which depend on the temperatures and thermionic properties of the electrodes. An iterative technique is described for obtaining solutions in specific cases, and some sample calculations based on hypothetical diodes are presented. The calculations indicate that the effect of the anode temperature is considerably more pronounced if the cathode-anode work function difference is large. I. INTRODUCTION IN the treatment of space charge effe8ts in vacuum thermionic diodes, it is usually assumed that the anode emission is sufficiently small to be neglected. An analysis based upon this assumption leads to the familiar Langmuir space charge equation.1,2 This equation has been used by several authors, for example, Webster,3 to determine performance characteristics of thermionic power converters. In many cases, however, neglect of anode emission cannot be justified; for instance, in space applications it is desirable to reject waste heat by radiation directly from the anode in order to minimize the weight of the system. A converter operating at 20% efficiency and producing 3 w/cm2 requires the anode to reject 12 w/cm2• Assuming black body conditions, the anode temperature would have to be greater than 1200°K. At this temperature, the anode emission can be appreciable and should, therefore, be considered in determining the expected performance. The effects of anode back current on the efficiency of a thermionic diode have been considered by Houston.4 In his paper, however, he did not consider the contri bution of the anode to cathode current to the space charge. The purpose of the present paper is to describe a method of calculating space charge when both cathode and anode are emi tting. II. ANALYSIS Figure 1 is a schematic representation of the potential energy of an electron in a thermionic power converter. tPc and tPa are the true work functions of cathode and 11. M. Langmuir, Phys. Rev. 21,419 (1923). 2 T. C. Fry, Phys. Rev. 17, 441 (1921). 3 H. F. Webster, J. AppI. Phys. 30, 488 (1959). 4 J. M. Houston, J. AppI. Phys. 30, 481 (1959). anode, respectively. Vo is the effective output voltage of the converter and is determined in actual practice by matching the external electrical load to the operating characteristics of the converter. V m is the potential energy maximum, and is determined by the distri bution of electrons between the electrodes. The follow ing derivation of the space charge distribution employs an analytical approach similar to that used by Fry.2 If a surface is at temperature T, it will emit electrons with a Maxwell velocity distribution characteristic of T. For a single surface, say the cathode, the number of FL c I I I I I I I eVm I I a '---......--- FL I I eVo ---.J __ t ______ l ____ _ -----l xmJ-1 .... --- FIG. 1. Schematic representation of an~electron's potential energy in a thermionic power converter. FL denotes the Fermi level. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Wed, 24 Dec 2014 04:55:431398 ANTHONY F. DUGAN electrons emitted per unit time per unit area with velocity components normal to the surface between Vo and vo+dvo is mvo dNe=Ne-exp( -mvo2/2kTc)dvo. kTe (1) The charge density at a point x where the potential is V can be found from pe=ei'" dNe(Vo), 'Vo' V (2) where [ 2e(V -4>e)]! v= vo2 m (3) is the velocity which an electron whose initial velocity was Vo has when it reaches x. Equation (2) must be integrated over the appropriate velocities. If x<xm, one finds at x, in addition to those electrons whose initial velocities were sufficient to carry them past x"" a reverse current consisting of those electrons whose initial velocities were sufficient to carry them past x, but not past xm. For X>Xm, one finds these electrons missing. The charge density is, therefore, Pe=eNe~ exp(-e(V -4>e»)[i'" exp(-mv2 )dV kTc kTe 0 2kTe i[2eCVm-V)/mJ' ( mV2) ] ± exp --- dv , o 2kTc (5) in which the upper and lower signs refer, respectively, to x<xm and x>xm• Equation (3) has been used to change the variable of integration Vo to v. The result of inte grating Eq. (4) is: pc=eNc(7rm/2kTc)1 exp[ -e(V-4>c)/kTc] X[l±erf{[e(V m-V)/kTc]!}], (5) where 2 iZ erf(z)=-- exp(-i)dy. (n')! 0 When the anode emits, the charge density will be increased by a term similar to that given by Eq. (5). (6) Poisson's equation relates the potential to the charge density: (7) [ J [e(V m-V)]!}] 1 X FFerf1 -kT:- f' (8) With the following changes of variables: 'I)=e(V m-V)/kTc, (9) a=Tc/T a, and some manipulations, Eq. (8) can be put into the form: (10) where 'l)c and 'l)a are the values of 'I) at cathode and anode, and the upper signs are to be used for ~<O and the lower signs are to be used for ~>O. Equation (10) is readily seen to reduce to Langmuir's equation if the anode temperature goes to zero. N a and N c are simply proportional to the saturation current densities which can be computed from Richardson's equation: 18=A J'2 exp( -e4>/kT), where A is the emission coefficient in amp/cm2 °K2. For the purpose of demonstrating the method, values of A and 4> were chosen which were obtained from Richardson plots for some materials. Strictly speaking, one should use the true work functions and set A = 120 amp/cm2 °K2. In the calculations described below, however, the empirical values of A and r/> were used throughout and no temperature dependence was considered. The calculations, therefore, describe hypothetical diodes for which the true work function and the Richardson work function are identical. The main features of the results are not seriously effected by this assumption, and the computational procedure is the same if the true work functions are used. A first integral of Eq. (10) can be obtained, but from there on one must use numerical methods. Letting 1'0 and I.e be the saturation currents from anode and cathode, and integrating Eq. (to), d'l)/ d~= {e~(l±erf'l)!) -[1± (2/7r!)r,!] + B(ea~[1 =r=erf(a'l)!]- [1 =r= (2/7r!) (a'l)t])}, (11) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Wed, 24 Dec 2014 04:55:43A NOD E EM ISS ION I N T Fl E R M ION I CPO W ERe 0 N V E R T E R S 1399 where B=(I.,;/I. c)a-ltexp('I1c-O!'I1,,). The net current drawn by the diode is (12) Since '110 and '11 .. involve V m, an iterative procedure is required to solve for V",. If the anode-cathode sepa ration is d, then (13) (14) with 18e in amp/cm2 and d in em. The procedure for finding V m is as foHows! Calculate "1c-1Ja=e(V o+cp,.-cpc)/kT c, Vo is the output voltage of the converter and is a known quantity. WebsterS assumes that the current is known, and calculates V(l. With the anode emitting, however, it is more con venient to assume Vi) and calculate the current. Now, 1Jc=e(V,,,- V)/kTc>'1.-'1Q' Choose, therefore, a value of '1c>'lc-1/al and find 71". Using Eg. 14, ~,.-~c can be calculated. Using the values of '11. and 71", Eq. (11) can be numerically integrated to obtain ~. and ~a. Next, compare the value of ~,,-~e obtained by integration of Eq. 11 with the value obtained from Eq. 14. Let o=(~a-~c)14-(~,,-~c)n where the subscript refers to the number of the equation by which the quantity is calculated. 0 will be zero for the correct value of 11 •• If o is not zero, fie should be improved by decreH.sing or increasing according to whether a is negative or positive. The process can usually be made to converge in five or six iterations. A program was set up for the IBM 709 to perform the calculations, and several performance curves were obtained. III. DISCUSSION It would appear worthwhile to be able to generate a family of solutions of Eq. 11 for various values of a and B. If B were a function of a only, it would be possible to generate such a family, but B depends on CPc, CPa, Te, and To, through l$c and I$(J' The result of this situation is that each converter under consideration must be treated as a special case. The usefulness of the method lies in the fact that it furnishes a way of determining under what conditions anode emission becomes a serious problem. This is demonstrated by some calculations based on the properties of some materials from which typical converters might be constructed. In the calculations described below, the cathode has the emission properties of the L-metal cathode, €Pc':::: 1.65 volt, and Ac=3 amp/cm2 °K2. The usual type of converter has an anode with a low work function, therefore, calculations were made for cP,,= 1.0 v and A,,=6.5 a.mp/cm2°K2. In order to determine the effects O~ __ ~_I __ ~ __ ~~~~~~~~~ 0." FIG. 2. Current voltage characteristics for t\VO types of ther mionic power converters. In the upper three curves, the anode has a work function of 1.0 v and emission coefficient of 6.5 amp/cm2 °K2. In the lower three curves, the anode has the same properties as the cathode, work function 1.65 v and emission coefficient 3.0 amp/em2 "K'. The electrode separation is 0.002 em and the cathode temperature is 1400°K, of changing the anode work function, some compu tations were made for the case in which the anode was the same material as the cathode. The effect of anode work function can be seen from Fig. 2 where current is plotted vs output voltage for various anode temperatures. The curves for which cPg,= 1.0 v start at higher currents than those for cp" = 1.65 v. As the anode temperature increases, the current drops more noticeably for the upper curves (cp,.= 1.0 v) than for the lower curves. The threshold t.emperature beyond which anode emission is appreciable is lower for a diode with a low anode work function. The drop in current as VQ is increased is attributable to two mechanisms. First, the forward current from cathode to anode decreases. Second, the back current from anode to cathode increases. Both these effects are readily understood from Fig. 1. Increasing V Q increases V m, and also causes V", to move doser to the anode. The limiting position of V m is at the anode surface. This implies that as Vo increases, (V",-CPa- Vo) decreases which in turn increases the anode emission. The forward current on the other hand decreases exponentially with increase in Vo. At highe.r tempera tures, the effect of increased back emission is greater than the effect of decreased forward emission. Figure 2 demonstrates this fact. If there is no anode emission, T ~ 8()()OK and T ~ l0000K for CPa= to v and 1.65 v, respectively, the current asymptotically goes to zero as V 0 is increased. At temperatures at which anode emission is appreciable, however, the current decreases more rapidly as Vo is increased, and becomes negative at some voltage which is not particularly large. Figures 3 and 4 are power output as a function of load voltage for the two types of converters. In Fig. 4, the anode has [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Wed, 24 Dec 2014 04:55:431400 ANTHONY F. DUGAN 0.1....----------....., 0.08 N E ~ 0.06 '6 ~ .: 0.04 " ~ Q. 0.02 Load, volts FIG. 3. Output power as a function of load voltage. Anode and cathode have the same thermionic properties </>= 1.65 v and A =3.0 amp/cm2 °K2. Electrode separation is 0.002 em and cathode temperature is 1400oK. the low-work function, and the power output is seen to be quite sensitive to the anode temperature. Even at Ta=900oK, the power peak has dropped to about two-thirds of what it was at Ta=O°K. With the high anode work function, an anode temperature of 9000K causes only a slight diminution in power, and then only at higher voltages. The effect of cathode-anode separation is quite interesting. It is usually stated in discussions of ther- 0.40,--------------, 0.35 0.30 N ~ 0.25 .... .. :: ~ 0.20 .. ~ 0.15 Q. 0.10 Load, volts FIG. 4. Power output as a function of load voltage for </>a= 1.0 v, A =6.5 amp/cm2 °K2, </>c= 1.65 Y, Ac=3.0 amp/cm2 °K2. Electrode separation is 0.002 em. Cathode temperature is 1400oK. mionic power converters that one way to eliminate space charge is to uS,e very small separations. Figure 5 shows the effect of spacing on the power output for the low-work function anode type of converter. There is a crossing of the power curves. This is understandable by the same argument that was used to explain the fact that increasing Vo increases the anode emission. Reducing the spacing causes a reduction in the height of the potential energy maximum, and also reduces the values of Vo for which the maximum reaches its limiting .4 II:: LI.I ~ .3 D.. .2 .1 .2 .3 .4 .6 LOAD. VOLTS FIG. 5. Comparison of power output for spacings of 0.001 em and 0.002 em. The anode properties are </>a= 1.0 v, Aa=6.5 amp/cm2 °K2, Ta=l000°K. For the cathode, </>.=1.65 Y, Ac=3.0 amp/cm2 °K2, and Tc= 1400oK. Whereas the peak is considerably higher for d=O.ool em, note the much sharper drop as the peak voltage is passed . position-the anode surface. As a result, for higher voltages, the net current drops as the spacing is de creased. This occurs however, at such a voltage that the net power output is too low to be of interest. The anode emission problem, it can be concluded, becomes most serious at voltages beyond that which yields peak power output. Radiation can, therefore, still be effectively used to reject waste heat, but if the anode work function is much lower than the cathode work function, one is limited to low operating voltages. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Wed, 24 Dec 2014 04:55:43

1.1744459.pdf

MolecularOrbital Treatment of Isotropic Hyperfine Interactions in Simple Aliphatic Radicals Donald B. Chesnut Citation: The Journal of Chemical Physics 29, 43 (1958); doi: 10.1063/1.1744459 View online: http://dx.doi.org/10.1063/1.1744459 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/29/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Molecular Orbital Calculation of the Isotropic Hyperfine Interactions in Triatomic Nitrogen Radicals J. Chem. Phys. 57, 2212 (1972); 10.1063/1.1678556 Theory of Isotropic Hyperfine Interactions in PiElectron Free Radicals. I. Basic Molecular Orbital Theory with Applications to Simple Hydrocarbon Systems J. Chem. Phys. 50, 511 (1969); 10.1063/1.1670829 MolecularOrbital Theory of Geometry and Hyperfine Coupling Constants of Fluorinated Methyl Radicals J. Chem. Phys. 48, 4802 (1968); 10.1063/1.1668077 Configuration Interaction in MolecularOrbital Theory J. Chem. Phys. 47, 4611 (1967); 10.1063/1.1701673 Isotropic Hyperfine Interactions in Aromatic Free Radicals J. Chem. Phys. 25, 890 (1956); 10.1063/1.1743137 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 01:00:52CLEANUP OF ATOMIC HYDROGEN 43 nitrogen, and the tube was then allowed to warm up very slowly. Measurements were made every 10 min, but not all the data are shown on the curve, so some of the points correspond to measurements made 30 min apart. With measurements made at this slow rate no large amount of hydrogen was recovered until after the temperature had risen above -150°C. THE JOURNAL OF CHEMICAL PHYSICS ACKNOWLEDGMENT The author wishes to acknowledge the help she received from Dr. Irving Langmuir who recommended the problem and made many suggestions, and to Lloyd B. Nesbitt who took charge of filling the Dewar flash with liquid hydrogen. VOLUME 29, NUMBER 1 JULY, 1958 Molecular-Orbital Treatment of Isotropic Hyperfine Interactions in Simple Aliphatic Radicals* DONALD B. CHESNUT Department of Physics, Duke University, Durham, North Carolina (Received March 12, 1958) The problem of isotropic hyperfine interactions in the EPR spectra of aliphatic free radicals is approached from the molecular-orbital picture of hyperconjugation. The ethyl, methylethyl, and 1,l-dimethylethyl radicals are treated by this approximation; with a reasonable choice of parameters, the results can be cor related rather well with our present knowledge of aliphatic radicals. The calculated coupling constants of methyl group hydrogens are of the order of 15 to 25 gauss, do not decrease radically with the presence of additional methyl groups, and are very nearly proportional to the molecular-orbital unpaired electron density at the central carbon atom. I. INTRODUCTION THE occurrence of isotropic hyperfine interaction in the electron paramagnetic resonance (EPR) spectra of aromatic free radicals is well known. Weiss manl and McConnelJ2 have shown that the coupling mechanism can be explained in terms of configuration interaction, and Jarrett3 has shown that this postulate leads to interaction terms of the order of magnitude of those observed experimentally. McConnell and Ches nut4 have given a molecular orbital (MO) treatment of the configuration interaction problem and have ex plained the apparent proportionality of the coupling constant for proton i to the MO unpaired electron density at the adjacent carbon atom and thus the observation that the total spread of spectra is approxi mately independent of the number of coupling protons.5 The occurrence of isotropic hyperfine interaction in the EPR spectra of aliphatic free radicals is also well known, but not so readily understood. Contrary to the case of aromatic radicals, the total spread of certain series of aliphatic radicals is very sensitive to the number of coupling protons, at times being apparently almost directly proportional with a constant of pro portionality of about 20-25 gauss per proton.6 Also * This research was supported by the U. S. Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AFI8(600)-497. 1 S. I. Weissman, J. Chern. Phys. 25, 890 (1956). 2 H. M. McConnell, J. Chern. Phys. 24, 764 (1956). 3 H. S. Jarrett, J. Chern. Phys. 25, 1289 (1956). 4 H. M. McConnell and D. B. Chesnut, J. Chern. Phys. 28, 107 (1958). • See, however, H. M. McConnell and D. B. Chesnut, J. Chern. Phys. 27, 984 (1957). 6 See, e.g., Gordy, Ard, and Shields, Proc. Nat!. Acad. Sci. U. S. 41, 983, 996 (1955). spectra are often obtained which are indicative of equivalent protons when the radical felt to be present contains chemically nonequivalent protons. In the absence of a multi-?r system, one means of achieving multiple isotropic electron-proton spin-spin interactio.ns in aliphatic free radicals is through hyper conjugation. Crawford and Coulson7 have treated hyperconjugation in methylated benzenes and have given numerical parameters which, in simple MO theory, lead to a satisfactory correlation of certain molecular properties, e.g., the dipole moment, with experimental observation. Bersohn8 has treated iso tropic hyperfine interaction through hyperconjugation in methylated semiquinones by a perturbation method and has achieved good agreement with the observed results. With the necessary data available, it would seem worthwhile to treat isotropic hyperfine interaction through hyperconjugation in a few of the simpler aliphatic free radicals by simple MO theory. The molecular systems chosen for the present calculations are the ethyl radical (I), the methylethyl radical (II), and the 1, 1-dimethylethyl radical (III). H "" / H ·C-CH3 (I) H "" / H3C ·C-CH3 (II) 7 V. A. Crawford and C. A. Coulson, J. Chern. Soc. 1953, 2052. 8 R. Bersohn, J. Chern. Phys. 24, 1066 (1956). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 01:00:5244 DONALD B. CHESNUT (III) II. THEORY Molecular-Orbital Treatment The treatment of hyperconjugation in MO-LCAO theory entails the use of hydrogen group orbitals.9 These group orbitals are listed in Table I along with TABLE 1. Hydrogen group orbitals for the methyl group. Hydrogen group orbItal" 4>,+4>2+1>3 XI = (3+6SHH)'/2 24>,-4>2-4>, X2= (6-6SHH)I/2 4>2-4>, Methyl-carbon wave function p, " 4> i = hydrogen Is atomic orbital of hydrogen i. SHH={ 4>il4>j). i",j the methyl carbon atom wave functions of similar symmetry. The 1 axis is the threefold symmetry axis of the methyl group and axes 2 and 3 are at right angles to it and to one another. The orientation of axes 2 and 3 is such that they are parallel to the pseudo-p orbitals Xl! and Xa. One may picture the bonding in the methyl group as follows: one of the two SPI hybrids of the carbon bonds with Xl forming a II bond, while the carbon P orbitals P2 and Pa form 7r-type bonds with X2 and X3, respectively; the second SPI carbon atomic orbital is available for II bonding the methyl group to some other atom or molecular group. If the methyl group is attached to a 7r system, the hydrogen group may be thought of as a pseudo-atom X contributing one electron to the total pseudo-7r or hyperconjugated system; likewise, the methyl-carbon will be part of this system, also contributing one electron. Coulomb and resonance integrals are assigned to the various centers and the calculations made in the usual way. Thus, if if;i denotes the 7r orbital centered on atom i, the mole cular wave function for the Xth energy level is given by where, in the case of hyperconjugation here considered, some of the if;. will correspond to 7r orbitals of methyl carbon atoms and the pseudo-atoms X. In the present calculations overlap of all but adjacent atomic orbitals is neglected. It is assumed that the central carbon atom (that atom holding the odd electron in classical valence structures) is sp2 hybridized, and that resonance integrals may be taken as proportional to the corresponding overlap integrals.lO Thus, following the notation of Crawford and Coulson,7 the quantities of interest are 5ii= (if;ilif;i), H ii= (if;iIHIif;i)=a+Ei, Hii= (if;iIHIif;i)=I'i{=Piil'O, I'ii 5ij p"=-=-' '1 1'0 50' (2) where 1'0, (30, and 50 are the values of these parameters for unsubstituted benzene. The calculations may be illustrated by considering the ethyl radical (I). A schematic picture of hypercon jugation for this radical is shown in Fig. 1 where two "static" geometrical configurations of optimum con jugation are shown. In configuration (a) the central carbon pz orbital conjugates with the pseudo-7r system composed of P2 and X2, while in configuration (b) the pseudo-7r system is made up of Pa and Xa. Now it can easily be shown that (3) so that our calculations will not be affected by taking the ethyl radical, for purposes of simplicity, to have either configuration (a) or configuration (b) so far as the molecular wave functions or energy levels are concerned. It will, however, prove useful in the later problem of calculating the coupling constants if we picture each configuration as contributing an equal amount to the "real" state of the system. In a sense what one is doing then is to allow the methyl group to "rotate." Let the 7r centers in the ethyl radical be labeled as 'lIx= L,C;if;i, (1) The secular determinate is then --------------------------------------- a+EI-W o It may easily be shown with the use of group theory that the calculations for the methylethyl (II) and 1,1-dimethylethyl (III) radicals-so far as the eigen- 9 C. A. Coulson, Valence (Oxford University Press, London, 1952),310. o P23[{30+ 50(a-W) ] = O. (4) value for the unpaired electron level is concerned are identical to Eq. (4) if one replaces P12 by (m)!P12, where m= 2, 3 is the number of methyl groups attached 10 R. S. Mulliken, ]. chim. phys. 46, 497 (1949). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 01:00:52H Y PER FIN E I N T ERA C T ION SIN S IMP LEA LIP HAT I C R A D I CAL S 45 FIG. 1. A sche matic representa tion of hypercon jugation in the ethyl radical. to the central carbon atom. It is assumed that the molecule may be represented by a single electronic configuration and that the odd or unpaired electron occupies the lowest energy level remaining after the lowest levels have been filled with electron pairs. The parameters in Eq. (2) were given the following values (radicals II and III as well as I) . El=O, E2= -0.1/30, Ea= -0.5/30, P13= 0, P2a= 2.5, P12=0.7 or P12=0.93. These are taken from Crawford and Coulson7 with the exception of P12=0.93, which was the value of P12 used by Bersohn8 in his work. Calculations were made for both of these values of P12, the results appearing in Table II. Normalization and Unpaired-Electron Density The inclusion of overlap requires the normalization condition to be where 1= (\}fxi\}fx)= LUCiXCiA(ifiiiifij) = Li.iCiXCiASij = LiCiAYiA, (5) If'l'o is the MO wave function of the odd electron, the odd-electron density at atom k, qk, is given byll (7) The foregoing definition of unpaired-electron density as given by Chirgwin and Coulson is the same as that given here by McConnell and Chesnut's5 spin density, Pk,nin, which, in the present approximation, reduces to Pk8nin= (\}foiAki\}fo) = LiiC;oCiO(ifiiiAkiifii) =CkOY kO, where the operator Ak satisfies the equation tJ.kifi i= ojkifi i· (8) (9) It should be noted that the present approximation requires values of the spin density to be positive, a situation which is not always necessarily SO,4.5 and which may not be true in the present problem. Calculation of Isotropic Hyperfine Interaction The Fermi contact Hamiltonian for isotropic hyper fine interactions has been discussed. 4 We consider here only the coupling of the methyl-group protons; it is assumed, as discussed later, that a knowledge of the electron density distribution will allow us to predict the coupling of protons bonded to the central carbon atom. Assuming the strong field case, the Fermi term for proton I reduces in the MO approximation to the evaluation of (10) where Il(rl) is the Dirac delta function for the distance r I measured from the lth proton. 01= Li';CiOC iO (ifiiio (rl) [ifii) (11) where (12) TABLE II. Results in the treatment of isotropic hyperfine (6) It is at this point that our model of the radical being composed of two equally contributing "static" con figurations is useful. It is immediately obvious that had we chosen one static configuration the three protons of a hyperconjugating methyl group would be mathe matically non equivalent and would acquire different ° values, whereas they are chemically equivalent. In our simple model, however, each configuration con tributes to the Il value of each proton. Thus, we must modify the definition of ri;' in Eq. (12) to be interaction through hyperconjugation for two values of the parameter PI'. Coupling Odd-electron density constant q, qx jAm,1 IAm,1 IAm.1 Central Radical (gauss) carbon Pseudo-atom q, qx P12=0.70 I 16.9 0.9160 0.0815 18.48 207.6 II 15.5 0.8433 0.0743 18.33 208.0 III 14.2 0.7825 0.0684 18.19 208.2 P12=0.93 I 27.8 0.8677 II 24.2 0.7669 III 21.5 0.6897 0.1307 31.99 0.1137 31.57 0.1008 31.17 212.4 212.9 213.4 (13) 11 B. H. Chirgwin and C. A. Coulson, Proc. Roy. Soc. (London) A201, 196 (1950); see also R. S. Mulliken, J. Chern. Phys. 23, 1833 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 01:00:5246 DONALD B. CHESNUT so that r i/ represents the average value of the quantity in brackets averaged over the equally contributing "static" configurations. For example, r231 of the ethyl radical will be It is easily shown that this procedure makes the r i/ equal for all I, thus giving each proton the same value of O. It may be mentioned that had one chosen the static conjugating system PzP2X2 (as is usually done) and taken an average 0 by averaging the individual oz's over this one configuration, the results would be the same. This latter approach, however, has the disad vantage that it apparently initially requires the methyl group to remain fixed in order to present a picture of "static" conjugation as in benzene, and then requires a rapid rotation to achieve proton coupling equivalence. Explicit calculations show that only the matrix ele ments r is (i= 1, 2, 3) are important in the evaluation of 0 for the ethyl radical (and also the analogous terms in radicals II and III). Thus, for the ethyl radical we may calculate () to a very good approximation as 0= LiC ;oCaor i3(2-OiS) , (14) where Oi3 is the Kronecker delta. Values of 0 are con verted to coupling constants (in gauss) for the methyl protons, IAmel, by the equation () IAmel = ¢2(0) 506.2, (15) where ¢ is the hydrogen 1s atomic wave function, and 506.2 the coupling constant in gauss for atomic hydro gen. The values of the important r matrix elements are: r3s=0.96580, r23= 0.22050, r13= 0.00811, (4)(0) = 1.4707). In evaluating these elements the following distances and angles were assumed: H / ·C-C-H '" H ·C-C=1.54 A, C-H=l.09 A, L ·CCH=109°28', LHCH=109°28'. Hydrogen-like wave functions were used with (Z.ff) carbon = 3.25 and (Z.ffhYdrOKen = 1.00. III. DISCUSSION Table II shows the results for the two values of P12. Bersohn 8 initially tried P12= 0.7 in his treatment of the semiquinone problem, but found that by using P12= 0.93 one could obtain much better agreement with the experimental data. Here, too, the larger value of P12 gives coupling constants closer to the often observed splitting of 20-25 gauss. The results of interest in the present work are not, however, in the exact numerical values obtained-which, due to the extremely crude approximations involved, are of dubious significance- but in the following: (1) The coupling constants are of the observed order of magnitude. The choice of a value for P12 is quite arbitrary. If one accepts P12=0.7 for the methyl benzenes, it seems reasonable to expect P12 in the present problem to be somewhere between 0.7 and 1.0. (It is assumed that the other parameters may be kept the same.) The coupling constant is rather sensitive to this parameter, varying essentially in a linear manner in this region; a value for P12 of unity gives a coupling constant of about 31.5 gauss for the ethyl radical. The order of magnitude is, however, preserved. One's only justification for perhaps putting more emphasis on P12=0.93 is that it gives good agreement with experi ment in another similar problem.8 The isopropanol radical postulated by Ingram et at.,t2 containing equivalent protons (assuming the hydroxyl proton is prevented from coupling by the shielding of the oxygen), has a line spacing of about 20 gauss and a total spread of 120 gauss, close to the results one would predict from the present calculations. (2) The coupling constant is not radically di minished when more than one methyl group hyper conjugates with the central carbon atom. This helps to explain why the total spread of the EPR spectra of aliphatic radicals often seems to increase linearly with the number of coupling protons, as opposed to the case of aromatic free radicals where, as mentioned pre viously, the total spread remains roughly constant. (3) The coupling constant, to a good degree of ap proximation, is proportional to the MO odd-electron density at the central carbon atom. If this is in fact true, it allows the "accidental" equalization of proton coupling resulting from two apparently different mechanisms. It has been proposed4 that for aromatic systems in which the carbon atoms are sp2 hybridized, the coupling constant for an sp2-bonded hydrogen is approximately -22 (±5) gauss per unit of spin density on the bonding carbon atom.IS Table II shows that for a specific value of P12 the ratio of the methyl-hydrogen coupling constant to the odd-electron density on the central carbon atom is essentially constant, varying 12 Ingram, Gibson, Symons, and Townsend, Trans. Faraday Soc. 53, 914 (1957). 13 See reference 8 and H. M. McConnell and H. H. Dearman, J. Chem. Phys. 28, 51 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 01:00:52H Y PER FIN E IN T ERA C T ION SIN S IMP LEA LIP HAT I C R A D I CAL S 47 with PI2 from about 18 to 32 in the range 0.7 :S;PI2:S; 0.93. t Thus, it is not inconceivable that within current experi mental accuracy it would be possible for chemically nonequivalent protons to give rise to hyperfine spectra indicative of equally coupling protons. For example, the ethyl radical, while containing two sets of chemi cally nonequivalent protons, might give rise to a sym metric six-line spectrum with a constant line spacing of about 20-25 gauss and a total spread of 100-125 gaussY For the methylethyl radical one might expect to find an eight-line spectrum with a spacing of 17-22 gauss and a total spread of 120-150 gauss. Accepting McConnell and Chesnut's value of -22 gauss per unit of spin density for sp2-bonded hydrogens, to achieve equal coupling for all five protons in the ethyl radical the value of PI2 would have to be close to 0.75, yielding a coupling constant of about 18 gauss in this particular case. The validity of the numerical results obtained from the approach used in this problem lies in the justifica tion of the numerical parameters used and, of course, in the validity of the molecular-orbital approximation in general. Unfortunately, at present there is no way to tell just how different the radical is from the neutral molecule, so that one must assume them to be similar and approach the problem from a semiempirical point of view. It is gratifying, however, that the solution of the present problem leads to agreement with our experi mental observations in more than one aspect, as illu strated in (2) and (3) above. In valence-bond language the presence of isotropic hyperfine interaction is explained in terms of the rela tive contributions of structures such as HI H '" / ·C-C-H / '" HI H (g) HI ·H '" C=C-H / '" HI H (e) to the ground state of the system.t Here, as III the t The fact that the ratio of the methyl-hydrogen coupling con stant to the odd-electron density on the pseudo-atom X is essen tially constant and independant of the choice of Pl2 follows from the fact that the term of greatest weight in the evaluation of IJ [Eq. (14)1 is C2aoraa, and that CaoCao=CaoYao=qa. 14 See W. Gordy and C. G. McCormick, ]. Am. Chern. Soc. 78, 3243 (1956). According to Professor Gordy (private communica tion), more recent work has placed the observed spread in the range of 100--130 gauss. t A very simple treatment of the present problem in valence bond language is given here. Each of the radicals above (I, II, molecular-orbital treatment, one must guess at the necessary parameters, in the valence-bond case the relative "weights" of the contributing structures. In the case of the ethyl radical, Table II shows us that each of the three equivalent structures of type e must have a weight of about tx (0.1307) X100%=4% to agree with the MO calculations. IS The treatment of more complicated radicals such as 'CH(R) (R') 'C(R) (R') (R") , where R is something other than a methyl group has been omitted in the present considerations. It is hoped that a better understanding and mathematical treat ment of the simpler cases and the approximations in volved will lead to a satisfactory treatment of these more complicated molecular systems. Since the ma jority of the aliphatic radicals which have been ob served by paramagnetic resonance to date have been radicals "trapped" in the solid or pseudo-solid state, a detailed analysis of a given spectrum will require a correspondingly detailed knowledge of g-factor aniso tropies and dipole-dipole interactions which may be present as well as the isotropic interactions. ACKNOWLEDGMENT I would like to thank Professor Walter Gordy for many helpful suggestions and stimulating discussions. III) receive contributions from the two types of structures g and e. If W is the weight of a given structure, the normalization con dition for n (= 1, 2, 3) methyl groups is ~iWin=1=Won+3nW",. (i) Denote the coupling constant of protons of type H' by Q and that for protons of type H by QO (the coupling constant for atomic hydrogen). If one now requires that in the radical the two sets of protons have identical coupling constants we may write QWon=QoWw=An, (ii) so that in the general case of n methyl groups surrounding the central carbon, the combination of (i) and (ii) requires the molecu lar coupling constant to be An= [Q/1 +3n(Q/QO)]. (iii) If we now set Q equal to 32 gauss (Qo= 506 gauss), the coupling constants for the three radicals I, II, and III are AI = 26.9 gauss, Au=21.7, Arrr= 19.1. (iv) in moderately good agreement with the results in Table II for PI2=0.93. 15 See Gordy and Shields, Gordy, and McCormick [J. Am. Chern. Soc. (to be published) ] for a discussion in terms of valence bond language of the observed hyperfine spectra in a number 0 amino acids and peptides. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.155.225 On: Mon, 24 Nov 2014 01:00:52

1.1730581.pdf

Electrical Conductivity of Single Crystals of MgO S. P. Mitoff Citation: The Journal of Chemical Physics 31, 1261 (1959); doi: 10.1063/1.1730581 View online: http://dx.doi.org/10.1063/1.1730581 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/31/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Injection and accumulation of electric charge in MgO single crystals J. Appl. Phys. 91, 5296 (2002); 10.1063/1.1459615 The effect of electric current on the conductivity of MgO single crystals at temperatures above 1300 K J. Appl. Phys. 49, 5971 (1978); 10.1063/1.324564 Creep in Vacuum of MgO Single Crystals and the Electric Field Effect J. Appl. Phys. 36, 2309 (1965); 10.1063/1.1714469 EFFECT OF AN ELECTRIC FIELD ON CREEP OF MgO SINGLE CRYSTALS Appl. Phys. Lett. 3, 160 (1963); 10.1063/1.1753913 Electronic and Ionic Conductivity in Single Crystals of MgO J. Chem. Phys. 36, 1383 (1962); 10.1063/1.1732744 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:21A R SO R P T ION C 0 E F F I C lEN T S FRO 1\1 I N F R ARE D DIS PER S ION 1261 disagreement with those previously reported for CS23. The results for CS2 indicated that the directly deter mined absorption coefficients were essentially equal and in turn agreed with the dispersion value for A g. On the other hand, A I calculated from the dispersion data was greater by the expected factor of 1.40. With CH212, a fundamentally different situation THE JOURNAL OF CHEMICAL PHYSICS exists in that the directly measured values for A I and A. are again equal but agree with the dispersion value for A I rather than A •. This apparent discrepancy can not be resolved at the present time but must await similar measurements on a number of other com pounds or more accurate measurements on CS2 and CH2h VOLUME 31, NUMBER 5 NOVEMBER, 1959 Electrical Conductivity of Single Crystals of MgO S. P. Muon General Electric Research Laboratory, Schenectady, New York (Received May 13, 1959) The electrical conductivity of magnesium oxide at temperatures in the region of 1300°C is observed to depend upon the partial pressure of oxygen surrounding the sample. The conductivity increases at oxygen pressures both higher and lower than 10-5 atmospheres. At this pressure the conductivity is a mini mum. This effect is increased as the iron content is increased and is almost absent in the purest samples. The conductivity is electronic rather than ionic and the number of charge carriers is controlled by the number of lattice vacancies. The dependence of conductivity on oxygen pressure may be satisfactorily explained by changes in stoichiometry and thus lattice defects in magnesium oxide. These changes in stoichiometry are larger when the magnesium oxide is contaminated with a variable valence impurity like iron than when it is pure. If an explanation based on iron changing valance state is accepted, then it may be demonstrated by an analysis of the chemical equilibria involved that anion deficiencies must appear at low oxygen pressures, cation deficiences at high oxygen pressures, and that all defects must lie close to the variable valence im purity ions. INTRODUCTION THE electrical conductivity of magnesium oxide at temperatures in the range of 800 to 1400°C has been attributed to electrons due to excess magnesium,! to positive holes due to excess oxygen,2 and to intrinsic valence electrons and holes.3,4 The absolute values for the conductivity reported by various investigators at a given temperature differ by as much as an order of magnitude (from one investigator to another), but there is general agreement in a value for the thermal activation energy of about 2 ev. No one has reported ionic conductivity as being important. Transport experiments at this Laboratory and by D. W. Magee5 have indicated that electronic conductivity predominates in this temperature range and also at higher temperatures (in excess of 1700°C). As is true for most oxides, the effects of impurities on the electrical properties of magnesium oxide has not been adequately explored. It has, therefore, not been 1 E. Yamaka and K. Sawamoto, J. Phys. Soc. Japan 10, 176 (1955). 2 R. Mansfield, Proc. Phys. Soc. (London) B66,612 (1953). 3 A. Lempicki, Proc. Phys. Soc. (London) B66, 281 (1953). 4 G. F. J. Garlick, JIandbllcit dey Physik (S. Fliigge, editor, Springer-Verlag, Berlin, Germany, 19S6) , Vol. 19, p. 317. 5 D. W. Magee, Lamp Development Department, General Electric Company, Cleveland, Ohio (private communication). resolved whether the observed conduction results from charge carriers from magnesium oxide and the ther mally stable defects in it, or from impurities and those defects which result from the impurities. If we knew, for example, the first and second ioniza tion potentials of trapped electrons and positive holes from vacant ion sites, and the energy for vacancy creation, we could make reasonable guesses of the expected electrical behavior of pure oxides and thus deduce whether impurities were important. Unfortu nately, there is too much latitude in the reasonable values to be assigned to these energies, so that such estimates are very uncertain. More positive evidence of the role of impurities may, of course, be obtained if new properties are observed on crystals of higher purity. THEORY Schottky and Wagner6 have shown that point defects and the conduction electrons and/or positive holes which result from them may all be treated as chemical species, so that their concentrations are given by 6 C. Wagner and W. Schottky, Z. physik. Chem. B11, 163 (1931); C. Wagner, Z. Elektrochem. 39, 543 (1933); W. Schottky, ibid. 45, 33 (1939); W. Schottky, editor, Halbleiter Probleme (Friedrich Vieweg and Sohn, Rrausc~weig, Germany, 1954), Vol. 1, p. 139. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:211262 S. P. MITOFF TABLE I. Concentration of Concentration of Schottky defects Reduced neutrality electrons in condition conduction band Negligible with respect [oJ = [OO'J to nonstoichiometric defect concentration Equal to nonstoichio- 2[MgO"J=[00'J [oJ= (KIK2)2!3X metric defect concen- (K3/2K4)1!3X tion PO,-1!3 Much larger than non-[MgO"J= [OO"J [oJ= (KIK2K3) 1!2X stoichiometric defect (K4)-i/4PO,-1!4 concentration equilibrium constants according to the law of mass action. With this approach, equations may be derived to relate electrical conductivity to the large number of variables which may be expected to influence con ductivity at high temperatures. Schottky's notation is used: 8 Electron in the conduction band. OOx Oxygen vacancy containing two trapped electrons-overcharge equals zero (difference in charge from that normally at the lattice point) . 00' Oxygen vacancy containing one trapped electron-overcharge equals plus one. 00" Oxygen vacancy-overcharge equals plus two. MgO" Magnesium vacancy-overcharge equals minus two. [ J Brackets indicate concentrations. P 02 Oxygen partial pressure. A reasonable ideal model for a metal excess oxide, such as that proposed for MgO by Yamaka and Sawamot01 at high temperatures and low oxygen pres sures would be one in which the following equilibria and equilibrium constants are considered: 1. The loss of oxygen creates oxygen vacancies with two trapped electrons: Lattice=i02(gas)+00x, K1=[00xJP o}. (1) 2. The first and second electrons are excited from the trap to the conduction band, leaving effective positive charges at the lattice site: 00·=00'+8, [00-][8 J K2= [OOxJ ' [00"J[8J Ks= [00-] . (2) (3) 3. Assume that Schottky defects are most stable, and are created according to the equilibrium reaction: Lattice=OO·+MgO", K4= [OO .. J[MgO"], (4) A Schottky defect is thus defined as a missing cation, MgO", and a missing anion, 00". It is possible in MgO that the defined Schottky defect does not exist in thermal equilibrium. It would not if the absolute sum of the ionization potentials for the second electron from the anion vacancy and the second hole from the cation vacancy were greater than the band gap. This would result in "Schottky" pairs of a different type which would contain one trapped electron and one trapped hole each, i.e., "Schottky" pair= MgO' +00·. The equilibrium constants may be evaluated theo retically. For instance, assuming that there is no degeneracy, K2 and Ks are the products of the effective density of states in the conduction band and the frac tional occupation of these states,1 and hence: K2= 2 (27rme *kT/h2)! exp[ -(Ec-E1) /kTJ, (5) Ks= 2 (27rme *kT/h2)i exp[ -(Ec-E2) /kTJ, (6) where Ec-E1 and Ec-E2 are, respectively, the energies necessary to excite the first electron and the second electron to the conduction band from the trap, and me * is the effective mass of an electron. K1 is a function of the total deviation from stoichiometry (the ionized vacancies must also be included). A fifth equilibrium reaction which occurs, that for the association of the Schottky defect pairs, need not be considered for the following reasons: The effect of this reaction in reducing the concentration of unassociated pairs may be in cluded in the constant K4, and the possibility of the associated defects acting as traps for holes and/or electrons must certainly be negligible at the high temperatures for which these equilibria are valid. Equations (1)-(4) may be combined with Eq. (7) which is a balance of electrical charge [8J+2[MgO"J=[00-]+2[00"J, (7) to yield the quartic [8 J4+[8 J3(K~i~s)P02-L[8 J(K1~i:K3)p02-1 (K1K2Ks)2p -1=0 (8) K4 02 • The simplified solutions to this equation for small concentrations of conduction electrons are summarized in Table r. The solutions are given in terms of the equilibrium constants and the partial pressure of oxy gen surrounding the sample. The three solutions are for different equilibrium concentrations of Schottky de fects relative to concentrations of the nonstoichiometric (oxygen vacancy) defects. The appropriate reduced neutrality condition is given for each case. The reduced conditions result from choosing the most important terms in Eq. (7) for the defect conditions set in the first column. 7 E. Spenke, Electronic Semiconductors (McGraw-Hill Book Company, Inc., New York, 1958), see especially p. 4S ff. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:21E L E C T RIC A L CON D U C T I V I T Y 0 F SIN G L E CRY S TAL S 0 F M g 0 1263 Kroger and Vink8 have generalized this approach by using some approximations to handle even larger numbers of variables including the effect of impurities. The effect of impurities are discussed later for the case of MgO, but a different treatment is necessary. Examination of Table I and Eq. (8) reveals that no single dependence of the conductivity on the oxygen pressure is to be expected even from this simplified model. From Table I, which is applicable at low con centrations of conduction electrons (i.e., where Ec &«kT) , the conductivity varies between the minus one-fourth and minus one-third power of the oxygen pressure. At higher conduction electron concentrations a simple power dependence no longer prevails. Also, the variation in the exponentials of K2, Ka, and K4 in Table I is a direct factor of the difference between the thermal activation energy for conduction which is a function of the Fermi level and the optical absorbtion energy of the trapped electrons. The exponentials indicate the position of the Fermi level for the relative concentrations to which Table I applies. For example, if the conduction electron concentration is small and the Schottky and nonstoichiometric defect concentra tions are equal, from the table, [8 J oe K22/3 KaliS, and from Eqs. (5) and (6), [8 Joe exp[ -(Ec-2/3E 1-1/3&)/kT]. From the simplified Fermi Dirac distribution function, Finally, solving for the Fermi energy, EF, Of course, the choices of metal excess MgO rather than oxygen excess MgO, vacancies rather than interstitials, and Schottky pairs rather than Frenkel pairs were entirely arbitrary. However, models based on any combination of the above possibilities yield similar equations and oxygen pressure dependencies (but with positive exponentials of the oxygen pressure for oxygen excess compounds). These models serve to define the limits of conductivity behavior expected from an oxide in which no impurities are present. EXPERIMENTAL The results reported here are direct-current con ductivity measurements made on single crystals of magnesium oxide by using both temperature and partial pressure of oxygen as parameters. The experi mental methods are discussed in the light of the diffi culties one may expect in making such measurements. 8 F. A. Kroger and H. J. Vink, Solid State Physics, F. Seitz and D. Turnbull, editors, (Academic Press, Inc., New York, 1956), Vol. 3, p. 307. Sample 15 15 15 16 16 TABLE II. Conductivity ohm-1cm- 1 Oxygen Temperature pressure four-probe two-probe 1306°C 1 atmos 5.0XlO-s 3. OX 10-6 1334°C 1 atmos 6.7XI0-6 4.3XlO-6 1343°C > 10-12 at mas 7.7XlO- 5 2.0XlO-5 1320°C 1 atmos 5.5XI0-6 3.8XI0-6 1320°C > 10-12 atmos 1.8XlO-5 2.0XI0- 5 1. Contact Resistance, Space Charge, and Polarization Measurements were made by employing both flat plates with sputtered platinum electrodes and bar shaped samples by using a four-contact scheme so that no current was allowed to flow through the measuring probes. Table II presents a comparison of the con ductivities calculated from two-probe versus four probe measurements in which the probes were merely wrapped around single crystal bars of MgO. The results show that the change in conductivity with oxygen pressure is not a result of contact effects. The measuring potential was varied between 0.1 volt and 250.0 volts. No substantial difference was noted at any potential below about 50 volts; at higher potentials, however, the current increased with time. 2. Surface Conductivity The size and geometry of the samples were altered so that the ratio of surface to bulk conductivity was changed with no detected difference in the calculated specific conductance. 3. Extraneous Gas Effects High and low partial pressures of oxygen were obtained by diluting flowing argon gas with water vapor and small amounts of. oxygen or hydrogen. The oxygen partial pressure was then calculated by using the thermodynamic data for the decomposition of water. Intermediate partial pressures were obtained by flowing mixtures of CO and CO2 over the sample. Low oxygen pressures were also obtained by using an atmosphere of argon which was purified by passing it over heated Cu foil, followed by heated oxidized copper foil, and finally through a cold trap. The oxygen pressures thus obtained are low but unknown. Therefore, the results are not quantitatively reported but they eliminate possible interfering effects of hydro gen or other gases, because they qualitatively agree with the results obtained by using hydrogen and water. 4. Equilibrium In some oxides, notably quartz, the dc conductivity is observed to change with time. An explanation9 9 H. E. Wenden, Amer. Mineralogist 42,859 (19.17). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:211264 S. P. MITOFF > >-:;: ~ '" ~ -I U 10 la' 1O'~'-:-6---0=-'7=---~0-\-~--~0 9=----~--' 10'/ T' K FIG. 1. Conductivity of optical quality MgO ploited against reciprocal temperature. The points represented by circles were obtained with a surrounding oxygen partial pressure of 10-2 atmos, and the points represented by triangles at an oxygen pres sure of less than 10-8 atmos. The slopes indicate activation energies of 2.3 ev and 2.5 ev, respectively. for this is that the current is being carried by impurity ions. The conductivity in single crystals of MgO has also been reported1o to change with time, possibly for the same reason. In this work, however, these effects were observed to disappear after heating the sample to temperatures of about 1300°C, and the results reported were reproducible and independent of time. 5. Ionic Conductivity In order to determine if an important fraction of the current was being carried by motion of ions, the follow ing experiment was performed. A flat sample of MgO was heated to 1600°C with a total surrounding gas pressure of 1 millimeter of argon for fifty minutes. The weight loss from evaporation was determined by removing the sample and weighing it. The sample was returned to the furnace and a 12-volt dc potential was applied to the sample, other conditions being un changed. The total charge transferred by the crystal would be sufficient to decompose 8.1 milligrams of MgO to magnesium and oxygen gas if the current was carried by either ion. However, the evaporation loss from the first (no current) run was 7.2 milligrams and 10 E. G, Rochow, J. Appl. Phys. 9,664 (19311). the total weight loss from the second (current) run was 5.2 milligrams. Therefore, no increase in weight loss was detected and we conclude that the majority of the current is not carried by ions. Further support of the conclusion that ionic con ductivity is relatively unimportant comes from the appearance of the crystals after passing the current through them. Ionic conduction without weight loss would require vapor transport of magnesium or oxygen and recombination at one or the other (or both) electrodes. One would expect that decomposition and regrowth at the exposed parts would distort the shape of the crystal. The crystals did not have any such appearance. RESULTS 1. Activation Energy Figure 1 is a plot of log conductivity as a function of reciprocal temperature for optical quality single crystals of MgO. * The results are given for two differ ent oxygen partial pressures. In Fig. 2 the results are compared with those of previous investigators. In Fig. 3 the temperature dependence of conductivity of the MgO crystal is compared with that of a MgO crystal, R-58, of higher purity especially with respect ~ ~ ., 5! 0 >->-;; TEMPERATURE I'C) 10-5r---,--12nOO,----,--1,00-0-~---80rO--____r___, \ \ \ \ \ \ , \ , 10,6 \ \ , \ , \ \ \ \ \ \ \ \ 10,7 \ \ \ \ \ \ \ , \ \ \ \ , \ \ \ \ t; 10-8-\ \ ~ \ \ \ \ 8 \ \ \ \ \ \ \ 10,9 \ \ - - - LEMPICKI \ --- MANSFIELD \ ------ YAM AKA AND \ SAWAMOTO \ MITOFF FIG. 2. Results of previous investigators compared with those obtained on optical quality MgO at an oxygen partial pressure of 10--2 atmos. * Obtained from the Norton Company, \Vorchester, :\Iassa chusetts. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:21E L E C T RIC A L CON D U C T I V I T Y 0 F SIN G L E CRY S TAL S 0 F M gO 1265 to iron contend The conductivity is less in the R-58 crystal and the activation energy between 1050°C and 1500°C is larger, 3.5 ev. compared with 2.3 ev for the optical quality MgO, giving indication that the 2.3-ev activation energy process is one which results from impurities. At temperatures lower than 1050°C the R-58 MgO takes on a much lower activation energy (0.9 ev) which probably results from an impurity. 2. Oxygen Pressure Dependence The increase in conductivity with decreasing oxygen pressure (see Fig. 1) is the behavior one expects from an n-type semiconductor in which the electrons result from a stoichiometric excess of metal. If this is the case, MgO may be represented by a model such as that pre viously mentioned, containing oxygen vacancies, or a similar model employing interstitial magnesiums. From Table I the oxygen pressure dependence of the number of conduction electrons and the conductivity must be somewhere in the range rrrx; PO,-1I3 to Po 2-114. Other models, i.e., ones involving interstitials, Frenkel TEMPERATURE ('C I 1O.5..-----.--~I20~O~---.------.-:';:00:::.0 ----r-----T''---~~ JO-~L.6 ----:0:l=7----:0""8----,0;;-';9:-----;';;----' 10'/T'K FIG. 3. Comparison of conductivity of optical quality MgO with higher purity R-58 MgO. The activation energy for optical quality is 2.3 ev and the energies for R-58 are 3.5 ev and 0.9 ev. t The crystals were made by Lewis Schupp of the Lamp De velopment Department, General Electric Company, Cleveland, Ohio, by fusion of MgO powder between carbon arcs. They were analyzed by R. P. Taylor of the same laboratory. The analysis indicated an iron content of 7.9 parts per million, strontium 5 ppm and lithium 3 ppm. In addition spectroscopic analysis indicated calcium and several other trace impurities. -; 1'l .. -6.0 i5 -50 ~ ti 5 8 ~ 3 -4.0 \~ 0 ":-. \, ° \" \" \ " \ "-\ , o \ " \" .1/, \ ' a' CPo, "/, \ Y a, CPo " ,~ " \ " -30 L--.':!--!.4:---.-:-6 -.-J,B,--'--:_I~O -.0':;-I'--!.14;---.-!;:-16-.-;';,B:----;;.';;.0 -.-;-;;-" LOG OXYGEN PRE5SURE (ATM I FIG. 4. The points show the variation of optical quality MgO conductivity with oxygen pressure at 1300°C. This is compared with the limits predicted for a cation vacancy model. defects, or Schottky defects with one electron and one hole per pair rather than none, also yield a pressure dependence in this range. Figure 4 shows the oxygen pressure dependence of the conductivity of MgO measured by the four-probe method. The points observed do not fall within or even close to the pressure dependence range predicted by the model for pure MgO. We take the lack of agreement between the model and the experiment as evidence that the conduction mechanism in these crystals is not that resulting from deviations from stoichiometry in a pure crystal. Support for this conclusion is the observa tion that the higher purity R-58 crystals exhibited even less variation of conductivity with oxygen pressure (Fig. 5) than did optical quality MgO. Experiments were, therefore, also made on magnesium oxide crystals which analysis showed to contain 1300 ppm iron. Averaged conductivity results from several experiments for samples of three different iron contents at three different oxygen pressures are given in Fig. 5. The following correlations were found with increasing iron content: The conductivity increased, the reversible changes in conductivity with changes in oxygen pres sure were greater, and the conductivity increased at high as well as at low oxygen pressures, compared to a minimum value at intermediate oxygen pressures. DISCUSSION 1. Conduction in Pure Magnesium Oxide The high thermal activation energy and reduced conductivity and oxygen pressure dependence of con ductivity in the pure MgO crystals indicate that the conduction mechanism in absolutely pure MgO is different from that in ordinary crystals. The possible ways in which a pure crystal might conduct are: (1) transport of cations, anions, or both, (2) transport of intrinsic electrons and holes, and (3) transport of extrinsic electrons or holes which arise from deviations from stoichiometry. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:211266 S. P. MITOFF FIG. 5. Averaged values of conductivity of MgO at three oxygen pressures at 1300°C. Circles are for MgO with 1300 parts per million iron, triangles about 100 parts per million iron, and the squares for R-58 containing 8 parts per million iron. There is experimental or theoretical evidence against each of these mechanisms. For ionic conductivity, the evidence is entirely experimentaL Ionic conduction has been ruled out by transport measurements as men tioned previously. Curiously, however, if one calculated the ionic conductivity to be expected from magnesium transport alone, by using the diffusion data of Lindner and Parfitt,ll and the Nernst-Einstein equation, D=O.249 exp[ -3.37 ev/kTJ (in the range of 1400°C to 1600°C), u=Dnz2e2/kT, (where D is the diffusion constant, z in the valence, e is the electronic charge, and n is the number of ion pairs per cma) , the result is a conductivity-temperature curve which is a nearly exact extrapolation of the experimental conductivity data for the purer MgO crystals. Barring the possibility that the experimental conductivity values are too low or the diffusion values too high, interpretation of the data requires that the self-diffusion of magnesium ions occurs largely by a neutral pair mechanism. Intrinsic electronic conductivity, as observed in semiconductors, is unlikely because the mobilities one calculates from the observed conductivity and esti mates of the band width are far too large to be reason- 11 R. Lindner and G. D. Parfitt, J. Chern. Phys. 26, 182-185 (1957) . able. However, the observed thermal activation energy (3.5 ev) is not much lower than one would expect for intrinsic conduction. Nelson12 has deduced from optical reflection measurements that the width of the band gap in MgO is about ten electron volts; it would then be expected that the activation energy for elec trical conduction is somewhat less than one-half this value (less because of the lattice relaxation which subtracts from the thermal activation process and not from the optical absorption). Finally, extrinsic conduction because of deviations from stoichiometry is also unlikely on the basis of re quiring unreasonably high mobilitiest if the thermal activation energy for conduction results only from the ionization of traps. The experimental evidence that the conductivity is relatively insensitive to oxygen pressure again makes any model based on metal excess or deficiency unlikely. All of the simple models, therefore, meet with some objection, whether it is the objections or the models which are in error cannot be resolved by the experi ments so far performed. 2. Conduction in Norton MgO and High-Iron MgO At this point it is prudent to state those points which are proven by the data, because that which follows in subsequent paragraphs is largely speculation which one can make with our present knowledge of the semi conducting properties of ionic compounds. In crystals of typical purity «99.9% MgO and in the order of 0.01% Fe) the conductivity is higher and the activation energy for conduction is much less than in purer (R-S8) crystals. Increased conduction occurs both at high and at low oxygen pressures, and the magnitude of this dependence upon pressure is lower than we expect for compound semiconductors with deviations from stoichiometry. Crystals with large iron contents show an increase in the oxygen pressure dependence of conductivity and a further increase in total conduc tivity. A reasonable deduction is that the charge carriers in magnesium oxide arise from impurities or from defects caused by the presence of impurities. By pursuing this, one may speculate as to the mechanism involved. Iron in the plus-two state is both more readily oxidized to higher valency and reduced to lower valency than Mg2+, thus a model may be proposed in which iron ions in magnesium oxide are responsible for oxygen deficiency at low oxygen pressures and oxygen excess at high pressures. Any other impurity ion of variable valence would do as well for the model. At high oxygen pressures the magnesium oxide lattice may contain cation vacancies by virtue of some of the 12 J. R. Nelson, Phys. Rev. 99, 1902 (1955). tAssuming T=1673°K, u=1.5X1O-v, [OJ=1O '8, and E/2= 3.5 ev, solving for the mobility results in 2X104 cmz/volt sec, whereas one would consider even 1()2, as being high at this tem perature. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:21E LEe T RIC ALe 0 N Due T I V I T Y 0 F SIN G LEe R Y S TAL S 0 F M g 0 1267 iron ions existing in the plus-three state, and at low oxygen pressures anion vacancies may be present with reduced iron ions (Fe or Fe+). High Oxygen Pressure Model At oxygen partial pressures of greater than 10-5 atmosphere we assume that cation vacancies and Fe+3 ions will be present in significant quantities. Two important points must be made before proceeding with the model. The first and more obvious is that a cation vacancy (or other excess of anions) must necessarily be present to explain the increase of con ductivity with increasing oxygen partial pressure. The second is that cation vacancies must be physically associated with the Fe3+ ions. Qualitatively, we can see this because conductivity, and therefore sources of charge carriers which must involve the vacancy, in crease in proportion to the iron concentration. The defect concentration is limited by the iron concentra tion. We know from the oxygen pressure insensitivity of conductivity in the purer R-S8 MgO that the equilibrium constant for the reaction, (9) is so small that it is unimportant to the conductivity. In a less pure crystal the identical equilibrium constant will hold except in the immediate vicinity of the iron impurity ions. Therefore, we expect an increased vacancy concentration near the iron ions only. The inadequacy of a model in which the defects are inde pendent of the iron is demonstrated in greater detail, and more rigorously in the Appendix. The equilibria which we consider are that for the creation of an iron ion plus cation vacancy complex, Fe2++1/20 2= (Fe-CatO)4++0--, Ks= [(Fe-CatO)4+ J/[Fe2+ JP02t, (10) and that for the ionization of the complex to form a positive hole, and iron balance, [Fe( total) J= [Fe2+ J+[(Fe- CatO) 3+ J +[(Fe-CatO)4+J, (14) to give th~ quadratic [EEl J2[(1/K 6) + (l/K5K6PO}) J+[EEl J= [Fe( total)]. (15) When the hole concentration is small with respect to the other concentrations this reduces to [EEl J""'[Fe( total]!K 5!K6!Po/14. (16) The hole concentration will then be proportional to the 1/4th power of the oxygen pressure until the oxygen pressure becomes large enough to make the vacancy concentration significant with respect to the total iron. Low Oxygen Pressure Model At very low oxygen pressures we propose that anion vacancies are formed. Again association with impurity ions is a necessary condition for the validity of the model. We may take an iron plus anion vacancy com plex to be formed according to the equation Fe2++0--= 1/202+ (Fe-AnD), Po2t[(Fe-AnO) J K7 [Fe2+J For the ionization, (Fe-AnD) = (Fe-AnO)++ e, (17) Ks=[(Fe-AnO)+J[eJ. (18) [(Fe-AnD)] Here, the iron ion makes possible the creation of the vacancy by increasing the stability of one or both of the trapped electrons from the vacancy. The conditions for charge balance (Fe-CatO)4+= (Fe-CatO)3++EEl, [e J= [Fe-AnO)+J, (19) K6= 1 [(Fe-CatO)3+ J[EEl Jl /[(Fe-CatO)4+]. (11) and iron balance The (Fe-CatO)4+ complex is an iron ion in the three-plus valence state having lost one electron to the associated vacancy. The vacancy now has a net charge of plus one. The iron may take on an electron from the valence band to form a charge carrier and the (Fe CatO)3+ complex. An alternative, but perhaps less likely, assumption is that two three-plus iron ions lose electrons to a single vacancy completely canceling its effective positive charge: (12) Equations (10) and (11) may be combined with the relationships, for charge balance, [(Fe-CatD)3+ J= [EEl J, (13) [Fe (total) J= [Fe2+ J+ [(Fe-AnD) J +[(Fe-AnD)+J, (20) lead to the quadratic [e J2[(1/ Ks) + (POHK7KS) J+[e J= [Fe( total)]. (21) For small electron and complex concentrations this is (22) At low oxygen pressures this predicts that conductivity will increase with decreasing oxygen pressure, as was demonstrated for MgO of typical purity. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:211268 S. P. MITOFF ::IE u >I:;: E is Z -. 10 810-1 10 -'0 -2 -4 LOG OXYGEN FIG. 6. Conductivity vs oxygen pressure at 1300°C for 13--58 (triangles) and high iron (dots) MgO. The curves are explamed in the text. The total conductivity as a function of the oxygen pressure may be approximated by the addition of the contributions from electrons from cation vacancies, holes from anion vacancies, and electrons or holes from other sources UX• The last contribution is not a function of the stoichiometry or impurities of multiple valences. We have q total = UCat+U An+Ux- From the previous models u= e[Fe]!{J.LEllKstK6!Po/14+J.LoK7!Ks!Po2-1/4j +ux. (23) Shoshea, Dekker, and Sturzl3 have recently proposed a model to explain the x-ray induced color centers in MgO. In this model positive holes result from cation vacancies which are present by virtue of iron (or similar ion) impurities, and electrons may be ionized from Fe2+ ions. However, their model is not sufficient to explain the electrical conductivity of MgO as a function of oxygen pressure. The following equations very nearly apply for the concentrations of electrons and positive holes according to their model: K6[Fe,otal] (1+1/K5Po21), (24) 13 Shoshea, Dekker, ancl Sturtz, Phys. and Chern. Solids 5, 23 (1958). where Ks is the equilibrium constant for the excitation of a hole from the vacancy and K9 is the equilibrium constant for the excitation of an electron from Fe2+. From the above equations it can be seen that there will be a change in total conductivity only through the narrow pressure range in which KSPo2t is quite close to unity, whereas the conductivity was observed to change over more than ten orders of magnitude of pressure variation. In the model proposed in the present paper in which both cation and anion vacancies are assumed to exist, there are two different constants corresponding to the term K5 in Eqs. (24) and (25). Test of Model The agreement of Eq. (23) with the experimental results may be empirically tested in the form u= CIPoF4+C2P02-1/4+ux. (26) Figure 6 shows the agreement for R-58 MgO and high iron MgO. The points are the experimental con ductivity results at 1300°C and the solid curves were computed from Eq. (26) by using a total of three experimental points to fix the constants for both curves. The dashed lines are the best fit to the equation u= CIPo/16+C2P02-1/6+ux. (27) This corresponds to the assumption that both the cation vacancy complex and the anion vacancy com plex contain two iron atoms per vacancy, as in Eq. (12). The best fit appears to be something between a one-fourth and a one-sixth power dependence. A rough test of the model may be obtained by seeing if Eqs. (15) and (21) yield reasonable conductivity values at their limits. At oxygen pressures sufficient to oxidize all the iron ions, Eq. (15) approaches (28) Chemical analysis of the high iron sample showed [Fe]=5X1018 cm-3• K6, approximated from the activation energy for conduction and Eq. (6), is equal to 4X 105 cm-3 at 1300°C. Therefore, Eq. (15) predicts a maximum number of hole carriers of l.4X 1012 cm-3 to result from the iron-cation vacancy complexes at 1300°C. By assuming a mobility close to that measured for electrons in BaO by Pell,J4 in the order of 5 cm2/volt sec, the maximum conductivity expected is 1.1X 10-6• The highest observed conductivity at high oxygen pressure was 4X 10-6• The agreement is not as good at low oxygen pressures where the same calculation applies and the conductivity was as high as 2 X 10-5 without evidence of saturation. However, for this type of calculation order of magnitude agreement is en couraging. 14 E. M. Pell, Phys. Rev. 87, 457 (1952). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:21E LEe T RIC ALe 0 N Due T I V I T Y 0 F SIN G LEe R Y S TAL S 0 F M g 0 1269 CONCLUSIONS A different conduction mechanism occurs in crystals of magnesium oxide of higher purity than those ordi narily available. This is evidenced by lower conduc tivity, higher thermal activation energy for conduction, and diminished oxygen pressure dependence of con ductivity in the temperature region studied (800°C to 1400°C). The source of charge carriers in high purity magnesium oxide is not known. Conductivity in crystals of ordinary purity is domi nated by impurities. There is a minimum in conduc tivity at a partial pressure of oxygen of 10-5 atmos pheres at 1300°C, and increasing conductivity at both higher and lower oxygen partial pressures. The effect of oxygen pressure on the conductivity may be ex plained on the basis of a tentative model in which (1) the impurities are ions of multiple valence states, pos sibly iron; (2) cation vacancies are created at high oxygen pressures and anion vacancies at low oxygen pressures; and (3) the vacancies are associated with the impurity ions. ACKNOWLEDGMENTS The author is indebted to J. E. Burke, R. E. Carter, and Werner Kanzig for helpful discussions and review of the manuscript. APPENDIX If the iron impurities are not associated with the vacancies, the following equilibrium equation will describe the effect of oxygen to produce cation vacan- Cles: tions down to the following four equations: 1/202 (g) =0--+MgD", MgOx= MgO' + $, K1= [MgOx][O- -]/ P021/2, K1'=[MgOx]/P021/2, (31) K2= [MgO'][EB ]/[MgOx], (32) MgO' = MgO" + EB, K3= [MgO"][EB ]/[MgO'], (33) 2Fe3+= 2Fe2++2EB, K41/2= K/ = [Fe2+ ][EB ]/[FeH]. (34) The material balance for iron is [Fe total] = [Fe2+ ]+[FeH], (35) and that for excess charge is [FeH]+[EB]=[MgO']+2[MgO"], (36) combining (34) and (35) [FeH] = [EB ][Fe,ot.1J! K/ +[EB]. From (31) and (32) K'KP l/" [M O'J= 1 2 02- g [EB J ' and from (33) Substitution of the last three concentrations into (36) 2Fe2++ 1/202(g) = 2FeH+0--+ MgO". (29) and rearranging gives The positive holes may be released by the iron to form charge carriers: (30) It is thermodynamically possible to break these reac-[EB J4+[EB J3([Fetot.1J+ K/) -[EB J2Kl'K2K3P02112 -[EB J2K/K2K3P021/2_2K/K~~/ Po/12=0. (37) Inspection of Eq. (37) reveals that the effect of in creasing iron concentration will always be to decrease the hole concentration and the conductivity. This is the opposite of what is observed experimentally. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 141.212.109.170 On: Tue, 09 Dec 2014 18:28:21

1.1730774.pdf

Mean Square Amplitudes and Force Constants of Tetrahedral Molecules. I. Carbon Tetrachloride and Germanium Tetrachloride Yonezo Morino, Yasushi Nakamura, and Takao Iijima Citation: The Journal of Chemical Physics 32, 643 (1960); doi: 10.1063/1.1730774 View online: http://dx.doi.org/10.1063/1.1730774 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/32/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Vibronic Effect on the MeanSquare Amplitudes of Internuclear Distances in a Tetrahedral Molecule J. Chem. Phys. 45, 4536 (1966); 10.1063/1.1727534 Force Constants of Tetrahedral Molecules J. Chem. Phys. 28, 514 (1958); 10.1063/1.1744177 LeastSquares Calculation of OVFF Force Constants for XY 4 Tetrahedral Molecules J. Chem. Phys. 23, 1727 (1955); 10.1063/1.1742428 Bending Force Constant of the Carbon Tetrafluoride Molecule J. Chem. Phys. 23, 1563 (1955); 10.1063/1.1742381 The Mean Amplitudes of Thermal Vibrations in Polyatomic Molecules. II. An Approximate Method for Calculating Mean Square Amplitudes J. Chem. Phys. 21, 1927 (1953); 10.1063/1.1698719 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13THE JOURNAL OF CHEMICAL PHYSICS VOLUME 32, NUMBER 3 MARCH,1960 Mean Square Amplitudes and Force Constants of Tetrahedral Molecules. I. Carbon Tetrachloride and Gennanium Tetrachloride YONEZO MORINO, YASUSHI NAKAMURA, AND TAKAO IIJIMA Department of Chemistry, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan (Received September 11, 1959) The atomic distances and mean square amplitudes of carbon tetrachloride and germanium tetrachloride were measured by the sector-microphotometer method of electron diffraction. The force constants were calculated by use of the mean square amplitudes thus obtained, combined with the normal frequencies. The results indicate that a force field of the Urey-Bradley type is approximately realized in these molecules. INTRODUCTION THE forces acting between atoms in a molecule hav usually been obtained by the analysis of vibra tional spectra. The number of frequencies observed is, in general, not enough to determine all the force con stants in general quadratic form of potential energy: for example, even for such a simple molecule as carbon tetrachloride, which has Td symmetry, three force constants are necessary for the F2 type vibrations, but we have only two frequencies of this class. Thus, in molecular spectroscopy, some special form is tenta tively assumed for the force field to reduce the number of unknown parameters. The simplest one is the so called valence force system, in which only the forces along the valence bonds and those against the deforma tion of valence angles are taken into account. This assumption is, however, too simple to have actual significance. The first improvement to be made in this assumption is to add force constants which represent the interaction between nonbonded atoms. The result is a force field of the Urey-Bradley typel which has been applied to many molecules and proven to be successful by Mizushima, Shimanouchi, and others.2 In this situation any phenomenon can be used to fill up the deficiency of the data mentioned above, if it has a close connection with the force field of the molecule. In fact, isotopic shifts of vibrational spectra,3 Coriolis coupling constants in rotation-vibration spectra,4 and centrifugal distortion constants in microwave spectra,5,6 etc. have been used heretofore. It is the purpose of this article to show that the mean square amplitude ob- I H. C. Urey and C. A. Bradley, Phys. Rev. 28, 1969 (1939). 2 T. Shimanouchi, J. Chern. Phys. 17, 245, 734, 848 (1949); S. Mizushirna and T. Shimanouchi, ibid. 17, 1102 (1949); J. Am. Chern. Soc. 71, 1320 (1949); I. Nakagawa, J. Chern. Soc. Japan 76, 813 (1955); 77, 1030 (1956); T. Miyazawa, ibid. 77, 366 (1956) , etc. 3 Cf. G. Herzberg, },folecular Spectra and Molecular Structure (D. Van Nostrand, Princeton New Jersey, 1945). 4 J. H. Meal and S. R. Polo, J. Chern. Phys. 24, 1119, 1126 (1956) . • D. Kivelson and E. B. Wilson, Jr., J. Chern. Phys. 20, 1575 (1952); 21, 1229 (1953); D. Kivelson, ibid. 22, 904 (1954); L. Pierce, ibid. 24, 139 (1956); P. H. Verdier and E. B. Wilson, Jr., ibid. 30, 1372 (1959). 6 Y. Morino and E. Hirota, Bull. Chern. Soc. Japan 31, 423 (1958) . tained by electron diffraction offers another basis for determining the force constants of molecules. A few words should be added here concerning the previous studies of electron diffraction on carbon tetra chloride and germanium tetrachloride. Since the pioneer work by Mark and WierF in 1930, carbon tetrachloride has been used as a reference substance for electron diffraction. It was on this molecule that Karle and Karle8 first demonstrated the usefulness of the sector method in 1949, and the most reliable measurement at present by the sector-microphotometer method must be that by Bartell, Brockway, and Schwendeman9 on the same substance. Morino and KuchitsulO used the substance for a test of their new sector apparatus. The reason why the same substance was again in vestigated was to get the most reliable values of the mean square amplitudes, by focusing attention upon their determination. The molecular structure of germanium tetrachloride was studied by Pauling and Brockwayll in 1935 by the visual method with the result Ge-Cl=2.08±0.02 A. EXPERIMENTAL The diffraction apparatus used is the same as that reported in the previous paper by Morino and Kuchitsu.lO An ,a sector was used with the shorter camera length, 11.8 cm, and an ,2 sector with the longer camera length, 27.9 cm. The accelerating voltage was about 45 kv, the fluctuation and the drift of the voltage being automatically regulated within 0.1 %. A pure sample of germanium tetrachloride was kindly pro vided by Mr. H. Oikawa, of the Coal Research In stitute of Mitsui Chemical Industrial Company, Ltd. About 10 cc of the sample was consumed in this work. The gaseous sample was kept in a l-liter glass bulb at a pressure of 30 to 35 mm Hg and was led to the diffraction camera through a fine nozzle. The pressure decrease of the sample holder during exposure, 2 to 4 7 H. Mark and R. Wier!, Naturwiss. 18,205 (1930). 81. L. Kar!e and J. Karle,]. Chern. Phys. 17, 1052 (1949). 9 Bartell, Bruck way, and Schwendeman, J. Chern. Phys. 23, 1854 (1955). 10 Y. Morino and K. Kuchitsu, J. Chem. Phys. 28,175 (19581. II L. Pauling and L. O. Brockway, J. Am. Chem. Soc. 57, 2684 (1935) . 643 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13644 MORINO, NAKAMURA, AND IIJIMA min., was a few mm Hg. The diffraction pattern was recorded on Fuji Process Plates which were developed with FD-131 developer at 20°C. The wavelength of the primary electron beam was calibrated by a transmission pattern of gold foil. Microphotometer traCing was carried out either with a Riken B type microphotometer using a photographic recording system or with a Rigaku Denki MP-3 microphotometer using a pen-recording system. They are both equipped with a rotating disk on which the diffraction photograph is mounted and spun rapidly in order to average out irregularities in the photographic plate. Both types of photometers gave results con sistent with each other. Six pairs of diffraction photo graphs including two pairs for small-angle region (q=8-30) were selected from about 40 plates and analyzed by the procedure described below. In order to avoid difficulties in the density-intensity calibration,12.13 the plates of optical densities less than 0.5 were used for the analysis. Having been well estab lished by experiment, a linear relation was assumed between the density and intensity in the above region. The molecular intensities were obtained by drawing a smooth background line which was iteratively cor rected with the criterion of minimum ghost in the radial distribution curve. ANALYTICAL PROCEDURE In order to obtain mean square amplitudes ac curately enough to allow a useful determination of the force field, every possible correction should be taken into consideration. The errors involved in the measure ment of mean amplitudes were thoroughly criticized by KuchitsuY He concluded that the largest error had its origin in the finite gas spread at the diffraction center. He found a method for estimating the distribu tion of gas molecules by the measurement of the devia tion of the slope of the background curve from the theoretical one. Another source of uncertainty is the failure of the Born approximation which was first pointed out by Schomaker and Glauber.lb We can roughly correct it by using Ibers and Hoemi's table.16 Anharmonicity of intramolecular vibration of atoms and the displacements of atoms perpendicular to the bond direction provide serious difficulties in the deter mination of equilibrium distances and mean square amplitudes. In this section the procedure of the analysis shall be presented, with an attempt to eliminate as many uncertainties as possible. 1. Molecular Intensity As was shown by Morino and Hirota,17 the intensity of the coherent beam diffracted by a tetrahedral mole- 12 J. Karle and I. L. Karle, J. Chem. Phys. 18,957 (1950). 13 L. S. Bartell and L. O. Brockway, J. App!. Phys. 24, 656 (1953). 14 K. Kuchitsu, Bull. Chem. Soc. Japan 32,748 (1959). 16 V. Schomaker and R. Glauber, Nature 170, 290 (1952). 16 J. A. Ibers and J. A. Hoerni, Acta Cryst. 7, 405 (1954). 17 Y. Morino and E. Hirota, J. Chem. Phys. 23, 737 (1955). cule is given by () ,,1 F ,,(s) II F/(s) I exp( -as2) sM s = L..J COS~l1--=---- i.i IB r. .( sinsr.-(s/r.) [(,iz2)_t( (~X2)+ (~y2»)] cossr.), (1) where s has the usual meaning of (47r/X) ·sinO/2 and IB is the sum of coherent and incoherent atomic scattering. F ,,(s) denotes the atomic structure factor of the ith atom for electron beam, * and COS~l1 is due to the imaginary part of F ,'(s) F/(s), that is, to the failure of Born approximation. Here 2a= (,iz2) is the mean square amplitude parallel to the equilibrium direction of the distance r., and (~X2) and (~y2) are the mean square amplitudes of the atomic displacements perpen dicular to the equilibrium direction. This formula was derived based on the probability function Ph(r) of the distance r, for which the intramolecular potential is assumed to be harmonic, Ph(r) = (l/47ra)![l +y(r-r.)] exp[ -(r-r.)2/4a], (2) where Now, considering the fact that s(,iz2)(1-'Yr.)/r. is much less than unity in the region s;£30, we have a simple form of sM(s) = L(I F,'(s) II F/(s) I/IB) COS~l1 ·exp( -as2) (l/r.) sinsra, (3) where the apparent distance ra is related to '. by the relation of (4) and the background intensity I B can be given by the equation,18-2O IB= L IF ,"(s) 12+ LS;(s). (5) i i 2. Anharmonicity of Vibration The effect of anharmonicity of vibration upon M(s) was discussed by Bartell21 for the ground state of a diatomic molecule, and was calculated by Reitan for Hz, H20, and CO2 molecules.22 For an atom in a poly atomic molecule the following expression of the dis tribution function seems to be adequate: Pa(r) = [l+/1t(r-r.) +;J2 (r-r.)2+/3s(r-r.) 3] Ph (r) , (6) * F,e(s) =Zi-j,(S) [or the first Born approximation. 18 L. H. Thomas and K, Umeda, J. Chem. Phys. 26, 293 (1957). " H. Viervoll and O. l)grim, Acta Cryst. 2, 277 (1949). 20 L. Bewilogua, Physik Z. 32, 740 (1931). 21 L. S. Bartell, J. Chem. Phys. 23, 1219 (1955). 22 A. Reitan, Acta Chern. Scand. 12, 131, 785 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13AMP LIT U DES AND FOR C E CON S TAN T S 0 FCC I 4 AND Gee I • 645 where Ph (r) designates the distribution function of a harmonic case given by Eq. (2). A manipulation similar to that carried out in the case of a harmonic potential provides an expression for the molecular intensity corresponding to Eq. (3), sM (s) = L(I F ,·(s) II F/(s) II I B) cos6.7j i, i ·exp( -a's2) (l/r.) sins(ra-Ks2). (7) where ra = re-2a' /r.+2 (131+1') a' + 12 (133+1'132) a'2, (8) a' = a+4a2 (132+1'131) , (9) and K=8a'3(!33+'Y!32) . (10) The information obtained by the analysis of the intensity curve is ra and a', and, if a precise measure ment is performed, one more parameter K shall be obtained. In order to get r., the values of 131, 132, 133, and l' are required. Although l' can be calculated from spectroscopic data,11 other parameters cannot be esti mated without precise expressions of them. Similarly, the correction term of the mean square amplitude, 4(!32+'Y!31)a 2, is left out of the measurement. In this connection it may be mentioned that ro= ~aJrPa(r)dr / ~aJ Pa(r) dr, = r.+2 (131+1') a' + 12 (133+1'132) a'2. (11) Therefore, comparing it with Eq. (8), we have (12) that is, ro,t the mean of the distance with regard to the anharmonic probability function, is given in a straightforward way by ra and a'; it is independent of the !3's which are necessary for calculating reo It would be worthwhile to consider here the precise meaning of the mean square amplitude. In particular, for the anharmonic case we must consider the effect of anharmonicity on the mean square amplitude. Since 2a= (6.z2) is the mean of the square of the deviation of the atomic distance from the equilibrium position, averaged with regard to the distribution function for a harmonic potential function, we find However, as was pointed out by Reitan22 and discussed by Kuchitsu,14 there are a number of slightly different definitions of the mean square amplitude, for example, t r. is the same as r.(O) defined by Bartell (reference 21). + Exactly speaking, (t1z2) is the mean of (Z_Z.)2, but it coin cides with ((r-T.)2) in the first approximation. and where 1.2 is the mean of the deviation from the equi librium position of the atom pair, II that from the distance corresponding to the center of gravity of the distribution function, both being averaged with respect to the distribution function for the actual anharmonic potential function. The definition of 1/ is quite ana logous with (6.z2), the difference being only in the respect that the mean is made with regard to the anliarmonic, rather than the harmonic probability function. If we assume Eq. (6) for Pa(r) , they are easily calculated as follows: and (16) (17) Comparison of Eq. (16) with Eq. (9) reveals that 1/=2a'. (18) Thus it is I. that is obtained by the analysis of the intensity curve, because, as will be described in the following section, a' is treated as a parameter to be determined by least squares calculation. This is in sharp contrast to the fact that for the equilibrium dis tance, rg is directly obtained by the measurement, whereas r. is too difficult to be calculated. 3. Method of Least Squares The least squares method has the advantage that it is not influenced by the termination effect which is encountered in the radial distribution method. At the same time it makes it possible to get the most reason able values of the parameters with their standard errors. The procedure applied here is essentially the same as that reported by Bastiansen, Hedberg, and Hedberg.23 The observed values of molecular intensity qM(q) at each q value,§ denoted by Fo, are fitted by Fc, the theoretically calculated values of qM(q) , so as to reduce the square sum of residuals as much as possible, by changing the molecular parameters which specify the Fc. For the analysis of the diffraction data of GeC4 and CC4, the following formula corresponding to Eq. (7) is assumed for Fc: Fc=i{ C1(1/ral) exp( -at' S2) sins(ral-Kls2) +C2(1/ra2) exp( -ab2) sins(ra2-K2s2)}, (19) in which the subscript 1 designates the Ge-Cl or C-Cl pair and 2 the Cl-Cl pair. The ro's in the denominators 23 O. Bastiansen, K. Hedberg, and L. Hedberg, J. Chem. Phys. 27, 1311 (1957). § The analysis in this article was carried out by use of the q scale (q= 10s/ ... ) though the description of the article is pre sented by the term s. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13646 MORINO, NAKAMURA, AND TOBIA TABLE 1. Results of the least squares analysis of the molecular intensity curves of GeCLI at 23°C. i rol(Ge-Cl) ro2(Cl-Cl) (2a/)1 (Ge-Cl) (2a2')i(CI-Cl) Kl(Ge-Cl) XI05 K2(Cl-Cl) XI06 With five unknown With seven unknown parameters parameters 0.97.±0.02 3 A 2. 112dO.ooh 3.440 9±0.004o 0.048 6±0.0020 O. 0995±0. 0033 0.97dO.023 A 2. 110D±0.oola 3. 4433±0. 006. 0.048 6±0.OO2 o 0.099.±O.OO3 3 -0.S3±0.SO 1.86±3.1 in Eq. (7) were safely approximated by ra's .• The coefficients 81 Fo:(s) II Fd(s) I L: I F;'(s) 12+ L:Si(S)-COS~1), (20) and are numerically calculated at each q by use of a number of tables16.18.19,20 and inserted into Eq. (19). The parameters to be determined by least squares are ra, a', and K for each atom pair and the index of resolution i. The preliminary values of the parameters for the starting model of least squares are given by the analysis of the radial distribution curve. The result of the radial distribution curve was so close to the final one that only one trial was necessary to get well-converged values. As for the weight to be multiplied to the difference of Fo-Fe at every q in the calculation of least squares, we have no exact knowledge except that it decreases at both ends of large and small q values. An approximate form was tentatively assumed for each case so as to realize the damping feature of the observed curves. All the numerical computations were carried out by the use of the parametron digital computer PC-1, at the Department of Physics. 4. Radial Distribution Function The modified radial distribution function which offers the starting model is provided by the Fourier transformation of the ideal molecular intensity with an artificial damping factor exp ( -bs2) , that is, f(r)=foSMi(S) exp(-bs2)·sinsrds. (22) The ideal molecular intensity SMi(S) is that of a fictitious model which has constant scattering factors instead of the true atomic structural factors depending ons, . "" Z ;,z i sinsr a sM'(s) = t'L:Z,(Z,+l) .----;;-exp[ -(B+a') s2]. (23) This is the very function that was introduced by Bartell, Brockway, and Schwendeman,9 except for a term exp( -BS2) which expresses the phase shift due to the failure of Born approximation. The factor B can be properly assumed14 by the condition that the function exp( -BS2) expresses the term COS~1) with minimum deviation in the required range of s, 18m"[coS~1)(s) -exp( -BS2) Jexp[ -(a' +b)S2Jds=0, o (24) where b is an artificial temperature factor for eliminating the termination effect. The value of B was estimated to be 0.464X 10-3 for the Ge-CI pair, by the value of COSA1) obtained from the table of Ibers and Hoerni and corrected for the accelerating voltage 45 kv by the use of their transformation formula. The correction of sM(s) to SMi(S) was made by the procedure proposed by Brockway, Bartell, and Schwen deman,9 that is, the difference between Mi(S) and M(s) is calculated for an approximate model, the difference AM(s) =Mi(S) -M(s) is added to the observed molecular intensity, and the structural param eters of the model are successively corrected with the result of the analysis of the radial distribution curve which is obtained by the Fourier transformation of the SMob8(S) +s~M(s). RESULTS AND DISCUSSION GeC14 Four curves of the observed molecular intensity were separately analyzed by the method described in the preceding. They gave results sufficiently consistent with one another, the average of which are listed in Table I with the standard deviaticms. The weight function w(q) in the least squares was tentatively chosen as w(q) =N1q exp( -alq2) for 8~q~25, w(q) = 1 for 25~q~43, w(q) =N2q exp( -a2q2) for 43~q~97, (25) {) 20 40 60 80 100 FIG. 1. Calculated and observed molecular intensity curves of GeCI •. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13AMP LIT U DES AND FOR C E CON S TAN T S 0 FCC I ~ AND G e C I 4 647 where Cll=8.00XlO-4 and Cl2=2.70X10-4. Normaliza tion coefficients Nl and N2 are taken so that the weight function w(q) is continuous at q=25 and 43. The least squares treatment was applied to the molecular in tensity curves, first with five unknown parameters, i, al, ~', ral, and ra2, neglecting the parameters Kl and K2, and then with seven parameters including Kl and K2. Equation (19) shows that the anharmonicity of vibra tion produces a phase shift -KS2, in the molecular intensity curve, so that it seems better to carry out the calculation with seven parameters. It was re vealed, however, that the values of the mean amplitudes obtained with seven parameters were almost the same as those obtained with five parameters and the dis tances were only slightly affected within the magnitude of the standard error. In addition, as the standard error of the phase shift parameter K is of comparable order with the value of K itself, the values of K obtained Ge-Cl (a) A_ CI-Cl 1\ b J. o 1 2 3 4 5 A (b) "" = , -==-= b o 1 2 3 4 5 A FIG. 2. Radial distribution curve of GeCl •. Curve B indicates the difference between the experimental RD (curve a) and synthetic RD for the final model. are meaningless. Thus there is no substantial difference whether the analysis is performed with five or with seven parameters. The calculated molecular intensity curve is com pared with the observed one in Fig. 1, and the experi mental radial distribution curvef(r) is shown in Fig. 2. It is well fitted by Gaussian peaks computed with the molecular parameters determined by the least squares. The difference between the experimental and the calculated f(r) is plotted in Fig. 2. The maximum difference is about 3% of the height of the Ge--Cl peak; it is comparable to the level of the ghost signals. The observed background line is compared with the theoretical one in Fig. 3. The theoretical background intensity for an 1'3 sector was calculated by the equation I(q) = (l/q) (l-o(q» {L:F i'(q)2+ L:s .(q)}, (26) i i where o(q) = (313200)}.2q2, a constant coefficient beimg omitted. Equation (26) was derived by taking into account the intensity of o 20 40 60 80 100 q FIG. 3. Background line of GeCI •. The deviation beyond q= 70 is due to the finite spread of gas molecules at the nozzle. scattered electrons which hit a unit area of a plate placed at right angles to the primary beam. The distribution of gas molecules around the nozzle was estimated by the discrepancy of the slope of the observed background line from the theoretical one, in the region of q larger than 70 as shown in Fig. 3, and the corresponding corrections of the root mean square amplitudes were found to be -0.0013 A for the Ge--Cl and -0.0016 A for the Cl-Cl pair, respectively. In Table II the final results of the analysis are tabulated, in which the distance parameters Yg were computed by Eq. (12) and the mean amplitudes were corrected for the finite sample size. The standard errors were determined by considering the standard devia tions of the least squares as well as other possible systematic errors. The effect of uncertainty in the values of cosA'1 for the Ge--CI pair was assumed to be about 2%. 1. Equilibrium Distance In order to obtain the equilibrium distance Ye, we have to have the correction terms in the expression of 1'e= Ya+2a' 11'.-2,},a' -2fJra' -12a'2(!13+y!12). (8') As for ,,/, the perpendicular amplitudes (AX2) and (Ay2) must be evaluated, in addition to the parallel amplitude (AZ2). They are easily obtained by the relation17 «(X.-Xj)2)= [Di/(M-IB')]'U' (~-l)' <Q2)~-lU . [Di/(M-IB')], (27) The ~ matrix elements were calculated with the force constants obtained by the consideration described TABLE II. Final result for the molecular parameters of GeCIt at 23°C. rg I. Ge-Cl 2. 113±0.003 A O.047.±0.003. CI-Cl 3.444±0.006 A 0.097 .±O. 0033 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13648 MORINO, NAKAMURA, AND IIJIMA TABLE III. Parallel and perpendicular mean amplitudes of GeCl. in A computed by the use of the force constants obtained by the present study (at 296°K). Pair Ge-Cl Cl-Cl (AZ2) 0.002028 0.009708 (Ay2) 0.007798 0.003571 below. The results of the calculation are listed in Table III. The term -2a'.B1-12a'2(.Ba+'Y.B2) is estimated to amount to -0.006 A, if a Morse type potential func tion is assumed V(r) =Dl1-exp[ -tl(r-r.) JP, (28) with tl=2.0 A-I. The correction is comparable to the term of 2'Ya', or rather larger than that. For the present as the exact values of the anharmonicity parameters are not available, it is impossible to get the r. with suffi cient accuracy. 2. Mean Square Amplitude As mentioned in the preceding section, the mean square amplitude obtained by the experiment is a', whereas the quantity obtained by the theoretical calculation is 2a= (&2), that for the harmonic po tential. In order to obtain the mean square amplitude which may be compared with the experimental result, we must make a correction for the anharmonicity of the potential function, which is expressed by the term 4a2(.B2+'Y.Bl) in Eq. (9). To make the argument simple, a case of diatomic molecules is taken as an example. In this case 1'=0, and .82 has the following relation to the Morse function parameters: (29) (30) 0.009790 0.010711 r 2.113 3.444 2a/r 0.00096 0.00282 0.00416 0.00207 mean square amplitude must be corrected by an in crease of about 1 %. In this connection it must be mentioned that the theoretical calculation should be performed by the use of the frequencies 11. but the actual frequencies we have for GeCl4 and CCl4 are 1I0'S: we have no means of secur ing the anharmonicity factor of these molecules. As the mean square amplitude is approximately inversely pro portional to the frequency, as indicated by Eq. (32), the true mean amplitude is obtained by multiplying the factor 110/11. to apparent mean amplitude l02 which was calculated by substituting 110 for 11., Therefore 2a' = 102(1 +Xe) (1-2x.) = 102(1-xe). (35) Thus the correction due to the 1I0'S would be partially canceled if we used a' instead of a in the comparison of the observed mean amplitude with the calculated one. This consideration is valid only for diatomic mole cules. However, even in polyatomic molecules, the error would be of the order of Xe, and the comparison of 2a' with 102 would have the smallest deviation. Considering the accuracy of the measurement of mean square amplitude by electron diffraction at present, we can safely neglect the correction and directly compare 2a', the factor obtained by electron diffraction, with the values calculated by the use of fundamental frequen cies 110. 3. Force Constants and xe=hlle/4De• The general quadratic form of potential energy (31) expression of a tetrahedral molecule is given by For the calculation of the correction term, 4a.B2, an approximate relation, (32) which holds exactly only in the ground state, can be safely used, and the result is (33) Therefore, the correction is of the order of Xe. In fact, Reitan22 calculated the effect for a number of diatomic molecules and arrived at a similar conclusion that root 2V= L)r(Mi)2+ L:2krr(Mi) (Mj) + L:ka(re~aij)2 + L:2kaa(re~aij) (re~ajk) + L:2kaa' (re~aij) (r.~akl) + L:2kraMi(r.~aij)+ L:2kra'~ri(r.~ajk), (36) where ~r denotes the displacement of bond distances and . ~a the change of valency angles. This equation has seven force constants, but the symmetry of the molecule Td reduces the number to five: 2V = FllS12+F22 (S2a2+S2b2) +Fa3(Sa}+Sab2+ Sa}) +2F34(SaaS4a+S3bS4b+S 3cS4c) +F44(S4a2+S4b2+S4c2), (37) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13AMP LIT U DES AND FOR C E CON S TAN T S 0 FCC I. AND G e C I 4 649 TABLE IV. Fundamental frequencies PO of GeC4 in em-I. P2(E) 396 132 453 172 where Sl=Z-I(ilrl+ilr2+ilr3+ilr4) , S2a= (lZ)-1r.(Zila12- ila23-ila13- ila14-ila24+ZilaM) , S2b= 2-lr.(ila23- ila13+ila14- il(24), S2a=6-1(ilrl+Ar2- 2ilr3), Sab= (12)-1(ilrl+Ar2+Ara-3Ar4), Sac = Z-1 ( -ilrl+Ar2), S40.= (12)-1r.(Zila12- Aa23-ilala+ ila14 + Aa24-2 A(34) , S4b = 6-1r. (Aa12+ ila2a+ ilala -ila14 -Aa24 -ilaM) , S4c = Z-lr.(Aa23- AalS-Aa14+il(24) ' S& = 6-1r. (ila12+ ila23+ ila13+ ilal4 + ila24 + ilaM) . (38) The elements of the F matrix have the following rela tions to the constants in Eq. (36): (39) The symmetry coordinates relate to the normal co ordinates by the transformation matrix L, S=LQ. (40) I, FIG. 4. Correlation curve of lJ(Ge-CI) and MCI-Cl) of GeC4. The shaded area indicates the observed values. The boundary lines are drawn to show the standard errors. The cross corresponds to the Urey-Bradley force field. TABLE V. Force constants of GeC4 in millidyne/A. For symmetry coordinate system Fll 3.276 F22 0.121 F$3 2.63±0.30 F$4 O.12±0.20 F44 O.IS±O.Ol For internal coordinate system k. krr k,,-kaa' k""-k,,,,' k.,,-k.,,' 2.79±O.23 O.16±O.OS O.IS±O.01 0.03±O.Ol 0.09±0.14 Now, the method to calculate mean square ampli tudes was previously presented by Morino et al.24j the mean amplitude of the vibration of a pair of the atoms whose displacements are given by the matrix relation .lR=AS, is given by (.lR2) = A'L (Q2)L' A, (41) where (Q2) is a diagonal matrix having the elements of (42) For a tetrahedral molecule Eq. (41) provides the following equations: 1l2(Ge-Cl) =iGu (Q12)+!L332(Qa2)+!LM2(Q42), (43) k2(CI-Cl) =jGll (QI2)+tG 22 (Q22) +H2L33+ L43)2(Qa2) +H2L 34+L44)2(Q42), (44) where the G.;'s are the elements of the well-known Wilson G matrix and are given by the masses of the atoms and the geometry of the molecule. L33, ~3, and L44 are calculated for every set of Fa3, F34, and F44 which satisfy the two observed frequencies, Va and V4, through the secular equation I GF-AE 1=0. As infinite sets of force constants are allowed to express the A3 and A4, there must be a correlation between the values of 11 and k which satisfy the two frequencies. They were calculated by the use of the fundamental frequencies measured by the Raman effect (Table IV),2& The result is shown in Fig. 4, where the observed values of the root mean square amplitudes are indicated by the TABLE VI. F matrix elements of GeC1 in millidyne/A calculated with the assumption of Urey-Bradley force field. 2.656 0.116 0.173 a With an assumption of 9'=0. b With an assumption of F'=-F/lO. 2.639 0.106 0.174 Present experiment 2.63±0.30 0.12±0.20 0.lS±0.01 24 Morino, Kuchitsu, and Shimanouchi, J. Chem. Phys. 20, 726 (1952). .. Delwaulle et al., J. Phys. radium 15, 206 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13650 MORINO, NAKAMURA, AND IIJIMA TABLE VII. Results of the least squares analysis of molecular intensity curves of CCl. at 22°C. Observer ra1(C-Cl) Ta2(Cl-Cl) (2a1,)1(C-Cl) (2~')1(Cl-Cl) K1(C-Cl)X1()5 K2(Cl-Cl)X1()5 1"1 1. 769±0.OO31 2.888±0.0016 0.0531±0.0020 0.0736±0.OOl1 -5.7±3.1 -1.2±1.9 99.9±1.5 1"1 1.768±0.OO31 2.887±0.0019 0.0499±0.OO22 O. 0723±0. 0012 -4.7±4A -4.8±2.6 93.7±1.6 1"1 1. 7 59±0. 0033 2. 882±0. 0013 0.0497±0.0024 0.0697±0.0013 3.1±3.5 0.9±2.1 99 A±1. 7 I 1. 769±0.0062 2.899±0.0031 0.0503±0.0042 0.0737±0.OO22 2.8±6.2 1.5±3.8 94.5±3.1 AI 1. 763±0.0036 2.888±0.0018 O. 0525±0. 0024 0.0718±0.0013 -3.0±3.6 3.9±2.2 95.5±1.8 Mean 1. 7653±0.002o 2. 8864±0.0021 0.0513±0.OO14 0.072b±0.0007 -2.12±2.6. 0.lo±1.4o 97.07±1.3 3 TABLE VIII. Final results for the molecular parameters of CCl4 at 22°C. Present measurement C-Cl ra 1. 7653±0.002o 2alr 0.0014 Tg 1. 7667±0.003 (2a')~ 0.0513±0.001 t.l -0.0012 I. 0.0505±0.OO2 Cl-Cl 2. 8864±0. 0021 0.0017 2.888J±0.003 0.0721dO.0007 -0.0025 0.0696±0.001 shaded area, the boundaries being drawn by the standard errors. The values of the F matrix elements corresponding to the line belonging to this area are shown in Table V. It should be noted that each of the force constants listed in Table V is not independent but must have a mutual correlation by the restriction that the set of force constants is consistent with the observed fundamental frequencies. The values of force constants in the usual potential energy expression of Eq. (36) were calculated by Eq. (39) and are listed in Table V. The Urey-Bradley force field 2V= LK(Mi)2+ LH(reAai;) 2+ LF(Aqii)2 + L2H're2Aaii+ L2F'qe(Aqii) , (45) has the following relations to the F matrix expression: Fll=K+4F, F22=H+!F-!F'+[1/(8)1]H', F33=K+tF+tF', F34=1(F+F') , F44=H+!F-iF'-[3/(8)!]H'. (46) The F matrix elements corresponding to the Urey Bradley field which satisfy the observed frequencies are listed in Table VI, one with the assumption H' = 0/6 and the other with F' = -F /10.27 Both sets of mean amplitudes which were calculated with these constants are indicated by the cross in Fig. 4. Thus it can be concluded that the molecular field of germanium tetra chloride is very close to the Urey-Bradley force field. 26 T. Shimanouchi, J. Chern. Phys. 17, 245 (1949). 27 T. Shimanouchi, J. Chern. Soc. Japan 74, 28 (1953). Bartell, Brockway, and Schwendeman C-Cl 'RD= 1. 769±0.003 rRDo=1. 766±0.003 0.058±0.005 Cl-Cl TRD=2.887±0.004 0.068±0.003 It should be mentioned that Cyvin28 also arrived at the same conclusion, using his new secular equation method for calculating mean square amplitudes. The same treatment was applied for carbon tetra chloride. The least squares calculation was carried out with seven unknown parameters. The weight factor for the measurement was assumed to have a similar form to that for GeC4, with the slight modification that the weight was assumed to be unity from q=21 to 60, both sides of which were spliced by two Gaussian functions: w(q) =exp[ -0.040(q-21)2], for 19~q~21, w(q) = 1, for 21~q~60, w(q) =exp[ -0.002 (q-60)2], for 60~q~95. (47) This function is somewhat arbitrary and of no par ticular significance, of course, but it was found that the result was not sensitive to the assumption about the weight factor. In order to check personal errors made by observers and fluctuations of photographic plates, three persons performed independent measurements on different plates. The results are shown in Table VII, in which the figures after ± signs indicate the standard devia tions in the least squares, not including systematic errors such as those due to finite sample size. Table VII clearly shows that the variation of results from person to person is of the same order as that arising in the course of one person's measurement and of the order of the standard error. Judging from the magnitude 28 S. Cyvin (private communication). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13AMP LIT U DES AND FOR C E CON S TAN T S 0 FCC I 4 AND G e C I 4 651 of the standard deviations, the values of K are meaning less, just as in the case of GeC4. The mean values listed on the lowest line are the weighted mean of the five observations, by taking into account their standard deviations obtained by least squares. The distribution of gas molecules around the nozzle was found to require a correction of -0.0012 A for the root mean square amplitude of the C-Cl pair and -0.0025 A for that of the Cl--Cl pair. The error in the measurement of the scale factor which might give a direct influence on the atomic distance was estimated to be about 0.1 %. It gives to the mean amplitudes an error of the same order but it is negligible compared with the standard errors. The final values corrected for the finite sample size are shown in Table VIII. It is noted that the results are in excellent agreement with those of Bartell, Brockway, and Schwendeman,9 except the root mean square amplitude of C-Cl: 0.0505±0.002 A is defi nitely smaller than their value, 0.058±0.005 A.II As for the calculation of the force constants, the same procedure as that applied to GeC4 was found to give the result shown by curve A in Fig. 5. In this calcula tion the frequencies observed by the Raman effect in the gaseous state29 were adopted for the frequencies of the normal vibrations (Table IX). It is well-known that the 113 is perturbed by a Fermi resonance with 111+114 to yield a doublet, 751.1 and 790.6 cm-I• The frequency of the stronger line, 790.6 cm-I was used as the frequency of 113. It will be recognized that the shaded square for the observed values cuts the theo retical curve A only with an edge. It seems likely that one of the reasons for this discrepancy comes from the ambiguity of the frequencies due to the Fermi reso nance and other causes. If small deviations are allowed I, (CI-Cl) A 0.070 0.065 0;050 0.055 A I, (C-CI) FIG. 5. Comparison of the observed and calculated root mean square amplitudes of CCL.. Curve A indicates the calculated values with the observed frequencies, point B designates the calculated values for the most probable set of force constants, and cross C corresponds to the U rey-Bradley force field for the set of frequen cies of point B. II Private communication. Bartell obtained 0.058 A in place of 0.060 A. which he reported in reference 9, by recalculating the correction for the failure of the Born approximation. 29 Morino, Watanabe, and Mizushima, Sci. Papers Inst. Phys. Chern. Research, 39,348 (1942). TABLE IX. Comparison of the frequencies and root mean square amplitudes of CCI. at 295°K. Observed Calculated Difference Pl(A) 455.7 cm-1 454.0 cm-1 +0.4% p.(E) 213.5 211.2 +1.1% pa(F2) 790.6 794.9 -0.5% P4(F2) 309.7 304.4 +1.7% It(C-CI) 0.0505 A 0.0515 A -1.8% l.(C1-CI) 0.0696 0.0688 +1.2% for the frequencies as well as for the mean square amplitudes, it might be interesting to select a set of force constants which represent both the frequencies and the root mean square amplitudes with the smallest deviations in the sense that L:[ (1I,.ob.-lI(alc/II,.ob. J2+ L:[ (l,ob'-l,calc) /l,.ob. J2 =min, (48) where equal weight is tentatively assumed for II. and l;. ~ The best set of the root mean square amplitudes and the frequencies thus obtained are compared with the observed ones in Table IX, and the elements of the F matrix corresponding to this choice are tabulated in Table X. In the latter the figures after the ± signs indicate the ranges of the force constants found only when the root mean square amplitudes have devia tions which are the same as the observed standard errors. Point B in Fig. 5 designates this best choice. Though it does not coincide exactly with the observed point 0, it is positively within the shaded area of the standard errors of the observation. The force constants in the usual potential expression (36) in terms of the displacements of bond distances and bond angles were calculated from the F matrix elements thus obtained; the result is listed in Table X. They compare well with those of germanium tetra chloride shown in Table VI. One of the main objects of this study is to examine the availability of the Urey-Bradley force field, as already mentioned. The mean amplitudes of the force set of the Urey-Bradley type** which satisfies the four frequencies of the best set is indicated by cross C in Fig. 5. It does not fall in the shaded area of the observed point 0, but it is located very close to the borderline of the area, and it is surely within the same range from the point B of the best set, which is shown by dotted lines. Judging from the nature of the estimated errors, this point can not be excluded as an unreasonable choice, but rather it may be preferable to say that the Urey-Bradley field is approximately ~ It may be most reasonable to assume the weight of each quan tity to be proportional to its accuracy, but the accuracy of the frequencies can not be evaluated because of the Fermi resonance and other unavoidable causes. ** With the assumption of F' = -F /10. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13652 M 0 R I NO, N A K A M U R A, AND I I J I M:A TABLE X. Force constants of CC4 in millidyne/A, For symmetry For internal coordinate coordinate system system Fa 4.305 k. 3.59±0.56 F22 0.311 k., 0.24±0.19 Faa 3.35±0.75 ka-kaa I 0.38±0.03 Fat 0.52±0.21 kaa-kaa I 0.03±0.01 F44 0.38±0.03 k.a-k.a I 0.36±0.15 realized. The conclusion is also supported by the fol lowing comparison: the values of the constants, K, H, F, F', and H' of Eq. (45) are calculated by use of the relations (46) from the F matrix elements obtained above, without any restriction imposed upon F' or H'. The values thus obtained, which are listed in Table X, are compatible with the assumption of F' = -F /10 or H'=O, which has frequently been assumed for the usual Urey-Bradley type of force field. The set of force constants obtained by Shimanouchi,27 though they were adjusted to the frequencies of liquid carbon tetrachloride, are also included in the range of errors. Looking at Table IX, it will be noticed easily that the discrepancies of the frequencies are not confined to the "3 which suffers a perturbation by Fermi resonance but they are equally distributed over all the frequencies. Thus it seems likely that the discrepancies are not only due to the Fermi resonance but also due to the anharmonicity of the frequencies. In fact, in a tetra hedral molecule three fundamental frequencies are degenerate so that the effect of anharmonicity would be exaggerated two or three times. Moreover, the location of curve B corresponding to given values of "3 and "4, is sensitive to the frequency of the 1'2. In this For Urey-Bradley force system Present authors Shimanouchi K 2.32±0.86 1.81 H 0.34±0.20 0.10 F 0.50±0.22 0.64 F' 0.28±0.38 -0.06 H' -0.31±0.37 -0.03 sense it would be most indispensable to get exact values of lIe, the frequencies of normal vibrations, not the "0 of the fundamental frequencies. In this connection it is supposed that the anhar monicity is more serious in CC4 than in GeC4, because the calculated values of the mean square amplitudes are closer to the observed ones in GeC4 than in CC4. It is likely that the compact packing of chlorine atoms in the carbon tetrachloride molecule would produce a force field with high anharmonicity, whereas in germanium tetrachloride larger atomic distances would bring the molecular field to a more regular one. ACKNOWLEDGMENTS The authors wish to express their gratitude to Professor H. Takahashi and his co-workers, in the Department of Physics, for the use of the PC-l com puter, and to Mr. H. Oikawa, of the Coal~Research Institute of Mitsui Chemical Industrial Company, Ltd., for his kind offer of germanium tetrachloride. They wish also to thank Mr. Y. Murata for reading one of the microphotometer tracings. They are in debted to the Ministry of Education of Japan for the re!';earch grant. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Sun, 21 Dec 2014 03:01:13

1.1777001.pdf

Dynamic Properties of the Polarizability in BaTiO3 Crystal Kazuo Husimi Citation: Journal of Applied Physics 30, 978 (1959); doi: 10.1063/1.1777001 View online: http://dx.doi.org/10.1063/1.1777001 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase transitions in BaTiO3 thin films and BaTiO3/BaZrO3 superlattices J. Appl. Phys. 116, 184102 (2014); 10.1063/1.4901207 Evolution of structure in Na0.5Bi0.5TiO3 single crystals with BaTiO3 Appl. Phys. Lett. 105, 162913 (2014); 10.1063/1.4900547 Magnetic properties and electronic structures of (YTiO3)2/(BaTiO3) n superlattices J. Appl. Phys. 115, 17D710 (2014); 10.1063/1.4863489 Large and electric field tunable superelasticity in BaTiO3 crystals predicted by an incremental domain switching criterion Appl. Phys. Lett. 102, 092905 (2013); 10.1063/1.4795330 Structure, magnetic, and dielectric properties of (1-x)BiFeO3-xBaTiO3 ceramics J. Appl. Phys. 109, 07D907 (2011); 10.1063/1.3554253 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.138.73.68 On: Sun, 21 Dec 2014 15:09:36JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 7 JULY, 1959 Dynamic Properties of the Polarizability in BaTi0 3 Crystal KAZUO HUSIMI Electrical Communication Laboratories, Nippon, TelegraPh and Telephone Public Corporation, Tokyo, Japan (Received November 21, 1958) The second harmonic distortion by a BaTi0 3 crystal of a small high-frequency sinusoidal electric field, superimposed on a low-frequency switching field, is studied by the filter method. From the phase relationship of the second harmonic distortion with respect to the measuring signal, together with the increase of the fundamental component of the capacitive current, it is concluded that the polarizability increases markedly for the backward direction during switching. I. INTRODUCTION FERROELECTRICS have well-known technical applications as nonlinear dielectrics. A measure of nonlinearity is the second harmonic content in the capacitive current response to a small sinusoidal electric field. In the experiment concerning nonlinearity of a BaTi0 3 crystal, an abnormal increase of the second harmonic distortion was observed during switching. This phenomenon has been studied by measuring it with a small high-frequency measuring signal superimposed on a low-frequency switching field by means of the filter method. II. EXPERIMENTAL ARRANGEMENT The measuring circuit is shown schematically in Fig. 1. The switching voltage is applied to the crystal through the resistance RI. The liberated charge due to switching charges the 1-Jlf standard capacitor C8• The terminal voltage of C. is amplified and is displayed on the oscilloscope C.R.O.-l· for hysteresis indication. At the same time, the output voltage developed across R2 is separated into the fundamental and second harmonic components of the capacitive current by means of band-pass filters. The two components are amplified separately, with the fundamental appearing on C.R.O.-2 and the harmonic appearing on C.R.O.-3, both against the switching voltage. For determination of the phase relationship, the Lissajous trajectory of the second harmonic distortion is also indicated for the measuring signal on the oscilloscope C.R.O.-4, which is unblanked by the switching voltage to be able to see only positive or negative half-cycle of switching. S .. -··SWITCH FOR BRIGHTNESS CONTROL FIG. 1. Experimental arrangement. This circuit is devised to accommodate standard instru ments with 7SQ impedance. High-frequency signals are indicated by the level difference with respect to the standard O-db level (0.274 v-7SQ) ordinal. Instead of 0.05 Jlf Ca in Fig. 1, larger coupling capacitors are used for measuring frequencies below 1 Mc. III. RESULTS In Fig. 2(a) is shown the ordinary SO-cps hysteresis loop on the C.R.O.-1. The second harmonic signal pattern of 400 kclsec due to the 200 kclsec measuring signal is also shown in Fig. 2(b). It is clear from these photographs that the second harmonic distortion in creases markedly in the vicinity of the steep rise portions of the SO-cps hysteresis loop and diminishes with the completion of switching. The fundamental component measured at the same time is shown in Fig. 2(c). In this case, the gain of the amplifier is several times smaller than that of Fig. 2 (b). This picture shows that the capacitive current also increases considerably over the same portions of the hysteresis loop. The second harmonic distortion is caused by the difference of the" polarizability of the crystal between positive and negative half-cycles of the measuring (a) (b) (c) FIG. 2. Photographs on the oscillographs. (a) 50- cps hysteresis loop; (b) second harmonic compo nent of the capacitive current; (c) fundamental component of the capacitive current. 978 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.138.73.68 On: Sun, 21 Dec 2014 15:09:36DY:\AMICS OF POLARIZABTLITY I:'\J BaTi0 3 979 signal. \\ie may imagine four possibilities in accounting for the difference in polarizability during switching. First, polarizability increases for the forward direction (case 1); second, it increases for the backward direction (case 2); third, it decreases for the forward direction (case 3); and fourth it decreases for the backward direction (cast! ~). If we assume that the polarizability change occurs only in the half-cycle of the measuring signal, the current response to the measuring signal v sin",/ with nonlinear distortion due to this polarizability change can be expressed by the following Fourier series: CASE I CASE 2 i= iu+ Lim cosmwt, i 1 = (1 +~)wCov, 20 i·,=-wCliv -37f ' DIRECTION OF SWITCHING WAVE FORMS PHASE RELATION j , i, //Z)yQy ¥ J. \[J'J[) V .,: \ \.' ! ';, \ \ .. ~.:! '", I \ I \ I v I \ I V \.j -~v MEASURING SIGNAL CAPACITIVE CURRENT (1) (2) (3) SECOND HARMONIC COMPONENT FIG. 3. Phase relation of the second harmonic component of the measuring signal for cases 1 and 2. where Co is the normal small signal capacitance and 0 is the fraction of the polarizability change. If the polariza bility decreases during switching, 0 is negative; hence Eq. (2) shows that the fundamental component should also decrease during switching. Figure 2(c) shows that such a decrease does not occur; the third and fourth possibilities are, therefore, rejected. We must next determine whether the polarizability increases for the forward direction or for the backward direction during switching. This can be decided from the Lissajous trajectory of the second harmonic distortion of the measuring signal. In Fig. 3, the phase relations between the second harmonic component and the measuring signal are shown both for forward direction and backward direction during negative half cycle switching. The V-type pattern corresponds to forward direction and the inverted V-type pattern to bachvard direction. The experimental results are shown in Figs. 4(a) and (b). Figure 4(a) is the pattern without brightness FIG. 4. Photographs of the Lissajous trajectory of the second harmonic com ponent to the measuring signaL (a) Signal pattern without brightness control; (b) signal pattern with negative half-cycle of switching unhlanking. (a) (b) control in which two patterns due to positive and negative half-cycles of switching are superimposed. Figure 4(b) is the pattern for the negative half-cycle alone. The inverted V-type pattern shows that the forward direction should be rejected. The peak output voltages of the second harmonic component V2p across R2 under the 50-cps switching field are shown in Fig. 5 for the measuring frequency range from 30 kc/sec to 14 J\Ic/sec. In this figure, the output voltage across R2 due to the fundamental component of the capacitive current is shown without switching ~'I and at its peak value during switching ~'Ip. Both are shown for the 20-db measuring signal level (2.74 v-7SQ) together with the maximum second harmonic output voltage without switching (max V2), the value of which is influenced by the remanent state. The samples used here, produced by the KF method, has a thickness of 0.22 mm and an electrode area of 1 mm2• Figure 5 shows that the second harmonic output voltages V2p is nearly independent of the measuring frequency. VI due to the capacitive current itself in creases linearly with the measuring frequency; hence the harmonic content V2p/VI decreases as l/f. The amplitude dependence is shown more clearly in Fig. 6 where the measuring signal level is taken as a parameter. At a lower measuring signal level, the second harmonic -70 --60 -" ~ -50 W <'l ~-40 -' ° >-30 t- ii: -20 t- => °_10 ""'- -<>-~ V,p "" -- V,p - SWITCHING VOLTAGE MEASURING SIGNAL LEVEL ~ mClX 7Jz. I"- o-r- ~~ ........ ....., "" 15 v 20 db "- 0001 0.02 004 0.1 0.2 0.4 I 2 r-,~ 3 4 10 20 (Me I MEASURING FREQUENCY FIG. 5. VI, VIp, V2p, and max V2 versus measuring frequency. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.138.73.68 On: Sun, 21 Dec 2014 15:09:36980 KAZUO HUSIMI 20 20db .......... • . 25 db 10 db ........ ....... r-. ....... ~ o db-o... ....... j'-., ~ ~ -Iodb .........., r--..... ~ ~"""'" r--.. 10 o -10 r--r--t--R ~ -.. r---.. r--SWITCHING VOLTAGE 15 V 1'-... f', ...... -30 0 MEASURING SIGNAL PARAMETER -5 0 0.01 0.02 0.04 0.1 0.2 0.4 2 4 6 10 20 MEASURING FREQUENCY (MC) FIG. 6. Second harmonic contents V2p!Vl versus measuring frequency. content itself and its frequency dependence become small and at the -10-db level (0.087 v-75r2), it is nearly proportional to II!!. If, on the other hand, measuring signal level exceeds 20 db, the capacitive current increases more rapidly because the dielectric constant increases for the larger measuring signal and, therefore, v2plvj decreases. In Fig. 7, the relationship between the second harmonic content V2p/Vl and the switching current density is shown for the 20-db level with the measuring frequency taken as a parameter. The second harmonic distortion increases linearly with the increase of switch ing current density for a low-intensity switching field, but after saturation its increase becomes more gradual. IV. DISCUSSION The increase in pol ariz ability will be represented by o. The values of 0 which are calculated from the di electric constant increase and the second harmonic distortion using the coefficients of the Fourier series are denoted as OJ and 021> respectively. These values are nearly in accord, but 02j is slightly larger than OJ for measuring frequencies above 100 kc/sec. This discre pancy comes from the assumption that the polarizability 2.5 2.0 1.5 ;:S ~ ;::S 1.0 0.5 o. SWITCHING FIELD lv/em) 500 1000 1500 2000 SWITCHING CURREN DENSITY 100 KC 200 KC 500 KC 1000 KC 0 5 15 20 25 30 35 SWITCHING VOLTAGE ( V) FIG. 7. Switching current density and V2p!Vl versus switching voltage. :;e ~ « 100 ~ ~ BOr iii Z 60~ r -40~ a:: a:: 20~ 0 40 does not change for the forward direction. If we furthermore assume that the polarizability decreases by E for the forward direction, and increases by 0 for the backward direction, Eqs. (2) and (3) can be modi fied as follows: 2 i2=-(0+E)WC OV. 371" (4) (5) The values of E and the modified 0 calculated by this method are shown in Fig. 8. As for the fundamental component, the real and imaginary parts of complex polarizability can be separated by the bridge method. The second harmonic distortion due to nonlinearity of the loss component is, however, 1800 out of phase with that of nonlinearity of polarizability; so we have no suitable method to separate them. 30 \ SWITCHING VOLTAGE 15V \ MEASURING SIGNAL LEVEL 20db 25 20 0 1\ \ 15 10 t\ E~ 1\ 5 1\ ~ "" ~1,- o -5 i 10 20 40 100 200 400 1000 MEASURING FREQUENCY (KC) FIG. 8. 5 and < versus measuring frequency. Drougard et aU have measured this dielectric con stant increase with their bridge method and shown that it can be expressed by a relaxation formula similar to the Debye type. According to their results, the fact that the frequency dependence of the second harmonic distortion is of the form 1/! rather than l/P means that, at high frequency the contribution of the non linearity of the loss component becomes dominant. In this case, from the phase relationship, it should be concluded that the conductivity increases for the forward direction during switching. Landauer et al.2 tried to explain their results by the e-a/E type field dependence of the switching rate, pointed out by Merz,3 but they were unsuccessful in 1 Drougard, Funk, and Young, J. Appl. Phys. 25, 1166 (1954). 2 Landauer, Young, and Drougard, J. Appl. Phys. 27, 752 (1956). 3 M. J. Merz, Phys. Rev. 95, 690 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.138.73.68 On: Sun, 21 Dec 2014 15:09:36DYNAMICS OF POLARIZABILITY IN BaTi0 3 981 explaining the dielectric constant increase during switching. Previous results clarify this question. The polarizability increase for the backward direction is a kind of hysteresis phenomenon in switching rate which has been observed by the ultrasonic method.4 Similar results have also been obtained with ferromagnetic materials, indicating that this is a common property of materials with domain structure. 4 K. Husimi and K. Kataoka (to be published), also Electrical Communication Laboratories Research Rept. No. 662 (1958-(}2). V. ACKNOWLEDGMENTS The author wishes to express his hearty thanks to the Head of Communication Research Section, Mr. Z. Kiyasu, for his constant encouragement and valuable suggestions for this work. He is also indebted to Mr. H. Minami,* a student of Osaka University, with whom this work was done as a part of his summer exercise in 1957. Thanks are also due Mr. S. Waku for the good quality crystal and Mr. K. Kataoka for his kind assistance. * Now at Tokyo Shibaura Electric Company, Kawasaki, Japan. JOURNAL OF APPLIED PHYSICS VOLUME 30. NUMBER 7 JULY. 1959 Spectrum of Thin Target Bremsstrahlung Bounded by a Forward Circular Cone J. H. HUBBELL NationaJ Bureau oj Standards, Washington, D. C. (Received November 12, 1958) The Schiff expression for the cross section per nucleus, for thin-target brem_sstrahlung into the angular and energy ranges dOo and dk, is integrated analytically over angle from zero to (10. Results are shown for the case Z=78, Eo-mc2=40 Mev and x,:,0.5, 1,2,4, and 8, where Eo-mc2 is the kinetic energy of the incoming electron and x is the reduced angle OoEo/mc2• The fraction of the total cross section included in a cone of angle 80 can be approximated to within 20% by ;t2/(1+;t2). At x=d, or 80=0.723°, the ratio of energy delivered by photons above (Eo-mc2)/2 to that below is 9% greater than for the limiting spectral shape at (10=0, or that given by the Schiff expression at 80=0, and 13% greater than for the spectrum integrated over all angles. For a target of finite thickness, multiple electron scattering should partially suppress the spectral dependence on t. THE Schiff! expression for the energy-angle dis tribution of thin-target bremsstrahlung has been widely used because of ease of evaluation in closed form and reasonable agreement with experiment.2 Targets used with present and proposed high-intensity, medium energy electron linear accelerators may in practice be sufficiently thin for this expression to be applicable. However, to obtain the energy distribution incident on an object of finite extent, integration over the angular aperture of the utilized beam should be performed. This paper reports the result of such integration over a forward circular cone of arbitrary aperture. The notation used in this paper will follow reference 1, according to which Eo= incident electron total energy, E= scattered electron total energy, k=Eo-E=photon energy, JJ=mc2=electron rest energy, 80= angle in radians between photon and incident electrons directions, and x= ErlJo/ JJ= the "reduced angle." Equation (1) of reference. 1 was obtained by inte gration of the Born-approximation Bethe-Heitler 1 L. I. Schiff, Phys. Rev. 83, 252 (1951), Eq. (1). 2 N. Starfelt and H. W. Koch, Phys. Rev. 102, 1598 (1956). differential cross sectionS over the angle of the outgoing electron, using an approximate screening potential of the form (Ze/r) exp( -r/ a), where a is taken to be C,,/mcZt. For convenience it is repeated here where (Eo+E)2 (X2+1)2Eo? (1) l/M(x) = (JJk/2EaE?+ [Zt/C (x2+ 1)]2. (2) The constant C was evaluated by integrating Eq. (1) between the limits X=O and x= OCJ [reference 1, Eq. (3)] and comparing numerically with a result derived by Bethe4 in which complete screening and the Thomas-Fermipotential6 were assumed. This yields the value C= 111. Neglect of higher order terms in obtaining Eq. (1) 3 W. HeitJer, The Quantum Theory of Radiation (Oxford Uni versity Press, London, 1954), third edition, p. 244. 4 H. A. Bethe,. Proc. Cambridge Phil. Soc. 30, 524 (1934). • L. H. Thomas, Proc. Cambridge Phil. So('. 23, 542 (1926); E. Fermi, Z. Physik 48, 73 (1928). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.138.73.68 On: Sun, 21 Dec 2014 15:09:36

1.1722317.pdf

Review of Germanium Surface Phenomena R. H. Kingston Citation: J. Appl. Phys. 27, 101 (1956); doi: 10.1063/1.1722317 View online: http://dx.doi.org/10.1063/1.1722317 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v27/i2 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJournal of Applied Physics Volume 27, Number 2 February, 1956 Special Article Review of Germanium Surface Phenomena R. H. KINGSTON Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington, Massachusetts (Received August 27, 1955) Germanium surface behavior has become of great interest recently, chiefly for its importance in the understanding of diode and transistor technology. In general, the surface may be treated as an assemblage of allowed electron states occurring in the normally forbidden energy range. A review of the measurements of the electrical properties suggests that there are two distinct types of state. The "fast" state has a hole or electron capture time not greater than a microsecond and is chiefly involved in the recombination process. The "slow" state has capture times from a millisecond to several minutes and determines the density and type of carrier at the surface. "Fast" states are believed to occur at the interface between the germanium and the oxide layer, and their density of about 10" cm-' is determined by the initial surface treatment. "Slow" states are associated with the structure of the oxide layer and the gaseous ambient and have a density greater than 1013 cm-'. Since these states determine the conductivity type at the ~urface they contribute to surface "leakage" in diodes and transistors and, because of their long equilibrium ti~es to low-frequency noise. The adsorption of gases such as water vapor, not only controls the density and ene~gy of the "slow" states but also leads to possible electrolytic conduction along the surface, in addition to the normal electron flow in the bulk semiconductor. INTRODUCTION IN 1947, Bardeenlt considered the properties of a free semiconductor surface and also a metal-to semiconductor contact in terms of the possible occur rence of extra allowed energy states at the boundary of the material. These surface states were postulated to have energies in the forbidden region or energy gap of the semiconductor, and their inclusion in the theory successfully explained previous results of Benzer2 and Meyerhoff3 on contact potential and rectification in semiconductors. Shortly after this work further studies of semiconductor surfaces led to the discovery of transis tor action by Bardeen and Brattain4 in 1948. Following this development, research into semiconductor behavior and transistor development led to rapid advances in the understanding of the bulk properties of germanium as well as the technology of transistor fabrication. Al though this work stemmed from studies of the ger manium surface, it has not been until recent years that a commensurate effort has been placed on the problem of the surface behavior. This interest in the surface has arisen largely from the practical problems posed by transistor fabrication, namely the failure of the bulk properties of the semiconductor to account for the anomalous behavior of diodes and transistors. Such effects as surface recombination of minority carriers, "leakage" current in diodes, and excess noise are found to be sensitive functions of the surface preparation and • This Laboratory is supported jointly by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology. t References which have "A" following the page number refer to abstracts of papers presented orally. ambient, and the present understanding of these phenomena will be of chief interest in this paper. 1 J. Bardeen, Phys. Rev. 71, 717 (1947). • S. Benzer, Phys. Rev. 71, 141 (1947). a W. E. Meyerhoff, Phys. Rev. 71, 727 (1947). 'J. Bardeen and W. H. Brattain, Phys. Rev. 74, 230 (1948). Since the majority of the surface effects to be dis cussed may be considered in the light of the surface state picture of Bardeenl the work may be conveniently 101 Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions102 R. H. KINGSTON divided into four major sections. First, experiments will be described which measure the electronic properties of the surface and which may be interpreted by a general ized distribution of surface states. Second, the present data available from these experiments will be reviewed leading to a semiquantitative description of the states. The third section will discuss the possible relation be tween the surface state distribution and the chemistry and atomic structure of the surface, while the last section will deal with the practical embodiments of these surface phenomena in semiconductor devices. I. PHYSICAL MEASUREMENT OF SURFACE EFFECTS In general the surface measurements to be described may be analyzed in terms of the energy level diagram of Fig. 1. This represents the equilibrium situation at the surface of a semiconductor, where Ec and Ev are the energies at the bottom of the conduction band and the top of the valence band, respectively. The dashed line, EF, represents the Fermi energy, which is a measure of the relative concentration of holes and electrons in the respective bands, while the dotted line, Ei, is the value of EF for intrinsic material, having equal densities of holes and electrons. For convenience the position of the Fermi energy will be defined with respect to Ei by the quantity, This leads to the expressions for the hole and electron densities at any point in the semiconductor,· n= ni exp(qrP/kT), p=ni exp( -qrP/kT) where ni is the density of either carrier in intrinsic material. The value of rP in the bulk, far from the surface, is determined by the bulk impurity density and will be designated rPB, while the value of rP at the surface will be called rPs. In this notation, positive values of rP correspond to excess electron or n-type conduction; negative values, to excess hole or p-type conduction. In • • E.------------ FIG. 1. Energy level diagram of germanium surface. 6 W. Shockley, Electrons and Holes in Semiconductors (D. Van Nostrand Company, Inc., New York, 1950), p. 304. addition to the allowed energy levels in the bulk of the material, a general distribution of states will be assumed to exist at the surface represented by the squares in Fig. 1. Now, under equilibrium conditions, the net charge at the surface must be zero, assuming no external field. If there were no surface states, this criterion would be satisfied if rP were constant and equal to rPB right up to the surface. If, now, states are added at the surface as in Fig. 1, they will contain a net charge which is a function of the position of the Fermi level or the quantity, rPs. In the case shown, the states would con tain too much negative charge if rPs were equal to rPB, and therefore the energy bands must bend up at the surface producing extra holes in the bulk material, and reducing the negative charge in the surface states. The equilibrium value of rPs is reached when these two ex cess charge distributions are equal and opposite in sign. (There will, of course, be a constant charge contribution associated with the states which is independent of rPs. This quantity depends upon the net charge of the state when empty, that is, whether it is a donor or acceptor.) In the presence of an external field applied perpendic ular to the surface, the neutrality criterion will be modified such that the net charge in the bulk plus the charge in surface states is of proper magnitude and sign to terminate the field. The charge in the bulk, the space charge, may be calculated from the values of rPB and rPs, and the appropriate value of ni for germanium at the temperature considered.6•7 Such a calculation also yields the expected depth of penetration of the space charge region into the semiconductor, which under normal conditions is the order of 10-6 cm, many orders of magnitude greater than that expected in a metallic conductor. This result, stemming from the much smaller free carrier density in the semiconductor, leads to a correspondingly greater potential change between the bulk and the surface, and thus explains why the semiconductor is much more sensitive to accumulated charge on the surface than a metal. 1. Surface Conductance The surface conductance of germanium may be measured by preparing a thin slab of the material such that changes in the conductance at the surface are a reasonable percentage of the over-all sample conduct ance. Measurements of conductance as a function of surface treatment give, in theory, a direct measure of rPs, since the density of holes and electrons in the space charge region is a unique function of rPs and rPB, the latter quantity known from the bulk resistivity. Strictly speaking, one can only measure changes of conductance in such an experiment; however, if the surface treatment is varied such that rPs moves over a large enough range, a minimum in conductance will be found which may be 6 W. Shockley, Bell System Tech. J. 28,435 (1949). 7 R. H. Kingston and S. F. Neustadter, J. AppJ. Phys. 26, 718 (1955). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsGERMANIUM SURFACE PHENOMENA 103 ~ 30 " ~ ~ " .r: E : 20 w <..> z <t I- U ::> 0 z 0 U 10 8 FIG. 2. Room temperature surface conductance 'ilS q,s for several values of q,B. used as a reference. This minimum occurs near CPs, equal to zero, or when the surface is near intrinsic, and is dependent upon CPB and the carrier mobilities. Curves of surface conductance vs CPs for different values of cP B have been calculated by Schrieffer,8 taking into account reduction of the free carrier mobility in the presence of a potential well at the surface. The curves of Fig. 2, taken from this work, show the conductance as a func tion of cps, with the zero value chosen arbitrarily at cfJs=O. Data obtained in this type of experiment, relating CPs to surface treatment, thus lead to a relation ship between surface state distribution and surface condition. 2. Surface Recombination Velocity The surface recombination rate of excess minority carriers is measured by the recombination velocity, defined by s=J/ap where J is the recombination current in hole-electron pairs per unit area per unit time and ap is the excess minority carrier density at the surface just inside the space-charge layer, that is, where cP equals CPB. Brattain and Bardeen9 first treated the surface recombination process by considering transitions through an inter mediate state in the gap similar to the bulk process treated by Hallio and Shockley and Read.1! Considering states at one discrete energy, Stevenson and Keyesl2 have generalized this treatment to give s= cpc"N I (po+no)[c n (n8+n81)+cp(P.+ P.l)J-1 8 J. R. Schrieffer, Phys. Rev. 97, 641 (1955). 9 W. H. Brattain and J. Bardeen, Bell System Tech. J. 32, 1 (1953). lOR. N. Hall, Phys. Rev. 83, 228 (1951); 87,387 (1952). 11 W. Shockley and W. T. Read, Phys. Rev. 87 835 (1952). 12 D. T. Stevenson and R. J. Keyes, Physica 20, 1041 (1954). where: N I is the number of states per unit area, cp and Cn are the capture probabilities per state per unit time for holes and electrons and are given by the product of the respective capture cross sections and the thermal velocity, p. and n. are the free carrier densities at the surface, P8I and n.l are the surface densities, if the Fermi level passes through the state, and po and no are the bulk carrier densities. This function is plotted in Fig. 3, for the case where the state energy EI, is greater than Ei. The recombination velocity is seen to be constant over a middle range, falling off rapidly where qcps equals (EI-Ei) and qcps equals kT In(cp/c n)-(Et-E,). For the case shown, the states being near the conduction band, the maximum recombination rate is determined by the capture probability of the holes by the surface state. The region of constant s for cps near zero results from the constancy of the product, ntp., where nt is the trapped electron density; that is, because of the Boltzmann statistics, the number of electrons in surface states is inversely proportional to the number of holes in the valence band. If the Fermi level is at the trap energy or beyond, then, the states become saturated, and although p. is decreasing, nt is now fixed and the rate then falls off with p •. At the other extreme, the Fermi level approaches the valence band, and the rate becomes limited by the flow of electrons into surface states, since p. is now large enough to maintain equilib rium with the state. The rapid decrease of s in this region is thus associated with the decrease in the surface electron density with decreasing CPs. There are a variety of techniques for measuring the recombination velocity, some of which will be discussed in the following section, along with the information that has been obtained by application of the foregoing equation. 3. Field-Induced Surface Conductance One of the earliest attempts to utilize a semiconductor for electrical amplification was made by Shockley and Pearsonl3 in 1948. They measured the change in con~ ductance of a thin evaporated film of germanium as a function of the electric field applied perpendicular to s FIG. 3. Surface recombination velocity, s, 'liS q,s for a recombination center near the conduction band. 13 W. Shockley and G. L. Pearson, Phys. Rev. 74, 232 (1948). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions104 R. H. KINGSTON I-- N P N 1"'r'1 + 1,- -,1+ I' 'I FIG. 4. Sectioned view of junction transistor with n-type channel on base region. the surface. Since the film was thin, the increased carrier density required to neutralize the field was ex pected to produce an appreciable change in film con ductance, leading to possible power amplification. This experiment, the "field effect," established that the charge drawn to the surface was largely immobilized in surface states, with only about 10% consisting of free carriers. Since the material was polycrystalline, it was difficult to interpret the measurements; however, this first direct experimental evidence of surface trapping led to the experiments which evolved the transistor. Application of this experiment to single crystal ger manium leads to two types of information about the surface states. First, the change in conductance with induced charge determines whether the surface is n-type or p-type and what fraction of the free carriers are trapped in surface states, while second, the time dependence of the conductance after application of the field yields information about the capture probability of these states. Schrieffer8 has calculated the expected behavior in the absence of surface states but including reduction of the mobility in the potential well at the surface. A general treatment is extremely difficult without some simplifying assumptions and these will be discussed in connection with the experimental data. 4. Contact Potential The contact potential of a semiconductor may be measured by the Kelvin method,9 and with a known value of work function for the reference electrode, gives the work function of the germanium, which, from Fig. 1, may be written W.F.=4>.-4>s. Unfortunately, there is no unique way to determine the reference electrode work function in a gaseous ambient, so that the measurements are limited to a determination of the changes in the germanium value with ambient, assuming that the reference work function remains constant. Since the resultant changes in work function are made up of two terms, one in 4>" and one in 4>s, interpretation of the data requires either a relationship between the two, or an independent determination of one of the quantities. Another quantity of interest is the change in contact potential with light, which, subject to the experimental restrictions discussed later in the paper, is a measure of the changes in CPs with light. If the germanium is illu minated, then the density of minority carriers in the bulk will be increased above its equilibrium value, re sulting in a proportional increase in this carrier density at the surface. As discussed by Brattain and Bardeen,9 the fractional increase in carrier density must be the same at the surface as in the bulk; that is, the quasi Fermi levelS will be continuous across the space-charge region. Therefore, unless the bulk and surface carrier densities are equal at equilibrium, there will be a net change in charge in the space charge layer. For a p-type surface on n-type, for example, doubling the hole density in the bulk by illumination will also double the hole density at the surface, while the electrons, which are the majority carrier, will not have their density changed appreciably. The same increase in positive charge will occur in the surface states, the net result being a deviation from the charge neutrality criterion. Actually, since the net charge must be zero, there will be a compensating shift in the potential energy of the surface which will restore the carrier distribution to a value producing zero net charge. In the example cited, the potential will decrease, thus increasing the electron density and decreasing the hole density. The work function therefore will decrease, corresponding to an increase in 4>s. Brattain and Bardeen9 have treated the theory of this effect considering only the charge balance in surface states; while more recently, Garrett and BrattainI' have considered the general case including the charge in the space charge region. The measurement of the change in contact potential with light, with certain assumptions, gives the type and magnitude of the carrier density at the surface and is thus an inde pendent method of determining 4>s. 5. Surface Conduction "Channels" One of the earliest practical surface problems with junction transistors was studied by Brownl5 in 1953. This involved the occurrence of an anomalous conduc tion path between the emitter and collector of an n-p-n transistor under conditions of reverse bias. This phenomenon, called "channeling" was found to be the result of an n-type surface layer on the normally p-type base material, producing a high resistance ohmic path between emitter and collector. In addition to its practical significance, it turns out that a study of this effect is a valuable tool in the basic understanding of the surface. Consider the structure of Fig. 4, which is a sectioned view of an n-p-n junction transistor. It is found that under the proper surface conditions, the surface of the germanium will become n-type and even with application of the reverse bias shown, will main tain its n-type conductivity, leading to a reverse biased p-n junction between the bulk material and the If C. G. B. Garrett and W. H. Brattain, Phys. Rev. 99 376 (1955). ' 16 W. L. Brown, Phys. Rev. 91, 518 (1953). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsGERMANIUM SURFACE PHENOMENA 105 surface. Since this junction has a high impedance it is possible to measure accurately the conductance ~f the thin n-type region by passing a small current between the emitter and collector. The expected value of the conductance may be predicted from a knowledge of IPs at the surface and the applied bias; thus measurements of the channel properties lead to information about the behavior of the surface states. In addition, information about the capture times may be obtained by varying the reverse bias, which is equivalent to varying the applied field in the field effect experiment. Similar re~~~s can be obtained with p-n-p structures by ~tillZl~g surface treatments which produce a p-type mverSlOn layer. II. EXPERIMENTAL RESULTS One of the first detailed studies of germanium surface properties was carried out by Brattain and Bardeen9 in 1953, where they measured the contact potential, change of contact potential with light, and surface recombinati~n. s.imultaneously on the same sample.The surface was Imtlally etched and then exposed cyclically to the three ambients, dry oxygen, ozone, and wet oxygen. By this treatment it was found that the contact potential could be shifted by about 0.5 volt, the positive e~treme in wet oxygen and the negative, in dry ozone; wIth the values relatively independent of the bulk resistivity. This corresponds to the highest work func tion in ozone, and the lowest in wet oxygen. The mean value of the contact potential between these two ex tremes was also found to drift in the positive direction during the cycling process, changing by about 0.2 vol~ after the freshly etched surface had been exposed to the ambients for several hours. The surface recombination velocity was found to be constant over this range of ambients, within the precision of their measurements a~d the change in contact potential with light agreed wIth the proposed surface model. This model predicted that the changes in IPS were approximately 20% of the observed change in contact potential, the remaining part consisting of changes in IP. of Fig. 1. Later, Morri sonl6 measured surface conductance in the same ambient cycle and found that the changes in IPs required to explain the data were more nearly equal to the accom panying change in contact potential. As discussed by Bardeen and Morrison,17 this indicates that the charge in the space-charge region changes more rapidly with IPs than the charge in the surface states, contrary to the original Brattain-Bardeen conclusions. 9 Attempts to reproduce the original contact potential measurements have cast doubt on the assumption that the reference electrode is stable, and can give anomalous results, apparently as a result of charge collection on the high impedance reference electrode.ls 16 S. R. Morrison, J. Phys. Chern. 57, 860 (1953). 17 J. Bardeen and S. R. Morrison, Physica 20,873 (1954). 18 H. Kolm and G. W. Pratt, Jr., Phys. Rev. 99, 1644(A) (1955). TABLE I. Surface Fermi level vs ambient. H20+N, H20+air H,O+O, N, (dry) Air (dry) O,(dry) H20, (peroxide) 03 (ozone) n-type <1>8 positive p-type <1>8 negative Further measurements of IPs as a function of surface treatment have been carried out both by surface con ductance and "channel" conductance measurements.I9--26 The results are summarized in Table I, which shows the behavior of an etched surface in terms of the value of IPs. The range of this quantity is 0.3 to 0.4 volt from the lowest to the highest value, with the ambients in the middle of the list corresponding to IPs near zero. It is not possible to list quantitative values of IPs, since as described in the contact potential measurements 'the value is also a function of surface history, IPs incre~sing by as much as 0.2 volt with prolonged exposure to air or oxygen after etching. In addition to the data of Table I, it is also found that n-type surfaces are produced by alcohol vapor,9 ammonia gas,24 and ultraviolet illumi nated mercury vapor ;22 p-type, by chlorine9.24 and ultra violet radiation in vacuO.22 The general conclusion from all these measurements is that the carrier type and density at the surface, or the quantity, IPs, is independ ent of the bulk resistivity of the germanium. Before considering the significance of these results in terms of the surface state structure, it is best to first consider the information on recombination velocity. Brattain and Bardeen's9 measurements, leading to a value of s which was independent of the ambients used, were performed by measuring the diffusion length of the minority carriers by the light-spot scanning technique.26 This method is not the most accurate since an inversion layer on the surface can produce anomalous results. (See Sec. IV.) More recent measurements, using the decay of photoconductivity in a thin slab,12 indicate that s may change by as much as a factor of two or three in the Brattain-Bardeen ambient cycle. These later results, however, tend to verify the original observation that the density of recombination centers is relatively independent of ambient, since the changes in s may be attributed to changes in IPs as shown in Fig. 3. Experi ments in which the recombination velocity and the Ig E. N. Clarke, Phys. Rev. 91, 756 (1953). 10 E. N. Clarke, Phys. Rev. 95, 284 (1954). 11 H. Christensen, Proc. lnst. Radio En~rs. 42, 1371 (1954). 22 H. Christensen, Phys. Rev. 98, 1178(A) (1955). .. deMars, Statz, and Davis, Phys. Rev. 98, 539 (1955). Z4 R. N. Noyce, Meeting of the Electrochemical Society, Cincinnati, May, 1955. ,6 R. H. Kingston, Phys. Rev. 98, 1766 (1955). ,6 F. S. Goucher, Phys. Rev. 81, 475 (1951). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions106 R. H. KINGSTON surface conductance are measured simultaneously24.27 do not give results in exact agreement with this curve, however a reasonable distribution of surface states in energy would fit the results within experimental error. Unfortunately the available data are not complete enough to determine the energy of the states, and lack ing a value for the capture probabilities, Cp and Cn, it is not possible to measure the density, Nt, by this tech nique. Other measurements to be discussed below in dicate that this density is the order of 1011 per cm2 for a well-etched surface, and since s is relatively constant over the middle range of r!>s, the states are probably located some distance from the center of the energy gap. Attempts to determine the energy by measurements of the temperature coefficient12.28.29 have not been too successful, since the variation of r!>s with temperature complicates the interpretation of the data. The variation of s with bulk resistivity has been considered by Schultz,30 and found to be in essential agreement with the variation expected from the term, (po+no) in the theory. In addition, measurements of s vs applied field31.32 are also consistent with the theory. Although the density of states is apparently independent of ambient, it is found to be a definite function of initial surface treatment. A careful etching procedure can produce a value as low as 20 cm/sec while sandblasting raises the value to the order of 106 em/sec, higher values being difficult to measure by present experimental techniques. As described in the previous section, one expects the value of r!>s to be determined by the distribution of surface states, taking into account the bulk resistivity, or r!>B. Since the surface conductance measurements indicate that r!>s is a function of ambient only, then one must conclude that the states associated with this am bient are dense enough to completely overpower the effect of the space charge region. Bardeen1 finds that such a condition would require more than 1013 states/ cm2. In addition, since the variation of r!>s with ambient requires a change in state density and energy, one must conclude that the surface state density is both large and ambient sensitive. This would, at first, seem to be in conflict with the conclusions reached from the surface recombination velocity measurements. This inconsist ency may be resolved by considering two different types of states; "fast" states which have a high capture probability and a density of the order of 1011 cm-2 and "slow" states with a capture probability many orders of magnitude smaller and a density greater than 1013 cm-2• Bardeen and Morrisonl7 have suggested that the "fast" states, which are involved in the recombination process, are at the interface between the germanium and the oxide layer while the "slow" states, which con- 27 D. T. Stevenson, Phys. Rev. 98, 1566(A) (1955). 28 Y. Kanai, J. Phys. Soc. Uapan) 9, 292 (1954). 29 W. H. Brattain and T. M. Buck (to be published). 00 B. H. Schultz, Physica 20, 1031 (1954). 31 Henisch, Reynolds, and Tipple, Physica 20, 1033 (1954). 32]. E. Thomas, Jr., and R. H. Rediker (to be published). trol r!>s, are on the outside of the oxide layer. The chem ical aspects of this model will be considered later. The most direct measurement of the capture times of the states is obtained from the field-effect and channel experiments. If an external field is applied to the semi conductor surface, a change in surface conductance will be produced as a result of the net change in charge. The behavior of the surface conductance after application of the field is generally found to exhibit two easily sepa rable transient phenomena. First, in a time comparable to the over-all recombination lifetime of the thin slab (of the order of 10 microseconds in a typical case), the conductance is found to shift from its initial instan taneous value to a new quasi-static value. Then, if observed for a longer time, from a millisecond to several minutes, the conductance returns to the same value observed before application of the field. Considering the process in detail, the initial transient may be associated with the approach to equilibrium of the holes and electrons by means of surface and volume recom bination. This equilibrium is initially disturbed since, except, in the unique case of r!>S=r!>B, injection or ex· traction may occur between the bulk and the surface due to the different hole and electron densities. LOW,33 in such an experiment, finds that, apparently because of an inversion layer, the instantaneous conductance change may be opposite in sign to the quasi-static value occurring after recombination equilibrium occurs. Considering an n-type surface on p-type material, the initial current flow with an induced positive charge will consist of holes flowing into the surface space charge region and electrons flowing from the surface into the bulk. After recombination has occurred the net distri bution will correspond to a decrease in surface electron density, this decrease supplying the required induced positive charge. On this basis, the conductance will initially increase, while after recombination, it will be less than the value before field application. After a much longer time, the conductance returns to its zero-field value, indicating that the surface densities have re turned to normal, or that r!>s is restored to its equilib rium value. Thus one concludes that the induced carriers are eventually captured by surface st~tes, the state density being so dense that there need be no perceptible change in r!>s to produce the change in state occupation. One may say that the charge in the surface states now shields the free carriers from the effect of the field. Although the initial transient has not been studied to obtain information about the capture time of the "fast" states, the quasi-static value of the conductance before the slow decay has been used by Brown and Montgomery34.36 to measure the density of these states. 33 G. G. E. Low, Proc. Phys. Soc. (London) B68, 10 (1955), see also S. G. Kalashnikov and A. E. Yunovich, Zhur. Tekh. Fiz. 25, 952 (1955). 34 W. L. Brown, Phys. Rev. 98, 1565(A) (1955). 36 H. C. Montgomery and W. L. Brown, Phys. Rev. 98, 1565(A) (1955). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsGERMANIUM SURFACE PHENOMENA 107 They used an ac field having a period short compared to the long time decay but long compared with the recombination time. With a sufficient field amplitude it was possible to produce enough induced charge density to move CPs through the minimum conductance point shown in Fig. 2, thus obtaining a reference value for CPs. By comparing the experimental conductance vs induced charge curve with a theoretical curve based on all the charge being free carriers, the amount of trapped charge was obtained as a function of CPs. This experi ment indicated that approximately two-thirds of the charge was trapped for the range studied. Statz, deMars, and Davis36 have also measured the charge in "fast" states by means of the channel conductance on a p-n-p transistor structure. If the bias between the n-type material and the p-type channel is changed abruptly, the charge terminating the internal field at the surface must be supplied by a change in the free and trapped charge in the channel. Since the conductance of the channel may be measured directly, and the field may be calculated, the change in free carrier charge may be compared with the total change, the difference being the charge in "fast" states. Here CPs may be calculated directly from the measured channel conductance taking into account the reduced mobility in the channel,s as discussed below. The curve obtained in this experiment may be fitted by assuming a density of lOll states per cm2, at an energy of 0.20 volt above the valence band. The exact number is somewhat sensitive to the surface treatment. The charge vs CPs curve found by Brown and Montgomery did not level off at a particular value of CPs, corresponding to the Fermi level passing through the trap. Consequently, no absolute state density or energy is available. There is reasonable agreement, however, where the curves overlap, and an indication that there are additional traps near the conduction band of. about the same density. Substitution of these values into the recombination velocity equation gives a capture cross section of the order of 10-15 cm2, which is a reason able value. This seems to support the conclusion that these states are the recombination centers and this is probably valid; however, it should be emphasized that, although such a state comes to rapid equilibrium with the free carriers, it is not necessarily a recombination center. If the capture probability is high for only one type of carrier, the above transient phenomena would still apply, but the state might have no effect on recom bination. Since there is evidence for the existence of such trapping states37-39 at the surface, this possibility should not be eliminated. If such were the case, then it is possible that the recombination centers are even closer to the valence or conduction band than the transient experiments indicate. Lacking more detailed evidence, 36 Statz, Davis, and deMars, Phys. Rev. 98, 540 (1955). 37 R. Lawrance, Proc. Phys. Soc. (London) B67, 18 (1954). 38 Lawrance, Gibson, and Granville, Proc. Phys. Soc. (London) B67, 625 (1954). 39 H. A. Gebbie, and K. Blodgett III, Phys. Rev. 100, 970(A) (1955). it will be assumed that the "fast" state density is the order of lOll cm-2, bearing in mind this uncertainty. The decay of carriers into the "slow" states has been observed both in the field-effect17.25.4o-43 and channel conductance experimen ts. 36.25 In the channel experimen t, a sudden change in bias results in a change in channel conductance followed by a gradual return to a new steady conductance value. This final value is found to correspond to the same CPs at the surface as that before the bias change. In fact, it is well established that, over the range of biases considered, the "slow" state density is large enough to completely control the surface Fermi level, just as in the field effect. The value of CPs may be calculated from a knowledge of the field at the surface, which is a function of the bias and the bulk resistivity, and the variation of mobility with effective thickness of the channel.s The values so obtained are in excellent agreement with the surface conductance data and are incorporated in Table 1. The "slow" state decay, as observed in the field effect, is not a simple exponential except in a few special cases. A study of the transient40-43 indicates that the mean decay time is extremely sensitive to surface treatment and ambient. McWhorter42 has postulated that the capture probability of the states varies from one point to another over the surface, while Morrison41 attributes the nonexponential character to the variation of the capture probability with occupation of the states. This problem will be mentioned later in connection with the noise properties of the surface. A re-examination of the contact potential measure ments of Brattain and Bar"cieen9 indicates that the sur face states in their model are the "fast" states, while the fixed ionic charge, which they attribute to the ambient may be treated as charge in "slow" states. This is con sistent with the observation by Garrett and Brattain14 that the change in contact potential with light slowly disappears after illumination with a time constant the same as that of the slow decay in the field effect. Since Brattain and Bardeen used chopped light in their experiments, the charge in the "slow" states remained constant and since CPo may be associated with these states, the change in contact potential may be correctly identified with changes in CPs. III. CHEMISTRY AND STRUCTURE OF THE SURFACE It is generally agreed that, under the normal condi tions of surface preparation, a germanium crystal is covered by a thin layer of oxide ranging in thickness from 10 to SO A.9 Because of this, it is reasonable to assume that the "fast" states are at the interface be tween the bulk germanium and the oxide layer, and that the "slow" states are either in the oxide layer or on its 40 R. H. Kingston and A. L. McWhorter, Phys. Rev. 98, 1191 (A) (1955). US. R. Morrison, Phys. Rev. 99, 1655(A) (1955). 42 A. L. McWhorter, Sc.D. thesis, Dept. of Elec. Eng., M.LT., May, 1955. a R. H. Kingston and A. L. McWhorter (to be published). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions108 R. H. KINGSTON E, --E 1----- GERMANIUM o o OXIDE FIG. 5. State structure at germanium surface. surface.17.26 This situation is shown schematically in Fig. 5, where the "fast" or interface states are desig nated by circles and the "slow" states by squares. The difference in capture times between these two types of state may then be associated with the barrier to flow of electrons into the "slow" states. Considering first the interface states, the results of the previous section indicate that their density is a function of the original surface treatment and not the ambient to which the surface is exposed. This fact may be seen to be consistent with the structure of Fig. 5 since the oxide layer should effectively shield the interface from any chemical effects of the ambient. Thus, one might expect the density to be largely dependent upon the structure of the first layer of oxide or the way it bonds to the germanium lattice.36 There are a variety of surface treatments available which produce a low surface recombination velocity and consequently a low interface state density. These treatments generally con sist of a mechanical polish followed by a chemical or electrolytic etch, leading to surface recombination velocities of about 100 cm/sec.44•29 The chemical etchant usually consists of hydrofluoric acid, in which the oxide is soluble, oxidizing agents such as hydrogen peroxide or nitric acid, and "moderating" agents such as acetic acid, metal salts, or bromine. The low value of s with this type etch is consistent with the assumption that the mechanically disturbed oxide and germanium must first be removed and the surface reoxidized in a uniform manner. The electrolytic process behaves in a similar way. The electrolyte is usually sodium hydroxide and the germanium is made positive with respect to the solution. Such an anodic process evidently combines the removal of the disturbed layer with the deposition of a u~ifo~ oxide ~oating in one operation. High recom bmatlOn veloClty surfaces, or high interface state densities, are obtained on mechanically treated surfaces befor~ etching, and in addition may be produced by chemIcal treatments. A "sand-blasted" or abrasive polished surface could have a high "fast" state density (l;5{/' McKelvey and R. L. Longini, J. Appl. Phys. 25, 634 due to the irregularity of the oxide structure' however .. " smce the reductlOn of s requires removal of as much as a micron of material,44-46 the recombination centers are undoubtedly associated with mechanical imperfections many lattice spacings into the bulk material. Clarke,47 for instance has found that sandblasting produces a high acceptor density many microns into the germanium resulting in a low resistivity p-type layer on the surface. High recombination velocities are also observed if an initially low s surface is etched in hydrofluoric acid,48 cathodically etched in most electrolytes 48 bombarded with ions,49 or heated in vacuO.48 With th~ exception of the vacuum heating, the increase in recombination velocity could arise from either a disturbance of the germanium-oxide bonding or mechanical damage to the germaniu~ lattice. In the acid and electrolyte cases, not only IS the oxide removed, but it is quite probable that the atomic hydrogen evolved could produce lattice damage many atomic layers below the surface. DO The inc~ease of s with heating in vacuo is not as easily ex plamed, other than invoking some evaporation mecha nism such as the loss of the last tightly bound water molecules which could have an effect on the germanium oxide bonds.61 It is apparent from all the data that the recombination centers are not Tamm states, which would be expected to occur with a density of about 1016 cm-2, or one state per lattice site. Although the Tamm treatment is strictly applicable to clean surfaces, Pratt62 has considered similar states at the boundary between two semiconducting crystals, and finds that the same number would be expected. In all probability then, the recombination centers are associated with th~ lack of perfection of the interface rather than the change in the periodic structure at the crystal boundary. Since the oxide layer on an etched germanium surface is at least several monolayers thick it seems reasonable to consider t~e "~low" states as represented in Fig. 6, where the OXIde IS assumed to behave like a semicon ductor having an energy gap of about 3 electron volts. It is not possible to predict how the energy bands of the two materials will line up, so that it is assumed that the c~nters of the respective gaps are at the same energy. FIgure 6(a) shows the possible states in the oxide which rna! b~ due to imperfections or chemical imp~rities, whIle FIg. 6(b) shows the possible potential energy of an electron in the presence of an adsorbed atom or mole cule. Either or both types of state may be important in ~he .surface behavio~. The states in the oxide may be Justified by comparIson with nonstoichiometric semi- 'A P. R. Camp, Meeting of the Electrochemical Society Chicago May, 1954. ' , '6. T. M .. Bu~k an.d F. S. McKim, Meeting of the Electrochemical Society, Cmcmnatl, May, 1955. :: E. N. Clarke, Phys. Rev. 91, 1566 (1953). Green, Keyes, and Stevenson (unpublished data) (1;5~). Holonyak, Jr., and H. Letaw, Jr., J. Appl. Phys. 26, 355 I!O J. E. Thomas, Jr., and M. Green (private communication). iii J. T. Law, J. Phys. Chem. 59, 67 (1955). H G. W. Pratt, Jr., Phys. Rev. 98, 1543(A) (1955). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsGERMANIUM SURFACE PHENOMENA 109 conducting oxides, while the adsorption states have already been considered in connection with catalysis, 63 and adsorption,64,66 theories. With either type of state an electron transition from the germanium would have to occur by thermionic emission over the barrier presented by the oxide energy gap, or by quantum mechanical tunneling, through it. Either process would explain the long time constants observed in the field effect and "channel" experiments. The density of such states could be much greater than the value of 1013 per cm2 required by the experimental data; in fact, it would be probable that, at least in the adsorption case, the density is as high as 1015, or one state per lattice site for a monolayer adsorption. Since the value of <fos is a measure of the position of the Fermi level required to maintain charge neutrality at the surface, the states associated with the different ambients would be ex pected to be concentrated in the vicinity of the ob served position of the Fermi level. Thus, on the basis of 0 0 Eo Ec GERMANIUM GERMANIUM 0 E. E. (a) (b) FlO. 6. Physical model of "slow" states at surface: (a) oxide states, and (b) adsorption states. the chart in Table I, one expects the average energy of the states in or on the oxide to increase as the ambient is changed from ozone through the successive gases to wet nitrogen. The actual chemical behavior is not clear; for instance, water vapor might produce adsorption levels near the conduction band, or its high dielectric constant might decrease the binding energy of other levels in the oxide thus raising the average energy level of these states. Oxygen or ozone by themselves, since they have a high electron affinity, would certainly be expected to produce low lying states, thus acting as electron acceptors.66 It is important to point out that the concept of an adsorption state is not limited to atoms adsorbed in both a neutral and ionized state. Actually, the adsorbed species could be completely ionized and the empty or un-ionized state might exist in i3 D. A. Dowden, J. Chern. Soc. (London) 242 (1950). M P. Aigrain and C. Dugas, Z. Elektrochem. 56, 363 (1952). 66 P. B. Weisz, J. Chern. Phys. 20, 1483 (1952); 21,1531 (1953). 58 N. F. Mott, Semiconducting M a/erials (Butterworth Scientific Publications Ltd., London, 1951), p. 1. the gas phase. If a field were now applied which pro duced extra electrons at the surface, then a neutral atom striking the surface could adsorb an electron and after successive ionizations the surface would be re turned to its original value of <fos. Since the rate at which atoms strike the surface is much greater than the rate at which electrons can cross the oxide layer, this con sideration will not invalidate the assumption of fixed adsorption states. Stated in a different manner, one may consider the gas as producing a set of "virtual" states, since, the electron transition being so slow, there is always a large number of atoms available to capture the electron. A complete understanding of the energy states associated with the gaseous ambient will depend upon further knowledge of the chemical structure and the mechanism of the adsorption processes. LawS! has studied the adsorption of water on a vacuum-fired surface and finds a multilayer process with the first layer irreversibly adsorbed. A comparison of his data with that obtained for <fos vs partial pressure of water vapor25 on a well-oxidized surface is shown in Fig. 7. The results tend to indicate that only the first and second layers of water have a pronounced effect, but further interpretation is difficult, since the surface treatments were different. Law67 has also measured the adsorption of carbon monoxide, carbon dioxide, nitrogen, and hydrogen and finds that the first three are physically adsorbed, while the last, hydrogen, is both physically and chemically adsorbed. Since chemisorption requires electron transfer, one would expect that of these four, only hydrogen would have any electrical effect, al though there is no information available at present on this point. Another phenomenon which has not been interpreted at this time is the increase of <fos with time over many hours during exposure to air or oxygen. It has been proposed that this gradual increase may be related to the slow growth of the oxide layer.23•25 Recently, how ever, Green and Kafalas68,&\l have attempted to measure 0.16 0.1 .. ~ c: Q ~ 0.0 ~ .. ... 2 8 0.04 0 I --x x-X ~-- / 2 3 4 5 6 7 MONO LAYERS OF ADSORBED WATER+- FIG. 7. Surface Fermi level, <(Is vs rnonolayers of adsorbed water. 67 J. W. Law, J. Phys. Chern. 59, 543 (1955). 68 M. Green and J. Kafalas, Phys. Rev. 98, 1566(A) (1955). 69 M. Green and J. Kafalas, Meeting of the Electrochemical Society, Cincinnati, May, 1955. Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions110 R. H. KINGSTON T W.F. GE T W.F. METAL E<------1 ! E f - - - - ---'-___ ---L_ 1------ EF £.------1 (a) E.----- (b) FIG. 8. Metal to semiconductor contact behavior with no surface states: (a) before contact, and (b) after contact. the oxygen uptake on a freshly etched germanium sur face and although they could detect less than a mono layer, found no observable oxidation. Since their measurements were not made until 30 minutes after the etch, it is possible that most of the oxidation occurred during this time; however, the contact potential results of Brattain and Bardeen9 indicate that the surface drifts appreciably for many hours. In another experi ment: utilizing the dissolution of the oxide in aqueous solutIOn, Green and Smythe60 have found that the oxide is approximately three layers thick five minutes after etching, and increases to six layer~ after a three day exposure to room air. This is in substantial agreement with Heidenreich's electron diffraction results as re ported by Brattain and Bardeen.9 This result is consis tent with the previous observation if one assumes that the oxidation occurs during the first 30 minutes of exposure, however, it is obvious that much more experimental data is necessary to understand the long time surface changes. Another approach to the chemistry of the germanium surface is to consider the behavior of a "clean" ger manium surface. Heretofore, only an etched surface has been :onsidered, which, in all cases, is expected to have an OXide layer plus other impurities introduced in the preparation. There have been two recent approaches to this problem, one utilizing a freshly fractured surface a?d the other depending upon ultra-high vacuum tech mque~ to remove all foreign materials. The first type of exper~ment, a study. of the oxygen adsorption after crushmg of a germamum sample, has been carried out by Green and Kafalas.68,69 A comparison of the total 60 M. Green and R. Smythe (unpublished data). oxygen uptake with a separate measurement of the surface area of the crushed sample leads to the number of oxygen atoms per lattice site. The preliminary results of this work indicate that oxidation occurs only to the extent of about 1.S layers. This is quite surprising in comparison with the known information on an etched surface, where as many as six layers of oxide are ob served. Statz et al.36 have suggested that the oxidation process is similar to that suggested by Cabrera and Mott.fil In this model the layer forms by the migration of positive metal ions through the oxide to the surface under the influence of the large electric field produced by the adsorbed negative oxygen ions. Such a mecha nism, however, leads to thicknesses of from 6 to 60 layers which indicates that the freshly fractured surfaces d~ not behave in this manner. Whether the Cabrera-Mott model is valid for an etched surface, as suggested above, cannot be answered at this time. It would be interesting to compare the electrical properties of a cleaved surface with those of an etched surface, since the oxidation properties seem to be markedly different. Although such experiments might be feasible, the vacuum preparation of a clean surface now seems to offer the best possibilities in this direction. Farnsworth and co-workers62 have found, by slow speed electron diffraction measurements that the surface of germanium may be cleaned by bom~ bardment with argon and subsequent annealing. Meas urements of hydrogen and oxygen adsorption63 and contact potential 64,fi5 have already been carried out on such surfaces and experiments of the type discussed in Sec. II should yield valuable information about the surface structure. The low-speed electron diffraction techniques might also reveal the difference between the oxide on an etched and a clean surface. It is quite pos sible that this difference may lie in the crystal structure of the oxide. The occurrence of different crystal forms has been suggested by Clarke66 and Kmetk061 as a possible explanation of some of the electrical anomalies. IV. PRACTICAL EMBODIMENTS 1. Surface Contacts As mentioned in the introduction, Bardeen's treat ment of surface states was developed to explain the anomalies of metal-to-semiconductor rectification. In particular, it was found that the rectification properties of the contact were not markedly dependent upon the work function of the metal. In Fig. 8, the expected (1;4~'. Cabrera and N. F. Mott, Repts. Progr. Phys. 12, 163 62 Farnsworth, Schlier, George, and Burger J Appl Phys 26 252 (1955). ,. . ., 63 Burger, Farnsworth, and Schlier Phys Rev 98 1179(A) (1955). ' . . , ... F .. G. Allen, Meeting of the Electrochemical Society Cin- cmnatI, May, 1955. ' 86 J. A. Dillon and H. E. Farnsworth Phys Rev 99 l""(A) (1955). ' . ., \r.tJ 86 E. N. Clarke, Phys. Rev. 98, 1178(A) (1955). 67 E. Kmetko, Phys. Rev. 98, 1535(A) (1955). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsGERMANIUM SURFACE PHENOMENA 111 behavior is shown, where the value, (cPS-cPB) for the semiconductor, is equal to the difference in work func tions between the metal and the semiconductor. This model attributes the curvature of the energy bands at the surface to the effect of the contacting metal; how ever, Bardeen pointed out that, if the density of surface states is large enough, cPs is determined by these states, and the induced charge associated with the difference in work functions (or the contact potential), is supplied by the surface states with no appreciable change in Fermi level at the semiconductor surface. The impor tant properties of the contact, such as its rectification and minority carrier emission efficiency are determined by cPs and the bulk resistivity, or cPB. For a metal point contact, it is generally found that the surface is p-type or cPs is negative, resulting in good rectification and emission efficiency on n-type germanium. Unfortunately an understanding of the point contact is complicated by the geometry and the fact that "forming," or the passage of large currents through the point, has a profound effect upon its behavior. There are a large number of theories about the behavior of point con tacts, especially the minority carrier multiplication at a reverse-biased collector contact. Since these theories are more directly related to the bulk behavior of the semiconductor they will not be discussed here; the reader is referred to two of the more recent articles, on rectification, by Simpson and Armstrong,68 and on collector multiplication, by Hogarth.69 Basic to all of the theories is the height of the surface barrier or the value of cPs, which is apparently determined by the germanium and not by the metal, except in the case of forming. It should be emphasized that although the work function of the metal does not determine cPs, the metal is directly involved, since the point undoubtedly damages the surface and changes the properties of the oxide. One of the important considerations is that cPs under the metal may be entirely different from that of the free surface around the periphery of the contact. There is, at present, no clear explanation of the behavior of cPs under the point other than the suggestion that the physical damage could possibly produce acceptor levels similar to those observed by Clarke47 and it is probably academic to attempt a distinction between the "fast" and "slow" states. A metal-to-semiconductor contact which is more clear cut in its behavior is that considered by Bradley et al.70 and Gunn.71 This is an electroplated or evaporated contact, whose area is large enough so that the geometrical and mechanical problems of the point contact no longer arise. These contacts exhibit a voltage-current characteristic much nearer to that of an ideal rectifier and, as a consequence, it is quite easy to obtain a value of cPs from the experimental data. As in 68 J. H. Simpson and H. L. Armstrong, J. App!. Phys. 24, 25 (1953). 69 C. A. Hogarth, Proc. Phys. Soc. (London) B67, 636 (1954). 70 W. E. Bradley et al., Proc. lnst. Radio Engrs. 41, 1702 (1953). 71 ]. B. Gunn, Proc. Phys. Soc. (London) B67, 409 (1954). the point contact case, the surface is generally found to be p-type, or cPs, negative, and the behavior is relatively independent of the work function of the metal. Boccia rellj72 and Hartig73 have studied the effects of different surface treatments and gaseous ambients on the height of the surface barrier both before and after application of the contact. There does not seem to be a direct correlation between the observed values of cPs in the diode and those reported in experiments on free surfaces. (See Sec. II.) Evidently the surface structure beneath the metal layer is sufficiently modified by the evapora tion or electroplating process so that the data on etched surfaces no longer hold. Bradley74 has considered the metal-semiconductor boundary conditions in terms of the chemical potential for the electrons and concludes that the value of cPs may be controlled by the proper choice of impurities in the oxide layer between the semiconductor and the metal. Another type contact which has been studied in de tail is that between germanium and an electrolyte. In this case, Brattain and Garrett7li-77 find that the surface conductivity type is determined by the polarity of the germanium with respect to the solution. Thus if the germanium is anode or positive with respect to the elec trolyte, the voltage-current characteristic indicates that the surface is p-type; that is, negative ions attracted to the germanium produce an excess hole density beneath the surface. By measurements of the change in electrode voltage with illumination and independent measure ments of minority carrier density at the surface, it was also established that the anodic current was limited by the supply of holes from the bulk and that the cathodic current depended on the flow of electrons to the surface. Thus, n-type germanium rectifies when biased positive with respect to the electrolyte and conversely p-type, when biased negative. In the anodic n-type case, it is also found that the minority carrier multiplication occurs with a factor of approximately two. The be havior is generally the same for any electrolyte, the ones studied in detail by Brattain and Garrett, being potassium hydroxide, potassium chloride, and hydro chloric acid. In addition to the interesting chemical and physical behavior of these contacts, the results yield information which is pertinent to the discussion of diode leakage, below. 2. Surface Leakage One of the most serious problems in semiconductor junctions is the occurrence of excess current flow in the reverse direction due to the surface behavior. Since the 72 C. V. Bocciarelli, Physica 20, 1020 (1954). 73 P. A. Hartig, Meeting of the Electrochemical Society, Cincinnati, May, 1955. 74 W. E. Bradley, Meeting of the Electrochemical Society, Cincinnati, May, 1955. 75 W. H. Brattain, Semiconducting Materials (Butterworth Scientific Publications, Ltd., London, 1951), p. 37. 78 W. H. Brattain and C. G. B. Garrett, Physica 20, 885 (1954). 77 W. H. Brattain and C. G. B. Garrett, Bell System Tech. ]. 34, 129 (1955). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions112 R. H. KINGSTON saturation current is determined by the rate of genera tion of minority carriers, the most obvious problem is associated with abnormally high surface generation in the vicinity of the junction. This may be induced on a normally prepared diode by ion bombardment49•78 or heating in vacuO.79 Although these effects associated with the increase in surface generation or recombination rate are important in transistors80 as well as diodes, a much more serious problem arises from true leakage along the surface, both of an ionic and an electronic nature. The electronic leakage effect will be considered first. As described in Sec. I, one of the first direct observa tions of surface inversion layer conductance was made by Brownl5 in studies of the channel effect in n-p-n grown junction transistors. Although the original quan titative data were taken at low temperatures more re cent work on both n-p-n26 and p-n-p'l:d structures indicates that the room temperature surface conduct ance may be treated theoretically by assuming that the value of IPs is independent of the applied bias between the bulk germanium and the surface region. Considering the potential plot of Fig. 9, which represents the channel on an n-p-n structure, such as in Fig. 4, the conduct ance per square of the surface may be calculated, given the quantities, IPs, IPB, and V A, the applied bias. The re sult, which utilizes Schrieffer's8 calculations of mobility in a narrow well, is consistent with the experimental data, if IPs is made a constant which is a function only of the surface treatment and not the applied bias or bulk resistivity. Under these conditions the conductance is found to decrease with increasing bias, approaching a V [1 relationship at large VA. This may be seen qualita tively, in Fig. 9, from the fact that the field at the sur face increases as V A!, causing the effective thickness of the channel to decrease inversely with this quantity. Since both the number of electrons and also the mobility for such a thin region decreases with the thickness, the resultant conductance variation approaches V A-I. Ec------------------ __ '----:q!----------- E.--7+~r+~--+---+----++--- FIG. 9. Energy level structure for n-type channel on p-type germanium. 78 H. Baldus, Z. angew Phys. 6, 241 (1954). 79 D. P. Kennedy, A.I.E.E. Winter Meeting, New York, January, 1954. 110 See, e.g., K. F. Stripp and A. R. Moore, Proc. Inst. Radio Engrs. 43, 856 (1955). The effect of inversion layers or channels on rectifica tion was first considered by Bardeen and Brattain81 and later by Aigrain et al.82 A model of the diode behavior is shown in Fig. lO(a), where a p-type inversion layer exists on the surface of normally n-type material. If the diode is now reverse biased, minority carriers may flow into the p-type region over a much larger area than that of the point itself. The effective collection area will be determined by the radial voltage drop along the surface away from the point. When this drop becomes com parable with the applied reverse bias then the p-n junction will no longer collect carriers and the limits of the active area will be reached. Since the active col lecting area is thus a function of applied bias, the diode no longer saturates and the reverse characteristic may have a low dynamic resistance. This type of leakage has been studied in great detail over the past few years, mostly on grown junction diodes. In Brown'slO original channel paper, it was suggested that the occurrence of the channel could be detected by measuring the photocurrent of the diode as a point source of light was moved along the surface near the junction. In Fig. lOeb), one would expect the photo response to fall off exponen- ,., ~-rYPE CH.o.Nr.I£L ,,, FIG. 10. Channel formation on diodes: (a) point contact, (b) grown junction, and (c) fused junction. tially to the left of the true junction while to the right, the n-type region would act as an efficient minority carrier collector and thus the photoresponse would remain high for an appreciable distance along the surface. Eventually, the active region would end, since the voltage drop along the channel would decrease the bias so that the surface junction was no longer a sink for majority carriers. The direct observation of the channel by this technique was first carried out by Christensen83•21 and later work by McWhorter84 established the relation between excess reverse current, channel length, and reverse bias. In this work it was found that the addi tional current beyond the normal saturation value was directly proportional to the measured channel length and could be attributed to bulk and surface minority carrier generation. It was also possible to predict the variation of channel length with bias by using the known channel conductance data for n-p-n structures25 in a treatment of the voltage drop along the surface. The 81 J. Bardeen and W. H. Brattain, Phys. Rev. 75, 1208 (1949). 112 Aigrain, Dugas, and Etzel, Semiconducting Materials (Butter worth Scientific Publications Ltd., London, 1951), p. 102; P. Aigrain, Ann. phys. 7, 140 (1952). 83 H. Christensen, Phys. Rev. 96, 827 (1954). 84 A. L. McWhorter and R. H. Kingston, Proc. Inst. Radio Engrs. 42, 1376 (1954). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsGERMANIUM SURFACE PHENOMENA 113 resultant voltage current relationship showed that the excess current should increase roughly logarithmically with the bias, as observed in the experiments for voltages up to ten volts. Similar channel observations have been carried out by Noyce24 with a novel direct observation technique utilizing a modified television system, and theory and experiment for a fused junction similar to Fig. 1O(c) have been given by Cutler and Bath.86 One of the complicating effects in these experi ments is the transient behavior of the surface states wh.ich ~aintain CPs constant. As discussed previously, a fimte tIme, of the order of seconds to minutes is re quired for the surface states to come to equilibriu~ with the bulk. Therefore the calculations of excess current are strictly valid for only the dc case. As soon as the voltage is varied at an appreciable rate, then the con stancy of CPs no longer holds and the reverse character istics may exhibit strong time-dependent effects. Such drift or "creep" of the current is observed not only in junction structures but also in point contact diodes.86.87 Another problem presently under study is the behavior n·TYPE CHANNEL J J--------- I I : LOW-RESISTIVITY .-TYPE I I I I FIG. 11. Channel on low resistivity material leading to possible breakdown. of a diode where the channel occurs on extremely low resistivity material. In Fig. 11, for example, an n-type channel is shown on low resistivity p-type germanium, which might be the case in the diode of Fig. 10(c). It turns out that the field between the channel and the bulk. is in general a function only of the bulk resistivity, and m this case, one would expect a breakdown voltage from channel to bulk much lower than that across the true junction.84 Such an effect has been observed by Miller88 in transistor experiments and by Bernstein89 in measurements of photomultiplication at the channel to-bulk junction. Despite the success of the simple channel theory in explaining many of the anomalies of reverse biased diodes, it is apparent that the behavior in the presence of water or other electrolytes is much more complicated. Over the range of voltages from 10 to 100 volts, the current is found to be approximately linear with volt- Ii M. Cutler and H. Bath, J. App!. Phys. 25, 1440 (1954). 88 M. Kikuchi and T. Onishi, J. Phys. Soc. Gapan) 9, 130 (1954). 87 M. Kikuchi, J. Phys. Soc. Gapan) 9, 665 (1954). 88 S. L. Miller, private communication. 811 H. Bernstein and R. H. Kingston, Phys. Rev. 98, 1566(A) (1955). -9 p-TYPE ~;t.e _c....:9~_..!!~"-~~_..!!9~_ 9!;!._..!!~~_.!!.9 _-r.!!.....!L.!!-.!!.-..!-...!!.....!!~Ln -TYPE CHANNE~ CHANNE~ FIG. 12. Channels on a junction in contact with an electrolyte. age,90 which rules out the simple channel explanation which should give a logarithmic variation. In addition, more recent work by Law and Meigs91 and Statz92 indicates that channel length measurements by the light-spot technique show poor correlation with the excess current at these voltages. By comparing the leak age current in water vapor with that produced by a field-induced channel, Clarke93 concludes that this extra current is due to nonuniformities in the channel. On the other hand, Law90 has made a direct correlation between the amount of adsorbed water and the leakage current in grown-junction germanium diodes and attributes the current to ionic conduction. One of the difficulties in this electrolytic conduction theory is the lack of any apparent by-product of the reaction. If true electrolysis were occurring, one would expect either a gaseous product or mass transfer along the surface. At this writing there is no unequivocal evidence of either of these cases. It is possible, of course, that some electro chemical process could occur with no measurable product; however, any such system is speculative at this time. In addition to the complications of the electro chemical process, there is the additional difficulty of treating the electrolytic process quantitatively. On the basis of Brattain and Garrett's work,7fr-77 one would expect the situation depicted in Fig. 12, where positive and negative ions will collect as shown. As in the above authors' studies, it is probable that rectification will occur between the electrolyte and the underlying mate rial thus limiting the flow of current into and out of the solution. Thus the leakage current should be limited both by the semiconductor-electrolyte barrier and by the high resistance of the thin film of electrolyte. Again, from Brattain and Garrett's results, it is apparent that the leakage current will be photosensitive, due to the semiconductor-electrolyte barrier. Compounding all these difficulties is the additional fact that the behavior of the electrolyte in Fig. 12 is such as to produce a channel on both sides of the junction. Thus, even though one is certain that any electrolyte in contact with the germanium has a high enough resistance to produce negligible ionic leakage, the slow drift of ions in the til) J. T. Law, Proc. lnst. Radio Engrs. 42, 1367 (1954). II J. T. Law and P. S. Meigs, Meeting of the Electrochemical Society, Cincinnati, May, 1955 (to be published). H H. Statz (private communication). D3 E. N. Clarke (to be published). Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions114 R. H. KINGSTON medium under reverse bias conditions may lead to slow creation of a channel. This last phenomenon is an alternative explanation of the slow drift of reverse current observed in diodes and transistors. 3. Excess Noise Experiments by Montgomery,94 and Gebbie et al.95 on germanium filaments and Kennedy79 and Slocum and Shive96 on p-n junctions indicate that excess or 1/f noise is dominantly a surface effect. This type of noise97 is characterized by a power spectrum which increases in amplitude with decreasing frequency as Ilfn where n is usually between 1 and 1.5. The chief difficulty in explaining the noise has been the lack of known physical phenomena in the germanium having correlation times or "memory" of the order of minutes to hours, since the spectrum is known to extend to frequencies as low as 10-5 cps. Several theories98 have been proposed to relate the surface properties to the noise; however,the ones of immediate interest here are those suggested by M c Whorter99•42 and Morrison,41 since they directly relate the noise properties to the "slow" states. Since the re laxation of the surface conductance in the "field-effect" experiment takes a time comparable to the times re quired to explain Ilf noise, one might expect a relation ship between the time dependence of this decay and the noise spectrum. The two noise theories differ in their interpretation of the nonexponential character of the decay. McWhorter, on the basis of ac measurements of the field-effect response,40·43 assumes that the capture times of the "slow" states vary from point to point over the surface. An analysis of the data with this model gives l/f noise, if one associates simple shot noise with each state and weights the noise according to the observed state distribution, which is roughly inversely proportional to the capture time. Morrison, in contrast, attributes the nonexponential behavior to a change in capture proba bility with time, due to changes in barrier height as 114 H. C. Montgomery, Bell System. Tech. J. 31, 950 (1952). 16 Gebbie, Maple, and Bess, J. App!. Phys. 26,490 (1955). 116 A. Slocum and J. N. Shive, J. App!. Phys. 25, 406 (1954). 87 For a general discussion of semiconductor noise, see R. L. Petritz, Proc. lnst. Radio Engrs. 40, 1440 (1952). .8 See L. Bess, Phys. Rev. 91, 1569 (1953) and reference 97. 118 A. L. McWhorter, Phys. Rev. 98, 1191(A) (1955). charge enters or leaves the "slow" state. By considering the frequency content of the associated decay transient, he also obtains a l/f noise spectrum. Tlius, although both theories relate the noise directly to the flow of carriers into "slow" states, the chief difference is the proposed mechanism for the decay process. McWhorter42 has calculated the magnitude and spectrum of the noise in germanium filaments based on the "field-effect" data and finds consistent agreement with the experi mental noise measurements. His theory, as applied to p-n junctions, is less satisfactory apparently as a result of leakage currents associated with the electro lytic process, and also the complicated geometry of the junction-surface region. The exact nature of the electron transition process from the germanium to the "slow" states is not clear; however, since l/f noise is relatively temperature independent, a quantum-mechanical tun neling process42 has been suggested as the most satis factory explanation. Much more experimental data will be required before these questions can be resolved. CONCLUSION On the basis of the results discussed in this paper, it seems fair to say that the electronic behavior of the bulk semiconductor as influenced by the surface is fairly well understood, in principle, if not in detail. The major unsolved problems lie in the realm of chemistry and atomic structure, and it is hoped that future work will resolve many of the difficulties associated with the state structure and the possible electrochemical processes. Little is known about the behavior of silicon surfaces at this time, however there is reason to believe that the general approach as presented in this paper will be valid in the case of silicon, but possibly with different orders of magnitude for the effects. ACKNOWLEDGMENTS It is obvious that a paper of this scope depended upon discussions and suggestions from a large number of workers in the field in addition to those cited in the references. The author wishes to acknowledge their help and in particular is indebted to J. E. Thomas, Jr., now at Wayne University, for his continuing encouragement and criticisms. Downloaded 12 Oct 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1735261.pdf

Effect of Variations in Surface Potential on Junction Characteristics J. H. Forster and H. S. Veloric Citation: Journal of Applied Physics 30, 906 (1959); doi: 10.1063/1.1735261 View online: http://dx.doi.org/10.1063/1.1735261 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of crystallites on surface potential variations of Au and graphite J. Appl. Phys. 71, 783 (1992); 10.1063/1.351358 Macroscopic variations of surface potentials of conductors J. Appl. Phys. 69, 7126 (1991); 10.1063/1.347601 Substitution effect on formaldehyde photochemistry. Potential surface characteristics of HFCO J. Chem. Phys. 72, 6800 (1980); 10.1063/1.439171 Effects of Variations of the PotentialEnergy Surface on the Attributes of Simple Exchange Reactions: Classical Calculations J. Chem. Phys. 56, 3200 (1972); 10.1063/1.1677679 Effect of Hydrostatic Pressure on pn Junction Characteristics and the Pressure Variation of the Band Gap J. Appl. Phys. 38, 4454 (1967); 10.1063/1.1709147 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09JOURNAL OF APpLIED PHYSlts VOLUME 30. NUMBER 6 JUNE. 1959 Effect of Variations in Surface Potential on Junction Characteristics J. H. FORSTER AND H. S. VELORIC* Bell Telephone LabMatories, Inc., Murray Hill, New Jersey (Received October 27,1958) A filamentary structure has been used to compare the electrical properties of a germanium surface with those of an adjacent p-n junction intersecting the same surface. Surface charge is varied by field effect plates in the isolated portion of the filament and near the junction. An orderlv relation can be found between surface potential variations and changes in the reverse currents across the jun"ction. At low bias, the junction current varies with surface recombination velocity, and for bias near breakdown, the breakdown voltage varies with induced charge at the surface. For inverted surfaces, the low bias current varies rapidly as expected from channel length variations. With inverted surfaces, channel growth leads to large reverse current variations with surface potential, but breakdown voltage becomes independent of surface charge. These variations are considered in terms of simple theory, and device implications are discussed. I. INTRODUCTION PROPERTIES of semiconductor surfaces have been the subject of numerous investigations.1,2 Of in terest here are studies of semiconductor device surfaces which can be divided into two categories: physical studies of surface structure on a single conductivity semiconductor and studies of surface problems related to device characteristics and their stability. In general, these two kinds of study have been carried out sepa rately; for example, changes in surface potential have been inferred from changes in p-n junction charac teristics, and changes in junction characteristics have been predicted from single conductivity filamentary measurements. It is the purpose of this paper to describe experiments which utilize a filamentary structure for direct compari son of the electrical characteristics of a p-n junction with physical properties of the semiconductor surface intersecting the junction. In this way, the two kinds of study mentioned above can be carried out simultane ously with the same material and surface preparation. In these experiments, a single filamentary germanium specimen is employed. The specimen consists of an extended single conductivity (p-type) portion, with an n+ region grown on one end to form an n+ -p junction. Using an ac field effect technique,· surface conductance4 and surface recombination velocity5 are determined as functions of surface charge on the single conductivity portion of the specimen. Using the ac field effect tech nique to vary the surface charge at the surface inter secting the n+-p junction, the nature and magnitude of the reverse bias junction current has been investigated for corresponding values of surface charge. At low junction bias, the reverse current variations * Now at RCA Semiconductor Division, Somerville, New Jersey. 1 R. H. Kingston, J. App!. Phys. 27, 101 (1956). 2 R. H. Kingston, editor, Semiconductor Surface Physics (Uni versity of Pennsylvania Press, Philadelphia, Pennsylvania, 1956), • W. L. Brown and H. C. Montgomery, Phys. Rev. 98, 1565(A) (1955). 4 W. L. Brown, Phys. Rev. 100, 590 (1955). 5 C. G. B. Garrett and W. H. Brattain, Bell System Tech. J. 35, 1019 (1955). are calculable in terms of the measured changes in surface recombination velocity, provided the surface conductivity type remains the same as that of the body. For inverted surfaces, an additional current appears, which is approximately calculable in terms of channel growth.6,7 At higher bias voltages (with noninverted surfaces), surface avalanche breakdown8,g is found to be an addi tional source of reverse current. The junction break down voltage increases with increasing magnitude of negative surface charge, reaching the body value soon after the surface becomes inverted. This measured variation of breakdown voltage with surface charge is not calculable on the basis of simple theory.9 The expected relations between an n+-p junction characteristic and the surface properties of the low conductivity side of the junction are discussed in more detail in the following section. II. RELATIONS BETWEEN n+ -p JUNCTION CHARACTERISTICS AND PHYSICAL PROPERTIES OF SURFACES A. Physical Properties of Surfaces Some of the properties of semiconductor surfaces have been successfully interpreted in terms of electronic energy level diagrams similar to the one in Fig. 1 which represents an idealized equilibrium situation at a semi conductor surface boundary. Ec and Ev represent the lowest energy in the conduction band and the highest energy in the valence band, respectively. Ei represents the Fermi energy for an intrinsic semiconductor and Er represents the Fermi energy in a p-type specimen. A quantity 'P may be defined by (1) • W. L. Brown, Phys. Rev. 91, 518 (1953). 7 A. L. McWhorter and R. H. Kingston, Proc. lnst. Radio Engrs. 42, 1376 (1954). B A. J. Wahl and J. J. Kleimack, Proc. lnst. Radio Engrs. 44, 494 (1956). • C. G. B. Garrett and W. H. Brattain, J. App\. Phys. 27, 299 (1956). 906 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09SURFACE POTENTIAL VARIATIONS AND JUNCTIONS 907 where q is the electronic charge. Referring to Fig. 1, ip EC varies from a value ipb in the semiconductor to a value CPs at the surface. The corresponding hole and electron densities vary from Pb and nb to ps and n., and are given by p=nicf3<P, n=nicf3<P. (2) (3) The density of holes and electrons in the space charge region is a unique function of CPs and ipb.4 If we assume the surface carrier mobilitieslO are the same as the bulk mobilities (i.e., small magnitudes of CPs), then the surface conductance is a unique function of ips and CPb. As the carrier concentration at the surface varies, the conductance of the semiconductor will vary. This con ductance change is given by (4) !1P and !1N are the changes in the hole and electron concentration per unit surface area summed over the region in the filament where cP~ ipb. The!1P and !1N are functions of CPs and CPb and have been tabulated by Kingston and Neustadterll and by Garrett and Bra ttain.'2 Brown4 has shown that measurements of changes in !1G produced by changes in an applied field directed normal to the surface can be used to determine the values of ips, provided that the boG range includes the minimum value of surface conductance. The surface recombination rate of excess minority carriers is defined as S=J/!1n, (5) where J is the recombination current and !1n is the excess minority carrier concentration near the surface, but far enough inside the semiconductor so that ip= CPb. Recombination at surfaces has been considered by Brattain and Bardeen '3 in terms of intermediate states in the gap (fast surface states), in an analogous manner to the bulk recombination process considered by Shockley and ReadY Using this kind of analysis, con sidering states at one discrete energy, Stevenson and Keyes'• have arrived at the following expression for S: S= N tCpC n(Pb+nb)/C n(n8+nsl)+C p(Ps+PsI), (6) where N t= the density of recombination centers (per unit area of surface), and Cp= the capture probability per center per unit time for holes if all centers are filled, Cn=capture probability per center per unit time for electrons if all centers are empty, psI = hole density at 10 J. R. Schrieffer, Phys. Rev. 97, 641 (1955). 11 R. H. Kingston and J. F. Neustadter, J. App!. Phys. 26, 718 (1955). 12 c. G. B. Garrett and W. H. Brattain, Phys. Rev. 99, 376 (1955). 13 W. H. Brattain and J. Bardeen, Bell System Tech. ]. 32, 1 (1953). 14 W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952). 15 D. T. Stevenson and R. ]. Keyes, Physica 20, 1041 (1954). E I ---1-------------------------" ", -Q'Pa ' Ef _____ ~ ________________________ _;~Q'Ps EV------------------------~ FIG. 1. Energy level diagram for a p-type semiconductor. the surface if Ef=E t, where Ee is the effective energy level of the center, and nsl = the electron density at the surface if Ef=E t• According to this model, S has a maximum value, SM, at a value of {3ipM given by {3CPM=t 10gCp/C n and is symmetrical in cP about ipM. Thus, for a ratio of Cp/Cn close to 1, S generally in creases, goes through a maximum, and decreases as CPs is varied from negative to positive values.16 B. Low Bias Reverse Junction Current The term "low bias" is used here to denote reverse bias values low enough to preclude avalanche multi plication effects at the junction (to be discussed in the following section), but substantially large compared to 1/{3. The current across a p-n junction at low reverse bias depends on the minority carrier generation on both sides of the junctionY For a junction between heavily doped n type and lightly doped P type, i.e., an n+-p junction, most of the reverse current is carried by electrons generated in the body and at the surface of the p-type material. For a rectangular geometry, in which surface generated carriers are appreciable, the current density is given by j=qn{:r(Cf3V-l), (7) where V is the applied voltage, D is the diffusion con stant for electrons in the p-type semiconductor and TE is an effective lifetime given by 2S 1 l/TE=-+-. t Tb (8) Here t is the thickness of the filament and T b is the body lifetime. This expression applies for rectangular fila- 16 Many, Harnik, and Margoninski in reference 2, page 89. 17 W. Shockley, Bell Syst. Tech. J. 28, 346 (1949). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09908 J. H. FORSTER AND H. S. VELORIC ments for which t is small compared to width and length and is valid for Sf/ D«1. It is to be expected that the reverse current will be related to changes in CPs near the junction, since S is related to CP8 by a function similar to the one given in Eq. (6). Variations in the saturation current of p-n junctions attributed to variations of S with <Ps have been reported.s,ls In the case of 2S/t»1/ Tb the reverse junction current primarily results from surface generation. According to Eqs. (6), (7), and (8), the low bias current increases as (3cps is varied from negative towards positive values, reaching a maximum near (3<P8=O if Cp/Cn~1. However, when (3<ps becomes positive, the surface becomes n type, and the inversion layer, or channel, tends to increase the area of the junction effective in collecting minority carriers. Thus, although for ~uffi ciently positive (3<p. Eq. (6) indicates a decrease in S, Eqs. (7) and (8) are not expected to apply for i3<Ps>O, and the junction current continues to increase with f1<P •• An approximate expression for the channel current7 as a function of f3<P. is given in Sec. B of the Appendix. C. Breakdown Voltage Wahl and Kleimack s have shown that changes in <p, may produce large changes in the collector breakdown voltage of alloy junction transistors. Garrett and Brattain9 have carried out a more detailed study of the breakdown of reverse biased germanium alloy junctions. Their experiments indicate that reverse biased p-n junctions can exhibit a multiplicative breakdown near -10 20 III ~ o > III > 10 <I -6 -5 -4 -3 _20 J3rps -5 -'4 -'3 -'2 ~I <> Os x 10-8 COULOMBS FIG. 2. Change in V BO as a function of surface charge. 18 J. E. Thomas and R. H. Rediker, Phys. Rev. 101,984 (1956). the surface at considerably lower bias values than those necessary to produce a multiplicative breakdown in the body of the semiconductor. Thus the observed break down voltage of a p-n junction, V BO, can be less than or equal to the body value, VB. The magnitude of the change in breakdown voltage, ~ V B, defined as I V B-V BO I is found to be a function of the fixed charge covering the surface in the vicinity of the junction. The value of ~ V B for p-n junctions in which the doping is substantially greater on one side (i.e., n+-p or p+-n junctions) is sensitive to the surface charge on the higher resistivity side of the junction. If this charge is of that sign which tends to induce a channel on the high resistivity side of the junction ~ V B is essentially zero. If the surface charge is of opposite sign, and sufficiently large, ~ V B becomes appreciable. Using the simplified model proposed by Garrett and Brattain, the value of ~ V B has been calculated as a function of (3<ps for the particular case of an n+-p germanium step junction with a resistivity value of 2.5 ohm-cm on the p-type side of the junction. This varia tion of ~ VB is indicated in Fig. 2. Indicated on the lower horizontal scale are values of Q" (the surface 2 3 p ':':':::;.:.:.;,:.:.:::;::.;.;.:.:.::.::;. A El FIG. 3. Experimental filamentary germanium diode. charge density required to neutralize the charge density just inside the semiconductor surface, neglecting charge in "fast" surface states) corresponding to indicated values on the (3<p., scale. As indicated in the figure, the theory predicts that at small positive values of (3<ps, the junction should exhibit body breakdown, and should do so until (31/'8 reaches negative values large enough in magnitude that a substantial enrichment layer has begun to develop. Then breakdown begins at the surface, and the observed ~ V H increases rapidly as -f3 <Ps increases. Ill. EXPERIMENTAL The brief discussion of surface and junction properties indicates that the junction saturation current and breakdown voltage should be uniquely related to <Ps. In order to investigate such relationships, the experimental structure described below can be used. Referring to Fig. 3, the filamentary grown junction germanium diode is placed on a thin mica spacer, sup ported by two similar flat metal plates cemented to a glass microscope slide. Electrodes 1 and 2 permit longitudinal flow of current through the p-type side of the junction and electrodes 1 and 3 permit current [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09SURFACE POTENTIAL VARIATIONS AND JUNCTIONS 909 flow across the n+-p junction. One of the plates at A, extending only along the p-type filament, serves to apply a field directed along the normal to the surface of the p-type material, and the other plate, at B, serves to apply a field in the same direction but at one of the surfaces where the junction current is generated. It is assumed that the effect of the field at the n+ surface produces changes which are unimportant compared to those produced at the high resistivity side of the junction. Electrical measurements are made with the circuit shown in Fig. 4. By passing direct current down the filament by means of electrodes 1 and 2, and modulating with an ac voltage on the field effect plate A, a measure ment of the ac field effect can be made in the manner described by Montgomery and Brown.3,19 After proper balancing out of the capacitive currents in the plate circuit by adjustment of R}, changes in surface con ductance (as changes in voltage across Rc) appear on the oscilloscope against changes in plate voltage. Using electrodes 1 and 3, a reverse dc bias is applied to the junction. The plate B is used to modulate the surface potential near the junction, and the effect on the reverse H o FIG. 4. Large signal field effect measurement circuit. current of the junction is observed as a function of the plate voltage. This provides a plot of I J, the reverse junction current change against QT, the charge per unit area on the field effect electrode. By increasing the bias sufficiently, the junction may be biased to breakdown voltage, defined as the voltage at 50-~amp reverse current. The current may be held constant at this value by increasing Re. The voltage changes induced across Re by the modulating plate voltage are essentially changes in breakdown voltage, and these changes are plotted against plate voltage on the oscilloscope. Thus, a plot of ~ V B against QT may be obtained. In theory, for sufficient applied fields, this equipment allows a determination of the variation in CPs and QT on the single conductivity p-type portion of the specimen, for direct comparison with variations in I J and ~ V B at the junction. Such a comparison should be a valid one, since the whole specimen has been subjected to the same surface prepara.tion and is exposed to the same ambient. 19 H. C. Montgomery and W. L. Brown, Phys. Rev. 103, 865 (1956). n 6G vs Or (LIGHT) -70 -60 -50 -40 -30 -20 ·/0 QX 10-9 COULOMBS 40 o X 24 :liE ::t C> <l 16 -4.5 -4 -3 -2 -I 0 I 2 3 4 4.5 ~¢Js 20 FIG. 5. Surface conductance as a function ot surface charge (2.5 ohm-em p-type germanium). IV. FIELD PRODUCED VARIATIONS IN <p. AND S In the interest of brevity, the single conductivity p type portion of the specimen between the electrodes 1 and 2 will be referred to as the "filament," and measure ments made on this portion of the specimen as "fila mentary measurements." In this section, the data from a typical filamentary measurement is presented. The specimen was etched in CP-8, and during the course of the experiment, was exposed to a dry oxygen ambient. A plot of the experimental ~G as a function of QT is indicated in Fig. 5 (curve I). The zero values for the ~G and QT scales are arbitrarily taken at the position of the conductance minimum. The frequency applied to. the field effect plate was 80 cycles per second (the trace was insensitive to frequency in the range of 60 to 500 cycles per second). The total excursion in plate voltage, V p, is about 400 volts which represents peak to peak fields on the order of 106 volts per centimeter. Curve II represents ~G as a function of QT during illumination by chopped light. The separation of these curves, after a correction has been made to account for the fact that the plate modu lates S on only one side of the filament, is proportional to liS". The data are obtained in a dry oxygen ambient. Ambient cycling is not necessary since the voltage sweep in the large signal field effect contains the con ductance minimum. Then it is not necessary to make the assumption that changes in ambient will vary the surface charge but not the nature of the surface states. t A calculated curve of ~G as a function of Q, is superim posed on the experimental dark curve so that the minima coincide. (Curve III, Fig. S.) From these three curves, the position of the energy bands at the surface,4,19 CP" and the variation of surface recombination velocity" can be determined as function,; t In several experiments we have observed that ozone ap parently changes the nature of the surface states as well as the surface charge. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09910 J. H. FORSTER AND H. S. VELORIC 1.0 I \ / I \- V O.S 0.6 04 I V ,/ I 02 - - -2 4 6 FIG. 6. Surface recombination velocity as a function of /3'Ps. of QT. The values of /3ifJs, corresponding to values of QT are indicated on the lower horizontal scale. The relative surface recombination velocity, SjSM, where SM is the maximum value observed, is plotted as a function of /3ifJs in Fig. 6. S has its maximum value at /3ifJs near 2. These filamentary data are qualitatively similar to those observed by other workers.20.21 These quantities will be used in later sections in the analysis of the characteristics of the filamentary diode as a function of the same values of surface charge. V. FIELD PRODUCED VARIATIONS IN JUNCTION CHARACTERISTICS A. Small Signal Variations The reverse junction current, I J, is observed to be sensitive to modulation by voltage changes on the field plate. For a given surface preparation and ambient, these effects are reproducible. In ambients favoring surface breakdown, the direction and magnitude of the changes in I J produced by a given change in plate voltage are found to depend on the dc bias applied to the junction. The field produced change in reverse current and the dc reverse junction current are compared for increasing bias on the junction in Fig. 7. While these measurements were made, the specimen was exposed to an ozone ambient. Curve I is a plot of the junction current as a function of the dc bias on the junction, V J. Curve II represents Al J+ as a function of V J where AI J+ is the reverse current change produced by a positive change in the field plate voltage, AV p' Curve III represents AI J as a function of V J, where AIr is the reverse current change produced by applying -A V p to the plate. It is apparent that Ah+ and Ah-are independent of V J until the reverse characteristic begins to soften (i.e., for V J near 60 volts). If we assume that the current col lected at the junction at low bias results from the 20 Many, Harnik, and Margoninski in reference 2, p. 85. 21 Garrett, Brattain, Brown, and Montgomery in reference 2, p. 126. generation of carriers at the surface, then Ah+ and AI J-should be independent of bias. For voltages ap proaching the surface breakdown voltage, the effect of the plate is to modulate the multiplication factor, which is in itself bias sensitive. Therefore, at these voltages the AI J+ and AIr are bias sensitive. Field effect conductance measurements on the single conductivity end of the specimen in this ambient indi cate that for the surface conditions maintained during this experiment (tending in the direction of an enrich ment layer on p-type material), S increases with in creasing positive charge on the field effect plate. Thus the AI J+ and AI J-at low V J values have the signs expected if the junction current is supplied by surface recombination. The fact that AI J-and AI J+ increase rapidly and change sign as V J is increased toward 2 8 24 20 8 4 o o II(IIIH) 0---0-_ 20 r I I ~ I I ~I ~ -\ 1/ ll(t.IJ-l ~ J I(J/ 40 60 80 100 VJ IN VOL TS 4 3 2 I !II W a:: w Q. ::E c( o a: <) o i ~ -I -2 -3 120 ., :;; FIG. 7. Comparison of field produced current changes with dc junction characteristics. breakdown voltage is in qualitative agreement with the trends discussed by Garrett and Brattain.9 A positive increase in V p produces a more n-type surface, a de crease in the surface mUltiplication factor, and a decrease in current. The filamentary field effect measurement indicates that the sign of the equilibrium value of /3ifJs favors surface breakdown. Photoconductance measure ments indicate the current at large V J is multiplicative and ambient variations of measured breakdown voltage indicate surface breakdown. These data therefore indicate that the field produced changes in junction current are related to the dc junction characteristic in a qualitatively appropriate manner if it is assumed that the junction current at low bias values arises from surface generation of minority carriers, and at higher bias values, can arise from multiplicative surface breakdown. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09SURFACE POTENTIAL VARIATIONS AND JUNCTIONS 911 Larger signal variations in the low and high bias junction currents will be discussed more quantitatively in the following two sections. B. Low Bias Reverse Current Variations Field produced junction current changes are shown in Fig. 8 for the low bias case. Experimental conditions are the same as those described in Sec. IV. The lower solid curve (I) is the experimentally determined dark reverse current as a function of QT. The upper curve (II) is the reverse current as a function of QT measured with the filament illuminated. The relative junction current as a function of QT is obtained by using the field effect plate on the junction side of the specimen as described in Sec. III. The absolute value of reverse current is determined by dc measure ment with V p=O. The field effect plates are identical and experimental conditions are the same at the junc tion as they are at the filament. If the surface conditions near the junction are the same as on the filament, then ··+···CALCULATED FROM S -EXPERIMENTAL -oOo-CALCULATED CHANNEL CURRENT JI-____ ~ -70 -60 -50 -40 -30 -20 -10 QTX 10-9 COULOMBS t , , I I I 4 10 20 -4.5 -4 -3 -2 -I 0 +1 +2+3 +4 +4.5 (3 CPs FIG. 8. Field produced current changes for an n+-p junction. the abscissa of Fig. 5 is the same as that of Fig. 8. The junction current for all values of QT is relatively in sensitive to junction bias up to values near the break down voltage. For a filamentary junction of the dimensions used (see Appendix), with equal S on both sides, the dark reverse current density is given by Eq. (7). However, the field effect plate can modulate S on one side only. The variation of the dark current with S changing on one side only is given by Eq. (14) derived in the ap pendix. The solid points shown on curve I, Fig. 8, are calculated from Eq. (14), using the values of S given in Fig. 6 corresponding to the appropriate values of IPs. The points on curve II are calculated from Eq. (19) in the Appendix, using the same values of S, after normal izing one theoretical point to fit the data at V p= O. Both light and dark curves agree well with theory for IPs less than O. When IPs becomes positive, an inversion layer begins to form at the surface, and the collecting area of the junction increases. Thus, for IPs> 0 it is ex pected that I J no longer depends simply on S. Curves I and II should begin to diverge from the calculated values as they are observed to do. The lower set of open points indicate values of the excess dark channel current, calculated using Eq. (25) in the Appendix. The upper set of open points represent the illuminated channel current, calculated from Eq. (26), using an experimental value of the light current L obtained with the help of Eq. (27). It is evident that the excess current at the large values of (3cps is of the order expected from channel formation. Since the experi ment is performed in a dry ambient with a field induced channel,22 ionic surface currents,23 and anoma lous channel conductances of the kind considered by Statz and associates24 are not encountered. C. Breakdown Voltage Variations Figure 9 shows the field produced variations in break down voltage. These data were obtained under the same experimental conditions as used in obtaining the data in Figs. 5 and 8. ~ V B is the magnitude of the change in breakdown voltage produced by a charge QT on the field effect electrode. ~ V B is small for positive values of IPs and increases as CPs decreases. Thus, as indicated in Sec. II C on breakdown voltage, a decrease in surface breakdown is favored by a negative surface potential. However, for this junction the theoretical plot of Fig. 2 indicates no substantial increase in the magnitude of ~V B until (3IP, is near -4.5. Therefore, the observed values of ~ VB for (31P,> -4.5 are inconsistent with the theory of Garrett and Brattain. This trend has previ ously been inferred by these authors from measure ments on alloy junction transistors. VI. DISCUSSION OF EXPERIMENTAL RESULTS The dual specimen has permitted direct examination of the variations in surface potential in the single conductivity portion for use in calculating variations in junction characteristics. Variation of (3lPs from about -4.5 to +4.8 on this part of the specimen produces a regular change in S with IPsanda maximum near (31P8= 2. -4 -3 -2 I , -I 0 l3f/Js 12 i 2 ;; 4.5 FIG. 9. Field produced variations in breakdown voltage. 22 Field induced channels have been studied under dc conditions by E. N. Clarke, Phys. Rev. 99, 1899 (1955). 23 J. T. Law, Proc. lnst. Radio Engrs. 42, 1367 (1954). 24 Statz, deMars, Davis, and Adams in reference 2, p. 139. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09912 J. H. FORSTER AND H. S. VELORIC This variation does not contradict the assumption of a recombination center at an energy level roughly SkT above the center of the gapP Our data are insufficient to confirm this unambigously. Comparison of the two curves, tlG as a function of Qr and tlG as a function of Q., indicates a buildup of the density of fast states from values of about 1010 cm-2 near the middle of the gap to values on the order of 1011 cm-2 at energies about ±SkT from the center of the gap.24.25 Thus, our fila mentary measurements indicate a regular variation of ((J. and S with surface charge corresponding to a unique system of fast surface states with similar properties to those observed by other experimenters on higher re sistivity germanium. The junction measurements indicate that in principle the properties of this system can be used to calculate the variation in junction current with ((J •. In a range of interest for practical device design, variations in low bias junction current are calculated in terms of variations in S for (3((J8<0. For (3((J.>O, varia tions are on the order expected for channel currents dependent on the rate of surface generation. The reverse current at high bias, although varying reproducibly with ((J" is in qualitative, but not quantitative agree ment with the assumption of multiplicative surface breakdown. If nonuniform surface conditions exist near the junc tion (patches), they would have a tendency to be averaged out in the measurement for the low bias case. However, the same conditions would produce appreci able effects in the measurement of surface breakdown, acting to reduce the observed breakdown voltage. The high dc bias voltages and the accompanying high field across the body junction subject the breakdown voltage measurement to another possible source of error. This is the shift in ((J. near the junction (for Vp=O) produced by the application of dc bias to the junction. The "clamping" of ((J8 by slow surface states (observed by deMars et al.26 and Kingston27) tends to suppress this effect, provided time is allowed for stabilization of the surface before the ac measurement is made. In any event, this effect tends to shift ((J. (on the p-type side of the junction) in the direction of positive increase, the wrong direction to account for observed values of tl V B for /3((Js> -4.5. VII. DEVICE IMPLICATIONS The results discussed above provide a basis for pre diction of surface dependent properties of junction devices. Device parameters of importance are reverse currents, breakdown voltage, and transistor current gain (a). The data indicate that these parameters can be discussed in terms of a single parameter ({J •• Referring to Fig. 8, the low bias reverse current can be relatively 26 Similar results have been reported by Bardeen, Coovert, Morrison, Schriefier, and Sun, Phys. Rev. 104,47 (1956). 26 deMars, Statz, and Davis, Phys. Rev. 98, 540 (1955). 27 R. H. Kingston, Phys. Rev. 98, 1766 (1955). small for (3((J.<O. However, as shown in Fig. 9, break down voltage decreases as ((J. decreases, and in fact, if body breakdown is desirable, ((J, must be positive. The drop in VB becomes substantial for ({J, approaching ((Jb· Thus, to insure low reverse currents and reasonable breakdown voltage, ((J. must be kept close to the interval ((Jb< ((J,<O. For high resistivity material, ((Jb-> 0 and the desired working range becomes small. For devices where current gain is influenced by surface recombination, confining ({Js to this interval tends to exact a further price of lower current gain. (See Fig. 6.) However, for surface sensitive transistors, probably the most satisfactory compromise is a value of ({J8 near zero. The experiments reported by Wahl and Kleimack seem to lead to a similar conclusion. VIII. CONCLUSIONS Modulation of both low bias saturation current and breakdown voltage of filamentary n+-p junctions by an electric field applied at the surface is possible. By combining this "junction field effect" with conventional field effect measurements, a filamentary structure has been used to compare surface properties of a free surface with those of an adjacent n+-p junction. The measure ment allows direct comparison of field induced varia tions in ({J, and S with field induced variations in junc tion current. This comparison indicates that junction current variations can be calculated from variations in S meas ured on the filament, provided the surface conductivity type remains the same as that of the body. When the surface conductivity becomes opposite in type to that of the body, an additional current appears which is roughly calculable in terms of channel growth. However, the variations in junction breakdown voltage are not quantitatively predictable from measured variations in ({Js, at least on the basis of the theory of Garrett and Brattain. These variations are qualitatively predictable, however, in the sense that shifting ({J, in the direction of an inverted surface tends to suppress surface breakdown. The observed dependence of junction characteristics on surface potential indicates that, for many surface sensitive devices, an optimum value of surface potential can be specified. ACKNOWLEDGMENTS The authors acknowledge the assistance of C. G. B. Garrett in the form of many stimulating discussions and suggestions. The encouragement of R. M. Ryder is also acknowledged. Many of the experiments were carried out by W. C. Meyer and A. R. Tretola. APPENDIX A. Filamentary Junction Currents (1) Dark Current The rectangular geometry of the filamentary diode is illustrated in Fig. 10. The shaded boundary at x= 0 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09SURFACE POTENTIAL VARIATIONS AND JUNCTIONS 913 FIG. 10. Geometry for the filamentary junction. indicates the n+-p junction, with the n+ material ex tending in the negative x direction. The ratios tlw (filament thickness to width) and wll (width to length) are small. It is, therefore, assumed that the filament is infinite in the z and x directions. We consider only the portion of the filament in the region x>O, bounded by the planes y= ° and y= t, on which the values of surface recombination velocity are S1 and S2. Under these conditions, t:J.n=n-no must satisfy the boundary condi tions, 11 at:J.n D--=S1.6.n, y=O, (9) ay at:J.n D--=-S2t:J.n, y=t. (10) ay The fractional change in t:J.n across the filament in the y direction must be less than Stl D, where S is the larger of S1 and S2. In our experiment, values of S encountered are less than 103 cm secI, and t is less than 8X 10-3 cm. Thus StlD is small. Under these conditions the equation for the steady state is, neglecting recombination in the body of the filament, S1+S2 d2t:J.n(x) 0= --~t:J.n(x)+D----. (11) t dx2 For reverse bias »kTlq, t:J.n(O)~-no, the solution of physical interest is [ (51+52)tJ t:J.n= -no exp -x --m ' (12) and the electron current density across the junction at x=O is d1ZI' (Sl+52)! jD=qD- =qno(D)~ -- . dx x=o t (13) Experimentally, a field effect plate serves to modulate S2 only. If So is the value of S2=S1 for zero voltage on the plate and the field produces a change t:J.S in S2, then the reverse dark junction current, I JD, for a junction area A, is given by (2So+t:J.S)~ J J[)=Aqnn(D)! --t -. (14) (2) Light Current We consider the case where the filament is weakly illuminated on the surface y=O. For generation of L electron-hole pairs per cm2 per sec near one surface by an external light source, the continuity equation is, approximately, (lS) The important solution is ( L) [ (Sl+S2)!] t:J.n= -no+--- exp -x --- S1+S2 Dt (16) the current density is (17) and if I JL is the junction current with illumination, L( Dt ! t:J.IJL=IJL-IJD=Aq- ---). t S1+S2 (18) The value of t:J.I J L produced by a change from 0 voltage on the field effect plate is then MJL=Aq!:...( Dt )1. t 2So+t:J.S B. Channel Current (1) Dark Channel Current (19) With a large positive voltage on the field effect plate (on the surface y= t) {3 CPs is positive, the surface is n-type (see Fig. 11), and an n-p junction is formed on this surface. With reverse bias applied to the n+-p junction, a current I(x) flows down the inversion layer. The n-p junction is biased to a potential Vex), and will act as a collector of electrons from the p-type filament until Vex), drops to a value Vel) which is insufficient to ensure collection. The distance l is defined as the channel length and depends on the value of I., the magnitude of the current density collected by the channel. McWhorter and Kingston23 have given the following expression for the excess reverse, or channel current, I CD: (Va )i ICD=I.tp 2goln-- , Vel) (20) where P is the perimeter of the junction, and Va is the voltage applied to the junction. They assume the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09914 J. H. FORSTER AND H. S. VELORIC ~: )(=1 Hl() 1 : ---+ 1.1 <It--- N P IsWctI 1=O----~------L- __ _ FIG. 11. Channel on the filamentary junction, channel conductance, Gc, is given by Gc= go/Vex). (21) Garrett and Brattain6 give an appropriate expression for go. With an n surface on p-type germanium this is where Ld= lAX 10-4 em, and In is the mean free path for electrons. For our experimental case, g~4.5X1O-9 exp(!1""s)=K exp(!1.,.,.). (23) If we make the approximation that all carriers gener ated at the surfaces are collected by the channel, then 1.= qnO(Sl+S~) = qno (2So+ AS), (24) and for P=w (Va )t I CD= (2So+AS)! exp(!1.,.,./2)w qnoK In- . V (I) (25) Equation (24) can only be accurate for values of channel length I large compared to (Dt/2So+AS)l, a condition not too well fulfilled in our experiments, even when !1""8 approaches 4 or S. (2) Channel Current with Illumination If the surface y=O is illuminated with nonpenetrating light, the effect is to increase the channel current to a value approximately given by (18+L)! lCL= L lCD. (26) The value of L can be estimated from the measurement of I J made when p.,.,. is negative and the channel does not exist. Using Eqs. (14) and (19) (27) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 15 Dec 2014 11:33:09

1.1735699.pdf

Stable, High Density Field Emission Cold Cathode E. E. Martin, J. K. Trolan, and W. P. Dyke Citation: J. Appl. Phys. 31, 782 (1960); doi: 10.1063/1.1735699 View online: http://dx.doi.org/10.1063/1.1735699 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v31/i5 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions782 M. T. WEISS for detailed understanding of these instabilities. The remarkable flat-topped nature of the absorption curves [Fig. 6(g)] at very high power levels also needs further investigation. IV. CONCLUSIONS The experimental results described in this paper clearly demonstrate the importance of the scattering of the uniform precession at impurities and imperfec tions in manganese ferrite single crystals. This in not surprising since these crystals were grown by the flame fusion process. The validity of the Suhl analysis of the JOURNAL OF APPLIED PHYSICS saturation of the main resonance is, of course, also demonstrated. The above experiments also reveal various unexpected high power effects such as an asymmetry in the line shape, an increase in the magnetic field required for resonance, bistabilities and jitter arising at power levels of about S db above critical, and flat-topped absorption curves at very high power levels. V. ACKNOWLEDGMENTS The author is indebted to H. Suhl for many helpful discussions and to J. F. Dillon, Jr. for kindly supplying several oriented spheres. VOLUME 31, NUMBER 5 MAY, 1960 Stable, High Density Field Emission Cold Cathode* E. E. MARTIN, J. K. TROLAN, AND W. P. DYKE Linfield Research Institute, McMinnville, Oregon (Received October 28, 1959) The practical application of the field emission electron source has heretofore been impeded by insufficient reliability. Instability (Le., changes in emitted current at a fixed applied voltage) results from changes in the cold cathode surface associated with contamination and sputtering. The cold clean cathode is shown to be electrically stable at dc emission densities up to 107 amp/cm2• Techniques are discussed which permitted stable operation of a single needle tungsten cathode during 1000 hr at an average beam power of 35 w (cor responding to a beam power density of 35 billion w per unit cathode area). A simple method is described which permits reconditioning of the cathode surface when required, and apparently extends life indefinitely; oper ating periods in excess of 12 000 hr are reported. An explanation is suggested for the small, gradual residual changes observed in the emitted current. INTRODUCTION ELECTRONS are emitted from metals in the pres ence of a strong electric field, a process called field emission which has properties suited to several elec tronic devicesl; however, the field emitter has heretofore exhibited insufficient reliability for applications. In the present paper the smooth clean cathode is shown to have electrical stability and longevity; contamination and sputtering are recognized as sources of earlier difficulties. Improved environments described herein have en abled a cold, microscopic, needle shaped field emitter to yield stably a current density of 107 amp/cm2 during 1000 hr of unattended dc operation in a sealed off vacuum tube. An average beam power of 3S w from a cathode area of 10-9 cm2 can be exploited for either extreme miniaturization or very high power. The field emission gun can be as small as a hypodermic needlel; a related emission process has recently been used in a 300 megawatt flash x-ray tube.2 * This work was sponsored by ARDC, United States Air Force, through The Electronic Components Laboratory, Wright Air Development Center. 1 W. P. Dyke, Proc. Inst. Radio Engrs. 43, 162 (1955); IRE Trans. on Military Electronics, MIL-4, 38-45 (1960). 2 F. J. Grundhauser, et al.., paper presented at Field Emission Symposium, 1959 (to be published). Field emission was discovered in 1897 by R. W. Wood.3 Millikan and Lauritsen4 recognized the dependence of stability upon high vacuum, and achieved sufficient reliability to determine empirically the field current voltage relationship. More recently, field emission mi croscopy5 has been used to identify the clean emission surface and conventional electron microscopy has re vealed the emitter geometry. Haefer6 used these tech niques to develop sufficient stability to draw stable current densities up to 106 amp/cm2 for short periods. The application of pulse techniques and very high vacuums7 led to field current densities of the order of lOB amp/cm2 and, at lower densities, to cathode life of the order of 100 hr.B Elinson and co-workers9 have in~ vestigated ion trapping in an attempt to obtain stability at conventional pressures; while stable operating periods were increased by a factor of about 100, the reported 3 R. W. Wood, Phys. Rev. 5, 1 (1897). 4 R. A. Millikan and C. C. Lauritsen, Proc. Natl. Adac. Sci. (U.S.) 14, 45 (1928). 6 E. W. Muller, Z. Physik 106, 541 (1937). 6 R. H. Haefer, Z. Physik 116, 604 (1940). 7 R. T. Bayard and D. Alpert, Rev. Sci. Instr. 21, 571 (1950). 8 W. P. Dyke and W. W. Dolan, Advances in Electron Physics, 8,89 (1956). 9 M. I. Elinson, V. A. Gor'kov, and G. F. VasiJiev, Radiotekh. i. Elektron 2, 205 (1957). Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsS TAB L E, II I G H DEN SIT Y FIE L D E MIS S ION COL DCA THO D E 7R3 N~ 8 "(J) ~ 6 ~ ~ 4 2-2 o ~r 0 4> • 6.3 -2~O----~~----~--~~----~~~ FIG. 1. Graph of the modified Fowler-Nordheim equation for field emission, showing field current density J as a function of applied surface electric field F, for several values of the work function cf> in ev. The vertical line at the upper end of each curve corresponds to a field sufficient to lower the top of the surface potential barrier down to the Fermi level. performance was considerably less than that required in most devices. In the present work, further improvements in vacuum are combined with ion and electron trapping to yield the stability mentioned above together with emitter life in excess of 12000 hr. It appears that sufficient stability and life are thus available for several applica tions including a new tetrode structure which will amplify microwave signals.tO CAUSES OF INSTABILITY The term "electrical stability" is used herein to de scribe the condition in which the relationship between field current and applied voltage remains essentially unchanged with time. It will be helpful first to examine the causes of instability (i.e., effects which alter the current-voltage relationship), and then to note how such effects may be reduced or avoided. Field emission instability usually arises through changes in two variables on which the electron current density has a sensitive dependence, namely ¢, the work function of the emitting surface, and /3, a geometric factor having units of cm-l and defined as the ratio of the electric field F at the cathode to the applied po tential V. An indication of the dependence of this ratio on the electrode geometries can be gained from the expression 11 /3=2[r In(4R/r)J-r, (1) which is based on the approximation of the field emitter geometry as a hyperboloid of revolution. Here r is the emitter tip radius, and R is the anode to cathode spac ing, both in cm. The importance of ¢ and /3 is seen in the following variation of the Fowler-Nordheim equation expressing the emission current density J at zero or low 10 F. M. Charbonnier, J. E. Henderson, and W. P. Dyke (to be puhlished). 11 C. F. Eyring, S. Mackeown, and R. A. Millikan, Phys. Rev. 31, 900 (1928). 10 A '" .., ~ 8 0. 10 g ~ 6 10 ~ ~ 4 Q: Q: :> u 2 '",. A "'-8 ''''''. ~. B c -----. c -------.-. FIG. 2. A graph showing decrease of emission current with time at constant voltage when work function is increased by adsorption of an electronegative gas such as oxygen; emission patterns show preferential adsorption on different surfaces. temperaturel2,13 : where KI and K2 are constants, V is the applied voltage, and fey) is an elliptic function which takes account of the image force. A good treatment of the theory leading to this relationship is given by Sommerfeld and Bethe/4 and tables of current density as a function of F and ¢ have been published by Dolan.15 Figure 1, which presents a graph of Eq. (2), further illustrates the strong dependence of current density on /3 and ¢. For a typical field strength of 5 X 107 V / cm and a work function of 4.5 ev (the central region of the figure), a 1 % change in /3 results in about a 15% change in J, ¢ and V remaining constant. For the same field and work function, a 1 % change in ¢ results in a change of about 20% in J, /3 and V remaining constant. Clearly, both ¢ and /3 must be carefully controlled in order to stabilize the emission mechanism described by Eq. (2). In the present work, stability was achieved by holding the ratio ¢3/2//3 constant by maintaining a clean smooth cathode surface of constant configuration. Changes in the work function of metal surfaces caused by the adsorption of various contaminants are well established. Sources of such contaminants are residual gases, materials adsorbed on electrode surfaces, de composition products formed at tube envelope surfaces during electron bombardment, etc. The corresponding rate of contamination observed during operation of a field cathode is large at conventional tube pressures, e.g., 10-7 mm of Hg. In fact, even in tubes having very low static residual gas pressures, contamination may occur when an electronic discharge is drawn from the cathode. A typical example of instability arising from changes in 12 R. H. Fowler and L. W. Nordheim, Proc. Roy. Soc. (London) A119, 173 (1928). 13 L. W. Nordheim, Proc. Roy. Soc. (London) A121, 626 (1928). 14 A. Sommerfeld and H. Bethe, IIandbuch der Physik (Springer Verlag, Berlin, Germany, 1933), Vol. XXIV, Part 2, p. 441. 15 W. W. Dolan, Phys. Rev. 91, 510 (1953). Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions784 MAR TIN, T R 0 L 1\ N, AND D Y K E emitter work function is shown in Fig. 2. The current voltage relationship and emission pat terns were ob served during the operation of a cOllventional diode elec tron projection tube consisting of a needle shaped tung sten cathode at the center of a hemispherical aluminized phosphor anode. Localized changes in cathode work function, caused by the selective adsorption of electro negative gas such as oxygen, resulted in changes in the emission pattern and a decrease of total current with time. Changes in {3 are also frequently observed in practice. At the highly curved field emitter, both strong surface forces and large opposing electrostatic forces are present, so that deformation of the surface is possible when energy is added. The deformation may merely alter the current-voltage relationship, as for example when a heated cathode is dulled in the absence of electrostatic forces, or it may also lead to a destructive vacuum arc16 when increasing surface roughness causes localized in creases in {3 and, correspondingly, excessive values of current densityY A common deformation mechanism is surface migra tion18 which is appreciable at intermediate temperatures for tungsten emitters of conventional size. The time rate of change of length dz/dl of a small projection of length z and tip radius r is given by -Qo2~oeXp(-Q/RT)['Y F2] dz/dt=-- ----, All r2kT r 87l' (3) where Qo is the atomic volume in cm:J/atom, 'Y is the surface tension in d/cm, Ao is the surface area in cm"/ atom, ~II is the diffusivity constant in cm2/sec, Q is the activation energy in cal/mole, T is the temperature of the projection in OK, and F is the applied electric field in v / cm.19 At the field strengths needed for appreciable emission, and with tungsten emitters of conventional size (i.e., F>3XlOi v/cm and r>3XlO-6cm), the term in brackets is negative; hence dz/ dt is positive and a surface projection tends to grow outward or extrude when either the temperature is high enough to cause appreciable surface migration or when surface atoms gain energy from ionic bombardment. It is observed that extrusion usually increases (3, and hence current density at a given voltage; this in turn may further in crease the emitter temperature through such processes as resistive heating and bombardment by ions formed in the residual gas. The process is thus regenerative, leading to further increases of both the extrusion rate and current density and ultimately to a low impedance 16 W. P. Dyke, et al., Phys. Rev. 91, 1043 (1953). 17 W. W. Dolan, W.P. Dyke, and J. K. Trolan, Phys. Rev. 91, 1054 (1953). 18 J. L. Boling and W. W. Dolan, J. App!. Phys. 29,556 (1958). 19 J. P. Barbour, F . .:\I. Charbonnier, W. W. Dolan, \V. P. Dyke, E. E. Martin, and J. K. Trolan, Phys. Rev. 117, 1452 (1960). 60 50 (/) w ~ 40 ~ o '" ~ 30 ::;; A B O~~L-~L-~L-~L-~L-~L-~L-~~~ o 20 40 60 80 100 120 140 160 180 TIME IN MINUTES FIG. 3. A graph shO\yin" the increase of emission current with time at constant voltage, attributed to cathode sputtering by helium ions. vacuum arc and cathode damage. It is convenient to refer to this process as extrusion breakdown.2l1 Extrusion breakdown can be initiated when suffi ciently large surface projections are formed by cumula tive cathode sputtering. This occurs for example when energetic helium ions impinge on the cold tungsten surface in the presence of a high field, a phenomenon which is commonly observed in tubes having envelopes of Corning 77 -!O Pyrex glass, or other similar glasses, and results from cliuffsion of helium from the atmosphere. Figure 3 illustrates effects which can be attributed to this mechanism. At a constant applied voltage, emission from the smooth, clean cathode (at A) is stable except for a gradual increase of current with time which is accompanied by changes in the emission pattern (as at B and C). Small areas develop (bright spots) at which current density is increased by several orders of mag nit ude as estimated from emission pattern analysis. \Vhen the current increase exceeds about a factor of five, the current-voltage relationship usually fluctuates erratically and cathode damage is likely. Sudden current decreases are accompanied by the disappearance of the brightest spots whose corresponding surface projections arc presumably "burned off" during localized vacuum arcs initiated by resistive heating. The criteria which determine whether such an arc merely removes the ini tiating projection or spreads to the entire emitting tip have not been fully determined. In experiments with a tube having a Corning 7740 glass envelope which was sealed off with an ion gauge, the relative time rate of increase of the current at fixed voltage was found to be proportional to helium pressure and approximately to the current level, as may be seen in Fig. 4 which presents data taken in the range 1O-8<p< 10-6 mm Hg, 10-7 <I <5X 10-5 amp. Insta- 20 Extrusion is complicated by the dependence of surface mi gration on crystallographic detail, a fact which is ignored in Eq. (3) without serious effect on the foregoing generalization. In practice, extrusion leads to a polyhedral cathode form whose planes are crystal surfaces with low :Vliller indices. B is increased at the ridges formed by the intersections of these planes, leading as before to extrusion breakdown. Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsS TAB L E, H I G H DEN SIT Y FIE L D E MIS S ION COL DCA THO D E 785 IOO.--------r--------~------~ 10 7U1 ~ ~ ~I~ -1-1.0 .. 314 ~ ~ 4 D ~ ~ p(IJ • WITH I IN AMPERES AND P IN mm HQ FIG. 4. A graph of the relative time rate of change of field emis sion current at fixed voltage as a function of the partial pressure of helium within the tube and emission current level; the solid line graphs the empirical equation I-ldI/dt=1011p(I)i. (Pressure values are equivalent nitrogen pressure.) bility was appreciable at pressures as low as 10-9 mm of Hg, a pressure which is supplied by the diffusion of helium from the atmosphere in a period less than 1 hr in a one liter sealed envelope made of Corning 7740 Pyrex glass.21 Although ionic pumping helpfully reduced the helium pressure, it was not found adequate to main tain stability in such envelopes. EXPERIMENTAL METHODS Envelope Material The diffusion of atmospheric gases into the tube en velope was effectively reduced at room temperatures by use of high density glass selected on the basis of data given by Norton,22 which identifies a number of glasses having helium permeation rates several orders of mag nitude lower than that of Corning 7740 Pyrex. In the present series of tests, both Corning 0120 lead glass and Corning 1720 alumino-silicate glass were satisfactory in this respect. However, the latter material is con siderably superior for ultra-high vacuum work for two reasons: first, the rate of release of gas from the heated glass is very low as compared to that from alternate glasses23; and second, its annealing point of 712°C makes possible high bakeout temperatures. Except where otherwise noted, all tests reported herein were in tubes having envelopes of this material. Some difficulty was initially experienced with fabrication of envelopes and electrical inseals from the alumino-silicate glass; how ever, it was found that it could be satisfactorily sealed to molybdenum leads and, if thoroughly clean and free of fingerprints, could be readily worked in a slightly re ducing hydrogen-oxygen flame. Detailed procedures for working alumino-silicate glass of different manufacture 21 D. Alpert and R. S. Buritz, J. App!. Phys. 25, 202 (1954). 22 Francis J. Norton, J. App!. Phys. 28, 34 (1957). 23 B.~LTodd,~J. App!. Phys. 27,1209 (1956). have been outlined and are generally applicable to the above material.24 Evacuation Experimental tubes were sealed off following evacua tion with a three stage, 70 liters/sec, all glass mercury diffusion pump with liquid nitrogen traps. Metal elec trodes were fabricated from molybdenum or tungsten and during evacuation were heated to a temperature of approximately 2000°C until a pressure of the order of 10-9 mm Hg could be maintained while the metal was hot. The glass was baked and the metals were heated alternately. Satisfactory evacuation of alumino-silicate glass envelopes was achieved using both 500° and 650°C oven temperatures. During the latter treatment, the tubes were contained in a stainless steel muffle filled with nitrogen derived from the boiling liquid. Such an atmosphere protected the exposed portions of the molybdenum leads from excessive oxidation and was found to be almost completely free of helium. The re sultant low level of dissolved helium in the envelope walls reduced the subsequent diffusion of that gas into the tube when the envelope became heated during either post-seal-off heating of electrodes or operation of the tube at higher power levels. In most cases evacuation was continued for periods as long as 50 or 100 hr; how ever, equivalent results can probably be achieved in a much shorter period if glass and metal are treated simul taneously insofar as possible and if bake periods are not unnecessarily prolonged. Pressure prior to seal-off was in most cases at or below the x-ray limit of a Bayard-Alpert type ionization gauge, i.e., of the order of 10-10 mm Hg. The pressure increased momentarily to 10-8 mm at the instant of seal-off, but returned to the lower value thereafter. Gettering by evaporation of titanium from a previously degassed filament and brief heating of the electrodes established the final environment. In most cases no change in the electrical characteristics of initially clean tungsten field emitters was observable after inoperative shelf periods of 1000 hr or more. A conservative estimate of residual pressure of chemically active gas based on kinetic theory reveals that the static pressure could not have exceeded 10-13 mm Hg during that period. Absence of pronounced sputtering of the cathode during pro longed operation, as reported later herein, indicates that the partial pressure of inert gases was similarly low. Tube Design Although some work was done in more complex struc tures, as for example in field emission tetrodes, the stability tests reported herein were made in simple diodes employing well outgassed metal anodes and some form of secondary electron trapping to inhibit bom bardment of envelope surfaces. In tubes of the type shown schematically in Fig. 5 24 M. Hillier and R. L. Bell, British J. App!. Phys. 9, 94 (1958). Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions786 MARTIN, TROLAN, AND DYKE FIG. S. Schematic diagram of experimental field emission diode tube which provides electrostatic trapping of secondary electrons. Envelope of Corning type 1720 glass; A, tungsten anode; B, bom bardment filament; C, cathode; G, getter. secondary electron trapping was achieved through use of a coaxial anode-cathode configuration in which secondary electrons were returned to the anode by the radial electric field between the cylindrical anode skirt and the emitter support filament. The effectiveness of such trapping can be estimated by calculation of elec tron trajectories. In the present series of tests anodes having a length to diameter ratio of approximately five were used, a configuration which insured that even those elastically deflected primary electrons having optimum direction for escape interacted a minimum of three times with the anode. Envelope bombardment is believed to have been reduced by about four orders of magnitude from that which would have occurred if a planar anode had been used. A second tube was developed to provide similar elec tron trapping with an additional provision to view the emission pattern periodically in order to judge the con dition of the emitter surface; the electrode configuration and typical electron trajectories for this tube are illus trated in Fig. 6. The anodes consisted of two parallel S-mil tungsten plates each 3XS em and spaced 2 em apart. The field emitter was placed midway between the plates with the emitter axis normal to an aluminum backed willemite phosphor screen. During prolonged periods of operation, a one kilogauss magnetic field supplied by a permanent magnet constrained the elec trons to helical paths which terminated near the mid points of the plates and inhibited secondary electron bombardment of the envelope walls. The use of a mag netic field also led to reduced bombardment of the cathode by anode formed ions. The emission pattern could be viewed at will by removing the magnetic field. Instrumentation Stability was judged on the basis of the current re sponse at a fixed value of the applied voltage during both steady state and pulsed operation. The sensitivity of field current to small changes of either cf> or {3 has been previously noted; since current was similarly sensitive to the value of the applied voltage, several precautions were taken to prevent its fluctuation. Steady voltage in the kilovolt range was supplied from a power supply controlled by a series array of gaseous voltage regulator tubes. As an additional precaution, a Sola constant voltage transformer was used in the ac circuit and output voltage was monitored periodically as a safe guard against possible drift. The emitted de currents were continuously recorded with a Leeds and Northrup Speedomax G chart recorder. Pulsed voltages of 1 j.l.sec duration were supplied at a repetition rate of 120 pulses/sec by a pulseforming network with resonant charging. Both the current and voltage pulses were monitored by Tektronix type 511 oscilloscopes. A Sorensen regulator controlled the input voltage of these circuits. EXPERIMENTAL RESULTS First, it may be of interest to note the general effect of electron trapping on emitter stability which was demonstrated by the following simple experiment. Two nearly identical emitters were operated in the same Corning 0120 glass envelope, which contained a tubular molybdenum anode, closed near one end by a partition as shown in Fig. 7. The emitters were so positioned, one inside and one outside the anode enclosure, that the entire primary emission of each would fall on the cor responding and opposite sides of the partition; thus they were presumably exposed to the same anode con ditions but to different sources of ions and gas at other surfaces since secondary electrons were trapped in one case and not in the other. Figure 8 compares the sta bility observed during the independent operation of the enclosed emitter (curve A) and the exposed emitter (curve B), i.e., emission was drawn first from one and then from the other. Curve C displays data from the enclosed emitter which was taken at the same time that its companion was yielding curve B, i.e., both emitters were operated simultaneously. It is seen that the current from the enclosed emitter was, in both cases, relatively stable while that from the external emitter decreased rapidly. These data illustrate the improvement of field emission stability by use of electron trapping. SCREEN TUNGSTEN ~ __ I---..... ~ ANODE PLATES FIELD EMITTER ELECTRON TRAJECT ORY IN MAGNETIC FIELD ELECTRON TRAJECTORY WITHOUT MAGNETIC FIELD EMISSION PATTERN FIG. 6. Schematic diagram of electrode configuration and elec tron trajectories in magnetically switched tube, which allows optional viewing of the emission pattern. Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsS TAB L E, HI G H DEN SIT Y FIE L II E 1\1 ISS I 0 J\' COL [) CAT II 0 D E 787 Second, even in the presence of electron trapping, there are apparently residual mechanisms which affect the emitter stability at a lower rate and over a longer time period; the data may be used to identify and evaluate such residual mechanisms. Figure 9 presents typical data obtained from fixed voltage operation of electrostatically trapped field cathodes in tubes of the type shown in Fig. S. The data are presented in semi logarithmic form in order both to compare performance over a wide range of currents and to relate the current changes to the ratio cf}12/{3 on which they depend ex ponentially in view of Eq. (2). These data are char acterized by a small initial current increase of short duration followed by a long period of essentially stable operation during which the current gradually decreased in some tubes and increased in others. The behavior of a given tube was reproducible, i.e., its operation could be repeated after cleaning the emitter by momentarily heating it to 2000°C; similar behavior was also noted over a wide range of currents. However, in a given tube, the rate of change of current was usually found to depend directly on the average current during both the initial transient and the later gradual change, provided that the anode tem perature ,vas less than about lOOO°C, i.e., when the power was limited to a level of 5 to 10 w in tubes of the type shown in Fig. S. This observation led to the as sumption that the rate of arrival of foreign material at the cathode of a given tube was proportional to the emit ted current. This suggests two possible mechanisms: ionization of residual gas and release of surface material by electron bombardment through a momentum trans fer process. The following experiment was performed to evaluate the relative contributions of these mechanisms. Two emitters were mounted in a field emission mi croscope tube having an aluminum backed willemite phosphor anode. During operation of the tube both c, c, FIG. 7. Schematic diagram of anode used in experimental tube to determine the effectiveness of electrostatic trapping of secondary electrons by comparing the stability of cathodcs C1 and C,. ~164~~~====A====:===============~== ~ c lo6L-L_--J,--_~--!::----::'_~~~~,?:,=~= o 5 10 15 20 25 30 35 40 45 50 HOURS OF OPERATION FIG. 8. Comparison of the operation of the enclosed emitter C\ of Fig. 7, curve A, \,ith that of the exposed emitter C2, curve B; curve C shows performance of C, during simultaneous opcration of Ce. t--3 10 ~ -5 .................... 3014 . . ............................................... . ........................................................ ~RlI.~ .• 3 100L----5'0-0--10~0-0--15LOO-~2~0~00~-2~5~0~0-~3~00 OPERATING TIME IN HOURS FIG. 9. Graphs of the current emitted at fixed voltage in two test diodes of the type shown in Fig. 5. emitters were equally exposed to contaminants origi nating at the anode; however, one emitter was subject to bombardment by ions formed in the residual gas while the other was relatively isolated from such bom bardment. By a proper choice of emitter radii, and hence electric field strengths at a common voltage, one emitter could be operated while the other, although also cathodic, was quiescent. During the test the smaller cathode was operated and its current-voltage relation ship was continuously monitored; the larger cathode was quiescent, except during brief intervals when current was drawn to check its characteristics. For that purpose the sharper emitter was electrically connected to the anode and the voltage was increased. Because of the configuration of the electric field in the vicinity of a needle-shaped emitter, only ions formed ncar the needle axis follow trajectories which terminate on the emitting portion of the needle and thus affect subse quent emission. In view of the divergence of the elec tron beam and the decrease in ionization probability with increased electron energy, the majority of such ions originate ncar the electron source. Thus, the quiescent needle was considerably less subject to bom bardment by ions formed in the gas than the emitting needle, resulting from the use of an inter-emitter spacing which was large compared to the emitter dimensions. Figure 10 indicates that the two emitters exhibited __ --~--------------B r------------- ____ D 01234567 OPERATING TIME IN HOURS A B 11 o I'll:. lO. Graphs and emission patterns obtained from two emit ters held at the samc voltage in a comillon tube; upper curve ( .. j to B) was oiJtained during operation of the smaller tip while thc lower curve (C to D) was obtained by periodic monitoring of the normally Ilonemitting larger tip. Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions788 M A l{ TIN, T R 0 L A ~, 1\ ,'" 0 D Y K E similar stability; thus, the observed current changes were apparently due primarily to contaminants arriving at the cathodes from the anode instead of from the residual gas in the tube. The curves of Fig. 10 arc sutIiciently similar to those of Fig. 9, although the rate of change in the former case is much higher, to suggest that similar mechanisms may have been involved in both cases. The mottled appear ance of the emission patterns obtained during the present test indicated that the surface was roughened by ionic bombardment (note the similarity to the patterns from the sputtered emitter shown in Fig. 3). That this was indeed the case was confirmed by a second experiment during which one needle was, except for periodic monitoring, maintained at anode potential. The curves and patterns obtained during this test are shown in Fig. 11; the behavior of the emitting tip wa3 essentially the same as before, while both the emission pattern and current-voltage relationship for the non emitting needle underwent changes characteristic of adsorption of electronegative gas (note the similarity to Fig. 2). Thus, in this tube and probably in the metal anode tubes which yielded the curves of Fig. 9 as well, both sputtering and adsorption processes were active at the emitting surface. Therefore, both neutral and ionized material appear to have been released during electron bombardment of the anode. It is interesting to note that the relative effects caused by these two con stituents remained essentially constant over long periods of time, over an appreciable range of current, and furthermore, apparently did not vary greatly bet ween different anode surfaces. The electrical stability of cold tungsten emit ters in electro-statically trapped tubes was also studied during pulsed operation. Curve A of Fig. 12 describes 5000 hr of operation at a peak current of approximately 45 ma from a single needle at an applied voltage of 17.1 kv. The overall rate of current drift during this test did not exceed 1 %/100 hr. Curve B shows the similar operation of another single needle emitter at a pulsed current in 10 ~O--~5~~IO~-7,15~~20~~2~5~~3~O~~35' OPERATING TIME IN HOURS aK;JBiI ABC D FJ(;. 11. Graphs and emission patterns obtained \\ith the same tube as in Fig. 10 but with the larf(er emitter held at anode potential. (Note. on the vertical axis, the upper coordinate should read 10-4 instead of -4.) the order of 100 ma and an applied voltage of 24.9 kv. An over-all current drift at the rate of 3%/100 hr was observed in this case and the test terminated with failure of the cathode after 850 hr of operation. The reason for this failure has not been established; however, the pulsed current density in both of these tests was of the order of 107 amp/cm2, i.e., approaching the resistive heating limit for tungsten emitters of this size. The curves of Fig. 11 show random current variations of the order of 10%, superimposed on the systematic drift. It is felt that these variations are not related to changes in q}/2/ /3, but rather to small fluctuations in applied voltage (less than 1 % voltage fluctuation would account for the observed current fluctuation). The limited accu racy of the voltage measurement by oscillographic tech niques did not allow this assumption to be verified. In more recent tests with magnetically switched tubes, having the electrode configuration shown in Fig. 6, a further increase in the milliampere hours of ~ -2 .... 10 ~ '" ::> u W2BI4 17.1kv W2B27 24.9kv -3 IOO~~~=-~~12~O~O~2~O~OO~~2~8~OO~~3~6~O~O~744~O~O~~52~O~O~~60'OO OPERATING TIME IN HOURS FIG. 12. Graphs of the pulse currents obtained during pulsed operation of tungsten emitters in tubes of the type shown in Fig. 5; micro-second voltage pulses of fixed magnitude were applied at a repetition rate of 120 pps. unattended operation has been observed during dc op eration. Figure 13 shows the observed current during two 1000 hr tests of such a tube at two different fixed levels of applied voltage. The tungsten emitter used in these tests had a terminal radius of approximately 1.5 X 1O~5 cm, hence, an emitting area of approximately 7X10~lo cm2; an average current of 7.5 ma was emitted during the first 1000 hr. This corresponded to an emis sion current density of 107 amp/cm2, a level which was heretofore achieved only during pulsed operation25; it also approaches the upper limit set by resistive heating and vacuum arc initiation.16 Emission pattern changes were noted at this extreme current density as can be seen in Fig. 13, A and B. However, during the subse quent operation at 2 ma, the current varied by only 10% in 1000 hr and the initial and final pattern pictures, C and D, are nearly indistinguishable. The greatly improved performance observed in mag net ically trapped tubes has not yet been completely 25 W. P. Dyke and J. K. Trolan, Phys. Rev. 89, 799 (1953). Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsS T I\. B L E, H r G H DEN SIT Y F r E L n E!VI ISS r 0 ~ C () L DCA THO D E 7R9 explained; however, the following generalizations can apparently be made: increased anode area led to im proved heat dissipation; increased anode to cathode distance (by about a factor of 4) led to decreased proba bility of neutral material reaching the tip from the anode, and the combined influence of magnetic and electric fields apparently reduced the incidence of ions on the emitting surface. The latter effect is also assumed to have accounted for the reduced initial transient current increase which was reported earlier in the curves of Fig. 9. Thus, the unattended operation of the cold field emission cathode has been extended to a level of ap proximately 104 ma-hr (Fig. 13); however, the useful life of the cathode is much longer. The initial cathode condition may be restored and the operation repeated when the cathode is resmoothed and cleaned by heating ~15 ffilo ~ 8 .• ..••.••••••••.••••••.•••••••••••.••••••••••••••• 8 ::; ..J :i4 ~ 3 .-2 ... ·········································.····0 ~ 'c !E a I "'0-;;10"'0'""'201>0"'0'"'31";;0"'07140"'0"'50=0"6"'00"'7"'0"'0"8"'0""0""9';;0""0"1000 OPERATING TIME IN HOURS A ~ ABC D FIG. 13. Graphs and emission patterns obtained during fixed voltage operation of a magnetically switched tube having elec trodes as shown in Fig. 6. it briefly at say 2000°K. Contaminants are evaporated and roughness is smoothed by surface migration during a heat flash of a msec duration which can be supplied by discharging a condenser through the cathode support filament. Provided that the period of heating is short, the reconditioning may be done without interrupting the electron emission; surface migration is preferential at small projections, which can be smoothed in a time short compared with that required for buildup, i.e., the deformation of the heated tip by electrostatic forces.2 By periodic reconditioning, cathodes have been op erated for periods up to 12 000 hr without any indication of a limit on further operation. It is probable that the actual cathode life is much longer than any period yet reached. Clearly, this observed longevity of the field emitter is due at least in part to its ability to yield useful currents at low temperatures where a number of TUBE MS 5 IOOOhrs.. 7.5ma, 35W.~/ 3000hrs., 300/La, 3w. ,/ /:3500 hr., 300/La,3w 3300 hrs., 80/La,04w .• /'" • IOOhrs.,4ma, 30w /' /' /.IOOOhrs.,80fLO, O.7w. ,'""' iCoo hrs., 20/La, 0.08w. UJ ~ ". /'" • 200lvs., BfL 0, O.06w u::: "e/ iOhrs., 25fLo, O.2w ~ 0.1// ii5 :;19:f54~';;:195~""-':1956~-~19:;;:57;---;1958~--;:195""'9 --;1""96"'0.--0.'1961 ~ !/l YEAR F!G. 14. Improvement with time of the stability figure of merit Q=It for a single needle cold field cathode operated de at fixed applicd voltage. I is the dc emitted currcnt, and t is the period of continuous operation without cathode reconditioning. The cathode current density in tube MS 5 exceeds 10' a/cm2• The rate of progress over the past 5 years has heen exponential with a time constant of ! year. disturbing mechanisms are minimized, e.g., chemical changes, transport mechanisms, etc . In summary, the clean, smooth, cold tungsten field emitter is electrically stable; it is possible to maintain stability during periods of the order of 1000 hr of un attended operation at useful levels of current and power. By a simple reconditioning process, the operation may be repeated; cathode life thus appears to be indefinite; operating periods in excess of 12 000 hr have been demonstrated. Longevity and stability have been observed at a dc current density of 10; amp/cm2 which corresponded to 35 billion w of beam power per unit cathode area. At tention has been called to the possibility of utilizing this cathode in an electron gun having either extreme minia turization at conventional power levels or unusually high peak power26; several possible applications were noted in the same reference. In general, the emission current level and the period during which the cathode can be operated stably without attention are found to be reciprocal quantities. Thus, the product of these two variables serves as a useful fIgure of merit for judgment of stability. In Fig. 14 are shown values of this product which had been obtained during the course of the work for de operation of cold tungsten field emitters. If the per formance of the cold cathode continues to increase at the rate shown, a number of additional applications may soon be possible. 26 w. P. Dyke, "Field Emission A Newly Practical Electron Source" (to be published) ; this is a reference to footnote 1. Downloaded 27 Feb 2013 to 129.8.242.67. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1723260.pdf

Field Modulation of Liquid Induced Excess Surface Currents on Germanium pn Junctions W. T. Eriksen Citation: Journal of Applied Physics 29, 730 (1958); doi: 10.1063/1.1723260 View online: http://dx.doi.org/10.1063/1.1723260 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/29/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Giant magnetoresistance modulated by magnetic field in graphene p-n junction Appl. Phys. Lett. 105, 193108 (2014); 10.1063/1.4901743 Current injection induced terahertz emission from 4H-SiC p-n junctions Appl. Phys. Lett. 103, 221101 (2013); 10.1063/1.4832824 Growth, electrical rectification, and gate control in axial in situ doped p-n junction germanium nanowires Appl. Phys. Lett. 96, 262102 (2010); 10.1063/1.3457862 Currentvoltage characteristics of amorphous silicon PN junctions J. Appl. Phys. 51, 4287 (1980); 10.1063/1.328246 EXCESS CURRENT GENERATION DUE TO REVERSE BIAS PN JUNCTION STRESS Appl. Phys. Lett. 13, 264 (1968); 10.1063/1.1652602 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Tue, 23 Dec 2014 08:13:55JOURNAL OF APPLIED PHYSICS VOLUME 29, NUMBER 4 APRIL, 1958 Field Modulation of Liquid Induced Excess Surface Currents on Germanium p-n Junctions W. T. ERIKSEN Research Division, Raytheon Manufacturing Company, Waltham, Massachusetts (Received November 15, 1957) The field modulation of liquid induced surface currents on germanium surfaces has been investigated. The field induced current is shown to conform to the behavior seen by liquid induced currents on germanium surfaces. The modulation increases with decreasing temperature down to the melting point of the liquid and then decreases radically to nearly zero. The modulation effect is seen only when polar liquid ambients are used. A qualitative explanation of the effect is discussed in terms of a model for current conduction in which the charge carriers move in the liquid outside of the semiconductor. The increased current with external field is thought to arise from the orientation of the dipoles in the liquid giving increased mobility to the charge carriers. INTRODUCTION IT has been shown recently that the surface currents which flow on a reverse biased germanium diode can be conveniently divided into two contributions.' One of these, conduction by inversion layers, has been ex tensively investigated and is reasonably well under stood.2 The second contribution, which will be called liquid layer conduction, is exhibited when p-n junctions are placed in ambients of certain condensible vapors.3 These include water vapor and the vapors of many or ganic liquids. If the condensible vapor molecules have a permanent dipole moment, the total surface current can be modulated by an external field. This phenomenon has been used to make an amplifying device called the Fieldistor.4 The purpose of this paper is to show that the modulation takes place in the liquid layer and to discuss some further aspects of the phenomenon. EXPERIMENTAL Two experiments were performed. One large area grown junction germanium diodes with the resistivity of the p and n sides of 6 ohm-cm were used. The bars were 1 inch long, 0.1 inch on edge, with the junction in the center. The units were etched until they exhibited a flat reverse characteristic in dry nitrogen. In all units v the dry reverse saturation current was less than 10 p.A at 25°C, In the second experiment grown junction p-n-p transistor structures were used. The resistivities of the p-n-p regions were 8, 6, 4 ohm-cm, respectively. The bars were 1 inch long, with the n region 2S mils wide, situated approximately in the center. These were pre pared in advance by etching until the conductivity be tween p regions across the base n region was less than 10-8 mhos in dry N 2. Two liquids, nitrobenzene and dioxane, were chosen for the experiments. Both of these give liquid layer conduction on germanium. They were chosen because nitrobenzene has a large permanent dipole moment (4.27 D.D.) while dioxane is a nonpolar liquid. Figure 1 shows a diagrammatic scheme of the experi ments from which data were taken. In (lA), diode re verse characteristics as a function of field were obtained. In (lB), surface conduction of the base (n) region was measured as a function of reverse bias and field. The samples and the field electrode were immersed directly in the liquid, which had previously been purified by distillation, and which had been dried by standing over a dehydrating agent (Dehydrite). A microammeter was in field circuit in all experiments and the current in this circuit was always less than 0.05 p.A. The external field had no effect on the measurements in dioxane; however, /000 CYCLE FIG. 1. Circuit diagrams of TUNED experimental setups. AMPLIFIER DIODE CHA~ACTERISTIC SURFACE CONDUCTANCE 1 Eriksen, Statz, and DeMars, J. Appl. Phys. 28, 133 (1957). 2 Statz, DeMars, Davis, and Adams, Phys. Rev. 101, 1722 (1956). 3 J. T. Law, Proc. lnst. Radio Engrs. 43, 1367 (1954). 4 O. M. Stuetzer, Proc. Inst. Radio Engrs. 40, 1377 (1952). 730 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Tue, 23 Dec 2014 08:13:55L I QUI DIN Due E D SUR F ACE CUR R E N T SON G E R MAN I U M 731 240 220 ~200 ~ ',:~ a 140 i 120 110 80 INVERSION LAYER SATURATION CURRENT J 2 ~ A 5 ~ 7 • • w u ~ ~ ~ ~ ~ cr REVERSE BIAS VOLTS FIG. 2. Reverse current-voltage characteristics of germanium diode under different ambient conditions. the effects of external field were quite pronounced in nitrobenzene. Figure 2 shows typical reverse current-voltage be havior at various external field strengths for the diode in nitrobenzene. A curve is included which shows the behavior of a diode with pure inversion layer conduc tion. The final curve is the saturation current of the diode in dry nitrogen, prior to immersion in nitroben zene. An external field had no observable effect on the reverse characteristics in dry nitrogen. Figure 3 illus trates the variation of surface conductance with applied bias and field. The effect of the external field on reverse character istics of a diode can be expressed as the ratio of current with field to current with no field at the same reverse bias. The ratio expressed is IF=Mlo, where IF=current with field 10= current with no field M = modulation coefficient. ~ ::;: W o '" ;= 010-6 :> o '" o o & NITROBENZENE 5 'feN F"lELD • NITROBENZENE NO FIELD 10" '---+-+--+-~H-!-H----+--+-+-+-I-HH 2 34.5.6.7.8.91 3456789 REVERSE BIAS VOLTS FIG. 3. Conductance vs reverse bias for germanium p-n-p grown junction transistor in nitrobenzene. °O~~J~2~'~4~5~£~7~~.~3~LO~"~'.~2~'~'~1~4~1~5~L6~ REVERSE BIAS VOLTS FIG. 4. Modulation coefficient vs reverse bias for germanium diode at various field strengths in nitrobenzene. Figure 4 shows a typical plot of M 'Vs bias at three values of field at 30oe. M 'Vs bias was also measured at selected temperatures down to below the freezing point of nitrobenzene. The effect of temperature is shown in Fig. 5 where values of M at a constant bias of 0.6 v are plotted against temperature. DISCUSSION OF RESULTS The results presented in the last section will be dis cussed in terms of the qualitative mechanism of liquid layer conduction proposed in reference 1. For the pur pose of this discussion it will be valuable to restate the main features of that mechanism in a form more imme diately applicable to the results presented in this paper. LIQUID LAYER CONDUCTION The system germanium-liquid can be considered from a physical chemical point of view as a two phase system with one common component, electrons. The interface is a membrane impermeable to both germanium and liquid but perfectly permeable to electrons. The con dition for thermal equilibrium is that the electrochemi- ·5"tern FIELD • TEMPERATURE OC ,. • FIG. 5. Modulation coefficient vs temperature for germanium diode in nitrobenzene. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Tue, 23 Dec 2014 08:13:55732 W. T. ERIKSEN >- THERMAL EQWBRlUM ---, :; I :t I P -+ I :t I ------I-! I r· L.J. --j ~, = I 11.1 =+ I Z P -I ~ + :: I ::; e-L- N ~it ! ~ CONDUCTION BANO NORMALLY EMPTY /' "BAND" IN WATER ~ WATER FILM (thlckn, .. exaggerated) -+ 1 :t I N ::1: I :t I :t I -+ I -+ I __ J SEMICONDUCTOR WOUlD -+ + I :::1:: I . -t+ I r -+! I __ oJ SEMICONOUCTOO WOUlD FIG. 6. Germanium surface-nitrobenzene system at thermal equilibrium and under reverse bias. cal potential (Fermi level) for electrons is equal in both phases. The attainment of this equilibrium will require charge transfer with a resulting space charge at the interface. This space charge is the seat of the surface conduction exhibited by the semiconductor. The liquid layer conduction is assumed to take place in the half of the double layer on the liquid side of the interface. Under the influence of the bias vol tage, charge will migrate along the surface of the semiconductor. On the side of the p-n junction where this charge accumulates, the equilibrium between liquid and semiconductor will be displaced, resulting in charge transfer across the interface. The same will apply on the side of the junc tion which becomes charge deficient except that the equilibrium is displaced in the opposite sense and the resulting charge transfer will be in the opposite direc tion. This is illustrated in Fig. 6 where transfer of elec trons across the interface is shown. This qualitative discussion has neglected the half of the space charge on the semiconductor side of the inter face. It mayor may not consist of mobile charge, depending on the number of localized states at the germanium surface. In those cases where it is mobile, inversion layer conduction will result and both con tributions to surface conduction will be present simultaneously. It has been found experimentally and confirmed theoreticallyl that when inversion layer conduction pre dominates, the current on a grown junction germanium diode varies as log v. In the presence of condensible vapors (when liquid layer conduction predominates), one observes a linear dependence of voltage on current. The difference in voltage dependence affords a means of identifying the dominant process. In Fig. 2 the linear behavior of the reverse characteristic at all values of field provides evidence that the liquid layer conduction is the dominant process. The no field characteristic shows some nonlinearity at low values of bias, which indicates that liquid layer conduction has not yet become the dominant process. This is further shown in Fig. 2, where the conduction vs bias shows typical in version layer behavior at low bias and liquid behavior at higher bias. The characteristic at 7.5 v/cm field in .+/++++ VALENCE BAND FERMI LEVEl. •• + .// t//, /,,'l,-NORMALLY FILLED "BANO" IN WATER )( FIG. 7. Energy diagram of germanium surface-water film system Fig. 1 indicates some decrease from linear behavior at higher bias, which seems to be because of a current saturation. The same effect is shown in Fig. 3, where the conduction at no field and at 5 v/cm approach one another at the high bias value. The discussion of this will be postponed until a later section. MODULATION COEFFICIENT The absolute value of the surface current decreases with decreasing temperature but M, the modulation coefficient, gives a relative measure of the ability of the external field to induce extra current into the diode cir cuit at different temperatures. Figure 4 plots the varia tion in M with bias at constant temperature. The data at 20°C is typical of that obtained at other tempera tures. The characteristic feature is the maximum in M occurring at a bias of about 0.5 v. This is a further indi cation of a current saturation occurring at higher bias values. Figure 5 shows the very striking behavior of M with temperature at a constant value of bias. The in verse dependence on temperature, plus the fact that in nonpolar liquids M is zero under all conditions, indicates that the orientation of the liquid molecules in the ex ternal field is responsible for the modulation effect. CURRENT SATURATION Referring to Fig. 6 as a model of what is occurring at the surface during liquid layer conduction, one can see that the total current at any bias is a function of at least three factors: 1. The rate at which charge can move across the p-n junction in the liquid layer. 2. The rate at which charge can transfer across the liquid-semiconductor interface. 3. The magnitude of the space charge at thermal equilibrium, which depends on the difference in Fermi levels of the phases making up the system before contact. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Tue, 23 Dec 2014 08:13:55L I QUI DIN Due E D SUR FA C E CUR R E N T SON G E R MAN I U M 733 Current saturation can be a result of either of the first two processes being a rate limiting one. In view of previous work on electrolyte semiconductor behavior, 6 it does not seem likely that charge transfer across the interface will be rate limiting at the low current levels used in these experiments. One would expect, however, a rather low mobility of charge carriers in the liquid film and it seems most probable that saturation occurs because of this. A much more exaggerated example of current saturation can be seen in Fig. 3 of reference 1. The above consideration points out one way in which the external field can cause an increase in conductivity of the liquid film. Figure 7 is a model of the semicon ductor liquid system, using water as the specific ex ample. The bands in the liquid are shown as well formed. Obviously the degree of order in the liquid is low and the bands will be hazy and ill defined. This will resul t in a low mobility of carriers in the liquid band. The orientation introduced by the field further orders the liquid and results in an increased mobility of the charge carriers. The total current which will flow at any bias will increase; subject to the rate at which charge can transfer across the p-n junction and the liquid semi conductor interface. 6 W. H. Brattain and C. G. B. Garrett, Bell System Tech. J. 34, 129 (1955). FIELDISTOR EFFECT The field modulation of a germanium diode in nitro benzene has been used to make a device called the fieldistor.4 One of the main characteristics of the device is the low frequency at which it gives appreciable gain. The poor frequency response can be understood in terms of the time necessary to readjust the charger distribu tion at the liquid semiconductor interface after appli cation of the field. It should be mentioned that all the data reported here are steady-state values. Long tran sients were observed after a rapid change of external field. This is in agreement with the work of Statz et al.2 who reported that the outer surface states require a long time to come into equilibrium after a change in bias. CONCLUSIONS The field modulation of surface current on a germa nium diode immersed in nitrobenzene is due to liquid layer conduction. The field induced excess current flows in the liquid next to the semiconductor surface and is the result of the increased order introduced into the liquid by the external field. ACKNOWLEDGMENT The author wishes to thank H. Brawley for assistance in the experimental work. Cover Photo The cover photograph was taken by Dr. W. H. Bennett of the U. S. Naval Research Laboratory, using the StOrmertron which is a laboratory tube in which can be produced scale models of the forms assumed by streams of protons from the sun as they enter the earth's magnetic field. The stream forms consist of bundles of StOrmertron orbits, i.e., the orbits of individually charged particles in the earth's magnetic field. The earth is simulated with an approximately uniformly magnetized sphere whose magnetic field has approximately the same shape as that of the earth. Electron streams are used to simulate proton streams with the direction of the magnet reversed. The tube contains mercury vapor in a low enough density to avoid appreciable falsification of the electron stream shapes due to gas focusing or other ionization effects. See also W. H. Bennett and E. O. Hulburt, Phys. Rev. 91, 1562 (1953), Phys. Rev. 95, 915-919 (1954), and J. Atmospheric and Terrest. Phys. 5, 211-218 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Tue, 23 Dec 2014 08:13:55

1.1744711.pdf

Correlation Correction Study of CH, NH, and OH M. Krauss and J. F. Wehner Citation: The Journal of Chemical Physics 29, 1287 (1958); doi: 10.1063/1.1744711 View online: http://dx.doi.org/10.1063/1.1744711 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/29/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in How are CH3OH, HNC/HCN, and NH3 Formed in the Interstellar Medium? AIP Conf. Proc. 855, 86 (2006); 10.1063/1.2359543 Theoretical study of conjugation, hyperconjugation, and steric effect in B2D4 (D=H, F, OH, NH2, and CH3) J. Chem. Phys. 105, 1046 (1996); 10.1063/1.471949 Microwave Studies of CollisionInduced Transitions between Rotational Levels. VI CH3OH and CH3NH2 J. Chem. Phys. 51, 3027 (1969); 10.1063/1.1672452 Some IntraAtomic Correlation Correction Studies J. Chem. Phys. 33, 840 (1960); 10.1063/1.1731272 ProtonTransfer Studies by Nuclear Magnetic Resonance. II. Rate Constants and Mechanism for the Reaction CH3NH3 ++OH2+NH2CH3 in Aqueous Acid J. Chem. Phys. 33, 556 (1960); 10.1063/1.1731183 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:10THE JOURNAL OF CHEMICAL PHYSICS VOLUME 29, NUMBER 6 DECEMBER, 1958 Correlation Correction Study of CH, NH, and OH M. KRAUSS AND J. F. WEHNER* National Bureau of Standards, Washington, D. C. (Received June 26, 1958) The semiempirical intra-atomic correlation correction (ICC) of Hurley and a configuration-interaction (CI) calculation are applied to previous single configuration calculations of the ground states of CH, NH, and OH. Results are reported for the 2II, 4~-, and 2~ states of CH, the 3~-state of NH, and the 2II state of OH. The binding energies and dipole moments obtained for the hydrides are compared with experimental values and with the results of a similar calculation by Hurley of the ground states of these molecules. Results of a calculation of the Fermi contact term of the magnetic hyperfine interaction are also given. I. INTRODUCTION THE results of extensive applications of the Roothaan1 self-consistent field (SCF) procedure contain con siderable information regarding the accuracy of the calculations based on the single configuration approxi mation.2-4 For many of the significant properties of the simple molecules that have been treated, the results have been somewhat disappointing. This is especially true of the calculation of binding energies which are obtained as the difference of two large quantities, the calculated molecular and atomic energies. A straight forward method of improving the calculation by means of configuration interaction (CI) employs a linear combination of configurations formed from some basic set of one-electron functions as a trial function. However, it has been found that significant improve ment would be obtained only if the basic set is in creased over the few Slater or Hartree-Fock orbitals usually chosen for molecular calculations.5 Computa tional difficulties have precluded such a solution and are likely to prevent such an attempt for some time in the future. This situation has led Moffitt to introduce certain semiempirical techniques.6 Moffitt observed that the energy of atomization of any molecule is a small fraction of the total binding energy. It was further noted that the errors in the molecular calculations are due to the poor representation of the dominant terms of the atomic type. The errors in a molecular or atomic calculation have been ascribed by Moffitt and by Hurley7 to essentially two causes. The first is the correlation error8 implicit in the almost universal choice * Now at the Chemical Engineering Department Catholic University of America, Washington, D. C. ' 1 C. c. J. Roothaan, Revs. Modern Phys. 23, 69 (1951). 2 C. W. Scherr, J. Chern. Phys. 23, 569 (1955). 3 R. C. Sahni, J. Chern. Phys. 25, 332 (1956). 4 Various other studies have been published of which a large number are described in the Solid-State and Molecular Theory Group, Massachusetts Institute of Technology, Progress Reports of the last three years. 5 A. J. Freeman, J. Chern. Phys. 28, 230 (1958). 6 W. Moffitt, Proc. Roy. Soc. (London) A210, 245 (1951). 7 A. C. Hurley, Proc. Phys. Soc. (London) A69, 49 (1956). 8 Correlation energy or error is defined as the difference between the correct energy and the energy obtained from Fock's equa tions. See P. O. Lowdin, Advances in Phys. 5, 31 (1956). of an antisymmetrized product of single electron orbitals as a trial function. The second error is due to the use of nonoptimum values for the parameters in the trial functions. There have been few attempts at varying the effective nuclear charges. It must be assumed that the usual choice of Slater shielding parameters is sufficiently accurate so that the larger error is the correlation one. These considerations lead to a scheme that permits the construction of the energy matrix with the terms atomic in character separated from the interatomic ones. In one formulation the energy matrix is corrected to give asymptotically the experimentally observed energies of the atoms. The Hurley correction is of the asymptotic type and also recognizes the importance of variation of the effective nuclear charges (all param eters) in constructing the energy matrix for infinite atom separations.9 The error to be eliminated can be identified with the correlation energy for the separated atoms, since the computed energies in the cases con sidered are reasonable approximations to Hartree-Fock energies. Because of this Hurley has designated the correction the intra-atomic correlation correction (ICC) . Hurley has applied the ICC correction to several molecules with very good results.9-11 Somewhat more disappointing results were obtained by Hurley and Freemanl2 for OH with a basic set of Hartree-Fock functions. Toward the end of this research we received a manuscript from Hurley reporting results for a con sistent treatment of the ground states of the hydrides BH, CH, NH, OH, and FH.13 Our results for CH, NH, and OH are in essential agreement with Hurley's and are being reported to support and supplement values obtained by a different procedure. Results are also reported for a straightforward CI calculation. 9 HF, A. C. Hurley, Proc. Phys. Soc. (London) A69, 30 (1956). 10 N2, A. C. Hurley, ProC. Phys. Soc. (London) A69, 767 (1956). 11 BH, A. C. Hurley, Quarterly Progress Reports, Solid-State and Molecular Theory Group, Massachusetts Institute of Tech nology, (January 15, 1957). 12 A. C. Hurley and A. J. Freeman, Quarterly Progress Report, Solid-State and Molecular Theory Group, Massachusetts Insti tute of Technology, (July 15, 1957). 13 A. C. Hurley (to be published). 1287 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:101288 M. KRA USS AND J . F. WEHNER TABLE I. Molecular orbital determinantal eigenfunctions." CH2JI CH2A e. Coefficient 10-20-30-4u 'Ir+ 'Ir_ e, Coefficient 10-20-30-40- 'Ir+ 'Ir_ 1 1 +-+-+- + 1 1 +-+-+ +- 2 1 +-+-+-+ 2 1 +-+- + +- 3 1 +-+-+-+ 3 1 +-+ +-+- 4 Ij.J2 +-+-+ + 4 1 +-+-+ +--1/ ..J2 +-+- + + 5 1 +-+ +-+- 5 Ij.J2 +-+ +- + 6 1 +- + +- +--1j.J2 +-+-+ + 7 2j.J6 +-+ + +- 6 Ij.J 2 +-+ +-+ -l/..J6 +-+ + +- -1/..J 2 +- + +-+ -1j.J6 +- + + +- 7 1 +-+- +-+ 8 Ij.J 2 +-+ + +- 8 1 +- +- +-+ -1/..J 2 +- + + +- 9 1 +- +- +- + 10 Ij.J 2 +-+ +-+ NH3~- -l/..J2 +- + +-+ Ij.J 2 +-+-+- + 11 Ij.J 2 +-+ +-+ -1/ ";2 +-- + +-+ Ij.J 2 +-+-+- + 12 Ij.J2 +- + +-+ 2 Ij.J 2 +- +- +- + -1/..J 2 +- + +-+ Ij.J 2 +- +- +- + 13 2j.J6 +-+-+ + 3 Ij.J 2 +-+-+-+ -1I..J6 +-+-+ + Ij.J 2 +-+-+- + -1/ ..J6 +-+- + + 4 Ij.J 2 +-+-+ + 14 2j.J6 +-+ +-+ -1/..J 2 +- +- + + -1/..J 6 +-+ +- + 5 Ij.J 2 +-+ +-+ -1/..J 6 +-+-+ + -1/";2 +- +- + + 15 2j.J6 +-+ + +-6 Ij.J 2 +-+ + +- -1/..J 6 +-+ +-+ -1/";2 +- +-+ + -1j.J6 +- + +-+ 7 1/2 +-+-+ + 16 2j.J6 +-+ + +-1/2 +- +- + + -1/..J 6 +-+ +-+ -1/2 +-+- + + -1/ ";6 +- + +-+ -1/2 +-+- + + 17 2j.J6 +-+ + +-8 1/2 +-+ +- + -1j.J6 +-+ +-+ 1/2 +-+ +- + -1j.J6 +-- + +-+ -1/2 +- +- + + -1/2 +- +- + + 18 2j.J6 +- + + +- 9 1/2 +-+ +-+ -1j.J6 +- + +-+ 1/2 +-+ +- + -1/ ..J6 +- + +-+ -1/2 +- + +-+ CH4~--1/2 +- + +- + 1 1 +- +- + + + OH2JI 2 1 +- +- + + + 1 1 +-+-+- +- + 3 1 +-+ +- + + 2 1 +-+-+-+-+ 4 1 +-+-+ + + 3 1 +-+-+- +- + 5 1 +-+ +-+ + 4 1 +- +- +- +- + 6 1 +- + +-+ + 5 Ij.J 2 +-+-+ +-+ 7 2j.J6 +-+ + + + -1/";2 +-+- + +-+ 6 Ij.J 2 +-+ +-+-+ -1j.J6 +-+ + + + -1I..J 2 +-+-+ +-+ -1/..J 6 +-+ + + + 7 Ij.J 2 +-+ +-+-+ 8 3/ ..J 12 +- + + + + -1/..J 2 +- + +- +- + -1j.J 12 +-+ + + + 8 2j.J6 +-+-+ + +- -1j.J12 +-+ + + + -1/ ..J6 +- +- + +-+ -1j.J 12 +-+ + + + -1j.J6 +-+- + +-+ 9 4/..J 20 +-+ + + + 9 2j.J6 +-+ +-+ +- -1/..J 20 +- + + + + -1/..J 6 +-+ +- +- + -1j.J6 +-+-+ +-+ -1j.J20 +-+ + + + 10 2j.J6 +-+ + +-+--1j.J20 +-+ + + + -1/..J 6 +-+ +- +- + -1j.J20 +-+ + + + -1/..J 6 +- + +- +- + " In this and subsequent tables (+) ( -) denote orbitals with plus or minus spins, respectively. II. CONFIGURATION INTERACTION for all orbitals. The solution of the Roothaan SCF The basis atomic orbital set chosen for the CH, NH, equations for the ground-state configurations14 of CH, and OH molecules consists of the ls, 2s, 2p., and 2p± NH, and OH therefore give four orbitals in addition carbon, nitrogen, and oxygen orbitals, and the 1s to the doubly degenerate 7r orbital, which is determined hydrogen orbital. Slater screening factors were chosen 14 M. Krauss, J. Chern. Phys. 28, 1021 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:10CORRELATION CORRECTION STUDY OF CH, NH, AND OH 1289 TABLE II. Atomic orbital determinantal eigenfunctions. ClPII Carbon dissociation 1/ti Coefficient c s z h 11'+ 11'_ product 1/..J 2 +-+- +- + 'Do, C- 1/..J 2 +-+- +- + 2 0.9745 +-+- +- + 'po, C -0.2243/..J2 +- +-+- + 0.2243/..J2 +- +- +- + 3 1/..J 2 +- +-+- 'Do, C 1/..J 2 +- +-+- + 4 0.9888/..J 2 +-+- + + ID,C -0.9888/..J2 +-+- + + 0.1488/..J 2 +- + + +--0. 1488/..J2 +- + +- + 5 1/2 +- +- + + IPO, C -1/2 +- + +- + -1/2 +- + +- + 1/2 +- + + +- 6 1/..J 2 +- + + +- 'D, C -1/..J2 +- + +- + 7 0.9814/..J2 +- +- +- + 'po, C- -0.9814/V2 +-+- +- + 0.1918 +- +- +- + 8 -0.1918/V2 +- +- +- + 'po, C- 0.1918/V2 +-+- +- + 0.9814 +- +- +- + 9 0.2243 +- +- +- + 'po, C 0.9745/..J 2 +- +- +- + -0. 9745/..J 2 +- +-+- + 10 1/..J 2 +- + + +- 'D, C- -1/..J2 +- + +- + 11 1/2 +- +- + + IDo, C 1/2 +- + +- + -1/2 +- + +- + -1/2 +- + + +- 12 -0.1488/V2 +-+- + + ID,C 0.1488/..J2 +-+- + + 0.9888/..J 2 +- + + +- -O. 9888/..J 2 +- + +- + 13 1. 9776/V 6 +- +- + + 3P, C -0.9888/..J6 +- +- + + -0.9888/V6 +-+- + + 0.2976/..J6 +- + +- + -0. 1488/..J6 +- + +- + -0. 1488/..J 6 +- + + +- 14 2/..J 12 +- + +- + 3PO, C -2/..J 12 +- + +- + -1/..J 12 +- +- + + -1/..J 12 +- + +- + 1/"; 12 +- + +- + 1/..J 12 +- + + +- 15 2/..J6 +- + +- + 'P, C- -1/..J6 +- + + +- -1/..J6 +- + +- + 16 2/..J6 +- + +- + 'P, C- -1/..J6 +- + + +- -1/..J6 +- + +- + 17 2/..J 12 +- + +- + 3Do, C 2/..J 12 +- + +- + -1/..J 12 +- +- + + -1/..J 12 +- + +- + -1/..J12 +- + +- + -1/..J12 +- + + +- 18 -0.2976/..J6 +-+- + + 3P, C O.1488/..J6 +-+- + + 0.1488/..J 6 +- +- + + 1. 9776/..J 6 +- + +- + -0.9888/..J6 +- + +- + -0.9888/..J6 +- + + +- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:101290 M. KRAUSS AND J. F. WEHNER TABLE II.-Continued. CH4~- Carbon dissociation 1/;i Coefficient c Iz 1r+ 1r_ product 1 1 +- +- + + + 4S0, C- 2 0.9888 +- +- + + + 'P, C 0.1488 +- +- + + + 3 1 +- + +- + + 4P, C- 4 0.9888 +- +- + + + 'P, C -0.1488 +-+- + + + 5 1 +- + +- + + 4P, C+ 6 1 +- + +- + + 4S0, C+ 7 2N6 +- + + + + 'Do, C -IN6 +- + + + + -l/v6 +- + + + + 8 3/ v 12 +- + + + + 'So, C -INI2 +- + + + + -INI2 +- + + + + -INI2 +- + + + + 9 4/v2O +- + + + + 6S0, C -IN20 +- + + + + -IN20 +- + + + + -IN20 +- + + + + -IN20 +- + + + + CWt. 1 1 +- +- + +-2Do, C- 2 0.9888 +- +- + +- ID, C -0.1492 +- +- + +- 3 1 +- + +- +- 2D, C- 4 0.1492 +-+- + +- ID, C 0.9888 +- +- + +- 5 1 +- + +- +- 2D, C+ 6 1 +- + +-+-2Do, C+ ·7 2N6 +- + + +- 'Do, C -IN6 +- + + +--IN6 +- + + +- 8 IN2 +- + + +- IDo, C -IN2 +- + + +- NHa~- Nitrogen dissociation 1/;i Coefficient c z 11 1r+ 1r_ product IN2 +-+- +- + 'P, N- IN2 +-+- +- + 2 0.1344N2 +-+- +- + ap,N+ 0.1344N2 +- +- +- + 0.9909N2 +- +-+- + 0.9909N2 +- +-+- + 3 -0. 1344N 2 +- +-+- + 'P, N+ -0.1344N2 +- +- +- + 0.9909N2 +-+- +- + 0.9909N2 +-+- +- + 4 IN6 +-+- + + 4S0, N IN6 +- +- + + IN6 +- +- + + -1/v6 +- +- + + -IN6 +-+- + + -IN6 +- +- + + 5 IN6 +- + +- + 4P,N IN6 +- +- + + IN6 +- +- + + -IN6 +- +- + + -1/v6 +- + +- + -IN6 +- + +- + 6 IN6 +- + + +- 'So, N+ IN6 +- + +- + IN6 +- + +- + -IN6 +- +- + + -IN6 +- + +- + -IN6 +- + +- + This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:10CORRELATION CORRECTION STUDY OF CH, NH, AND OH 1291 TABLE Ir.-Continued. NH~- Nitrogen dissociation .pi Coefficient s It 'Ir+ 'Ir_ product 7 1/ V 12 +-+- + + 'Do, N 1/ V 12 +- +- + + -2!V 12 +-+- + + -1!V 12 +-+- + + -1!V12 +-+- + + 2/ V 12 +-+- + + 8 1/ V 12 +- +- + + 'P, N 1/ V 12 +- +- + + -2!V12 +- + +- + -1!V 12 +- + +- + -1!V 12 +- + +- + 2/ V 12 +- +- + + 9 1/ V 12 +- + +- + 3Do, N+ 1/ V 12 +- + +- + -2!V12 +- + + +--1!V 12 +- + +- + -1!V 12 +- + +- + 2/ V 12 +- +- + + Own Oxygen dissociation .pi Coefficient 0 z II 'Ir+ 'Ir_ product 0.1786 +- +-+-+- + 'PO, 0+ 0.9839!V 2 +- +- +-+- + -0.9839!V2 +-+- +-+- + 2 1 +- +- +- +- + 'po,O- 3 1!V2 +- +- +-+- + 'DO, 0+ 1!V2 +-+- +- +- + 4 0.9839 +- +- +- +- + 'PO, 0+ -0. 1786!V 2 +-+-+- +- + 0.1786!V2 +-+- +-+- + 5 2!V6 +- +- + +- + 'P,O -1!V6 +-+- + + +- + -1!V6 +-+- + +- + 6 2!V6 +- + +- +- + 'PO, 0 -1!V6 +- + +- + +- + -1!V6 +- +- + +- + 7 2!V6 +- + +- +- + 'P,O+ -1!V6 +- + + +- +- + -1/-oJ6 +- + +- +- + 8 1!V2 +-+- + + +- + ID,O -1!V2 +-+- + +- + 9 1!V2 +- + +- + +- + IDO, 0 -1!V2 +- +- + +- + 10 1!V2 +- + + +- +- + 'D,O+ -1!V2 +- + +- +- + by symmetry. The rr orbitals are obtained as linear tions of determinants of atomic orbitals, are listed combinations of the basis atomic orbital set, in Table II along with the symmetry states of C, N, irr= LaiiXi> (1) or 0 atoms into which the state, 'It;, decomposes at infinite internuclear distance. The molecular wave function <P is chosen as a linear with Xi= is, 2s, 2p., iSH and i= 1, 2, 3, 4. In each case combination of the 8i. the 4rr orbital is not occupied in the ground state, but <P= LCl8;. (2) must be considered in enumerating all functions of a ; given symmetry. In Table I we list the symmetry The variation of the coefficients in the energy expres-states, 8;, as linear combinations of determinantal sian eigenfunctions formed from the molecular orbitals for the following molecular states: CR, 2II, 4~-, 26.; NR, E= J <p*H<pdp= ~ J;CkO*HklOClO (3) 3~-; OR, 2II. The states, 'It i, which are linear combina- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:101292 M. KRAUSS AND J. F. WEHNER TABLE III. CI Coefficients. Ci CH'1l CH4Z- CH'.!l NH'Z- 1 0.9765 0.9650 0.9893 0.9893 2 -0.0808 0.0409 0.0061 -0.0437 3 -0.0479 0.2178 0.0829 -0.0968 4 -0.0110 -0.0299 -0.0148 0.0336 5 -0.0098 -0.0505 -0.0361 -0.0793 6 -0.0536 -0.0658 -0.0618 0.0009 7 0.1207 -0.0968 0.0184 -0.0023 8 0.0799 -0.0009 -0.0934 0.0028 9 -0.0190 -0.0509 -0.0499 10 -0.1011 11 -0.0028 12 -0.0017 13 0.0065 14 -0.0523 15 0.0003 16 0.0058 17 -0.0079 18 0.0040 leads to the usual variation equation L (Hk1L EOk1) C10= 0, i where OH21l 0.0176 0.9913 -0.1100 -0.0324 -0.0152 -0.0070 -0.0423 0.0182 0.0369 -0.0061 (4) (S) It is to be noted that the basic functions, i(l, are ortho normal and, therefore, so are the 8i. The matrix ele ments are relatively easily determined and solutions of the secular determinants were obtained on the IBM 704 computer. The coefficients of the wave functions are given in Table III and in Table IV the total energy is listed. The CI and ICC calculations were both performed for the following distances: CH, 2.1240 a.u.; NH, 1.9614 a.u.; OH, 1.8342 a.u. with 1 a.u.= 0.S293 A. The Slater parameters chosen for the atomic orbitals used in the molecular calculations were as follows: H, 01=1.0; C, 01=S.7, ~=1.62S; N, 01=6.7, 02=1.95; 0,01=7.7,02=2.27S. When constructing the SCF equations we do not independently vary the space parts of the molecular orbitals assigned different spins. All equations are obtained, essentially, from a variation of the closed shell configuration 1(122(123(12 which interacts with the potential of the fixed 7r electrons. These SCF equa tions may be considered as modified by an average equivalence restriction.I5 As a result, Brillouin's theorem16 is of little use, as single substitution energy matrix elements are not necessarily zero or even small. It is also assumed that states arising from excitation of the 1(1 electrons do not interact strongly with the states listed here. For this reason no states arising from excitation of the 1(1 orbital are considered for the 8's and similarly for the \[I's with regard to the 1s orbital. 1. R. K. Nesbet, Proc. Roy. Soc. (London) A230, 312 (1955). 16 L. Brillouin, Actualites sci. et indo No. 159 (1934). In Table V we list the CI molecular energies, atomic energies calculated from Slater atomic orbitals, and the binding energies for the CI and single configuration cases. For the CH case we have also considered the interaction of the C2S2p2 and c2p4, 3 P, configurations of the carbon atom in computing the CI binding energy. It is seen that the corrections obtained here are of the same order as those found in other works.5 As noted before, the CI calculations do not appreciably improve the agreement with experiment over that obtained for a single configuration. III. ICC CORRECTION The energy matrix for any basis set \[I, for which it is asymptotically diagonal may be decomposed into two Hermitian matrices.6 H=!(MW+WM)+!(V+ vt), (6) where vt is the Hermitian conjugate. The W matrix has the observed atomic energies for diagonal elements. We also see that V=H-MW', (7) where W' is the asymptotic form of H for increasing internuclear separations. Therefore the corrected energy matrix in the 'It basis is given by H=H+!!M(W-W')+(W-W')M}, (8) where Hii= !\[IiH'ltidv M ii= ! \[I i\[lidv W= diag (WA, W B, ... ) W'=diag(W'A, W'B, ... ). (9) The \[I's were so chosen that the asymptotic form of H and W' is diagonal as prescribed above. In the process of molecule formation a change in the coupling of the orbitals is usually involved. Another and perhaps more important change from atom to molecule is that of the charge density of the orbital of each electron. Both effects lead to different correlation energies for the atoms and the molecule composed of these atoms. In Moffitt's treatment such changes were assumed unimportant. They are discussed more fully by Hurley7 and Arai.I7 In the earliest applications the separation to infinite distance was assumed to occur in a nonadiabatic fashion. The orbital parameters determined at the finite (usually equilibrium inter nuclear) distance were used to evaluate all the atom states in the W' matrix.I8 Hurley and RahmanI9 have 17 T. Arai, J. Chern. Phys. 26, 435, (1957). 18 W. Moffitt, Proc. Roy. Soc. (London) A2tO, 224 (1951). 19 A. Rahman, Physica 20,623 (1954); A. C. Hurley, Proc. Phys. Soc. (London) A68, 149 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:10CO RRELAT I ON CORRE CT I ON STU DY OF CH, NH, AN D OH 1293 TABLE IV. Total and binding energy.& E(CI) Es(ICC)b ER(ICC)c E(exp) B(ICC)d B(exp)e CH,2fl -38.203 -38.464 -38.489 2.97 3.64 CH, 'z- CH,2LJ. -38.205 -38.068 -38.449 -38.364 -38.384 NH,2z- OH,2fl -54.810 -75.086 -55.240 -75.764 -55.232 -75.753 -55.259 -75.778 3.27 3.93 4.01 4.58 & Atomic units used for all but binding energies which are in ev. 1 atomic unit (a.u.) =27.210 ev. b Slater shielding parameters in atomic aclculations. C Roothaan shielding parameters in atomic calculations. See reference 21. d ER(ICC) results used for NH and OH and Es(ICC) for CH. e Includes zero-point energy. Atomic energies from C. E. Moore, Nat!. Bur. Standards (U.S.) Circ. No. 467, (1949). Molecular energies from reference 20. shown that this procedure is in error. Hurley has emphasized that the W' elements should be the best atomic energies which can be obtained from the chosen basic set when all parameters are varied as the atom separations go to infinity. This is the prescription for the construction of W' in Eq. (7). It is assumed that the molecular correlation errors may be attributed to the atomic correlation errors with no allowance for orbital deformation on molecule formation. However, Hurley showed that the earlier approach to the con struction of the W' matrix also included errors due to non optimum parameter values which should not be considered with the correlation error. For our purposes the solution of the secular deter minant (10) was more practicable than that of the determinant which would be based on the energy matrix of Eq. (8). We have in the 8 basis. The transformation matrix is given by 'lr=8S, (12) where (13) We now see that (W -W/)O= S(W -W') S-1. (14) We must, however, justify the use of Eq. (12), for neither 8 nor'l' is complete; they do not include states arising from excitation of the 10" or 1s functions. S "hould then be an mX n matrix where m equals the total number of 8 states and n is the number of 'lr states chosen for the problem. We note, though, that in the construction of the correction matrix in the 8 basis the nXn part corresponding to the restricted 8 basis arises from the application of only the nXn part of S in Eq. (14). The correction matrix in the'lr basis is an nX n diagonal matrix and no consideration need be made for those 'lr functions which would complete the basis. The neglect of certain of the 8 states gives rise to two errors. st S is no longer equal to M and the energy matrix elements corresponding to functions arising from excitation of the 10" orbital are neglected. The latter error is very small and essentially is identical with the original one incurred by not considering excitation of the 1s orbitals. The correction terms will not change previous considerations of second-order perturbation theory that contributions of such excited states are negligible. The errors in M or concomitant errors in the coefficients of the states due to improper normalization will lead to small errors in those relations such as the populations which concern themselves with the charge distribution. However, direct computation of M shows that such errors amount to only several parts in the ten thousandth place. The calculated and observed atomic energies required for the correction matrix are given in Table VI. The eigenvalues and eigenvectors of the secular deter minants are given in Tables IV and VII, respectively. Secular determinants arising only from the more strongly interacting states were solved for the CH 2II, NH3~-, and OH2II cases. The results for the 81> 82, TABLE V. Hydride binding energy. CH NH OH CI Molecular energy (a.u.) -38.203 -58.810 -75.086 Slater Atomic energy (a.u.) -38.138b -54.765 -75.033 a Zero-point energy included. b CI energy included. c See reference 13. CI Binding energy (ev)& 1.7 1.2 1.4 Single configuration Binding energy (ev)a This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:101294 M. KRAUSS AND J. F. WEHNER 84,87,88, and 810 states of CH were essentially identical to that obtained by considering all 18 states. For NH and OH the difference was greater, but we see that a TABLE VI. Experimental and calculated atomic energies (a.u.). State Experimental- Theoretical (Slater) C S2p2 3p -37.855 -37.6383b 'D -37.809 -37.5697b Sp3 .50 -37.701 -37.5218 3Do -37.563 -37.2932 3pO -37.512 -37.2248 'DO -37.409 -37.0418 350 -37.373 -37.0190 'po -37.309 -36.9732 p' 3p -37.144c -36.7863b 'D -37.097c -36.7177b C+ S2p 2p -37.441 -37.2853 b Sp2 .p -37.245 -37.1125 2D -37.100 -36.8999 2p -36.937 -36.7027 p3 ·5 -36.794 -36.6346 2D -36.756 -36.5206 2p -36.672 -36.4001b C-S2p3 ·5 -37.895d -37.5609 2D -37.843 c -37.4691 2p -37.819 c -37.4393 b sp' .p -37.637c -37.2351 2D -37.551c -37.0603 2p -37.429c -36.8943 po 2p -37.234c -36.5867 b Theoretical Theoretical State Experimental- (Slater) (Roothaan) N S2p3 '5 2D sp· .p 2p N S2p3 3p Sp2 3D 35 p' 3p N-S2p' 3p H-h2 '5 H h 25 H o S2p' 3p 'D sp. 3p Ip o S2p3 2D 2p sp' 2D 2p p. 2p o s2p. 2p -54.612 -54.524 -54.210 -53.892 c -54.077 -53.657 -53.371 -53.079c -54.602e -0.52756 -0.5 o -75.109 -75.037 -74.534 -74.245 -74.487 -74.424 -73.853 -73.640 -73.163c -75.1641 -54.2651 -54.1417 -53.8025 -53.3515 -53.8209b -53.3628 -53.0146 -52.6987 b -54.0268 -0.47266 -0.5 o -74.5330 -74.4370 -73.8309 -73.4809 -74.0579 -74.0022b -73.3349 -73.0674 -72.5120b -74.2863 -54.2689 -54.1485 -53.8114 -53.3694 -53.8347 b -53.3727 -53.0411 -52.7117b -54.0431 -74.5404 -74.4467 -73.8442 -73.5007 -74.0778 -74.0191b -73.3509 -73.0945 -72.5316 b -74.3040 • All values not otherwise designated taken from C. E. Moore, see Table IV, reference d. b CI included. c Quadratic extrapolation. d C. R. Lagergren, thesis, University of Minnesota (1956). • Arbitrarily chosen 0.01 a.u. above '5 of N. See Branscomb and Smith, J. Chem. Phys. 25, 598 (1956). 1 Branscomb and Smith, Phys. Rev. 98, 1127 (1955). smaller number of trial functions can be chosen by a second-order calculation to give essentially the same results as would be obtained from the solution of the TABLE VII. ICC coeflicients. Ci CH2IT CH42:- CH2~ NH22:- OH2fl 1 0.9771 0.9757 0.9854 0.9838 0.0130 2 -0.0452 0.0796 0.0940 -0.0110 0.9813 3 -0.0049 0.1737 0.1074 -0.1020 -0.0417 4 0.0786 -0.0106 -0.0188 0.0187 -0.0081 5 -0.0395 0.0124 0.0115 -0.0505 0.1727 6 -0.0066 -0.0188 -0.0041 -0.0619 -0.0508 7 0.1339 -0.0865 0.0755 0.1075 -0.0089 8 0.0785 -0.0011 -0.0481 -0.0555 0.0117 9 -0.0057 0.0577 -0.0177 0.0465 10 -0.0985 -0.0171 11 -0.0195 12 0.0109 13 -0.0021 14 -0.0311 15 0.0002 16 0.0028 17 0.0000 18 0.0035 complete energy matrix. The energy computed by a second-order calculation also differed inappreciably from that obtained by solution of the complete secular determinant. However, such computational short-cuts would lose significant information which, we shall see later, resides in the energetically less important states. IV. DISCUSSIONS In Table VIII we list the ICC results that have been obtained for various hydrides. Although the results are very definite improvements over those values obtained by SCF of CI calculations, it is stilI not possible to choose between experimental values sepa rated by O.S ev or less. In NH and OH the binding energies computed from ER and Es, listed in Table IV, differ by about 0.3 ev. Assuming a smaller change for CH we then obtain approximately 2.7 ev for the CH binding energy which should be compared with the value reported by Hurley who used optimum two parameter Slater functions. The 2II_2~ separation for CH is found to be 2.7 ev for the ICC calculation. This agrees quite well with the TABLE VIII. ICC binding energies.- Molecule ICC Hurleyb ICC This work Experimentalc BH CH NH OH FH -In ev. 2.72 2.94 3.22 4.00 2.45d 5.59 b See reference 13. c See reference d of Table IV. d See reference 12. 2.97 3.27 3.93 3.15±0.4 3.64 4.01 4.58 6.08 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:10COR R E LA T ION COR R E C T ION STU D Y 0 F C H, N H, AND 0 H 1295 experimental20 value of 2.87 ev but, of course, the 2~ level is in error, absolutely by the same amount as the 2II level. However, the agreement for this level separa tion indicates that some faith may be put in the 2II -41;-separation of 0.4 ev. A CI calculation shows the 2~ state 3.7 ev above the 2II (Table IX). Hurley has obtained extremely accurate vertical excitation energies for all the excited states18 of CH, NH, and OH; the 2II_2~ separation for CH is reported as 2.83 ev. Although such results extend the feeling of con fidence in the method, any judgment valid for the ground state should apply here. We can only state that more accurate single configuration calculations and extensive parameter variation in the molecular ICC calculation are probably required before the method can be thoroughly checked. A discussion of the dipole results will offer some insight into the effects of the correlation correction (Table X). The values computed in the molecular orbital basis agree closely with those found by Hurley who used a restricted valence bond basis. This IS another indication that such a restricted basis is adequate for most purposes. TABLE IX. ICC excitation energies of CH.· CI ICC This work This work Hurleyb Experimentalc • In ev. o -0.0 5 3.7 b See reference 13. c See reference 20. o 0.60 2.83 3.36 3.97 o 2.87 3.27 3.94 The increase in the OH moment over that found for the SCF or CI cases is due mainly to a large admixture of 8.; that of NH arises mainly from the 81 state. In both cases these states give no contribution in the second order in the uncorrected energy matrix, but are the second most important states in the corrected matrix. The contributions to the correction matrix have been broken down with reference to the individual atom state corrections for the 1,1 and 1 ,7 elements of NH, the 2,2 and 2,5 elements of OH, and the 1,1 ele ment of CH. These are exhibited in Table XI. The importance of the negative ion states has already been noted by Hurley and the basis for this can be seen in their contributions to the corrections. It must be noted that the ICC procedure as applied here empha sizes these states, which are very poorly represented by 20 G. Herzberg, Molecular Spectra and Molecular Structure, I. Spectra of Diatomic Molecules (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1950), second edition. TABLE X. Dipole moment." Molecule BH CH NH OH FH SCF 0.99b 0.93c 0.90c 0.920 0.87b 2.655d CI 0.99b O.60b 0.53b 0.82b 0.96h 0.77 0.85 0.78 2.285e ICC 1.26b 1.61b 1.94b 2.16h 2.35b 1.54 1.63 2.05 2.759" Experimental 1.541 1.91- " Direction X-H+ in all cases and in units of Debyes. b See reference 13. c See reference 14. d See reference 5. o See reference 12. 1 R. P. Madden and W. S. Benedict, J. Chern. Phys. 23, 408 (1955) . -N. Hannay and C. Smythe, J. Am. Chern. Soc. 68, 171 (1946). the two or three21 parameter functions and therefore lead to large corrections. For CH the largest term in the correction arises from the 8 P ground state of the carbon atom. The ground state of the carbon negative ion, the 4S, plus a proton do not correlate with a 2II molecular state but apparently this leads to no anoma lous break in the trend of dipole moments or other similar quantities. Hurley has noted that another striking feature of the ICC calculation is the reduction of the antibonding of TABLE XI. (W -W')iko=~Si;(W -W') ;,;S;k-1 CH NH; NH OH ",(i,kH,i) 11 11 (1,7+7,1) 22 j", 2' , 2 ' 1 -0.0472 -0.1952 -0.1793 -0.0319 2 -0.0343 -0.0003 0.0037 -0.2794 3 -0.0005 -0.0541 0.0971 -0.0401 4 -0,0257 -0.0813 0,0295 -0.0098 5 -0.0025 -0.0060 -0.0304 -0.1968 6 -0.0034 -0.0230 -0.0074 -0.0192 7 -0,0462 -0.0446 0.0162 -0.0122 8 -0.0031 -0.0039 -0.0199 -0.0683 9 -0.0098 -0.0101 -0.0033 -0.0069 10 0 -0,0037 11 -0,0028 12 -0.0009 13 -0.0698 14 -0.0066 15 -0,0115 16 0 17 -0,0062 18 -0.0026 ]; -0.2731 -0.4185 -0.0938 -0.6685 ; OH (2,5+5,2) 2 0.0649 -0.3086 0,0740 0.0051 0.0729 -0.0466 -0,0117 0.0253 -0,0169 -0.0035 -0.1451 21 C. C. J. Roothaan, Technical Report, Laboratory of Molec ular Structure and Spectra, University of Chicago (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:101296 M. KRAUSS AND ]. F. WEHNER TABLE XII. Atomic and overlap populations. Calculation N(k) N(s) N(u) CH CI 2.001 1.840 1.104 1.063 ICC 2.001 1.827 1.255 1.069 NH CI 2.001 1.908 1.159 2.000 ICC 2.001 1.933 1.320 2.000 OH CI 2.000 1.919 1.203 2.999 ICC 2.000 1.938 1.443 2.999 the s orbital, as measured by the overlap population,22 from that found in the single configuration. The popu lations in the molecular orbital basis are in qualitative agreement with those obtained by Hurley with the most significant difference in the case of CH. The functions we employ allow for U-'>1r excitation and we note a small transfer of 0.069 electrons in the ICC case. However, the total popUlation N(p) of all p electrons is 2.324 which is in very good agreement with Hurley's valuel3 of 2.228. For completeness these values are given in Table XII. In NH the following valence states are neglected by Hurley, N(SZ2xy, i2P+14P)+H, N+(szxy, i3D+13S) +H-, and N+(Z2xy, 3P)+H-, whose occupation numbers, as defined by Hurley,23 are -0.0003, 0.0004, and -0.023, respectively. In OH the occupation numbers of the valence states 0+ (szx2y, PP+!2D) +Ho (SZ2x2y, !ID+t3P)+H, 0+ (S2z2y, !2D+!2P)+H-, and 0+(Z2x2y, 2P)+H- are 0.0001, -0.000 0, O.OO~, and -0.0002, respectively. These are to be compared with the weights of the negative ion states which are 0.443 and 0.405 for 0-and N-, respectively. We see that the neglect of these states is justified. Another important way of assessing the merits of the CI or ICC calculations is from a study of the Fermi contact term of the magnetic hyperfine interac tion.24 Unfortunately, an experimental determination of the contact term for any of the radicals studied here is lacking; the contact term has not been separated from the dipole-dipole terms.25 Matheson and Smaller26 find a resonance absorption doublet at g= 2.0 in ice irradiated at 4 OK which they attribute to the OH radical. The splitting of about 30 Mc, however, arises 22 For the ICC or CI cases the population analysis of R. S. Mulliken, J. Chern. Phys. 23, 1833 (1955) can be generalized by using the equations of P. O. Lowdin, Phys. Rev. 97, 1474 (1955) on the first-order density matrix i,n the case of CL See Hurley, reference 13. 23 A. C. Hurley, J. Chern. Phys. 28, 532 (1958). 24 Kotani, Mizurno, Kayama, and Ishiguro, J. Phys. Soc. Japan 12, 707 (1957). 'Ii Dousrnanis, Sanders, and Townes, Phys. Rev. 100, 1735 (1955). 26 M. S. Matheson and B. Smaller, J. Chern. Phys. 23, 521 (1955). N(h) n(kh) n(sh) n(uh) n 0.991 -0.010 -0.300 0.647 0.367 0.848 -0.008 -0.138 0.644 0.497 0.932 -0.008 -0.287 0.581 0.286 0.746 -0.007 -0.210 0.547 0.330 0.878 -0.006 -0.253 0.526 0.267 0.617 -0.004 -0.111 0.478 0.363 from both the contact term and the dipole-dipole term since at 4 OK rapid rotation of the radicals, by which the dipole-dipole term averages to zero,27 is not to be expected. Nevertheless, it is of interest to calculate the quantity 167r /3 (I-'Ol-'r/ I S.)\jt2 (0) . (15) 1-'0 is the Bohr magneton and I-'r and I are the nuclear magnetic moment and spin, respectively. 'lF2(0) is the spin density at the magnetic nucleus. Usually, \jt2(0) is assumed to be zero as this corresponds to the zero order eigenfunction with the free spin localized on the nonmagnetic nucleus in a 1r orbital. As has been noted by a number of authors, CI is one means of determining free spin at the hydrogen atom.2B For OH the values obtained for the contact constant are 1.0 and 15.0 Mc for the CI and ICC cases, respectively. Two points are of interest in connection with this computation. The most important terms in the calculation of the contact term are the cross terms between the most important or ground state and the higher states for which the matrix elements are nonzero. The higher states required for OH and CH have been those which could be neglected without serious error in the energy calculation. We see that complete calcula tions must be attempted if quantities other than the energy are of interest. For OH we also observed con siderable cancellation among the matrix elements which raises a serious question as to the accuracy of the OH results. For NH we note that the contact interac tion vanishes identically for the nine states considered here. The results for CH may be of more direct interest as this radical provides the simplest case of a phe nomenon widely exhibited in the aromatic radicals and ions, the delocalization of the free spin initially as sociated with a 1r orbital, and rough calculations for analogous CH fragments have been employed to elucidate the more complicated situation.26 We have 27 S. I. Weissman, J. Chern. Phys. 22, 1378 (1954). 28 H. M. McConnell and D. B. Chesnut, J. Chern. Phys. 28, 107 (1958). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.216.129.208 On: Mon, 24 Nov 2014 21:42:10( 0 R R E L A T ION COR R E C T ION STU D Y 0 F C H, N H, AND 0 H 1297 obtained -75.2 and -34.2 Mc for the CI and ICC cases, respectively, which agrees very well with the results of McConnell and Chesnut. However, we note that only approximately 50% of the splitting arises from the matrix elements connecting the ground state and singly excited states; the CH system cannot be treated quantitatively by only a first-order perturba tion treatment. We must also recognize that a first order treatment is very dependent upon the exact form chosen for the SCF equations by the application of equivalence restriction considerations as this strongly THE JOURNAL OF CHEMICAL PHYSICS affects the value of the single substitution energy matrix elements.15•16 It must be noted also that some 20 to 30% of the foregoing values arise from contribu tions of the unpaired oxygen or carbon fT electron density at the proton. ACKNOWLEDGMENTS The authors are indebted to Mr. P. J. Walsh and Dr. E. Haynesworth for their invaluable assistance in facilitating the computations and to Dr. A. C. Hurley for his very helpful comments and criticisms. VOLUME 29, NUMBER 6 DECEMBER, 1958 Theory of Orientational Effects and Related Phenomena in Dielectric Liquids A. PIEKARA AND S. KIELICH Institute of Physics, Polish Academy of Science, A. Mickiewicz University, Poznan, Poland (Received January 13, 1958) Formulas are developed for the molar constants of five non linear effects of molecular orientation in liquids; namely, the di electric saturation in electric, magnetic, or optical fields and the electric as well as the magnetic birefringence. No assumption is made concerning the nature of the intermolecular forces, and no special model of molecular interaction is introduced. If the molecules may be considered to possess axial symmetry, four correlation factors Rp , ReM, Rj{, and Rs can be derived. These factors appear in the formulas of the molar constants of the Cotton-Mouton and Kerr effects, and of the effect of dielectric saturation in an electric, magnetic, or optical field. The correlation factors have been calculated as functions of ()(pq), the angle between the axes of 1. INTRODUCTION ORIENTATIONAL nonlinear effects, or phenomena of molecular orientation, are phenomena in which the external electric or magnetic field produces align ment of anisotropic molecules causing a change in electric, magnetic, optical, or other properties of the medium. There are five nonlinear phenomena of molecular orientation which the authors intend to consider in the present paper, namely: (1) dielectric saturation in the electric field or the effect of the electric field on the electric permittivity of the medium, (2) dielectric saturation in the magnetic field, (3) dielectric saturation in the optical field or the effect of an intense light beam on the electric permittivity, (4) electric birefringence or the electro-optical Kerr effect, (5) magnetic birefringence or the Cotton-Mouton effect. In dense gases and liquids these phenomena are influenced by intermolecular forces, among which directional forces causing short-range molecular orien tation are of importance. Because of intermolecular forces, such phenomena as the scattering of light or the lowering of the freezing point of solutions are intimately related with the orientational effects. symmetry of the pth and qth molecules, in the absence of a bias ing field. The theory makes it possible to predict the value of the magnetodielectric saturation effect in diamagnetic liquids, as well as the photodielectric saturation, relating these phenomena to magnetic or electric birefringence. Moreover, the meaning of this theory consists in the fact that it gives a quantitative explanation of the inverse saturation effect appearing in some polar liquids and increasing their dielectric constant. Satisfactory results have been obtained by applying this theory to such phenomena as light scattering in liquids or the lowering of the freezing point, in which the orientationally dependent intermolecular forces play an im portant part. One of us (A. P.) had previously given a theory of molecular orientational effects in polar liquids and their solutions in nonpolar solvents.1-7 That theory, however, which was based on the assumption that the directional forces due to the effect of the momentarily nearest molecule playa greater part than other inter molecular forces, had but a restricted field of applica tion. Nevertheless, very good agreement with experi mental results had been obtained for nitrobenzene and its solutions in nonpolar solvents,6.7 and it had been supported by cryoscopic measurements. s In particular, the theory yielded a quantitative explanation of the rather peculiar phenomenon of the positive or inverse saturation effect consisting in an increase in the permittivity of polar liquids, when placed in an electric 1 A. Piekara, Acta Phys. Poloruca 4, 53, 73, 163 (1935). 2 A. Piekara, Acta Phys. Poloruca 6,130,150 (1937); Physik. Z. 38, 671 (1937). ~ A. Piekara, Z. Physik 108, 395 (1938). 4A. Piekara, Compt. rend. 204, 1106 (1937); 208,990, 1150 (1939). • A. Piekara, Proc. Roy. Soc. (London) Al72, 360 (1939). 6 A. Piekara, Nature 159, 337 (1947). 7 A. Piekara, Acta Phys. Poloruca 10, 37, 107 (1950). 8 A. Piekara, Acta Phys. Poloruca 11, 99 (1951). This article is copyrighted as indicated in the article. 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1.1735065.pdf

``OpenCircuit'' Voltages in the Plasma Thermocouple H. W. Lewis and J. R. Reitz Citation: J. Appl. Phys. 30, 1838 (1959); doi: 10.1063/1.1735065 View online: http://dx.doi.org/10.1063/1.1735065 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v30/i11 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 16 Jul 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 30. NUMBER 11 NOVEMBER. 1959 Letters to the Editor "Open-Circuit" Voltages in the Plasma Thermocouple* H. w. LEwIst AND J. R. REITz~ Los Alamos Scientific Laboratory. Los Alamos, New .I'vI exico (Received August 3. 1959) INa previous note,I we have discussed the characteristics of the plasma thermocouple from a thermodynamic point of view. We here rederive and extend our results on the open-circuit voltage of the cell, so as to clarify the mechanism involved, and to take into account the circumstance that few experimental measurements are made at zero current. We begin by considering the equilibrium, in vacuum, between a hot thermionic cathode, and the associated electron gas. This equilibrium is determined by the equality between the Richardson current flowing out of the cathode, jsat = jout = A T"e-W/kT, and the current into the cathode from the electron gas jin= lnev= (nkTe)/(21rmkT)', (1) (2) where n is the density of the electron gas. In equilibrium, (1) and (2) are equal, which determines the pressure p, of the electron gas. The result is p,= (jout/e) (21rmkT)! or, roughly, between 20000K and 3000oK, pc(mm Hg)"" (l/5000)j,at(amp/cm'). (3) (4) This result is, of course, equally well obtained from the equality of the electrochemical potential inside and outside the cathode. For plasma densities leading to electron pressures greater than that given by (3) or (4), the electron gas is neutralized, so that, in equilibrium, a barrier B is erected against jiD, over a distance of a Debye length. Then the pressure becomes an independent parameter, and B is determined by equating jin= (nkT)/(21rmkT)!e-B/kT (5) to jout, still given by (1). In the regime governed by (5), consider a cathode at tempera ture TI, and an anode at temperature T" in contact with the plasma (see Fig. 1), at constant pressure. Suppose further that FERMI LEVEL I } ~ " }. j \. PLASMA SHEATHS) CATHODE ANOOE FIG. 1. Potential diagram for plasma diode during passage of a trickle current. ! ,is the chemical potential of electrons in the plasma i =kT In[h'n/ 2 (2 .. mkT).J. there is no temperature drop across the Debye sheath near'either cathode and anode. Then Bc= We +kTI InLTI2(2:mkTl)l] (6) and similarly for the anode. The emf of the cell is given by E=Ba-Bc+W c-Wa= -kTllnLTI2(2~mkTl)'] +kTzlnLT2'(2;mkT2)J (7) With the correct value of A, this is exactly the formula given in the previous note. It is clear that it is only correct for a genuine open circuit, since the current balance at the anode is important, and the emission current of the anode is generally small. § Suppose now that we draw a current of density j through the cell, large compared to the anode emission current, as we will normally do with even a good high-resistance voltmeter. We shall suppose it small compared to the emission of the cathode, since we do not propose here to discuss the full characteristic of the cell. Then Ba is determined by setting (5) equal to j: Ba(j) =kT,lnC(21r:Ze kT2)J (8) so that E(j)~Ba(j)-Bc(O)+Wc-Wa = kT'lnC(h~ekT,)I] -kT1lnLTl'(2:mkTl)l] -Wo. (9) This formula represents, in effect, the fact that the cathode and anode are acting as junction rectifiers, with the cathode operating in the direction of easy current flow, and the anode reversed. Consider now a special case, that for which Tl = T2• By this we mean that the electron gas in the vicinity of the anode is at the cathode temperature, not that the cathode and anode are at the same temperature. There are reasons for believing that this is a good approximation in some cases, when the current drawn is not too large (or too small); other cases are outside the intended scope of this note. Then (9) becomes (10) It is important to note that the j appearing in this formula is the current density at the anode, which may differ from that at the cathode in, for example, a cylindrical device. This can now be written in an interesting form by bringing the work function of the anode into the logarithm, and observing that A T"e-Wa/kT is the saturation current the anode would have, if its temperature were T. Call this jo(T). Then E(j; T1=T2)=kTln[jo(T)/jJ (11) which is our final result for this case, which we now discuss. It is valid only for a uniform temperature of the plasma, and for a trickle current j large compared to the actual emission current of the anode. First, it is clear from (10) that, for a given trickle current j and temperature T, the voltage depends upon the anode work function Wo. This explains in a simple way the increase in "open circuit" voltage observed by Pidd et al.,' when 10-6 mm of cesium was released into their cell at constant temperature. The cesium coated the anode, reducing its work function by a volt or two, and the output voltage increased accordingly. It therefore pays to have a low work function anode, for a high-voltage output. At the higher cesium pressures we believe that coating of the insulators, and another geometric effect, to be discussed below, have affected the results. The second point to be noticed is that, as mentioned before, the current density j appearing in (11) is the current density at the anode. This means that, for a given trickle current, a large area anode increases the "open-circuit" voltage. For example, a cylindrical geometry, with the anode on the outside, will give an increased voltage t:.E given by t:.E=kT In (R/r) (12) over the corresponding plane case, where Rand r are the outer and inner radii of the plasma volume. This is simply a consequence 1838 Downloaded 16 Jul 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsLETTERS TO THE EDITOR 1839 of the fact that a larger anode area requires a biggcr barrier voltage to limit the total current. \Ve want also to point out that we must be circumspect about quantitative analysis of the voltages in geometries such as those used by Pidd et al.,' in which an emitter button is welded to the center of a plate. In such a geometry, very large leakage currents flow from the button to the surrounding colel plate, through the plasma, even when no current is drawn from the anode. Indeed, the button is probably always loaded nearly to saturation by the surrounding short~circuit load. In such a case, one can believe the effects that depend upon the anode, but not, in detail, those depending on the current balance at the cathode. It is much easier to analyze quantitatively either cylindrical or more nearly plane geometries. * \\'ork perfornwd under the atl~pic('s of the {T. S. Atomic Encrgy Commission. t Permanent addres,",: UniVf'fsity of \\,iscoll::;in, Madbon, \Viscon:.;in. t Permanent address: Case Institute of Technology, Clevelanrl, Ohio. I II. W. Lewio amlJ. R. Reitz. J .. -\ppl. I'hys. 30,1439 (1959). § There is, in addition, a small arlditional voltage in the cell, which has been determined by SpitZf'f and by LanrlsllOff to be 0.7/;:('1'1 ~T2). \Ve have ignored this term, which amounts to a COli pIe of tenths of a volt in the case of interest to ll~. 2 Pidd,'.Gro\·rr, Salmi, Roehling, and Erickson, J. Appl. Phy.<. 30, IS7S (1959). • On Graphite Whiskers* P.\PL J. BJ{Y.\~'[, T. HRITE DA;-;n:r., A-"'l"D FRED R. ROLl.I:--.lS, JR. .~lid'W('st Researrh Institute, Kansas City 10, lfiss()llri (Rccciwo July 1,1959) THE structure of graphite whiskers is being studied in connection with some experiments on the mechanism of friction. Whiskers were selected for these experiments because it was thought that they were single crystals. The whiskers grow inside a solid boule which is deposited by a high-pressure dc carbon arc. I-a '\licroscopic examination of (he whiskers showed damage which may have occured when the houle was broken apart and the whiskers pulled out. A technique for removing them from the boule without causing damage was developed.4" \Vhiskers have been removed from a boule by heating to 525°C for 2 min in a stream of oxygen moving about 1 cm/sec. The method also works when the boule is heated in still air although more time is required. Selective oxidation takes place for two reasons: first, the whiskers have a more stable crystal structure than the matrix; second, they are preferentially cooled because of their large surface to volume ratio. A practical advantage of FI(;. 1. Graphite whiskf'r siJ()\ving se\'f'ral cn:~tal segl1leIlt~. ScalP marker, lOp. thermal e,traction is that large numbers of whiskers can be caught on a screen placed downstream from the boule. Figure 1, taken through an oil immersion objective, shows a graphite whisker which was removed from a boule by the method described in (he foregoing. The whisker shows several crvstal segments each having a polygonal cross section. - We wish to express our appreciation to Dr. Roger Bacon of the ;\;'ational Carbon Company for providing us with several houles. * Thiti research \vati supporU'd by the U. S. Air Force under Contract No. A.F33(616)-6277, monitored by the Materials Laboratory, \Vright Air Df'w'lopment CE'nter, \Vright-Patterson Air Force Hase, Ohio. 1 R. Bacon and J. C. Bowman, BIlII. Am. Phys. Soc. Ser. I 1,2, 131 (1 ()S7). 2 R. Bacon, Bull. Am. Phys. Soc. Ser. II, 3, lOS (1958). ;) R. Bacon in Growth (lild Perfection 0/ Crystals (John \Yiley & Sons, InC'., Kew York, 1958). 4 P. J. Bryant, Bull. Am. Phys. Soc. Ser. II, 4,265 (1959). " T. B. DaniC'1 in ProaetiinRs of the First Symposium on ,r..,'urJarr Effects on ,f..;pace Craft ,\4a/erials (John \Viley & Sons, Inc., .:'\P\\' York) (to b(' Pllbli~hed late iu 1959). Theory of Coehsive Peeling of Adhesive Joints FRA:-.IKLl~ S. C. CHA~C Jlystih Adhesive Proriucls, Inc., Sorthjuld, Illinois (Rf'cE'ivE'd 11arcb 9, 1(59) WHEN an adhesive joint is subjected to peeling, the adhesive film breaks into tiny strings. In the first phase of peeling the number of strings increases and the strings are stretched longer and longer. As a result the peeling force required increases accord ingly. When the first layer of strings reaches the maximum elongation allowed by the circumstance, a "steady state" is reached: the peeling force is a maximum, the first layer of strings starts to break, and a new layer of strings comes into being at the other end of the strained rcgion. When the first layer of strings is broken, the peeling force drops down, but climbs up again to the maximum as the peeling goes on. The cycle repeats itself again and again until the joint is completely peeled. The maximum peeling force just before the breaking of strings will be dealt where in this report. The treatment in this rcport is restricted to the case of 90° cohesive peeling, in which no interfacial break is allowed and the joint or the "glue line" remains perpendicular to the line formed by the free ends of the adherenc1s as shown in Fig. L Sincc the adhesive film is assumed to be homogeneous, the strained region is symmetrical to a line bisecting the joint. Thus only one~half of the joint needs to be considered. From the theory of elasticity one has dO/dL=Jf / IiI, where 0, M, l~, anel] are the bending angle, bending moment, Young's ,modulus, and moment of inertia of the adhcrend, respectively, anu dL is an infinitesimal length of the adherend. It follows that (d'o/dV) = (1/ HI) (d2M /dD) = (1/ FJ) (dV /dL) , (I) where dV is the load on dL. When the number of strings per unit area of adherend is designated by m, the contact area bet ween the string and the adherend is l/m. The stress S is referred to the projection of this area on the x axis, which is cosO/m. The component of the strctch~ ing force perpendicular to the adhercnd is S cos'fl / m and the number of strings on the area W dL is mW dL, W being the width of the joint. So the bending force on this area is d l' = -WS cos2lJdL. Putting this quantity into] and expressing 0 in terms of the elongation y by use of the relation dy/dL=sinO, we have Y4 (1-y,2)2+3YlY2y:;(1- y,2) +yi+2 yI2y,' WS + JU (l-y,2)'/'=O, (II) where Yi is the ith derivative of y with respect to L. To compute the stress S, a 3-element mechanical model,l as shown in Fig. 2, is used to represent the behavior of the adhesive, which consists of an elastic spring of Young's modulus l~, con~ nected in series with a parallel combination of a viscous flow Downloaded 16 Jul 2013 to 18.7.29.240. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.1744665.pdf

1,2 versus 1,4Addition to 1,3Butadiene by a Simple LCAOMO Method Frank L. Pilar Citation: The Journal of Chemical Physics 29, 1119 (1958); doi: 10.1063/1.1744665 View online: http://dx.doi.org/10.1063/1.1744665 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/29/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase Diagrams and Thermodynamic Properties of Binary Organic Systems Based on 1,2, 1,3, 1,4 Diaminobenzene or Benzidine J. Phys. Chem. Ref. Data 23, 295 (1994); 10.1063/1.555959 Radiative and radiationless transition phenomena in 1,4, 1,3, and 1,2diazanaphthalene vapors J. Chem. Phys. 61, 3895 (1974); 10.1063/1.1681682 Photoionization of 1,3Butadiene, 1,2Butadiene, Allene, and Propyne J. Chem. Phys. 49, 2659 (1968); 10.1063/1.1670466 Errata: 1,2 versus 1,4Addition to 1,3 Butadiene by a Simple LCAOMO Method J. Chem. Phys. 30, 591 (1959); 10.1063/1.1729999 Further Considerations of a Previous LCAOMO Study of the AllylicType Transition State in 1,3Butadiene J. Chem. Phys. 30, 375 (1959); 10.1063/1.1729959 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 13:06:25THE JOURNAL OF CHEMICAL PHYSICS VOLUME 29, NUMBER 5 NOVEMBER, 1958 1,2-versus 1 ,4-Addition to 1 ,3-Butadiene by a Simple LCAO-MO Method FRANK L. PILAR Department of Chemistry, University of New Hampshire, Durham, New Hampshire (Received June 9, 1958) The addition of various reagents such as halogens and hydrogen halides to 1,3-butadiene by an ionic mechanism is ideally visualized as proceeding in two steps, viz., the electrophilic attack of the 1-or 2-position followed by nucleophilic attack to form either the 1,2-or 1, 4-addition product. A simple LCAO-MO ap proach was applied to a proposed transition state arising from the first step of the reaction. Atom stabilization energies and frontier electron densities were calculated for the various positions of attack involved during the final step of the addition. The effect of the electronegativity of the group adding during the first step relative to the electronegativity of the carbon atoms in the conjugated system was used to alter certain of the LCAO-MO parameters. Agreement of the results with experiment is discussed briefly. INTRODUCTION WHEN a reagent RA (such as Br2 and Hel) adds to 1 ,3-butadiene by an ionic mechanism, it is customary to depict the addition as occurring in two steps: (\) R+ +CH~-CH-CH ~CH,----> [ ~~/ CH -CH~CH, r (2) A-+ [~H2/ CH-CH =CH2l+~~H2-CH =CH-~H2 R R A Wheland1 has shown that both 1,2-and 1, 4-addition are compatible with molecular orbital theory but that "We are unable to predict which of the two possible products will predominate." This paper will suggest an approach whereby it appears possible to predict the predominant product in certain simple cases. The exact nature of the bracketed transition-state carbonium ion is not known although the author's depiction as a cyclic complex has found some favor.2 However, it will be assumed that at some point prior to attack by the nucleophilic species A-the configura tion of the transition state may be represented by [R-C-C-C-C]+ 0' [C-J-C-C r I II where the remaining two pi electrons are confined to three 2jnr orbitals of carbon (the hydrogen atoms have been omitted for clarity). The first step in the problem is to choose the more likely transition state by some useful criterion of 1 G. W. Wheland, Resonance in Organic Chemistry (John Wiley and Sons, Inc., New York, 1955), p. 459. 2 K. Mislow and H. M. Hellman, J. Am. Chern. Soc. 73, 244 (1951). and chemical reactivity, i.e., is the R-atom more stable in the I-position (transition state I) or in the 2-position (transition state II)? This paper will suggest how this choice may be made and, furthermore, how the chosen transition state may be treated to predict the position attacked during the second step of the addition. OUTLINE OF CALCULATIONS I The molecular orbitals (MO's) for the 1, 3-butadiene molecule are formed by (1) where the cp's are 2P7r atomic orbitals of the rth carbon atom and the subscript j refers to the particular energy levels obtained by solution of the secular equations (2) r.' for r values of the pi electron energy E. The following assumptions are made: Hr.(1 r-s 1 = 1) =(3; Hr.(1 r-s 1 >1)=0 the standard C-C resonance integral; Hr.(r=s) =a; 1119 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 13:06:251120 FRANK L. PILAR TABLE 1. Atom stabilization energies and frontier electron densities of 1,3-butadiene.a r=1 r=2 0.6708 1.2526 a Atom numbering scheme RC1C2CaC •. b In units of {3. o In units of electron charge e. 0.725 0.550 the standard carbon coulombic integral; i.e., all overlap integrals are neglected (or.= Kronecker delta). Solution of the secular Eqs. (2) using the above assumptions and the normalization condition, (3) leads to the well-known MO's (lowest to highest) and corresponding values of E (in units of fJ) : 1/11 = 0.3718(</>1+</>4) +0.6015 (4)2+<pa) 1/12= 0.6015 (</>1-<P4) +0.3718 (4)2-<t>s) 1/Ia= 0.6015(</>1+</>4) -0.3718 (c1>2+<t>s) 1/14= 0.3718 (<pl-<P4) -0.6015 (c1>2-<Pa) E1= 1.6180 E2=0.6180 Ea= -0.6180 E4= -1.6180. From these MO's and pi electron energies it is possible to calculate quantities useful for predicting the relative reactivities of the various carbon atoms in the 1 ,3-butadiene molecule. For the purposes of this paper only two of these quantities will be considered, viz., the atom stabilization energy of the rth carbon atom defined by Nakajima3 as oee. 1:,:1-'= l:.2Cr/Ej±crle" i (4) where the superscript (+) refers to electrophilic attack and (-) to nucleophilic attack, and the frontier elec tron density of Fukui et al.4 (5) In both of the latter equations the subscript f refers to the frontier orbital (the lowest unoccupied orbital for electrophilic attack and the highest occupied orbital for nucleophilic attack). For alternant hydrocarbons, Eq. (4) leads to Er-=€r+. The atom stabilization energy is regarded as the energy required to promote the pi electrons of atom r from the conjugated state to the valence state (er= 0) , and it is apparent from Table I that transition state I would be favored since the lowest atom stabilization energy arises when the R-atom is attached to carbon atom 1. It will thus be assumed that it is this particular 3 T. Nakajima, J. Chern. Phys. 23, 587 (1955). 4 Fukui, Yonezawa, and Shlngu, J. Chern. Phys. 20,722 (1952). transition state which will be attacked by the nucleo philic species A-. Note that the frontier electron densi ties lead to the same choice of transition state since attack by R might be expected to occur at the position of highest frontier electron density. The next step of the problem is to utilize the criteria (4) and (5) to predict the most reactive position of transition state I with respect to nucleophilic attack byA-. OUTLINE OF CALCULATIONS n Transition state I consists of two pi electrons re stricted to the 2P7r orbitals of carbon atoms 2, 3, and 4. This transition state is conveniently represented by [Q-C-CJ+, where Q=R-C 1-C2 and may be regarded as a pseudo heteroatom whose electronic characteristics have been modified by the presence of the R-atom in the 1-posi tion. The atom numbering system Q2CaC4 will be used for this transition state. The presence of the R-atom will have an effect upon both the resonance and Coulombic integrals of carbon atom 2. If R is more electronegative than carbon, the electron density of the 2-atom will be enhanced at the expense of the charge density of the 3-atom. Since Hrs is related to the pi electron densities of the atoms r and s, i.e., it is a property of the r-s bond, then H23> H34• Similarly, the Coulombic integral is a rough measure of the electronegativity of an atom and is directly proportional to the charge density on that atom. Thus, if the R-atom is more electronegative than carbon, then H22> Ha3• In the case that R is less electro negative than carbon, the converse effect occurs. The MO's for the transition state I are formed by (1) as before but now c1>2 refers to the modified 2P7r orbital of the second carbon atom. The parameters in the secular Eqs. (2) are now redefined in the light of the preceding paragraph by H23=kfJ, where k> 1 if R is more electronegative than carbon and k< 1 if R is less electronegative than carbon, Sr.= 0", where c>O if k> 1 and c<O if k< 1, It is assumed that the R-atom has no effect upon the Coulombic integrals of the carbon atoms 3 and 4, al though this approximation cannot be strictly true. H34 in this case cannot be the same resonance integral as used previously since the values of H23 and H34 must necessarily be mutually dependent. However, this will not affect the relative values of the final calculated pi electron energies. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 13:06:251 , 2 - V E R SUS 1, 4 -ADD I T ION TO 1, 3 -BUT A DIE N E; THE 0 R Y 1121 The secular determinant to be solved is k -e 1 =0, (6) where a= 0 is used as the zero for expressing the pi electron energies. If the value of (3 is arbitrarily set equal to unity (the e values are then expressed in units of (3) then expansion of (6) yields (7) This expression could be solved analytically for the three values of e, but it is convenient to resort to some simplification at this point. If c is not much different from zero, which is not too unreasonable in view of the fact that the inductive effect of R is transmitted through carbon atom 1, then all terms containing c may be dropped. The resulting expression readily yields the solutions and leads to the normalized MO's (lowest to highest), '1/;1 = (2k2+ 2) -l[ k¢2+ (k2+ 1) IcP3+cP4], '1/;2= (k2+1)-l[¢2+kcP4], '1/;3= (2k2+2)-~[k¢2- (k2+1)}cP3+cP4]. It must be borne in mind that if the assumption of negligible c is invalid, then Eq. (7) must be solved analytically. Although this task is relatively easy in principle, the resulting expressions for the MO'5, er's and qr/S become rather intractable. Calculation of the atom stabilization energies Er yields Examination of the atom stabilization energies shows that for (1 ,4-addition predominates), (1, 2-addition predominates) , (1 ,2-and 1, 4-addi tion equal) . It is interesting to note that in all cases carbon atom 3 is the least probable point of attack as would most certainly be expected. The frontier electron densities for the lowest un occupied MO '1/;2 are Thus if k= 1 q42= q22> q32 (1,2-and 1 ,4-addition are equally probable) . If the most probable point of attack is the position of highest frontier electron density, then this gives results in agreement with the atom stabilization energies. In the light of present-day knowledge it appears impossible to decide ab initio whether atom stabilization energies or frontier electron densities are better criteria of chemical reactivity. The final test of such criteria will have to lie in the agreement of theory and experiment. DISCUSSION Coulson and Longuet-Higgins5 have shown that the use of self-polarizabilities obtained by means of pertur bation theory appear to be better criteria of chemical reactivity than are charge densities. Nakajima3 has shown that in many cases predictions of chemical reactivity by use of atom stabilization energies agree well with predictions obtained by the use of self polarizabilities. Greenwood6 has shown that the predictions based on the frontier electron densities do not always parallel the predictions of the Coulson and Longuet-Higgins perturbation treatment. Since the results of the calculations show agreement between atom stabilization energies and frontier elec tron densities, it can be shown that the treatment in this paper leads to results not in conflict with the experi mental data reported in the literature. For Br2 addition k> 1 and the atom stabilization energy predicts a predominance of l,4-product. The experimental data on this reaction are not completely unambiguous but appear to support the result pre dicted.! Mislow and Hellman2 have shown that the analogous addition of Cl2 results in the 1, 4-product. For hydrogen halide addition k< 1 (since the initial attack involves addition of a proton) and addition should be mainly 1,2-, a prediction in accord with 5 C. A. Coulson and H. C. Longuet-Higgins, Proc. Roy. Soc. (London) A191, 39 (1947); A192, 16 (1947). 6 H. H. Greenwood, J. Am. Chern. Soc. 77, 2055 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 13:06:251122 FRANK L. PILAR experiment.7 In the addition of HCI to isoprene the main product is reported to be 1,4-,8 however, Ultee9 states that the main addition product is the 1, 2-pro duct which then rearranges in the presence of HCI to the more thermodynamically stable 1, 4-product. On the basis of the above few examples it appears as if both the atom stabilization energy and the frontier electron density are valid criteria of chemical reactivity. A more detailed and rigorous examination of this prob lem is necessary before it is possible to draw definite conclusions concerning this matter. At present we are using the facilities of the MIT computation center for 1 W. J. Jones and H. W. T. Chorley, J. Chern. Soc. 832 (1945). 8 M. S. Kharasch et at., J. Org. Chern. 2, 489 (1937). 9 A. J. Ultee, J. Chern. Soc. (London) 530 (1948). THE JOURNAL OF CHEMICAL PHYSICS programing a more detailed treatment of this problem and will publish the results when they are obtained. * ACKNOWLEDGMENTS The writer expresses his gratitude to a former teacher, Professor Hans H. Jaffe of the University of Cincinnati, for inspiring interest in this problem and to Mr. M. Donald Jordan, Jr., for helpful discussion. * Note added in proof.-Numerical solutions of Eq. (7) show that the inclusion of c increases the highest qr{ and 'r-and decreases the lowest qr' and 'r-, thereby emphasizing the predicted direction of attack. It should also be mentioned that consideration of total electron density leads to the same prediction as given by the qrj and .,-. This is readily verified from the Cri, keeping in mind that only >/;1 is occupied and that the most reactive position is the most positively charged. VOLUME 29, NUMBER 5 NOVEMBER, 1958 On the Statistical Mechanical Theory of Solutions* R. M. MAZot Enrico Fermi Institute for Nuclear Studies, University of Chicago, Chicago, Illinois (Received July 1, 1958) A perturbation expansion of the excess free energy of a binary mixture based on the theory of composition fluctuations in an open system is presented. The theory is applied to classical mixtures and quantum mechan ical isotope mixtures, and comparisons with conformal solution theory and Chester's theory of isotope mix tures are made. The advantage of the present theory is that it contains in summed form, in each order of perturbation, terms which in the direct expansion of the partition function would be considered to be higher order. INTRODUCTION IN recent years there has been a great interest in what may be called perturbation theories of solu tions.t-8 These are theories which attempt to express the properties of a solution in terms of those of some suitable pure substance plus correction terms. The theories are usually couched in terms of a Taylor's series development of the partition function with respect to some appropriate parameters. * This research supported by the U. S. Atomic Energy Com mission. t Present address: Gates and Crellin Laboratories, California Institute of Technology, Pasadena, California. I H. C. Longuet-Higgins, Proc. Roy. Soc. (London) A205, 247 (1951). 2 W. B. Brown and H. C. Longuet-Higgins, Proc. Roy. Soc. (London) A209, 416 (1951). a W. B. Brown, Proc. Roy. Soc. (London) A240, 561 (1957). • W. B. Brown, Phil. Trans. Roy. Soc. (London) A250, 175 (1957). & R. L. Scott, J. Chern. Phys. 25, 193 (1956). 6 Prigogine, Bellemans, and Englert-Chwoles, J. Chern. Phys. 24, 518 (1956). 7 Salsburg, Wojtowicz, and Kirkwood, J. Chern. Phys. 26,1533 (1957). 8 Wojtowicz, Salsburg, and Kirkwood, J. Chern. Phys. 27, 505 (1957). There are also several exact theories of solutions,9-11 which have, until now, been mainly useful for providing power series expansions in the concentrations of the various solutes. The object of this paper is to demon strate that one of these exact theories, that of Kirk wood and Buff/o is a particularly convenient starting point for a perturbation expansion which has the ad vantage of including, in a given order of perturbation, some of the terms which would be considered as higher order in the conventional treatments. In this paper we develop the theory for both classical solutions and quantum mechanical isotope mixtures, to the lowest nonvanishing order of approximation in each case. This is the first order in the classical case, and second order for the isotope mixtures. Part I contains the general theory, and Parts II and III the specializations to the two cases considered. We restrict ourselves to binary mixtures, and assume that all internal partition functions are independent of the state of aggregation. 9 W. G. McMillan and J. E. Mayer, J. Chern. Phys. 13, 276 (1945). 10 J. G. Kirkwood and F. P. Buff, J. Chem. Phys. 19, 774 (1951). 11 T. Hill, J. Am. Chem. Soc. 74, 4885 (1957). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.120.242.61 On: Sun, 23 Nov 2014 13:06:25

1.1723040.pdf

Theoretical Surface Conductivity Changes and Space Charge in Germanium and Silicon V. O. Mowery Citation: Journal of Applied Physics 29, 1753 (1958); doi: 10.1063/1.1723040 View online: http://dx.doi.org/10.1063/1.1723040 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/29/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Characteristics of surface states and charge neutrality level in Ge Appl. Phys. Lett. 95, 252101 (2009); 10.1063/1.3270529 Modeling of negatively charged states at the Ge surface and interfaces Appl. Phys. Lett. 94, 012114 (2009); 10.1063/1.3068497 Electrical conduction of Ge nanodot arrays formed on an oxidized Si surface Appl. Phys. Lett. 91, 123104 (2007); 10.1063/1.2784181 Amorphization/templated recrystallization method for changing the orientation of single-crystal silicon: An alternative approach to hybrid orientation substrates Appl. Phys. Lett. 87, 221911 (2005); 10.1063/1.2138795 Theoretical phonon thermal conductivity of Si/Ge superlattice nanowires J. Appl. Phys. 95, 682 (2004); 10.1063/1.1631734 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 03:48:34JOURNAL OF APPLIED PHYSICS VOLUME 29. NUMBER 12 DECEMBER. 1958 Theoretical Surface Conductivity Changes and Space Charge in Germanium and Silicon V. O. MOWERY Bell Telephone Laboratories, Whippany, New Jersey (Received May 21, 1958) Graphs are presented displaying predicted surface conductivity changes as a function of resistivity for various values of surface potential, and space charge as a function of resistivity for the same range of surface potential, in germanium and silicon. The curves display the trends of the two properties over wide resistivity ranges and are particularly convenient for obtaining theoretical field effect curves. A brief discussion of values for the electrical properties of germanium and silicon is included. INTRODUCTION SINCE charge neutrality is maintained deep within the bulk of semiconductor, any electric field appearing at the surface must be terminated in an in duced charge. Part of this charge can be distributed throughout a space charge region near the surface; in addition there can be bound charge localized at the surface in surface states having energies within the energy gap. These surface states could be due to the termination of the crystal lattice as predicted by Tamm1 and further examined by Shockley2 and others, or they could. result from surface imperfections or absorbed impurity atoms. Because of these surface states there can also be a charge double-layer with zero external field. Under these conditions the potential at the sur face can differ from that within the bulk beyond the space charge layer, and a difference in conductivity can result from an excess or deficiency of mobile carriers appearing in the space charge layer. This description of a semiconductor surface has been useful in theoretical work, such as explanation of contact potential differ ence,3 dependence of surface potential on chemical environment,4 channel conductance,6 field induced con ductivity changes,6 etc., and it is becoming extremely useful in studying surface behavior in semiconductor device technology. Using results of the analyses of Bardeen,3 Kingston and Neustadter,7 and Garrett and Brattain8 for prop erties of a semiconductor surface, and assuming reason able estimates for the necessary constants of germanium and silicon, condensed charts have been obtained for the amount of space charge as a function of bulk resis tivity for various values of surface potential, and change in surface conductivity as a function of bulk resistivity for various values of surface potential. The data in this 11. Tamm, Physik Z. Sowjetunion 1, 733 (1932). 2 W. Shockley, Phys. Rev. 56, 317 (1939). . 3 J. Bardeen, Phys. Rev. 71, 717 (1947). 'W. H. Brattain and J. Bardeen, Bell System Tech. J. 32, 1 (1953). 5 W. L. Brown, Phys. Rev. 91, 518 (1953). 6 For an extensive list of germanium field effect measurement references see R. H. Kingston, J. Appl. Phys. 27, 101 (1956). 7 R. H. Kingston and S. F. Neustadter, MIT Lincoln Lab. Rept., (August, 1954) p. 10; J. Appl. Phys. 26, 718 (1955). 8 C. G. B. Garrett and W. H. Brattain, Phys. Rev. 99 376 form, Figs. 1-5, conveniently display behavior over wide resistivity ranges. ANALYSIS Electric field in a semiconductor can be found by integrating Poisson's equation in which charge density is expressed in terms of mobile hole and electron den sities and bound ionized donor and acceptor densities. Normally the assumption is made that all donors and acceptors are ionized throughout the material up to the surface regardless of the bending of the bands. Assuming also that Fermi statistics apply throughout the semi conductor up to the surface, we have, using the notation of Shockley,9 p=nie{3('Po-t), n=nl/p. (1) Figure 6 shows the relationship between energy levels near the surface of a semiconductor and defines the potentials. Far enough from the surface, where 1/1=1/10, charge neutrality exists and therefore the donor and acceptor densities can be expressed in terms of bulk hole and electron densities. If we assume there are no surface states, then the charge terminating the electric field appears in a space charge layer and can be written in terms of a "surface excess"8 (charge per unit surface area) from Gauss' law ~ •• = _KEO(dl/l) dx t=t, = -2yqniKEo/2fj{ ±v'2[ coshfj(1/I- cpo) -coshfj(I/Io- cpo)+fj(1/I-1/Io) sinhfj(1/Io- cpo) r L =>/1, = -2qniLDF[ (1/1,-cpo), (1/10-<PO)]. (2) where LD is the Debye length for the semiconductor,9 LD= [KEo/2qn;,BJI (rationalized units). If a surface potential, in units of 1/ fj= kT / q, is defined relative to the bulk potential by Y=fj(1/I.-1/I0), then for (1955). ' 9 W. Shockley, Bell System Tech. J. 28, 435 (1949). 1753 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 03:48:341754 V. O. MOWERY RESISTIVITY IN OHM-CM 200 P-TYPE I 10000 .. 2 )( ~ ::IE u ~ j z 4( 3 ::IE II: ::> 1&1 0 " 0 ~ u . I-i iii I ( II 10 P-TYPE 1.0 ;;-b FIG. 1. Space charge x ~ .. as a function of re ;; sistivity for vanous ~ values of surface poten ;;; z tial Y in germanium and ~ 8 silicon. The signs of Y o J should be reversed for 5 iii n-type material. Sign to 8 be used for 2:.. is ex ~ plained in the analysis. ~ 0.02 I 0.1 I I I! , I !! 10 I 0.01 0.1 "-TYPE GERMANIUM RESISTIVITY IN OHM -eM a positive surface potential the energy hands are bent downward at the surface and the positive square-root branch of the F function must be used giving a negative space charge. Similarly, for a negative surface potential the negative square-root branch is used giving a positive space charge. 80Or~--------~--------r-----r---' \ rn o :z: ::IE 600 400 IE 200 o -5 "" -4 ~ --"2 ~14 +19 p-TYPE GERMANIUM --I 0_+ 1 Surface excesses r p and r n of holes and electrons in the space charge layer result in a change of conductivity ~G=q.up(rp+brn). (3) When the potential at the surface is shifted from 1/;0 to 1/;., the change in holes per unit area is the integral of the p -TYPE GERMANIUM FIG. 2. Change in sur face conductivity AG as a function of resistivity for various values of sur face potential Y in p-type germanium. -4000.02 0.1 05 -5 ~~I~.O-----L--~~~~~IO~--- RESISTIVITY IN OHM -eM RESISTIVITY IN OHM -eM [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 03:48:34SURFACE CONDUCTIVITY AND SPACE CHARGE IN Ge AND Si 1755 800.------:\:------,\:------' 300..-,\----:-,--,--:-\--:,-..., +4 +5 Y·+S +6 -IS +7 -14 250 400 <J) <J) 0 0 :z: :z: ::E FIG. 3. Change in ::E 0 0 It: surface conductivity t!J.G It: 0 0 200 i as a function of resis-i tivity for various values ~ ~ of surface potential Y C!> C!> in n-type germanium. <I <I n= TYPE GERMANIUM n-TYPE GERMANIUM 0.1 RESISTIVITY IN OHM-CM difference between the hole density where the potential is 1/; and hole density in the bulk where the potential is 1/;0 from the surface to far within the bulk. Using Eg. (1) and dif;/ dx as in Eg. (2) the result is i1/l0 e-{J(1/I-"'0'-e-{J(1/Io-"'0' rp=n;/3LD d1/; 1/1. F[(1/;-cpo), (1/;-1/;o)J S RESISTIVITY IN OHM-CM and by symmetry the change in electrons per unit area is given by the same expression with a change of sign for the potentials. The F and G functions have been plotted by Kingston and Neustadter7 for limited ranges of the variables UB=/3(1/;O- cpo) and u.=/3(1/;.- cpo). Seiwatz and Green10 have shown that the expressions simplify for extended ranges. Validity of the assump tions used in the analysis have been discussed =n.LDG[(1/;.- 1<'0), (1/;.-1/;0)]' (4) previously.8-10 FIG. 4. Change in surface conductivity t!J.G as a function of resis tivity for various valeus of surface potential Y in p-type silicon. 7·r--7,-------~\-~~ / / -6 -7 +19 -8+18 6 \ S - 5 p-TYPE SILICON 200 1000 RESISTIVITY IN OHM-CM 10 R. Seiwatz and M. Green, J. Appl. Phys. 29, 1034 (1958). \ I 14 Y=-6 +14 \ / -7 +13 \ / -8+12 / 1.2 P -TYPE SILICON 10000 50000 RESISTIVITY IN OHM -CM [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 03:48:341756 V. O. MOWERY 0.8 2 0.6 n • TYPE SILICON / " / n-TYPE SILICON FIG. 5. Change in sur· face conductivity tlG as a function of resistivity for various values of sur· ' face potential Y in n· type silicon. 2000 10 000 R(SISTIVITY IN OHM-CM CONSTANTS In order to evaluate the expressions for space charge and change in "surface conductivity," it is necessary to assume reasonable values for the constants in ger manium and silicon. These constants and estimates of their numerical values are listed in Table I. Empirical evaluations have been made at room temperature (3000K) where lattice scattering should predominate, at least for low impurity content material. The mobilities used in resistivity and change in sur face conductivity calculations have been assumed to be lattice-scattering mobilities and agree with the experi mental expressions of Morin and Maita for siliconll and germanium.12 Intrinsic carrier concentrations ni have been determined from the experimental expressions obtained by Morin and Maita from their mobility measurements. The values of the dielectric constant are those commonly used and agree with the refractive indices reported by Briggs.13 DISCUSSION The most important contribution to drift mobility other than lattice scattering is usually impurity scat tering.14 Ionized impurity scattering has been investi gated by Conwell and Weisskopf, Brooks and Herring, and Sclar.16 The Conwell-Weisskopf treatment is a classical calculation using the Coulomb scattering 11 F. J. Morin and J. P. Maita, Phys. Rev. 96, 28 (1954). It F. J. Morin and J. P. Maita, Phys. Rev. 94, 1525 (1954) 11 H. B. Briggs, Phys. Rev. 77, 287 (1950). . It M. B. Prince, Phys. Rev. 92, 681 (1953) and Phys. Rev 93 1204 (1954). . , 16 E. M. Conwell and V. F. Weisskopf, Phys. Rev. 77, 388 (1950)i.H. Brooks, Phys. Rev. 83, 879 (1951); N. Sclar, Phys. Rev. 1114, 1548 (1956). RESISTIVITY IN OHM-CM cross section suitably cut off. Since this is a classical calculation it requires that the scattering trajectory be well defined. This implies the restriction kta» 1 where kl is the wave number for the charge carrier and 'a is the cutoff distance assumed for the scattering potential. The Brooks-Herring treatment uses the cross section for scattering from a screened Coulomb potential calcu lated using the Born approximation. The requirement that the phase shift in the wave function of the charge carrier be small is equivalent also to the condition k1a»1. In both treatments, mobility due to ionized impurity scattering varies inversely with impurity density. Because of the restriction on a, the results are less valid for very high impurity densities. Sclar's treatment uses the partial wave technique with square-well or square-barrier potential character istics for the scattering by an attractive impurity or a repulsive impurity, respectively. The results are valid for k1a«1 and at low temperatures and high impurity densities predict a mobility higher than obtained by the Brooks-Herring or Conwell-Weisskopf formulas. These analyses assume the simple model of isotropic quadratic energy dependence of the charge carriers on momentum resulting in spherical energy surfaces in momentum space and have not been revised to account for more intricate band structures. Results of these investigations could be used to modify the data given here. However, in using impurity scattering formulas, the effective mass of the charge carriers must be known and the total density of ionized impurities must be approximated. If the ionized impurity density is assumed to be the same as the density of free charge carriers, then a relationship (empirical or otherwise) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 03:48:34SURFACE CONDUCTIVITY AND SPACE CHARGE IN Ge AND Si 1757 --------------Ec ---------------E v X=O FIG. 6. Diagram of levels of energy near a semiconductor surface, n type with Y>O. between charge carrier density and resistivity must be used. Determination of mobility for a given resistivity is further complicated in compensated samples where both donors and acceptors are present. If only one type of impurity is present in any appreciable density, the minori~y carriers will see about twice as many charge scattermg centers as do the majority carriers. Conse quently the minority carriers will have decreased mobility which means that the ratio b for heavily doped material should be larger in an n-type semiconductor. Measured mobilities for germanium decrease by about ten percent at one ohm-cm (greater for p type, less for n type) and by about twenty or thirty percent at a tenth ohm-cm.14 The ratio of electron mobility to hole mobility increases for decreasing resistivity. Mobilities for germanium listed in Table I are less than those reported by Prince14 and greater than those repo:ted by Haynes and Shockley,l6 but are nearly wlthm the probable errors of both. Applying the fore going orders of magnitude correction for p.p and b de creases the conductivity changes in germanium at the lower resistivities by about one-fourth in n type of 1& J. R. Haynes and W. Shockley, Phys. Rev. 81, 835 (1951). TABLE I. Semiconductor constants. Numerical value Constants Silicon Germanium /(= dielectric constant n;=intrinsic carrier density (em-a) ILn = electron mobility (cm2/volt-sec) ILp=hole mobility (cm2/volt-sec) b=ratio of mobilities ~/ILp) LD= Debye length (em) 12 1.56X101O 1500 SOO 3.0 2.33X1O-s 16 2.56X101s 3800 1800 2.11 6.7XIo-6 Fig. 3 and one-half in p type of Fig. 2. Measured mo bilities in silicon decrease very little in the resistivity range considered here; therefore, the near intrinsic values of lattice scattering mobilities are probably good approximations for silicon. Values listed in Table I are the extrapolated values given by Prince.14 The ratio b may decrease for decreasing resistivity in silicon. Charge carriers confined to a narrow layer near the surface of a semiconductor could have reduced mobili ties due to scattering from the surface,l7 Since the space charge layer is two or three Debye lengths for the higher surface potentials on high resistivity material,7 the correction for germanium is of the order of a few percent for the higher surface potentials. The correction is even less for silicon because of the longer Debye length. ACKNOWLEDGMENTS The author wishes to thank T. M. Buck for use of his extended curves of the F and G functions as computed on the IBM 650, and E. J. Scheibner and M. M. Atalla for numerous discussions. 17 J. R. Schrieffer, Phys. Rev. 97, 641 (1955). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 03:48:34

1.1735514.pdf

Experimental Investigations of the Cesium Plasma Cell W. A. Ranken, G. M. Grover, and E. W. Salmi Citation: Journal of Applied Physics 31, 2140 (1960); doi: 10.1063/1.1735514 View online: http://dx.doi.org/10.1063/1.1735514 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental investigations on the propagation of the plasma jet in the open air J. Appl. Phys. 107, 073302 (2010); 10.1063/1.3369538 Experimental Investigations of Dusty Plasmas AIP Conf. Proc. 799, 3 (2005); 10.1063/1.2134567 Experimental investigation of electron emission of a tungsten cathode in a cesium plasma Appl. Phys. Lett. 21, 283 (1972); 10.1063/1.1654379 Experimental Investigation of the Normal Modes in a Warm Cylindrical Plasma J. Appl. Phys. 40, 3680 (1969); 10.1063/1.1658257 Investigation of the KelvinHelmholtz Instability in a Cesium Plasma Phys. Fluids 9, 309 (1966); 10.1063/1.1761674 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09JOURNAL OF APPLIED PHYSICS VOLUME 31, NUMBER 12 DECEMBER, 1960 Experimental Investigations of the Cesium Plasma CeU* W. A. RANKEN, G. M. GROVER, AND E. W. SALMI Los Alamos Scientific Laboratory, Los Alamos, New Mexico (Received December 16, 1959; and in final form July 25, 1960) Some aspects of the performance of a cesium plasma cell with tantalum emitter are evaluated in terms of experimental determinations of the effects of variations in such parameters as cesium vapor pressure, emitter temperature, and emitter-collector separation distance. Experiments relating to the effect of collector serrations and to the feasibility of radiation shielding are described. Voltage-current characteristics are presented for several emitter temperatures and for a wide range of cesium vapor pressure. I. INTRODUCTION THE direct conversion of heat to electricity through the use of what has been called either a thermionic converter or a plasma thermocouple, depending pri marily on the pressure of cesium vapor used to neutral ize the electronic space charge, has been reported by several research groupS.l-5 Basically, the device is a simple diode with a hot filament or button serving as an electron emitter and a cold surface serving as an electron collecting electrode. The region between the electrodes is filled with cesium vapor which is ionized either upon contact with tne hot high work function emitter, or by other processes, and hence serves to neutralize the electronic space charge so that saturated Richardson emission currents can be obtained. The source of the voltages observed across the cell terminals has been attributed to the contact potential difference existing between the high work function emitter and the low work function collector by Hernqvist et al,2 when cesium pressures are sufficiently low so that electrons making the transit from the emitter to the collector do so with few or no collisions with ions or atoms in the gas. In applying this model it is assumed that the low work function collecting electrode is maintained at a tempera ture sufficiently low for its thermionic emission to be negligible compared to emission from the hot electrode. For cesium pressures where the number of collisions made in the region between electrodes is large, the situation becomes more obscure. Lewis and Reitz6 have treated the cell as a plasma thermocouple and when no current flows have used transport theory and thermo dynamical considerations to calculate the thermoelectric voltage generated when an electron temperature gradi ent exists. Pidd et al.4 have demonstrated that the zero current voltages of the cell follew roughly the behavior predicted from this theory for cesium pressures of the order of 0.3 mm Hg and above. This paper describes experiments performed with cell configurations where tantalum is used as the emitter. The general purpose of these experiments was primarily to determine optimum operating conditions for such a cell and secondarily to obtain a measure of under standing of the physical mechanisms involved in its performance at high (on the order of 1 mm Hg) cesium pressure. Reported herein are investigations relating to the effects on cell performance of modifications of the geometrical configuration of the cell, including the effect of varying the separation distance of the emitter and collector, the effect of serrations of the collector electrode (designed to double the effective collecting area exposed to the cesium plasma), and the effect of introducing radiation shielding between the emitter and collector. In the process of these investigations the behavior of the voltage-current characteristics and power outputs of the cell were studied over a wide range of cesium vapor pressure and emitter temperature. II. EXPERIMENTAL CELL CONFIGURATIONS The basic design features of the cell have been described in an earlier reportl but for the sake of clarity these will be presented again insofar as they relate to the present set of experiments. Figure 1 shows the more important aspects of the design of the cell with adjustable emitter-collector spacing. A tantalum button i in. thick and i in. in diameter is welded into a 0.Ol2-in. thick tantalum disk and the assembly is mounted between two circular knife edges thus dividing the cell into two chambers. The upper chamber, which is continuously evacuated, houses the electron gun used to heat the tantalum button. In the normal range of operation the gun * This work was performed under the auspices of the U. S. bombards the tantalum button with electron currents Atomic Energy Commission. of from 0.3-0.8 amp at voltages up to 2600 v. Consider- 1 G. M. Grover, D. J. Roehling, E. W. Salmi, and R. W. Pidd, ing that the resistance of the button and disk is less than J. App\. Phys. 29, 1611 (1958). • K. G. Hernqvist, M. Kanefsky, and F. H. Norman, RCA Rev. 10-a ohm it can be seen that the extraneous voltages 19, 244 (1958). associated with button heating are very small. 3 V. C. Wilson, J. App\. Phys. 30, 475 (1959). The quartz viewport mounted on the insulator at the 4 R. W. Pidd, G. M. Grover, E. W. Salmi, D. J. Roehling, and G. F. Erickson, J. Appl. Phys. 30, 1861 (1959). top of the gun chamber allows pyrometric observation of 5 R. Fox and W. Gust, Bull. Am. Phys. Soc. Ser. II, 4, 322 the button temperatures. For this purpose, also, small (1959). 6 H. W. Lewis and J. R. Reitz, J. AppJ. Phys. 30, 1439 (1959). holes are drilled four-fifths of the way through the 2140 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09EXPERIMENTAL INVESTIGATIONS OF CESIUM PLASMA CELL 2141 tantalum button. The length to diameter ratio of these holes is 5 to 1, and hence only small emissivity correc tions to the pyrometrically observed temperatures are required. Of more importance, however, is the fact that temperatures can be measured without regard to changes in emissivity of the button surface brought about by prolonged bombardment heating. In addition, the effect of reflected light, from the tungsten filament in the electron gun, on the observed temperature is negligible. Temperature drops of up to 200°C occur from one face of the button to the other and hence a calibra tion must be made of the temperature of the lower face of the button (the emitting surface of the cell) vs observed hole temperatures. The emitting area of the cell is taken to be that of the tantalum button (2 cm2). Justification for this assump tion is given in Fig. 2, which shows a typical temperature profile for the button and disk geometry. Since electron emission falls off exponentially with temperature, the To SUPPORT DISK '-f.l\F==~t==Jt'cu COlLECTOR THERMOCOUPlE COlLECTOR- EMITTER ~ C)------1l-'SEPARATION ADJUST MENT SCREWS Sf\-----1l--Cu BELLOWS DIFFERENCE THERMOCOUPlE OIL BATH C. RESERVOIR INSULATOR l!::::::::~!~~~~~~·COOUNG OIL LINES FIG. 1. Schematic illustration of experimental cell showing assembly details and electrical circuits. error made in neglecting the emission from the disk is small insofar as this emission is characterized by the work function of tantalum. For cesium pressures on the order of 10-2 mm Hg and higher, the possibility exists that annular regions of the disk are partially coated with cesium with the resulting formation of low work func tion surfaces.3 In this instance it is likely that the emission per unit area of portions of the disk is com parable with the emission per unit area of the button and the emittet area is no longer well defined. This effect will be further discussed in Sec. III. The region of the button and disk where positive ion formation takes place is not generally well defined. The lower chamber depicted in Fig. 1 forms the thermoelectric cell. The collector is a i-in. diam capped copper tube through which silicone oil is circulated. Thermocouples are inserted in the oil stream, as shown, to determine the oil inlet and outlet temperatures, thus providing a somewhat rough calorimetric measure of the total heat flow to the collector. The oil is circulated by <> . iii I~OO II: :;) ~ II: ... ... 1000 :. ... .., 500 w-~30~~2~O~~I~O~~0~-L~~~L-~ DISTANCE FROM CENTER OF BUTTON (mm) FIG. 2. Temperature profile of tantalum button and mounting disk as viewed from cesium side of cell. a centrifugal pump driven by an air motor. The latter is used so that no problem arises from pickup voltages appearing across the cell terminals when open circuit voltages are being measured. The spacing between the emitter and collector may be changed by means of three adjusting screws which expand or contract the copper bellows. The cesium is encapsulated in glass and placed in a side tube as shown. After the cell has been thoroughly baked out at a temperature of about 350°C the copper pumping tube is pinched off in a hydraulic press, forming a reliable cold weld joint, and the cesium capsule is crushed. The whole assembly is immersed in a thermostatically controlled silicone oil bath by means of which the cesium pressure in the cell can be controlled over a range of 10-"-6 mm Hg. The insulator through which the collector leads pass is maintained about 50°C hotter than the oil bath to combat the formation of a cesium coating. When this precaution is taken the leakage resistance across this insulator is generally in the range of 1 X 105 to 3X105 ohms. The cell configuration shown in Fig. 1 is that used to obtain data on the effect of emitter-collector spacing on o "4 I.tz ~ , , ! ,I ! SCALE Ta EMITTER Cu RETAINER ALUNDUM INSULATOR Fw. 3. Schematic illustration of radiation shield mounting. Scale is in inches. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:092142 RANKEN, GROVER, AND SALMI ----,------r---l--'-:"-"-----J--- f-:----r-------,-- r---i------r-----,---r- (0): . (b) : 1 I -3 10 ~ •• 10-4mmHQI ~.'1.6x10 mm HQ (/) 100. IIJ II: IIJ -I Q.. 10 2 I I ~I~--~+-~ I I r n m c( -2 : TE I 10 CURVESr! 25400K I-CURVES:][ 2300 OK CURVESrJ[ 20100K Z -3 CURVESrlll 11200K ~ I 0 ~"'----'---'---;--'--'----'-- ~--'----L.--'.---+_'--.l- II: g 101 o IIJ 0 ~ 10 IIJ (/) ~ 101 I r n FIG. 4. Current-voltage charac teristics of cell with tantalum emitter and serrated copper col lector. Curves I through IV cover the emitter temperature range of 2540 to 1720°K. Figures 4(a) through 4(f) cover the cesium vapor pressure range of 10-4 to 2 mm Hg. -3t I 1Q4;--'--_';;2--.J'--0;;..1 --'-----,:2;--L-_4 -2 0 2 -4 -2 a 2 4 CELL TERMINAL VOLTAGE (RELATIVE TO EMITTER) cell performance. The experimental configuration used in the experiments on the value of collector serrations is identical to that shown in Fig. 1 with the exception of circular serrations in the face of the copper collector. These circular grooves were 0.032 in. deep and served to double the surface area of the collector face. The experimental arrangement used to obtain some idea of the feasibility of improving the cell efficiency through the use of radiation shields is sketched in Fig. 3. The radiation shield consists of 10 circular 1-mil thick tantalum foils spot welded together at several points. Through the center of the stack is a k-in. diam hole. This arrangement is mounted on a tantalum support disk which is mounted, in turn, on an alumina insu lator. The shield is allowed to float electrically. The collector used in this experiment was fabricated from stainless steel. III. VOLTAGE CURRENT CHARACTERISTICS OF THE SERRATED COLLECTOR CELL Of the various cell configurations, that which was studied most extensively was the cell with the serrated copper collector described in Sec. II. This cell had a collector-emitter separation distance of 2.5 mm. Cur rent-voltage characteristics were obtained for a voltage range of -24 v to +24 v except where cell currents exceeded 30 amp. The voltage source consisted of a 6-ohm rheostat placed across two 12-v storage batteries. Such a voltage source has the disadvantage of variation of the load across the cell as the voltage is changed, but this is of consequence only in the region of high cesium pressure and positive applied voltages where arc discharges are encountered. In the working region of the cell the voltage current relations obtained by this method duplicate exactly those obtained by varying a load resistor. In measuring most of the voltage-current charac teristics of the cell it was not found necessary to correct the emitter temperature for the variation in Peltier cooling occurring as electron currents were varied from zero to short circuit values. The geometry of this cell is sufficiently inefficient for the amount of heat removed by Peltier cooling to be small compared to the amount lost by radiation and conduction from the button and disk. As a result, changes of emitter temperature occurring as the cell current was changed from zero to values up to about 20 amp were too small to be detected with the optical pyrometer, (less than about lOOK). When cell currents significantly larger than 20 amp were obtained (Sec. IV) it was necessary to vary the power supplied to the emitter in order to obtain a constant emitter temperature current-voltage characteristic. Voltage-current characteristics obtai·ned for the ser rated collector cell in the range of net electron current are presented in Fig. 4 for emitter temperatures (T E) ranging from 17200K to 25400K and for cesium pres sures PCs covering the range of 10-4 to 2 mm Hg. For the two lowest cesium pressures [Fig. 4(a) and 4(b)] the curves are similar to those observed by Hernqvist et al.2 Even for these low cesium pressures the electron currents are completely space-charge neutralized. In fact, with the exception of the curve for emitter tem perature equal to 25400K and cesium pressure of 10-4 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09E X PER I MEN TAL I N V EST I GAT ION S 0 FeE S I U M P LAS MAC ELL 2143 mm Hg, the electron space charge is over-neutralized; more positive ions are being produced than are required for space charge neutralization. The observed currents rise from zero with a behavior dependent upon electron temperature until they essentially level off at a voltage close to the contact potential difference between the cesium-coated copper collector and the uncoated tanta lum emitter. The magnitudes of the currents at the points where current saturation occurs are slightly more than half the expected thermionic emission from tanta lum. At these low pressures this discrepancy can be partially ascribed to the fact that some of the electrons from the emitter travel to the walls of the vacuum chamber rather than to the collector. Except for the curve with TE=2070oK and Pc.=1.6XlO-3 mm, the currents increase slowly with applied positive voltage in the range of 4-24 v not shown in Fig. 4. The behavior of the excepted curve is unusual in that the cell ap parently goes into a thermionic arc discharge at a lower applied voltage (3.5 v) than the voltages (>24 v) for which breakdown occurs at higher emitter temperatures. For a cesium pressure of 2.3 X 10-2 mm [Fig. 4( c) ] the voltage at which the transition from the normal operat ing mode of the cell to the discharge mode occurs de creases with emitter temperature in accord with early experiments on thermionic arcs in cesium vapor.7 The most striking behavior at this pressure occurs for T E= 1720oK, where an abrupt increase of current of more than three orders of magnitude takes place in the voltage range of 1-3 v positive. The currents obtained in the discharge region show very little dependence on emitter temperature. It is difficult to explain the magnitude of the electron currents obtained for positive cell voltages on the basis of effects occurring at the button surface. It is more probable that these high current values result from the coating of the support disk with cesium. This formation of fractions of a monolayer of cesium on tantalum produces low work function surface areas even if the tantalum is at a relatively high temperature, provided the cesium pressure is sufficiently high. Hence, portions of the emitter support disk may be prolific electron emitters. However, these surface regions, because of their low work function, are not good ion producers. Hence the emitted electrons cannot reach the collector until the collector is made positive relative to the emitter and the accelerating field between the electrodes be comes large enough to enable ionization by impact of electrons with neutral cesium atoms to take place. Extension of the emitting area beyond the confines of the emitter button apparently has little effect on the shape of the current voltage characteristics in the work ing region of the cell, at least for the higher emitter temperatures. This conclusion is based on a series of measurements in which a small, cooled, Langmuir plane probe was mounted flush and central with respect to 7 F. H. Newman, Phil. Mag. 50, 463 (1925). 100 10 z o -.01 UJ > ~ Ul o a. .001 ...... • • CURVE 1 2 3 4 5 6 • Pes TE(OKl 1.6 x 10-32540 MM 2300 Hg 2070 2 2540 MM 2300 Hg 2070 .: .OO~~ __ ~ ____ ~ __ -7.~ __ ~ __ ~~ __ ~ __ ~ o -4 -8 -12 -16 -20 -24 28 CELL TERMINAL VOLTAGE - FIG. 5 .. Positive ion current versus negative potential applied to cell termmals. Curves 1 through 3 were obtained for a cesium vaI;l0r pressure of 1.6X 10-3 mm Hg. Curves 4 through 6 are for a cesIUm vapor pressure of 2.0 mm Hg. the collector surface in a cell with tantalum emitter and a 3-mm collector-emitter separation distance. The collector diameter was 0.625 in. When the collector and probe were maintained at the same voltage the collector served as a guard ring for the probe and shielded it from any effects of emission from the support disk. Under these conditions, and for an emitter temperature of 2540oK, the ratio of short circuit currents to the probe and the collector was just the ratio of their areas for values of cesium pressure up to 0.5 mm Hg, the highest pressure at which probe measurements were made. This result is not to be expected if the collector is drawing appreciable current from the support disk. Measure ments of this type were not made at the lower emitter tempera tUres. The behavior of the current-voltage characteristics m the region of reverse (positive ion) currents for pressures on the order of 10--1 mm and less is typified by curves 1, 2, and 3 which are presented in Fig. 5. These curves were obtained at a cesium pressure of 1.6 X 10-3 mm Hg. Above 10-1 mm pressure, arclike discharges are en- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:092144 RANKEN, GROVER, AND SALMI CD CL 2 ~. t Z ... II: II: EMITTER TEMp, c230O'K CURVE C. PRESS\JRE (MM. HvJ 1.0x 10-4 2.0XI0-Z 1.6X10-1 .OO!5~~-~4'----~3-----*2-----·I--~0~--~--~2 TERMINAL VOLTAGE Z 5 0.1 Z o II: t- EMITTER TEMP •• 2540·K o ... ...J II.! CURVE Co PRESSURE (mm "V) -4 1.0)(10 ... ~ 1.6 X 10-5 Z 1-0.01 3 1.6 X 10'" o.OOI~~J- __ ~ ____ ~ __ -L __ ~~ __ J-__ -J -5 -3 -2 -I 0 TERMINAL VOLTAGE (a) (b) FIG. 6. Plot of true electron current (observed current corrected for magnitude of positive ion current) versus cell terminal voltage. (a) Emitter temperature = 23000K, cesium vapor pressures of 1 X 10-', 2X to .... , and 1.6X 10-1 mm Hg. (b) Emitter temperature = 2540oK, cesium vapor pressures of 1 X 10-4, 1.6X to-3, and 1.6XIO-I mm Hg. countered, with breakdown voltages decreasing with increasing pressure until at 2-mm pressure the behavior demonstrated by curves 4, 5, and 6 in Fig. 5 is en countered. The reverse current increases from zero (at equivalent open circuit voltage) to 25 amp with no sign of a transition region. Low and intermediate pressure ion curves show the behavior expected from a plane Langmuir probe in a plasma.s Carrying the similarity a little further, one can extrapolate the flat region of the ion curves to less negative voltage values and add the magnitudes of the extrapolated ion currents to the observed electron currents to obtain curves of true electron current vs collector voltage. This is, of course, the procedure followed in constructing Langmuir probe characteristics. When collector characteristics are plotted in this manner, curves of the type shown in Fig. 6(a) and (b) are obtained. The similarity to probe characteristics is obvious. The initial increase of true electron current with voltage is indeed exponential. For a true probe the slope of this portion of the characteristic is (on a semi logarithmic plot) proportional to the reciprocal of the electron temperature in the plasma. Electron tempera tures obtained from the curves in Fig. 6(a) and (b) for a pressure of 10-4 mm are within 100 deg (Kelvin) of the emitter temperatures. This result has been predicted by Hernqvist et af.2 However, as the cesium pressure increases, the behavior of the collector characteristics in Fig. 6(a) and (b) suggests that the electron tempera ture increases to values on the order of twice the emitter temperature. That such high electron temperatures should exist in the cell plasma is reasonable enough for large currents flowing through the cell, but the collector characteristics do not give evidence of electron tempera ture depending on the current. The relatively abrupt change in slope of the charac teristics is identified, in the case of true probes, with the point at which probe voltage is equal to the plasma potential. Ideally this is the potential for which all electrons in the plasma are able to diffuse to the probe and further increase in probe voltage will not increase the electron current. In the case of the collector the change in slope (pseudosaturation of electron current) occurs, at low pressures, when the electron current reaches the value of Richardson emission of the emitting surface. The voltage at which this break occurs is not necessarily related to the plasma potential. For high cesium pressures diffusion limiting may also falsify ap parent space potentials. (Diffusion limiting refers to the case where the collector drains electrons from the plasma adjacent to it at a rate limited by the electron replacement by field free diffusion from more remote regions in the plasma.) In any case, measurements of plasma potentials and electron densities obtained from the collector characteristics in the usual manner are likely to be in error. Plasma densities may, however, be estimated from the saturation values of the positive ion 8 A complete discussion of the Langmuir probe and probe characteristics is given in L. B. Loeb, Basic Processes oj Gaseolts Hectronics (University of California Press, Berkeley, California, 1955), pp. 332-370. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09EX PER I MEN TAL I N V EST I GAT ION S 0 FeE S I U M P LAS MAC ELL 2145 currents.9 That these estimates are reliable, at least for cesium pressures below 10-1 mm Hg, has been verified by results obtained with the Langmuir probe mentioned earlier. In principle, if the electron velocity distribution is indeed Maxwellian then once the electron densities are established the plasma potential may be obtained. In most cases this procedure involves large extrapola tions of the initial portions of the collector charac teristics and hence is subject to considerable error. The general trend of plasma potential with cesium pressure is from a value of about zero for a Cs pressure of 10-4 mm Hg to values of about 2 v positive (relative to the emitter) for to-I mm Hg. It is perhaps worthwhile to point out that although the short circuit currents for a given emitter tempera ture increase with cesium pressure, as can be gleaned from the curves in Figs. 4 and 6, the maximum power does not follow this behavior. Figure 7 depicts the dependence of maximum power output on cesium pres sure for several emitter temperatures. The power maximum is seen to decrease slightly with increasing cesium pressure until at a pressure of about 2X to-I mm Hg a rapid rise occurs. This pressure is in the range for which cesium coating of the emitter is expected to become appreciable. It can be seen that the rate of increase of power with pressure is greater for lower emitter temperatures. The improvement of the power density of the cell brought about by doubling the effective area of the collector through the use of serrations was about a factor of two for high cesium pressure and high emitter temperature operation. At the lower emitter tempera tures little or no improvement was observed. Hence the evaluation of the effect of the serrations is not straight forward. For low pressure operation, where the cell currents are known to be emission limited, there was no increase in power density nor does one expect any. '" m 0 w w > > II: II: ::> ::> 0 0 12 2.4 W t'l j~ Jo.6 ,,11'1 1111I]'i i i\lilil i i tlCilll CURVE TE (OK) A 2540 B 2300 C 2060 en 1O 2.0 0.5 l-i 8 1.6 0.4 :6 I. A 0.3 E II. 0.2 4 0.8 " i 2· Q4 / ----------- .. _---------.... __ .9 ____ ..-"''; 0.1 ° 1O' FIG. 7. Maximum output power in watts (Pm•x) versus cesium vapor pressure (Pc,). Note separate ordinate scale for each emitter temperature. .,J. E."Allen, R. L. F. Boyd, and P. Reynolds, Proc. Phys. Soc. (London) B70, 297 (1957). IV. EFFECT OF EMITTER-COLLECTOR SPACING ON CELL PERFORMANCE The two phenomena expected to influence the amount of current drawn at a given cell terminal voltage (in the working region of the cell) and at a given emitter temperature are the emission limit of the emitter and the effective impedance of the plasma, be it due to multiple scattering of electrons by ions or to negative space charge formation resulting from positive ion deficiency. It has been seen in Sec. III that for cesium pressures below about to-2 mm the electron currents drawn are definitely emission limited. For the higher pressures it was found that the current was a strong function of the collector voltage in the range of - 2 to o v and indeed, at the latter voltage, exceeded the expected Richardson emission for tantalum. At the low pressures the plasma impedance should be negligible since electron and ion mean free paths are large compared with the dimensions of the cell and the electron space charge is completely neutralized. For high pressures the interaction of electrons and ions with each other and with the neutral atoms can no longer be neglected and it becomes of interest to determine the relative importance of emission and plasma impedance in limiting the current output of the cell. The behavior of the current output of the cell as a function of emitter collector spacing can be used as an indication of the importance of plasma impedance in limiting the current flow, provided that this impedance is associated with the bulk of the plasma between the electrodes rather then confined to thin layers adjacent to one or both of the electrodes.4,6 Figure Sea) gives the dependence of maximum power vs electrode separation for a cesium pressure of 0.5 mm Hg and for four emitter temperatures. The curves are normalized to the value for T E= 25400K at 6.7-mm separation. These measure ments were obtained with the cell configuration shown in Fig. 1. Since for this pressure the cell voltages were nearly independent of spacing the behavior of zero voltage (short circuit equivalent) current with spacing is similar to that shown in Fig. Sea). It can be seen that as the spacing is decreased from 6.7 mm to 1.2 mm the maximum power is increased by an average of only 50%. Much of this effect can be attributed to increased loss of positive ions and electrons from the working region of the cell as the spacing is increased. It appears that the bulk impedance of the plasma has a negligible effect in limiting the cell's performance in this range of operation. Some experimental data were also taken at a cesium pressure of 2.0 mm Hg. The results are presented in Fig. S(b). Again the curves are normalized to a point at 6.7 mm and T E= 2540oK. As compared to the results in Fig. Sea) the increase of power with the reciprocal of electrode separation is somewhat more pronounced for an emitter temperature of 24200K and much more pronounced for the emitter temperature equal to [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:092146 RANKEN, GROVER, AND SALMI I!! 5 l e( ~ 4 . o e ... 3 CESIUM PRESSURE:O.~mm ti9 CURVE T[ ORDINATE SCALE 2~40 -K Pmax x 1.0 2420 -K Pmoa x 2.15 2300 -K Pmoll x 5.7 2175 'K PmG• 11. 9.6 00 2 3 4 5 6 1.0 6.0 5.0 I!! 4.0 I-; . ~ 3.0 ... 2.0 1.0 SEPARATION DISTANCE (mm) CESIUM PRESSURE: 2 mm H9 ORDINATE SCALE P",u x 1.0 Pmax x !5.6 0'O~--~--~2'-~3---7.--'5~-t---i SEPARATION DISTANCE (mm) (a) (b) FIG. 8. Effect of emitter-collector separation distance on maximum output power (P max) for several emitter temperatures. (a) Cesium vapor pressure=O.5 mm Hg. (b) Cesium vapor pressure=2 mm Hg. Power values have been multiplied by the factors given in the columns labeled "ordinate scale" in order to normalize the curves for different emitter temperatures to the same value for a separation distance of 6.75 mm. 2175°K. The latter behavior may result from a space charge limiting effect near the collector and will be discussed in Sec. VII. Electrode separation for the cell with the serrated collector was reduced to about 0.1 mm and the cell was operated at a cesium pressure of 6 mm Hg and an emitter temperature of 271OoK. This operating point represented the highest value of cesium pressure that could be safely attained with the silicone oil bath and the highest controlled emitter temperature attainable. The zero-voltage current was 110 amp and the maximum power produced in an external load was 60 watts, representing an approximate efficiency of 15%. The method of measuring a value for the efficiency is dis cussed in Sec. V. At this close spacing there was no indication that the power would not continue to increase with increase in cesium pressure or emitter temperature. V. CELL WITH RADIATION SHIELD The parallel requirements of high electron emission and sufficient ionization of cesium necessitate high temperature operation of the plasma cell emitter. At the temperatures required, radiation transport from the emitter to the collector constitutes a major effect in limiting cell efficiency. The spacing experiments de scribed in Sec. IV indicated that limitation of cell per formance by plasma impedance at the optimum operat ing point was not a major consideration. This being the case, it was of interest to investigate the feasibility of placing shields between the emitter and collector which would intercept radiant energy while allowing the flow of electrical current. The initial experiment to explore the value of the radiation shielding concept was done with the cell con figuration depicted in Fig. 3 and discussed in Sec. II. The tantalum shield was allowed to assume the wall potential (potential at which electron and positive ion currents to the shield are equal) determined by some average value of the parameters of the plasma in contact with it. The flow of r:adiation is restricted to that which passes through the l-in. diam hole in the center of the radiation shield. The temperature of the collector side of the radiation shield rides about SOOoK below that of the emitter . Observations made with this cell configuration re vealed some unexpected results. Perhaps the most sur prising was the fact that at high cesium pressures the maximum power obtained was equal to that of an un shielded cell with identical electrode spacing performing at the same pressure and the same emitter temperature. Hence the cell efficiency was considerably higher. The relationship of efficiency and emitter temperature for optimum cesium pressures for the shielded and com parable unshielded cell are shown in Fig. 9. It must be emphasized that the efficiencies shown are not typical of the values that can be obtained with the cell at lesser electrode separations and higher cesium pressures. The large value of the spacing used was necessitated by the crudity of the initial radiation shield design. Efficiency values quoted are those obtained by divid ing the maximum measured electrical power by the total power received by the collector. The heat transfer to the collector is measured through the use of a differential thermocouple measurement of the temperature rise of the cooling oil as it passes by the collector. This efficiency measurement neglects heat loss through the electrical (and mechanical) connections to the emitter. The efficiency curves demonstrate the effectiveness of radiation shielding. However, the maximum efficiency obtained with the shielded cell is still considerably below the maximum value attained with an unshielded cell (Sec. IV). It is to be presumed that an improved design for the radiation shield (one which, in particular, allows smaller electrode separations) will increase the cell efficiency. However, attention must be given to the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09EX PER 1M EN TAL I N V EST I GAT ION S 0 FeE S I U M P LAS MAC ELL 214i problem of exposing too much surface area to the plasma and hence draining its supply of positive ions through recombination on the exposed area. The behavior of the voltage-current characteristics of the cell for a cesium pressure of 0.5 mm Hg is demon strated in Fig. 10. It is seen that the introduction of the constriction between the electrodes has resulted in pronounced double mode behavior. The results of positive ion measurements indicate that for cesium pressures above 10-2 mm and emitter temperatures above 21000K the positive ion densities are of approxi mately the right magnitude to support the electron currents observed in the low current modes. There are not nearly enough ions to provide neutralization for the electron flow observed in the high current modes. Thus it is apparent that for these high current modes addi tional ions are being produced. It is reasonable to expect that as the cell terminal voltage is brought toward zero, electric fields are built up in the vicinity of the hole in the radiation shield so that eventually the cell current jumps to the high current mode, with the ion deficiency now removed by the formation of ions as a result of electron impact with neutral cesium atoms. In this regard it is of interest to note that the current voltage relations in the high current mode are quite similar, in shape and magnitude, especially in the range of -0.5 to 0.5 v, to those obtained for the unshielded cell ..: ... "'1.0 ...I ...I '" o /' ,I / B 2100 2300 2500 2700 EMITTER TEMPERATURE (OK) FIG. 9. Measured cell efficiency versus emitter temperature for tantalum emitter cells with and without radiation shielding. The cesium vapor pressure was 0.5 mm Hg for each curve. Efficiency measurements neglected thermal losses through the electrical connection to the emitter. The maximum measured efficiency for the unshielded cell is comparatively low because of the large emitter-collector spacing (10.8 mm). ii) IL :E ~ t-Z 1&1 II:: II:: :::I 0 Z 0 II:: t-o 1&1 ...I 1&1 1&1 :::I II:: t-100r----.-----r----.-----,----.-----,--~ 0.01 Ca VAPOR PRESS. -0.5 MM. HQ. CURVE EMITTER TEMP. I: 2385·C 2: 2150·C 3: 2025·C 4: 1905·C 0.001 '------_-!:-3-------:_2e-----_-"-, -------:Oe-----,L, ----..J2e----~3 APPLIED TERMINAL VOLTAGE (REL. TO EMITTER) FIG. 10. Current-voltage characteristics of cell with radiation shield for a cesium vapor pressure of 0.5 mm Hg and several emit ter temperatures. in the same range of high emitter temperature and cesium pressure. On referring to Fig. 3 it can be seen that a possible current path exists from the emitter, around the outside of the radiation shield support insulator, to the copper retainer. However, it is quite unlikely that electrons would choose this path because of its length and because the temperature of the cesium vapor in the region be tween the outside of the insulator and the vacuum chamber wall must be near that of the oil bath. Hence the plasma density in this region should be very low. On the other hand, since the emitter-radiation shield combination can be considered as an approximation to a cavity emitter, it is obvious that the distance from the hot plasma (at a temperature near that of the emitter) to the collector is only about 4 mm by way of the hole in the radiation shield. VI. POSITIVE ION CURRENT MEASUREMENTS Since the existence of a copious supply of positive ions is essential to the operation of the plasma cell, a certain amount of effort has been devoted to the study of the behavior of ion currents as a function of emitter temperature and cesium pressure for various cell configura tions. Another reason for studying these ion currents is that, as indicated earlier, a plot of saturated ion currents as a [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:092148 RANKEN, GROVER, AND SALMI CURVE T (OK) I 2660 2 2540 3 2415 4 2325 , 5 2180 FIG. 11. Saturation magnitudes of positive ion current versus cesium vapor pressure for several emitter temperatures as observed for a tantalum emitter cell with an emitter-collector separation distance of 13 mm. The areas of the emitter button and copper collector were each 2 cm2• Measured ion current values have been corrected for a background of photoelectron current from the collector. The dashed line indicates the behavior of the un corrected data obtained for an emitter temperature of 2540oK. function of cesium pressure is essentially a plot of the dependence of the plasma density at the collector on cesium pressure. The dependence of ion current on cesium pressure is shown in Fig. 11. The various curves were obtained for five different emitter temperatures in the range of 2180- 2660oK. The range of cesium pressure is from 2 X 1O-L 1 mm Hg. The cell used to obtain these results was of the type shown in Fig. 1. The copper collector was! in. in diam and was spaced 13 mm from the tantalum emitter. The cylindrical surface of the collector was coated with a 0.015-in. thick layer of alumina. In the pressure region of 1O-L1O-5 mm Hg it was necessary to wait 1 or 2 hr, for each change of emitter temperature, in order to allow the cesium vapor pressure in the cell to reach equilibrium with the cesium in the reservoir. A similar waiting period of 20-to 30-min duration followed each small change in the bath temperature. The observed ion currents have been corrected for background currents of photoelectrons ejected from the cesium coated collector by thermal radiation from the emitter. The dashed line in the pressure range of 10-6- 10-5 mm Hg (Fig. 11) shows the behavior of the un corrected data for the emitter temperature of 2540oK. The background corrections were first estimated roughly from the shape of the uncorrected data. Values thus obtained were modified slightly in order to give the corrected curves the expected linear dependence of ion current on cesium pressure in this pressure region. This correction procedure produced the result apparent in Fig. 11 where the data for each emitter temperature can be represented at the lowest cesium pressures by the same straight line. Further justification of the correction procedure arises from the fact that the photoelectron background current obtained in this manner had the correct dependence on the temperature of the emitter. , In the cesium pressure range of 1O-LIO-1 mm Hg it can be seen that the ion currents rise to values as high as 350 ma/cm2• There is little doubt but what this current magnitude represents a flow of ions to the collector rather than a flow of electrons away from it. In the latter instance the mechanism of electron produc tion is restricted to the possibility of significant thermi onic emission from the low work function collector (since the order of magnitudes of photoelectron currents have already been reasonably well established as much less than the reverse currents obtained at 10-1 mm Hg cesium pressure). Since the collector temperature is known it is possible to calculate the collector work func tion required to obtain a thermionic current of the magnitude observed for the reverse currents. This work function is found to be about 0.6 v. Even in the very unlikely event that the collector work function was this low (the current voltage characteristics indicate it is more like 1.4 v) the dependence of the reverse current magnitude on emitter temperature cannot be explained on a basis of thermionic emission because the variation in collector temperature with varying emitter tempera ture is not sufficiently great. When potassium is used in place of cesium in a cell of the same geometry used to obtain Fig. 11, the current voltage characteristics indicate a collector work function greater than 2 v. Yet the magnitude of the reverse current at the maximum is only about 25% less than that obtained with cesium when the emitter tempera ture is 25400K for both cells. Thus the invoking of thermionic emission to explain the observed reverse currents is not rewarding. For cesium pressures higher than about 5 X 10-2 mm Hg the ion current data is not very reproducible, both from one run to another and from one experimental cell to another. At the 13-mm emitter-collector spacing the data are consistent in that the decrease in ion current with cesium pressure invariably occurs. However, the pressure at which the decrease begins may vary by a factor of 2 or 3 from one cell to another and the slope of the decreasing portion of the ion current shows some variation. These difficulties arise from the instability of the cell at these high pressures for voltages at which saturation of the ion current appears (about - 5 to -6 v). A voltage transient, such as opening and closing a switch, may produce an arclike mode of discharge with currents on the order of tens of amperes being drawn. For emitter-collector spacings of 3 mm this discharge behavior occurs so readily (see Fig. 5) that it is im possible to obtain any realistic measurements of the true ion current for cesium pressures higher than 0.1 mm Hg. VII. DISCUSSION A full understanding of the behavior of the plasma cell as exemplified by the various experiments reported in the previous sections cannot as yet be said to exist. However, there are a number of features about cell [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09EXPERIMENTAL INVESTIGATIONS OF CESIUM PLASMA CELL 2149 operation which are understood qualitatively if not quantitatively. The role of potential barriers formed both at the emitting and collecting surfaces is an important con sideration in the interpretation of cell characteristics. Some of the effects of potential barriers at the emitting surface can be seen in Fig. 11. For the lowest cesium pressures the rate of ion production is insufficient to neutralize the electron emission. Hence, there is formed at the emitter surface a negative potential barrier whose magnitude is sufficient to limit the number of electrons entering the plasma per second to approxi mately the rate required to neutralize the ions. (For cesium this rate is about 500 times the ion production rate. However, when ion currents are being measured it drops to 250 times the ion production rate since elec trons are reflected from the collector back to the emitter whereas ions are neutralized at the collector.) Since more than enough electrons are generated to neutralize the ions, the ion currents measured at the collector are independent of the emitter temperature. As the cesium pressure in the cell is increased, with the emitter temperature held constant, the negative poten tial barrier decreases in magnitude and eventually becomes a positive potential barrier when the rate of ion production exceeds that required to neutralize the electron emission. This positive potential barrier forms because, in a plane parallel geometry and in the absence of collisions, the plasma density in the region between the emitter collector is established by the rate at which electrons are emitted. The emitted electrons are accelerated by the emitter potential sheath, reflected by the collector potential sheath (when ion current is being collected) and returned to the emitter, having suffered no net energy gain or loss in transit. The potential diagram for the plasma cell, for the case of a positive potential barrier at the emitter and non colliding electrons, is shown in Fig. 12 (a). In the dia gram the cell is at open circuit voltage so that the height of the collector barrier is established by the condition that the magnitude of the electron current reaching the collector must be limited to the magnitude of the posi tive ion current. This means that approximately 249 out of 250 electrons are reflected by the barrier and return to the emitter. For this particular case the energy of the electrons impinging against the collector barrier con sists of the usual thermal energy imparted to them by the hot emitter, plus the directed energy component imparted by the emitter potential sheath. The formation of the positive potential barrier at the emitter surface means that, even though the rate of formation of ions increases about linearly with increas ing cesium pressure, the ion current obtained at the collector should remain constant, at the value deter mined by the electron emission rate. As this rate is increased, by increasing the emitter temperature, the pressure at which the transition from negative to positive potential sheath occurs is naturally higher. (0) 4>E OPEN CIRCUIT FERMI LEVEL ____________ ::16E (b) I I EMITTER---l LCOLLEClOR SURFACE I I SURFACE 4>E OPEN CIRCUIT VOLTAGE FERMI LEVEL ___ 1 FIG. 12. Schematic diagram of electron potential energy versus distance for plasma cell at open circuit when ion production rate is significantly greater than that required to ~e~traJ!ze elec~ron emission. (a) Case where electrons make no colliSIOns m transltJo and from the collector. (b) Many-collision case. The type of behavior described in the foregoing ap pears to be occurring in Fig. 11 for cesium pressures between 10-6 and 10-4 mm Hg. However, the ion currents never become truly constant with increasing cesium pressure, possibly as a result of electron electron and electron-ion collisions not being entirely negligible. A change in the behavior of the saturation ion cur rents with cesium pressure can be seen to occur in Fig. 11 at a cesium pressure of 10-4 mm Hg. The slopes of the ion curves increase rather abruptly at this pressure. For the higher emitter temperature the ion currents once more approach a linear dependence on cesium pressure. This effect is not well understood. Certainly the discus sion of ion current behavior at low pressure does not apply when collisions become an important considera tion. In fact, for the many-collision case, the' simple procedure of equating both inward and outward going electron currents and inward and outward going ion currents at the emitter-plasma boundary suggests that, for a Maxwellian distribution of particles, the plasma density should increase as the square root of the pressure. If one takes the collision cross section for ions with neutral cesium atoms to be 0.14X 10-3 at O°C and 1 mm Hg,1O then one finds that, at l.4X 10-4 mm Hg, the mean free path of the ions is equal to the emitter-collector spacing.ll Presumably, the electron mean free path for 10 A. Von Engel, Ionized Gases (Oxford University Press, New York, 1955), p. 27. 11 In calculating this mean free path the temperature of ~he neutral cesium is taken to be that of the oil bath used to establish the cesium vapor pressure. This procedure is reasonable since only a small fraction of the cesium particles leaving the hot emitter are neutral atoms and until such time as the mean free paths for ion neutral and neutral:neutral collisions become considerably smaller [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:092150 RANKEN, GROVER, AND SALMI 10- 10- 10- CESIUM VAPOR PRESSURE FI(~. 13. Zer? current (open circuit) voltage of plasma cell as a functIOn of ceSlUm vapor pressure for several emitter temperatures. Cesium pressure is given in mm Hg. scattering by neutrals is comparable. Thus, it appears that for a pressure of 10-4 mm Hg collisions of ions and electrons with neutral cesium atoms can no longer be neglected. Hence, a correlation of the change in slope of t~e ion current curves with the onset of charged p~rtlcle-neutral atom collisions seems reasonable, espe CIally when note is taken of the fact that the pressure at which the change in slope occurs is independent of emitter temperature, and, thus, also independent of plasma density. Another effect correlated with the change of slope of the ion currents is the onset of a decrease of open circuit voltage with increasing cesium pressure. This behavior is shown in Fig. 13, which presents open circuit voltage as a function of cesium pressure for four emitter tem peratures. These data were taken with a cell having a 3-mm emitter-collector separation distance. The minima in the open circuit voltages which occur between 4 and 8X 10--2 mm Hg cesium pressure are considerably smaller in magnitude than these observed for the serrated collector cell with 2.5-mm spacing (Fig. 4). This difference may be an effect of the collector serra tions, but it appears more likely that it represents a difference in the emitter-collector spacings of the two cells. It is known that the magnitudes of the open circuit voltage minima are dependent on the emitter-collector spacing. For instance, at a spacing of 12.5 mm the open circuit voltage obtained for an emitter temperature of 25400K dips to a value of 0.6 v. The difference in the emitter-collector spacing of the two cells in question has been quoted as 0.5 mm. However, this value may be in error by an amount on the order of a millimeter since , the spacing is determined when the emitter is cold and procedures undertaken to correct for the change in than th~ emitter-collector spacing, the neutral cesium vapor must be conSidered as a Knudsen gas whose temperature is determined by the cell walls and the collector surface. As the number of free paths between the emitter and collector increases the average ~emperature. of the cesium vapor between the two ~Iectrodes will mc:ease untIl It approaches a value midway between that of the emitter a~d the collector. ~ote ~hat a consequence of this effect is that. a reglO!I of pressure will eXist for which the macroscopic cross s~ctIon for lon-neutral scattering will increase more slowly than lmearly with increasing cesium pressure. spacing as the emitter is brought up to temperature are not always accurate. In any case, Fig. 13 shows the general behavior of open circuit voltage as a function of cesium pressure. A direct comparison of Fig. 13 and Fig. 11 is not possible because of the difference in emitter-collector spacing. However, it is known from other experiments that the decrease in open circuit voltage begins at the same cesium pressure for which the ion current renews its almost linear rise with cesium pressure for a given emitter-collector spacing. The commencement of the decrease of open circuit voltage at the same pressure for which the ion currents change slope is expected on the basis of the development of a .:\laxwellian electron distribution when the effect of electron collisions with neutrals becomes important. Such collisions serve to increase the time a given electron spends in the region between the emitter and collector and, hence, the time during which this electron can exchange energy with other electrons. For cesium pres sures sufficiently low for collision effects to be treated as negligible, the voltage picture is that shown in Fig. 12(a). The collector barrier is larger than the emitter potential sheath by an amount required to turn back 249 of 250 electrons with a velocity distribution equivalent to electrons at emitter temperature. How ever, as the electron distribution becomes more random with increasing cesium pressure the anode barrier decreases. The component of velocity in the direction of the collector is gradually reduced until eventually the energy distribution of electrons in the plasma is completely Maxwellian with an electron temperature approximately that of the emitter. Although the plasma density is now much greater, the collector barrier con sists only of the kinetic energy term approximately equal to the voltage required to repel now on the order of 499 of 500 electrons. The fact that the rise in ion current in this pressure region is linear with pressure implies that the height of the emitter barrier has not changed. As a result, the open circuit voltage has decreased as shown in Fig. 12(b). While the mechanism suggested here can ac count for the correlation between the rise of plasma density and decrease in open circuit voltage, it can only account for a voltage drop of the order of a volt or less. Hence some other effect is required to explain the full magnitude of the voltage drop shown in Fig. 12. It is possible to relate the behaivor of the ion current with pressures above 10--2 mm Hg (Fig. 11) to the combined effect of ion diffusion and the recombination of ions and electrons. The calculational procedure in volves three assumptions. The first of these is that the source of ions in the plasma cell is restricted to the net numbe: per second which enter the plasma through the potentIal sheath at the emitter. The second is that the loss of ions from the plasma occurs only through re combination and through neutralization at the collector surf~ce. The third assumption is that the mean free path for lOn-neutral scattering does not vary in the region [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09EXPERIMENTAL INVESTIGATIONS OF CESIUM PLASMA CELL 2151 between the emitter and collector. This assumption is not very realistic, since the temperature of the neutral cesium atoms certainly varies in this region. However, it is required if the mathematical treatment of the problem is to be at all tractable. With the foregoing assumptions, the transfer of ions from the emitter to the collector can be represented by the steady-state diffusion equation (1) Here N is the ion or electron density and p is the recombination coefficient for ions and electrons. Da is the ambipolar diffusion coefficient for the ions and electrons. When the ion and electron temperatures are equal, Da is just twice the ion diffusion coefficient and can be represented by the relation Da= (Av)/3, where A is twice the ion-neutral scattering mean free path and v is the mean ionic velocity. When the electron tempera ture is greater than the ion temperature, Da is greater than twice the ion diffusion coefficient. This effect can be more than compensated for by calculating Da in the plasma region immediately adjacent to the emitter and, in keeping with the third assumption made above, taking this value as representative of the entire plasma region. In calculating the ambipolar diffusion coefficient, collisions of ions with electrons and with other ions are not considered. The neglecting of ion-electron collisions is of little consequence, and it seems apparent that the effect of an individual ion-ion encounter in impeding the flow of ions from emitter to collector is of much less importance than an ion-neutral encounter, since in the former instance there can be no distinction made as to which particle is which. An ion proceeding in the direc tion of the emitter may lose its momentum in this direction through collision with another ion, but the second ion will gain the momentum lost by the initial ion. In Eq. (1) it can be seen that the first term represents the net rate at which ions diffuse into a unit volume of plasma while the second term is the rate at which ions are lost as a result of recombination. For a plane parallel geometry the equation can be restricted to one dimen sion and integrated. This procedure yields dN =_(2PNJ +A)t, dx 3Da (2) where A is the constant of integration. As a first boundary condition, the diffusion current at the collector is set equal to the ion current received by the collector. This conditions may be written as (3) where nl refers to the ion density immediately adjacent to the collector sheath. This relation evaluates the constant A and Eq. (2) may be rewritten and integrated yielding -Inl [(nJv/4D,,)2- (2p/3Da)nJ3+(2p/3Da)N·J-!d~Y n = [XI dx, (4) where n is the ion density in the plasma region adjacent to the emitter. With the following substitutions Da=AV/3, a=32pAnl/9v, 17=a/(1-a), and z= 17! (N/nl), Eq. (4) may be rewritten as 3Xl a1(1-a)1/6-. 4A (5) The range of interest of values of the dimensionless parameter a is from 10-1 to 10-9 so that with little error 17t and at (1-a)116 can be replaced by al. The integral in Eq. (5) represents an elliptic function of the first kind which may be written as f dz -0.75984F(a,q,)+constant, (6) (1 +Z3)! where F(ex,q,) is given by <I> F(ex,q,)= ~ (l-sin2a sin2.J;)-tdlj;. In the solution [Eq. (6)J above, ex is 75° and q, is given by the relation z+1-v'J cosq,= , z+1+v'J where -l~z~ 00. The simplest calculational procedure is to evaluate the integral in Eq. (6) by making a binomial expansion of (1 + z3)-i for the two cases of z < 1 and z> 1 and combining the two solutions obtained by the use of the exact result for the value of the integral from zero to infinity, this value being obtained from tables of elliptic functions. Once a graph of the value of the integral in Eq. (5) has been plotted as a function of z, it is a straightforward procedure to obtain graphs of a(n/nl) as a function of !eX/A) for various values of a. These graphs form a set of master curves from which the behavior of the ion density as a function of !eX/A) may be obtained for various assumed dependences of the ion density at the emitter on cesium pressure. These master curves are shown in Fig. 14. If the ion density adjacent to the emitter is assumed to increase linearly with pressure (and A is assumed inversely proportional to pressure) then a(n/nl) is a constant. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:092152 RANKEN, GROVER, AND SALMI i ":7L"_~ ___ ~~,~" ~_,_~~ ~ ~ ~ ~ ~ 3 x 4"1: FIG. 14. Curves showing the behavior of the quantity a (n/nl) as a function of !(X/A) for several values of the parameter a=32pAnl/9v. The ordinate is essentially a constant times nA, where n is the ion density at the emitter and A is twice the mean free path for ion-neutral elastic scattering. Hence a constant value of the ordinate describes the case where the ion density at the emitter increases linearly with cesium vapor pressure if A is taken to be inversely proportional to this pressure. Various values of a(n/nl) then correspond to various emitter temperatures and curves of !(x/A)a vs !(X/A), obtained from Fig. 14, are essentially plots of the ion current versus cesium pressure. A set of such curves is shown in Fig. 15. It can be seen that the behavior of ion current versus cesium pressure is similar to that shown in Fig. 11. It has been found that the assumption of a linear increase of the ion density at the emitter with cesium pressure results in a considerably better fit to the observed data than does the assumption of a square root dependence of the ion density on cesium pressure. Fits to experimental curves such as shown in Fig. 11 require that A be evaluated as discussed previously, and that p be given a value of from 1 to 4XI0-12 cma/sec instead of the accepted value for cesium of 3 X 10-10 cm3/sec.12 The ion current curve for the highest emitter temperature shown in Fig. 11 (curve number 1) cannot be fitted with the same parameters which fit curves 2, 3, and 5 obtained at lower emitter temperatures. It may be that this discrepancy results from a breakdown of the assumption that the ion source is restricted to ions entering the plasma from the emitter, that is, photo ionization and ionization by electron impact with cesium atoms may be relatively more important for very high emitter temperatures. In Fig. 15, it is apparent that for high cesium pres sures the ion density at the collector becomes inde pendent of the ion density at the emitter. This limiting curve is given by the relationship nl= 14.6(v/ p)(A2/xB) for a«1. The equation indicates that the ion current at 12 See p. 141 of work cited in footnote to. the collector eventually falls off as the inverse square of the cesium pressure for a given emitter-collector spacing. In Fig. 4(e) and Fig. 10 there appeared instances of a double valued dependence of current on voltage. This double mode behavior of the plasma cell in the region of high pressure and relatively large electrode spacing can be understood, in part at least, in terms of positive ion deficiency and hence electron space charge limitation in the region near the collector. The results of the experiments with the radiation shielded cell (Sec. V, Fig. 10) demonstrated that the positive ion currents obtained at the high pressures are of about the right magnitude to neutralize the electron current obtained in the low current mode. The high current mode requires a source of ionization other than the formation of iom at the emitter. A similar situation exists for the cell with 12-mm electrode spacing at high cesium pressure. As a result of FIG. 15. Behavior of the quantity HX/A)a as a function of !(X/A) for various values of a(n/nl), corresponding to various emitter temperatures. The ordinate is equal to a constant times the ion density at the collector for the case of X=XI, the emitter collector separation distance. Hence these curves represent the behavior of collector ion current as a function of cesium vapor pressure when the ion-neutral scattering mean free path is taken to be inversely proportional to the cesium pressure. recombination, the ion density at the collector can fall below the level required to pass all the electron current from the emitter. Hence a space charge barrier will become established in the region near the collector and the cell current will satuarate at a level determined by the ion density at the collector. As in the radiation shielded cell, if an additional source of ions is provided, the current will increase to values apparently limited by the emission of the electron source. The double mode behavior which occurs for a tanta lum emitter cell with 12-mm spacing and a 51-mm diam collector is depicted in Fig. 16. With emitter tempera ture at 2175°K [Fig. 16(a)] two modes are not observed. At a pressure of O.S mm the current increases continu ously with voltage. For the two higher pressures the current levels off at values which decrease with pressure. Electron accelerating fields in the cell are not sufficient [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09EX PER 1M EN TAL I N V EST I GAT ION 5 0 FeE 5 I U M P LAS MAC ELL 2153 to increase the electron temperatures to the point where ionization by collision can occur frequently enough to maintain the high current discharge mode. At the next higher emitter temperature [Fig. 16(b)] a double mode exists for pc.= 1 mm. For the lower pressure only the high mode exists and for the higher pressure only the low mode is observed. Further increase in emitter temperature [Fig. 16(c)] brings about the appearance of double modes at pc. = 2 mm. For Tp=2540oK, PC8= 1 mm, a discontinuity in the slope of the current voltage characteristic is the only indication of the tendency for double mode operation. It can be seen that the minimum voltage for which the high current can be sustained increases with increase in cesium pressure and decreases (becomes more negative) with increasing emitter temperature. Other experiments have demonstrated that for a given emitter temperature the double mode behavior will first appear at a fixed Z 10 r--~------" -----,----,----,--r----,--..,.-,I -01 ---'1 ~ (f) t CURVES C. PRESSURE ( .... H,) .... I :r 0.5 (d) T .. 2540.K ffi 10' lr 1.0 IL JI[ 2.0 :I C I 0 ~ 10 (.) .... 2175 .K .... a:: a:: :::l of (,) 10 Q .... > a:: ·z .... 10 (f) CD o I~~~~ __ ~_~I ~I __ ~I __ ~I~ __ ~I __ L-~~ -3 0 -2 -I 0 -2 -I 0 -2 -I 0 -2 -I CEll TERMINAL VOLTAGE FIG. 16. Current-voltage characteristics at high cesium pressures for tantalum emitter cell having collector-emitter separation dis tance of 12-mm and 51-mm diam copper collector. Each set of curves represents the characteristics at the given emitter tem perature for cesium pressures of 0.5, 1.0, and 2.0 mm Hg. value of pressure times electrode separation distance, a result to be expected from similarity considerations. The general behavior of the transitions from low to high current modes is closely analogous to the transi tions to temperature limited arcs described by Malter et al.13 as occurring in rare gases. Malter also observed that such arcs could be maintained even when the voltage applied across the gas was reduced to negative values. 13 L. Malter, E. O. Johnson, and W. M. Webster, RCA Rev. 12, 415 (1951). The identification of cell behavior at high cesium pressures as arclike suggests a potential distribution observed by Compton and Eckart14 for a mercury vapor arc wherein the emitter barrier is about equal to the ionization potential (of cesium, in this case) and the plasma potential decreases from this maximum to values near zero at the collector. For the case where the emitter work function is higher than that of the collector the voltage at the collector can be negative.Ia,15 In the body of the plasma the electron current is sustained by forces of thermoelectric origin, i.e., electron temperature and concentration gradients. Confirmation that the plasma cell is generating a temperature limited arc discharge at high cesium pres sure and low cell voltage has recently been obtained from observations made in the Los Alamos Laboratory.16 These observations were made through the use of a tantalum emitter cell equipped with a sapphire window to permit visual (and spectrographic) observation of the plasma region between the emitter and collector. Visual inspection of this region demonstrated that the dis continuous transition from the low current to high current mode was accompanied by the abrupt appear ance of a yellow glow which filled the region between the emitter and collector. In the low current mode of cell operation no light from the plasma region would be distinguished from the background of scattered light originating from the emitter. In the high current mode increasing the cell voltage in a positive sense (to and beyond the condition of short circuit) resulted first in the intensification of the glow. In the region from zero to one volt positive the cell current increased rapidly with voltage while at the same time the light from the discharge could be seen to spread across the tantalum support disk. This indicates, as suggested in Sec. III, that the high currents obtained at high cesium pressure for positive cell voltages result from the fact that a large region of the tantalum support disk is furnishing elec trons to the plasma under these conditions. ACKNOWLEDGMENTS The authors would like to acknowledge the assistance of D. J. Kelly, R. E. Aamodt, D. J. Roehling, and G. F. Erickson in obtaining various portions of the experi mental data described herein. 14 K. T. Compton and C. Eckart, Phys. Rev. 25, 139 (1925). 15 See p, 254 of work cited in footnote 10. 16 L. E. Agnew, Los Alamos Scientific Laboratory (private communication). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.193.164.203 On: Sun, 21 Dec 2014 20:26:09

1.1742600.pdf

ProtonDeuteron Hyperfine Structure in Paramagnetic Resonance: A πδ Interaction Balu Venkataraman and George K. Fraenkel Citation: The Journal of Chemical Physics 24, 737 (1956); doi: 10.1063/1.1742600 View online: http://dx.doi.org/10.1063/1.1742600 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Measurement of the protondeuteron radiative capture process AIP Conf. Proc. 768, 143 (2005); 10.1063/1.1932893 Polarized Structure Functions: Proton/Deuteron Measurements in Hall C AIP Conf. Proc. 747, 95 (2005); 10.1063/1.1871645 Electron Spin Resonance of Irradiated Single Crystals of Alanine: Proton—Deuteron Exchange Reaction of a Free Radical in the Solid State J. Chem. Phys. 40, 3328 (1964); 10.1063/1.1725002 Oxygen17 Hyperfine Structure in Electron Paramagnetic Resonance J. Chem. Phys. 37, 1879 (1962); 10.1063/1.1733382 Hyperfine Structure in Paramagnetic Resonance Absorption Spectra J. Chem. Phys. 25, 1289 (1956); 10.1063/1.1743211 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.88.90.140 On: Tue, 02 Dec 2014 13:55:01CELL AND CELL-CLUSTER MODELS FOR LIQUIDS 737 If one neglects contributions from multiply occupied cells, I1(a)1(b) f f exp{ -(3(if;1(a)+if;1(b»)}dXl(a)dX1(b) (50) This integral will be evaluated approximately by ex panding gl(a)l(b) in powers of (3(<I>l(a)l(b)-W1(a)l(b) -W1(b)l(a)+U 1(a)l(b»), neglecting cubes and higher powers. The linear term vanishes when inserted in (50), because of the definitions in (18) and (24) ofW1(a)l(b) and U1(a)l(b). Hence Kab/(KaKb) is approximately the average value of !(32{<I>I(a)l(b)2- U1(a)l(b)2}, the average being calculated with the weight factor exp{ -(3(W1(a) +W1(b»)}, or, again approximately, the average value THE JOURNAL OF CHEMICAL PHYSICS TABLE III. (v.lv)' 0.7 0.6 0.5 0.4 0.3 0.2 0.1 (J(").. •• -)...) 0.028 0.033 0.029 0.022 0.016 0.013 0.009 p(p'-P)!i* +0.076 -0.042 -0.067 -0.045 -0.015 -0.003 -0.002 of !f32{Wl(a)1(b)2- U 1(a)1(b)2}, the average being calcu lated with the weight factor exp( -(3Wl(a»). The values of (3 CAao-Aa) calculated in this way for (3A = 10 are shown in Table III. In the last row are the corrections con tributed by clusters of two cells to the values of (3pv* calculated in the previous section. It will be noticed that the quantitative effect of these corrections is to this approximation not great, and to this extent the cell model is vindicated. But the introduction of cell clusters is necessary if one wishes to achieve a self consistent model for a particular cell size. Actually this self-consistency remains rather theoretical in the above calculation: one still cannot use (39) to calculate the pressure, owing to the fact that multiple occupation of cells has been disregarded. However, one is assured that if more detailed calculations were made, no mathe matical contradiction would be found. VOLUME 24, NUMBER 4 APRIL, 1956 Proton-Deuteron Hyperfine Structure in Paramagnetic Resonance: A 7C-a Interaction* BALU VENKATARAMAN AND GEORGE K. FRAENKEL Department of Chemistry, Columbia UniVersity, New York 27, New York (Received July 18, 1955) The mechanism of the hyperfine splitting observed in paramagnetic resonance spectra of certain aromatic free radicals containing protons adjacent to the aromatic ring is examined. It is shown that the magnitude of the interaction between an unpaired electron in a 7r orbital and a vibrating hydrogen atom is insufficient to account for the observed splitting. Such a vibrational mechanism is considered to be untenable: (i) on the basis of quantitative calculations, (ii) by the failure to observe lines corresponding to the first excited vibrational state, and (iii) by a comparison of the splitting caused by protons and deuterons. It is sug gested that the unpaired electron is not in a purely 7r state and that the splitting may be accounted for by configuration interaction between 7r and u states. RECENT studies of the paramagnetic resonance spectra of solutions of aromatic free radicals have shown the existence of small but complex fine structure,I-3 It was tentatively concluded that this structure was caused by intramolecular hyperfine interactions between protons and the unpaired electron, and a confirmation of this hypothesis was provided by performing complete analyses of the spectra of methyl substituted and unsubstituted p-benzosemiquinone * This research was supported in part by the United States Air Force through the Office of Scientific Research. 1 Weissman, Townsend, Paul, and Pake, J. Chern. Phys. 21, 2227 (1953). 2 S. I. Weissman, J. Chern. Phys. 22, 1135 (1954); Lipkin, Paul, Townsend, and Weissman, Science 117, 534 (1953); Chu, Pake, Paul, Townsend, and Weissman, J. Phys. Chern. 57, 504 (1953). 3 H. S. Jarrett and G. J. Sloan, J. Chern. Phys. 22, 1783 (1954). ions.4.6 One interesting feature of the spectra, which is considered in detail in the present article, is the existence of a splitting, of the order of two gauss, attributable to the protons on the aromatic ring. Clearly, since the unpaired electron in a conjugated system is in a 11' state, and since the splittings measured in solution are caused by an unpaired-electron density at the proton in question,6 a proton in the plane of the aromatic ring, which is the nodal plane of a 1J' orbital, should not give rise to any interaction. Weissman and co-workers have suggested that the requisite splitting 4 B. Venkataraman and G. K. Fraenkel, J. Am. Chern. Soc. 77, 2707 (1955). 6 B. Venkataraman and G. K. Fraenkel, J. Chern. Phys. 23, 588 (1955). 6 S. 1. Weissman, J. Chern. Phys. 22, 1378 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.88.90.140 On: Tue, 02 Dec 2014 13:55:01738 B. VENKATARAMAN AND G. K. FRAENKEL could arise from the zero-point vibrations of the protons,! and Jarrett and Sloan have proposed an indirect coupling between the electron and proton moments through the chemical bond,3 of a type similar to the coupling between nuclear moments observed in nuclear magnetic resonance spectra.7,s An account is given in the following of three different approaches that have been used to study the nature of this inter action: the magnitude of the splitting has been calcu lated on simple theoretical grounds; a search has been made for the spectrum that should arise, assuming the existence of a vibrational mechanism, from the first excited vibrational state; and a study has been made of the spectrum of the deuterated p-benzosemiquinone ion. The calculation of the magnitude of the interaction caused by a vibrational mechanism proceeds as follows: The Hamiltonian contains a term that depends on the nuclear spin of the form L i a;S .1;, where S and I; are electron and nuclear spin angular momenta, respec tively, in units of h/2'1T', and the summation is extended over all nuclei possessing magnetic moments. The expec tation value of the parameter ai is given by a formula due to Fermi9: ai=(8'1T'/3)g{3gi{3N!,p(i)J2Av, where g is the spectroscopic splitting factor of the electron (g= 2.0023 for a free electron), {3 is the Bohr magneton, gi is the nuclear g factor of the ith nucleus, and (3N is the nuclear magneton. !,p(i) !2AV is the density of the unpaired electron at the ith nucleus averaged over the vibrational motions; the average introduced here is an appropriate modification of Fermi's result. The calcu lation of the hyperfine splitting thus requires an evaluation of ! ,p(i) !2AV' We first calculate the interaction for one 'IT' electron on a carbon atom adjacent to a proton. It is assumed that the carbon orbital is adequately represented by a Slater-typelO 2p orbital (the Hartree-Fock wave function of Torrance gives similar resultsll) and that the vibration is equivalent to a one-dimensional oscillator performing simple harmonic motion in a direction parallel to the direction of the p-orbital. The average over the vibra tional motion of the density of the 2p orbital at a distance x from the carbon atom, in a direction per pendicular to the orbital, is given by where n is the vibrational quantum number, m is the effective reduced mass of the appropriate normal mode with frequency II, and ao is the Bohr radius. For sim plicity, we assume that m is the mass of the proton and II is the frequency of the perpendicular C-H bonding 7 Gutowsky, McCall, and Slichter, J. Chern. Phys. 21, 279 (1953). 8 N. F. Ramsey, Phys. Rev. 91,303 (1953). 9 E. Fermi, Z. Physik 60, 320 (1930). 10 J. C. Slater, Phys. Rev. 36, 57 (1930). 11 C. C. Torrance, Phys. Rev. 46, 388 (1934). mode in benzene (II/C= 671 cm-I),12 Although the proper decomposition of the motion into normal modes is not carried out, the result will not be seriously in error. By using this formula for ! ,p2p !2AV, the value of the splitting constant for one 2p-carbon orbital and a proton exe cuting zero-point vibrations at a distance of 1.08 A (C-H bond distance) is found to be ai= 0.68 gauss. To determine the interaction constant for a semi quinone molecule, the coefficient of the 2p wave function for a carbon atom adjacent to a proton in the total orbital of the unpaired electron must be determined. An estimate of the coefficient was made by performing a rudimentary molecular orbital calculation in which all but nearest neighbor exchange integrals and all overlap integrals were neglected,13 and in which the Coulomb integral for oxygen was assumed to differ from that for carbon by the resonance integral {3. (Hoo-Hcc={3 in Coulson's notation.) The unpaired electron occupies the fifth orbital (energy E=Eo+0.2541(3) and the wave function is ,p6=0.4700(4)0-4>0,)+0.3505 (4)1-4>4) -0.2795 (4)2-4>3-4>6+4>6), where 4>1 to 4>6 are the six 2p-carbon orbitals numbered according to the usual chemical convention, and 4>0,4>0' are the 2p-oxygen orbitals. The calculated splitting constant for a proton in the p-benzosemiquinone ion is therefore aH= (0.68)X (0.2795)2=0.053 gauss. This estimate of 0.053 gauss is about one-fiftieth of the experimental value of 2.4 gauss. In all probability, because of the tendency for the 'IT' orbital to follow the vibrational motion and thus remain perpendicular to the C-H bond,t4 this calculated value is an over estimate. Conventional ideas about 'IT' orbitals and vibra tional motion cannot therefore account for the experi mentally observed splitting. It is conceivable that a vibrational mechanism is possible but that for some unexplained reason the mag nitude of the wave function of the unpaired electron at the protons is in error. If this were the case, the spectrum of molecules in the first excited vibrational state should be detectable, and the splitting constant for the excited molecules should differ from the constant for molecules in the ground state. Assuming a frequency of 671 cm-t, the population of molecules in the first excited vibrational state at room temperature is about 4% of the population of the ground state. Since by this mechanism the splitting constant is determined by the mean-square vibration amplitude, the constant for the excited state should be three times the constant for the 12 G. Herzberg, Molecular Spectra and Molecular Structure: II. Infra-Red and Raman Spectra of Diatomic Molecules (D. Van Nostrand Company, Inc., New York, 1954), first edition, p. 363. E. B. Wilson, Phys. Rev. 45, 706 (1934). 13 C. A. Coulson, Valence (Oxford University Press, New York, 1953), first edition, p. 238. 14 P. ]. Wheatley and J. W. Linnett, Trans. Faraday Soc. 45, 897 (1949); D. F. Eggers, Jr., J. Chern. Phys. 23, 221 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.88.90.140 On: Tue, 02 Dec 2014 13:55:01H Y PER FIN EST Rue T U REI N PAR A MAG NET I eRE SON A NeE 739 ground state, and therefore all but the central line of the spectrum of the excited state of the p-benzosemi quinone ion should occur outside the spectrum due to the ground-state vibration. A search, which was made under conditions of spectrometer sensitivity such that a 1 % rather than a 4% contribution of the excited state should have been detectable, failed to reveal the presence of these lines. The above argument assumes that the line width in the excited state is the same as in the ground state. The line width is probably determined by the lifetime of a spin state, and if a vibrational mechanism were respon sible for the hyperfine interaction, the lifetime of a spin state would be determined by the lifetime of the vibrational state. Since these lifetimes are controlled by collisions, it is unlikely that the lifetime of an excited vibrational state should be markedly different from the lifetime of the ground state and, therefore, the widths of lines from ground and excited states should be of the same order of magnitude. Furthermore, since the lines have a width of only 0.3 gauss or about 1 Me/sec, a spin state persists for the order of 107 oscillations of a vibrational state with frequency 671 cm-I. Since a life time of 10-6 sec for a vibrational state in solution is excessively long, the narrowness of the observed ab sorption lines also points to the impossibility of a vibrational mechanism for the hyperfine structure. A third prediction can be made from the theory of the vibrational mechanism: since the ratio of the splitting constants for protons and deuterons (aH and aD, respectively) will be (aH/aD)= (gH/gD)(!,p(H) 12Avll,p(D) !\,), a measurement of the experimental ratio would give the ratio of the average odd-electron density at the protons and the deuterons. If a vibrational mechanism is responsible for the splitting, the mean odd-electron density is determined by the mean-square vibration amplitude which, in turn, is inversely proportional to the square-root of the mass of the vibrator. Neglecting the proper normal modes and reduced mass, this implies that, since gH/gD=6.514,t5 aH/aD"-'V2 6.5=9.2. If a vibrational mechanism is not primarily responsible for the splitting, a ratio of aH/aD=6.514 would be expected. The deuterated p-benzosemiquinone ion was prepared from the deuterated hydroquinone by atmospheric oxidation of the hydroquinone in alkaline ethanol solution. The deuterated hydroquinone was obtained by allowing ordinary hydroquinone to exchange with 0.1 M NaOH in D20 at 150°C for three days16; the exchange was carried out in a sealed tube in the absence of oxygen to avoid oxidation of the hydroquinone. The deuterated hydroquinone was recovered by freezing the tube in dry 15 N. F. Ramsey, Nuclear Mommts (John Wiley and Sons, Inc., New York, 1953), p. 78. 161. P. Gragerov and A. I. Brodskii, Doklady Akad. Nauk S. S. S. R. 79, 277 (1951); 1. P. Gragerov and G. P. Miklukhin, Zhur. Fiz. Khim. 24, 582 (1950). FIG. 1. Paramagnetic res onance spectrum of partial ly deuteratecl p-benzosemi quinone ion. Upper half: entire spectrum. Lower half: the low field portion of the spectrum including part of the central peak. ice before opening the tube to the atmosphere, adding sufficient HeI to the opened tube to neutralize the alkali, and then evacuating the tube before the mixture was allowed to thaw. The slightly acidic solution thus obtained was dried under reduced pressure, the residue was extracted with ether four or five times, and the residue obtained by evaporating the ether extracts was recrystallized from 99% D20Y The predicted spectrum of the completely deuterated p-benzosemiquinone ion consists of nine equally spaced lines with intensity ratios 1:4:10:16:19:16:10:4:1. The predicted spectrum for the deuterated ion with three deuterons and one proton consists of fourteen lines which, taking the central line of the completely deuterated species as the origin, would be spaced at ± (3aD+aH/2), ± (2aD+aH/2), ± (aD+aH/2), ±aH/2, ± (aD-aH/2), ± (2aD-aH/2), ± (3aD-aH/2), and would have intensity ratios 1:3:6:7:6:3:1:1:3:6:7: 6: 3: 1. If the ratio aH/ aD"-'6.5, all but four lines (the two extreme pairs) of the partially deuterated molecule containing three deuterons and one proton would lie very close to the lines of the completely deuterated species and as such would not be distinguishable if the partially deuterated compound were at very low con centration. The actual spectrum contains thirteen lines, in agreement with a prediction based on a mixture of the fully deuterated and the 3-1 partially deuterated molecules. The eleven strongest lines are shown in Fig. 1; the two outside lines are too weak to be seen on the scale used in the photograph. The other partially deuterated molecules containing fewer deuterons could not be detected in our sample. The failure to detect these other partially deuterated 17 Ordinary water probably could have been used for the recrystallization. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.88.90.140 On: Tue, 02 Dec 2014 13:55:01740 B. VENKATARAMAN AND G. K. FRAENKEL molecules is consistent with the relative concentra tions predicted by assuming equilibrium between all protons and deuterons in the reaction mixture. If the isotope effect is neglected, the concentrations of the molecules containing 4, 3, 2, 1, and 0 deuterons in the ring would be, respectively, in the ratio d4:4d3h:6d2h2: 4dh3:h4, where d and h are the fraction of hydrogen isotopes in the reaction mixture in the form of deuterons and protons, respectively. The value of d calculated from the stoichiometric proportions used in the reaction mixture is 0.986; while that estimated from the spectrum (assuming the ratio of the 4-deuteron compound to the 3-deuteron compound is given by d/4h) is 0.97. This discrepancy is presumably caused by incomplete equi librium and by the neglect, in the calculations, of the isotope effect. The value of d=0.97 means that the mixture contained at least 80% of the completely deuterated species. The hyperfine splitting constant for the deuteron in the p-benzosemiquinone ion, aD, was found to be 0.365 gauss. This value is somewhat uncertain because of overlap between the hyperfine components and con tamination of the spectrum by the 3--1 partially deu terated species. To obtain this value the spacings between the different peaks of the spectrum had to be corrected for overlap. The spectrum was assumed to consist of individual lines arising from the fully deu terated and 3-1 partially deuterated molecules and an estimate of the ratio of the concentration of the two types of molecules was made. This estimate was ob tained by comparing the observed spectrum, using successive approximations, with a computed derivative spectrum assuming that the lines were of Lorentz-type shape with identical widths.4 Since the line width of the deuterated compound depended slightly on the concen tration of the radical, which in turn decreased with time, only an approximate estimate of the line width could be made. The width between extrema on the derivative of the spectrum was found to be approximately 0.18 gauss, corresponding to a full width at half-intensity for a Lorentz line of 0.32 gauss. The uncertainties in the line width and concentration of the partially deuterated species made the corrections for overlap somewhat uncertain, and we estimate that the splitting constant an may be in error by as much as 2%. The splitting constant of the protons in the p-benzo semiquinone ion was remeasured and found to be 2.366±0.006 gauss. The ratio is therefore aH/ aD= 6.48 to within about 2%. Since the value of gH/gD=6.514, these measurements indicate that the unpaired-electron density at either deuterons or protons is the same within experimental error. The value for this ratio of about 9 predicted on the basis of a mechanism involving vibra tion of the hydrogen atoms is clearly incompatible with the experimental observations, and therefore such a vibrational mechanism can only contribute a minor amount to the splitting. In the search for a satisfactory explanation of the hyperfine splitting, the suggestion of Jarrett and Sloan3 that an indirect coupling between the electron and proton moments through the chemical bonds, similar to the coupling between nuclear moments observed in nuclear magnetic resonance spectra,8 has to be con sidered. Not only have no quantitative estimates of this mechanism been made but, in addition, it is not clear that such a mechanism is qualitatively correct: no demonstration has been given that a nonvanishing S . I interaction of this nature could arise from an electron in a 'If' orbital. We are thus at present forced to find another mechanism. It is suggested that the unpaired electron is partly in a 0' state; this hypothesis is a contradiction of the previous assumption that the odd electron is in a purely 11" state. Just as the hyperfine interaction between the unpaired electron and the methyl-group protons in methyl-substituted semiquinones is believed to be due to hyperconjugation,6 it is suggested that the inter action between the unpaired electron and the ring proton is due to configuration interaction between 0' and 'If' states. Configuration interaction can give rise to an odd-electron density at the ring protons either by uncoupling one C-H 0' bond, promoting one of the 0' electrons to an excited 0' state, or by unpairing two such bonds simultaneously and promoting one electron from each bond to 11" states. Altmann has pointed out that the usual Huckel approximation for the treatment of the energy levels of conjugated systems, in which the 'If' electrons and 0' electrons are considered independently, is adequate for the ground state, but that the effect of 'If'-0' interactions is of great importance in computing exited state energies.1s Although such a 11"-0' interac tion would leave a finite odd-electron density at the ring protons, only quantitative calculations along the lines suggested by Altmann for the ethylene molecule19 could show whether or not this effect is of sufficient magnitude to account for the interaction constant observed. ACKNOWLEDGMENTS The authors gratefully acknowledge numerous helpful discussions with Professor George E. Kimball and Professor Richard Bersohn. 18 S. L. Altmann, Proc. Roy. Soc. (London) Al10, 327 (1951)_ 19 S. L. Altmann, Proc. Roy. Soc. (London) A2l0, 343 (1951). This article is copyrighted as indicated in the article. 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1.1735414.pdf

Dipole Mode of Minority Carrier Diffusion with Reference to Point Contact Rectification B. R. Gossick Citation: Journal of Applied Physics 31, 29 (1960); doi: 10.1063/1.1735414 View online: http://dx.doi.org/10.1063/1.1735414 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Point contact Andreev reflection from semimetallic bismuth—The roles of the minority carriers and the large spin-orbit coupling J. Appl. Phys. 113, 17C718 (2013); 10.1063/1.4796049 Epitaxial silicon minority carrier diffusion length by photoluminescence J. Appl. Phys. 104, 054503 (2008); 10.1063/1.2973461 Imaging transport for the determination of minority carrier diffusion length Appl. Phys. Lett. 88, 163509 (2006); 10.1063/1.2196236 Effect of Vacuum Heating and Ion Bombardment of Germanium on Point Contact Rectification J. Appl. Phys. 27, 525 (1956); 10.1063/1.1722416 Some Problems in the Diffusion of Minority Carriers in a Semiconductor J. Appl. Phys. 25, 99 (1954); 10.1063/1.1721530 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:49JOURNAL OF APPLIED PHYSICS VOLUME 31, NUMBER 1 JANUARY, 1960 Dipole Mode of Minority Carrier Diffusion with Reference to Point Contact Rectification B. R. GOSSICK Physics Department, Arizona State University, Tempe, Arizona (August 5, 1959) The dipole mode of minority carrier diffusion about a spherical emitter is presented. The current-voltage relationship, and frequency characteristics of this mode are determined. Compared with the unipole mode, which has been treated extensively, the dipole mode offers superior high-frequency performance, which is partially offset by an inferior dc characteristic curve. A representative numerical example gives 40 mllsec mean response time with the unipole mode and 0.5 mllsec with the dipole mode. It is proposed that the reproducibility of special diode characterictics might be improved by designing diodes to suppress either the unipole or dipole mode. Methods are suggested for the suppression of either mode. I. INTRODUCTION THE behavior of rectifying point contacts which inject minority carriers has been analyzed by solving the heat equation with only one space coordi nate, viz., the radial distance about the point.1-6 The result is that in the work reported to date, all spherical harmonics in the general solution have been ignored save one. The intention here is to obtain an answer to the question: What is the relative importance of the dominant higher mode of excitation, as compared with the fundamental mode? The answer happens to have a significant bearing on diode performance, and therefore it is important. Although we refer here to point contact rectifiers, the discussion applies not only to rectifiers with a metal point contact on n-type germanium, but to hemispheri cal alloy junctions, "gold-bonded" diodes, and in fact any semiconducting device with a rectifying barrier of the type indicated in Fig. 1. In the discussion hereafter, minority carriers will be represented by holes. II. FEATURES OF UNIPOLE AND DIPOLE MODES OF HOLE INJECTION Although the unipole mode has been used as a model for point contact rectification, it appears impossible to design a practical experiment to illustrate the case. We resort to a "dedanken experiment," in which an imaginary electrical connection couples a central sphere with one terminal of a generator, and a real electrical connection couples a surrounding concentric shell with the other terminal of the generator (Fig. 2). The central sphere is an injecting contact. The surrounding concentric shell is an ohmic contact, and the concentric layer in between is an n-type semiconductor. With such * Work supported by U. S. Atomic Energy Commission Contract. 1 P. C. Banbury, Proc. Phys. Soc. (London) B66, 833-840 (1953). 2 Melvin Cutler, Phys. Rev. 96, 255-259 (1954). 3 H. L. Armstrong, J. App!. Phys. 27,420 (1956). 4 B. R. Gossick, J. App!. Phys. 27, 905-911 (1956). 6 E. Hofmeister and E. Groschwitz, Z. angew. Phys. 10, 109-114 (1958). 6 Heinz Beneking, Z. angew. Phys. 10, 216-225 (1958). geometry, the flow of holes is radial from the spherical injecting contact, which typifies the unipole mode of minority carrier diffusion. The excess concentration of holes about the injecting sphere obeys, for the unipole mode, the relation p(r)-p.= p.(eqV/kT -1)ro/r, (1) in which V represents applied voltage; k, T, and q denote respectively Boltzmann's constant, absolute tem perature, and electronic charge; p. denotes equilibrium hole concentration; ro, the exterior boundary of the space charge region (Fig. 1); and r, the radial coordi nate. The hole current through one hemisphere is given by the well-known Wagner7 relation I(V)= I. (eqV/kT -1). (2) The saturation current I. stands for the expression 27rrolT pkT / q in which IT p gives the conductivity due to holes. The unipole mode of hole diffusion has been applied to rectifiers with geometric design typified by Fig. 3. The n-type semiconducting die is either a rectangular parallelopiped or right cylinder, with an ohmic contact across the bottom surface, and a rectify ing point contact on the center of the opposite surface. 29 'ERi N-TYPE SEMICOMlUCTOR i I I I I I I I I I I I I v.o-}i-~~ __ ~~~~~~BA~ND~ ____ ~I ~------,------------------------I I FERMI LEVEL 1 I I I I: I I I VALENCE BAND I I I : I I I I I I t-o ,arm "'0 r=~ FIG. 1. Profile of potential energy of electrons shown with the diode in equilibrium. The radii rm, To, and T1 refer to the radius of the metal contact (rm) , the exterior radius of the space charge layer (ro), and the distance between the center of the injecting contact and the boundary of the ohmic contact Crt) as shown in Fig. 4. 7 C. Wagner, Physik. Z. 32, 641 (1931). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:4930 B. R. GOSSICK I I I I I , I , _______________ J FIG. 2. Hypothetical arrangement to illustrate spherically sym metric diffusion of minority carriers, i.e., by the unipole mode. The flow pattern of hole current injected from the point contact has been sketched to conform reasonably to the pattern required for the unipole mode shown in Fig. 2. However, such a pattern can be attained only with either a very large surface recombination velocity on the sides of the die, or by extending the ohmic contact to the sides. On the contrary, the free surfaces are normally etched to minimize leakage current about the point contact. Therefore, in practice, the current density is nonuniform across the surface of the emitting contact, less current being emitted from the sides and more being emitted from the center. It becomes evident in the next paragraph that such a departure from the unipolar flow pattern leads to the characteristic pattern of the dipole mode. One can, at least in principle, design an experiment to illustrate the dipole mode of hole diffusion. Such an arrangement is shown in Fig. 4. The injecting contact floats electrically in the center of the semiconducting sample. While one hemisphere of this contact emits holes, the opposite hemisphere collects them. In order to discuss this case, it is necessary to introduce another coordinate, viz., the angle e between the applied field E and the radius vector r. The excess concentration of holes about the central sphere obeys, for the dipole mode, the relation p(r,e) -pe= Pe(eQV cos8/kT -1)rNr2. (3) As defined here, the term applied voltage V refers to one hemisphere in the case of the dipole mode (Fig. 4). The purpose for so defining applied voltage is to make it serve for the practical scheme illustrated in Fig. 6. The density of radial hole current at the edge of the space FIG. 3. Conventional point contact-semiconductor die arrangement with current How pattern typical of theunipole mode. charge layer may be written J(ro,e) = 20'pkT(eQV cosB/kT -1) (qrO)-l. (4) We have also for the hole current through one hemisphere, J(V)= 2I8[kT(eQV/kT -1) (qV)-L 1]. (S) The current-voltage characteristic (5) is indeed dif ferent from the Wagner law (2), as may be seen in Fig. S. The ultimate reverse current with the dipole mode is twice as great as that with the unipole mode, and the forward current with the dipole mode is somewhat lower as compared with the unipole mode. The dipole mode of hole diffusion is applicable to rectifiers with the geo~ metric design sketched in Fig. 6. Once more, the n-type semiconducting die may be either a rectangular parallelopiped or right cylinder, with an ohmic contact across the bottom surface, and a rectifying contact on the center of the opposite surface. The flow pattern is just what might be expected with a well-etched die. T v 1-T -1 FIG. 4. Schematic arrangement to illustrate the diffusion of minority carriers by the dipole mode. A small spherical injecting contact floats electrically in the center of a semiconductor. The lower hemisphere of the central contact emits, while the upper hemisphere collects holes. The hole current which bypasses the spherical contact is not shown. Therefore, from the viewpoint of the practitioner, the dipole mode of minority carrier diffusion merits as much study as the unipole mode. The dipole mode exhibits a shorter injection recovery time than the unipole mode for the following reason. The excess holes obey the relation l/r with the unipole mode, but are distributed as 1/r2 with the dipole mode. Hence the total injection is localized nearer the contact with the dipole mode, and can be collected faster when the diode is cut off. This advantage of the dipole mode is partially offset by a longer time constant for the space charge layer with forward bias, because the slope of (S) is less than the slope of (2) with V>O. In order to illustrate briefly the main features of dipolar diffusion, some details have been brushed aside which will be considered in the following sections. III. DIPOLE SOLUTION OF THE HEAT EQUATION The analysis which follows employs a set of initial postulates and corollaries stated in an earlier paper.4 Only minority carriers will be discussed. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:49DIPOLE MODE OF MINORITY CARRIER DIFFUSION 31 The equation of continuity for excess hole concentra tion [pl(1',B)= p(1',B)-P.] provides the basis from which the hole current will be determined: (6) Symbols D and r' denote, respectively, the diffusion coefficient and lifetime of holes. The initial statement (6) will be solved for the region outside an injecting sphere (1'0<1') as illustrated in Fig. 4. subject to the boundary conditions (7) and FIG. 6. Conventional point contact-semiconductor die arrangement with current flow pattern typical of the dipole mode. and 1 d d8 (10) pl(ro,B,t) = p.(eQV(t) cos8/kT -1). (8) with --sinO-+l(l+1)8=0 sinO dO dfJ ' The first boundary condition (7) specifies that the flow 18 • I I I :1 I ~ 6 I I I I 14 I , I , 12 , , ____ un/llOle mode I I 10 I _dipole mode I , , 8 , I , , , , FIG. 5. Current-voltage characteristic curves for the unipole and dipole modes. of holes from the injecting sphere must be stable. The second boundary condition (8) comes from the Boltz mann distribution.4 For example, when the injecting sphere (Fig. 4) is polarized, it is surrounded by an electric dipole field. The dipole potential about the exterior surface of the space charge layer obeys the familiar relation V (1'0, B) = V cosB, which has been used with the Boltzmann distribution to give (8). Taking the Fourier transform of Pl(r,B,t) as the product of two functions R(r,jw) 8 (B,jw), then it follows from (6) that these functions separately satisfy the differential equations (9) 1/A2= (1+ jwr)/(Dr), (11) and l= 0, 1, 2,3,' . '. The solution of (10) consists of the well-known Legendre polynomials 8 I of the first kind . An equation of the same form as (9) was solved by Poisson8 in the course of investigating the asymmetric propagation of heat in a sphere. It was demonstrated later by Lommel8 that, except for a factor (A/ j1')l, Poisson's solution consists of Bessel functions of order (l+!). For example, the function R(r,jw) may be written Rl(1',jw)= {AlHz-H(jr/ A)+ BlH I-t-t(jr/ A)*}(A/ jr)t, (12) in which HI-t-!(jr/A) is a Bessel function of the form for complex arguments introduced by Hankel, HI-t-!(j1'/A)* is the conjugate, A I and Bl are constants to be evaluated from the boundary conditions. The constants Bl vanish immediately from the boundary condition (7). Therefore, the Fourier trans form of the excess hole concentration reduces to eigen functions of the form (13) Take now the dc case, which is formally obtained by making V (t) a constant in the boundary condition (8) and forcing the angular frequency w to vanish in the diffusion length A. The constants A I are determined by expressing the boundary condition (8) by an expansion of (13), e.g., 00 p.(eQVCo88/kT_1)= (A/ jro)! L A lHI-t-!(jro/A) 81• (14) l=1 The unipole mode, with l=O, for which 80= 1, has been discussed in an earlier paper.4 It is excluded from the expansion (14) because it gives a spherically symmetric distribution of holes about the injecting sphere, and therefore cannot contribute to the solution for the dipole mode. The following procedure may be used to evaluate each constant A I in (14). One mUltiplies both sides of (14) by 81" and then integrates over the solid 8 G. N, Watson, Bessel Functions (The Macmillan Company Inc., New York, 1948), second edition. ' [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:4932 B, R. GOSSIeK angle 4n-. By employing the well-known orthogonality relation (14a) and the integral formula after Gegenbauer8,9 (14b) one obtains the constants. They are given by the relation (21+1) Az= -2-(j)l+l (27rrOkT)i JZ+1'( -jqV/kT) Xp. -- . AqV HIH(jro/A) (14c) The diffusion current, as a function of applied voltage, is obtained by integrating the radial current density over one hemisphere of the space charge layer. The current, as presented here, is based only on the term for 1= 1, but, as the integral fob 82dQ vanishes, we have neg lected only the terms for which l~ 3. The result may be written ( ro 1 ) I(V)=iI. 1+-+-- A 1+ (ro/A) (qV qV qV) (kT)2 X -cosh--sinh- - , kT kT kT qV (14d) which takes the form I(V)=V.(1+ ro + 1 ) A 1+ (ro/A) x[--=-qV +~(qV)3+~(qV)5+ .. 'J, (14e) 3! kT 5! kT 7! kT with qV/kT«1. Note that the dipole mode, as based on Fig. 4, gives a current which is an odd function of applied voltage, in contrast to the unipole mode which displays an asym metric dependence between current and voltage with the well known "forward" and "reverse" characteris tics. However, the symmetry in the dipolar current voltage function would no longer remain in a hemi spherical system applicable to point contact rectifiers (see Fig. 6), but would give way to a "forward" and "reverse" characteristic typical of rectifiers. A mathe matical model for analyzing such a hemispherical system is required, and the model proposed here 9 The function h+i' (-jz) in Gegenbauer's formula is a Bessel function of the first kind. proceeds from the boundary condition (8). As a starting point (8) gives the excess hole concentration at the exterior surface of the space charge layer. Since the dependence of excess holes on the coordinate 8 is contained in the factor (eq v 0008/ kT -1), and the de~ pendence on the radius can be approximated by the factor 1/r2, therefore the excess hole concentration may be written as (3) which leads to the dipolar rectifier formula (5). Consider small voltage fluctuations of amplitude V«kT/q, about a fixed bias Vo. This case requires only a single term (1= 1) of (13), e.g., the boundary condition (8) may be expressed as p.eqVOooS8/kTqV cos8/kT which immediately gives the constant A 1, and permits expressing the transform of the fluctuation in hole concentration. qV Pe( A+r) (ro)2 p1(r,8,jw)=-- -- - kT A+ro r, Xexp[(ro-r)/A+qV o cos9/kT]' (16) The following expression for the small amplitude fluctuation of diffusion current density comes directly from (16): let) =G(Vo)[1+ro/ A+ (1+ro/ A)-1JVeiwt/2. (17) The conductance G(Vo) represents the slope of the current-voltage characteristic (5) evaluated for a bias voltage Vo. The admittance for the dipole mode may be written Y(jw) = jwC(Vo) +G(Vo)[1 +ro/ A + (1 +ro/ A)-1]/2, (18) in which C(Vo) is the barrier capacitance with bias Vo. The transform Y(jW)-1 may be regarded as a transfer function, and its inverse transform as a weighting function jet), after James and Weiss.lO The physical significance of the weighting function jet), in this instance, is that it gives the voltage response to a unit impulse in current density. As before,4 we use a method described by Elmore and Sandsll to determine the mean response time, as weighted by the function jet), and obtain the relation (t)AV= T d+ (rN2D), (dipole mode) (19) in which T d is the time constant of the space charge layer C(Vo)/G(Vo). The term rND will be called the 10 H. M. James and P. R. Weiss, Theory oj Servomechanisms, edited by James, Nichols, and Phillips (McGraw-Hill Book Company, Inc., New York, 1947), Chap. 2. 11 W. Elmore and M. Sands, Electronics (McGraw-Hill Book Company, Inc., New York, 1949), p. 137. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:49DIPOLE MODE OF MINORITY CARRIER DIFFUSION 33 diffusion time hereafter in this paper. The mean response time with the unipole mode, as reported earlier,4 may be written (t)AV= Tu+ (ro/2) (r/D)t (unipole mode). (20) The second term of (20) is one-half the geometric mean of the diffusion time and the bulk lifetime, which is (Dr)l/ro times as great (a large factor) as the corre sponding term in (19). However, this advantage of the dipole over the unipole mode is somewhat offset by the ratio of the charging time constants Td/T" which increases with increasing forward bias. At frequencies for which the relation wr02«2D holds, the admittance reduces to Y(jw)=G(V o)[l+ jW(t)AV], (21) which has the frequency dependence of a simple relaxation process. At high frequencies (wr02»2D) the admittance obeys the relation { (wr02)1 [ (wr02)1]} Y(jw)=G(V o) 1+ 2D +j wTa+ 2D ' (22) which has the same form as the corresponding admit tance with the unipole mode. IV. COMBINATION OF BOTH MODES In practice it would be very difficult to confirm or disprove a formal solution for both modes operating simultaneously. The justification for sketching such a solution is that it reveals an important dependence of both frequency and amplitude response on variations in surface conditions. To treat both unipole and dipole modes simultaneously, it is necessary to rewrite the boundary condition (8) as follows h(ro,fJ,t) = p.(exp{ q[1-/3+/3 cosfJ] V (t)/kT} -1), (23) in which /3V is the fraction of the applied voltage taken up by the dipole mode. By expressing (23) by the expansion of (13), viz., p.(exp{q[l-/3+/3 cosfJ] V (t)/kT} -1) 00 = (A/jro) 1 L: A1H1-H(jro/A)E>I, (24) 1~0 the constants A I can be evaluated, and, in principle, the hole concentration and, in turn, the hole current are determined. The essential difficulty which impedes the practical utilization of such a solution is the evaluation of /3. The fraction /3 depends in a subtle way on the geometrical design of the point contact diode and the surface recombination velocity at all free surfaces of the semiconducting die. Most important of all, /3 is not truly constant, but varies with injection level. There fore, the constants A I which determine the relative magnitude of the modes contributing to the current can themselves vary during the operation. In this way, (a) (b) FIG. 7. (a) Point contact-semiconductor die arrangement de signed to suppress the unipole mode. (b) Point contact-semi conductor die arrangement designed to suppress the dipole mode. variations in frequency response might occur which would be unjustifiable by the theoretical expressions for dynamic behavior of the individual modes listed in the previous section and in an earlier paper.4 Also, as surface condition is difficult to control, it might be expected that current-voltage characteristics representing various combinations of (2) and (5) might appear in diodes made under more 'or less similar conditions. It would appear that more reproducible characteristics might be obtained if diodes were designed deliberately to suppress either the unipole or dipole mode, as dictated by the application. Two simple geometric designs (Fig. 7) are shown, one to suppress the unipole mode, and the other to suppress the dipole mode. The solutions for the higher modes (1) 1) of minority carrier diffusion about a point contact should be un important as long as the radius of the space charge layer ro is small compared with the wavelength of the applied field. With the present size of injecting contacts and the frequency bands in use, the higher modes should be of minor concern. v. DISCUSSION The validity of approximations (3), (19), and (21) requires that the radius of the space charge layer ro be small compared with the diffusion length (Dr)!, i.e., the dc value of A. Furthermore, the radius of the space charge layer ro enters into the mean response time (t)AV with both unipole and dipole modes. Therefore, the magnitude of ro is of practical concern, and we are led to ask questions such as the following. Are there limits to the miniaturization of ro? If so, what are the limits, and what do they indicate for attainable operating frequencies? The limits on ro may be inferred from a study of the space charge surrounding disordered regions in germa- [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:4934 B. R. GOSSICK nium.12 According to that study, if the radius of the metal contact r m is small compared with the Debye Htickel length (kTE)!(qW)-!, in which N is the con centration of ionized donor impurities, then we have a space charge layer with radius ro= (3 ipErm/qN) 1, rm«(kTE/qW)! (25) in which ip is the equilibrium barrier height. On the other hand, if the Debye-Huckel length is small compared with the contact radius, or of the same order, then we have Equation (25) gives a radius ro of extent 3500 A for a metal contact with radius r m of 150 A on an n-type germanium die with 10 ohm-cm resistivity, while (26) gives a radius ro of 1400 A for the same contact on a 0.03 ohm-cm die. One might remark that ro is not a sensitive function of resistivity, because a reduction by more than two orders of magnitude in resistivity produced, in this instance, a reduction in ro of less than one order of magnitude. A reasonable conclusion is that ro cannot be reduced significantly below around 3000 A with n-type germanium suited for diodes. The diffusion length (Dr)! may be as much as 2500 times as great as the radius ro taking material with bulk lifetime 200 J.Lsec, and diffusion coefficient 40 cm2/sec. Therefore, the condition (Dr)t»ro, required for the validity of (3), (19), and (21), can be met. Furthermore, the same figures for ro and D give a diffusion time r02/Drv3XIO-11 sec, which indicates that diodes could be made to obey (21) up to a frequency of about 5 kMc/sec. At zero bias, the time constant of the space Equations (27)-(30) become inaccurate with increasing forward voltage because the hole concentration within the range rm<r<ro has been neglected in the deter mination of these expressions. However, in the range of applied voltage over which motion of the space charge layer noticeably influences the hole current, Eqs. (27)-(30) are accurate. As the space charge boundary is relatively insensitive 12 B. R. Gossick, J. Appl. Phys. 30, 1214 (1959). charge layer is about 5X 10-10 sec with n-type germa nium having resistivity of 10 ohm-cm regardless of the mode, which, with ro=3500 A, gives attainable values of response time (t)AV"-' 5 X 10-10 sec with the dipole mode as compared with (t)Av,,-,4X 10-8 sec with the unipole mode. The mean response time of the dipole mode is about two orders of magnitude smaller with this example, and by consideration of the general expressions for (t)AV, Eqs. (19) and (20) in Sec. III, it is evident that the dipole mode is inherently superior to the unipole mode for high frequency applications. The motion of the boundary of the space charge layer with variations in applied voltage has been neglected thus far. As the boundary of the space charge layer is at the same time the bounding surface for current flow, the current for a given applied voltage is restricted by the area of this bounding surface. Therefore, motion of the space charge layer is of interest in connection with the current-voltage characteristics. While the size but not the shape of the space charge layer varies in the case of the unipole mode, both size and shape vary with applied voltage in the case of the dipole mode. As only the size of the space charge layer varies with the unipole mode, its dependence on applied voltage can be treated by expressing the exterior radius as the function ro(V). The departure from spherical form with the dipole mode is treated here by considering the space charge layer as a sphere with bounding radius equal to the average radial coordinate (ro(V) )AV of the exterior boundary. The average is taken over the range of the angular coordinate 0<0<71/2. The expressions ro(V), for the unipole mode, and (ro(V) )AV, for the dipole mode, may be approximated by the following modifications of (25) and (26): rm« -(27)) (kTE)t fN (28) to applied voltage, its effect on the expressions for dc current (2) with the unipole mode, and (5) with the dipole mode, can be accounted for by direct substitution of the appropriate radius, ro(V) or (ro(V) )AY from (27) through (30) for ro in the saturation current (I.). This has been done with the current-voltage characteristic curves for reverse voltage, sketched in Figs. 8 and 9. These curves apply to diodes made of the same material, viz., n-type germanium with 10 ohm-em resistivity. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:49DIPOLE MODE OF MINORITY CARRIER DIFFUSION 35 A 0.3 v equilibrium barrier (cp) has been assumed in each example. Figure 8 refers to a low-frequency diode with 33:3-tL contact radius (rm), which by (26) gives a 35-tL space charge bounding radius (ro) with zero bias. Taking 200 tLsec bulk lifetime (T) and zero bias gives 4 tLsec for the mean response time (tjAV with the unipole mode, and 0.15 tLsec with the dipole mode (with reference only to Fig. 8). The motion of the space charge layer with reverse bias qV jkT= -2000, which is -50 v at room tempera ture, makes the reverse current increase in magnitude from -1. to -1.61. with the unipole mode, and from -21. to 1.4(21.) with the dipole mode (Fig. 8). Figure 9 refers to a high-frequency diode with 150 A contact radius (r m) which by (25) gives a 3500 A space charge qV iT FIG. 8. Reverse current-voltage characteristics of both unipole and dipole modes for a large contact radius suitable for low frequency applications. layer radius (ro) with zero bias. Taking 200 tLsec bulk lifetime (T) and zero bias gives 40 mtLsec for the mean response time (tjAY with the unipole mode and 0.5 mtLsec with the dipole mode. In this case, the motion of the space charge layer with reverse bias qVjkT= -2000, makes the reverse current increase in magnitude from -1. to -5.51. with theunipole mode, and from -218 to -4.1(21.) with the dipole mode (Fig. 9). The percent- agewise variation in saturation current through its !!. kT -'103 -3.1,} -2.103 -.103 -5 1: -6 1:s -7 -8 -9 -10 -II FIG. 9. Reverse current-voltage characteristics of both unipole and dipole modes for a small contact radius suitable for high frequency applications. dependence on the space charge boundary is less with the dipole mode in both examples, the physical explana tion being that the motion of the boundary with the dipole mode is restricted by a constraint at (J=7rj2 where it is held stationary. A final conclusion, which comes from a comparison of Figs. 8 and 9, is that a smaller (i.e., faster) contact inherently gives a larger magnitude of reverse current. On the basis of this discussion it is clear that the dipole mode has characteristics which are distinctly different from and in certain respects superior to those of the unipole mode. Therefore, the experimental investigation of the dipole versus the unipole modes of minority carrier diffusion is obviously warranted. ACKNOWLEDGMENTS It is a pleasure to thank both H. C. Schweinler and A. T. Wager for helpful comments. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Sat, 20 Dec 2014 15:48:49

1.1743211.pdf

Hyperfine Structure in Paramagnetic Resonance Absorption Spectra H. S. Jarrett Citation: The Journal of Chemical Physics 25, 1289 (1956); doi: 10.1063/1.1743211 View online: http://dx.doi.org/10.1063/1.1743211 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/25/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Oxygen17 Hyperfine Structure in Electron Paramagnetic Resonance J. Chem. Phys. 37, 1879 (1962); 10.1063/1.1733382 Detailed Hyperfine Structure in the Electron Paramagnetic Resonance Absorption Spectrum of the Phenazine Semiquinone Ion J. Chem. Phys. 36, 2237 (1962); 10.1063/1.1732869 Direct Field Effects in Electron Paramagnetic Resonance Hyperfine Spectra J. Chem. Phys. 35, 1410 (1961); 10.1063/1.1732060 Nuclear Hyperfine Structure in the Paramagnetic Resonance Absorption of Triarylaminium Perchlorates J. Chem. Phys. 23, 1540 (1955); 10.1063/1.1742344 Paramagnetic Resonance Absorption: Hyperfine Structure in Dilute Solutions of Hydrazyl Compounds J. Chem. Phys. 21, 761 (1953); 10.1063/1.1699023 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 136.167.3.36 On: Mon, 01 Dec 2014 06:33:20LETTERS TO THE EDITOR 1289 splittings as (Al2+AI3); the other is (Al2-A13). In this manner the relative signs of A I' and A 13 are found, as well as approximate numerical values. The numerical values can be refined by a de tailed analysis of the fluorine spectrum. The best example we have found is 1,4-difluoro, 2,5,6-trichloro benzene for which the proton and fluorine spectra were first observed by Holm.7 Inspection of the line intensities in the fluorine spectrum shows that the quartets are 1357 and 2468; the assign ment is confirmed by calculation of the relative intensities. This assignment of the fluorine spectrum gives AOHF +AmHF= 14.7 cps, where the subscripts designate ortho-and meta-orientations. There fore, the 2.3 cps splitting in the proton spectrum is AOHF -AmHF. The final result for the absolute values is AeHF= 8.5 cps and A mHF = 6.2 cps; the signs of the coupling constants are alike. Similar analyses have been made of the spectra of several other fluorobenzenes. The general method can be used to determine relative signs of the coupling constants in CF 2 = CFCI and several other fluoroethylenes,8 work on which is in progress. A more detailed report, including determination of numerical values of H-H, H-F and F-F coupling constants for ortho-, meta-, and para-orientations,7 is being prepared. We are indebted to Dr. G. L. Finger and Dr. R. E. Oesterling of the Illinois State Geological Survey for the samples and to Dr. C. H. Holm for the original observations of the high resolution spectra. * Assisted by the U. S. Office of Naval Research and by Grants-in-Aid from E. l. du Pont de Nemours and Company and The Upjohn Company. 1 Gutowsky. McCall. and Slichter, J. Chern. Phys. 21. 279 (1953). 2 N. F. Ramsey. Phys. Rev. 91. 303 (1953). 3 H. M. McConnell. J. Chern. Phys. 24, 460 (1956) . • McConnell. McLean, and Reilly, J. Chern. Phys. 23, 1152 (1955). 'E. L. Hahn and D. E. Maxwell, Phys. Rev. 84, 246 (1951); 88,1070 (1952). • W. A. Anderson, Phys. Rev. 102, 151 (1956); this article also notes the possibility of determining relative signs of coupling constants. 7 C. H. Holm. "Structural applications of radiofrequency spectroscopy," Ph.D. thesis, University of Illinois (1955). 'McConnell, Reilly ,and McLean, J. Chern. Phys. 24, 479 (1956). Anisotropic Chemical Shielding and Nuclear Magnetic Relaxation in Liquids H. M. MCCONNELL* AND C. H. HOLM Shell Development Company, Emeryville, California (Received August 9, 1956) THE purpose of this note is to suggest that anisotropic chemical shielding can sometimes provide significant nuclear spin-lattice relaxation in liquids. We consider first the theory of anisotropic chemical shielding. Let Ho be the external field applied to a molecule containing a nucleus N; Ho is the z direction of a system x, y, z fixed in the laboratory. The field at N, HN, is related to Ho by the shielding tensor 0', (1) An equation for (1 can be obtained by extending Ramsey's shielding theoryl to the anisotropic case. In Ramsey's notation,l (1= (1/2m) (e/c)'(o 1 ~i(IrrLririri-3) 10) -2 ~n 1/(En-Eo){(nl ~i m;"rr310)(01 ~i m;"ln) +(01 ~im;"ri-3In)(nl ~im;"lo)}, (2) where I is the unit tensor. For simplicity consider a molecule axially symmetric at Nj (2) becomes (3) where 0'.1. is the shielding perpendicular to the symmetry axis l (a unit vector) and tJ.0' is the difference between the shielding parallel to the symmetry axis and 0'.1.. The Fourier spectrum of the x, y components of HN, i.e., HNz=sint'J(t) cost'J(t) cos",(t)tJ.O'H o (4) can have components of the proper frequency, Vo= (2 ... )-I-yHo, to give spin-lattice relaxation. t'J and '" give the orientation of l relative to x, y, z. The procedure of Bloembergen, Purcell, and Pound' gives for T1, for nuclei of spin !, TI-l= (8 ... '/15) (tJ.0')'vo'Tc(1 +4...'VO'Tc')-I. (5) From (2), and from solid-state studies on anisotropic chemical shielding,3 we infer that tJ.0' is sometimes of the order of 0' itself, i.e., 10--2-10-6• Numerical estimates using (2), (5), and empirical shielding data can thus support the idea that anisotropic tensor coupling of the x, y components of the nuclear magnetization to the external field Ho yields strong spin-lattice relaxation in some molecules. According to (5), TI always decreases with increasing vo. Exactly the opposite behavior is expected for magnetic dipolar relaxation. The foregoing discussion is further supported by our finding of experimental evidence for (a) a short T 1 (",60 sec) for C13 in un enriched oxygen-free CS2, and (b) an apparently long Tl for CI3 in CCl4• For (a) the chemical shift relaxation must be dominant because of the virtual absence of magnetic nuclei in CS2• In (b) the chemical shift is isotropic by symmetry and cannot contribute to TI-l. * Address after August 24, 1956: Department of Chemistry, California Institute of Technology, Pasadena, California. 1 N. F. Ramsey, Phys. Rev. 78, 699 (1950). 2 Bloembergen, Purcell and Pound, Phys. Rev. 73, 679 (1948). 3 N. Bloembergen and T. J. Rowland, Phys. Rev. 97, 1679 (1955). Hyperfine Structure in Paramagnetic Resonance Absorption Spectra H. S. JARRETT E. I. du Pont de Nemours and Company, Wilmington 98, Delaware (Received September 6, 1956) ISOTROPIC proton hfs in paramagnetic resonance absorption spectra of solutions of aromatic organic free radicals has been shown in order of magnitude calculations by McConnell1 and Bersohn' to arise from configuration interaction. It is the purpose of this note to show that a calculation similar to that of McConnell, if carried to completion without neglect of certain terms, leads to a value of the hfs splitting in excellent agreement with experiment. Valence bond calculations were made on the three-electron system consisting of two 0' electrons and one unpaired ... electron of an aromatic CH fragment. In second order, other orbitals do not give rise to hfs and were not considered. Configuration interaction between the two doublet states in conjunction with the Fermj3 isotropic hyperfine interaction yields a hyperfine splitting H _ 16... [h2(rH) -u'(rH) J>. tJ. -3 IJ.HIJ.. 1-Suh' ' where h2(rH) and u'(rH) represent the squares of the magnitude of the hydrogen electron wave function and the carbon 0' wave function at the position of the proton and Suh is the overlap inte gral. The configuration interaction constant, >.= (J n-J rh)/ tJ.E, where the energy separation between the doublets tJ.E= V1{ (J h1r -Ju,..)'+ (Jh1r-Juh)'+ (Ju.--Juh)'}t. The part of tJ.H involving h'(rH) is just the hfs of a free hydrogen atom and is equal to 510 gauss. The 0' bond contribution at the proton can be determined from the self-consistent field calculations of Torrance4; viz., 0'2 (rH)/h'(rH) =0.17, and tJ.H=423>./(1-S uh') gauss. The matrix element, Jh.-=0.745 ev, which is a two-center integral, was obtained by Altmann6 with use of Slater wave functions. Although Slater wave functions are not a faithful representation of the actual radial wave functions, they represent the outer parts of the actual wave functions where an appreciable contribution to this integral is expected. J u.-, which is an atomic integral, is also given by Altmann. However, his value is too large because the Slater functions do not represent the inner parts of the actual radial wave function where This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 136.167.3.36 On: Mon, 01 Dec 2014 06:33:201290 LETTERS TO THE EDITOR there is a strong contribution to this atomic integral. The value of In was obtained from the relation, J .... =i(G I+2F2)=0.88 ev, where GI and F2 are the radial exchange and Coulomb integrals, respectively.6 Voge7 obtained the values of GI and F2 empirically by adjusting these parameters to fit the electron energy levels of carbon. J"h= -2.3 ev was obtained by Van Vleck.s S"h=0.80 was obtained by fitting a series of Slater wave functions to the self-consistent field calculations of Torrance and then computing from the tables of Mulliken et al.,9 the uk overlap integral at the CH equilibrium bond distance. Substitution of the values of the matrix elements yields t:J.E=6.2 ev and A=0.024. The hyperfine interaction then becomes t:J.H = 28 gauss. It has been assumed that there is unit unpaired electron density in the 11' orbital. Actually, in an aromatic free radical, the unpaired electron is distributed over many carbon atoms and the magnitude of the hfs would be expected to be proportional to the actual unpaired electron density on the carbon. The value of 28 gauss per unit unpaired electron density is in excellent agreement with experimental results.lO This result does not imply that t:J.H is constant for all free radicals. Higher order terms that take into account the symmetry of the molecule and the contributions from other orbitals could cause variations of several gauss in this calculated value. Calculations for CI3 hfs become increasingly difficult since many more orbitals contribute to the hfs in second order. The author wishes to acknowledge several helpful discussions with Professor S. I. Weissman. * Contribution No. 400 from the Chemical Department, E. 1. du Pont de Nemours and Company, Wilmington, Delaware I H. M. McConnell, J. Chern. Phys. 24, 764 (1956). 'R. Bersohn, J. Chern. Phys. 24. 1066 (1956). 'E. Fermi, Z. Physik 60, 320 (1930). • C. C. Torrance, Phys. Rev. 46, 388 (1934). • S. L. Altmann, Proc. Roy. Soc. (London) A210, 327, 343 (1951-1952). • E. U. Condon and G. H. Shortley, Theory oj Atomic Spectra (Cambridge University Press, New York, 1951) p. 177. , H. H. Voge, J. Chern. Phys. 16,984 (1948). • J. H. Van Vleck, J. Chern. Phys. 2. 20 (1934). 'Mulliken, Rieke, Orloff, and Orloff, J. Chern. Phys. 17, 1248 (1949). lOH. M. McConnell, J. Chern. Phys. 24, 632 (1956); E. De Boer, J. Chern. Phys. 25,190 (1956); H. S. Jarrett (to be published). Bond Localization and the Hyperconjugative Effect in the Aromatic Carbonium Ions TOSHIFUMI MORITA DePartment oj Chemistry, Faculty oj Science, Tokyo Metropolitan University, Fukasawa-cho, Setagaya-ku, Tokyo, Japan (Received September 24, 1956) THE absorption spectra of methylbenzene carbonium ions indicate the "blue-shifts" of the corresponding absorption bands, with increasing number of methyl substituents.1 To inter pret this, the wavelengths of the first transition for benzenium (E), toluenium (T), and mesitylenium (M) ions are calculated, using the semiempirical LCAO MO method including overlap. As the molecular models, the following two are adopted; (1) the hyper conjugation model, in which the pseudo-atom H2 formed by attached proton together with hydrogen atom already present at the position of proton attack conjugates with the ring; (II) the bond localization model, in which the H2- C bond is localized with respect to the rest of the molecule and other things are identical with (I). Table I shows theoretical and experimental data, ac cording to which, assuming that the singlet-triplet separations are not the same but do not differ so much among these substances, the blue-shifts in question can be explained by neither model. If model (II) is accepted for (B) and (T), whereas model (I) for (M), the self-consistency of the theoretical data in relation to the experi mental results will be achieved. Table II shows the conclusive theoretical data. It is likely that these substances do not fix to TABLE 1. First excitation energies calculated for the two models and experimental values. ForHCJ For B.L. Absorption Oscillator Carbonium modelb model' wavelength strength ions (-{J') (-{J') obs (A) obs Benzenium 0.870829 0.801925 4170 0.15 Tolueniuma 0.879022 0.815739 4000 0.15 Mesitylenium 0.818010 0.814447 d 0.760181 3900 0.27 • For only the type which belongs to C" symmetry group, computation is carried aut. b HCJ =Hyperconjugation. The values in this column are those calcu lated using 5 = -0.3. See Y. I'haya, J. Chern. Phys. 23, 1165 (1955). 0(3 is the resonance integral including overlap between the adjacent carbon atoms in benzene molecule, and adopted the value of -60 kcal/mole. See C. C. J. Roothaan and R. S. Mulliken, J. Chern. Phys. 16, 118 (1948). d This is the value obtained using, besides 5 = -0.3, the auxiliary induc tive parameter, l5ind = -0.1. e B.L. = Bond localization. The values in this column are those calculated using, besides 15 = -0.3, the auxiliary inductive parameter, l5ind = -0.1. either model. If the eigenfunction of such molecules is written as <I>=a<l>I+b4>rr (<1>1, <I>/I: eigenfunction of (I) and (II), respectively; a, b; coefficients), the behaviors of these substances are understood in the unified point of view. In the case of isomeric xylene carbonium ions also, this idea is adequate to interpret the basicity of parent hydrocarbons. Our calculation shows the 11'-electron energies of m-, 0-, and p-xylene carbonium ions in the bond localization model are -11.4819, -11.4914, and -11.4905 (in -fj), respectively.2 From Gold and Tye's equation,3 we obtain for KI/KII (KI, KIl; equilibrium constants for carbonium ion formation of m-and p-xylene, respectively), InKI/KIl= -{ -x+c+0.696}/RT, where, -X=EI_EI.O; -c=EIl_Err,o; EpI_EpIl= -0.18 kcal/ mole4; EI.O_O.516 (kcal/mole)=Ell,o; EI, Ell; EI,O, EIl.O; 11' electron energies for the actual and bond localization states of m and p-xylene carbonium ions, respectively; EpI, EpIl: 11'-electron energies of m-and p-xylene. Admitting the actual state of carbonium ions to be the hybrid of hyperconjugation and bond localization states, the magnitudes of x and c indicate the stabiliza tion energies by the hyperconjugation of H2 pseudo-atom in m-and p-xylene carbonium ions compared with their bond localization states, respectively. Taking KI / Kll = 26 at 20°C· into account, the TABLE II. Conclusive theoretical data on the aromatic carbonium ions. Singlet- Longest triplet absorption Carbonium separations wavelength Oscillator ions Model assumed (ev) (A) strength Benzenium B.L. 1.77 4168 0.31 Toluenium B.L. 1.95 3999 0.20 Mesitylenium HCJ 2.10 3909 0.32 percent fraction of the hyperconjugation state of carbonium ion of m-xylene relative to that of p xylene may be obtained by 100 X (x-c)/5.9, on the assumption that the stabilization energy through the perfect hyperconjugation is 5.9 kcal/mole.6 Our calculations indicate that the extent of the hyperconjugation state in the carbonium ion of m-xylene is larger by 44% than that of p-xylene carbonium ion, which is almost identical with that of o-xylene carbonium ion. ! C. Reid, J. Am. Chern. Soc. 76, 3264 (1954). 2 These values are those for the most stable of all possible different types produced, depending on the position of proton attack. As for m-xylene carbonium ion, the next stable type is accepted owing to the consideration on the experimental facts of nitration reaction of m-xylene. 3 V. Gold and F. L. Tye. J. Chern. Soc. 1952,2184. 4 C. A. Coulson and V. A. Cra,,{ord, J. Chern. Soc. 1953.2052. 'M. Kilpatrick and F. E. Luborsky, J. Am. Chern. Soc. 75, 577 (1953). • Muller, Pickett, and Mulliken, J. Am. Chern. Soc. 76. 4770 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 136.167.3.36 On: Mon, 01 Dec 2014 06:33:20

1.1735246.pdf

Interdiffusion in Binary Ionic Semiconductors R. F. Brebrick Citation: J. Appl. Phys. 30, 811 (1959); doi: 10.1063/1.1735246 View online: http://dx.doi.org/10.1063/1.1735246 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v30/i6 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 28 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsPYROELECTRIC HYSTERESIS LOOPS IN BaTiO. 811 asymmetry in the electric field direction, and this has been confirmed in the present experiments. It is usually easier to reverse the polarization from +p to -p, where +p refers to the polarization vector pointing from the inside to the outside of the original BaTiOa butterfly wing. Husimi gives no quantitative data on this asym metry; however, in the present experiments, the field asymmetry is much less than that sometimes observed by Chynoweth.2 Thus, while the present results would seem to explain the previously measured pyroelectric asymmetry without resort to space-charge layers, the large bias with respect to the electric field axis observed by Chynoweth cannot be attributed to the edge effects considered in the present paper. There is also the question of the origin of residual pyroelectric signals observed at temperatures above the Curie point of the bulk material.2 Are these residual signals affected in any serious manner by electrode fringe effects? Several samples were appropriately masked and measured pyroelectrically as they were heated up through the Curie point. These experiments show that residual signals are indeed observed above the Curie point with masked, as well as unmasked samples, and the data are in qualitative agreement with the results given by Chynoweth.2 However, in general, the residual signals measured in the present research, with masked or unmasked samples, are about one order of magnitude smaller than those reported by Chynoweth. The residual signals are found to be of the order of 1% of the room temperature pyroelectric signals,19 The wide variation of the characteristics of the residual pyroelectric pulses observed by Chynoweth, and in the present experiments, leads one to suspect that these signals are not an inherent characteristic of BaTi0 3, but that they may depend in an important way on the particular electrode-BaTi0 3 structure under study. It is clear that much more work in the neighborhood of the Curie point is required to explain the observed phe nomena in detail. In any event, the present results support the conclusions given by Chynoweth which indicate the presence of a polarized surface layer above the Curie point of the bulk material. ACKNOWLEDGMENTS The authors would like to take this opportunity to thank Dr. A. G. Chynoweth and Dr. H. L. Stadler for many informative discussions on the subject of this paper. 19 This result is in agreement with unpublished data obtained by Chynoweth after the publication of his paper given as reference 2. The residual signals reported by Chynoweth in reference 2 were perhaps something like an order of magnitude larger than those observed in later work. JOURNAL OF APPLIED PHYSICS VOLUME 30. NUMBER 6 JUNE, 1959 Interdiffusion in Binary Ionic Semiconductors R. F. BREBRICK U. S. Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland (Received October 27,1958) Interdiffusion in a nondegenerate, exhaustion range, binary ionic semiconductor is investigated using the recently improved theory for the concentrations of defects in crystals and Wagner's phenomenological flow equation. The composition dependence of the interdiffusion constant is found to be determined by the ratio of the ion jump frequencies, the Schottky constant, and the intrinsic concentration of conduction band electrons. The removal of the restriction to an exhaustion range semiconductor and the replacement of the basic assumption of local electroneutrality are discussed. I. INTRODUCTION SOME binary ionic semiconductors have been shown to be stable over a small but measurable com position range; the excesses of the metallic and non metallic components behave, respectively, as donor and acceptor impurities. We are concerned with inter diffusion in pure crystals of this type, i.e., the propa gation of composition changes by the diffusion of the crystal components. The theoretical basis of inter diffusion in binary salts was first given by Wagner in connection with an analysis of tarnishing reactions.l,2 1 C. Wagner, Z. physik. Chern. B21, 25 (1933). 2 C. Wagner, Z, physik. Chern. B32, 447 (1936). Semiconductors were treated as a special case charac terized by the electronic transport number being essentially unity. The early experimental techniques consisted mainly of exposing a metal to an electro negative gas or vapor and following the weight change or uptake of gas. The experimental techniques of semi conductor physics now offer the possibility of more refined diffusion experiments.3 It is therefore worthwhile to re-examine the basic assumptions of Wagner's phenomenological theory. Moreover an extension of • W. C. Dunlap, Jr., Progress in Semiconductors (John Wiley & Sons, Inc., New York, 1957), Vol. II, p. 167. Downloaded 28 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions812 R. F. BREBRICK his results can be made using the recently improved theory4,. of the concentrations of defects in crystals. We shall be mainly concerned with binary ionic semiconductors which are non degenerate and in the exhaustion range. The analysis is then as simple as possible and the results available in analytical form. The essential meaning of ionic, as used here, is that place exchange between the two atomic species is not an important type of defect. Although the arguments given and the results obtained here are more general, we shall consider only crystals in which vacancies are the predominant point defects and in which movement by vacancies is the predominant diffusion mechanism. II. CRYSTAL MODEL We consider a homogeneous uni-univalent ionic crystal which can deviate from the stoichiometric composition by the inclusion of vacancies in the cation and anion sublattices. The vacancies distribute them selves at random over the lattice sites and increase the internal energy of the crystal proportional to their concentrations. Each cation (anion) vacancy has associated with it an acceptor (donor) level in the electronic energy band structure with a degeneracy of two and a maximum occupancy of one. If there is no electron (hole) in the associated level, the anion (cation) vacancy is said to be ionized. This model contains the essential features of those used to correlate the optical properties of CdS crystals4 and the electrical properties of PbS crystals· with the conditions of preparation. It represents an extension over the earlier treatments in that it takes better account of the electronic band structure of solids. A statistical mechanical analysis has been given by the author which can be consulted for details.6 Provided the deviation from the stoichiometric composition is small the internal variables of the crystal are related by mass action law equations. For those equations containing the concentration of conduction band electrons, n, or that of the valence band holes, p, to be valid, it is also necessary that the electron chemical potential, Ji.e, be a few units of kT away from both the bottom of the conduction band and the top of the valence band (nondegenerate semiconductor). Two pertinent and well-known expressions are given by (1) (2) where V c and V A are the concentrations of ionized cation vacancies and ionized anion vacancies, respec tively. The constants k; and ni are the Schottky constant and intrinsic carrier concentration, respec tively. They are functions of temperature only and 4 Kroger, Vink, and van den Boomgaard, Z. physik. Chern. B203, 1 (1954). 5 J. Bloem, Philips Research Repts, 11, 329 (1956). 6 R. F. Brebrick, J. Phys. Chern. Solids 4, 190 (1958). are a measure of the intrinsic ionic and electronic disorder. The chemical potentials of the metallic component, Ji.m, and that of the nonmetallic component; Ji.x, are given by6,* Ji.m=kT In(SjVc)+Ji..+ h(T) Ji.x=kT In(S/VA)-Ji.e+ h(T) (3) (4) where S is the concentration of sites in both the cation and anion sublattices and is of the order of 1022• The first terms of the right-hand members of these equations are the cation and anion chemical potentials. At this point we narrow our considerations to a nondegenerate, exhaustion range semiconductor. Only a negligible fraction of the vacancies of each kind are then un-ionized. The electroneutrality condition reduces to Nc-NA=n-p (5) where N c and N A are respectively the concentrations of cations and anions. Defining the deviation from stoichiometry, del, as the difference in the cation and anion concentrations, i.e., (6) the electron chemical potential can be written as follows7 : Finally the conservation of lattice sites gives Nc+Vc=NA+VA=S. (8) Equations (1)-(8) define the crystal model sufficiently for our purposes. It is to be noted that all of the internal variables of the crystal are fixed when the temperature and deviation from stoichiometry are given. Moreover the chemical potentials of the cations, anions, and electrons do not correspond to an ideal solution. They depend much more strongly on the crystal compositionl This is in contrast to the chemical potentials of im. purities in elemental semiconductors which show non ideal behavior at very small concentrations only through the electron chemical potential. 7 In the case of an inhomogeneous crystal it is assumed that local thermodynamic equilibrium exists. Because of the different mobilities of the charged particles present, a position-dependent space charge density and electrostatic potential can be present. It is assumed that the effect of the electrostatic potential is to shift the energies of the ions and the electronic energy levels proportional to this potential. Then if the space charge density is everywhere zero (local electroneutrality), the above equations for a homogeneous crystal are still * For an exhaustion range semiconductor these expressions are the same as those given by Wagner (reference 2). In their appli cation, however, Wagner only considered limiting cases. 7 R. L. Longini and R. F. Greene, Phys. Rev. 102,922 (1956). Downloaded 28 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsIN T E R D IFF U S ION IN BIN A R Y ION I C S E M leo N Due TOR S 813 valid. Otherwise Eq. (5) must be replaced by 6.=n-p+p/e, (9) where p is the space charge density and e is the electronic charge, and Eq. (7) for the electron chemical potential must be modified by replacing 6. by 6.-p/e in the right hand member. Therefore in the absence of local electro neutrality, the temperature and deviation from stoichi ometry, are not sufficient to determine the local values of the internal variables. III. PHENOMENOLOGICAL FLOW EQUATION Following Wagner2 the flow of the deviation from stoichiometry, which is the difference of the cation and anion flows, is given by J 6 = J e -J A = -D 6 grad6.. (10) For a semiconductor the interdiffusion constant is given by (11) The flows, J, are given in terms of number per cm2 per sec; and the mobilities be and bA are the velocities per unit force of the cations and anions, respectively. If the crystal is not a semiconductor the electronic trans port number must be included as a factor in the right hand member of Eq. (11). In the absence of an over-all composition gradient the rate of self-diffusion of either cations or anions is determined by the corresponding ion mobility as given by the following relations: De*=kTb c DA*=kTb A (l2a) (12b) where De* and DA* are the self-diffusion constants for the cations and anions, respectively. Since the ion mobilities are non-negative, comparison of Eqs. (11) and (12) shows that a minimum value for the ratio of the interdiffusion constant to either self-diffusion constant is given by the product of the concentration of lattice sites by the derivative of the chemical potential. As can be seen from the last section the latter is of the order of the reciprocal of the total vacancy concentration, i.e., (Ve+ V A)-I. The interdiffusion constant is therefore 10L 106 times as large as the larger of the self-diffusion constants. The fundamental assumptions upon which Eqs. (10) and (11) are based are: (a) local thermodynamic equilibrium previals; (b) local electroneutrality pre vails; (c) the mobile charged species are cations, anions, conduction band electrons, and valence band holes; (d) the flow of each charged species is given by the negative product of the concentration of that species by its mobility by the gradient in its electrochemical potential. The assumption of local electroneutrality is an approximation which is better the shorter the Debye length (the larger the concentration of conduction band electrons).8 A better approximation for diffusion in semiconductors is obtained if one assumes that the gradient of the electron electrochemical potential (Fermi level) is negligible relative to that of the ion electrochemical potentialst and discards the assump tion of local electroneutrality. Then one again obtains the flow of the deviation from stoichiometry as Now, however, the chemical potential of the metallic component is no longer a function of the temperature and deviation from stoichiometry alone as discussed in Sec. II. As a result the diffusion boundary value problem is not complete with Eq. (13) and the boundary con ditions. Poisson's equation and the equation for conservation of charge [Eq. (9)J must be included also. Unless explicitly stated otherwise the assumption of local electroneutrality will be made in the following discussion. IV. THE VACANCY MECHANISM It is assumed that the predominant diffusion mechan ism is that of vacancies. The flows of cation and anion vacancies are opposite and equal to the cation and anion flows, respectively. It is therefore unnecessary to treat explicitly the vacancy flows in the phenomeno logical approach. It can be shown by a kinetic treatment that the vacancy mechanism of diffusion is consistent with the phenomenological equations of the last section.9 Moreover the composition dependence of the ion mobilities can be obtained and is given by kTbc= kCX2(V c/ S) kTbA = kAX2(V .1/ S) (14a) (14b) where X is the ion jump distance and the k's with subscripts are the ion jump frequencies which are independent of composition but depend exponentially on the temperature. Although we are primarily interested in the case where all the vacancies are ionized (exhaustion range semiconductor), the discussion in this and the previous section remains unchanged in the more general case if the un-ionized vacancies are assumed to be immobile. We shall not take account of the correlation effect in vacancy diffusion.lo This results in the self-diffusion being smaller by a factor of the order of one to two than predicted by Eq. (14). 8 W. Schockley, Bell System Tech. J. 28, 435 (1949). t If, as is the case in a semiconductor, the electronic transport number is essentially unity, then local electroneutrality implies this. The converse of course is not necessarily true. 9 J. Bardeen, Phys. Rev. 76, 1403 (1949). 10 J. Bardeen and C. Herring, Atom Movements (American Society for Metals, Cleveland, Ohio, 1956), p. 87. Downloaded 28 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions814 R. F. BREBRICK V. COMPOSITION DEPENDENCE OF THE INTERDIFFUSION CONSTANT The phenomenological expression for the inter diffusion constant given by Eq. (11) can now be written more explicitly for a nondegenerate, exhaustion range semiconductor. The ion mobilities are given by Eqs. (14a) and (b), and the required derivative of the chemi cal potential can be obtained from Eqs. (1) through (8). The result is given by (15) where j is the fraction of vacancies which are cation vacancies, i.e., 2j= 2V cI (V c+ V A) = 1-b./ (<l2+4k.) 1. (16) The first factor in braces in Eq. (15), A, is essentially the sum of the ion mobilities multiplied by the deriva tive of the cation chemical potential with respect to del. Provided the range of stability of the crystal is large enough the fraction of cation vacancies, j, appearing in this factor can vary from near zero on the excess metal side of the stoichiometric composition (n-type semiconductor) to near unity on the excess nonmetal side (p-type semiconductor). In this case A can vary by a factor which is the ratio of the ion jump fre quencies. In general the ratio of ion jump frequencies is expected to differ considerably from unity. On the other hand, if the range of stability of the crystal is small enough and the Schottky constant large enough A will be nearly constant. ' The second factor in braces in Eq. (IS), B, is the derivative of the chemical potential of the metallic component divided by that of the cation, both deriva tives being taken with respect to del. The composition dependence of B is symmetrical about the stoichiometric composition. For large deviations from stoichiometry (.<l2»4k., 4ni2), B approaches the value two asymptoti cally. At the stoichiometric composition its value depends on the "intrinsic disorder ratio," k.l/ni, and is greater or less than two as this ratio is respectively greater or less than unity. It is not to be expected that the ion transport number will remain negligible, as the intrinsic disorder ratio increases indefinitely. In practice, therefore, the contribution of B to the com position dependence of the interdiffusion constant of semiconductors is probably limited to a factor of ten or less. The ratio of the interdiffusion constant to the cation self-diffusion constant, evaluated at the stoichiometric composition, is given by Dd/Dc*= 1/2(1 +kA/kc)(S/k.!+S/ni); .<l=O. (17) Since the square root of the Schottky constant and the intrin~ic carrier concentration are generally orders of magmtude smaller than the total concentration of lattice sites, this ratio can be quite large. The behavior of this ratio with crystal composition depends upon the ratio of the ion jump frequencies, the Schottky constant, and the intrinsic concentration of conduction band electrons. Its asymptotic values for large deviations from stoichiometry (large in the above sense) is no less than 2S/.<l and is therefore still quite large. The ratio of the interdiffusion constant to the anion self diffusion constant is analogous. Equation (15) for the interdiffusion constant is not valid when the deviation from stoichiometry becomes large enough that the semiconductor crystal is no longer in the exhaustion range. One easily visualized reason stems from the fact that the type of vacancy (cation or anion) in excess is then largely un-ionized. Since a repulsive force exists between an ion and an un-ionized vacancy in its sublattice, the mobility of the ion is smaller for a given vacancy concentration when a significant fraction of these vacancies are un-ionized. The dependence of the chemical potential of the metallic component on composition is also not as strong as when the crystal is in the exhaustion range. The net effect is to decrease the diffusion constant for large deviations from stoichiometry below the values given by Eq. (IS). The general approach used in this paper can be applied with analogous results to crystal models with interstitial rather than vacancy point defects and in which the interstitial rather than the vacancy diffusion mechanism predominates. In essence we have accepted Wagner's phenomeno logical flow equation [Eqs. (10) and (11)] but have used expressions different than his for the ion mobilities and the chemical potential. He neglected the com position dependence of the ion mobilities and used crystal models with only one type of point defect. The latter is not important for large values of the deviation from stoichiometry (.<l2»4k8, 4ni2) but is for smaller values. As a result Wagner's values of the interdiffusion constant2 for large deviations from stoichiometry agree with the asymptotic values of Eq. (1S). His values at the stoichiometric composition, however, only include the case in which the intrinsic disorder ratio k!/ n· and one of the ion jump frequencies are zero. ' 8 " Finally for completeness the flow equation is given for the case in which the basic assumption of local electroneutrality is discarded as discussed in the last part of Sec. III. The flow of the deviation from stoichi ometry then depends not only on the gradient in that concentration but also on that of the space charge density, p, and for a nondegenerate exhaustion range semiconductor is given by ( [ .:l2+4k. )1 ) X grad.:l+ grad[.:l-p/eJ . (18) (.:l-p/e)2+4n;2 Downloaded 28 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsINTERDIFFUSION IN BINARY JONIC SEMICONDUCTORS 815 Equation (17) reduces to our previous results when the space charge density is zero. VI. APPLICATION Our final results CEq. (1S)J apply only to a non degenerate, exhaustion range semiconductor. In order that a compound semiconductor meet this restriction over a significant range of composition, it is necessary that the donor and acceptor levels be relatively close to respectively the conduction and valence bands com pared to the width of the energy gap itself. This is the case over a wide temperature range for lead sulfide and probably also for lead selenide and telluride. The inter diffusion constants have been measured in lead sulfidell and lead telluridel2•13 by electrical means although experiments were not carried out to investigate their composition dependence. An experimental study of the lead-sulfur phase dia gram in the neighborhood of the composition, PbS, has established the composition range of stability of crystal line lead sulfide and the temperature variations of the Schottky constant and intrinsic carrier concentration.5,14 The crystal model used to analyze the experimental data is substantially the one used here. We, however, consider lead sulfide to be a uni-univalent compound since each excess lead or sulfur atom yields only one current carrier. The intrinsic disorder ratio, k81/ni' varies from a value of 0.05 at 8000K to 0.7 at 1350oK. Factor B in Eq. (15) for the interdiffusion constant varies with composition by a factor of two or less, being smallest at the stoichiometric composition. Over this same temperature range the fraction of cation vacancies varies from near unity to near zero between the com position limits of stability. Factor A in Eq. (15) there fore varies by the ratio of the ion jump frequencies which has not yet been determined. Since the jump frequency for the sulfur ion is most likely smaller than that for the lead ion, the interdiffusion constant is smaller for n-type lead sulfide than for p-type. 11 R. F. Brebrick and W. W. Scanlon, Phys. Rev. 96, 598 (1954). 12 Edward L. Brady, J. Electrochern. Soc. 101,466 (1954). 13 B. 1. Boltaks and Yu N. Mokhov, Zhur. Tekh. Fiz. 26, 2448 (1956). 14 J. Bloem and F. A. Kroger, Z. physik, Chern. 7, 1 (1956). The diffusion of radioactive lead in pressed lead sulfide powders has been measured by Anderson and Richards.ls Assuming the tracer diffusion occurred with no gradient in the composition and in samples that were stoichiometric, their diffusion constant is that for the self-diffusion of lead. A minimum value for the interdiffusion constant can then be calculated for stoichiometric lead sulfide using Eq. (17) with the sulfur ion jump frequency set equal to zero. At 823°K the lead ion self-diffusion constant is 2XIO-11 cm2/sec, the Schottky constant is S.8X 1032, and the intrinsic carrier concentration is 4X 1017. Using the value, 2X 1022, for the concentration of lattice sites, the minimum value of the interdiffusion constant at 823°K is 9X 10-6 cm2/sec. The interdiffusion constant in initially near stoichiometric, n-type lead sulfide crystals has been measured at 823°K by following the pene trating p-n junction and is 2X 10-6 cm2jsecY The agreement with the value calculated above is sur prisingly good in view of the assumptions concerning the conditions during the tracer experiments. VII. IMPURITY DIFFUSION IN ELEMENTAL SEMICONDUCTORS Recently the diffusion of impurities in elemental semiconductors by the vacancy mechanism has been of experimental and theoretical interest.7.16.17 In many respects such diffusion and interdiffusion in binary ionic semiconductors are analogous. There are however two main points of difference. First, vacancies asso ciated with only one type of atomic species are present in the elemental semiconductors and these vacancies are associated with acceptor levels. Therefore the diffusion of n-type impurities tends to be favored by the coulombic interaction between the impurity ions and those vacancies that are ionized. Secondly, the chemical potential of an impurity atom in an elemental semi conductor is nonideal only through its dependence on the electron chemical potential. Consequently if the latter is fixed by heavy doping the impurity diffusion constant is independent of composition whereas the interdiffusion constant is not. 15 J. S. Anderson and J. R. Richards, J. Chern. Soc. 1946,537. 16 M. W. Valenta and C. Rarnasastry, Phys. Rev. 106, 73 (1957). 17 R. A. Swalin, J. App!. Phys. 29, 670 (1958). Downloaded 28 May 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1735392.pdf

Figure of Merit for Thermionic Energy Conversion N. S. Rasor Citation: Journal of Applied Physics 31, 163 (1960); doi: 10.1063/1.1735392 View online: http://dx.doi.org/10.1063/1.1735392 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Transducer figure of merit (L) J. Acoust. Soc. Am. 132, 2158 (2012); 10.1121/1.4747019 Enhanced figure of merit in thermal to electrical energy conversion using diode structures Appl. Phys. Lett. 81, 559 (2002); 10.1063/1.1493224 Figure of merit for thermoelectrics J. Appl. Phys. 65, 1578 (1989); 10.1063/1.342976 Figures of merit for energy conversion processes Am. J. Phys. 46, 637 (1978); 10.1119/1.11266 Dependence of the Thermoelectric Figure of Merit on Energy Bandwidth J. Appl. Phys. 33, 1928 (1962); 10.1063/1.1728871 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.22.67.7 On: Tue, 02 Dec 2014 05:06:06JOURNAL OF APPLIED PHYSICS VOLUME 31. NUMBER 1 JANUARY. 1960 Figure of Merit for Thermionic Energy Conversion* N. S. RASOR Atomics International, A Division of North American Aviation, Inc., Canoga Park, California (Received August 3, 1959) The optimum performance for emission-limited thermionic energy conversion is derived in convenient analytical form. The steps which are thereby indicated to reduce fundamental performance limitations are enumerated and briefly discussed. A figure of merit is defined with brief description of its usefulness and significance. A comparison of thermionic and thermoelectric conversion is thereby afforded by the analysis. 1. INTRODUCTION SEVERAL analyses of the direct conversion of ther mal energy to electrical energy by thermionic emis sion have been published.1-4 However, these analyses do not yield a simple single parameter as a figure of merit for comparing prospective materials, such as that extensively used5,6 in the development of thermoelectric conversion. In the following, a similarly useful figure of merit for the thermionic system is obtained by an analysis analogous to that used in the thermoelectric case. This parameter permits a quick estimate of the efficiency of a thermionic converter, quantitatively illustrates the relative importance of the various materials properties, and permits direct intuitive and quantitative comparison of thermoelectric and therm ionic systems. II. EFFICIENCY ANALYSIS Figure 1 is a diagram of negative electric potential for a thermionic diode. V c and Va are the potential differ ence between the maximum negative potential of the system and the Fermi levels of the cathode (hot electrode) and anode, respectively; cPo and cPa are the respective work functions, T c and T a the respective temperatures, and J c and J a the saturation electron emission current densities of the respective electrodes, and V the potential drop across the external load. The potential curve between the two electrodes depends on the space charge formed there by the emitted electrons. The nature of the space charge and means for its suppression are important problems which have received detailed attention elsewhere,1-l0 and concerning which significant progress is presently being made. While such effects are important factors in the performance of existing devices, there already exist means by which * Supported in part by the U. S. Atomic Energy Commission under Contract No. AT(1l-1)-GEN-8. 1 W. Schlichter, dissertation, University of Gottingen (1915). 2 M. J. O. Strutt, Proc. lnst. Radio Engrs. 40, 601 (1952). 3 H. Moss, Brit. J. Electronics 2,305 (1957). 4 J. M. Houston, J. Appl. Phys. 30, 481 (1959). • A. F. Ioffe, Semiconducting Thermoelements and Thermoelectric Cooling (Infosearch, Ltd., London, 1956). 6 C. Zener, Ind. Labs. 9, 538 (1958). 1 V. C. Wilson, J. Appl. Phys. 30, 475 (1959). 8 H. F. Webster, J. Appl. Phys. 30, 488 (1959). 9 Hernqvist, Kanefsky and Norman, RCA Rev. 19,244 (1958). 10 G. N. Hatsopoulos and J. Kaye, J. App!. Phys. 29, 1124 (1958). they can, in principle, be made negligibly small. The primary purpose of the present analysis is to display the fundamental system requirements and limitations with the greatest simplicity consistent with generality. Therefore, the absence of space charge limitation will be assumed in the analysis, with subsequent consideration of such effects as perturbations. This is similar to the omission of contact resistance, nonuniformity, and other troublesome but nonessential effects from the classic thermoelectric analysis, which indeed have been largely overcome as that technology has matured. The potential diagram for a thermonic diode deliver ing maximum power to its load, but without space charge limitation, is shown in Fig. 2. The Richardson Dushman equation for the saturation emission currents will be assumed Jc=ATl exp( -ecPc/kTc) la=AT} exp( -e¢a/kTa), (1) (2) where A = 120 amp/ cm2-°K2 is the theoretical con stant, e the electronic charge, and k the Boltzmann constant. This assumption neglects electrode patchiness, electron reflection, temperature dependence of the work function, and energy transfer between the electron gas and other gas present in the interelectrode space.4 The influence of these effects on Eqs. (1) and (2) as used herein is either negligible or can be made arbitrarily small in most practical applications. For simplicity it will be assumed that the anode temperature T a is so low that J a is negligibly small. The thermal efficiency of the emission-limited thermionic converter at its optimum output emf (VO=cPc-cJ>a) is NEGATIVE POTENTIAL IT '--D-IS-TA-N-C-E "'c CATHODE t ANODE v -L FIG. 1. Negative electric potential diagram for a space-charge limited thermionic diode with arbitrary load. 163 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.22.67.7 On: Tue, 02 Dec 2014 05:06:06164 N. S. RASOR NEGATIVE POTENTIAL DISTANCE T OATHODE Jc--Jo FIG. 2. Negative electric potential for an emission-limited thermionic diode with optimum load. then given4•1l by Jc(cf>c-cf>a- V w) 'T/ Jc(cf>c+2kTc/e)+PL' (3) where V w is the potential drop across the lead wires to the load. PL=Pr+Pm+(Pw-JcVw/2) is the net extra neous power loss per unit cathode emitting area due to radiation (Pr) heat conduction along the cathode lead wire (Pw), and other miscellaneous losses (Pm) such as gaseous heat conduction. By superposition, half of the Joule heat dissipated in this wire flows back to the cathode and thus subtracts from the conduction loss Pw• Thus the numerator of Eq. (3) is the electrical power per unit emitting area delivered to the load, while the denominator is the sum of the heat per unit emitting area required to emit the electrons from the cathode and supply its extraneous heat losses. Since P wand V ware both functions of the lead wire size, Eq. (3) can be maximized with respect to this variable. The optimum lead configuration may be shownt to be given by Aw ( P )l( 'T/)l -=J A 1--Xc. 'T/K(Tc-Ta) 2' (4) where Aw, X, p, and K are the area, length, average electrical resistivity, and thermal conductivity, re spectively, of the cathode lead wire, and Ae is the effective electron emitting area of the cathode. As will be shown later, even under ideal conditions the efficiency 'T/ is usually less than 50%, allowing 'T/«2 to be used as a very good approximation in the remaining analysis. Using this optimum lead size, the optimized efficiency is found from Eq. (3) to be F[( Be)l ]2 'T/o= B2 1+F -1 , (5) 11 G. N. Hatsopoulos, Ph.D. thesis, Massachusetts Institute of Technology (1956). t The optimum values given by Houston (see reference 4) are not strictly correct. His otherwise logical description of the opti mization may be corrected by replacing" in his Eqs. (12) through (15) by ,,/(1-,,) where F=pK(Tc-Ta)/cf>c2, C=1-cf>a/cf>c, and B=1 + 2kT cI ecf>c+ (P r+ P m)/ J ccf>c. As in the thermoelectric case, this fairly exact but formidable expression for the optimized efficiency may be greatly simplified by the use of a few generally valid but restrictive approximations, as follows. Up to this point it has been tacitly assumed that the anode is so cold that no electrons are emitted from it. However, for minimum PL in a given system, the temperature of the anode should be as high as the absence of appreciable back-emission permits. It may be seen from Eqs. (1) and (2) that the ratio of anode to cathode emission currents is negligibly small if the anode temperature is slightly less than the optimum anode temperature Tao defined by Tao Tc -=-=a. cf>a cf>c (6) As will be shown below, values for Jc within an order of magnitude of 1 amp/cm2 are required for efficient energy conversion. Therefore Eq. (1) gives to an adequate approximation for 1 <Jc<10 amp/cm2 at 1500oK. Furthermore, it may be shown that (BC/F)l»1 if the cathode lead wire is made of a typical metal obeying the Lorentz (Wiede mann-Franz) relation pK=L(Tc+Ta)/2, where the constant L=2.45X1O-s w-ohmsrK2. Assuming the anode to be operated at the optimum temperature given by Eq. (6), assuming for the moment o .., ~ :::: w °O~--------------~----------------~2 E 1/3 4>0 FIG. 3. Relation between anode work function CPa and cathode work function CPc for maximum conversion efficiency. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.22.67.7 On: Tue, 02 Dec 2014 05:06:06FIGURE OF MERIT FOR THERMIONIC ENERGY CONVERSION 165 TABLE I. Performance summary for thermionic conversion. Performance for arbitrary anode and cathode Performance for arbitrary anode, and optimum cathode From Eq. q,. <t>. Given anode work function Cathode work function q,c (given) FO!1!>l% (13) </>co= 0.92</>.+0.411 El Optimum anode temperature Required cathode temperature T.o==aif>. T.o=aq,. Tco=a<f>cQ (6) (6) (10) Maximum efficiency T,,=aq,c 1-",.1</>. 'fJo=f3 1+ .&pl[1- (<t>aI<t>c)4] nmax=---- and (14) where with that miscellaneous heat losses (Pm) may be neglected, and using the approximations discussed in the preced ing paragraph, Eq. (5) gives for the optimized efficiency (3 (8) where I1t=1-T,,/T e=1-<p,,/<pe=maximum thermody namic (Carnot) efficiency, D=lJ(1+2ka/e), and (3/D = 1-[2LDa2(1 + Ta/Te)! (1+ DP rIJ .4>.)Jl. The param eters D and PID are very insensitive to all variables (e.g., O.82<(3/D<1), a value of "'0.9 being typical for both. Finally, the radiative heat transfer term is given by (9) where Ar is the effective area for heat radiation from the cathode, t is the effective emissivity of the cathode anode system, and q is the Stephen-Boltzmann radiation constant. Using Egs. (6) and (9), Eq. (8) becomes where (11) Equations (10) and (11) yield the efficiency for arbitrary values of <Pe and tP",· However, for a given value of tP/J' it may be seen from Eq. (10) that an optimum value of ¢c exists at which '1)0 is a maximum. Physically, this maximum occurs when the gain in Carnot efficiency with increasing cathode temperature (increasing ¢c) is compensated by the rapid onset of radiation losses. 1+4.&p,o~ Specifically, this maximum occurs when (11) (7) (8) (12) where ¢cQ is the optimum cathode work function defined by this equation. Equation (12) is plotted in Fig. 3, and can be represented to a good approxi. mation by (13) for 0.1 < Ei<Pa <3, which covers the entire region of present practical interest. The corresponding maximum value of '1)0, which is thus the maximum attainable conversion efficiency, is found to be {3 (14) III. FUNDAMENTAL PERFORMANCE LIMITATIONS The results of the foregoing analysis are summarized in Table 1. Thus the problem of developing a highly efficient thermionic converter reduces to: (1) obtaining a surface with the lowest usable work function for the anode; (2) obtaining a surface for the cathode with a work function near the optimum value [given by Eg. (13) or Fig. 2J; (3) constructing and operating these electrodes to give the smallest obtainable value of the parameter E defined in Eq. (11); (4) reducing extra neous heat losses, including use of the optimum lead size given by Eq. (4). The maximum efficiency of the resulting converter is then given by Eq. (14) if the optimum cathode work function is achieved, and by Eq. (to) if it is not. Item (1) has a twofold critical importance. Firstly, for a low efficiency converter, the maximum obtainable efficiency is about inversely proportiona.l to the cube of [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.22.67.7 On: Tue, 02 Dec 2014 05:06:06166 N. S. RASOR the anode work function [Eqs. (13) and (14)]. Secondly, the temperature at which the cathode must operate for maximum efficiency is roughly proportional to the anode work function [Eqs. (6) and (13)]. The lowest usable work function is that which corresponds [through Eq. (6)J to the system heat sink temperature. Thus work functions as low as ! ev, corresponding to room temperature as the optimum anode temperature, are of interest for thermionic conversion. Since the lowest work function presently known is greater than 1 ev, and there appears to be no fundamental reason why it cannot be lower, considerably increased efficiencies and substantially lower temperature requirements could result from development of surfaces with lower work functions. Such surfaces must be compatible with the methods of space charge neutralization, however. Item (2) does not appear too formidable in itself since a variety of materials are available having work functions in the 1 to 5 ev region of interest for a con verter cathode. Also, the efficiency maximum is rela tively broad in CPe. However, in a practical application, the highest temperature available from the heat source may limit the cathode temperature, and thus its work function, to values smaller than Teo and CPeO. Item (3) permits further discrimination among those electrode materials most closely satisfying items (1) and (2). The material property affording the greatest latitude in the parameter E is the saturation current density Ie obtainable from the cathode surface. The relationship in Eq. (1) tends to obscure the important intuitive fact that this current density is an intrinsic property of a surface once a maximum permissible rate of its vaporization is imposed by a practical application. As is shown in the Appendix, the limitation on Ie is primarily determined for a surface by its ratio of cohesive energy to work function. Although existing data indicate that most known materials are incapable of emitting more than 10 amp/cm2 without serious volatization, there is apparently no fundamental reason why surfaces capable of substantially higher emission cannot be developed. As stated previously, in some existing devices Ie is limited principally by inter electrode space charge. The present analysis may be applied, using observed values of Ie, when space charge limitation is small enough to be considered as a per turbation (i.e., when Ve~e and Va::::CPa), and in any case gives the maximum performance obtainable. Having obtained the largest possible emission current I e, further reduction in the parameter E is possible by reducing the emissivity E, and the heat-radiating to electron-emitting area ratio AT/A •. The latter ratio can in principle be made arbitrarily small in gas filled devices where the electron mean-free-path is sufficiently small to allow the electric field to penetrate small cavities. Using such an approach, "heat-shielded" cathodes have been developed12 for high power rectifiers 12 E. F. Lowry, Electronics 6, 280 (1933). TABLE II. Fundamental correspondence of thermoelectric and thermionic converter performance. Thermoelectric Thermionic 1 1 Fraction of Camot efficiency 0.5----- 0.8------- 1+KdK, 1+0.9 (Pr/Jc</>,) Usable energy transport Extraneous energy transport Figure of merit pK and thyratrons with an electron-emitting area more than 100 times the effective heat-radiating area. If this principle could be similarly applied to existing gas-filled converters/·9 quite high fractions of Carnot efficiency could be obtained using existing materials. The only noteworthy extraneous heat loss relative to item (4) which has not been treated is that due to gaseous conduction. It is completely absent in the vacuum converter, and affects the efficiency to only a negligible extent in existing gas filled devices. For those cases where it might be important, an analysis similar to that subsequent to Eq. (8) may be carried out including F m, yielding similar but somewhat more complex results. Finally, for a thermionic converter operating in the range of practical power density and temperature implied by the typical value a=640oK/v, the ideal upper limit of efficiency may be seen from Eq. (14) to be "-'74% (i.e., for Fr= Ta=O). Of the ",26% intrinsic inefficiency, '" 16% arises from cathode lead losses, and "" 10% from the kinetic energy transported by the emitted electrons. IV. INTUITIVE SIGNIFICANCE OF PARAMETERS From Eqs. (10) and (14) it may be seen that the single dimensionless parameter 1 1 Aele M=-=-- ECPe3 Dua4 ATEc/Je3 (15) is a convenient figure of merit for the thermionic converter. Converters having maximum conversion efficiencies greater than 16% are distinguished from those of lower efficiency by a value of M greater or less than unity, repspectively. Indeed for low efficiency converters (i.e., M«1), the parameter M is approxi mately equal to the largest fraction of Carnot efficiency obtainable, and M/4 is approximately the maximum over-all efficiency obtainable by optimizing such a converter. Taking A./ A~1 and substituting numerical values for the constants in Eq. (15) (using a= 640oK/v), M is found to be essentially numerically equal to IciEc/Je3, for Ie in amp/cm2 and CPc in ev. A direct comparison between thermionic and thermo electric converters may readily be made using the results of the present analysis [Eqs. (8) and (15)J and the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.22.67.7 On: Tue, 02 Dec 2014 05:06:06FIG U REO F MER I T FOR THE R M ION ICE N ERG yeo N V E R S ION 167 analogous thermoelectric results obtained elsewhere.5,6 The correspondence is displayed in Table II. K L, K., and K, are the lattice, electronic, and total thermal conductivities, respectively, S the Seebeck coefficient and p the electrical resistivity of the thermoelectric material. T is the hot junction temperature. It should be recognized that in both cases the figure of merit is essentially the electrical power delivered to the load divided by the heat lost from the heat source, for an inefficient converter with its heat sink at absolute zero temperature. Specifically (TS)2/ -p-KT +-t A.l r!fJci ArO"ETc4. V. CONCLUSIONS It has been found that the use of a few generally valid approximations permits the optimum efficiency of thermionic energy conversion to be expressed in simple analytical form. The resulting expressions are similar to those extensively used in thermoelectric conversion technology, and should be correspondingly useful in comparing prospective materials and configurations for thermionic converters. The optimum relationship be tween the work functions of the electrodes, their required or optimum operating temperatures, and a figure of merit for convenient quantitative comparison of thermionic conversion systems have been defined. Examination of steps toward improving thermionic converter performance reveals no fundamental limita tion preventing achievement of conversion efficiencies as high as 74%. APPENDIX: MAXIMUM ELECTRON EMISSION AS AN INTRINSIC SURFACE PROPERTY The rate of vaporization of the emitting surface may be represented13 by q=Q exp( -eE/kT c), (16) where q is the flux density of evaporating atoms, E is the cohesive or adsorption energy, in electron potential units, of the emitting surface. Q is dependent on various factors such as temperature and the nature of the evaporating species. These factors do not in general cause Q to vary by more than an order of magnitude for present purposes, so that it may be considered constant compared with the extreme sensitivity of the ex ponential to its argument. 13 E. H. Kennard, Kinetic Theory of Gasses (McGraw-Hill Book Company, Inc., New York, 1938), pp. 409 and 69. TABLE III. Effect of E/</>c on emitter vaporization rate. 1.8 2.0 2.2 2.4 2.6 Evaporation rate (mm/yr) 1000 40 1 0.04 0.001 Combining Eqs. (1), (6), and (16) yields l~qP expr~(: -1) 1 (17) where P likewise may be considered constant for purposes of the present illustration. Thus, since in any practical application q is limited to some maximum value, leis likewise limited to a value determined by q and the surface property E/tPc. Indeed, since lc is enormously more dependent on the property E/tPc than on q, the suitability of a material as an emitter is primarily determined by this property. The importance of the property Ehc is illustrated in Table III for a typical value of lc required for efficient conversion (10 amp/cm2 at lS000K). The value of P used was obtained from observed data for tungsten. It would thus appear that surfaces having E/tPc<2.0 are totally un suitable for use as a converter cathode, while those having E/tPc>2.4 would be excellent. Using this criterion on the pure metals whose E/tPc is known, tungsten and tantalum are marginal and the rest are distinctly unsuitable. Dispenser cathodes14,15 and cathodes using an adsorbed film in equilibrium with its vapor7 allow substitution of a high adsorption energy for a low cohesive energy, and permit substantially higher values of q. Examples have been found which are suitable for continuous emission of 10 amp/cm2• Al though little suitable data are available for evaluating the large number of refractory compounds which might be considered for use as a cathode, some of the data which do exist imply spectacular emission capabilities. For in stance, the value of E/tPc for zirconium carbide (ZrC), computed from the reported work function16 (corrected for the temperature dependence of ¢c) and thermo chemical data, is found to be in the vicinity of S. 14 A. W. Hull, Phys. Rev. 56, 86 (1939). 16 A. Venema et al., Phillips Tech. Rev. 19, 177 (1957). 16 D. L. Goldwater and R. E. Haddad, J. App!. Phys. 22, 70 (1951). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.22.67.7 On: Tue, 02 Dec 2014 05:06:06

1.1742374.pdf

On the Application of the Kinetic Theory of Liquids of Born and Green to the Problem of the Calculation of the Volume Viscosity R. E. Nettleton Citation: The Journal of Chemical Physics 23, 1560 (1955); doi: 10.1063/1.1742374 View online: http://dx.doi.org/10.1063/1.1742374 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/23/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reference state for the generalized Yvon–Born–Green theory: Application for coarse-grained model of hydrophobic hydration J. Chem. Phys. 133, 124107 (2010); 10.1063/1.3481574 A String-theory calculation of viscosity could have surprising applications Phys. Today 58, 23 (2005); 10.1063/1.1995735 Application of Kinetic Theory to the Problem of Evaporation and Condensation Phys. Fluids 14, 306 (1971); 10.1063/1.1693429 Born–Green Pair Potentials for Liquid Lead J. Chem. Phys. 51, 547 (1969); 10.1063/1.1672032 Volume Viscosity of Liquid Antimony J. Acoust. Soc. Am. 42, 1161 (1967); 10.1121/1.2143942 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Thu, 25 Dec 2014 09:47:121560 LETTERS TO THE EDITOR On the Application of the Kinetic Theory of Liquids of Born and Green to the Problem of the Calculation of the Volume Viscosity R. E. NETTLETON* Department of Physics, Brown University, Providence 12, Rhode Island (Received June IS, 1955) BORN and Greenl have proposed to apply the superposition approximation of Kirkwood' to the second in the hierarchy of recurrence relations originally obtained by Yvon3 for the con figuration-velocity distribution functions in a fluid to yield a closed equation for the pair distribution function for two molecules in the twelve-dimensional space of pairs. Following the method suggested by Born and Green, a solution to this equation was assumed in the form of an expansion in the gradients of tempera ture and fluid velocity. On substitution of this solution into the integro-differential equation for the pair distribution function, equations are obtained for the coefficients in the assumed solution. From these coefficients, if they could be calculated, one could obtain the contribution of collisional transfer to the transport coefficients in a liquid. Unfortunately, the equations for the coefficients in the assumed expansion for the pair distribution function are found to be homo geneous, in agreement with Klein and Prigogine,4 so that while approximate solutions can be obtained by a Fourier transform method similar to that of Born and Green, they are undetermined to the extent of an arbitrary multiplicative constant. There ap pears to be lacking a principle of irreversibility. An attempt was made to estimate the bulk viscosity in a gas by going out to the next order in the calculation and comparing coefficients of second-order space derivatives of temperature in the expression obtained after substitution of the assumed expan sion into the equation for the pair distribution function. On the neglect of terms which should be of the order of the contributions of triple collisions, a formula was obtained yielding the value of 4 micropoises for the bulk viscosity of gaseous argon, zero for nitrogen, both at O°C, 1 atmos. The formula has the form: '10= 2Am (3P -Pav- 2pk)+~c. 3Pavk T m 3 Here '10= bulk viscosity, A= thermal conductivity, P= pressure, k=Boltzmann's constant, av=l/p(ap/aT)v, T=absolute tem perature, m=molecular mass, p=mass density, and 'Ic=contribu tion of collisional transfer to the shear viscosity. The latter quan tity was estimated by subtracting from the measured value of shear viscosity a value for the kinetic component calculated from a formula of Lennard-Jones.' The estimate for argon is probably too high because of inaccuracies in the available data. It is, how ever, still too small to show up in the acoustic absorption and streaming experiments which have been made.6 * Corrinna Borden Keen Research Fellow, Brown University, 1954-1955. 'Born and Green, Proc. Roy. Soc. (London) A190, 455 (1947). 'J. G. Kirkwood, and E. M. Boggs, J. Chern. Phys. 10,394 (1942). • J. Yvon, Actualites Scientifiques et Industrielles, No. 203 (1935). • Klein and Prigogine, Physica 19, 89 (1953). 'S. Chapman and T. G. Cowling, The Mathematical Theory of Non uniform Gases (Cambridge University Press, London, 1952), Chapter 12. 'See, e.g., H. Medwin, J. Acoust. Soc. Am. 26, 332 (1954). Reply to Tietz' Letter: "Approximate Analytic Solution of the Thomas-Fermi Equation for Atoms" H. C. BRINKMAN Central Laboratory T.N.D., Delft, Netherlands (Received April 21,1955) IN a recent letter to the editor, Tietzl gives a modification of my method of solving the Thomas-Fermi equation' applied to atoms. The following remarks may serve to compare the relative positions of Tietz' modification and my original method for atoms: Approximations Exact Brinkman Tietz Equation d' .. /dx' =x-i .. t d' .. /dx' = (x .. )!~ .. ' d' .. /dx'=-- X (X .. )! (x .. )! =c, =0.64 (x .. )+ =c, =0.576 .. =cx+k, (2c,+x+) 6c, Solution numerical .. /x+(6c,)+}' Asymptotic behavior x-+ "', .. -+144/x' exponential .. -+6c./x' The approximate solutions do not differ much numerically for intermediate values of x, while Tietz' solution has a simpler form. For x--> 00 my solution decreases too fast and Tietz' solution too slowly. However, the essential difference lies in the fact that my equa tion is linear, while Tietz' equation is not. Linearity greatly facilitates the application of the method to molecules, as was shown in my treatment of the H20 molecule.3 On the other hand, the extension of Tietz' method to molecules seems to be hardly possible. 'T. Tietz, J. Chern. Phys. 22, 2094 (1954). , H. C. Brinkman, Physica 20, 44 (1954). 3 H C. Brinkman and B. Peperzak, Physica 21, 48 (1955). Reply to Brinkman's Letter: Concerning My Letter "Approximate Analytic Solution of the Thomas-Fermi Equation for Atoms" T. TIETZ Department of Theoretical Physics, University of L6dz, L6dz, Poland (Received May 19, 1955) My approximatel analytic solution of the Thomas-Fermi equation for a free neutral atom <I>=_b_2 _= __ 1_. ~""b (b+X)2 (1+ax)" a (1) is the next homolog of the simplest form given by Kerner2 1/1 +cx. The general homolog l/(1+ax)" has not only sufficient founda tion in the modification of Brinkman's method3 given by me, but also has foundation through the following variational integral: (2) For the measure of the approximate degree of the approximate solution of the Thomas-Fermi equation for a free neutral atom, Umeda4 proposes to employ the nU[I1erical value of the variational integral evaluated by putting the given approximate solution in place of <1>. Since the minimization of I is equivalent to the integra tion of the Thomas-Fermi equation, the better approximate solu tion being the convergence limits. For Miranda's· exact solution, direct numerical quadrature of (2) gives the value 1=1.3625 which should be, of course, the limiting minimum value of I. The general homolog is now being studied on the basis of the approxi mation degree mentioned above. For the Sommerfeld approxi mation, Umeda4 has found for I the best value 1=1.3670. Ac cording to Table I, we see that the general homolog seems to be capable of extensive application by virtue of its simplicity and higher degree of approximation. The simplicity is of essential interest for the approximate solutions. The linearity of nonlinearity of the equation in different problems of this field has no great importance, because the simplicity of the approximate solution is of essential interest. The application of Brinkman's method to the following problems: free positive ion, compressed neutral atom, the solution of the Schriidinger equation for an approximate This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Thu, 25 Dec 2014 09:47:12LETTERS TO THE EDITOR 1561 TABLE 1. The functional dependence of I and a of the general homolog for a. I a a=1.0 1.3679 1.3679 a =1.2 1.3653 1.1565 aminimum=1.3939a a=7/S=1.4 1.3635 1.3636 0.88608 0.881206 a =3/2 =1.5 1.3637 0.80812 a =8/5 =1.6 1.3640 0.74566 a =9/5 =1.8 1.3650 0.64599 a=2 1.3663 0.56927 • The values of this column were privately communicated to the author by Professor K. Umeda for which I am very grateful. atomic field and so on, we see to be complicated or hardly possible. My method' seems to have an application to these problems. Brinkman's solution has two constants whose values we find by trial. For x---> 00, my solution decreases too slowly, Brinkman's solution too fast. But for x>20, my solution is more accurate than Brinkman's solution as we see in the following: x Fermi 20 0.0058 Sommerfeld 0.0056 Brinkman 0.0027 Tietz7 0.0065 Brinkman's solution has exponential behavior and my solution has the following behavior: . <I>->_1_ for x---> 00 • (ax)2 The application to Brinkman'ss method to molecules is much simpler than my method, which seems to be hardly possible. 1 T. Tietz, J. Chem. Phys. 22, 2094 (1954). , E. H. Kerner, Phys. Rev. 83, 71 (1951). 3 H. C. Brinkman, Physica 20, 42 (1954). 'K. Umeda, J. Phys. Soc. Japan 9, 290 (1954). 'C. Miranda, Mem. R. Acad. d'Italia 5,285 (1934). 6 T. Tietz, Nuovo cimento (to be published). 7 The value for a in Eq. (1) is given in Table 1. • H. C. Brinkman and B. Peprzak, Physica 21, 48 (1955). r-Centroids for Diatomic Molecules* R. W. NICHOLLS AND W. R. JARMAIN Department of Physics, University of Western Ontario. London, Canada (Received June 6, 1955) THE fairly general conditions under which it is possible to use a quantity fv'v" associated with the v'--->v" molecular band, where f'l)'1)" fif-'v'f1/;v"dr fif-'v'if-'v"dr and also under which it is possible to write (1) (2) have been discussed elsewhere by Fraser.I if-'v' and if-'v" are molecular vibrational wave functions and r is the internuclear separation. Because of Eq. (1), the name r-centroid is ascribed to fv'v" which is evidently an "average" internuclear separation with respect to the weighting factor if-"'if-'v", associated with the v'->v" transition. An array of r-centroids for a band system is a set of discrete values which lie across the range experienced by the molecules in all levels of both electronic states of the transition. fv'v" lies within or close to the range of r which is common to the regions between the classical turning points of motion in the upper (v') and lower (v") levels, respectively. The use to which r-centroids have so far been put is, through Eq. (2), to determine how the electronic transition moment f (r) in this case-varies with internuclear separation,l-4 and it is to be expected that similar uses may be found to determine other r-dependent quantities. A number of methods by which arrays of r-centroids may be evaluated for band systems have been developed and will be published shortly. Arrays of r-centroids are available for some band systems.' A further interpretation or fv'v" may be seen in the following: By definition, the average or expectation value of r experienced by the molecule in the level v' is (3) Now (4) where (v',v") = f if-'v'if-'v"dr. The Franck-Condon factor of the transition is (5) It strongly influences the relative probability that the transition occurs. From Eqs. (1), (3), (4), and (5) (6) Similarly (7) It may thus be observed from Eqs. (6) and (7) that the expecta tion values fv' and fv" may be considered as weighted averages of r-centroids with respect to the Franck-Condon factors and that qv'v"fv'v" is the contribution to each from the v'--->v" transition. Finally it may be remarked that a smooth variation has been observed, and established analytically, between fv'v" and the band wavelength Av'v'" If ro' and ro" are the equilibrium inter nuclear separations of the upper and lower states of the transition, fv'v" is an increasing function of Av'v" when r.'>r." and is a decreasing function of Av'v" when r 0' <r .". * This work has been made possible by the Air Force Cambridge Re- search Center through Contract AF 19(122)-470. 1 P. A. Fraser, Can. J. Phys. 32, 515 (1954). 'R. G. Turner and R. W. Nicholls, Can. J. Phys. 32, 468-474 (1954). • R. G. Turner and R. W. Nicholls, Can. J. Phys. 32,475-479 (1954). 'L. V. Wallace and R. W. Nicholls, J. Atm. and Terrest. Phys. (to be published). 'W. R. J armain and R. W. Nicholls, Scientific Report No. 20, Contract AF 19(122)-470, Department of Physics, University of Western Ontario, April, 1955. Lattice Energies of the Alkaline Earth Imides and the Heat of Formation of NH-2 AUBREY P. ALTSHULLER 2715 East 116th Street, Cleveland 20, Ohio (Received June 13, 1955) THE lattice constants of the alkaline earth imides, CaNH, SrNH, and BaNH, have been determined.1•2 They are found to have the NaCl type lattice as do the alkaline earth oxides and sulfides.2 The lattice energies of CaNH, SrNH, and BaNH can be calculated by the Born equation.' A more detailed calculation is unjustified since the compressibilities of the alkaline earth imides and the polarizability and "main frequency" energy< of NH-2 are unavailable. The minimum cation-anion distances for CaNH, SrNH, and BaNH are 2.58, 2.725, and 2.92 A, respec tively.l,2 The repulsion constants n used here for CaNH, SrNH, and BaNH, 8, 8.5, and 9.5, are the same as those used formerly for the corresponding alkaline earth oxides" The lattice energies calculated for CaNH, SrNH, and BaNH are 787, 752, and 711 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.138.73.68 On: Thu, 25 Dec 2014 09:47:12

1.1743212.pdf

Bond Localization and the Hyperconjugative Effect in the Aromatic Carbonium Ions Toshifumi Morita Citation: The Journal of Chemical Physics 25, 1290 (1956); doi: 10.1063/1.1743212 View online: http://dx.doi.org/10.1063/1.1743212 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/25/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantification of hyperconjugative effect on the proton donor X–H bond length changes in the red- and blueshifted hydrogen-bonded complexes J. Chem. Phys. 137, 084311 (2012); 10.1063/1.4748135 Electronic Spectra of Carbonium Ions J. Chem. Phys. 27, 826 (1957); 10.1063/1.1743855 The Energies of Strained Carbonium Ions J. Chem. Phys. 21, 550 (1953); 10.1063/1.1698944 Resonance Energy in Carbonium Ions J. Chem. Phys. 20, 744 (1952); 10.1063/1.1700534 Lack of Resonance Energy in Gaseous Carbonium Ions J. Chem. Phys. 19, 1073 (1951); 10.1063/1.1748473 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.55.97 On: Tue, 09 Dec 2014 03:40:581290 LETTERS TO THE EDITOR there is a strong contribution to this atomic integral. The value of In was obtained from the relation, J .... =i(G I+2F2)=0.88 ev, where GI and F2 are the radial exchange and Coulomb integrals, respectively.6 Voge7 obtained the values of GI and F2 empirically by adjusting these parameters to fit the electron energy levels of carbon. J"h= -2.3 ev was obtained by Van Vleck.s S"h=0.80 was obtained by fitting a series of Slater wave functions to the self-consistent field calculations of Torrance and then computing from the tables of Mulliken et al.,9 the uk overlap integral at the CH equilibrium bond distance. Substitution of the values of the matrix elements yields t:J.E=6.2 ev and A=0.024. The hyperfine interaction then becomes t:J.H = 28 gauss. It has been assumed that there is unit unpaired electron density in the 11' orbital. Actually, in an aromatic free radical, the unpaired electron is distributed over many carbon atoms and the magnitude of the hfs would be expected to be proportional to the actual unpaired electron density on the carbon. The value of 28 gauss per unit unpaired electron density is in excellent agreement with experimental results.lO This result does not imply that t:J.H is constant for all free radicals. Higher order terms that take into account the symmetry of the molecule and the contributions from other orbitals could cause variations of several gauss in this calculated value. Calculations for CI3 hfs become increasingly difficult since many more orbitals contribute to the hfs in second order. The author wishes to acknowledge several helpful discussions with Professor S. I. Weissman. * Contribution No. 400 from the Chemical Department, E. 1. du Pont de Nemours and Company, Wilmington, Delaware I H. M. McConnell, J. Chern. Phys. 24, 764 (1956). 'R. Bersohn, J. Chern. Phys. 24. 1066 (1956). 'E. Fermi, Z. Physik 60, 320 (1930). • C. C. Torrance, Phys. Rev. 46, 388 (1934). • S. L. Altmann, Proc. Roy. Soc. (London) A210, 327, 343 (1951-1952). • E. U. Condon and G. H. Shortley, Theory oj Atomic Spectra (Cambridge University Press, New York, 1951) p. 177. , H. H. Voge, J. Chern. Phys. 16,984 (1948). • J. H. Van Vleck, J. Chern. Phys. 2. 20 (1934). 'Mulliken, Rieke, Orloff, and Orloff, J. Chern. Phys. 17, 1248 (1949). lOH. M. McConnell, J. Chern. Phys. 24, 632 (1956); E. De Boer, J. Chern. Phys. 25,190 (1956); H. S. Jarrett (to be published). Bond Localization and the Hyperconjugative Effect in the Aromatic Carbonium Ions TOSHIFUMI MORITA DePartment oj Chemistry, Faculty oj Science, Tokyo Metropolitan University, Fukasawa-cho, Setagaya-ku, Tokyo, Japan (Received September 24, 1956) THE absorption spectra of methylbenzene carbonium ions indicate the "blue-shifts" of the corresponding absorption bands, with increasing number of methyl substituents.1 To inter pret this, the wavelengths of the first transition for benzenium (E), toluenium (T), and mesitylenium (M) ions are calculated, using the semiempirical LCAO MO method including overlap. As the molecular models, the following two are adopted; (1) the hyper conjugation model, in which the pseudo-atom H2 formed by attached proton together with hydrogen atom already present at the position of proton attack conjugates with the ring; (II) the bond localization model, in which the H2- C bond is localized with respect to the rest of the molecule and other things are identical with (I). Table I shows theoretical and experimental data, ac cording to which, assuming that the singlet-triplet separations are not the same but do not differ so much among these substances, the blue-shifts in question can be explained by neither model. If model (II) is accepted for (B) and (T), whereas model (I) for (M), the self-consistency of the theoretical data in relation to the experi mental results will be achieved. Table II shows the conclusive theoretical data. It is likely that these substances do not fix to TABLE 1. First excitation energies calculated for the two models and experimental values. ForHCJ For B.L. Absorption Oscillator Carbonium modelb model' wavelength strength ions (-{J') (-{J') obs (A) obs Benzenium 0.870829 0.801925 4170 0.15 Tolueniuma 0.879022 0.815739 4000 0.15 Mesitylenium 0.818010 0.814447 d 0.760181 3900 0.27 • For only the type which belongs to C" symmetry group, computation is carried aut. b HCJ =Hyperconjugation. The values in this column are those calcu lated using 5 = -0.3. See Y. I'haya, J. Chern. Phys. 23, 1165 (1955). 0(3 is the resonance integral including overlap between the adjacent carbon atoms in benzene molecule, and adopted the value of -60 kcal/mole. See C. C. J. Roothaan and R. S. Mulliken, J. Chern. Phys. 16, 118 (1948). d This is the value obtained using, besides 5 = -0.3, the auxiliary induc tive parameter, l5ind = -0.1. e B.L. = Bond localization. The values in this column are those calculated using, besides 15 = -0.3, the auxiliary inductive parameter, l5ind = -0.1. either model. If the eigenfunction of such molecules is written as <I>=a<l>I+b4>rr (<1>1, <I>/I: eigenfunction of (I) and (II), respectively; a, b; coefficients), the behaviors of these substances are understood in the unified point of view. In the case of isomeric xylene carbonium ions also, this idea is adequate to interpret the basicity of parent hydrocarbons. Our calculation shows the 11'-electron energies of m-, 0-, and p-xylene carbonium ions in the bond localization model are -11.4819, -11.4914, and -11.4905 (in -fj), respectively.2 From Gold and Tye's equation,3 we obtain for KI/KII (KI, KIl; equilibrium constants for carbonium ion formation of m-and p-xylene, respectively), InKI/KIl= -{ -x+c+0.696}/RT, where, -X=EI_EI.O; -c=EIl_Err,o; EpI_EpIl= -0.18 kcal/ mole4; EI.O_O.516 (kcal/mole)=Ell,o; EI, Ell; EI,O, EIl.O; 11' electron energies for the actual and bond localization states of m and p-xylene carbonium ions, respectively; EpI, EpIl: 11'-electron energies of m-and p-xylene. Admitting the actual state of carbonium ions to be the hybrid of hyperconjugation and bond localization states, the magnitudes of x and c indicate the stabiliza tion energies by the hyperconjugation of H2 pseudo-atom in m-and p-xylene carbonium ions compared with their bond localization states, respectively. Taking KI / Kll = 26 at 20°C· into account, the TABLE II. Conclusive theoretical data on the aromatic carbonium ions. Singlet- Longest triplet absorption Carbonium separations wavelength Oscillator ions Model assumed (ev) (A) strength Benzenium B.L. 1.77 4168 0.31 Toluenium B.L. 1.95 3999 0.20 Mesitylenium HCJ 2.10 3909 0.32 percent fraction of the hyperconjugation state of carbonium ion of m-xylene relative to that of p xylene may be obtained by 100 X (x-c)/5.9, on the assumption that the stabilization energy through the perfect hyperconjugation is 5.9 kcal/mole.6 Our calculations indicate that the extent of the hyperconjugation state in the carbonium ion of m-xylene is larger by 44% than that of p-xylene carbonium ion, which is almost identical with that of o-xylene carbonium ion. ! C. Reid, J. Am. Chern. Soc. 76, 3264 (1954). 2 These values are those for the most stable of all possible different types produced, depending on the position of proton attack. As for m-xylene carbonium ion, the next stable type is accepted owing to the consideration on the experimental facts of nitration reaction of m-xylene. 3 V. Gold and F. L. Tye. J. Chern. Soc. 1952,2184. 4 C. A. Coulson and V. A. Cra,,{ord, J. Chern. Soc. 1953.2052. 'M. Kilpatrick and F. E. Luborsky, J. Am. Chern. Soc. 75, 577 (1953). • Muller, Pickett, and Mulliken, J. Am. Chern. Soc. 76. 4770 (1954). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.248.55.97 On: Tue, 09 Dec 2014 03:40:58

1.1730881.pdf

Absolute Infrared Intensities of the Ammonium Ion in Crystals C. C. Ferriso and D. F. Hornig Citation: The Journal of Chemical Physics 32, 1240 (1960); doi: 10.1063/1.1730881 View online: http://dx.doi.org/10.1063/1.1730881 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/32/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Absolute Infrared Intensities of Stannane J. Chem. Phys. 46, 1176 (1967); 10.1063/1.1840787 Absolute Infrared Intensities in Mixed Hydrogen Halide Crystals J. Chem. Phys. 43, 4180 (1965); 10.1063/1.1696666 Infrared Studies of Crystal Benzene. V. Reflection Spectrum and Absolute Intensities J. Chem. Phys. 36, 62 (1962); 10.1063/1.1732319 Measurement of Absolute Infrared Absorption Intensities in Crystals J. Chem. Phys. 34, 1061 (1961); 10.1063/1.1731636 Infrared Studies of Crystal Benzene. IV. Absolute Intensities J. Chem. Phys. 33, 233 (1960); 10.1063/1.1731089 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:19tHE JOURNAL OF CHEMICAL PHYSICS VOLUME 32, NUMBER 4 APRIL,1960 Absolute Infrared Intensities of the Ammonium Ion in Crystals* c. c. FERRISOt AND D. F. HORNIGt Metcalf Chemical Laboratories, Brown University, Providence, Rhode Island (Received October 26, 1959) The absolute intensities of the two infrared active fundamentals of the ammonium ion, .3 and .', have been studied in several environments. Dilute solid solutions of NH.+ ions in KCl, KI, CsCl, and CsI were studied and compared with pure NH.I. After separating the effect of the net charge, the apparent bond moments ranged from 0.63 d to 0.80 d (av 0.71) and the bond dipole derivatives, ap./ar, from 2.06 d/A to 3.59 d/ A (av 2.80 d/ A). The bond moment is substantially the same as in NH3. INTRODUCTION Up to the present time no direct experimental studies of the absolute infrared absorption intensities of ions in crystals have been carried out. Deciusl-3 has estimated dipole moment derivatives (from which intensities may be calculated) from the magnitude of the frequency splittings caused by inter molecular coupling, basing the calculation on the assumption that only dipole-dipole forces were im portant. Haas and Hornig4 have computed intensities and dipole moment derivatives from the frequency difference between longitudinal and transverse waves in crystals, obtained either from Raman spectra or infra red reflection spectra. This calculation depends on the validity of the assumption of purely electrostatic interactions. It seemed important to us to obtain a direct measurement involving as few auxiliary assump tions as possible. Straightforward measurement of single crystal speci mens is exceedingly difficult because the thickness re quired to study fundamentals in ionic crystals is of the order of 10 J.I. or less. Techniques for producing such specimens have not yet been developed. In some cases, such as the ammonium salts, suitably thin samples can be made by sublimation. In order to measure intensi ties, the number of absorbing molecules per cm2 of beam must be known. This might be determined by measuring the thickness and index of refraction of the film optically. Alternatively, if it is established that the film is of uniform thickness this same quantity can be found by quantitative chemical determination of the amount of material in the film. We have used the latter method to study the absolute infrared intensities of the two infrared active bands of the ammonium ion in films of ammonium iodide. * Based on a thesis submitted by C. C. Ferriso in partial fulfill ment of the requirements for the Ph.D. degree, Brown University, 1956. This work was supported by the Office of Naval Research and presented at the Symposium on Molecular Structure and Spectroscopy, Ohio State University, Columbus, Ohio, June, 1955. t Present address: Convair Astronautics, Applied Research, San Diego, California. We have also studied the intensities of dilute solid solutions of ammonium ions in alkali halide crystals.5 The ammonium ions can thus be placed in crystalline environments which closely approximate all of the phases of pure ammonium halides, but any complica tions produced by intermolecular coupling are avoided since in a dilute solution the ammonium ions are far apart. In addition, the specimens can be made con veniently thick. The integrated infrared absorption intensities of the NH4+ ion were therefore measured in KCI and KI, which have the NaCl structure character istic of the high-temperature phases of the pure am monium salts, as well as in CsCI and CsI which have the same structure as the low-temperature phases. EXPERIMENTAL METHODS (a) Sample Preparation The NaI films were deposited on masked KBr plates by sublimation in a high vacuum. The KBr plates were cooled to about O°C and the NaI heated to about 100°. The growth of the films was followed by observing the color of reflected light when they were illuminated with white light and in this way the thickness could be estimated accurately enough to prepare films for the intensity measurements. The fact that the color was relatively uniform over the entire film surfaces indicated that they were homo genous. The amount of material deposited was deter mined later by dissolving the specimen and analyzing for NH4+. Since the mask determined the coated area accurately, this was sufficient to give the concentration of NH4+ per cm2• The solid solutions were prepared by adding an aliquot of stock NH~ solution to a saturated water solution of the corresponding alkali halide. Some of the material in the solution was then precipitated, either by evaporation or by the addition of ethanol, yielding crystals in which the ammonium ion replaced a small proportion of the alkali ions. No differences were noted between the spectra of samples prepared by the two methods. t Present address: Department of Chemistry, Princeton Uni- versity, Princeton, New Jersey. 6 All the systems studied form solid solutions at the concentra- 1 J. C. Decius, J. Chem. Phys. 22, 1941, 1946 (1954). tions used, e.g., R. W. Havighurst, E. Mack, Jr., and E. C. Blake, 2 J. C. Decius, J. Chern. Phys. 23, 1290 (1955). J. Am. Chern. Soc. 47, 29 (1925). This is also evident from our 3 W. C. Steele and J. C. Decius, J. Chem. Phys. 25, 1184 (1956). spectra which are characteristically different from those of the • C. Haas and D. F. Hornig, J. Chern. Phys. 26,707 (1957). pure phases. 1240 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:19A B SOL UTE I N F R ARE DIN TEN SIT I E S 0 F THE A M M 0 N I U M ION 1241 The sample, usually about 7 to 8 g, was dried over night at 115°C and then ground for 30 min in an agate mortar. The powder was then dried again overnight, ground once more for 15 to 30 min and stored in the oven until used. For study, this powder was pressed into pellets in a press of our own design and construc tion. The powder was placed in the mold and evenly distributed by inserting the plunger and rotating it several times. The mold was assembled, evacuated to about 20 J.I. Hg for 5 min, and the pellet formed by pressing at about 100000 psi for 1 min or so. The pressing was repeated two or more times, rotating the mold 120° after each pressing. The resulting pellet was transferred with tongs to the absorption cell or a desiccator. The pellets were of 20.6 mm diam and their thickness, measured with micrometers, was uniform to within 1 or 2% over the entire area. Their thicknesses were usually in the range 0.3 mm to 1.4 mm. The powdered alkali halides were pressed to within less than 1% of their theoretical densities. The pellets prepared in this way remained optically transparent for more than four months if stored in a desiccator. (b) Pellet Press Many evacuable molds which produce good quality pellets have been described in the literature ;6-8 how ever, some of them suffer from the fact that the pellet may be fractured and chipped when removed from the mold. The unique feature of our mold is the tapered lower die (3 degrees) which eliminates the problem of fracture when the pellet is removed from the mold. Figure 1 shows a section through the vacuum mold. The powder is pressed in the cylinder D between the polished surfaces of the quench hardened tool steel dies (E) and (F). Pressure is applied by means of the plunger A and supported by the lower anvil (C) which screws into the carbon steel body of the mold (B). The sample chamber (D) can be evacuated through the port (G) about 3/4-in. above the sample. The 0 rings on the plunger and anvil provide the vacuum seal. The pressure was delivered by a hydraulic ram which provided a total force of up to 55 000 lb, yielding a pressure up to 110 000 psi on the sample. (c) Analysis of the Pellets In order to measure the ammonium ion concen tration, a weighed pellet was dissolved in 100 ml of water. The NH4+ was then determined as NH3, using Nessler's reagent prepared as suggested by Winkler.9 The colorimetric analysis was carried out utilizing a 6 V. Schiedt and H. Rernwein, Z. Naturforsch. 7b, 270 (1952). 7:VI. A. Ford and G. R. Wilkinson, J. Sci. Instr. 31,338 (1954). 8 D. H. Anderson and N. H. 'vVoodall, Anal. Chern. 25, 1906 (1953). 9 L. W. Winkler, Z. Untersuch. Nahr. u. Genussrn. 49, 164 (1925). FIG. 1. Section through the vacuum mold. B spectrophotometer as described by Kistiakowsky et al.1O In order to obtain reproducible and accurate results it was found necessary to prepare fresh Nessler's reagent before each analysis and to carry out the spectro photometric determination as soon as the color was fully developed, approximately 5 min after the addition of reagent to the solution being analyzed. The calibra tion curves were constructed by analyzing standard solutions containing from 10-7 to 10-8 mole of NH4+ per cc of water in the presence of from 0.01 cc to 0.001 cc of alkali halide per cc of water. It must be noted that by analogy to liquid solutions concentra tions in the pellets were expressed in moles of N~+ per cc of alkali halide, the alkali halide being treated as if it were a solvent. The alkali halides used included KCI, KI, and CsCI; no effect on the calibration curve depending on the salt was found. It was concluded from the study of standard solutions that the absolute error was within ± LOX HJ-9 mole of N~+ per cc of water. This represents an error of 1.5% in the crystal solutions. (d) Measurement of Intensities The absolute intensity is defined by the relation A = (l/le) f In (IolI) truedv, band where 10 and 1 are the true values of the intensities of the incident and transmitted beams, I is the thickness and e is the concentration of NH4+ in the specimen. The quantity which can be measured is B= (llle) f In(ToIT)expt.dv, band which may differ from A because of the finite resolution of the spectrometer.ll Here T is the apparent trans mission measured by the spectrometer. It is readily shown that B----+A as le----+O. Consequently, if Ble is plotted vs te, B= A in that region of Ie where a straight line is obtained. Another test is also available in that 10 G. B. Kistiakowsky, P. C. Manglesdorf, Jr., A. J. Rosenberg, and W. H. R. Shaw, J. Am. Chern. Soc. 74, 5015 (1952). 11 E. B. Wilson, Jr., and A. J. Wells, J. Chern. Phys. 14, 578 (1946); A. M. Thorndike, A. J. Wells, and E. B. Wilson, Jr., ibid. 15. 157,868 (1947). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:191242 C. C. FERRI SO AND D. F. HORNIG NH! IN KCI 30 NH~ IN Kl ~ 20 .6 .8 1.0 1.2 Ie, (MIGR~~OLES X GM) FIG. 2. Blc. vs lc. of eopreeipitated mixtures in the 7p. band. when B7"'A, B must depend on the resolution of the spectrometer. The infrared spectra were obtained on a Perkin Elmer Model 83 monochromator which was modified for use as a ratio-recording double beam instrument.12 A CaF2 prism, calibrated by the procedure of Dowie et al.,13 was used. Intensities were measured for both infrared active fundamentals of NH4+, the triply degenerate stretching vibration, 113, and the triply degenerate bending vibration, 114. To obtain the base line corresponding to 10, a spectral region through the absorption band was scanned, with a blank pellet made from the same salt substituted for the sample. In first order this procedure should have eliminated both reflection and scattering errors for solid solutions. The same is not necessarily true for the NH4I films; 'e ~ .. 25 200 ~ 15 ..." .£ c ..::.. " 100 u II) NH! IN KGI Ics (M1CRg~OLES X CM) FIG. 3. Blc. vs lc. of copreeipitated mixtures in the 3p. band. 12 D. F. Hornig, G. E. Hyde, and W. A. Adcock, J. Opt. Soc. Am. 40, 497 (1950). 13 A. R. Dowie, M. C. Magoon, T. Purcell, and B. Crawford, Jr., J. Opt. Soc. Am. 43, 941 (1953). however, the reflectivity is sufficiently small in bulk specimens, even at the 113 and 114 peaks, so that the correction should be negligible for films whose thick ness is much smaller than the wave length of the radia tion, since the reflection is still further attenuated. Measured intensities (B) were obtained by replotting the measured TofT as In( TofT) vs frequency. The measured intensities were very little influenced by changes in spectral resolution. When the resolution was decreased two-or threefold, the change in B was only 2 or 3%, the decrease in peak intensity being roughly compensated by the increase in band width. The spectral resolution used in the 7-f.L region ranged from 6 to 15 cm-l and that in the 3-f.L region varied from 20 to 40 cm-l. As a second check on the extent to which B was equal to A, a series of pellets of varying thickness and con centration were prepared. These pellets were not analyzed but it was assumed that the concentration of NH4+ ion in each pellet was proportional to the ratio TABLE I. c lc System (moles/ee) (p.M/em2) B(va)a B(v,)a NH.+-KCI 8.346XlO-S 0.771 259.8 28.9 NH.+-KI 12.353X10--6 0.726 212.9 39.5 NH.+-CsCI 16.425X 10--6 1.257 117.9 20.9 NH.+-CsI 7.490X1O- s 0.770 106.4 22.4 NH.I 1. 732XlO-2 0.39 80 18 a Units: Darks=cm-1/p.M cm'XIO-' [D. G. Williams, W. B. Person, and B. Crawford, Jr., J. Chern. Phys. 23,179 (1955). of NH4+ and alkali halide concentrations in the solution (cs). The resul ting plots of BlcB vs lcs for both 113 and 114 are shown in Figs. 2 and 3. It is seen that in all cases the points fall on a straight line passing through the origin so throughout this range of Ie, it may be con cluded that B= A and the measured intensities are reliable . The final experimental values, given in Table I, were determined from a set of three-intensity runs for each ammonium ion-alkali halide system, and for the pure N~I films from a set of ten films ranging from about 0.5 to 2.3 f.L in thickness. All of these specimens were carefully analyzed to determine the NH4+ concentra tion. The magnitude of the absorption in each case placed them within the intensity range covered by Figs. 2 and 3. It should be noted that the intensity measured for 113 includes that of the combination band, 112+114, which is in Fermi resonance with itY We assume that the major part of the intensity of the entire band derives from the fundamental. 14 E. L. Wagner and D. F. Hornig, J. Chern. Phys. 18, 296, 305 (1950). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:19A B SOL UTE IN F R ARE DIN TEN SIT I E S 0 F THE AM M 0 N I U M ION 1243 DISCUSSION OF EXPERIMENTAL RESULTS The previous absolute intensities have been deter mined in a medium with refractive index differing from unity; they must be corrected for the fact that the effective field of the radiation is not that in free space. This problem has been considered by Polo and Wilsonl5 for the case of a molecule embedded in a spherical cavity in a continuous dielectric medium and experi encing an Onsager reaction field. Their result was that A i= N1r(n2+2)2(~)2, 3cn 3 aQi (3) where A i is the absolute integrated intensity of the ith normal mode; N the number of absorbing molecules per cc, c the velocity of light in a vacuum, n the re fractive index of the medium, and af.L/aQi the rate of change of vector dipole moment with normal coordi nate Qi. This expression should be a good approxi- TABLE II. Substance ± (ap./ aQa) ± (ap./ aQ4) NH4+-KCI 305 101 NH4+-KI 256 110 NH4+-CsCI 193 81 NH4+-CsI 173 79 NH4I 163 71 NH,Cla (-190°C) 99 NH,Brb (-190°C) 104 a Reference 4, Raman frequency difference between longitudinal and trans verse modes. b Reference 4, infrared reflection spectrum. mation for an ion embedded in a cubic crystal. The resulting ap,jaQ/s for the NH4+ ion in various lattices are given in Table II, together with the results of Haas and Hornig for comparison. The experimental intensi ties were divided by three since both modes are triply degenerate. The sign ambiguity is present because (ap,/aQi) appears as the square in Eq. (3). The general agreement between the results obtained with two different methods in the present work and two dissimilar methods in reference 4 is very satisfying. The intensities obtained from solid solutions in the NaCl-type lattices were generally greater than in the CsCI-type lattices, but those for NH4I, which has an NaCllattice, were lower than in any of the solid solu tions. It is hard to rule out systematic errors but pre sumably the differences reflect the differing local en vironment of the NH4 ion in the various lattices. Calculation of af.L/ aRi To obtain information about the N-H bonds m NH4+ the af.L/aQ's must be related to the change m moment during stretching and bending motions of 15 S. R. Polo and M. Kent Wilson, J. Chern. Phys. 23, 2376 (1955). FIG. 4. Orientation of atoms for calcu lation of ap./aR •. • Z I known amplitude. Using the orientation of atoms shown in Fig. 4, we have employed the symmetry coordinates (4) and R4a= (aI2+a23+aI3-a14-a24-a34)/(6)! (5) for the infrared active species, where T i is the change in length of the NHi bond and aij is the change in the angle HiNH j• The coefficients in the transformation (6) can be worked out by the method of Wilson.16 The details of the calculation for a tetrahedral molecule have been given previouslyY In order to carry it through it is necessary to know the force constants. Since there are three force constants, F33, ToF34, and T02 F44, in the most general potential function for the F 2 species of a tetrahedral molecule, the two frequencies of NH4+ are not enough to solve the problem. If the frequencies of ND4+ are also used, there are just enough frequencies to solve for all the force constants. The frequencies used in our calculation were as followsl4: NH4+ 3130 cm-l 1400 cm-l ND4+ 2350 cm-l 1067 cm-l• Because of the anharmonicity, these frequencies do not follow the Teller-Redlich product rule, the fre quency product differing from the harmonic oscillator TABLE III. Summary of results for the ammonium ion in KCI. Set I Set II (L-I)"X1Q--12 1.2439 1.2440 (L-I)4,XIQ--12 -0.0316 -0.0299 (L-I)34XlO- 20 0.1262 0.1238 (L-I)44X 10-20 0.8668 0.8643 f (a",jaQsL+ (a,u/ aR4) = 1. 265 d 1. 256 d or (a",/ aQ4) -(ap./aRa) =3.759 d/A 3.760 d/A f (ap./ aQ,) (a",/aR,) =0.497 d 0.501 d or (ap./aQ,) =-(ap./ aRs) =3.823 d/ A -3.822 d/A 16 E. B. Wilson, Jr., J. Chern. Phys. 7, 1047 (1939); 9, 76 (1941). 17 P. N. Schatz and D. F. Hornig, J. Chern. Phvs. 21, 1516 (1953). • This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:191244 C. C. I<ERRISO AND D. F. HORNIG TABLE IV. (ap.jaRi) for the ammonium ion in various alkali halides. KCI (ap./aR.) = 1.27 d 0.50 d (ap./aR 3) = +3. 76 d/A -3.82 d/A KI (ap./aR.) 1.28 d 0.60 d (ap./aR,) +3.16 d/A -3.23 d/A CsCl (ap./aR,) 0.95 d 0.46 d (ap./ aR3) +2.38 d/A -3.23 d/A CsI (ap./aR.) 0.91 d 0.47d (ap./aR 3) +2.14d/A -2.19 d/A NH.I (ap.jaR.) 0.82 d 0.41d (ap./ aR3) 2.00 d/ A -2.05 d/A 8 Preferred solution. value by about 3%. The anharmonicity resides largely in Va since the first overtone of the bending vibration, 2V4, is found at almost exactly twice the fundamental frequency. These frequencies therefore yield two sets of force constants. The first (I) is obtained from the two bend ing frequencies plus the NH4+ stretching frequency, v •. The other (II) is obtained from the two bending fre quencies plus the ND4+ stretching frequency, V3. Taking ro= 1.035 A from nuclear magnetic resonance studies18 the results obtained are as follows: Set I 5.380X106 dynes/em 0.506 0.539 Set II 5. 719XI05 dynes/em 0.536 0.541 The consistency of the two values of F44 and the differ ences in Fa3 tend to bear out the initial assumption that the anharmonicity resides in Va. In order to test the sen sitivity of the a}J./aR's to these uncertainties in the force field, a complete calculation was carried out with both sets for Na+ in KCI and the results are summarized in Table III. The two sets of force constants yielded such consistent results that only Set I was used in calculating the dipole derivatives in the other lattices; the results are given in Table IV. The two solutions result from the sign ambiguity arising because only (a}J./aQ)2 is measured. In order to choose between the formal solutions it is important to note that positive Ra and R4 result in proton motions in opposite directions. Hence any model with an effective charge located on the protons will predict that aJ.l./aRa and aJ.l./a~ have opposite signs. Consequently, we feel that the last column of Table IV is most probably the correct one. EFFECT OF DISPLACEMENT OF CENTER OF CHARGE The results given in Table IV cannot be interpreted immediately in terms of bond characteristics since the 18 H. S. Gutowsky, G. E. Pake, and R. Bersohn, J. Chern. Phys. 22, 643 (1954). dipole moment change includes the total charge times the displacement of the origin with respect to which dipole moment is measured. The magnitude of this effect (which does not occur in neutral species) can readily be estimated if the nitrogen atom is chosen as an origin for the definition of internal moments and the coupling of the internal vibrations to the lattice vibration is neglected. The extra apparent moment is then the electronic charge times the displacement of the nitrogen atom, yielding (aJ.l./aRa) = (a}J./aRa)0_ (2e/3) (mH/M) = (aJ.l./aRa) °-0.306 d/ A (aJ.l./aR 4) = (aJ.l./aR 4)0-(2ero/3) (mH/M) = (aJ.l./aR 4) °-0.319 d, (7) (8) where mH is the proton mass and M is the total mass of the ion. In Eqs. (7) and (8) it is assumed that the net effective charge on the protons is positive in both motions. Otherwise the sign of the second term, repre senting the displacement of the positive center of charge, must be changed. It would be possible to improve on these assumptions if the frequency and intensity of the Reststrahlen mode were also measured. If the additional assumption is now made of additive changes in bond moments, directed parallel to bonds when they are stretched and perpendicular when they are bent, we have also that (aJ.l./ar) = (v3/2) (a}J./aRa)0 }J.o= (v3/2) (a}J./aR 4)0. The resulting dipole derivatives and bond moments are given in Table V. Several points are worth noting. First, the bond moment is not high; interpreted as effective charge it corresponds to a charge +0.14e on each proton and +O.44e on the nitrogen.19 It is interesting that the NH moment found is identical with the bond moment found in NHa.20 Secondly, a}J./ar, which also has the significance of an effective charge, is very much larger, corresponding to over +0.6e. This is presumably the case because TABLE V. Bond moments and bond moment derivatives for ammonium ions in various crystals. KCI -4.13 0.82 3.59 0.72 KI -3.54 0.92 3.08 0.81 CsCI -3.54 0.78 3.08 0.69 CsI -2.50 0.79 2.18 0.69 NH,I (pure) -2.36 0.73 2.06 0.64 avo -3.21 0.81 2.80 0.71 19 L. Pauling, Nature of the Chemical Bond (Cornell University Press, Ithaca, New York, 1948). 20 D. C. McKean and P. N. Schatz, J. Chern. Phys. 24, 316 (1956). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:19A B SOL UTE I N F R ARE DIN TEN SIT I E S 0 F THE A M M 0 N I U M ION 1245 the motion involved in the anti symmetric stretch corresponds roughly to the process NH4+-.H++ NHa so that the polarity changes violently as the protons vibrate. Thirdly, the differences between the various environ- THE JOURNAL OF CHEMICAL PHYSICS ments appear to be real and outside the experimental error. On the other hand, no particularly systematic trend appears and no immediate explanations can be given. It seems clear, though, that there is considerable polarization of the NH4+ ions by the surrounding ions. VOLUME 32, NUMBER 4 APRIL, 1960 Unimolecular Decomposition of Chemically Activated sec-Deuterobutyl Radicals from D Atoms plus cis-Burene-2* R. E. HARRINGTON, B. S. RABINOVlTCH, AND R. W. DIESEN Department of Chemistry, University of Washington, Seattle 5, Washington (Received November 2, 1959) Chemically activated sec-deuterobutyl radicals were produced at 25°C by the reaction of D atoms with cis-butene-2. These vibrationally excited species contain an increm'ent of energy above that of the corre sponding light radicals as formed from H plus cis-butene-2 in a previous study [B. S. Rabinovitch and R. W. Diesen, J. Chern. Phys. 30,735 (1959)]. Apparent rate constants for the unimolecular decomposition to propylene of the deuterobutyl radicals were obtained as a function of pressure, relative to the collision induced stabilization process. Theoretical values for the rate constants at the limits of high and low pressures were calculated using a direct count for the density of vibrational energy levels. The calculated and experi mental results are compared with one another, and with the results of the previous study of the sec-butyl radical decomposition. The expected energy effect is observed; the deuterobutyl radicals appear slightly more monoenergetic than the equivalent nondeuterated species. VIBRATIONALLY excited alkyl radicals may be produced homogeneously in the gas phase by the addition of atoms or radicals to an olefin. The ensuing unimolecular decomposition and collisional stabilization processes can be studied concurrently under varying conditions of temperature and pressure. Although the method is restricted in some ways, it nevertheless provides a means of studying rapid unimolecular processes at low temperatures such that the excited species are nearly monoenergetic. Rabinovitch and Diesen (RD) 1 have described experimental and theo retical aspects of the method and its application to the formation of sec-butyl radicals, by addition of H atoms to cis-butene-2, and their decomposition to propylene plus methyl. Reference should be made to this work for details and for nomenclature used here. It is of value to employ different addition reactions to produce the same (or virtually the same) species in different states of vibrational excitation. In this paper we present the experimental findings for the uni molecular decomposition of chemically activated sec deuterobutyl radicals formed by addition of D atoms to cis-butene-2. The deuterobutyl radicals have a minimum energy, Emin, which is ,...",2 kcal/mole larger than that of butyl radicals produced with H atoms due to zero point energy factors. A comparison is made * Abstracted in part from a Ph.D. thesis by R. E. H. to be submitted to the Graduate School, University of Washington. 1 B. S. Rabinovitch and R. W. Diesen,}. Chern. Phys. 30,735 (1959). between this work and the earlier results. In the calculation of ka, the average observed rate constant for decomposition, the semiclassical approximation of RD for the evaluation of the density and weights of vibrational energy levels, based upon expressions of Marcus and Rice,2 has been replaced by direct count. EXPERIMENTAL Deuterium gas (purity 99.5%) was passed through a silica gel trap at -195°C and was used without further purification. Other experimental details were as previously. Corrections to the Data In addition to providing an experimental verification of the primary and secondary reaction mechanisms given by RD, the use of deterium atoms disclosed two additional complicating processes. The first was the contamination of the reaction products by CD4• This arose from "cracking" to CD4 of butene which effused into the discharge tube, followed by back-effusion of CD4 into the reactor. The effect was most marked at the higher run pressures. Methane analyses were performed mass spectrographically; only light CH4 was considered in the computation of stabilization products resulting from disproportion at ion of methyl and sec-butyl. 2 R. A. Marcus and O. K. Rice, }. Phys. Colloid Chern. 55, 894 (1951); R. A. Marcus, J. Chern. Phys. 20,352,359 (1952). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Fri, 05 Dec 2014 12:51:19

1.1722031.pdf

Diffusion, Static Charges, and the Conduction of Electricity in Nonmetallic Solids by a Single Charge Carrier. II. Solution of the Rectifier Equations for Insulating Layers Selby M. Skinner Citation: Journal of Applied Physics 26, 509 (1955); doi: 10.1063/1.1722031 View online: http://dx.doi.org/10.1063/1.1722031 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/26/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Single carrier devices with electrical doped layers for the characterization of charge-carrier transport in organic thin-films Appl. Phys. Lett. 97, 013303 (2010); 10.1063/1.3460528 Analytical solution for charge-carrier injection into an insulating layer in the drift diffusion approximation J. Appl. Phys. 104, 123708 (2008); 10.1063/1.3050297 Generalized equations for rectified diffusion J. Acoust. Soc. Am. 72, 1586 (1982); 10.1121/1.388494 Static SpaceCharge Distributions with a Single Mobile Charge Carrier J. Chem. Phys. 30, 212 (1959); 10.1063/1.1729876 Diffusion, Static Charges, and the Conduction of Electricity in Nonmetallic Solids by a Single Charge Carrier. I. Electric Charges in Plastics and Insulating Materials J. Appl. Phys. 26, 498 (1955); 10.1063/1.1722030 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33JOURNAL OF APPLIED PHYSICS VOLUME 26. NUMBER 5 MAY. 1955 Diffusion, Static Charges, and the Conduction of Electricity in Nonmetallic Solids by a Single Charge Carrier.*·t II. Solution of the Rectifier Equations for Insulating Layers SELBY M. SKINNER Department of Chemistry and Chemical Engineering, Case Institute of Technology, Cleveland, Ohio (Received June 15, 1954) The discussion of the thermodynamic equilibrium of an electron atmosphere in a dielectric is extended to the current-carrying steady state. Under the assumption that the experimental behavior at zero current is continuous with that when there are finite electrical currents, it is found that forward and reverse currents show distinct behavior. Expressions are developed for the electrostatic potential, field, charge density, the current-applied voltage relation, and the capacity and rectifying behavior of dielectric layers. The non-ohmic conduction exhibits rectification, and departures from the Wagner relation are predicted, of the type observed experimentally, including turnover voltages. The contact charging of a dielectric is the zero-current limit of the non-ohmic behavior which causes rectification. I. EFFECTS OF CURRENT FLOW 1. Current-Carrying State WHEN the insulator is no longer in a condition of electrostatic charge equilibrium only, ~but is transmitting current: (a) An applied voltage exists, so that U2-U1 must =eV¢O. (b) The thermodynamical state of the system is not an equilibrium one, but a steady state one. (c) The electron· gas outside the second electrode is at a potential 'P2-'Pl+eV with respect to that outside the first electrode. Appropriate changes must be made in the boundary conditions. (d) The temperature will rise because of the PR heat unless provision is made for cooling. Since constant temperature has been assumed, it is necessary to assume that suitable cooling is provided. For small currents, it may be assumed that with natural cooling the observed behavior will be sufficiently well represented by as suming constant temperature. (e) Since actually observed insulating layers (at least in small thicknesses) show rectifying properties, e.g., the copper oxide layer, the results should indicate such rectification. 2. Continuity of Physical Behavior The equilibrium sta te discussed in I (with zero current flow) is not essentially different physically from one with small current flow. The solution of Eq. (9) for small current should reduce to the solution at zero current as the current flow becomes zero.l If this is * Originally received, November 3, 1953. Received in final form June 15, 1954. Portions of this paper were presented at the same times and places as mentioned in reference 1 of the first paper of this series, J. Appl. Phys. 26, 498, 1955, hereafter referred to as I. t With the support of the Aeronautical Research Laboratory, Wright Air Development Center, U. S. Air Force. 1 There exist in the literature statements that the equilibrium possible, the validity of the whole range of solutions is intuitively more satisfying, since the two types of phenomena are thus directly related, and, for example, equilibrium charge densities at contact are the natural consequence and limiting result of the observed current flows at the interface between two materials when the net current becomes zero.2 The condition will therefore be applied that there be no discontinuity in the potential, field, or charge density, as j is continuously changed from any finite value through zero to a finite magnitude of opposite sign. 3. Solution of the Diffusion Equation for Nonzero Current3 Since the equation is nonlinear, considerably more complex interrelationships between the experimental parameters may be expected than in equations of the usual linear type. The experimental behavior of die lectrics varies sufficiently from sample to sample, and, for example, with thickness and temperature so that some such complexity is indicated. Equation (9) of I has been obtained by an integration with respect to x, and therefore g2 does not depend upon x, but it may well depend upon j, or T, or b. When there is a current, in analogy to the treatment in reference 3, condition at zero current may differ by finite amounts from that approached as the current decreases to zero. Such a metastable state is not considered here. With the equalizing tendency of diffusion and thermodynamic fluctuations, it would not be expected to persist. 2 While the considerations are applicable to a part of the phenomena in triboelectric charging, they are not the complete explanation; it is not at all clear that equilibrium (and certainly not, constant temperature) exists in the instances usually ob served. The subject is comprehensively covered by P. S. H. Henry, Science Progr. 41, 617 (1951). 3 A solution to Eq. (9) has been obtained by F. Borgnis, Z. Physik 100, 478 (1936). However, his treatment is not suitable for the present purpose since he determined the values of his constants to agree with Ohm's law at infinity, considering only the electro static potential. An approximate treatment has also been given by Mott and Gurney (I, reference 5), and R. C. Prim, Phys. Rev. 90, 753 (1953), Appendix II. 509 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33510 SELBY M. SKINNER Eq. (9) of I is modified by the substitutions or where 2kT F=-(3y; z=(3(x+XO)+,s-2 a2 e (33 = 27re2 j / Ebk2P becoming thereby a Ricatti equation dy -±i+z=O. dz (1) (2) The form of the solution depends upon whether z is positive or negative. If z>O, A L2/3(w)+ BI2/3(w) y= -zl -zlR(w), (3) A1l(W)+BLl(W) where W= (2/3)z! and the I" are Bessel functions of imaginary argument. If z<O, AJ_2/3(W)- BJ2/3(W) y= (z)i (z)lR(w), (4) AJl(W)+BLl(W) where: z= -z; w= (2/3)(z)l; the J" are the Bessel functions of the first kind, of real argument. Only one constant in y is significant. While separate solutions are not additive, analogous relations may be found in the mathematicalliterature.4 Defining the charge density by Poisson's equation, the solutions for the electric potential energy, field, and charge density, ne, are for z>O, 1/;=2kT In[zi{AIl(W)+BLl(w)}]+constl F= -(2kT/e)(3ziR(w) ~, n=(EkT/27re2)(32 z[R2(w)-IJ J and for z<O, 1/;=2kT InCZl{1J!(W)+BLl(W))J+const} F= -(2kT/e)(3z}R(w) . n= (EkT /27re2),82Z[R2(w)+ IJ (5.1) (5.2) By Eq. (1), the coefficient{3, and therefore, the quantities z and w, depend upon the value of the current. 4. Evaluation of the Constants A and B By the foregoing paragraph 2, Eqs. (5) must continu ously approach the solutions given in Eqs. (10)-(12) of I, as j approaches zero. The evaluation of A and B to satisfy this condition requires a mathematical strata gem, but can be shown to be valid by considering the 4 For example, G. N. Watson, Bessel Functions (The Cambridge University Press, New York, 1948), revised edition, Chap. IV. asymptotic or near-zero values, respectively, of the final expressions. The result of the development in Appendix I is that the three types of solution for g2~0, corresponding to Eqs. (10)-(12) of I, are: I. 1/;-1/;0= 2kT In[ wi~ 2(W)] I wOlH2(wo) F=-(2kT/e)(3z1G ' n= (EkT /27re2)(32Z[G2+ IJ (6) where the subscript zero refers to the value at x= 0, and [ 2 U ! Z=_Z=,s-2gL(3(x+xo); w=iz!=~ 1-3~] ; f= -t={3-2g2; ~=H!; u=g(x+xo); G=ii 1/ii2; iil=JiWJ2/3(W)+LlWJ-2/3(W) ; ii2=Jta)J -l(W)-J -tWJt(w). II. 1/;-1/;0=2kT In[ Wi H2(W)] I woiH2(WO) F= -(2kT/e){3z1G ' n= (EkT /27re2)(32z[G2-1J (7) where: z=,s-2a2+(3(x+xo); W=iZI={l+~~r a2=_g2j t=,s-2a2j ~=it!j u=a(x+xo)j G=H 1/H2j H1=LIWL2/3(w)-IIWI 2/3(w) ; H2=LiW1i(W)-I1WI_i(W). III. (a) For z<O: the expressions in Case I, just above, except that g=O; ii1=J-2/3(W) j ii2=Jl(W); w=i[ -(3(x+xo)JI. (8.1) (b) For z>O: the expressions in Case II, just above, except that a=Oj H1=L2/3(W) j H2=Ii(W) j w= i[{3(x+xo) J!. (8.2) These solutions reduce, when (3~, to those of (I), Section II, 4, as follows: Solution I II III Becomes solution in Section 4 of I, for g2=-a2>O g2=-a2<O g2= -a2=O and pass continuously into each other, if the constants in the potentials are properly chosen. Since the argu ments of the Bessel functions must be real or imaginary [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33RECTIFIER EQUATIONS FOR INSULATING LAYERS 511 and not complex, in order to carry physical significance, the valid solutions are those in which z or z is positive. This imposes the conditions: Solution I: g2/(X+XO) >,83> -OC) III. a: ,83<0 II: -a2/(x+xo) <,83< OC) b: ~3>0. The behavior of I in the negative current range or II in the positive current range is straightforward. From the point of view of an observer at x=O, the former is the forward current and the latter the reverse current as defined in rectification theory. At the limit of the other part of the range, the argument of the Bessel functions of either kind becomes zero and a continuous transition to the other type solution takes place. The values in Eqs. (6)-(8) are independent of the choice of origin. Ii The difference in the expressions for positive and negative current is indicative of the physical difference between the forward and reverse currents in rectifying contacts. However, the solutions may not be accepted uncritically exactly as given above; con siderations analogous to those in I must be used. 5. Electrochemical Potential The fact that the electrical potentials are represented by logarithmic functions of functions of the space coordinates yields for the electrochemical potentials, U=if;+kT In(n)+const: U-Uo=kT Inn, where n has the values w4/3[H 12(W)+Hl(w)] W04/3[HNwo)+H22(WO)]' W4/3[H 12(W) -H22(W) ] W04/3[H 12 (Wo) -H22(WO)]' (9) (9.1) (9.2) respectively, and corresponding terms in the cases in Eqs. (8.1) and (8.2). The space rate of variation of U must have a definite value to produce a particular current, since j= -nbVU.6 6 A change of origin which decreases the value of x will increase the value of Xo and leave unchanged their sum. This has been accomplished by the choice of the constant in Eq. (9) in I. Other wise a change of origin would have required a different value of g2 to return to a designation of the same eJqlerimental conditions. Independence of choice of origin is essential to physical reality, just as in the solution of a linear equation. Equation (9) likens the current flow, caused by diffusion and electric field, to that at a distance from a source plane of infinite charge density, whose distance behind the electrode is determined by the physical conditions of the problem, such as the energy gap at the interface; the description should be unaffected by whether the origin is taken at the electrode or away from it. 6 This value may be distributed in any way between the space variation of the electrostatic potential and the space variation of the logarithm of the concentrations without changing the com puted current. However, the considerations in reference 16 of I, and the assumption of the simultaneous validity of the diffusion equation and Poisson's equation provide the differentiation be tween how much of the gradient of the electrochemical potential is attributable to each of the two terms, and permit unambiguous Repeating Eq. (14) of I, the externally applied potential is related to the values of U at the ends of the dielectric by: (10.1) The relationship between applied voltage and current may be obtained from Eq. (10.1) either by inserting the expressions for the U's or as follows: dU j (10.2) -=--* dx nb' Equation (10.2) shows that the applied voltage is at least of the first order in the current. 6. The Constant, g2 or a2 Using Poisson's equation, the relation (Eq. 9 in I) for g2 becomes ( eF)2 (27re2n) 27re2j ---- =~2Z= __ (X+XO)_g2, 2kT ekT ebk2p (11) which is the generalization of Eq. (10.4) in I to the case of current flow. If both the field and charge density are known at a boundary for a particular current, g2 is thereby determined; the xo which enters into Eq. (11) is evaluated by the condition on n at the boundaries, and all other quantities are measurable. If (x+xo)¢O, current flow causes a change in the relation between field, charge density, and g2. Using the solutions of Eqs. (6) to (8), and Eq. (11), it may be shown that j= -nbdU /dx, identically. Experimentally, the charge density and potential at the boundary of the dielectric may be regarded as measurable quantities; the measurement of the electric field at the boundary, is, on the other hand, not a convenient procedure, and other means of determining the value of g2 are desired. 7. Asymptotic Development In Eqs. (6) to (9), the quantity ~ is proportional to j-I. Since the substances considered are insulators, only small currents need be considered, and development in powers of the current requires an asymptotic develop ment in terms of t1• Physically, it is obvious that the location of the inner potential maximum, and other eJqlressions for the field and charge densities within anyone material. The question is not one of the reality of the Galvani potential; Gibbs, Guggenheim, and van Rysselberghe have shown that the attempt to define the Galvani potential as apart from the electro chemical potential is without meaning; exhaustive recent attempts to improve the situation, such as that in the third reference below do not make the Galvani potential operationally significant. The inner potential used here is that discussed in I, Sec. III, 1, defined with the use of the eJqlerimentally determinable x. J. W. Gibbs, Collected Works (Longmans Green and Company, New York, 1928), pp. 429-30; E. A. Guggenheim, Thermodynamics (North-Holland Publishing Company, Amsterdam, 1950), second edition. International Committee on Electrochemical Thermodynam ics and Kinetics: Proc. of the IIIrd Meeting, Berne, 1951 (Carlo Manfredi, Milan, 1952), especially_Sec. IV, pp. 275-403. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33512 SELBY M. SKINNER "'&-Io,·eV FERMI LEVEL METAL FER.MI lEV£L FIG. 1. Energy levels in a dielectric with metal electrodes, when voltage is applied. parameters such as g, the xo's, and the charge densities at the boundaries may depend upon the current. The development under these conditions is indicated in Appendix II. The results are: U-Uo=kT In(A/Ao), t/t-t/to=2kT In(B/Bo),} F= -(2kT/e)C, n= (EkT/27re2)D, (12) where the quantities A, B, C, D, are power series in the current defined in Appendix II, and the subscript zero means that u is evaluated at the electrode-dielectric interface. 8. Finite Current-Carrying Dielectric Since the solutions pass continuously into those for zero current described in I, it is to be expected that the energy level differences at the boundaries will affect the observed current-carrying behavior. It is again neces sary to consider separately the case when the electro static potential is monotonic, and that when a potential maximum occurs in the dielectric. (a) Monotonic Potential Consider a dielectric specimen of thickness D, with, for example, identical energy gaps at each electrode. To be physically meaningful, it is necessary that the solu tion for small positive current at x=O shows the same behavior as the solution for negative current of the same magnitude does at X= D, provided the origin is changed from x=o to x=D, and the directions of the axis and the applied potential are reversed. If the electrodes are different, or for any other reason, the energy gap is different at the two electrodes, it is necessary to make appropriate adjustments of where the energy gaps are inserted into the boundary conditions when the origin and direction of the axis are changed. If the contact is between the dielectric and a metal, it is unnecessary to consider changes in the Fermi level of the metal due to the current, except the raising of the level in one metal relative to the other by the amount of the applied voltage. This also causes the conduction level in the dielectric at that boundary to be raised by eV, so that the same boundary conditions on n are applicable as before; this is illustrated in Fig. 1. If an equilibrium existed, and the Fermi levels in both metals were at the same height, a conduction level at height '1'2-x+ e V would imply an equilibrium charge density in the dielectric of Nc exp[-(tp2-x+eV)/kT]; the applied voltage raises the Fermi level and increases the charge density to Nc exp[ -(tp2-x)/kT]; the steady state flow of current is the attempt to reach thermo dynamic equilibrium by decreasing the charge density to the former value, unsuccessful because of the charges continually supplied by the source of external voltage. The boundary conditions, therefore, will be taken as: t/t",-o= tpI-Xj t/t",-D= tp2-X+eVj no,=N c exp[ -(tpi-x)/kT]; g= gO+,83gI+ .. '; xo.= XOO.+,83XH+ ... ; (13) with the definitions of p and VI: ,83=2pjj eV=2pjv IkT; in which only the first-order terms have been retained, and the electrostatic potential zero is at the Fermi level in the first metal. A single set of equations, Eqs. (12) suffice in the monotonic case. Applying these conditions: (1) the zero order terms are found to satisfy the zero current expressions given in I, and, (2) the first-order terms yield 4g04Xli= 4g02 (tanuo.- uo.) gl -2UOi tanuo.+sin2uo.+uor. (14) The evaluation of gi requires additional considerations similar to those used in I in showing that go is the same in the monotonic case as in the potential maximum case j it is not needed in the voltage-current relation however, since the same gl applies at both boundaries. The voltage-current relation is obtained by substituting the equations from Appendix II into Eq. (10.1), giving the analog of the Wagner equations, namely exp(eV/kT)-l=L: hd', i-I hI = pgo-S[sinuo cosuo-uO-4g02g1]I2= 2PVI, h2= p2(8go6)-I[16g02gI(u-sinu cosu-u sin2u) +3 (1-2u2+4u sinu cosu-2 sin2u) + 2u2 sin2u (15) -16go4XOI sin2u+2 sin4uJr2. The u's are evaluated at the boundaries as in Eq. (14), and the subscript and superscript on the brackets have the same meaning as in the expression of the value of a [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33RECTIFIER EQUATIONS FOR INSULATING LAYERS 513 definite integral. The coefficient h1 may be written e2D --[{sin(goD)/goD} cos(goS)-l], (16) go2ebk2J'2 where goS=1r(D+2 x01)/(D+x01+X02) ; goD = 1rD/ (D+X01+X02). The quantity in brackets is always less than zero; by Sec. 4, this solution is valid if j is negative; there fore, Eqs. (15) and (16) correspond to the direction of easy flow. A similar treatment of Eqs. (II) yields h1=pgo-2D[{sinh(goD)/goD} cosh(goS)-l] (17) the solution being valid for positive j. This, therefore, represents the current-applied voltage relation in the reverse direction. The expressions (16) to (17) may also be obtained by inserting the zero-current approximation for n into Eq. (10.2), and higher terms in the series result from use in Eq. (10.2) of the higher terms in the expression for n in Appendix II. The higher terms give promise of explaining the turnover voltage observed in rectification by, for example, copper-oxide rectifiers. This will be discussed in Sec. 13 below. The observed limit of resistivity of the material under steady-state conditions and uniform temperature as the current is decreased, is p= V / jD= (e/kT)(eb)-t[(D+x02+xol)2jr]~, (IS) where ~ is the expression in brackets in Eqs. (16) or (17). (b) Potential Maximum As in I, it is necessary to use separate solutions on either side of the maximum; on the right-hand side (side 2), x is replaced by D-x, and the negative sign in front of the expression for the electrostatic field becomes a plus sign. { sinu1 lh=2kT In -.-[1+ (pgo-Sj/2){Ul-Sg02gu SmU01 + (4go2guUl-U12+4v1) COtUl} ::~] }+lpl-X { sinu2 "'2= 2kT In -. -[1 + (pgo-a j/2) {U2-Sgo2g12 SmU02 + (4go2g12U2-U22+4v2) COtU2} ::~] } + 1,02-x+e V in which, as in Appendix II, Ul=gO(X+XOl); u2=go(D-x+X02); V1 = go4(xn+al) ; V2= go4(X12-al), (19) with analogous relations for the fields and charge densities. The boundary conditions are: at x=O: (1) "'1= 1p1-X; } (2) nOl=Nc exp[ -(1p1-x)/kT]; at x=D: (3) "'2= 1p2-x+eV; (20) (4) n02=Nc exp[ -(1p2-x)/kT]; at x=a: (5) "'1="'2; (6) F1=0; (7) F2=0; and in Eq. (19) the conditions on the potential at x=O, D, have been incorporated. The other five conditions applied to the zero order terms yield the same results as in the zero-current case treated in I. Applied to the first-order terms in the current (remembering that the term eV is of the first order in the current), they supply five relationships for the five quantities: Xu; X12; gu; g12; and a1. The term a1 accounts for the shift of the potential maximum with increasing current, and since this shift is the same whether viewed from the right or from the left, it must enter into the v's in the manner shown in Eqs. (19). Applying the boundary conditions, and utilizing the results from the zero order terms, there is obtained a set of five simultaneous equations: in which Cu = -C12 = Cn = Ca2 = 21rgo2 ; C13 = C14 = C16 = C22 = C24 = Cal = Caa= C42= C44= C45= C5l = C5a= C55= 0; C23= -C25= C34= C36= 4go4; C4l = 4go2(I-uo2 COtU02); C52= 4go2(I-uOI COtUOl); C43= _4go4 COtU02; C54= -4go4 COtUOl; lit = 1rg03V1-(1r /2) (sinuo2 COSU02 02=03= (r/4+1); 04=U022 cotuo2+sinuo2 cosUo2+2u02; 05 = U012 COtUOl + sinuol COSUOI + 2UOl. (21) The first is obtained from boundary conditions 2, 4, and 5; the remainder, in order are obtained from the conditions 6, 7, 4, and 2 separately. The five simultane ous equations suffice to determine the five unknown parameters. Since the electrical quanties are given by different expressions on the two sides of the maximum, the current-voltage relationship must be obtained as follows: exp[(U2-U ,,2)/kT] = 1+ [sinu cosu-u-4go 2g1],,2 exp[(U1-U "NkT] = 1+ [sinu cosu-u-4go2gl]1" (22) eV /kT=exp[(U 2-U1)/kT] = 1 + (1/3~o)[sinuo2 COSU02 +sinuol COSUOl-(U02+UOl)+1r] since U,,= 1r/2; U02= gOX02, UOl = gOXOl. Therefore exp(e V /kT) -1 = pgo-2jD[ (singoD/ goD) cosgot+ 1J, (23) where got=gO(X02-XOl). Recalling from I that the loca tion of the maximum is at x=a= [D+ (X02-XOl)]/2, it is possible to show that the current-voltage relationship Eq. (23) passes continuously into that given by Eq. (15) as the maximum passes either interface by variation of the xo's. The choice between Eqs. (15) and (23) is [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33514 SELBY M. SKINNER determined by whether the maximum is inside the dielectric at zero current. To examine the behavior of the potential maximum as the current is increased from zero, it is necessary to examine al. Solving Eqs. (21), there is obtained (24) where r = 4tlt2(7r+3sl-S2) +t2[27r (3U2-S2) +47r (Sl+S2) +4(1 +4ul2+Ulsl)+3r]- tl[27r(3ul-Sl) +4(1+4ul+u2 s2)+r]+[,r+ 2rsl +27r(S2U2-SlUl)+2(U22_Ul2)], 41= 16go4(t2+tl+7r), ti= tanuOi-uOi; S;= sinuoi COSUOi- UOi. If the electrodes are of the same material, U02 = UOl, and Eq. (24) becomes al = [(Ss+47r)t2+S7rst+4Tt +2rs+,r]/16g o4(7r+2t). (25) In this case, a large energy gap at the boundary produces a very large Xo, compared to which the thickness of the dielectric is negligible. The quantities Uo being 7rxo/(D+2xo) may be represented by (7r/2)-0 where 0'::::.7rD/4xo. The value of al therefore, in the case of a large energy gap and little zero-current charging of the dielectric is (25.1) If, on the other hand, the zero-current energy gap is small, so that the xo's are small, and considerable charge transfer occurs at equilibrium, the value of al is (25.2) Thus, in both limiting cases, the location of the maxi mum is found approximately from a=ao+ (2go4)-1{33= (D/2)+go-4pj, so that its rate of transfer from the center of the dielectric, with current is (25.3) In the former case go is approximately 7r/2, and in the latter case, it is approximately 7rxo/D. II. RECTIFICATION 9. Effect of the Boundary Energy Gaps on the Resistivity The great variety of behavior of different dielectrics follows from Eqs. (IS) and (23). The resistivity is inversely proportional to the charge carrier mobility, and depends both upon the energy gaps at the bound aries, and upon the thickness of the dielectric as related to the characteristic lengths Xo. Since, as in I, a small energy gap and large zero-current transfer of charge between electrode and dielectric implies a small Xo which may range as low as 10-7 em, and a large energy gap and a small charge transfer implies an Xo of astronomical length, there would be expected considerable ranges of behavior. If both xo's are small compared to D,7 whether or not they are equal to each other, Eq. (23) gives for the resistivity, p=---. kT Eb 7r2 (26.1) If both xo's are large compared to the thickness of the dielectric, and both about equal, the resistivity is e 2go-2 ~---= (47rnoeb)-l kT Eb (26.2) the second equation being obtained by use of Eqs. (17) of I. In this case, j a: noebF, (26.3) so that with a dielectric-metal combination in which large energy gaps exist between the electrode and the dielectric conduction levels, the observed behavior for small currents will approximate that in the case of a metal, but interpretation of the results as metallic conduction will give a slightly incorrect value of the product of charge density and mobility. If both xo's are large compared to the thickness of the dielectric, and quite different from each other, e 1 D2 p=---, kT Eb 6 (26.4) which is not essentially different from Eq. (26.1). Equation (26.4) also is valid if one Xo is large compared to D and the other is small compared to D. Intermediate values are more complex. The tempera ture dependence includes both a specific dependence upon r-1, and the dependence of parameters such as the mobility and the quantities Xo upon T, and requires special consideration for each case encountered ex perimentally. lO. The Applied Voltage-Current Characteristic When the electrodes are identical, a zero-current potential maximum exists, and Eq. (23) is applicable. The Wagner relation becomes exp(eV)_1=~ D[SingoD +1]j, kT Ebk2P go2 goD (27) showing that the voltage rises rapidly with current. The greatest slope occurs when goD is small, i.e., large energy gaps at the boundaries; the slope approaches zero, if g~. Physically this means that, for a given thick ness, a large initial charge transfer from the electrodes introduces a sufficient number of carriers so that the conductivity is large, whereas with a large energy gap at 7 This is the limit of the solution I, and corresponds essentially to the solution IlIa. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33RECTIFIER EQUATIONS FOR INSULATING LAYERS 515 the boundary, insufficient carriers exist to permit ap preciable current without large voltages. In the case of considerably different energy gaps at the boundaries, e.g., when the electrodes are of metals of quite different work functions, Eq. (17) is applicable. The slope of the analogous expression on the right-hand side is now negative, so that the voltage does not rise so rapidly with the current for the same absolute value of the coefficient of the current. However, the absolute value of this coefficient is now smallest when goD-->O, and got approaches zero. This occurs when XOi and Dare small compared to X02; the resistivity decreases not only with the thickness but also with a decrease of energy gap as before. If an essentially infinite dielectric is considered, as in the second paragraph of the next section, the voltage current relation becomes in the direction of easy flow: exp(eV/kT)-l=cij-cdl+ ... ; in the opposite direction: exp(eV /kT)-1=cij+C3)2, (28.1) (28.2) where Ci=i{U23-U13)P; C2= (p2/9)(U26+4u18U28_5u16); C3= (p2/9)(5u26_4u18U28_U16) and u2=D+xo, Ui=XO; Considering only the first term on the right in either case, If the energy gap 1P1-X, is large, so that the dielectric is not appreciably charged at equilibrium, xo»D, and remembering that for this case, no= EkT /27re2no2 there is obtained eV= jD/nob or its equivalent j=noebF. If, on the other hand, the energy gaps are small, and appreci able contact charging occurs at equilibrium, 7re2Vj jV eV = (2/3)--=--. EbkT 3bnoxo2 It was shown in I that the total charge in the dielectric is noexo ; letting this total charge equal CV where C is the capacity of the condenser formed by the electrodes and the dielectric, and substituting for Xo in terms of no by the equation first given, (30) where K= (V/3bC)[~Nc exp{ -(lPl-X)/kT}]. EkT Since the capacity may be taken as constant, there should be observed a current proportional to the square of the applied voltage for small currents.8 11. Rectification Ratio By the expressions developed, the slope on either side of zero current is the same, and the rectification ratio is 8 Compare Mott and Gurney, reference 5 of I, p. 172. one. This, however, does not mean that rectification cannot be obtained. In Sec. 4 it was shown that each type of solution, (I), and (II), is valid on both sides of zero current; on one side the solution is valid for any current, no matter how large, but on the other side the solution is valid only up to a particular current. At this current, the argument of the Bessel functions becomes zero, and further increase of current causes the argu ment to become pure imaginary, so that the analytic continuation of the solution involves Bessel functions of the other type. The slopes of the current-voltage curve are quite different on either side of this transition. When the transition occurs in the vicinity of zero current, the generally observed rectification behavior is predicted. Consider, first, an insulator which is, for all practical purposes, infinitely thick. What is implied is not so much the actual thickness as the ratio of the thickness to the xo's. In this case, go approaches zero, and the solutions (8.1) and (8.2) are applicable. These solutions have not been treated in detail, since they are continuously ap proached as go becomes zero, and accordingly all equa tions previously derived apply if go is allowed suitably to approach zero. Both represent the same zero-current solution, corresponding to the infinite dielectric case in I and departures therefrom with increasing forward or reverse current are in opposite directions. In the vicinity of zero current, they can be expanded in the usual infinite series for small argument. The voltage-current relations, Eqs. (28), imply slopes on either side of the equilibrium point of Ci-j(C12+2c2); Ci-j(CiL2c8). (31) The rectification ratio is therefore 1+2[(C2+C8)j]/Ci and increases with the current. If, on the other hand, the thickness of the dielectric is not large compared to the xo's the other solutions must be used. In this case, the transition occurs not at zero current but at z=O, or z=O, for which the current has the value, at x= 0, of EbPT2 7r j=-- . 2e2 (D= X02+XOi)2XO (32) Accordingly, in many cases, the transition occurs sufficiently near zero current so that for all practical purposes it is at zero current. The behavior in the vicinity of the transition will be treated in more detail later. The hyperbolic functions (and Bessel functions of imaginary argument) do not ever, at zero or near zero current, represent the same solution as the circular functions (and Bessel functions of real argument). The same zero-current solution must be used for the current in both directions; small departures therefrom in either direction must be made with the asymptotic solutions previously discussed. In one direction the current may increase indefinitely without changing the type of Bessel functions. In the other direction, as the current [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33516 SELBY M. SKINNER TABLE I. Limiting values of the rectification ratio. roD goS R 0 0 1 1.0 1r/2 cosh(1r/2)-1 = 1.5 1r [(cosh1r)-lJ/2= 5.3 21r 00 0 ~(Sinh1r)/1rj-l= 2.7 1r sinh211-/211" - 1 = 42 21r sinh1l" cosh21r/1r J-1 = loa increases, there comes a point at which the transition must be made to Bessel functions of the other type, with, however, the same values of the zero order parameters. If different initial (zero current) conditions are encountered, the values of the parameters change, and a different solution must be followed through analytically. An examina tion of the rectifica tion ra tio, R = Eq. (17) / Eq. (16) shows that if the dielectric is thin, goD may approach zero, and then the maximum possible range of rectification ratios is encountered. The limiting cases are given in Table 1. The largest rectification ratio is obtained when X02 is much less than XOl and D is small. Therefore, the rectification efficiency is greatest for thin layers of dielectric, and considerable difference in energy gaps at the two boundaries. This energy gap may be controlled by the use of proper electrodes, or by the use of one electrode (or a layer upon the electrodes) which exhibits nonmetallic properties. Thus, the use of semiconducting electrodes or the treatments such as "forming" of the copper oxide and other rectifiers can be expected to produce necessary differences in energy gaps at the boundaries. In this connection, reference should be made to the discussion on the nature of the energy gaps in actual contacts which was given in I, Sec. V, 4. It is obviously impossible to obtain zero thickness of useful dielectric, and if the dielectric thickness is reduced below about 10--5 or 10--6 cm, the tunnel effect enters so that complete diffusion equilibrium cannot be attained, and the considerations used here commence not to apply. However, such thicknesses are not desirable, even apart from considerations of chemical behavior and homogeneity, for it is known that the tunnel effect produces the opposite behavior from the diffusion equi librium and therefore rectifying properties would be decreased. A small layer of dielectric between a point contact and a semiconductor would increase the measured rectifica tion ratio; this could help explain some of the anomalies in slope and shape of curve observed in such cases.9 9 H. K. Henisch, Metal Rectifiers (Clarendon Press, Oxford, England, 1941); H. C. Torrey and C. A. Witmer, Crystal Rectifiers (McGraw-Hili Book Company, Inc., New York, 1948); Blake more, DeBarr, and Gunn, Semiconductor Circuit Elements, Reports on Progress in Physics (physical Society, London, 1953), Vol. XVI; M. F. Manning and M. E. Bell, Revs. Modem Phys. 12, 215 (1940); and many others. As the energy gaps at the boundary become more nearly equal, the rectification decreases, so that the initially prepared rectifier would not be expected to show efficient performance. This discussion has dealt with the ratio only; if the slope is large in the easy flow direction, a large voltage is required for a given current, and rectifier heating and other effects may make the dielectric unsuitable. 12. Capacity of the Rectifying Junction The capacity of the dielectric specimen per unit area may be defined as c= I Qel and since Q= fD ndx, eV ",=0 E IF2-Fl/ E IF2+Fl/ c=--- or---- 411" eV ",=0. D 411" eV ",=0. D' (33) which for small current, in the easy flow direction, becomes c= EgO[Si.n(U02~U01)], 411" smU02 SmUOl (34) in which the positive sign is taken if there is a potential maximum. The extension to the first term in j is obvious from Appendix II. 13. Turnover Voltage (Fig. 2) The current enters into the voltage-current equation in an infinite series. Considering only the first two terms, we have eV/kT= Ihdj± Ih2-(1/2)JzlIP, so that a voltage maximum in the current occurs if (h2/h1)-(h1/2)<O, and j=h1/(2h2-h12)I. This ex· plains the turnover voltage usually observed, Fig. 2. To determine the location of the maximum, it is neces- ----~----------~~------4----------~ FIG. 2. Schematic diagram of rectifying behavior showing the turnover voltage, V T. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33RECTIFIER EQUATIONS FOR INSULATING LAYERS 517 sary to insert the expressions for hi and h2, using the correct values for gl and ai, as the particular case re quires. This is a lengthy procedure, and since the succeeding terms in the series will modify the location of the potential maximum, it will not be undertaken here. The qualitative prediction of a voltage turnover, is, however, evident, and may be illustrated for the infinite dielectric, by Eqs. (31). In this case only one turnover voltage is found j this occurs for j= (3Ebk2']'2/27rt!)fJ, where for small Xo, fJ~IFs, and for large xo/D, fJ"'(6xoS)-I. In the former case the turnover voltage is approximately kT / e= 39 volts j in the latter, it is at (D/2xo)(kT/e)~0 volts. 14. Discussion The electric field, for a given current is nonlinear and has a steep gradient in the vicinity of the electrode with the small energy gap, the steepness being independent of the current at small currents. Most of the potential drop measured in the dielectric between zero-current probes applied to the rectifying junction will be in the immediate vicinity of the interface. The relationships pictured by the equations are those due to the behavior of space charges in thermodynamic equilibrium both throughout the dielectric, and with other space charges in the electrodes. The rectification is, of course, due to the piling up of the space charges towards one or the other electrode j the special behavior in the case of initially equal energy gaps is related to the symmetry of the initial electrical conditions. The energy gaps at the boundaries and the applied voltage enable the distinction between forward and reverse current behavior. Since only one carrier has been assumed, the equations are not applicable at large applied voltages when intrinsic charges appear. With the usual insulators, however, the intrinsic energy gap is so large that this is not likely to occur with applied voltage gradients short of internal breakdown. It has not been necessary to assume image forces or any other detailed characteristic of the interface, except the presence at either end of the dielectric of materials of suitable energy gaps. The energy gaps may be due to the contact potentials of the original metals and the conduction level of the original dielectric or may depend upon energetic considerations and chemical reactions between the metal and dielectric at the interface, but may be regarded as unambiguously specifiable. Certain conclusions follow: (1) The intrinsic gap should be large enough so that smoothing effects from the thermal release of intrinsic electrons do not wash out the rectification. (2) The best rectification is obtained when the initial energy gaps at the boundaries are quite different. (3) There is a particular thickness of dielectric which will give the best rectification, for each combina tion of dielectric and electrodes. (4) In general, the best rectification will be obtained with quite thin, but not too thin layers of dielectric. (5) Since the resistance of the dielectric becomes great if its conduction level is much higher than both initial Fermi levels of the metal electrodes, it is desirable that the energy gap be not too large at one boundary. (6) Observed contact charge distributions in 'insulators are the limiting case of the current-carrying behavior which exhibits rectification, and both are due to the non-ohmic thermodynamic be havior of the charges in the insulator. APPENDIX I. EVALUATION OF THE CONSTANTS IN THE NONZERO CURRENT SOLUTION Because of Eqs. (5) it is immaterial whether the evaluation of A and B is accomplished by 1f, F or n, and on account of its greater simplicity, the expression for 1f will be used. Considering first, Eq. (5.1), and defining u=a(x+xo), as in I, there results by Eq. (1) z= ({3u/a)+[r2 a2j w=jz., (Al) and A and B are to be evaluated so that Eq. (5.1) reduces to Eq. (12.1) of I as {3 approaches zero. Disre garding the additive constants for the time being, this requires that where limCPI = sinhu } 13--+0 CPI(U) = wl[AIl(W) + BLl(W)] (A2) An expansion in terms of (3 is not useful, but if CPI is expanded as a Taylor's series in u, there is obtained u" CPI=,E CPI(n)(O)'_j " n! where cpI(O)=~l[AIIW+BLIW], ~=j[rsaS=jr!, CPI' (0) = ~![AL2/SW+ BI 2/sW], cp/'(O) = (3/2)f({3/a)2~fcpI(0), cp/" (0) = ({3/ a)scpI (0)+ (3/2)f({3/ a)2~lcp/ (0), and, in general, if CPI(n-l) = X n-I' CPI+ Y n-I' CPI', then (A3) (A3.1) (A3.2) (A3.3) [ (3W)f dY n-I ] + ({3/a) 2 dw+Xn-I CPI'. From the foregoing expressions, there may be obtained by induction, and by the series expansion of sinhu, the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33518 SELBY M. SKINNER necessary conditions desired. That they are sufficient conditions also, may be shown by asymptotic expansion of the resulting expression for the electrostatic potential, namely Eq. (7). These conditions are and 'Pl(O) =0, 'PI' (0) = 1, lim C/3/a)2t = 1, limC/3/a)3=0. i->O i->O (A4.1) (A4.2) By the definition of t in Eq. (A3.2) (/3/ a)2t= 1, and by the definition of /3, limj-+o[,8/a]=O if arfO. Equations (A4.2) are therefore satisfied. Using the well-known Wronskian relation 2 sinn'll' Ln(w)I n-l(w)-I n(w)Ln+1(w)=--, 'Il'W there are obtained, after solving Eqs. (A4.1) for A and B A=c~ILiW } B=-c~lJiW ' (AS) where the constant is the same in both expressions and may be absorbed in the additive constant in Eq. (7). An analogous procedure and Wronskian yields, in the case of the Bessel functions of real argument, i.e., Eq. (5.1) A=Cl~fJ_lm }. B= -cl~IJim (A6) In the intermediate case, when a2=0, continuity with the cases a2rfO is obtained by the choice B=O. When these values of the constants are used, the expressions for all quantities, electrostatic and electrochemical po tential and potential differences, field and charge density, are continuous through j=O and a2=g2=0. APPENDIX II. ASYMPTOTIC DEVELOPMENT OF THE EXPRESSIONS FOR THE ELECTRICAL QUANTITIES Since the solutions of Eq. (7) may be obtained from those of Eq. (6), by the substitutions g= -ia, and ~= i~, only the former will be developed. With obvious adjustments, these become the expansions about j = 0 of Eqs. (8). The fact that the latter is a convergent series about j=O, suggests that Eqs. (12) and its analog may be a convergent series. This is not taken up here, but it may be noted that the expressions in (B2) below and its analog may be differentiated with respect to x, and the corresponding next lower expres sion is obtained, after multiplication by the appropriate constants which follow from the electrostatic laws. All Bessel functions are defined as in reference 10. Using asymptotic expansions of the Bessel functions Jt etc., of the Stokes type, there results H2=R cos(w-O-S sin(w-~), where (Bl) [ 5·7·13 5·7·11 5·7 ] 'Il'(w~)!P=3i 1+ + ... , 2 (72W)2 2(72~)2 (72w)(72~) 'Il'(w~)iQ=3i[(7 /72w) + (5/72~)+·· .], 'Il'(w~)tR= 3i[ (5/72w)- (5/72~)+· .. ], [ 5·7·11 5·7·11 5·5 ] 'Il'(w~)lS=3! 1 + + .... 2(72w)2 (72~)2 (72w)(72~) Next there are obtained appropriate expressions for w and ~; it is necessary to remember that since the location of the potential maximum may depend upon the cur rent, and boundary conditions will be encountered there, x must also be treated as a function of the current in the general development. Therefore, let g= go+/33g1+/36g2+· . '; X=X+/33 a1+···; Then XOi= oXOi+/33 X1+' . '; UOi= go (oXOi+X) ; V= go4(xl+al); ~o= i,B-3go3. (3~O)-I= pgo-3j where P='Il't?/Ebk2P. w= ~o[l + (1/ ~o) (2go2g1-UO) + ... ]; ~= ~o[1 + (1/ ~o) (2go2g1) + ... ]; w-~= -uo+ (1/6~o)( -4g02g1UO+uo2-4v), and after some manipulation, there is obtained for the quantities in Eq. (12): A = 1 +pgo-{sinuo cosuo -uo-4go2g1]j+· . " B = sinuo+ (pgo-3/2)[ (uo-8go2g1) sinuo + (4go2g1uo-uo2+4v) cosuo]j+ .. " go-IC=cotuo+(pgo-3/2)[(4go2g1-2uo) cotuo (B2) + (uo2-4v-4go2gIUO) csc2uo+ l]j+· . " go-2D= csc2uo[1 + (pgo-3){ (4go2g1-2uo) + (sin2uo+uo2-4go2g1uo-4v) Xcotuo}j+'" ]. At the metal dielectric interfaces both x or D-x, and al are taken as zero. 10 H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, Cambridge, England, 1950). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.7.27.132 On: Sun, 23 Nov 2014 23:08:33

1.1722004.pdf

Moderation of Neutrons in SiO2 and CaCO3 Jay Tittman Citation: J. Appl. Phys. 26, 394 (1955); doi: 10.1063/1.1722004 View online: http://dx.doi.org/10.1063/1.1722004 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v26/i4 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 03 Apr 2013 to 147.226.7.162. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 26. NUMBER 4 APRIL. 1955 Moderation of Neutrons in Si02 and CaC0 3* JAY TITTMAN Research Laboratory, Schlumberger Well Surveying Corporation, Ridgefield, Connecticut (Received June 23, 1954) The spatial distributions of indium resonance neutrons about a "point" Ra-Be source have been measured in pure CaCOa (limestone) and Si02 (sand) out to r",,91'. Both media show nearly Gaussian distributions corresponding to Fermi ages 1'(Si02) = 1906±90 cm2 and l' (CaCOa) =461±23 cm' over most of the range observed. Excess resonance flux near the source, more pronounced in CaCO, than in Si02, appears to be due to inelastic scattering although such interpretation is not unambiguous. Space integrals indicate ",10 percent more absorption in CaCO, than in Si02 during moderation. Data were corrected for 4.5-hr activity, 54-min activation by high energy neutrons, and edge effects; considera.tion was given to the 3.9-and 9-ev In levels in defining the mean detection energy and to the nonmonoenergetic nature of the Ra-Be source in affecting the shape of the indium resonance spatial distribution. I. INTRODUCTION THE moderation of fast neutrons has been studied experimentally primarily in the media of interest in pile design! and shielding,2 and for the purpose of examining source spectra and slow neutron resonance phenomena.a The present paper investigates how well age theory describes the moderation of Ra-Be neutrons to In resonance energy in two media of geologic interest, limestone (CaCOa) and silica (Si02), and measures the effective ages, 1'.4 The Boltzmann transport equation as applied to neutron moderation,6 is usually written to imply (a) spherical symmetry (in the center-of-mass system) of the elastic scattering, (b) a lack of inelastic scattering, and (c) the neglect of chemical binding. Reduction to the Fermi age equation involves the further assumptions: (a) nearly isotropic collision density, (b) small fractional energy loss per collision, (c) slow variation of scattering mean-free-path, X., with neutron energy, and (d) energy independence of ~, the mean logarithmic energy loss per collision. For a point monoenergetic source in an infinite medium the well-known age solution for the isotropic component of the neutron flux per unit energy interval is (1) The second-order approximation in anisotropy and in energy dependence of the collision density6 then corrects * For preliminary reports see J. Tittman and F. F. Johnson, Phys. Rev. 91, 452 (1953); J. Tittman, Phys. Rev. 95, 660 (1954). 1 D. J. Hughes, Pile Neutron Research tAddison-Wesley Press, Cambridge, Massachusetts, 1953); S. Bernstein, Phys .. Rev. 73, 956 (1948). 2 A. M. Munn and B. Pontecorvo, Can. J. Research A25, 157 (1947); P. C. Gugelot and M. G. White, Phys. Rev. 74, 1215 (1948). See also reference 6. a E. Amaldi and E. Fermi, Phys. Rev. 50, 899 (1936); Amaldi, Hafstad, and Tuve, Phys. Rev. 51, 896 (1937); R. D. O'Neal, Phys. Rev. 70, 1 (1946); J. H. Rush, Phys. Rev. 73,271 (1948). 4 Throuidlout this paper experimental 1"S refer to the param eters in tlie best fitting Gaussians..,i calculated 1"5 are evaluated from T-lf ~~~u/~(I- (C0s8)Av)J. 6 R. E. M , Revs. Modem Phys. 19, 185 (1947); Marshak, Brooks, and Hurwitz, Nucleonics, 4, No.5, 19; 4, No.6, 43; 5, No. 1,53; 5, No.2, 59 (1949). Eq. (1) by the factor G= 1 + (4u)-2F(r2/T) (2) where and where u=ln(Eo/E) is the lethargy, and the a's are positive constants depending on the mass of the nuclei in the medium. Age theory should fit CaCOa and Si02 quite well be cause of their relatively large average nuclear masses. Furthermore, chemical binding effects are negligible for In resonance detectors. However, strict comparison, even with second-order theory, is hampered by the broad energy spectrum of Ra-Be neutrons and by the perturbing effects of inelastic scattering. n. EXPERIMENTAL APPARATUS AND TECHNIQUES A 1.03 g Ra-Be source was used throughout these experiments. Earlier measurements using the same type of source in other media6 indicate that a single age, corresponding to an average source energy of roughly 5 Mev, dominates the moderation process. From the work of Roberts, Hill, and McCammon7 it is to be expected that the finite extension of the source (I inch diameter by 1 inch high) does not perturb the In resonance neutron distributions in the present measure ments. One-inch diameter In foils, 99.97 percent pure and 104 mg/ cm2 thick, were exposed in Cd cassettes of 20-mil thiCkness. These were attached to the ends of long aluminum "swords" which were placed in fs-inch wall aluminum "sheaths" in the silica or in slots milled in the limestone so that the foil centers were in the horizontal plane of the source. Experiments established that no significant 54-min activity was induced by the Ra ')'-rays. Subsequent to subtraction of background, S Dacey, Paine, and Goodman, Technical Report No. 23 (Laboratory for Nuclear Science and Engineering, Massachusetts Institute of Technology, 1949); C. W. Tittle, Ph.D. thesis, Massachusetts Institute of Technology, 1948; B. T. Feld, MDDC- 1437; E. Fermi, Nuclear Physics (University of Chicago Press, Chicago, 1950), revised edition, p. 191. 7 Roberts, Hill and McCammon, Phys. Rev. 80, 6 (1950). 394 Downloaded 03 Apr 2013 to 147.226.7.162. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsMOD ERA T ION 0 F N E U T RON SIN S I LIe A AND LIM EST 0 N E 395 TABLE I. Properties of the moderating media. Density Composition (g/cm l) Size Purity Principal impurities CaCO. 2.72±.01 5-ft cube >98% Al-o.l % Mg-o.6% Fe-o.05% Organic-o.06~ Si-o.I% H,O-<O.OO8 o SiO. 1.61±.O2 8-ft cube 99.89% Fe-o.OX% 1120-<0.007% which was < 10 percent of the total counting rate for all foil positions except those farthest from the source, the measured activities were corrected to zero time and front-and-back counting rates were averaged to give AM, one-half the 54-min saturation activity. Since the two media were studied one year apart, a 2.8 percent adjustment was made to one of the sets of data to compensate for decay of a Sr90 source used as a reference. The limestone moderator consisted of ground Vermont marble blocks (see Table I), most of which were eon-foot cubes, the remainder being 1 ft by 1 ft by 3 in. A l-inch hole was drilled vertically through the central column for insertion of the source. During all exposures, this hole was filled with a thin-walled alu minum tube packed with limestone powder to reduce the effect of this inhomogeneity in the medium. Density and spectrographic analyses showed a high degree of homogeneity as well as the absence of any impurities which might affect the slowing-down properties. The silica moderator consisted of an 8-foot cubical pit filled with silica sand (see Table I). Thin-walled aluminum tubes were shaped to provide t\-inch thick sheaths running vertically through the medium. In agreement with Kolbuszewski,8 there exists continuous variation in density of the poured sand from bottom to top, ranging from 1.63 to 1.59 g/cm3, with a median plane value of 1.61 g/cm3• Extensive examination of the sand indicated negligible adsorbed moisture. All exposures were made with a thin-walled tube of sand resting on the source to mjnimize inhomogeneity of the medium. III. CORRECTIONS The counting rate data were normalized to the srOO reference source and experimentally corrected for the 4.S-hr activity induced by 'Y-rays9 and high-energy neutrons.lO In both media this correction was found to be negligible for r2> r/2. The 54-min activation cross section extends into the Mev regionll so that activation by high-energy neutrons can affect "resonance" distributions, particularly in the vicinity of the source. Roberts, et al.7 have corrected for this effect, using data taken with Cd and with Cd+In 8 J. Kolbuszewski, Research (London) 3, 478 (1950). 9 Collins, Waldman, Stubblefield, and Goldhaber, Phys. Rev. 55,507, 1129 (1939). 10 S. G. Cohen, Nature 161, 475 (1948); A. A. Ebel and C. Good man, Phys. Rev. 93, 197 (1954); Martin, Diven and Taschek, Phys. Rev. 93, 199 (1954). 11 Neutron Cross Sections, AECU-2040, (Office of Technical Services, Department of Commerce, Washington, D. C., 1952); Hughes, Garth, and Levin, Phys. Rev. 91, 1423 (1953). Frn .9 /; ~. Sin. ~ / .8 flcm)-I- 10 20 60 FIG. 1. The experimental correction factor, FIn, for high energy neutron 54-min activa:tion. See Eq. (4). absorbers surrounding the detector, by means of the relation (4) where r is the position of the foil, r' is some distance from the source sufficiently large to insure negligible contribution to the 54-min activity from high energy neutrons, and u is the lethargy corresponding to In resonance. Equation (4) assumes that the only effective resonance is at 1.46 ev. It is now known that the 3.9 ev and 9.2 ev resonances are also due to Inur, and lead to this activity.12 However, estimates of the effect of these other levels on the present measurements indicate that it is quite small. (See Appendix.) We have thus ignored the presence of these levels in making the correction. Figure 1 shows the experimental values of F In as a function of position for the two media. In neither moderator does the value of the dominant 1" depend critically upon the points thus corrected. However, appreciably better fit of Eq. (1) to the data at small distances is brought about by taking this effect into account. The first-order correction to age theory, expressed by Eq. (2), was calculated for a range of mass values and is shown in Fig. 2. We assumed Eo= 5 Mev and E= 1.5 ev in evaluating u; appropriate averages over nuclei and energy were taken in determining the effective masses M(CaCO a) = 18.9 and M(Si0 2) = 18.7. The CaCOa pile was assembled on the floor of a large room, far from all walls. Since the source and detectors were in the median plane, the concrete floor had no effect on the data. However, the two foils farthest from the source on either side were sufficiently close to the surface to be affected by the loss of neutrons into the room. The two circled experimental points of Fig. 3 were thus moved upward by 4.6 and 2.7 percent for the smaller and larger r, respectively. The corrections were a V. L. Sailor, Phys. Rev. 87, 222 (1952); v. L. Sailor and L. B. Borst, Phys. Rev. 87, 161 (1952). Downloaded 03 Apr 2013 to 147.226.7.162. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions396 JAY TITTMAN +4.0 -r--- 1\ ;r- l\\ //~ ~ I r.=_ T +3.0 +5 +4 +2,0 +, +Z +10 0 ~ 4 S S 10 IZ ~~ if) 0/ -, \~ M;40 ---:;;;:: 1'j 4 ------/ - 0 I~ M~ c- O M120 ~ r- :~ - 0 -4 U 10 -, -Z -2. -, FIG. 2. First-order correction to age theory for several nuclear masses. Left-hand scale is the function of Eq. (3) which depends only on fliT for given M; right-hand scale is percent correction for present measurements, using tt= 15.1. calculated from the ratio of the age theory predictions of the slow neutron fluxes in infinite and finite media.Is The silica moderator was in a cubical pit, surrounded on five sides by cement walls backed by earth. The source and detectors were located in the median plane, about 4 feet from the cement bottom and an equal distance from the open surface. Experiments revealed no measurable edge effects in this medium. IV. RESULTS Indium resonance neutron distributions, extending to r2R:9T, are shown in Fig. 3. In each medium a single T is adequate to describe the distribution, within the experimental error, except in the immediate vicinity of the source. The excess of resonance neutrons at small r in CaCOa can be ascribed to a component (T s;too cmZ) of ::;; 1 percent strength in the source spectrum.14 In Si02, the fit at r2<300 cmz can be improved somewhat by assuming a low energy component of T::;; 100 cm2 and strength ::;;0.1 percent. The presence of a low energy component in the source spectrum is consistent with the results of other investigators.s.15 However, their re ported strengths or upper limits on strengths are con siderably larger. Some of the earlier work did not in clude the correction ofEq. (4) which may account for at least part of this difference. In the present measure ments, however, this component must be considered purely empirical since it is not known whether the smaller T'S in the two media correspond to the same source energy. Furthermore, the fact that the low energy source strengths required in CaCO g and in SiOz differ by an order of magnitude indicates that the excess resonance flux at small r depends upon the slowing down medium to an appreciable extent. The experimental result that Ra-Be neutrons produce )3 P. R. Wallace and J. Lecaine, Elementary Approximations in the Theory of Neutron Diffusion, MT-12 (National Research Council of Canada, Chalk River), p. 63. 14 The la.rger value of the source strength given in a preliminary report was due to a numerical error. 15 Ya]ow, Yalow, and Goldhaber, Phys. Rev. 69,253 (1946); D. L. Hill, AECD-1945. Gaussian spatial distributions in these media despite their broad energy spectrumHdS is to be expected theoretically. From Eq. (1) the spatial distribution of neutrons of energy E about a nonmonoenergetic source is proportional to fdS(T) exp( -r2/41') ._- dl' dl' T! ' where we see that the source spectrum-in-T is required rather than the specturm-in-E o. Now the fractional width of the Ra-Be spectrum-in-T for most moderators will be smaller than that for the spectrum-in-Eo by roughly the factor4 A.z![3Hl- (COsO)AV) TO] where 1'0 is evaluated at the peak of the Eo spectrum. Qualitatively one can see, for example, that all neutrons in a source spectrum which has a lower cut-off at, say, 1 Mev g g :;l . "'0 0:: 9 , E 8 " 7 .o. .£ 5 _. f--- I-- o ,0' 9 & ~ 7 ~ . 'c 5 ::::> , :::::210 "0 i <t 7 (j) • g 5 g 4 o If) l (j) 0:: E '" '6 c • J I CoCO, (2.72 g/cm') "-, -!..-- .$l.. "-.. ~ ~ :r-.. ~ ~ -t------."" -_ .. , ..... "-•. . --~ i-r-=- lOOO 2000 3000 4000 5000 r'{cm') !-SiO, (1.61 g/cm') 1 I '" l"- I I :'" ~III I I ' '" , ; , '" i l"- i ! I i I I I i I i I , -10Q j 2: -3 4 5 6 7 e 9 10 II Ii 13 14 15 16 17:11 10 r'(cm') FIG. 3. Indium resonance neutron distributions in SiOs and CaCO •. Vertical bars represent standard deviations of several independent measurements; horiwntal bars are estimated un certainties in fl. Solid curves are age theory Gaussians corrected by Eq. (2) and fit to last six experimental points in CaCO. and last five in Si02• Circled points were corrected for edge effects. 16 P. Demers, MP-204, Nat!. Research Council of Canada Ann. Rept. Downloaded 03 Apr 2013 to 147.226.7.162. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsMOD ERA T ION 0 F N E U T RON SIN S I LIe A AND LIM EST 0 N E 397 suffer energy degradation in common over roughly six decades in E. Hence, if the upper Eo cut-off of the spec trum is at !"V10 Mev, the fractional width of the spec trum-in-T is of the order 1/7 while that of the spectrum in-Eo is of the order unity. A detailed calculation of the expected spatial distribution of In resonance neutrons in CaCOa about a Ra-Be source yields a Gaussian (to within better than one percent from 1'2=0 to 1'2= 9To) having a T slightly smaller than TO' An average of the data of Hilll5 and Demers16 was taken for the source spectrum-in-E o. It can be shown rather gen erally17 that for a source spectrum-in-T describable by a peak at TO and a width w, a Gaussian spatial distri bution results for (1'2/ To)~<256( TO/W)2, a condition which is satisfied easily by the present measurements. Table II lists the measured dominant T'S for the two media, as well as the calculatedl8 values for an assumed Eo= 5 Mev. Wherever a gap in the cross-section data exists, a smooth connection to data on either side of the gap was made. Further, in the cases of Si and Ca, averages over resonances were used to smooth out Aa (E); in all cases it was assumed that Atotal = A8• Be cause of these approximations, and the likelihood that the appropriate Eo;;zf5 Mev, it is questionable whether the Si02 difference is significant. In view of the larger discrepancy in the case of CaCOa, however, it is possible that at least part of the difference is due to inelastic scattering which can reduce the effective age. A further possible manifestation of inelastic scattering is the intro duction of an apparent component in the source having a smaller T than the dominant component. This could account qualitatively for the excess resonance flux at small l' in CaCOa and also explain the apparent de pendence of the strength of the low-energy component in the source upon the moderator. The difficulty in making a unique assignment of causes for these effects arises from the lack of exact knowledge of the Ra-Be source spectrum and inelastic scattering data. The coefficient (4?rT)-! in Eq. (1) arises from normal ization, since the absence of absorption in the medium requires the space integral of the slowing-down density to equal the source strength. Since the same source and foils were used in both media, we should expect B(CaCO a) (Aa/~)cacoa B(Si02) (A./~)SI02' (5) where B= fo"'r2A54dr and ),. and ~ are evaluated at the detection energy. The B's were determined by ex tending the experimental data to infinity using Eq. (1) and the experimental T'S of Table II. It should be noted that the data at small r, which are subject to the largest 17 J. Tittman (unpublished). 18 Cross sections were taken from E. Melkonian, Phys. Rev. 76, 1750 (1949); Rainwater, Havens, Dunning, and Wu, Phys. Rev. 73, 733 (1948); Havens, Rainwater, Wu, and Dunning, Phys. Rev. 73, 963 (1948); AECU-2040. (See reference 11.) TABLE II. Measured and calculated slowing-down parameters. Parameter Measured Calculated" r~Si02) (em!) 1906::1::90 2130 r CaCOa) (cm2) 461::1::23b 560 B(CaC03)c 0.36 0.43 B(Si0 2) • Assumes E. =5 Mev, Ed •• = 1.6 ev (see appendix). b The slight discrepancy between this value and that of an earlier report is caused by the introduction of the correction factor G(u,r'lr) of Eq. (2) '" c B = h r2Ali4dr. corrections, contribute only weakly to the total area because of the r factor in the integrand. A comparison of the experimental left-hand side of Eq. (7) with the computed18 right-hand side yields a measure of the relative integrated absorption in the two media. The values thus obtained are listed in Table II, where we note that CaCOa shows a 14 percent reduction in In resonance flux relative to that of Si02• A calcula tion of exp[ -fou(J a/ «(J .~)du Ji9 for both media, using an assumed l/v absorption normalized to the known thermal values, accounts for only about 2 percent of the measured effect. The remainder is presumably due to excess resonance absorption in CaCOa during the slowing down process. Although the appropriate ),'s are not known with high accuracy,18 it seems unlikely that the remaining discrepancy is due to errors in these quantities. ACKNOWLEDGMENTS The author wishes to thank the Schlumberger Well Surveying Corporation for permission to publish these results. The measurements and calculations were carried out with the assistance of F. F. Johnson and A. H. Heim. Cooperation of the Vermont Marble Company is gratefully acknowledged. APPENDIX Indium Resonance Detection Energy Since the recent work of Sailor12 has shown that several In resonances contribute to the 54-min activity, it is of interest to examine how this affects the average energy at which neutrons are detected in these measure ments For this purpose, we took the average detection energy to be flfleVEf(r,E)Tcd[l-Tln(E)]dE o (E)Av=----------f1fleV f(1'2,E)Tcd[1-TIn(E)]dE o where f is the age theory flux, TCd is the transmission through the Cd cassette, and TIn is the transmission of 19 See reference 6, E. Fermi, p. 184. Downloaded 03 Apr 2013 to 147.226.7.162. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions398 JAY TITTMAN the In detector foil. In the calculation of TIn, twice the measured foil thickness was used to account for the isotropic nature of the flux in which the foils were placed; also the potential scattering cross section was subtracted from the total cross section20 for In so that [1-T In (E) ] is a fairly good representation of the activability of the foils. Since the quality of the slow neutron flux changes with " it is apparent that (E)AV is , dependent. Thus (E)AV was evaluated in each medium at r=O and at the , corresponding to the farthest foil. The results are the same for both media to within the precision of the cal- ~ A detailed experimental curve of q(E) for In was kindly provided by V. L. Sailor. culation: ,2=0, (E)Av=1.80 ev and ,2""9T, (E)Av=1.73 ev. The larger value at ,=0 is to be expected since the flux in the vicinity of the source is richer in the higher energy neutrons. Actually, the values given above repre sent rough upper limits on (E)AV since no account has been taken of the relative detectability of (3-rays originating at different depths in the foil. This latter effect will tend to weight more strongly the importance of the 1.46 ev level. We have thus arbitrarily assumed that the present measurements refer to an energy of approximately 1.6 ev rather than 1.46 ev, corresponding to a reduction in T by slightly less than one percent in both media. JOURNAL OF APPLIED PHYSICS VOLUME 26, NUMBER 4 APRIL. 1955 Re-Examination of the N onsteady Theory of Jet Formation by Lined Cavity Charges*t R. J. EICHELBERGER Department of Physics, Carnegie Institute of Technology, Pittsburgh, Pennsylvania (Received May 26, 1954) A new type of analysis is applied to observations on jets from lined cavity charges in order to test the nonsteady-state hydrodynamic theory of a jet formation. The results support previous findings that the theory is qualitatively accurate, but give evidence of departures from the ideal situation assumed in the theory. An attempt is made to assess one of the more likely sources of error, the neglect of acceleration of the liner. SUMMARY OF THE THEORY AND PREVIOUS TESTS THE nonsteady theory of jet formationl is based on the same steady-state hydrodyanmical concepts of streamline flow used by Taylor and Birkhoff in the original steady-state theory.2 The conservation laws lead to the same equations for jet velocity, jet mass, and other jet characteristics, for a given set of condi tions at the stagnation point where the jet is being formed. The important equations are Va coSa sinh=--- (Taylor's relation); (1) 2UD Vj= Vo csc(3/2 cos (a+o-(3/2) ; (2) dm./dm=cos2(3/2; (3) dmi/dm= sin2(3/2. (4) 8 is the angle between the direction an element of the liner travels after being struck by the detonation wave and the normal to the liner surface; Va is the velocity at which the liner element travels toward the axis; a is * Submitted in partial fulfillment of the requirements for the de~ee of doctor of philosophy at Carnegie Institute of Technology. r This work was performed under research cOI!tract with. ~e Office of the Chief of Ordnance, U. S. Army, and WIth the Ballistic Research Laboratories, Aberdeen Proving Ground. 1 Pugh, Eichelberger, and Rostoker, J. Appl. Phys. 23, 532 (1952). 2 Birkhoff, MacDougall, Pugh, and Taylor, J. Appl. Phys. 19, 563 (1948). the half-angle of the conical liner; U D, the detonation rate of the explosive; V;, the velocity of the jet element formed; {3, the angle between the collapsing liner wall and the axis; m., the mass of the slug; mj, the mass of the jet; and m, the mass of the liner. The masses m., mil and m are each functions of x. The mass m is that part of the mass of the liner that is included between the top (apex) (x= 0) of the cone and the plane per pendicular to cone axis at x= x. The masses m. and mj are the parts of m that end up in the slug and the jet respectively. Except for the differential form of Eqs. (3) and (4), these are identical with the results of the steady-state theory, but all parameters except a and U D vary with x, the original position of the liner element. The essential difference between the original and the nonsteady theories lies in the fact that varia tions of Va that are taken into account in the latter make the collapsing liner assume a far more complex form, and (3, instead of being a constant given by (3= 2h+a=a+ 2 sin-l (Va cosal2U D), (5) is variable and is given as a function of x by the far more complex formula sin(28+a)-x sina(l-tanh tan[a+8J) Vo'iVo tan(3 cos(26+a)+xsina(tan[a+hJ+taoo)Vo'IVo' (6) where V 0' = dVo/ dx. Downloaded 03 Apr 2013 to 147.226.7.162. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.3062648.pdf

Astrophysics I S. Flügge , C. C. Kiess , Citation: Physics Today 11, 7, 26 (1958); doi: 10.1063/1.3062648 View online: http://dx.doi.org/10.1063/1.3062648 View Table of Contents: http://physicstoday.scitation.org/toc/pto/11/7 Published by the American Institute of Physics26 Books Principles of Quantum Theory I. Vol. 5, Part 1 of Handbuch der Physik. Edited by S. Fliigge. 376 pp. Springer-Verlag, Berlin, Germany, 1958. DM 90.00 (subscription price DM 72.00). Reviewed by Freeman J. Dyson, Institute for Advanced Study. Volume 5 of the new Handbuch der Physik comes in two parts, one real and one imaginary (nomenclature due to Weisskopf). The real part, which is now before us, consists of Pauli's 1933 article on the general prin- ciples of quantum mechanics and a new article by Kallen on quantum electrodynamics. The imaginary part will consist of an article on field theory by Schwinger. Pauli's article has been reprinted without any im- portant changes except that the last 28 pages have been amputated. Those pages dealt in a preliminary way with the quantization of the Maxwell field and are now superseded by Kallen's fuller exposition. It is not nec- essary in a review to say much about Pauli's article. It remains as it was long ago described by Oppenheimer "the only adult introduction to quantum mechanics". It concentrates exclusively on the basic theoretical and conceptual framework, does not discuss examples or applications, and contains little that has become irrele- vant or incorrect during the 25 years since it was written. The famous error in the 1933 article, in which Pauli derived the 2-component neutrino equation only to dismiss it with the remark "These wave-equations are not invariant under reflection and are therefore not applicable to physical reality," has been rectified by the omission of the last eight words. Kallen's article is written in a quite different style from Pauli's. It is much closer to standard textbook form, pedestrian in its full development of computa- tional details, and suitable for students who are learning the subject for the first time. The subject matter is quantum electrodynamics in the narrow sense, that is to say, the study of radiation phenomena involving elec- trons, positrons, and photons alone. There is heavy emphasis on the applications of the formalism to spe- cific problems by means of second-order perturbation theory. In particular, the shift in the fine structure of the hydrogen spectrum, which was discovered by Lamb in 1947 and which stimulated the modern development of quantum electrodynamics, is calculated in full detail. Kallen begins with the quantization of the Maxwell and Dirac fields without interaction, continues with the formal theory of the interacting fields, covariant per- turbation theory, the calculation of radiative corrections to specific processes, the procedures of mass and chargerenormalization, and ends with a general discussion of the renormalization program. Most of this material is already to be found in various textbooks. Only the last section, containing Kallen's proof that at least one of the renormalization constants must be infinite inde- pendently of any perturbation-theory approximations, appears for the first time between hard covers. It is clear from the content of Kallen's article that it must have been written and essentially completed in 1954 or 1955. Its publication was delayed by the editors of the Handbuch in a vain attempt to avoid the division of Volume 5 into two parts. The delay is not Kallen's fault, but it is none the less regrettable. It has the effect of making the article look old-fash- ioned. Kallen has been careful to revise his numerical results and to put in references to electrodynamic cal- culations published as recently as 1957. But the point of view of the article is still definitely 1954. Of the "New Look" which came into field theory after 1954, with various attempts to abandon altogether perturba- tion theory and the renormalization program, the article reflects no trace. While the recent attempts to rebuild the theory upon a more secure mathematical foundation have not been notably successful, still it is a disappoint- ment not to find them subjected to Kallen's critical scrutiny. Astrophysics I: Stellar Surfaces—Binaries. Vol. 50 of Handbuch der Physik. Edited by S. Fliigge. 458 pp. Springer-Verlag, Berlin, Germany, 1958. DM 98.00 (subscription price DM 78.40). Reviewed by C. C. Kiess, National Bureau of Standards. This fiftieth volume of the Encyclopedia of Physics is the first of a series on astrophysics. The term "ency- clopedia" gives promise of articles containing a com- prehensive survey of the knowledge relating to the topics discussed, with ample references to the pertinent literature. This expectation is realized in all the articles except one. As indicated in the title, one set of articles deals with the surfaces of the stars, the other with binary stars. The light from stars, revelatory of the physical state of their surfaces, is investigated best with photometers and spectroscopes. Early fruits of this work revealed that many stars can be assigned to a sequence of spec- tral classes in which temperature is the principal physi- cal parameter. Later, when spectroscopic and photomet- ric details became more abundant a second parameter, luminosity, was needed to mark subdivisions of the classes in a 2-dimensional scheme. More recently still, the distribution of energy in the continuous spectra of stars has led to a 3-dimensional classification. All these subjects are treated fully in the article in French, "The Spectral Classifications of Normal Stars", by Charles Fehrenbach, director of the Marseilles Observatory. Not all stars, however, fit neatly into the normal sys- tems of classification. The article "Stars with Peculiar Spectra" by Philip C. Keenan, astronomer at the Per- kins Observatory, discusses groups of stars, usually PHYSICS TODAY"Should be read by everyone, scientist and non-scientist alike" R. T. Birge, Professor Emeritus, U. of California, Former President, AMERI- CAN PHYSICAL SOCIETY. K394. FADS AND FALLACIES IN THE NAME OF SCIENCE by Martin Gardner. Atlantis . . . Wllhelm Reich & orgone energy . . . L. Ron Hubbard & dianetics . . . Korzybskl and General Semantics . . . Bri- dey Murphy . . . psionics machines . . . Velikovsky . . . flying saucers . . . 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Please send me the following books as indi- cated below: K19 K113 K267 K37O K428 K876K27 K174 K295 K372 K431 K877 im enclosing $K81 K190 K361 K377 K442 K878 inK107 K193 K366 K387-S K443 full Da^K109 K256 K367 ( K394 K875 'ment. Pi ment in full must accompany all orders except those from libraries or public institutions, who may be billed. Please add 10c per book for postage and handling charges on orders less than $5.00. On orders of $5 or more Dover pays postage. PLEASE PRINT Name Address City Zone State GUARANTEE: Satisfaction guaranteed. All Dover books and records are returnable within 10 days for full, immediate refund.small in number, whose spectra are related to the nor- mal classes but exhibit some peculiarities. Although the basis for labelling a star "Pec" has shifted over the years, yet the designation carries with it recognition of abnormalities in the physical and chemical properties of the stars' atmospheres. "The Spectra of the White Dwarfs" is the title of the article by Jesse L. Greenstein, professor of astro- physics, California Institute of Technology. White dwarfs are stars of low luminosity, of small volume and mass, but of exceedingly high density, which tells us that the material composing them is in a very unusual state. Although more than 200 of these objects are known, yet owing to their faintness, spectroscopic data are available for fewer than half of them. These have proved sufficient, however, to set up several classes with distinctive spectral features, and differing in tempera- ture and color. Tenuous clouds of gas surrounding very hot stars appear under telescopic scrutiny as disk-shaped or ring- shaped objects. This has led to their designation as planetary nebulae. Two articles in German, "The Spectra of the Planetary Nebulae" and "Theory of the Planetary Nebulae", by Karl Wurm, professor and observer in the Hamburg-Bergedorf Observatory, are devoted to them. Logically these thorough articles be- long together and should be read together, but, for some reason not apparent, the editor has separated them by more than 200 pages. The spectral features of stars with surface tempera- tures above 7000° are those of gases in the atomic state. At lower temperatures some atoms combine to form diatomic molecules, hydrides and oxides, and their spectra will appear together with the atomic spectra. In the atmospheres of the coolest stars these molecular features become so prominent that their strengths and chemical nature are criteria for the classification of the cool, red stars. These and related topics are fully dis- cussed by P. Swings, professor in the University of Liege, in his article, written in French, "Molecular Bands in the Stellar Spectra". The spectral features described in the foregoing articles are the consequences of the interaction between the energy generated in the stellar interiors and the matter in the outer gaseous layers. How does this interaction take place? The attempt to answer this question has led to a vast amount of theoretical work which is thoroughly presented in the article "General Theory of Stellar Atmospheres", written in French by Daniel Barbier, astronomer at the Paris Observatory. The second group of articles, written in English, deals with the mechanical and physical properties of binary stars—pairs of stars that revolve around a com- mon center of gravity. Some of these pairs are sepa- rated widely enough to be seen telescopically as double stars. The problems concerning them are treated in the article "Visual Binaries" by Peter van de Kamp, pro- fessor of astronomy, Swarthmore College. The subject of "Spectroscopic Binaries" and their problems are discussed by Otto Struve, professor of astrophysics, andhis colleague Su-Shu Huang, research fellow, in the Department of Astronomy, University of California. Both articles are comprehensive in content and provide ample references to pertinent literature for further study. The article remaining for consideration is "The Eclipsing Binaries" by Sergei Gaposchkin, astronomer at the Harvard College Observatory. It is offered by him "in condensed form (as) a summary of the multi- farious known data on eclipsing binaries". As such it lacks the comprehensiveness that should be the main attribute of an encyclopedic article. Although the authorities for the tabulated data are named, there are only seven specific citations to the extensive literature covering this subject, none of them, however, referring to the epoch-making work, at the Princeton Observa- tory, that initiated this branch of astrophysics more than four decades ago. In composition, illustrations, and presentation of tabular matter, this book maintains the high standard for which the Springer-Verlag is noted. The trilingual indexes are detailed and run to nearly 40 pages. On the whole the standard set by the articles in this volume is high; and one hopes that it will be maintained in the succeeding volumes of the series. Nuclear Reactions III. Vol. 42 of Handbuch der Physik. Edited by S. Fliigge. 626 pp. Springer-Verlag, Berlin, Germany, 1957. DM 135.00 (subscription price DM108.00). Reviewed by E. R. Rae, A.E.R.E., Ear- well, England. The latest volume of this well-known reference series contains six monographs on important topics in nuclear physics in a text of 610 pages, together with an English- German and German-English subject index. The articles included in the present volume are as follows: (1) Nuclear Isomerism, by D. E. Alburger, 108 pages. The author starts with an account of the historical de- velopment of the idea of nuclear isomerism and pro- ceeds to discuss the methods available for obtaining information about the isomeric states from their life- times and from the radiations emitted in the decay of those states. The account is followed by fifty pages of level diagrams and tables describing the present state of knowledge of isomeric states. The review is com- pleted by a brief discussion of the systematics of the lifetimes of the isomeric states. (2) Alpha Radioactivity, by I. Perlman and J. O. Rasmussen, 96 pages. This article considers first the systematics of alpha-decay energies, the type of spectra observed from even-even and from odd nuclei and the interpretation in terms of nuclear models. This is fol- lowed by a consideration of the theory of the lifetimes of alpha-emitters for even-even and odd nucleon types and the article is rounded off with some illustrative examples of individual decay schemes. (3) The Transuranium Elements, by E. K. Hyde and G. T. Seaborg, 104 pages. The place of the heaviest PHYSICS TODAY

1.1734999.pdf

TwoCarrier SpaceChargeLimited Current in a TrapFree Insulator R. H. Parmenter and W. Ruppel Citation: Journal of Applied Physics 30, 1548 (1959); doi: 10.1063/1.1734999 View online: http://dx.doi.org/10.1063/1.1734999 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Space-charge-limited current involving carrier injection into impurity bands of high-k insulators Appl. Phys. Lett. 86, 203506 (2005); 10.1063/1.1935045 Onecarrier spacechargelimited currents in solids with two sets of traps distributed in energy: Two particular cases J. Appl. Phys. 49, 3353 (1978); 10.1063/1.325291 Spacechargelimited currents in insulators with two sets of traps distributed in energy: Theory and experiment J. Appl. Phys. 48, 3415 (1977); 10.1063/1.324185 Diffusion effects in onecarrier spacechargelimited currents with trapping J. Appl. Phys. 45, 2787 (1974); 10.1063/1.1663671 Theory of OneCarrier, SpaceChargeLimited Currents Including Diffusion and Trapping J. Appl. Phys. 35, 2971 (1964); 10.1063/1.1713140 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:001548 K. E. MORTE~SON fabricated diode performance and calculated values were obtained. In addition, maximum capacitor Q's in excess of 200 at 1 kmc are achievable when alloying on the (111) planes. IV. CONCLUSIONS The fabrication of high Q, large swing, diode capac itors using a double alloying technique has been demonstrated. Such diodes would be useful in par ametric amplifiers with "pump" frequencies in excess of X band. Finally, it is anticipated that capacitor Q's in excess of 500 at 1 kmc will be achieved with closer control of the alloying process. V. ACKNOWLEDGMENTS The author acknowledges the valuable assistance and suggestions of E. P. Teter in the actual fabrication of the diodes and is further grateful to E. 1\1. Pell and R. N. Hall for their helpful discussions on diode fabrication. The author also expresses his thanks to W. W. Tyler for the germanium crystals he provided. JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 10 OCTOBER, 1959 Two-Carrier Space-Charge-Limited Current in a Trap-Free Insulator R. H. PARMENTER* AND W. RUPPEL Laboratories RCA, Ltd., Zurich, Switzerland (Received January 20, 1959) The space-charge-limited electrical behavior of a trap-free insulator containing both mobile electrons and holes is investi gated under the approximation of neglecting diffusion current. When both contacts are ohmic, the dependence of two-carrier space-charge-limited current on voltage and interelectrode spacing remains the same as in the one-carrier case; i.e., the current varies as the square of the voltage and as the inverse cube of the spacing. For given voltage and spacing, however, the two carrier current is usually orders of magnitude larger than the one-carrier current. This is in striking contrast to the case of space-charge-limited flow of electrons and positive ions in a vacuum. The two-carrier current in an insulator can be reduced to the size of the one-carrier current either by making one contact highly blocking or by reducing the mobility of one of the carriers I. INTRODUCTION THE charge carriers by which a current is carried in an insulator are either generated in the volume of the insulator or injected from the contacts. While the volume generation of carriers-thermally or by incident radiation-leaves the volume of the crystal electrically neutral, the injection of excess carriers from the elec trodes or the extraction of carriers from the volume gives rise to a net space charge in the insulator. The importance of excess majority carrier injection for the conduction mechanism of an insulator has first been shown by Smith and Rose.l In general, when a voltage is applied to an insulator across ohmic contacts, appreciable number of excess carriers will be injected as soon as the transit time through the crystal reaches the order of the field-free dielectric relaxation time (i.e., the relaxation time associated with the thermal equilibrium carrier concentration). For example, con sider a trap-free n-type insulator having a dielectric * Now at RCA Laboratories, Princeton, New Jersey. 1 R. W. Smith and A. Rose, Phys. Rev. 97, 1531 (1955); A. Rose, ibid 97, 1538 (1955). to zero. While the first approach yields in detail the well-known one-carrier case, the second approach differs from the one-carrier case in that no net excess space charge can be injected into the insulator. The intermediate case of a slightly blocking contact shows that, while the current may be carried mainly by one sign of carrier, the net space charge of the insulator can have the opposite sign. When both contacts are blocking, the current saturates for sufficiently high voltages. The current is smaller than the saturation value by an amount proportional to the cube of the interelectrode spacing and the inverse square of the voltage. Near such saturation, the space charge is negligible and the electric field is uniform across the insulator. It is plausible that many of the above results will also be true for an insulator with traps. relaxation time 'Trel = EEO(nOeJ.Le)-l and a transit time (Here E is the dielectric constant, no the thermal equilibrium conduction electron density, J.Le the electron mobility, d the electrode spacing, and V the applied voltage.) These two times become equal at a voltage Vi= noed2( EEo)-l. Alternatively we could equally well define Vi as the voltage where the space-charge-limited (SCL) current becomes comparable with the ohmic current, or, what is equivalent, where the injected carrier density n becomes comparable with the thermal-equilibrium carrier density no. To see this, we note that, approximately n= EEO V (ed2)-l (this being obtained by setting the injected charge equal to the voltage times the geometrical capacitance). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:00SPA C E -C H A R G E -LIM I TED CUR R E N TIN A N INS U L A TOR 1549 Setting the foregoing expression equal to no, we see that we obtain the previous expression for Vi. If the insulator contains traps, then a certain fraction "of the injected electrons go into the conduction band; the remainder going into the traps. (The ratio" depends upon the position of the steady-state Fermi leveL), The SCL current density will be reduced by this factor '" so that the threshold voltage Vi will be increased by a factor ,,-1, ViK= noed2("HO)-I. To discuss this case by means of transit-time arguments' we see that we must redefine an effective transit time TTK=d2("V /-L.)-1 increased by a factor ,,-lover the trap-free case. If the cathode is ohmic for electron flow and the anode is ohmic for hole flow, then carriers of both signs can be injected into the volume of the insulator. The resulting current will be ohmic or SCL depending on the relative magnitudes of the dielectric relaxation time and the transit times for electrons and holes. There are three cases to consider: (1) Trel < Tre and Trh; (2) Tre<Trel<TrhOr Tre> Trel> Trh; (3) Trel> Tre and Trh• The first case is that of a semiconductor into which only minority carriers can be injected. No net space charge will be formed since the injected minority carrier charge is compensated after time T reI by the majority carriers. The increase in current due to minority carrier injection is ohmic, with an increase in conductivity of Au= e(/-Le+/-Lh)AP, Ap being the injected minority carrier density and /-L. and /-Lh the electron and hole mobilities. In the second case the current will be carried es sentially as a one-carrier SCL current by the carrier whose transit time is shorter than the relaxation time. The space charge of the other carrier will be compen sated by the majority carrier over most of the volume of the crystaL The total current will thus be a one carrier SCL current, having an "ohmic part" modified by the injected minority carriers. In the third case the contribution to the total current of each of the carriers will be SCL over large parts of the crystal volume, and the resulting current may be called a two-carrier SCL current. The three categories indicate the ranges of carrier injection into an insulator. They are not sharply separated from each other, but they will give an idea whether to expect ohmic or SCL behavior of an insulator and, in the case of SCL conduction, whether or not the contribution of the second carrier is important. Since the presence of traps raises considerably the voltage required to make the SCL current exceed the ohmic current, it might be possible that by varying the applied voltage one switches from one case to another. The problem of one-carrier SCL current in an insulator with traps has been treated by Rosel and by Lampert.2 By the detection of band-gap dc-electro luminescent emission from CdS and ZnS, Smith3.4 has shown, however, that both carriers can be injected into an insulator. The nonlinear current-voltage curve obtained3 for CdS points to SCL conduction. For this reason it seemed worthwhile to the writers to examine the problem of two-carrier SCL current in insulators. It was found that with the aid of three approximations one could obtain an analytic expression for the current-voltage curve. These three approxi mations are: (1) diffusion current neglected; (2) thermal equilibrium carrier densities neglected; (3) traps neglected. The first approximation is probably no more serious than it is in the one-carrier case." None of the results should be affected qualitatively by neglecting diffusion current. We shall return to this question when we discuss the results of the present calculation. The second approximation means only that one is unable to discuss in quantitative detail the transition from ohmic behavior at low voltages to SCL behavior at higher voltages. Since the thermal equilibrium concentration of electrons and holes is assumed to be zero, the dielectric relaxation time associated with the thermal carrier concentrations becomes infinite and the current contribution of both electrons and holes is SCL at any voltage. For the case of dark conductivity (which is what we consider exclusively here), this is not a serious deficiency. The transit-time discussion given earlier always allows one to determine the voltage above which the approximation will be valid. The third approximation is the most serious physically. It is fair to say, however, that the trap-free solution gives physical insight into the more general problem involving traps. Many of the conclusions obtained in this paper for the current flow through the crystal, the field distribution inside the crystal, and the net space charge accumulated in the crystal will also be valid for an insulator with traps. Qualitatively, the presence of traps will probably affect the two carrier problem just as it does the one-carrier problem. The solution obtained by the writers can be simply 2 M. A, Lampert, Phys. Rev. 103, 1648 (1956), 3 R. W, Smith, Phys. Rev. 105,900 (1957). 4 R. W. Smith, Phys, Rev. 100, 760 (1955). ;; For a discussion of this approximation in the one-carrier case, see reference 2. Forward diffusion currents playa predominant role in the case of a semiconductor p-i-n or p-i-metal junction. This case has been treated by A, Herlet and K Spenke, Z. angew. Phys. 7, 99, 149, 195 (1955); A. Herlet, Z. Physik 141, 335 (1955); K Spenke, Z, Naturforsch. 11a, 440 (1956), Throughout these papers in the intrinsic region space-charge neutrality is assumed. The two-carrier SCL current in an insulator corresponds to current flow at extremely high forward voltages in a semicon ductor p-i-n junction. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:001550 R. H. PARMENTER A~D W. RUPPEL generalized to include the case where there are shallow electron and hole traps but recombination occurs only between free carriers. This case, however, is less interesting physically than is the case where the traps assist in the recombination process. In addition to the three approximations already discussed, it should be mentioned that we shall assume constant electron and hole mobilities. This of course puts an upper limit on the size of the electric field for which our analysis is valid. For the present purposes such a restriction is not serious. A discussion of the effect of field-dependent mobilities upon one-carrier current flow has recently been given by Lampert. 6 II. BASIC EQUATIONS If we neglect the diffusive contribution, the current density J through the insulator becomes (1) Since the thermal equilibrium carrier concentrations are taken to be zero, nand p are the injected electron and hole concentrations. Both nand p and the electric field E depend on the position x in the crystal. We consider the current flow in the steady state where divJ= divJe+divJ h= 0, Je and Jh being the electron and the hole contributions to the current: Je=nefJ.eE, Jh=PefJ.hE. Neglecting traps in the recombination process, the recombination rate of the carriers becomes npvs, where v is the thermal velocity of the carriers and s their recombination cross section. Thus we have as continuity equation -fJ.e(d/dx) (nE) = +fJ.h(d/dx) (pE) = npvs. (2) A third equation is given by Poisson's law dE/dx= (e/HO) (n-p). (3) We have chosen the sign convention for the electric field E such that E is the positive gradient of the voltage. Taking x=O at the cathode and x=d at the anode, we see that E is never a negative quantity. The total current density J, the electron concentration n and the hole concentration p are likewise always positive or zero. If we wished not to neglect the tht;rmal equilibrium carrier densities no and po, then the right-hand sides of Eqs. (2) and (3) would be changed to (np-nopo) vs and e(Ho)-I(n-p-no+Po), respectively. The addi tional term nopovs represents the rate of thermal generation of carriers. The additional term e(HO)-1 X (po-no) results from the bound charge in the insulator. With these additional terms present, we see that when n= no and p= po we can immediately solve the differential equations to obtain the case of ohmic current flow. 6 M. A. Lampert, J. App!. Phys. 29, 1082 (1958). Let us define the constants fJ.o= Hovs(2e)-1 lIe=fJ.e/fJ.O IIh=fJ.h/fJ.O. (4) (5) (6) The constant fJ.o has the dimensions of a mobility. We shall refer to it as the "recombination mobility." Picking reasonable values for v and s (v= 107 cm/sec, s=3X10-19 cm2 in CdS4), one sees that fJ.o is of the order of 10-5 cm2/v sec. This means that the pure numbers lie and IIh are usually much larger than unity. The significance of these facts will become apparent in Sec. III. In order to integrate Eqs. (1)-(3) we must make a change of variables. Defining the constants a= fJ.efJ.hHO(2fJ.oJ)-1 (7) (3=efJ.e/J (8) "1= efJ.h/ J (9) and the variables A=aE2 (10) B=(3nE (11) C='YpE, (12) we may rewrite Eqs. (1)-(3) as B+C=1 (13) -B'=+C'=BC/A (14) A'=lIhB-IIeC, (15) the prime denoting differentiation with respect to x. The variables Band C represent at any point x the fraction of the total current carried by electrons and holes, respectively, at that point. At this point let us interject the remark that if we had included shallow electron and hole traps (traps unable to assist in recombination), then Eqs. (13)-(15) would still hold true provided we redefine lie as fJ.e(KefJ.o)-l (Ke being the fraction of electrons which are in the conduction band) and we similarly redefine IIh. (This implies that nand p are the mobile carrier densities.) Since we assume shallow traps, Ke and Kh are constants independent of position in the crystal. Although we see that there are no additional mathematical diffi culties with treating the more general problem, we shall not do so in order to keep the physical discussion as simple as possible. III. INTEGRATION OF THE BASIC EQUATIONS Substituting Eq. (14) into (15), we get (A'/A)=lI e(B'/B)+lIh(C'/C) (16) which can be immediately integrated to A = KB"C'h= KB"(1-B)'\ (17) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:00SPA C E -C H A R G E -LIM I TED CUR R EN TIN AN J N S U L AT 0 R 1551 K being an integration constant. Differentiating this, we get d [VhB-ve(1-B)] A'=KB'-[Bv'(1-B)Vh]= -B'A . dB B(1-B) But Eq. (15) implies that A'=v hB-v,(1-B). Thus it follows that dx= -A[B(I-B)]-ldB= -KBv,-l(l-B)vHdB. Integrating from the cathode (x=O) to the (x= d), we get [fBC ]-1 K = d Ba Bv,-I (1-B)vHdB (18) anode (19) where Be is the value of B at the cathode, Ba the value at the anode. Equation (19) serves to define the constant K in terms of the two (as yet unknown) integration constants Ba and Be. Combining Eqs. (10), (17), and (18), we have Rdx= -adA ![B(1-B)]-ldB = -c:c!KIBh-I(1-B)!vHdB. (20) Integrating (20) from the cathode to the anode and making use of (19), we find the applied voltage V to be (21) This is as far as we can go without saying something about the boundary conditions, since we need to determine the integration constants Ba and Be. Before doing so, however, we must point out the following difficulty. In the immediate vicinity of any real contact, the density of carriers being injected is never greater than the thermal equilibrium value.7 This means that Eqs. (2) and (3) are invariably incorrect at the contacts. In addition, in many cases the diffusion current is more important than the ohmic current at the contact. When such is the case, Eq. (1) is also incorrect at the contact. Thus the question of what boundary conditions to choose becomes somewhat ambiguous since we are integrating differential equations which are incorrect at the boundaries. There is one boundary condition which is reasonably unambiguous, however. We define a cathode to be blocking if the cathode is incapable of supplying to the insulator an electron current density greater than a saturation value J.e• If J.e is infinite, we say that the cathode is ohmic. Analogously, the anode 7 The injected carrier density may in fact be smaller than the thermal equilibrium value, as has been pointed out by M. A. Lampert, Bull. Am. Phys. Soc. Ser. 11,3,218 (1958). is either blocking or ohmic depending on whether or not there is a finite saturation hole current J.a connected with the anode. We shall assume that we are dealing with contacts where there are no further restraints. This of course is an idealization in that any real contact has the further restraint already mentioned with regard to the injected carrier density. We do not wish to invoke this further restraint since it is incompatible with our assumed differential equations. Because of the lack of any further restraints on our contacts, there is still a degree of freedom left with regard to specification of the boundary conditions. This degree of freedom can be removed with the aid of irreversible thermodynamics. Prigogine8 has proved a theorem which shows that the steady state of a system in which irreversible processes are occurring is that state in which the rate of entropy production is a minimum subject to any external constraints. For the present problem, the rate of entropy production is simply the power dissipation divided by the absolute temperature. The power dissipation in turn is simply IV times the contact area.t Thus for a given value of I we wish to minimize Eq. (21) with respect to Ba and Be, subject to the restraints imposed by the nature of the contacts. Although the writers have not found a general mathematical proof, they have convinced themselves empirically of the truthfulness of the follow ing conjecture: Conjecture: The voltage V, as expressed by Eq. (21), is always minimized by making Be as large as possible and Ba as small as possible, where O~ Ba~ Be~ 1. A discussion of the reasons for believing the conjecture will be found in Appendix A, where it is also proved that the conjecture holds whenever Ba and Be lie in the vicinity of zero or one. We will see shortly that this corresponds to the cases of greatest physical interest (the case of two ohmic contacts, and the case of one ohmic and one blocking contact). In Appendix B proofs are given for two auxiliary theorems used in Appendix A. One of these theorems (theorem I) shows that the conjecture holds whenever O<Ba ---> Be< 1. This corresponds to another case of considerable physical interest (the case of two saturated blocking contacts). Expressed in physical terms, the conjecture states that we wish to maximize the fraction of the current 8 I. Prigogine, Etude thermodynamique des phenomenes irreversi bies (Desoer, Liege, 1947). t This is not as trivial a statement as it may appear. In addition to the heat generated by the ohmic resistance, there is heat (or light) generated by the carrier recombination. This second source of power dissipation is not included in the expression JV. For the case of a band gap independent of position (such as we consider here), this second form of power dissipation is exactly cancelled by the heat absorbed in generating the carriers at the electrodes, so that the net power dissipation is still JV. This cancellation will not occur, however, if the band gap varies with position from cathode to anode. Under such conditions, there will be either thermoelectric heating or cooling, with a net electric power dissipation greater or less, respectively, than JV, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:001552 R. H. PARMENTER AND W. RUPPEL carried by the electrons at the cathode and the fraction of the current carried by the holes at the anode. By so doing, we minimize the accumulation of space charge in the crystal, and thus the potential drop across the crystal. IV. CURRENT FLOW (a) Ohmic or Nonsaturated Blocking Contacts If both contacts are ohmic, or more generally if the total current I is smaller than either 18c or 18a, then clearly the conjecture tells us to compose the cathode current entirely out of electrons (Bc= 1) and the anode current entirely out of holes (Ba=O). Making use of Eq. (7), the definition of a, and making use of the fact that m!n! (m+n+1)!' m and n> -1, (22) we may rewrite Eq. (21) ast I = (9/8)HOJ.!effd-3V2, (23) where the effective mobility J.!eff is defined as Equation (23) shows that the current depends upon applied voltage V and interelectrode spacing d just as in the one-carrier case.1.2 The only difference between the one-carrier case and the two-carrier case is that for the latter we use the effective mobility J.!eff. It was mentioned in Sec. II that II. and IIh are usually quite large numbers. This suggests invoking Stirling's approximation (25) for the factorials appearing in Eq. (24). If this is done, we find This shows that the current is enhanced (as compared with the one-carrier case) roughly by the square root of the ratio of the smaller carrier mobility to the recombination mobility. With a smaller carrier mo- t Note added in prooj.-This same result, with J.leff as given in Eq. (26) multiplied by a number of order unity, has recently been obtained by M. A. Lampert of the RCA Laboratories, using a semiquantitative approach which does not entail rigorous solu tion of the differential equations. His approach also makes it possible to handle situations where the drift velocity is field dependent and where the recombination is of monomolecular character. This work will be reported in the December, 1959, issue of the RCA RC1!iew. bility of 10 cm2/vsec this enhancement factor will be of the order of a thousand.9 This large enhancement factor is somewhat surprising since it was shown many years ago by LangmuirlO that the space-charge-limited current of nonrecombining electrons and positive ions in a vacuum is only about double the size of the space-charge-limited current when electrons alone are present. Even if we take the rather unrealistic case where the positive ions have the electronic mass (e.g., positrons), we still only get an enhancement factor of about four. The essential difference between the vacuum case and the insulator case lies in the fact that space-charge neutralization is much smaller in the former than in the latter. Near the cathode in the vacuum the positive ions are moving rapidly, the electrons slowly. Thus the positive-ion space-charge density is too small to effectively neutralize the electron space-charge-density. Interchanging the roles of electrons and positive ions, we may take similar statements for the anode. In the insulator, on the other hand, the carrier velocity depends only on the local field, not the total potential through which the carrier has dropped. This means that the densities of the two types of carriers can be more nearly equal in the insulator than in the vacuum. Returning to Eq. (24), let us now examine the situation where the smaller of the carrier mobilities is much smaller than J.!O. Once again J.!eff simplifies. Consider the case where J.!h«J.!O so that IIh«1. In general, for 5«1, we have (5-1) !=5 lj5""Olj5= 5-1, so that when J.!h«J.!O, Eq. (24) simplifies to (27) thereby becoming the one-carrier expression for the mobility. Similarly, when J.!e«J.!O the effective mobility becomes equal to the hole mobility. (b) One Saturated Blocking Contact In the previous section we discussed the case where the total current I is smaller than 18c or lsa, the satura tion currents associated with the cathode and anode, respectively. Next let us consider the case where I is larger than one of the saturation currents but smaller than the other. For definiteness, we take the case where I.a <I <Isc. Thus the cathode acts ohmic, while the anode is blocking. The voltage V is obtained from Eq. (21). Since the total current density I exceeds the saturation hole current l,w, the maximum value C can take at the anode 9 Such an enhancement may possibly be the explanation of the huge currents seen in copper-doped germanium at liquid nitrogen temperature by P. J. van Heerden, Phys. Rev. 108, 230 (1957). 10 I. Langmuir, Phys. Rev. 33, 954 (1929). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:00SPA C E -C H A R G E -L 1 1\1 r TED C 1.T R REX T T ~ A i\' I t\ S 1.T L A TOR 1553 is Ca=J.a/J, and hence In general, it is impossible to evaluate the integrals in Eq. (21) analytically. There are two limiting cases which we can handle, however. First we restrict ourselves to the case of a strongly saturated blocking anode where J.,a«J. Equation (21) is then approximated by v (28) Inserting Eq. (7) for a yields 1= (9/8)Ho,u ed-3V2, (29) so that once again we recover the one-carrier expression for the current. This case of the highly blocking anode thus has the same current-voltage relation as does the case of ohmic contacts with a very small hole mobility. Next we restrict ourselves to a weakly saturated blocking anode where Ba=l-I. a/I«l. Here Eq. (21) is approximated by [ I I-Jw/J ] d~ f B!ve-1(1-B)~vHdB-f . B~ve-ldB o. c v a{CBve-l(l-B)VHdB- ~l-J,"/J Bv.-ldB r (30) Neglecting the small terms involving (l-I.a/I), we obtain Eq. (23) for the current, so that the current voltage relation is the same for the weakly blocking anode as it is for the ohmic contact case. (c) Two Saturated Blocking Contacts We now consider the case where both contacts are blocking and saturated, so that J is greater than either I.e or I.a. It is obvious, however, that is is physically impossible for I to be greater than 18c+I8a• This means that the total current must saturate for sufficiently large voltage. We restrict the discussion to the case where we are close to saturation 50 that the ratio Now Be cannot be larger than 18e/ I, while Ba cannot be smaller than (l-I.a/I). Thus Be-Ba cannot be larger than 0, so that the range of integration of the integrals in Eg. (21) is very small. Treating the integrands as constants over this limited range of integration, we get v It is obvious that V is minimized by setting Bc-Ba equal to its maximum value o. This is also an immediate consequence of theorem I or Appendix A. Making use of the definition of a, Eq. (31) can be written as (32) We see that the current is smaller than the saturation value by an amount proportional to the cube of the interelectrode spacing and the inverse square of the voltage. V. POTENTIAL DISTRIBUTION AND NET SPACE CHARGE In principle, our analysis allows us to determine the potential V (x) at an arbitrary point x in the crystal. By integrating Eq. (18) between x=O and x=x, we get (33) This is an implicit relation for B (x), the value of B at the point x. Knowing B(x), we can integrate Eq. (20) between x= 0 and x= x to obtain (34) Making use of (33), this can be written in the suggestive form fBC xl B!ve-l (1-B) JvHdB B(x) (35) For the discussion of the potential close to the anode, [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:001554 R. H. PARMENTER AND W. RUPPEL it is convenient to write Eq. (33) in the form d-x (36) d and Eqs. (34) and (35) in the form V(d)-Vex) (37) and V(d)-Vex) (38) respectively. Unfortunately, it is in general impossible to evaluate the integrals appearing in Eqs. (33)-(38). For this reason we shall limit ourselves to the special cases discussed in the previous section. Even here we must content ourselves with calculating the potential in the vicinity of the contacts. (The only exception is the case where at least one of the contacts is blocking and strongly saturated, for which case we can evaluate the potential distribution throughout the crystal.) From the potential distribution, we can immediately calculate the field E(x) in the vicinity of the contacts. Alterna tively we can directly calculate E from Eqs. (10) and (17). Knowing the field at both contacts, we can determine the net space charge accumulated in the insulator. (a) Weakly Blocking and Ohmic Contacts We wish to evaluate the potential in the vicinity of a weakly blocking contact (the other contact being ohmic). To be consistent with the previous section we take the anode to be blocking. At a weakly blocking anode the current is not any more carried exclusively by holes as it was in the ohmic case, but the total current exceeds only slightly the anode saturation hole current. The ratio of the hole current to the total current at the anode is given by Jsa/ J = 1-0, where 0«1. 0=0 covers the ohmic case since the total current does not exceed the saturation current. The integral in the x-a CATHODE FIG. 1. Potential distribution for ohmic cathode, weakly blocking anode. (a) !J,h.«!J,Q and !J,e; (b) !J,e>!J,h»!J,O; (c) !J,h>!J,e»!J,O; (d)JI.,«!J,o and !J,h. denominators of Eqs. (36) and (37) becomes where CI3(Ve,Vh) is Euler's beta function given by Eq. (22). Since Vex) is evaluated only in the vicinity of the anode, the integrals in the numerators of Eqs. (36) and (37) are approximated by B(x) 1 Bv,-l(l-B)vHdB"-'Ve-l[Bve(x)-ov,] (40) B. and (41) respectively. With the aid of these approximations, we get expressions for x and Vex) as functions of B(x). Eliminating B(x), we get Vex) in terms of x, ( Jaa)Ve]il ( J,a)ilV'} + 1-- -1-- . J J (42) In Fig. 1, VeX) is plotted for various values of the mobilities in the case of a weakly blocking contact. For the field in the vicinity of the anode, we get [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:00SPA C E -C H A R G E - LIM I TED CUR R E N TIN A N INS U L A TOR 1555 .\t the anode the field reduces to (44) Since in this discussion the cathode is considered to be ohmic and hence the field at the cathode to be zero, the net space charge Q accumulated in the insulator per uni t contact area is just given by Q= HuE(d). (45) In Fig. 4 the charge Q in the crystal is plotted as a function of the degree of blocking of the anode to hole injection. The range described by Eq. (44) is indicated by the full line on the right-hand side of the figure. The crystal has a net negative charge when the anode is weakly blocking. This holds true even if the electron mobility is much smaller than the hole mobility, so that most of the current is carried by holes. Thus we have the rather striking situation where the sign of the net charge in the insulator is opposite to the sign of the carriers which predominate in carrying the current. Although we have examined only the case where the anode is blocking, analogous results hold if the cathode is blocking and the anode is ohmic. When both contacts are ohmic (or nonsaturated blocking), we can see that the field vanishes at both contacts. This is an immedaite consequence of the fact that A=aE2 vanishes at B=O and B=l, as may be seen from Eq. (17). This is consistent with Eq. (44) vanishing when I = I,a, the limit of the case of a nonsaturated blocking anode. This result leads to the conclusion that there is no net space charge in an insulator with ohmic (or nonsaturated blocking§) contacts. In the case where f.Lh«f.LO, this means that the electron space charge distributed throughout the bulk of the insulator is exactly cancelled by hole space charge lying very close to the anode. Although it was shown in Sec. IV(a) that the one carrier expression for the current is recovered when the smaller carrier mobility is much smaller than the recombination mobility, in this limit the one-carrier and the two-carrier cases are not in all respects identical, since in the two-carriers case quite generally the total injected space charge vanishes whenever both contacts are ohmic.11 The conclusion to be drawn from the § It should be pointed out that for a real blocking contact, the field at the contact may become finite before saturation sets in. This is a manifestation of the fact that actually there are con straints on carrier density as well as current density. For reasons already discussed, we are considering only idealized contacts in this paper, these idealized contacts being ones having restraints on current density but not on carrier density. This idealization of a blocking contact will have no effect on the results when the contact is saturated or when the contact is well below saturation (so that it is essentially ohmic). II One might ask how the inclusion of diffusion current in the theory would affect this result of vanishing electric field at the contact. The neglect of diffusion current is most serious in the regions of low electric field, since it is here that the diffusion current can become comparable to or larger than the ohmic X<O CATHODE X< d /lNODE FIG. 2. Potential distribution for ohmic cathode and anode. (a) !J,h«P.O and p.,; (b) P.,>P.h»P.O; (c) p.,,>p.,»p.o; (d) p..«p.o and P.h. discussion of this section is that the net charge in the insulator depends almost entirely on the nature of the contacts rather than on the bulk properties of the insulator itself. For the potential distribution near the anode in case of an ohmic anode, Eq. (42) reduces to V(x)= V(d)-1[2J/(Ho,uh)]!(d-x)f or v (x) = {1-(f.Leff/ f.Lh)![ (d-x)/ dJ} V (d). (46) The corresponding expression for the potential near the cathode is V(x)= (f.Leff/f.Le)!(x/d)lV(d). (47) The potential distribution in case of two ohmic contacts is plotted for some representative mobilities in Fig. 2. The vanishing of E at ohmic contacts will un doubtedly hold true for a more realistic model of an insulator, containing traps that assist in the recombina tion process. In fact, one might choose to define ohmic behavior of a contact as that behavior which results in vanishing electric field at the contact. For the purposes of the present paper, the writers feel it is better not to use such behavior as a definition but rather to show that such behavior follows automatically, when there is no restraint on the contact, from the concept of minimum entropy production. (b) One Strongly Blocking Contact Let us consider the case where the cathode is ohmic and the anode is strongly blocking, i.e., I8a/ I = 0«1. The integral in the denominators of Eqs. (36) and (37) current. However, one can argue that the inclusion of diffusion will affect the electric field at a given point only by tending to' decrease E (the idea being that since the total current is no longer entirely ohmic, E can afford to be smaller). This reasoning is not rigorous (since the effect on the carrier densities of including diffusion is ignored). The writers nevertheless feel that such reasoning is qualitatively correct in predicting that the vanishing of E at the contacts will be unchanged by including diffusion. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:001556 R. H. PARMENTER AND W. RUPPEL x-O CATHODE ANODE FIG. 3. Potential distribution for ohmic cathode, strongly blocking anode. becomes (48) The integrals in the numerators of Egs. (36) and (37) are appr9ximated by and 2 = __ [O~vh-C~vh(X)J, (50) 3Vh respectively. With the aid of these approximations, we get expressions for x and Vex) as functions of C(x). Eliminating C(x), we get Vex) in terms of x, V(x)= V(d) (x/d)!. (51) It is illustrated in Fig. 3. This potential distribution is valid throughout the insulator. It is the familiar one of the one-carrier case.1.2 The corresponding field distribution is (52) and the net charge in the insulator given by Eg. (45) Q [ cg~~Bl " \ o \ \ \ \ \ \ \ "" \ \ I ___ SATURATED HOLE CURRENT. ~ TOTAL CURRENT i FIG. 4. Net space charge in an insulator for ohmic cathode. Anode varying from blocking to ohmic. carrier treatment of Rosel and Lampert2 is approached for the trap-free case from the two carrier treatment by making one contact more and more blocking. (In the work of Rose and Lampert it is tacitly assumed that the anode is completely blocking for holes.) The approach of lowering the mobility of one carrier leads also to the well-known one-carrier current-voltage relation. The potential distribution and the net space charge in the insulator turn out, however, differently from the one-carrier case with one highly blocking contact, as has been discussed in the previous section. (c) Two Saturated Blocking Contacts When both contacts are blocking and the total current is near the saturation value of J.,,+J 8a, we have already seen in Sec. IV(c) that B is essentially constant over the entire thickness of the crystal. It immediately follows from Egs. (10) and (17) that the field E is constant over the crystal. Thus E may be obtained simply by dividing the total voltage drop by the interelectrode spacing d. Since E is the same at both contacts, it follows that the net space charge in the crystal is negligible. APPENDIX A The conjecture used in Sec. III states that in Eg. (21) Ba is to be minimized and Be is to be maximized in order to minimize the voltage drop across the insulator for a given current density. In other words, the conjecture claims that (A1) becomes and (53) This result Is shown as the full line at the left of Fig. 4. In the dse of a highly blocking contact both the current-voltage relation, Eg. (29), and the voltage distribution, Egs. (51)-(53), of the trap-free one carrier case1.2 are reproduced. It thus turns out that the one-aV;aBa~O (A2) whenever O~ Ba~ Be~ 1. We first point out that we need consider only that portion of the conjecture involving the upper limit Be, since Eg. (Al) implies (A2). This can be seen immediately by replacing B as an integration variable [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:00SPA C E - C H A R G E - LIM I TED CUR R E N TIN A N INS U L A TOR 1557 by C in the integrals in Eg. (21). Although we have been unable to prove the conjecture, in Appendix B we prove the following two theorems: Theorem I: Whenever Be?:. Ba such that Bc"(l-Be)'" :S;9j4Ba"(1-Ba)'\ then aVjaBc:S;O. Theorem II: If aVjaBe:S;O holds for O=Ba:S;Be:S; 1, then it also holds for O:S; Ba:S; Be:S; 1. Theorem II is particularly useful in checking the conjecture empirically, since it is much easier to examine av jaBe when Ba is zero than it is when Ba is finite. We have checked the conjecture for a number of special cases where Ve and Vk are integers. [The integrals appearing in Eg. (21) can be evaluated analytically for arbitrary integration limits whenever Ve and Vh are integers.] Fortunately we can show that the conjecture holds whenever Ba and Be lie in the vicinity of zero or one, which corresponds to the cases of greatest physical interest. First let us take Ba= Oa, Bc= l-oc, where oa and 0,«1. Then II foa (' (!ve-l) l(!vk-l)! 20a1', 20el'h B~'.-I(1-B)!'HdB- B~'.-ldB- Cl.HdC (![ve+vh]-l) ! o o· 0 3ve 3Vh a~d-lV""' (A3) [fl foa foe r [(ve-l) !(Vh-1)! Oa" oc'hf B·.-I(1-B).HdB- B'e-ldB- C'h-ldC (v.+vh- 1)1 o 0 IJ Ve Vh Here we have made use of Eg. (22). One can see by inspection that the right-hand side of (A3) achieves a minimum when oa and oe vanish. Next let us consider the case where both Ba and Be lie in the vicinity of one. By theorem I it follows that V is minimized by making Be=1. Thus we look at the case Ba=l-o, Bc=1. ° f O·H[l- (!v.-l)C]dC o ~-------------------- This is clearly minimized by making 0 as large as possible. In exactly the same fashion, we can show that the conjecture holds when Ba and Be both lie in the vici?ity of zero. APPENDIX B We prove here the two theorems mentioned in Appendix A. It is convenient to make some changes in notation as compared with Appendix A and the text. Define F(a,b) fbG'Y(B)D(B)dB L£bG(B)D(B)dB r (Bl) where G(B)=Bm(1-B)n D(B)=B-I(l-B)-I. (B2) (B3) We assume that a and b lie in the range zero to one. We take 1'> 1. (We are of course actually interested in the particular case I' = !.) We wish to determine under what conditions we have aF jab:S; O. Rather than look at aF jab, we shall look at M(a,b) = [G(b)D(b)]-1 (B4) x[ibG(B)D(B)dB r+1 (aF jab). (BS) Since [fb ]'Y+I [G(b)D(b)]-1 a G(B)D(B)dB ?:. 0 (B6) under all conditions, it follows that M and aF j ab will always have the same sign. We have M(a,b) =G'Y-l(b) fbG(B)D(B)dB a -I' fbG'Y(B)D(B)dB. (B7) a Let us pick some fixed value of b and ask how M varies with a. In the limit as a -> b, we have M(a,b) -> (l-'Y)(b-a)G'Y(b)D(b). (B8) Since 1'> 1, it follows that M(a,b):S; 0 in this region of a -> b. In general (B9) (aM jaa)=G(a)D(a)[ 'YG'Y-I(a)-G'Y-I(b)]. (BlO) We see that (aMjaa) can vanish at the point a=O. If [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:001558 R. H. PARMENTER AND W. RUPPEL b= 1, then (aMlaa) can also vanish as a -'; b= 1, the point considered previously. In addition, (aM laa) can always vanish at the point a= aI, defined such that [G(al)]/[G(b)]= (l/y)I/('Y-il. (Bll) Since 1'> 1, Eq. (Bll) is less than one. For the case of interest to us ('Y=!), we have [G(al) ]/[G(b)]= 4/9. (B12) From the definition of G(B), it is apparent that there is always just one value of al which satisfies (Btl) or (B12). This value of al will always be smaller than m(m+n)-t, the value of B for which G(B) achieves a maximum. [This follows from the facts that G(al) <G(b), al<b, and G(B) has a single maximum.] The significance of this fact is that it implies that G(al) ~ G(B) (Bl3) for all values of B lying between al and b. Thus M(al,b) =G-y-I(b) f"G(B)D(B)dB al -I' f"G'Y(B)D(B)dB~ [G'Y-l(b) -I'Gy-l(al)] al X f"G(B)D(B)dB=O. (B14) at Thus (B9) holds at a= al as well as at a= b. At a= al we have a2M ae -=['Y('Y-l)G'Y-1(a)D(a)-] . (BlS) da2 da a =al Since al<m(m+n)-t, it follows that (aGlda)~O. Therefore at a= al (B16) Since there are no minima or maxima of M as a function of a for al<a<b, since (B9) holds at a=al and a=b, and since (B16) holds at a= aI, it follows that (B9) must hold for all a lying between al and b. Theorem I follows immediately. In order to prove theorem II, we assume (B9) holds at a= O. Since there are no minima or maxima of M as a function of a for O<a<al, since (B9) holds at a=O and a= aI, and since (B16) holds at a= aI, it follows that (B9) holds for O~ a~ al. Since in the last paragraph we showed that (B9) holds for al~a~b, we see that (B9) holds for O~ a~ b. Theorem II follows im mediately. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 69.166.47.134 On: Fri, 12 Dec 2014 05:08:00

1.1735058.pdf

Plastic Deformation of InSb by Uniaxial Compression J. J. Duga, R. K. Willardson, and A. C. Beer Citation: Journal of Applied Physics 30, 1798 (1959); doi: 10.1063/1.1735058 View online: http://dx.doi.org/10.1063/1.1735058 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Deformation behaviors of InP pillars under uniaxial compression Appl. Phys. Lett. 101, 151905 (2012); 10.1063/1.4758479 Plastic deformation of gallium arsenide micropillars under uniaxial compression at room temperature Appl. Phys. Lett. 90, 043123 (2007); 10.1063/1.2432277 Determination of deformation potentials in strained InSb quantum wells Appl. Phys. Lett. 88, 171901 (2006); 10.1063/1.2198101 Plastic Bending of InSb J. Appl. Phys. 33, 169 (1962); 10.1063/1.1728479 Photoanodization of InSb J. Appl. Phys. 30, 1110 (1959); 10.1063/1.1776989 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:091798 R. I. BEECROFT AND C. A. SWE:\SON :::t 0.1 .... z '" U Q: '" "-z 0 Vi z .. "-x '" Q: .. '" z ::; -0.6 280 290 300 TEMPERATURE, oK 310 FIG. 4. Linear ther mal expansion of Teflon in the transi tion region. Points are shown only for the lowest tempera tures, where the maximum scatter oc cured. The expan sion is expressed as percent of the meas uredlengthat 298°K. Measurements were made with an aver age, but not con stant, temperature change of 0.2 deg/hr, with a maximum of 0.6 deg/hr. measurements was 0.002 mm, and readings were esti mated to lo division. The maximum scatter was less than 0.005 mm, so a discontinuity in length as small as 0.05% would have been easily detected. It is possible for Teflon to be anisotropic in the extruded form, a factor which would explain the difference between the values for A VI-II as obtained from the pressure meas urements and as calculated from the single linear ther mal expansion measurements. Anisotropy could also possibly mask the second room temperature transition in the thermal expansion measurements, although not in the pressure measurements where it should have been seen. It is apparent from zero pressure data that there is some modification of the structure of Teflon at about 3000 to 305°K, but from the pressure results and other evidence in the literature there is some doubt that it is a true first-order transition. It seems more likely that this temperature marks the beginning of some type of dynamic disorder similar to that occuring in amorphous polymers at the transition point. This view is supported by the stress relaxation experiments of Nagamatsu, Yoshitomi, and Takemoto,20 who found an activation energy peak at 298°K of the type which is character istic of such second-order transitions, and by the x-ray studies, which indicate gradually increasing disorder above this temperature. The effect of crystallinity on the transitions and x-ray studies at elevated pressures should provide in teresting additional information about the complex be havior in this temperature region. ACKNOWLEDGMENTS The authors are indebted to E. 1. du Pont de Nemours & Company for providing the sample, and they par ticularly wish to express their appreciation to Dr. M. 1. Bro, who graciously provided the sample analysis and much of the available literature on the properties of Teflon. 2() Nagamatsu, Yoshitomi, and Takemoto, ]. Colloid Sci. 13, 257 (1958). JOURNAL OF APPLIED PHYSICS VOLUME 3D, NUMBER 11 NOVEMBER, 1959 Plastic Deformation of InSb by Uniaxial Compression*t ]. J. DUGA, R. K. WILLARDSON, AND A. C. BEER Battelle Memorial Institute, Columbus, Ohio (Received March 12, 1959) Plastic deformation of InSb by uniaxial compression was found to produce decreases in both the electron mobility and magnetoresistance, but to have no effect on the Hall coefficient. Analyses of the temperature dependence of the conductivity mobility and the weak-field magneto resistance, in terms of mixed scattering by acoustic lattice vibrations and ionized impurities, suggest that the principal effect of this mode of de formation is the creation of ionized vacancies and interstitials in approximately equal densities. The analysis permits an estimate of the density of point defects, which can then be related to the total energy expended during deformation. Reference is made to the effects of plastic bending of InSb where the carrier concen tration is affected. This behavior is similar to results on silicon and germanium which have been analyzed in terms of the Shockley-Read trapping model. I. INTRODUCTION IT is well known that the electrical properties of both metals and semiconductors are altered by the pres ence of various imperfections in the crystal lattice. For * This research was supported by the Office of Naval Research. t A preliminary account of this work was presented at the Chicago Meeting of the American Physical Society in March, 1958. example, the resistivity of copper, following severe cold work, may increase by 2% over the value for a well annealed sample, and the resistivity of other metals may increase by as much as 18%.1 More pronounced changes occur in semiconductors where the charge carrier density I E. Schmid and W. Boas, Kristallplastizitat (Verlag Julius Springer, Berlin, Germany, 1930), p. 214. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:09DEFORMATION OF InSb BY UNIAXIAL COMPRESSION 1799 may change and lifetime and conductivity decrease by a factor of 10 or more.2 Thus the increased sensitivity to physical defects, which is exhibited by semiconduc tors, renders a study of such defects more amenable to investigation. Various treatments of the changes in the electrical properties have been proposed. In the case of metals, Koehler3 has considered the resistance due to the scatter ing of electrons by the distorted potential associated with an edge dislocation, the amount of lattice distortion having been calculated by elastic continuum theory.4 Assuming the validity of l\fatthiessen's rule, his calcula tions resulted in values of the dislocation-induced resis tance which were in good agreement with experiment. A similar calculation has been carried out by Dexter and Seitz· for the effects of dislocation scattering in semiconductors. The principal difference from Koehler's work is that they used the Maxwell distribution to re place the Fermi function. Their results show that the electron mobility should increase as the first power of the absolute temperature. The experiments on germanium have led to another model, proposed by Shockley6 and elaborated by Read.7 At an edge dislocation, there is one atom which has a covalent bond with only three, rather than four, nearest neighbors. Associated with each atom at an edge dis location, there is a "dangling unpaired electron" which introduces a deep-lying acceptor site, thereby reducing the density of conduction electrons and the lifetime of minority carriers. Furthermore, an additional scattering mechanism is present as a result of the linear charge distribution arising from the conduction electrons which are trapped by these sites. A change in carrier mobility thus results. This model seems to be applicable to the effects observed in germanium and silicon crystals which have been deformed by bending. Recent experiments by Greiner and co-work ers8 have shown that the electrical properties of germanium samples which have been deformed by uniaxial compres sion may be qualitatively explained in terms of two mechanisms; (a) scattering by neutral dislocations and (b) scattering from vacancies and interstitials intro duced during the deformation. The experimental data on InSb samples which have also been deformed by uniaxial compression suggest that the predominant effect of the deformation is the creation of ionized vacancies and interstitials producing acceptor and donor centers in approximately equal numbers. There are indications that this effect is promi nent initially and that scattering from neutral disloca- 2 G. K. Wertheim and G. L. Pearson, Phys. Rev. 107, 694 (1957); Pearson, Read, and Morin, Phys. Rev. 93, 666 (1954). 3 J. S. Koehler, Phys. Rev. 75, 106 (1949). 4 J. S. Koehler, Phys. Rev. 60,398 (1941). 5 D. L. Dexter and F. Seitz, Phys. Rev. 86, 964 (1952). 6 W. Shockley, Phys. Rev. 91, 228 (1953). 7 W. T. Read, Jr., Phil. Mag. 45, 775 (1954); 45, 1119 (1954); 46, 111 (1955). 8 Greiner, Breidt, Hobstetter, and Ellis, J. Metals 9,813 (1957). tions may contribute to the scattering as the deforma tion proceeds. The difficulties associated with the isola tion and analysis of dislocation scattering are discussed in the following. II. EXPERIMENTAL PROCEDURE A. Deformation of Samples Single crystal specimens having an excess donor con centration of 1 X 1016 cm-3 were grown by the Czochral ski technique and cut into parallepipeds (3X3X8 mm) such that the bounding planes were (110), (211), and (111). Samples of higher purity but of unknown orienta tion (excess donor concentration of 1 X 1014 cm-3) were cut from large crystallites in zone-refined ingots. Plastic deformation was carried out by uniaxial com pression at 300°C, at pressures of 500, 1500, and 2000 psi. These temperatures and pressures were chosen as a result of earlier experiments using dislocation etch pits as a means of observing deformation.9 In the case of the oriented crystals, the compression stresses were applied parallel to the [111J direction, in the same direction as the current for the electrical measurements reported below. Compression of the zone-refined specimens was in a direction perpendicular to the current. At intervals in the course of deformation, the samples were cooled and removed from the compression apparatus for elec trical measurements. B. Electrical Measurements To detect the presence of scattering mechanisms with a temperature dependence differing from that associated with lattice vibrations or ionized impurities, measure ments were made of the Hall coefficient and resistivity as a function of temperature in the extrinsic range be tween 77 and 300oK. The magneto resistance at 77°K was also determined as a function of magnetic field strength between 300 and 20 000 gauss. All measure ments were made by conventional dc techniques, the pertinent voltages being read on a Leeds and Northrup K-2 potentiometer. In Fig. 1 is shown the temperature dependence of the resistivity of a sample having an original excess donor concentration of 1 X 1016 cm-3• The different curves correspond to various stages of deformation, the total time under compression being indicated. Since the Hall coefficient remained constant, it is seen that the mobility jJ. (= RH .... "po) decreased steadily as the deformation progressed. A decrease is also observed in the mobility jJ.M which is calculated from the low-field limit of the magneto resistance (Fig. 2), although the change is not so pronounced. The lack of change in the Hall coefficient with defor mation indicated that if there were trapping of conduc tion electrons on dislocation sites, the density of such events was negligibly small compared to the original 9 Maringer, Duga, and Beer (to be published). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:091800 DUGA, WILLARDSON, AND BEER - TOIOI li~. (hrs) of ~m - 4 pressiOn (300oC,:> 2000Ps~21 3 ~ ~ ~ ~ ~ 2 ~ ...0-5 ~ ~I P .-rr- 9 ~ .c. ......... 8 Jt'.f' 1, ~,~ . r-.... ~ I---..... 0- 6, ( 5 41 ~ 4 12 14 FIG. 1. Resistivity vs reciprocal temperature at various stages of compression; N D -N A = 1 X 1016 em-a. carrier concentration. Such effects, if present, should be more readily observed in samples of higher purity. Con sequently a sample having an excess donor concentra tion of 1 X 1014 cm-3 was deformed at 300°C under a pressure of 500 psi. The temperature dependence of the mobility before and after one-half hour of deformation is shown in Fig. 3. The large decrease (by a factor of 6) in the mobility at 77°K resulting from small deforma tion is indicative of the sensitivity of InSb as a tool for studying effects of physical imperfections on electrical properties. As in the previous example, there was no change in the carrier concentration. Further deforma tion resulted in a continued decrease in the electron mobility and very slight decreases in carrier concentra tion (less than 5%). This concentration change is not considered significant. It could, for example, be ac counted for by the dissociation of the compound and the subsequent evaporation of antimony from the specimen.lO The field dependence of the magnetoresistance co efficient of this high-purity sample is shown in Fig. 4. It is noted that the behavior of the coefficient was changed only slightly following the first compression, but further deformation resulted in a steady decrease of the low-field values. III. DISCUSSION OF CHARGE-CARRIER SCATTERING MECHANISM In view of the constancy in the extrinsic Hall co efficient as the deformation progressed, (aside from small changes attributed to dissociation of the compound 10 Whether dissociation is of importance depends on the temper ature, the total time of heating, and the purity of the specimen. Preliminary investigations showed that for our experiments at 300°C, the effects were insignificant compared to those introduced by deformation. This was not true for n-type specimens at 450°C. and evaporation of antimony), no further consideration will be given to the Shockley-Read trapping mechanism as an explanation of the experiments reported here. It should be mentioned that the work of other investi gators has shown that plastic deformation by bending in silicon and germanium has resulted in changes of the electrical properties of the type considered by Read7; however, Greiner's experiments on compression were analyzed by consideration of the combined processes of scattering from neutral line imperfections and an nealable vacancies and interstitials. A similar approach is used here on the analysis of data from compressed samples. Deformation of InSb by bending, on the other hand, is found to affect the electrical properties in a manner which may be expected if the Shockley-Read mechanism is effective. This behavior will be discussed later. When scattering of current carriers arises from both lattice vibrations and ionized impurities, the tempera ture dependence of the carrier mobility in nondegenerate samples may be approximated reasonably well by 1 -=aLT!+ar T-!, p. (1) where the aL and ar terms describe the effects of scatter ing by lattice vibrations and ionized impurities, respec tively.u When the effects of a localized distorted poten tial as created by a dislocation are considered,5 the carrier mobility given in Eq. (1) is altered to the form 2.5 1 -=aLT!+a[l!+a DT-I, p. (2) 2 ff)tal time (tin) of compression IJIlf"C. 2000 .. ,) ,,::;r'0"---+ ___ ...... __ 1.,d"I----+---~---.:j::a,.,~-+---_l_-___l • 1 hl lad H,90USS FIG. 2. Magnetoresistance coefficient vs magnetic field at various stages of compression; N D-N A = IX 1016 em-a. II Strictly, the reciprocal relaxation times for the different scat tering processes should be summed before the result is averaged over energy. This has actually been done [see Eqs. (3)J for cal culations representative of mixed scattering by the lattice and ionized impurities. Where three scattering mechanisms are opera tive, however, the evaluation of the integrals is tedious, and for practical purposes one resorts to the approximation of Eq. (2). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:09DEFORMATION OF InSh BY UNIAXIAL COMPRESSION 1801 5xlcrt-t--t---+-+-II---~-I ~ .. ~ ~ 2xI05t--t---t----tT--+--f----~ He CJ 6XI~'--~--..J'--...L..--'--'- ___ _J 34 6810 20 Ict/ToK FIG. 3. Electron mobility vs inverse temperature before and after compression; N D-N A = IX 10" em-a. the (XD term describing the effects of scattering from neutral dislocation lines. The upper curve in Fig. 3, representing the unde formed specimen, is very well described by a function of the form of Eq. (1). From the values of (XL and CY.[, the impurity scattering parameter /3= 6f.J.d IJ.I was found to be in good agreement with that found by other tech niques. If it is assumed that the only factor contributing to the decrease in mobility after deformation is scatter ing from neutral dislocations, it should then be possible with those values of (XL and (XI representative of the undeformed specimen, to determine (XD so as to fit the lower curve in Fig. 3. However, this procedure was not successful. Any choice of aD resulted in much broader maxima than were observed experimenatlly. It was noted that the mobility at low temperatures decreased more rapidly than that predicted by a linear dependence on T. Actually the general shape of the curve is remi niscent of that obtained from samples which were known to be highly compensated. It was, in fact, found that the curve can be approximated fairly well by the scattering mechanisms represented by expression (1) with an appreciable increase in (XI as a result of the deformation. Since (XI is approximately proportional to the density of ionized impurities, this finding lends con siderable support to the hypothesis that the primary result of the compression is an introduction of ionized scattering centers, say vacancies and interstitials, in approximately equal numbers. These act as acceptor and donor sites so that the net density of charge carriers (and hence the Hall coefficient) is not changed. Analysis of the magnetoresistance data (Fig. 4) also suggests that such a mechanism is predominant. A more quantitative consideration of these concepts now follows. IV. THEORETICAL EXPRESSIONS FOR MOBILITIES The solution of the Boltzmann equation (neglecting the effects of heat currents and temperature gradients) leads to expressions for the current densities of the form12 Spherical energy surfaces, classical statistics, and a mixed scattering process have been assumed. The pa rameters indicating the degree of impurity scattering and the magnetic field strength are defined, respectively, by /3= 6f.J.d f.J.I and 'Y= (97r/16) (f.J.LH)2. The mobility f.J.L is that resulting from lattice vibrations; f.J.I, that from ionized impurities. The functions K(/3;y) and L(/3,'Y) have been evaluated13 for various choices of /3 and 'Y. The expression for the weak-field magnetoresistance may be found by expanding K and L in power series in 'Y, and neglecting terms containing the magnetic field in powers greater than HZ; whence we have14 /1p = 'Y[_-_K_' (_/3,_0) p K(/3,O) where K'(/3,O)=[aK(/3,'Y)/a'YJ'Y=o. From the definition 2 3 __ -..~.,11!. I Totol lime (hts) of --i~~t----f--; compression (300· C. 500 psI) .ar~ 1'109t--t---t--+--""'N1}--+--~ Ii. gous5 FIG. 4. Magnetoresistance coefficient vs magnetic field at various stages of compression; N n-N A = IX 1014 em-a. 12 See for example, V. A. Johnson and W. J. Whitesell, Phys. Rev. 89, 941 (1953). 13 Beer, Armstrong, and Greenberg, Phys. Rev. 107, 1506 (1957). " J. Appel. Z. Naturforsch. 9a, 167 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:091802 DUGA, WILLARDSON, AND BEER I o a 0. 0 8 6"" 4 2 I 8 00 0.06 004 1-1"- ~ I" ~ i I E I 1"-I 1,1 , I'L' K(~) ~I ~ [\ I' ,.I'L .~ 1 i I I , . fH· 'LK(M t r--( '~ -.' ~~H-O 91I(1~~/4j} = 'LF(~) 0.02 I 0.0 0008 0006 I f\ ,\ \ \\ i \ ~ i I 1\ ~\ I \ '\ 0~01 002 005 01 02 05 I 2 5 10 20 50 100 200 500 KlOO -FIG. 5. Relative conductivity and magnetoresistance mobilities as a function of the impurity scattering parameter {3=6Jl.dJl.I' of 'Y, the expression for lattice mobility is { ( I1p ) 16[-K' ((3,0) /LL= pH2 H ... O 911" K((3,O) (4) We define the magnetoresistaoce mobility J.LM in terms of the constants for (3 = 0, namely (5) where {[ K' ((3,0) 7r V ((3,0) J}" F((3) = K((3,O) 4 K2((3,0) . 1-11"/4 (6) In terms of the strong-field limit of the Hall co efficient, the actual conductivity mobility can be ex pressed as IL=RH-+oo(Jo. From (3) this is seen to be IL = ILLK ((3,0) = ILLK ({3). In Fig. 5 are shown K ((3) and F ((3) as functions of (3. The experimental results will be analyzed in terms of these curves. V. INTERPRETATION OF EXPERIMENTAL RESULTS It has been pointed out above that the electrical properties of plastically compressed InSb indicate that the primary electrical changes resulting from this mode of deformation were due to the introduction of ionized vacancies and interstitials in approximately equal den sities. In particular, the character of the electron mo-bility as a function of temperature, following deforma tion, is quite similar to that in specimens which are known to be well compensated or in specimens which have been subjected to radiation damage. In the latter case, many of the observed changes in the electrical properties are directly attributed to the formation of such ionized point imperfectionsl5 and their subsequent recovery upon annealing is well accounted for in terms of recombination and annihilation processes.16 From the curves of Fig. 5, we may qualitatively follow the experimental data by first considering two separate ranges of the impurity scattering parameter, (3. For values of (3 such that 1 </3 < 10, it is seen that although the mobility IL, as calculated from the Hall coefficient, decreases rather strongly with increasing /3, the magnetoresistance mobility, J.LM, remains essentially constant. However, for /3'230, both IL and ILM decrease steadily with increasing (3. Since (3 is approximately pro portional to the total ionized impurity concentration, a comparison of this qualitative description with the experimental curves (Figs. 1-4) lends further support to the hypothesis that the principal effect of the de formation is the creation of ionized vacancies and inter stitials in nearly equal densities. A more quantitative comparison is shown in Fig. 6 where the experimental data have been superimposed on the theoretical curves. In plotting these points for com parison to the theory, the value for (3 was determined by fitting the experimental value of IL=RH-+oo(Jo=ILLK((3) since this value is more unambiguously determined than is the weak-field magnetoresistance. The corresponding value of ILM was then calculated and plotted at this same value of (3. Although the experimental points are all somewhat above the theoretical curve, the agreement is quite good, especially considering the difficulties associated with accurate measurement of the weak-field magnetoresistance and the limitations of the theory. Thus far, it has appeared to be sufficient to account for the experimental curves by assuming that ionized vacancies and interstitials are created in the course of the plastic deformation. Although we shall not discuss the details associated with the production of the point defects, it appears that they are created by the mecha nism of the crossing of moving dislocationsP Since the samples were oriented in such a way that different slip systems would be operative, it is reasonable to assume that a random distribution of these defects would be generated. Thus the above data and conclusions suggest that one might make a semiquantitative determination of the energy dissipated in the creation of a vacant lattice site or an interstitial. Although exact measurements of the amount of deformation were not made in the course of 15 Brown, Fletcher, and Wright, Phys. Rev, 92, 591 (1953). 16 R. C. Fletcher and W. L. Brown, Phys. Rev. 92, 585 (1953). 17 F. Seitz, Advances in Physics 1,43 (1952). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:09DEFORMATION OF InSb BY UXIAXIAL COMPRESSION 1803 0 a o 6 Q 4r-- 2 I 0.0 a 006 004 00 2- 1-00 000 a 0006 1 I -+-+ .1. I'-. ~ --~ ~e Ie,' K(M "" '" r\ ~ r\ ~M/e" F(~) r-r i 1 i I I I 1 I I e ' ...s.. , " K(~I E,H , I, eM' {~zL 9T1U!%/4f 'e,F(~) I I - I I i -- I 1 I I i f\ie ~\ <1>, ,\ • 1 1 \\ I ~\\ 1\ \ 0004001002 005 01 0.2 0.5 I 2 ~ 5 10 20 50 100 200 500 1000 FIG. 6. Experimental determinations of III ilL and IlM I ilL, super imposed on the theoretical values. To facilitate interpretation of the data, the following legend applies: On the K ({3) curve, symbols of one kind (say, all squares) refer to one specimen. The open symbol represents conditions in the un deformed sample; the symbol with a vertical line through it corresponds to the com pressed sample; and that with the + sign inscribed applies to later compression of the sample. The experimental value determined by magnetoresistance for F ((3) lies directly above or below that for K((3) and is blacked-in completely. the experiments reported here, earlier data indicate that a compression on the order of approximately 2% re sulted in an increase in the number of scattering centers by about 2 X 1016 cm-3• This latter figure is calculable from knowing K (f3) necessary to account for the mo bility J.I., from which a value for {3 and hence J.l.1 is found. Using the Brooks-Herring relation for J.l.1, the total im purity density may be determined. The resulting calcu lations show that one lattice defect is produced for every 50 ev of energy expended in effecting the deformation. This value is in good agreement with the order-of magnitude calculations in experiments on copper and rock salt specimens which were deformed in a similar manner.18 18 The data on these, as well as other materials, has been re viewed and evaluated by Seitz (see reference 17). VI. CONCLUDING REMARKS We have seen that the behavior of the Hall coefficient, the transverse magnetoresistance, and the conductivity mobility suggest the predominating effect of small amounts of uniaxial compression of InSb at 300°C to be the production of ionized vacancies and interstitials in approximately equal densities. Such a hypothesisqualita tively accounts for the salient features of the electrical data. A more quantitative determination of the damage produced and energy expended in creating these defects would require not only a more precise measurement of the amount of deformation, but also a more involved treatment of the charge-scattering mechanism. For example, one would have to consider the effects of polar, rather than only acoustic-mode, scattering by the latticel9; an improved treatment of scattering by ionized impurities2o,21; and the effects of carrier degeneracy.22 In contrast with the behavior described above, pre liminary experiments on plastic bending of InSb show results entirely different from the effects described above for compressed samples.23 Even a very slight bending (radius of curvature> 2 m) was found to introduce an acceptor level near the middle of the band gap resulting in a decrease in carrier concentration and electron mo bility. This is precisely the behavior found in plastically bent germanium and silicon and a qualitative descrip tion in terms of Read's scattering mechanism is possible so long as impurity scattering is negligible. However, there are certain experimental data which suggest that the discrete character of the trapped charges must be taken into account. A more detailed description of these effects of plastic bending is being prepared for publication. ACKNOWLEDGMENTS The authors wish to thank 1\1r. R. E. Maringer for his helpful comments and suggestions during the course of the work reported herein. We also wish to thank Dr. Peter B. Hirsh for his helpful discussions on the mechanism of the formation of vacancies and inter stitials in the lattice. 19 See, for example, the comments made by Bate, Willardson, and Beer, J. Phys. Chern. Solids 9, 119 (1959). 2Q F. J. Blatt in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1956), Vol. 4, p. 199. 21 N. Sclar, Phys. Rev. 104, 1548 (1956). 22 Due to the low effective mass, degeneracy effects in InSb begin to appear at nOK for electron concentrations of around 7X 1014 cm-·. 23 J. J. Duga, Bull. Am. Phys, Soc. Ser. II 3, 378 (1958). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Wed, 26 Nov 2014 13:51:09

1.3060461.pdf

Statistische Thermodynamik Arnold Münster T. Teichmann , Citation: Physics Today 10, 8, 26 (1957); doi: 10.1063/1.3060461 View online: http://dx.doi.org/10.1063/1.3060461 View Table of Contents: http://physicstoday.scitation.org/toc/pto/10/8 Published by the American Institute of Physics26 tions are chosen to illustrate the principles and methods involved in the various types of chain processes. The most interesting section of the book deals with the mathematical treatment of chain reactions including nonsteady state conditions. The rigorous mathematical representation may at first appear to be an exercise in the solution of complex differential equations. How- ever the labor is well justified in the application of the theoretical results to an analysis of the characteristics of spontaneously explosive reactions, such as the hy- drogen-oxygen reaction and the oxidation of carbon monoxide. The final chapter is devoted to the kinetics of an unbranched chain reaction as encountered in ad- dition polymerization of unsaturated compounds. In general this book serves the very fine purpose of examining the theoretical principles of the kinetics of chain reactions, including the interplay of mass trans- port and chemical interaction. The treatment is not di- rected at the specialist engaged in research in chemical kinetics, rather it is a brief account of a rapidly ex- panding field of scientific endeavor. The keen interest in combustion processes and polymerization reactions coupled with the availability of modem computing de- vices promises major advances in chemical kinetics dur- ing the next twenty years. An Introduction to Junction Transistor Theory. By R. D. Middlebrook. 296 pp. John Wiley & Sons, Inc., New York, 1957. $8.50. Reviewed by R. Hobart Ellis, Jr., New York City. As engineers take over radar's microwaves and the nuclear chain reaction, the physicist's frontier seems to be shifting to the study of solids. The ten-year-old transistor is the most useful contribution to come from this study. It is still so young that to most physicists it is just a tiny substitute for a vacuum tube. This book will serve as a simple path to a more subtle under- standing. For convenience we can divide transistor study into semiconductor physics, electrical action of transistors, and transistor circuitry. Dr. Middlebrook aims at the second. He offers some semiconductor theory as a foundation, discusses electrical function thoroughly, and leaves circuitry to others. In terms of minority carrier density and migration he describes transistor action for us and develops in detail the equivalent circuit on which he has worked at Stanford University. The nonpragmatic scientist, curious about the nature of things for the fun in it, may be a little unhappy at the physics in the book. The relationships among elec- tron orbitals, holes, and conduction electrons are not clearly delineated. One hard-to-take model pictures hole migration in terms of a cluster of negative mass elec- trons that moves as a unit in a direction opposite to the force of the electric field and carries the hole along in the center. But the author frequently refers his reader to Shockley's basic Electrons and Holes in Semi- conductors, in which such matters are treated exactly. He assumes the Fermi-Dirac population formula with-out derivation and discusses Brillouin zone conduction in only a cursory manner. We must compliment author and editor on the plan- ning of the book. It has been wisely said that the way to teach a subject is to describe it completely in a page, then do it over again in a complete chapter, then at book length, and so on. This book follows this plan. Quantitative descriptions follow qualitative ones, and the reader is kept constantly aware of what is ahead of him. In a few years the term "circuit design" will not im- ply only vacuum-tube circuits as it does in current book titles. People will learn to use the transistor for its unique properties as a current amplifier, and this book will help them learn. Statistische Thermodynamik. By Arnold Miinster. 852 pp. Springer-Verlag, Berlin. Germany, 1956. DM 138.00. Reviewed by T. Teichmann, Lockheed Aircraft Corporation. While statistical and mechanical thermodynamics re- main among the most basic and unifying aspects of modern science, the development of new techniques and their application to new problems lead to an inevitable change of emphasis and approach. Professor Miinster has recognized this in his work which essays to satisfy the needs of the practicing physicist and physical chem- ist while yet retaining some of the aspects of a text- book and providing the student with a thorough foun- dation. Naturally such an ambitious approach has ne- cessitated certain restrictions in topics and methods— for example, only equilibrium states of matter are con- sidered and while the basic methods are thoroughly de- scribed, many possible variants are omitted—but the book remains the most comprehensive and palatable ac- count presently available. In order to make this great mass of material more accessible to beginners, the more advanced topics for application are "starred" and can be omitted at the first reading. The book consists of four sections dealing with the foundations of statistical mechanics, the theory of gases, the theory of crystals, and the theory of liquids. It is, of course, impossible in a review of this length to list all the significant subjects described but certain fea- tures of the treatment seem of particular interest. There are included a very thorough treatment of Gibbs' method and a discussion of the ergodic problem, and an ex- tended discussion of phase transition including the new methods of Lee and Yang, a description of the general theory of condensation, and discussion of the Born- Green theory of molecular distribution function. In the section on crystals, a detailed description of the Kram- ers-Wannier theory is given and Onsager's solution of the two-dimensional Ising problem is presented in the form first given by Montroll and Newell. The implica- tion of this method for three-dimensional problems is touched on, though not as thoroughly. The electron conductivity of metals and the Nernst heat theorem are also given thorough consideration in this section. The PHYSICS TODAYr = AN NCR PHYSICS RESEARCH TEAM examines the new N-400 cores. NEW FERRITE CORESIMPROVE MEMORY OF TRANSISTORIZED COMPUTERS AT NCR This NCR Physics Research team has developed a quality magnetic ferrite core — the N-400. The success which has resulted from such an integrated research program underscores the team spirit and high degree of com- petence you will find at NCR. Exploring new techniques and processes enabled this research team to develop toroidal cores that have significant advantages over other types of square-loop ferrites. The N-400 core has stability when used with low- current transistor circuitry that permits advances in the design of future core-memory computers and solid-state business machines. There is a team position open for you now at NCR where research is unlimited. Perhaps you are interested in magnetics or other solid-state research areas such as cryogenics, electro-luminescence, ferroelectrics, or photoconductors. NCR consistently earns national recognition as one of America's best managed companies. Our Research Division is growing steadily and NCR is a growing company within a growing industry. That three-fold growth factor can mean only one thing to you — opportunity.MAKE PLANS NOW to take an active part in one of the world's most successful corpora- tions. Write to the Director of Scientific Personnel, Section A, The National Cash Register Company, Dayton 9, Ohio. NCRTHE NATIONAL CASH REGISTER COMPANY28 final section on the theory of liquids deals with the problems of melting and of solutions of weak and strong electrolytes and more of solutions of macromolecules. The book is distinguished by many recent references and by the inclusion of most of the significant modern work in this field. In particular, the author has given careful attention to the work of Kirkwood and his col- laborators and has even included references to such an exotic item as Kirkwood's "Princeton notes of 1947". Great effort seems to have been made throughout to make the presentation both comprehensive and per- spicuous and at the same time to avoid overwhelming the reader with inessential mathematical minutae. The book has the excellent typographic format which the readers are led to expect from this series and is a worthy addition to the Springer collection. Annual Review of Nuclear Science. Vol. 6. Edited by J. G. Beckerley, M. D. Kamen, L. I. Schiff. 471 pp. Annual Reviews, Inc., Palo Alto, Calif., 1956. $7.00. Reviewed by S. F. Singer, University of Maryland. The present volume is the sixth in the series and takes in nuclear physics from its astrophysical aspects to its biological aspects. The variations of primary cosmic rays are discussed by Sarabhai and Nerurkar with par- ticular emphasis on the special interest of the authors, the solar diurnal variation. No universally accepted ex- planation exists, but the accumulation of data on the time variations, particularly during the forthcoming In- ternational Geophysical Year, should advance our un- derstanding of their causes. The polarization of fast nucleons is discussed by Wolfenstein with emphasis in the region 100 to 400 Mev. The article develops a for- malism which may be used in the analysis of experi- ments with polarized nucleons. Heydenburg and Tem- mer treat the Coulomb excitation or electric excitation due to a passing charged particle of low-lying nuclear excited states. Excitation by electrons is briefly touched upon but the main portion of the article is devoted to heavy particle excitation and includes a brief discus- sion of the theory as well as an account of experiments in the field. In particular the interpretation of the ex- periments in terms of the electric quadrupole moments of nuclei is described. Mack and Arroe give a brief dis- cussion on the isotope shift in atomic spectra. Way, Kundu, McGinnis, and Lieshout have a lengthy paper on the properties of medium-weight nuclei giving much tabular material on their ground state, spins, magnetic moments, quadrupole moments, levels, and gamma-ray lifetimes. Home, Coryell, and Goldring present a short paper on generalized acidity in radiochemical separa- tions. Mattauch, Waldmann, Bieri, and Everling give a detailed discussion with much tabular material on the masses of light nuclides. Brooks gives a very topical and comprehensive paper on nuclear radiation effects in solids. It discusses the theory of atomic displacements and includes such items as thermal spikes due to in- tense heating in a region of atomic dimensions, phase changes, and cold working. The rest of the chapterdeals with particular materials, such as graphite, ura- nium. The final portion discusses damage to various solids: semiconductors, metals, valence crystals, and alkali halides. Taube discusses some applications of oxygen isotopes in chemical studies. Oxygen unfortu- nately has no radioactive isotopes which makes the problem rather difficult. Recent advances in low-level counting techniques is the subject treated by Anderson and Hayes and deals with advances in the techniques for beta counting (Cu and H3), gamma-counting, dou- ble beta-decay, and the problem of detecting the neu- trino. One of the longest chapters is on nuclear reactors for electric power generation by Davidson, Loeb, and Young. It discusses a great variety of power reactor designs, 27 of them. Of interest is the economic discus- sion at the end of the chapter which compares the cost per kilowatt for different installations. Values as low as $250 per kilowatt are mentioned. The longest chapter is on cellular radiobiology by Gray. Over 380 papers are reviewed, most of them published in 1955, iadicat- ing the tremendous activity in this field. The review covers the radiobiology of the cell including the influ- ence of various environmental factors and the genetic damage problem. The second part deals with the radio- biology of various tissues. O'Brien has a chapter on vertebrate radiobiology which deals with the effects of ionizing radiations on the embryonic development of fish, amphibia, birds, and mammals. Relaxation Spectrometry. By E. G. Richardson. 140 pp. (North-Holland, Holland) Interscience Publishers, Inc., New York, 1957. $5.75. Reviewed by J. G. Castle, Jr., Westinghouse Research Laboratories. In this pleasant little book, printed on soft white paper, Professor Richardson surveys the experimental spectrometry of acoustical relaxation. His historical dis- cussions of experimental work, including much of his own, serve to outline the bibliographies and to occa- sionally describe the cardinal sample configurations, but are not often detailed enough to support the author's conclusions. Certainly the discussions serve well to out- line the work in the various areas. After an appropriate introduction of concepts of re- laxational behavior and their illustration by models, the author covers in order spectra in the infrasonic, sonic, and ultrasonic regions. He points out the use of analog simulation of the physical sample's relaxation processes as a considerable aid in the parametric interpretation of observed relaxation phenomena. Then under Dielectric Relaxation he describes the strong similarity between viscoelastic behavior and dielectric behavior, conclud- ing with graphs showing the "concurrence" of the di- electric and acoustic relaxation spectra of glycerin at — 28° C. In the final chapter, on Spectrum Analysis, he points up some of the roadblocks and useful detours on the way toward resolution and shape studies on relaxation spectra. The book was read without conscious inspection for accuracy because the reviewer is not an expert in the PHYSICS TODAY

1.1721791.pdf

DirectCurrent Transients in Polymethyl Methacrylate and in Polystyrene Paul Ehrlich Citation: J. Appl. Phys. 25, 1056 (1954); doi: 10.1063/1.1721791 View online: http://dx.doi.org/10.1063/1.1721791 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v25/i8 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 26 Jun 2013 to 131.170.6.51. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions1056 LETTERS TO THE EDITOR fields. With the approximate solution the maximum is no longer smooth; also it is necessary to relax one condition. A double layer or a charge sheet must be permitted at x=d; the latter introduces considerable error, the former negligible error, so it is chosen. Using these boundary conditions, the expressions in the paper are obtained, except that the right-hand side of (3) above appears in the constants in the potential expressions; the space dependence of the inner potentials, the fields, and charge densities, and the computed adhesive forces are those of the paper. The energy gap is kT InK'; K, independent of either 'P or x. The deisred quantities, namely charge transferred and adhesive forces, are accurate to within a fraction of experimental error. The difference from the solution in the paper does not affect the adhesion to the metal, but takes account of a transfer of electrons between the dielectrics themselves; this equalizes Fermi levels in them and produces an electrostatic attraction between them if they are solids. 3. See Morant's Sec. 5: By the expressions, the maximum inner potential difference (with respect to the interface) attainable within even a 2-cm thick layer of adhesive at room temperature and minimum Xo is 0.75 v and reasonable values are 0.1-0.7 v. The energy gap in insulators is from 2 to 7 volts. However a more fundamental point is involved: the charge density decreases as the inner potential increases, and the Fermi level remains constant; in the general nonmetal, it is known that electrons are raised thermally to conduction levels only if the Fermi level moves up. Field emission requires much higher (externally applied) fields and a decreasing rather than increasing inner potential. Com mercial insulators and plastics are insulators precisely because precaution has been taken to avoid impurity levels in the intrinsic gap. Thermal contributions are not therefore a limitation on the inner electrostatic potential in insulators, though they may be in semiconductors. 4. With respect also to Morant's Sec. 5, mathematically it is possible to assume insulator resistance to be due to small charge concentration or small mobility. It is quite possible that the mobility is smaller than in semiconductors, but to attribute the small conductivity entirely or even in major part to smaller mobility raises conceptual difficulties. (1) Astronomical periods are required to reach equilibrium. (2) Even amorphous substances have molecular structure. Since polystyrene, for example, has 10--18 the conductivity of germanium, attributing the insulating value to poor mobility requires assuming an electron mean free path 10--18 that in germanium, i.e., less than 1()--22 cm. Such a value has little physical significance; even reasonable modifications of assumed electron temperature do not correct the difficulty. Electron concentration in the conduction level, however, depends exponentially on the intrinsic energy gap; a gap of 0.7 to 2 ev would be sufficient to explain, on the basis of charge concentration, observed insulating properties. KimbalF has shown that in FIG. 1. Energy levels after infinite time. diamond the gap may be as great as 7 ev. The question can, of course, only be settled by measuring both nand b by experimental methods suitable to insulating materials. 5. Morant's Sec. 3 appears to refer to the legend on Fig. 5 (b) (2) of (I). This is the charge density at x=d not at the metal; at the metal, n goes to No not 0, though at d it reaches a fraction of No. We agree with him that the present treatment does not explain frictional electrification, and did not make such claim. 6. The intention of the article was to point out that the transfer of electrons from a metal into a dielectric adhesive could bring about a significant electrostatic contribution to the measured adhesion. Such transfer does not depend upon the properties of the valence bonds or energy levels of the dielectric alone (which determine the intrinsic energy gap), but upon the energy gap at contact, i.e., between the Fermi level of the metal and the conduc tion level of the adhesive; therefore the force cannot be predicted from the measured chemical properties of the adhesive alone. It is our understanding that in at least two commercial research laboratories the subject is now being investigated in detail; the results of such investigations together with ours should yield useful data on contacts between adhesives or similar materials and metals, and help to determine the range of applicability of the present considerations. * With the support of the Aeronautical Research Laboratory. Wright Air Development Center. U. S. Air Force. t Since much of what follows was in manuscript when the preceding letters were received, their acknowledgment was to be incorporated therein. A delay in the second article makes it desirable to treat the points raised, directly. For lack of space much of the contents of correspondence to Dr. West and Mr. Morant is omitted and will be included elsewhere. The first article by the present authors is designated by (n. I Treated by v. Laue (as mentioned above); Richardson (references in Fowler. Statistical Mechanics); Fowler himself; Mott and Gurney, Elec tronic Processes in Ionic Crystals (Oxford University Press, London, 1948) second edition; and F. Borgnis. Z. Physik 100, 117 (1936). • S. M. Skinner, J. Appl. Phys. (to be published). • This is discussed more fully in the second article on the contributiont of electron atmospheres to adhesion. Skinner. Savage, and Rutzler, which will be submitted in the near future to J. Appl. Phys. 'The additional surface charge density at the metal due to the dis· continuous charge sheet is at most 4 percent and usually of the order of 10-3 or less. Ii The electrostatic component of adhesion at room temperature between a metal surface coming up in air to one side of a thin dielectric film already adhering on its other side to a metal surface will be far less than one micro dyne cm -., whereas the electrostatic component of the adhesion of the film to the first metal could be 600-1000 Ib in.--. • See R. H. Fowler and E. A. Guggenheim. Statistical Thermodynamics (Macmillan Company, New York, 1939), p. 486. 7 G. E. Kimball, J. Chern. Phys. 3, 365 (1935). Direct-Current Transients in Polymethyl Methacrylate and in Polystyrene PAUL EHRLICH* Diamond Ordnance Fuze Laboratories. t Washington 25, D. C. (Received February 17. 1954) QUANTITATIVE measurements of the dc transients in materials with dc conductivities of the order of 10-18 mho/em and less have not been reported previously. This com munication reports on such measurements made with a commercial Vibrating Reed Electrometer1 on two widely used high polymer insulating materials. The procedure used was that of plotting on an automatic recorder the charge accumulated on the internal capacitor of the instrument. The voltage across this capacitor was 1 volt maximum which was only a small fraction of the battery voltage. The effect of the measuring instrument on the normal current flow through the specimen was therefore negligible. The samples were molded disks ft in. thick with an effective electrode diameter of approximately 3 in. Data are presented for two different brands of commercial polymethyl methacrylate, a sample of polystyrene vacuum-polymerized at the National Bureau of Standards, as well as a commercial sample of the same polymer. Figure 1 is an illustration of the circuit used, Downloaded 26 Jun 2013 to 131.170.6.51. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsLETTERS TO THE EDITOR 1057 -----------------------------------... 20~~ vLJ I07Jl o 10 : ~ o r" o __________________________________ ..J FIG. 1. Circuit used in conjunction with Vibrating Reed Electrometer (V.R.E.) for the measurement of very low dc conductivities. Figures 2 and 3 present data of charging and of discharge currents in polymethyl methacrylate and in polystyrene as a function of time. It is seen that in all cases measurable absorption currents continue to persist over the entire time scale and that, even at the longest times, it is impossible to report steady-state conductivities. In one of the polystyrene samples the steady-state conductivity must be only a small fraction of the smallest trans ient conductivity observed, as can be seen from the identity of charging and discharge curves over the entire time scale.2 In the other two cases, the charging curves bend upward and away from the discharge curves at the upper end of the time scale, indicating that the steady-state conductivities have become a substantial fraction of the total conductivities. There is a significant difference in the data for the two samples of polystyrene; data obtained for a different commercial brand of polymethyl methacrylate, however, gave, within experimental error, identical results with the ones presented, except that the charging curves did not bend away from the discharge curves at long times, indicating the absence of any substantial steady-state conductivity. It is of interest to examine the connection between the measure ments reported and the ac loss factor. It is well known that a relation exists between the frequency-dependent dielectric constant .' (w) the loss factor ." (w) and the dc transient current, i(t).3 Von Schweidler4 has calculated apparent capacitances and resistances for a decay function of the form i(t)=Arn and .'(w), -16,------------------------, -17 ~ -18 ~ o U co o .J -19 1.0 2.0 3.0 Log Time (Seconds) 4.0 FIG. 2. Logarithm of the dc conductivity of a sample of polymethyl methacrylate plotted as a function of the logarithm of the time after application (0,0) and removal (.6.,';;7) of the potential. The second symbol in each case refers to a run made under reversal of the battery terminals. -17r----------------------- U :::J -19 ~ u co o .J -20 1.0 2.0 3.0 Log Time (Seconds) 4.0 FIG. 3. Logarithm of the dc conductivity of a commercial sample of polystyrene plotted as a function of the logarithm of the time after applica tion (0.0) and removal (.6.,';;7) of the potential. The second symbol in each case refers to a run made under reversal of the battery terminals. The filled in symbols refer to a sample of vacuum-polymerized polystyrene. as well as ."(w), can be immediately obtained from this develop ment with the following result for the loss factor:~ ."(w)= 1/C.[Go/w+,sCown-1r(1-n) cos(n"./2)], (1) where C. is the capacitance of a sample of air which has replaced the dielectric, Co is the capacitance of the dielectric at very high frequencies, Go is the steady-state dc conductivity, and A, ,s, and n are constants. It is readily seen that the loss factor may increase, decrease, or remain constant with frequency, depending on the exponent n of the decay function. By a suitable variation of the exponent through the time scale, the loss factor may be made to assume any of the shapes ordinarily observed. Hamon has shown' that, subject to certain requirements almost invariably satisfied, Von Schweidler's method can be made the basis of an approximation which is valid even where the exponent is not constant over the entire time scale. Hamon obtains ."~i(O.1/f)/2"'fC. V, which may be written as ."~G(O.1/f)/2"'f<o, (2) (3) where i(O.l/f) and G(O.l/f) are transient current and transient conductivity, respectively, at a time equal to one tenth of the recipricol cps frequency, V is the applied voltage and <0 is the permittivity of free space (8.854X 10-14 farads/em). Table I presents values of the loss factor for polymethyl metha crylate and polystyrene calculated from the transient measure ments by Eq. (3), and values at higher frequencies obtained TABLE I. Loss factor of polymethyl methacrylate and polystyrene. Polystyrene Polystyrene Polymethyl Vacuum- commercial methacrylate polymerized sample Fre- Fre- Fre- quency Loss quency Loss quency Loss (cps) factor (cps) factor (cps) factor 10'" 8.1 XIO-' 10--' 3.2 XIO-' 10-' II. XIO-' 6.7 X 10-3 9.3 X 10-3 10-' 2.0 XIO-' 10-' 3.6 X 10-' 10' 2.1 XIO-l 10' 2.5XIO-· 10' 2.8 XIO'" 10' 1.4 XIO-l 10' 2.5 XIO-' 10' 2.8 X 10-' Downloaded 26 Jun 2013 to 131.170.6.51. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions1058 LETTERS TO THE EDITOR directly on a Schering bridge. Barring a sharp dispersion region between 1()2 and 10--2 cps, polystyrene would therefore appear to undergo no very decided change in loss factor at room temperature over most of the interval considered. Polymethyl methacrylate, on the other hand, has to undergo roughly a 20-fold drop in loss factor in a little over four frequency decades. Judging from the behavior of this polymer at other temperatures and frequencies,' this value would seem quite high; nevertheless, polymethyl meth acrylate has been shown to have a 100 cps loss factor maximum at about room temperature" and a marked drop in loss factor on the low-frequency side of this maximum should, therefore, be expected. The author gratefully acknowledges helpful advice given to him by Dr. A. H. Scott and Mr. N. Doctor in setting up the measuring circuit. He is also indebted to Dr. L. A. Wall for provid ng the vacuum-polymerized polystyrene. * Present Address: Department of Chemistry, Harvard University, Cam-bridge 38, Massachusetts. t Formerly a part of the U. S. National Bureau of Standards. I Palevsky, Swank, and Grenchik, Rev. Sci. lnstr. 18,298 (1947) . • A de conductivity of 10 ..... mho/em for a very pure polystyrene 45 minutes after application of the potential has just been reported in another investigation. Warner, Muller, Nordlin, J. App!. Phys. 25,131 (1954). • K. S. Cole and R. H. Cole, J. Chem. Phys. 10,98 (1942). 4 E. Von Schweidler, Ann. phys. 24, 711 (1907). , B. V. Hamon, Proc. lnst. Elec. Engrs. 99, Part IV, 151 (1952). 'D. J. Mead and R. M. Fuoss, J. Am. Chern. Soc. 64, 2389 (1942). Optimal Loading of a Thermoelectric Generator JAMES F. PERKINS Convair, Fort Worth, Texas (Received March 3, 1954) TELKESI has derived an equation for the over-all efficiency of a thermoelectric generator, for which it was assumed that the maximum over-all efficiency would be obtained by matching the load resistance R. to the generator internal resistance R. Papet2 has pointed out that, while for a given generator the maximum power output can be achieved by matching impedances, the optimal operating condition is one corresponding to a maximum over-all efficiency and that this can be obtained by using a load resistance, R" somewhat greater than the internal resistance R of the tenerator. Following Papet's suggestion, Greeff3 has calculated efficiencies for various loading conditions. Some of the calculated efficiencies are in error by more than a factor of two, however, as a result of neglecting the Peltier effect. For instance, if one analyzes generator II according to the reasoning applied to cases III, IV, and V, one concludes that, since generator II is only 1/25 as large as generator I, it will absorb only 1/25 as much heat at the hot junction. Since the outputs of I and II are the same as we have: Efficiency of I1=25XEfficiency of I=25X2.34 percent = 58.8 percent. By comparison, the actual efficiency of II was calculated by use of Telkes' formula to be 20.9 percent. If one generalizes Telkes' equation to include the value of the load resistance, one obtains where Efficiency A = l/(Carnot efficiency) = TH/ (TH-Te). B 4.92X1O-8(TH+Te) e2(TH-Te) The condition for maximum efficiency, which is easily obtained by differentiating with respect to R./R, is (R./R) optimal = (A+B)!/B'. In the limiting case of A/B->O, i.e., the case of vanishing efficiency, the Peltier term does not contribute to the heat transfer. Hence, maximum efficiency corresponds to maximum output, and thus is obtained by impedance matching, R,=R. In Greeff's Case I, for example, we find A/B=1.546/1O.3=0.15, and (R./R)optimal=1.075. The limiting case A/B->oo corresponds to a generator whose thermal partial efficiency, as defined by Papet, is independent of Re. Over-all efficiency is thus proportional to electrical efficiency 71.= R./(R+R.) and is maximized by choosing R. much larger than R, i.e., (R,/ R)optimal-> 00. In Greeff's case II we find A/B= 1.546/0.412=3.74, R. optimal = 2.18R=4.36 ohms, EfficiencYoptimal = 24.0 percent as compared to Efficiency R. = R = 20.9 percent. At still higher efficiencies the improvement in efficiency obtained by proper loading becomes more significant. For a generator with e= 1000", V rC, operated at the temperature difference of 500°C considered by Greeff, we have Efficiency R. = R= 28.6 percent, EfficiencYoptimal = 39.0 percent, (R./ R)oPtimal = 4.0. 'M. Telkes, J. App!. Phys. 18, 1116 (1947). , R. M. Papet, J. App!. Phys. 19. 1180 (1948). 'M. B. Greeff, J. App!. Phys. 21, 943 (1950). Power Output of Thermoelectric Generators MARIA TELKES Research Division, College of Engineering, New York Unhtersity, New York. N. Y. (Received May 25, 1954) THERMOELECTRIC generators~theoretically~may offer numerous advantages, when compared with conventional electric power generators. There are no moving parts, nothing to wear out, if thermal deterioration at the hot junction could be avoided. The efficiency of the modern power generating stations is around 30 percent. The writer's calculations' indicated that such efficiencies may be approached, if thermocouple materials are available giving a thermoelectric power of 1000 microvoltrC (one millivolt per 0c), operating at a temperature difference of 500°C, provided that the thermocouple materials have "normal" Wiedemann-Franz-Lorenz relation. This relation is expressed by k/u=k·p=2.451O-8T, where k=specific heat conductivity, p=resistivity, u=specific electric conductivity, and T=absolute temperature. When the thermocouple is operated in a temperature range with Tk hot junction and Tc cold junction temperature, the corresponding k, u, or p values should be used. The writer has constructed numerous experimental thermo electric generators, operated at various temperature ranges. The results have been summarizedl and another article on "Solar Thermoelectric Generators" will appear in the near future.2 The highest efficiency obtained by the writer was around seven percent, using thermoelectric materials of 300 to 400 microvoltrC with moderate deviations from the normal Wiedemann-Franz Lorenz relation. In addition many materials have been used, some of them with thermoelectric power values in excess of 1000 microvoltrC (but unfavorable k·p values), but it was impossible to obtain higher efficiencies, except the values predicted by the theoretical calculation. Papet3 and Greeff4 published additional calculations indicating that the efficiency of thermoelectric generators could be increased by structural changes and by variations in the external load resistance, R. as compared with the internal resistance Ri. Anyone who has constructed thermoelectric generators would naturally wish to obtain the maximum power output and for this reason would change the R./R. ratio in the widest possible range. The writer has naturally done this, and found that the maximum power output could be obtained when R./Ri was 1, or between 1.05 and 0.95. Perkins,6 in a recent letter to the editor, calculated that the R./R. ratio change may increase the theoretically calculated efficiency of thermoelectric generators, provided that thermo- Downloaded 26 Jun 2013 to 131.170.6.51. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.1743011.pdf

Soft XRay Absorption Edges of Metal Ions in Complexes. I. Theoretical Considerations F. Albert Cotton and Carl J. Ballhausen Citation: The Journal of Chemical Physics 25, 617 (1956); doi: 10.1063/1.1743011 View online: http://dx.doi.org/10.1063/1.1743011 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/25/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Use of oneelectron theory for the interpretation of near edge structure in Kshell xray absorption spectra of transition metal complexes J. Chem. Phys. 73, 3274 (1980); 10.1063/1.440523 XRay KAbsorption Edge of Niobium in Niobium Metal and Its Oxides J. Chem. Phys. 52, 4093 (1970); 10.1063/1.1673616 Soft XRay Absorption Edges of Metal Ions in Complexes. III. Zinc (II) Complexes J. Chem. Phys. 28, 83 (1958); 10.1063/1.1744085 K XRay Absorption Edges of Cr, Mn, Fe, Co, Ni Ions in Complexes J. Chem. Phys. 26, 1758 (1957); 10.1063/1.1743624 Soft XRay Absorption Edges of Metal Ions in Complexes. II. Cu K Edge in Some Cupric Complexes J. Chem. Phys. 25, 619 (1956); 10.1063/1.1743012 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Sat, 22 Nov 2014 07:57:27THE JOURNAL OF CHEMICAL PHYSICS VOLUME 25, NUMBER 4 OCTOBER, 1956 Soft X-Ray Absorption Edges of Metal Ions in Complexes. I. Theoretical Considerations F. ALBERT COTTON* AND CARL J. BALLHAUSENt Mallinckrodt Laboratory, Harvard Uni~'ersity, Cambridge 38, Massachusetts (Received December 5, 1955) The effect of crystal field symmetry on the appearance of K x-ray absorption edges of metal ions in com plexes is considered. In particular, the splitting of the Is-4p absorption for first row transition elements is related to the splitting of the degeneracy of the 4p energy levels in the ion by the crystal field. INTRODUCTION THE intention of this and following papersl is to study the relation of symmetry and bond type in complexes of transition metals to the structure of the soft x-ray absorption edges. The central theme will be that in complexes where the interaction of the metal ion and ligands can be regarded as primarily ionic an adequate description of the essential features of the fine structure of the edge may be obtained by consider ing the crystal field splitting of the p orbitals of the metal ion. That this may be so has been suggested to us by the fact that the "visible" absorption spectra of such complexes can be accounted for by considering the perturbations of the partially filled 3d orbitals by the crystal field resulting from the surrounding ligands. THEORY The K x-ray absorption edge of an assembly of noninteracting identical atoms is assumed to result from the absorption by these atoms of x-ray quanta with the energies required to cause the Is (K) electrons to jump first to the lowest available p orbital, and with increasing energy to higher-lying levels which finally converge to a limit representing complete removal of the K electron from the atom. For argon gas the requirements of noninteracting atoms and good spectrometer resolution are well satisfied, and in this instance it has been shown by Parratt2 that the ob·· served edge structure is precisely in accord with the above description. For an isolated metal ion of the first transition series the K edge would begin with a sharp rise to a maximum corresponding to the Is-4p transi tion followed by a series of converging maxima corre sponding to Is-np (n= 5,6) transitions. In practice only metal ions in salts, complexes, and other com pounds can be studied and usually only the first strong maximum is well resolved. In some cases a second maximum probably assignable as Is-5p is discernible, but usually little or no other well-defined structure is observed until, in solids, the region of the Kronig structure is reached. * Present address: Denartment of Chemistry, Massachusetts Institute of Technology, Cambridge 39, Massachusetts. t Permanent address: Chemistry Department A, Technical University of Denmark, Copenhagen, Denmark. 1 Part II is the following paper. Parts III and IV will shortly be submitted to this Journal. 2 L. G. Parratt, Phys. Rev. 56, 295 (1939). We wish to determine the effect of the crystal field provided by the ligands of the complex upon the appearance of the Is-4p transition. If these ligands may be regarded as ions or dipoles whose major effect upon the metal ion is to cause perturbation of the energy levels of the metal ion, then we may attempt to predict the appearance of this absorption maximum by considering how these ions and/or dipoles split the 4p levels of the metal ion.t The expressions for the energies of the perturbed p orbitals are derived, to the first order in a manner quite similar to that of an earlier work.3 We shall be considering octahedral coordina tion. The zero-order wave functions for an unperturbed p electron are the following combinations of hydrogen like wave functions {1/V2i(1/;l+1/;_l) 'l'= 1/\12 ('/II-1/;-l)' 1/;0 (1) The subscript on the 1/; refers to the magnetic quantum number m, with 1/;m=R(r)Yml(o,~)-I~m~l, (la) where yml is a spherical harmonic normalized to unity and R(r) is the radial wave function also normalized to unity. The first-order perturbation matrix then contains the elements in which 1 00 I 4r ~I X=-L L D(m,q)-- (2r)~ l~ m~l 21+1 r>l+l t The appearance of the Is-4p absorption will not correspond with the splitting of the 4p band if the presence of the crystal field places other levels to which Is electrons may jump at energies near the 4/) band (vide infra). 3 C. J. Ballhausen, Kgl. Danske Vid. Selskab. Mat.-fys. Medd. 29, No.4 (1954). 617 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Sat, 22 Nov 2014 07:57:27618 F. A. COTTON AND C. J. BALL HAUSEN The perturbed energies are then EI,2(1)=t[Hl1+H_I-I]±HI-I E3(1)=Hoo. (3) In order to evaluate the integrals (2) containing products of three spherical harmonics, we utilize the tables in Condon and Shortley4 for the values of ck given by ck(lm,l'm') = (2/2k+1)t f'/l' o X8(k, m-m') 8 (I,m) 8 (l',m') sinOdO. It is also necessary to evaluate the functions of charge distribution denoted in (2) as D(m,q) and F(q). D(m,q) is a function only of the equatorial charges and of m in Eq. (2). We number the equatorial charges ql-q4, placing ql and q3 on the y axis, q2 and q4 on the x axis, and placing the remaining charges, q6 and q6, on the z axis as in Fig. 1 in reference 3. It is readily shown3 that { + if m=O D(m,q) = (ql+q3)± (q2+q4) _ if m= 2 F(q) = (q6+q6). (4) Inspection of Eq. (4) reveals that only three cases need be considered since insofar as the matrix elements depend upon the arrangement of charges (or dipoles) in the octahedral coordination sphere, it is only the sums of the charges on each axis which are important. We shall henceforth use Qx, Qy, and Qz to denote these sums. Upons solving Eq. (2) the matrix elements are found to have the following values, in all of which there should be a constant factor of 8/45(Z/3) which we have omitted for convenience. Hoo= (Qx+Qy)[GO-iG2]+Qz[GO+~G2] Hl1=H_I-I= (Qx+Qy)[G O+ioG2] +Qz[GO-iG2] (5) HI-I =H-11 = 130 (Qy-Qx)G 2 with These expressions are valid when the qi are point charges. For dipole ligands the qi are to be regarded as point dipoles, the common factor then being 8/45 X (Z/3)2 while each Gl is replaced by another related radial integral.3 . A qualitative prediction of the splittings can, of course, be made on the basis of symmetry alone. When all q/s are identical, the complex belongs to symmetry group Ok in which the p orbital transforms as TI,.. It will be seen from Eqs. (3) and (5) that more generally 4 E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, England, 1953). TABLE. 1. Symmetry group whenever Qx=Qy=Qz no first-order splitting occurs. If Qx=Qy;6-Qz the complex will have symmetry D4h if ql=q2=q3=q4;6-q6=q6 or symmetry C2v if ql=q2;6-q3 = q4 and q6= q6 (it is unimportant whether or not q6=ql or q3). In these two cases the representations of the p orbitals contain the following irreducible repre sentations shown in Table I. From Eqs. (3) and (5) it is seen that for either point group symmetry a splitting into only two levels separated by HQz-Qx)G 2 is to be expected. While for D4h the splitting predicted from symmetry is in agreement with that calculated, there might appear to be a contradiction in the C2v case. However, this is not so, for although not of course rigorously degenerate, the A I and B2 levels are ac cidentally degenerate in the crystal field approxima tion. § 6 Finally in case ql = q3, q2 = q4 and q6 = q6 with no other equalities present the complex has symmetry D2h and for the representation of the p orbitals we have r p=Blu+B2u+B3u with splitting into three sublevels to be expected. From Eqs. (3) and (5) the energy differences are found to be EI(1)-E2(1) = 2H1_1 =HQy-Qx)G 2 EI(I)-E3(1) =H11+HI_I-Hoo= HQy-Qz)G 2. In order to ascertain the order of magnitude of the splitting of the 4p orbitals the integration of the radial portion of the wave function was performed for the case of a four-coordinated square planar complex. Using reasonable values of the various parameters a value of ,,-,7 ev was obtained. The nature of this approximation is such that the value is probably a few volts higher than would be expected for octahedrally coordinated complexes. It may be noted that this value of 7 ev is about three times as great as the over-all splitting for a 3d electron under the same conditions. Thus far we have taken no account of possible interactions of the p level with other orbitals. For example, in cubic symmetry, the 41 orbitals will split into components of symmetry A2, TI, and T2• Since the p level in cubic symmetry transforms as T I there may be significant interaction under a crystal field of this symmetry if the 41 and 4p levels are not too far separated in energy. Similarly, under crystal fields of lower symmetry, 1 and p levels will again have com ponents with the same transformation properties which may interact. Owing to a lack of information about § This has been stated earlier6 by saying that so far as the crystal field symmetry is concerned both cis-and trans-isomers have "tetragonal" symmetry. 6 Basolo, Ballhausen, and Bjerrum, Acta. Chern. Scand.9, 810 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Sat, 22 Nov 2014 07:57:27SOFT X-RAY ABSORPTION EDGES.!' THEORETICAL 619 the energies of 41 levels in elements of interest here, we have not considered such effects but have preferred to wait and see whether forthcoming experimental data require considerations of this kind. Subsequently, experimental results for complexes of Cu (II) (Part II) and for other metal ions will be presented and analyzed on the basis of the theory developed here. In addition some results will be re ported for complexes in which the ligand to metal THE JOURNAL OF CHEMICAL PHYSICS bonding is presumed to be highly or even completely covalent. ACKNOWLEDG MENTS Thanks are due Professor Geoffrey Wilkinson for his interest and encouragement and Professor H. P. Hanson of the University of Texas for interesting dis cussions. This work was supported by the U. S. Atomic Energy Commission. VOLUME 25, NUMBER 4 OCTOBER. 1956 Soft X-Ray Absorption Edges of Metal Ions in Complexes. II. Cu K Edge in Some Cupric Complexes* F. ALBER.T COfTON,t Department of Chemistry, Harvard University, Cambridge 38, Massachusetts AND HAR.OLD P. HANSON, Department of Physics, University of Texas, Austin, Texas (Received December 5, 1955) It is shown that the crystal field splitting of the 4p orbitals of Cu (II) in some complexes can be correlated with the splitting of the 1s-4p transition observed in studies of the K absorption edges of these complexes provided the ligand-metal bonding is not appreciably covalent. ' INTRODUCTION THE gross features of x-ray spectroscopy such as the diagram lines are understood in complete detail. The situation for nondiagram lines is not quite so clear, but the explanation in terms of multiple ioniza tion seems to account for most of these satellites in a satisfactory fashion. However, our understanding of the radiation associated with phenomena near the ab sorption edge is still incomplete; this applies to both the absorption and emission processes. There are essentially only two types of experiments that have yielded results which seem amenable to simple and consistent interpretation. First, the K-ab sorption edge structure for argonl was explained in terms of the excitation of the is electron to np states of a potassium-like atom. Since there are no perturbing influences due to neighboring atoms, one expects that absorption will be restricted to ls-np transitions. Thus Parratt found that on analyzing the edge into such transitions, reasonable values for the transition proba bilities were obtained. The second type of experiment yielding results which are fairly predictable in terms of a general theory is the emission of very soft x-rays from the light ele ments, principally the metals.2 The valence electrons * Part I is the preceding paper. t Present address: Department of Chemistry and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge 39, Massachusetts. 1 L. G. Parratt, Phys. Rev. 56, 295 (1939). 2 H. W. B. Skinner, Repts. Progr. Phys. 5, 257 (1938). in a metal are presumed to occupy a band of energies in contrast to the discrete levels of the individual atoms. The theoretical predictions of the variation of the density of electronic states with the energy in the band and of the sharp cut off at the Fermi level are well verified in the x-ray emission spectrum. Thus one can explain with fair confidence the absorption spectrum of isolated atoms in the Angstrom range and the emis sion spectrum of solids in the hundred Angstrom range.3 The explanation of the K-absorption spectra of salts with edges of the order of angstroms in terms of solid state concepts has not been particularly successful. Several recent articles4 have discussed the structure of K edges as an example of exciton formation. This may be a perfectly valid approach to the problem, but it would seem more circuitous than necessary. First of all, the hole associated with the exciton is essentially im mobile since it is a K electron which has been excited. Furthermore, exciton levels are usually discussed in relation to the bands of the solid. In the energy range of K edges of the elements of the first transition series, at least, one finds that experimentally this relation ship is not an obvious one. 3 The interpretation of edges of the transition metals advanced in a series of papers by Beeman, Bearden, and Friedman [Phys. Rev. 56, 392 (1939); 58, 400 (1940); 61, 455 (1942)] undoubtedly have considerable validity. Since, however, lack of knowledge about transition probabilities does not permit one to analyze the edge structure into a plot of density of states versus wave length, one cannot be certain that all factors have been considered or explained. 4 See, for example, L. G. Parratt and E. L. Jossem, Phys. Rev. 97,916 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Sat, 22 Nov 2014 07:57:27

1.3059807.pdf

Physics of Semiconductor Surfaces H. K. Henisch Citation: Physics Today 9, 10, 48 (1956); doi: 10.1063/1.3059807 View online: http://dx.doi.org/10.1063/1.3059807 View Table of Contents: http://physicstoday.scitation.org/toc/pto/9/10 Published by the American Institute of Physics48 OPERATIONS RESEARCH ANALYSTS To apply training or experience in: Pure or Applied Mathematics Mathematical Statistics Management Science Econometrics Psychophysics Toward solution of challenging and widely diversi- fied military and industrial problems. Currently expanding activities include: Analytical Statistics and Design of Experiments Air and Road Traffic Control Studies Weapons Systems Evaluation Communication System Analysis Inventory and Production Control Equipment Reliability Analysis Counter measure Development These are full time positions offering salaries com- parable with those in industry and educational bene- fits for graduate study. Qualifications include ad- vanced degree, prior operations research experience,' familiarity with application of electronic computers. J. A. Metzger ARMOUR RESEARCH FOUNDATION of Illinois Institute of Technology 10 West 35th Street Chicago 16. Illinois ULTRA LOWcapacitance & attenuationWE ARE SPECIALLY ORGANIZED TO HANDLE DIRECT ORDERS OR ENQUIRIES FROM OVERSEAS SPOT DELIVERIES FOR U.S. BILLED IN DOLLARS- SETTLEMENT BY YOUR CHECK CABLE OR AIRMAIL TODAYTYPE Cl Cll C2 C22 C3 C33 C4 C44uuT/h 7.3 6.3 6.3 5.5 5.4 L 4.8 4.6 4.1IMPED.n 150 173 171 184 197 22O 229 252O.D. .36 .36' .44' .44' .64' .64* 1.03" 1.03' 'MXam/SM SUBMIMATURE CONNECTORS Conttant 50n 63n 70n Impedances TRANSRADIO LTD. 138A Cromwell Rd. London SW7 ENGLANDClllft Tilt SIMS.eetings Physics of Semiconductor Surfaces THE current interest in the surface properties of semiconducting materials has its origin largely in earlier attempts (e.g., 1938-47) to understand rectifica- tion phenomena at contacts between semiconductors and metals. To account for the lack of correlation be- tween rectification properties and the thermionic work functions of the surfaces concerned, Bardeen suggested in 1947 that the free surface of a semiconductor is as- sociated with a potential barrier arising from the pres- ence of surface states. Electrons accommodated in these states would act as an electrostatic screen and would protect the barrier (in varying degree, depend- ing on the density of surface states) from interaction with external charges. This would make the barrier in- dependent of (or, at any rate, insensitive to) differ- ences between the work functions of one contacting metal and another. Since then, surface states and asso- ciated phenomena have been the subject of intensive research, and the interest is no longer confined to con- tact rectification. Surface properties influence the per- formance and, notably, the stability of a variety of semiconductor devices, and independent arguments have established the importance of surface states in a whole range of catalytic phenomena and oxidation processes. The investigations are thus supported by important practical as well as theoretical interests. To survey the progress made during the last ten years or so, a "Conference on the Physics of Semi- conductor Surfaces" was held at the University of Pennsylvania, June 4th-6th, 1956. It was sponsored by the University, by the Office of Naval Research, and by the Lincoln Laboratory of MIT and was or- ganized by a committee under the co-chairmanship of J. L. Jackson, R. H. Kingston, and P. H. Miller, Jr. 123 research workers participated by invitation, in- cluding many from abroad. The detailed proceedings of the Conference are being published by the University of Pennsylvania Press under the general editorship of R. H. Kingston. The present report is intended as a summary and is, as always in such circumstances, in- escapably personal and necessarily incomplete. Since surface properties are sensitive to contamina- tion, two complementary lines of research have de- veloped. In the conference program these were desig- nated respectively as "clean surfaces" (free from con- taminants and approaching perfect structure) and "real surfaces" (as encountered in practice and covered at PHYSICS TODAY49 least by an oxide film). Most of the investigations have been carried out on germanium (though certainly not all of them) since the bulk properties of this material are reasonably well understood. The session on "clean surfaces" was under the chair- manship of J. Bardeen. C. Herring gave a general in- troduction to the theory of surface states, distinguishing between those which arise from surface imperfections and those associated with the perfect lattice. He dis- cussed the origin of Tamm and Shockley levels and dealt with some of the practical complications which have to be envisaged, e.g., strain and the interaction of surface imperfections with one another. Among the important problems under discussion was, of course, that of obtaining a really clean surface. This can be done, for instance, by cleaving a crystal in high vacuum, but experimental results of this kind were not available in time for the conference. Alterna- tively, a "real" surface can be subjected to a number of sputtering and annealing cycles, as described by R. E. Schlier and H. E. Farnsworth, who also showed how the resulting surfaces can be sensitively examined by means of low-energy electron diffraction. Some recent measurements of the thermionic work function, photoconductance, surface conductance, and (transverse) field effect in ultra-high vacuum were dis- cussed by P. Handler. A model was presented which ac- counted for the strong dependence of the observed effects on chemisorbed oxygen in terms of the un- filled germanium orbitals at the surface. R. H. Kingston then gave a brief historical review of the critical ex- periments which have served to establish our present picture of surface structure: the discovery of the in- version layer and injection, the demonstration of field effects, first studies of the oxide layer on germanium, and investigations of the surface conductance under static as well as transient conditions. The sessions on "real surfaces" were under the chair- manship of A. F. Gibson and H. K. Henisch. Studies of lateral conduction within the inversion layer of a barrier call for a knowledge of the effective carrier mobility. This important quantity has to be calculated, and J. R. Schrieffer reviewed his analysis of the scatter- ing and averaging problems involved. P. C. Banbury, G. G. E. Low, and J. D. Nixon dealt with the effect of capacitively applied fields on the surface conductance and surface recombination, with particular reference to the changes occurring within the first few hundred microseconds. The results have led to an evaluation of the capture cross section of surface states for majority carriers in w-type and p-type material. The surface states involved in these phenomena are designated as the "fast" states, to distinguish them from those which give rise to changes over a period of minutes or so. A. Many, E. Hamik, and Y. Margoninski described ex- periments in which the surface recombination velocity was measured as a function of barrier height, giving good agreement with theoretical expectations. Capture cross sections could again be evaluated. B. H. Schultz presented results on hole storage in a semiconductingANALYTICAL ENGINEERS At Hawthorne, in Southern California, Northrop Aircraft has a continuing need for experienced en- gineers seeking new opportunities. There are attractive positions open in the following fields: Aerodynamics, Dynamics, Thermodynamics, Stress, Loads, Performance Analysis. In Northrop's superbly equipped multi-million- dollar engineering and science center, now near- ing completion, you will be given constantly fresh and challenging assignments. Present programs include Northrop's new supersonic trainer air- plane, the Snark SM-62 intercontinental guided missile, plus advanced aircraft and missile pro- jects yet to be revealed. You'll be associated with a high-calibre engi- neering team that has established an outstanding record in aeronautical design and development. Your initiative and ideas will be recognized, en- couraged and rewarded, for at Northrop Aircraft the progress of personnel is as important as the progress of projects. Besides attractive remuneration, you will enjoy other benefits unexcelled in the entire industry- retirement plans, health and life insurance, col- lege educational reimbursement plan, regular vacations plus extra year-end vacations with pay. Easily-reached mountain, desert and beach re- sorts in sunny Southern California offer year 'round attractions for you and your family. You will find the career opportunity you are seeking at Northrop, pioneer in the design and pro- duction of all weather and pilotless aircraft. If you qualify for one of these attractive positions, con- tact the Manager of Engineering Industrial Rela- tions, Northrop Aircraft, Inc., ORegon 8-9111, Extension 1893, or write to: 1015 East Broadway, Department 4600 W, Hawthorne, California. NORTHROPNORTHROP AIRCRAFT, INC., HAWTHORNE, CALIFORNIA Produces ol Scorpion F-89 Interceptors and Snark SM-62 Intercontinental Missiles OCTOBER 195650 IScientists for basic research IPhysicists, Physical Chemists & Physical Metallurgists THE HONEYWELL Research Center offers physi- cal scientists with graduate training or equiv- alent experience the opportunity to do funda- mental research in semiconductors, solid state, and magnetic and dielectric materials. You will be encouraged to participate in planning the research program and to publish results of your individual research in professional journals. Typi- cal of the topics of interest: • electric conduction and electron scattering mechanisms • imperfections • infrared • surface studies • band theory of solids • ferrites and ferromagnetic alloys • high temperature materials • radiation damage • thermodynamics and differ- ential thermal analysis • thermoelectric properties • oxidation studies If you are interested in a career with a company whose sound growth is based on research, send a resume to Dr. Finn J. Larsen, Director of Re- search, Dept. PY-10-236, Honeywell Research Center, Hopkins, Minnesota. Honeywell (>cfcu>&- r LIAISON ENGINEER for Weapons Systems $12,000 Major manufacturer of highly engineeredelectronic equipment is organizing a team ofsystems evaluation engineers to support aweapons system evaluation program. 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Experiments of this kind, together with recently obtained information on the capture cross sections of surface states should soon make it possible to analyze the transient behavior of the surface recombination process in detail. There is as yet only partial agreement as regards the energetic distribution of these states, which implies some un- certainty as to whether the surfaces used in different experiments are really of identical structure. To clarify the problem of energy distribution, W. L. Brown, W. H. Brattain, C. G. B. Garrett, and H. C. Montgomery added capacitance measurements to the techniques more commonly employed, making use of the fact that the capacitance is not influenced by the carrier mobility. Capture cross sections of "fast states" were again deduced, in this case from measurements on transients performed throughout an oxidation cycle. H. Statz, G. A. de Mars, L. Davis, Jr., and A. Adams. Jr. reported experiments on the static and transient surface conductance of germanium and silicon and estimates for the energy distribution of the fast states which are also believed to be responsible for surface recombination. Certain interesting anomalies were ob- served in the presence of organic vapors. Experiments on the "slow states" which are thought to be located on the outside of the oxide film were reviewed by S. R. Morrison. The observed time relations are generally nonexponential and an analysis can be made in terms of the barrier height which governs the probability of charge transfer from the bulk material to these "slow states". The time constants involved have been studied in detail by M. Lasser, C. Wysocki, and B. Bernstein and were found to increase with increasing oxide thick- ness. The surface can in fact be stabilized by growing a thick oxide film. Problems connected with noise gen- eration at semiconductor surfaces were reviewed and discussed by A. L. 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The presence of impurities and of surface imperfec- tions, e.g., places where dislocations terminate, plays an important part in the formation of oxide nuclei, as, indeed, it does in etching treatments and perhaps also in certain forms of catalysis. M. Green, J. Kafalas, and P. H. Robinson described experiments in the course of which germanium crystals were first crushed in high vacuum and in which the subsequent oxygen uptake was kept under observation as a function of time, tem- perature, and pressure. The results have led to a pro- posed model for the structure of the first oxide layer. Its quantitative confirmation depends on the accuracy with which surface areas can be measured by inde- pendent methods, e.g., krypton adsorption. J. T. Law and P. S. Meigs have studied the oxidation rate of dif- ferent crystal faces on germanium and have found that a protective surface film is formed after a few minutes at 700°C, after which the oxygen uptake ceases. 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1.1731152.pdf

Osmotic Pressure of Moderately Concentrated Polymer Solutions Marshall Fixman Citation: The Journal of Chemical Physics 33, 370 (1960); doi: 10.1063/1.1731152 View online: http://dx.doi.org/10.1063/1.1731152 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/33/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Osmotic pressure of isotropic solutions of rodlike polymers J. Chem. Phys. 118, 3904 (2003); 10.1063/1.1539843 Osmotic pressure of ring polymer solutions: A Monte Carlo study J. Chem. Phys. 113, 11393 (2000); 10.1063/1.1326908 Osmotic pressure of ringpolymer solutions J. Chem. Phys. 87, 4201 (1987); 10.1063/1.452924 Theory of Moderately Concentrated Polymer Solutions J. Chem. Phys. 43, 1334 (1965); 10.1063/1.1696924 The Flow of Moderately Concentrated Polymer Solutions in Water Trans. Soc. Rheol. 8, 3 (1964); 10.1122/1.548976 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28THE JOURNAL OF CHEMICAL PHYSICS VOLUME 33, NUMBER 2 AUGUST, 1960 Osmotic Pressure of Moderately Concentrated Polymer Solutions MARSHALL FIXMAN Mellon Institute, Pittsbltrgh, Pennsylvania (Received March 17, 1960) ~y t'"':o ~eth.ods, a Iin.eari~ation and a va~ational principle, the Born-Green-Kirkwood equation for the radial distributIOn functIOn IS solved approximately and the osmotic pressure of chain polymer solutions ~omputed at arbitrary concentration. The gaussian intermolecular potential energy of Flory and Krigbaum IS used, and this restricts the range of validity of the theory to volume fractions less than one-tenth. It is shown how the distribution of polymer molecules in the solvent becomes random as the concentration is increased. For good solvents, the quantity [(P/c2)-RT/Mc], where P is the osmotic pressure and M the molecular weig~t, is predict~d to incre~se rapidly with concentration c, and then to level off rapidly, the whole ~ffect bemg accom?hs~ed .at qUite low concentrations as the molecules are forced to overlap. Some experimental corroboratIOn IS displayed. Severe doubt is cast on the practicality of the virial expan sion of P, and possibly=on the validity, beyond quite low concentrations. I. INTRODUCTION "S the first stage in a calculation of the changes in .ft polymer dimensions with concentration, and for its own interest, we present here a discussion of the osmotic pressure of chain polymer solutions of moderate concentration. As the concentration of polymer molecules increases through the range of interest here, a particular polymer molecule begins to overlap the domain occupied by other molecules until at sufficiently high concentrations the solution satisfies one of the assumptions of the Flory-Huggins theory, and the centers of the polymer molecules become almost randomly distributed in the solvent. The behavior of thermodynamic or dimen sional properties in the transition range of concentration is exceedingly interesting both from the experimental and theoretical side.! The bearing on polymer science needs no comment, but the model of polymer interac tions in dilute solution, devised by Flory2 and to be applied here, also furnishes a very interesting applica tion of the general statistical mechanics of fluids and solutions. The intermolecular potential is very soft and weak as measured by the usual standards of this field. The centers of two or more molecules may occupy the same volume element with appreciable probability, and moreover, an increase in concentration, instead of leading to a more ordered structure, as is common, may produce a more random distribution of molecules! In the model we use, the polymer molecule is pictured as a continuous, spherically symmetric distribution of segments, the average number of segments in a unit volume being peaked at the center of mass and spherical 1 L. Kotin, Doctoral Dissertation, Harvard Cniversity 1959. Kotin has given an extensive discussion of the observed ~ffects, and several approaches to the theory. Occasional observations of apparent discontinuities in the rate of change of thermodynamic or dimensional properties with concentration have been made. In this work we shall attempt neither a critique of nor extensive comparison with experiment. 2 P. J. Flory and W. R. Krigbaum, J. Chern. Phys. 18, 1086 (1950). symmetry of the molecule. The interaction energy of one polymer molecule with another is assumed to be an integral over an energy density, which is in turn as sumed to be proportional to the product of segment densities (number of segments per unit volume), at each point. The assumption by Flory and Krigbaum that the segment density is a gaussian function of the distance from the point of observation to the center of mass, has as its consequence that the intermolecular potential is a gaussian function of the distance between molecular centers. It is this potential that we shall use in subsequent calculations. When we speak of intermolecular potentials here, it is no doubt clear that we refer to potentials of average force, the average being with respect to the configura tion of all solvent molecules. Many molecular formulas for thermodynamic functions of single species systems may be used in the theory of mixtures if the potential of "instantaneous" force is replaced by the potential of average force. This was demonstrated by McMillan and Mayer3j see also Hil1.4 It is true of the cluster expansion of the pressure, in particular, which becomes the osmotic pressure under the transformation from intermolecular potentials to potentials of average force evaluated at infinite dilution. Unfortunately, it is not true that the potential of average force between solute molecules is in general independent of solute concentration. The nature and distribution of the solvent molecules implicitly affects the potential of average force, and the distribution will change with solute concentration. We shall however assume as a first approximation· that this ~otential i; independent of concentration and pairwise additive, 3 W. G. McMillan and J. E. Mayer, J. Chern. Phys. 13 276 (1945). ' 4 T. L. Hill, Statistical ),[ecitanics (McGraw-Hill Book Com pany, Inc., New York, 1956), Chap. 6. 6 The second approximation, which we only anticipate and do not use here, will take into account the change in potential caused by the change with concentration of polymer dimensions. The latter change is in !a.ct .our. ultimate object, and presumably may be evaluated by mlmmlzatlOn of the free energy of the solution. 370 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28OSMOTIC PRESSURE OF POLYMER SOLUTIONS 371 and we may expect the assumption to be fairly good up to some finite concentration. Both experimental and theoretical arguments may be adduced a priori for the belief that the limit of validity is a solute volume fraction of about 0.1, and the quantitative comparison with experiment seems to confirm this expectation. An experimental observation in favor of this limit comes from the behavior of osmotic pressure as a function of concentration under conditions such that the second virial coefficient vanishes. For an intermolecular po tential which is a monotonic function of intermolecular separation (the kind we consider), the second virial coefficient may be made to vanish only by making the potential vanish, and in this situation all higher virial coefficients vanish. But in practice6,7 the osmotic pressure in these solvents begins to deviate noticeably from ideal solution behavior at volume fractions of about 0.1. On the theoretical side, we refer to Flory's derivation of the intermolecular potentia1.2 The energy density was calculated from the Flory-Huggins free energy of mixing, with the suppression of terms arising from the entropy of mixing solute molecules (these being at specified positions). In the course of the derivation an expansion of the entropy term In(1 -V2), where V2 is the local solute volume fraction, was made In(1-v2) = -t'z -(vN2) -(vN3) -""". Terms of order vz3 and higher were suppressed.8 It may be supposed as a very rough consideration that this assumption is adequate at a bulk volume fraction V2 such that (2V2/3) «1, and that it breaks down when the inequality becomes feebler, at say V2>0.1. In sum, we here take the intermolecular potential VCR) to be independent of concentration, pairwise additive, and VCR) =kTX exp( -BR2) , ( 1) where a relating of the parameters X and B to the dimensions of the polymer chain and to the segment segment excluded volume will be deferred to part V. With the choice of potential established, there now remains the choice of a statistical mechanical method of calculation. Two broad categories are lattice model calculations, and integral-differential equations for the radial distribution function g(R) that are based on the superposition approximation. We have excluded lattice calculations for three reasons: first, we are interested in concentrations sufficiently high that a restriction to nearest neighbor interactions only would not suffice, and so the major simplification of lattice calculations is lost; secondly, because the possibility of rapid changes in thermodynamic properties in the transition concen- 6 W. R. Krigbaum and D. O. Geymer, J. Am. Chern. Soc. 81, 1859 (1959). 7 P. J. Flory and H. Daoust, J. Polymer Sci. 25, 429 (1957). S See T. A. Orofino and P. J. Flory, J. Chern. Phys. 26, 1067 (1957), where the second virial coefficient is discussed without this assumption. tration range would seem so innate in a lattice model that their finding could easily be regarded as an artifact of the calculation, and third, because of the well-known inadequacy of lattice models at very low concentra tions.9 From the radial distribution function approaches we select the Born-Green-Yvon-Kirkwood4 method, which begins with kT\\g( 12) +g( 12) v\ V (12) +p jg(l, 2, 3)v\V(13)dR a=0, (2) an equation rigorous under the assumption of a pairwise additive potential. Equation (2) relates the radial distribution function g(12) to the triplet correlation function g(l, 2, 3) for assigned number density p (polymer molecules per unit volume), and intermo lecular potential VCR). In Eq. (2) the integers 1,12 stand for R1, R12=R2 -RI, and so on. The equation is purely formal until the superposition approximation g(1, 2, 3) =g(12)g(23)g(13) is employed.lO Its use in Eq. (2) yields kT"V1lng( 12) + "VI V (12) (3) +p f g(13) g(23)"VI V(13)dRa=O, (4) which is sufficient to determine g(R) and through it the thermodynamic properties. The exact solutions of Eq. (4) seem to be quite good, though not perfect determinations4 of g(R); but to obtain these exact solutions is a'numerical problem of the first magnitude and does not seem warranted here. We will look into two other possibilities. First, Eq. (4) may be line arized,1l,12 and when so modified, its solution becomes tractable (after a few other approximations). We will follow the linearization procedure in part II, but we originally regard the method with distrust in the context of this work. For the basic assumption of the linearization method is that g(R) ",-,exp[ -V(R)/kT], the latter being the exact solution at low concentrations, whereas we wish to be able to follow g(R) through the concentration range in which the molecules become randomly distributed, that is g(R)",-,l for all R. We desire, at least, that this be a possible outcome of the calculation, and the linearization would seem to exclude it. However, the actual outcome of part II is qualita- 9 See work cited in footnote reference 4, Chap. 8. 10 It should be understood that the additivity of average forces has been assumed in two different connections: first for V, where the average is with respect to the positions of all solvent molecules, and second in Eq. (3), where the additional average is over the positions of all but two solute molecules. II M, Born and H. S. Green, A General Kinetic Theory of Liquids (Cambridge University Press, New York, 1949), Chap. II. 12 A. E. Rodriguez, Proc. Roy. Soc. (London) A196, 73 (1949). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28372 MARSHALL FIXMAN tively quite similar to the result of the more extended calculation of part IV, probably because of the lineariza tion of exp( -V/kT) in part II. Our second approach to Eq. (4) entails the construc tion of a variational principle equivalent to Eq. (4), and use of a suitably simple trial function for its solu tion. The trial function that we use is unrealistic for large X, and the number of variational parameters in g(R) is restricted to two for small X, and one for large X. Nevertheless, the trial g(R) is most strained, and the thermodynamic properties most sensitive to the choice of parameters in the very low concentration range, where the calculation of the second virial coefficient2 is available as an encouraging check. The calculations are carried to the numerical stage only for positive deviation of the osmotic pressure from ideal behavior, that is X;::::O. II. LINEAR THEORY A. General Derivation The original derivationl1•12 of the linear approxima tion to Eq. (4) proceeds through an equation derived from Eq. (4) by a very complicated integration by parts. As far as the requirements of the linear theory go, this was a pointless detour. By an application of the desired approximations directly to Eq. (4), the linear theory can be derived almost as quickly as the approxi mations can be stated. To demonstrate this, and to make this section self-contained, we here proceed with such a derivation. Put g(R)=exp[j(R) -V(R)/kTJ (5) a(R)=[exp -V(R)/kTJ -1, (6) as the definitions of feR) and a(R), respectively. The motivation of Eq. (5) is the hope that feR) is small, so that when the definitions are entered into Eq. (4) the expansion g(R)::[1 +a(R) J[1 +f(R) ] (7) may be used on the rhs. The substitutions of (5), (6), and (7) being made, Eq. (4) becomes Vrf(12) -p / [1+f(13)][a(23) +lJ X[1+f(23)]V 1a(13)dR 3=0. (8) A further approximation is now introduced, motivated by the desire that Eq. (8) be simply soluble by Fourier transformation, which is possible only when the integral has the form of a convolution; that is, the integrand is a product of a known or easily evaluated function of R13 and f(23). This form is obtained by the assumption that wheref(R) is multiplied by a(R) or its derivative, these being very short range functions, feR) may be replaced by the constant t -1, an average value in the neighborhood of the origin f(R)~€ -1 = jf(R)a(R)dR / / a(R)dR. (9) This step and the observation that because a(R) is spherically symmetric, short range and bounded, bring Eq. (8) to the form Vlf(12) -pel [j(23) +ea(23) ]Vla(13)dR3=O. (to) Fourier resolutions of feR) and a(R) are now intro duced into Eq. (to) f,.=(27r)-3/ feR) exp-ik·RdR; feR) = j fk expik·Rdk (11) a(R) = jak expik·Rdk (12) are the definitions, while j f(23)a(13)dR3= (27ryj flA!k expik.Rl~k. (13) Substitution of Eqs. (11), (12), and (13) into Eq. (10) gives The factor in curly brackets must be zero to within an arbitrary additive multiple of a () function of k, since j{)(k)k expik·Rl~k=O. (15) But a 0 function in fk contributes a constant term to fCR), as may be seen in Eq. (11), and this constant term must be zero, since g(R)~l as R~oo. There fore, the arbitrary multiple of ()(k) is a zero multiple and Eq. (14) gives fk=81i3p(Eak)2/(1-81i3PEak)' (16) Equations (7) and (16) constitute the Greenll Rodriguez12 linear approximation to g(R). B. Application to Chain Polymers We will first evaluatefk explicitly, and then use it to obtain the osmotic pressure. Even with all the approxi- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28OSMOTIC PRESSURE OF POLYMER SOLUTIONS 373 mations made to obtain Eq. (16), a computation of the transform ak with VCR) given by Eq. (1) cannot be effected in finite terms. Consequently we employ the device used by Stockmayer and Casassa13 in their calcu lation of the third virial coefficient. They make the approximation (17) and for given X and B choose Xo and Bo so as to give the correct second virial coefficient, and by a graphical comparison, an adequate similarity of a(R) to ao(R). The graphical agreement is thought to be adequate for X <4. From Eqs. (12) and (17), ak= -Xo(27r)-3! exp( -BoR2 -ik·R)dR. (1S) The integration is effected by completion of the square in the exponential, and gives Next in Eq. (16) we must get E as a function of p, XG, and Bo. We substitute in Eq. (9) ja(R)dR= -Xo(1r/Bo)l, (20) and from (11) and (12), we get j !(R)a(R)dR= ! !kOf.(R) expik·RdRdk which gives E-1= -8(7rBo)IXOljak!I:dk. (22) Substitution of Eqs. (1S) and (16) into (22) gives E -1 = (pE2XNSBo3) 1"" 41rk2 exp( _k2/4Bo)dk o 1+tpXo(1r/Bo)! exp( -k2/4Bo)' (23) which has to be solved for E as a function of p, Xo, and B. The following scheme accomplishes this, though only numerically. This, however, is sufficient since the calculation is now at a late stage. Put 'Y= (1r/Bo)'EpXO, and also in Eq. (23) change variables to k=2Botx. (24) 13 W. H. Stock mayer and E. F. Casassa, J. Chern. Phys. 20, t560 (1952). Equation (23) becomes or where e -1 = (4eXo!1ri)-yI("Y), I(-y) =[OOx2 exp -3x2dx. o 1+yexp-x2 (25) (26) The procedure now is to obtain I (I') in terms of tabu lated functions, tabulate t as a function of "y for a specified Xc using Eq. (25), and then divide the values of I' by EXo to get from (24) Vo= ('Y/EXo) = (7r/Bo)tp. (27) The result amounts to a tabulation of E against the dimensionless measure of concentra tion Vo for the desired XG. First, then, expand the integrand of I (1'), Eq. (26), in partial fractions, 'Y exp( -x2) ] + x2dx. (28) 1 +1' exp( -x2) The first two terms integrate readily, and in the last term, the substitution x2=y permits the integral to be written as a tabulated function where ~=ln'Y Fp(~) =1'" yPdy. (30) o 1 +exp(y -~) The functions F p (.i') occur in the Fermi-Dirac statis tics,14 and have been tabulated by McDougall and Stoner15 and Beer et al.16 The numerical completion of the procedure described in the foregoing furnishes!k, Eq. (16). We now compute the osmotic pressure P from4 P=pkT _(21rp2j3) £,0 R3g(R)aV(R)joRdR (31) =PkT{ 1+ (21rpj3) LX> R3(l+!(R)]aa(R)/oRdR}, (32) the second form deriving from Eqs. (6) and (7). We may note that the status of Eq. (31) for the osmotic 14 A. H. Wilson, The T~eory oj Metals (Cambridge University Press, New York, 1953), p. 332. 15 J. McDougall and E. C. Stoner, Phil. Trans. Roy. Soc. (Lon don) A237,67 (1938). 16 A. C. Beer, M. N. Chase, and P. F. Choquard, Helv. Phys. Acta 28, 529 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28374 MARSHALL FIXMAN pressure is somewhat lower than the equivalent form for the hydrostatic pressure, as Eq. (31) requires4 pairwise additivity of the potentials of average force V(Rii) . The problematical integral in Eq. (32) is the one involving f(R). It may, however, be reduced by a procedure analogous to that used for € L= j R3f(R)aOl(R)/aRdR = (41f')-1/f(R) R· \7ROI(R)dR after introducing the Fourier representation of OI( R). The divergence theorems may be used in k space to rewrite the k integral, giving L= _(41f')-1j[301k+aOlJ./akJf(R) expik·RdkdR, or with Eq. (11) L= -2r ![301kfk+klkaOik/akJdk. (34) Equation (22) provides the first term on the rhs, in terms of E -1. In the second term, with Eqs. (16) and (19) for fk and OIk, respectively, the identical changes of variables used subsequent to Eq. (22) here give where or -2r jkfkaOik/akdk= -2€X 02Bo-!J('Y), (35) () jx4 exp -3x2dx J 'Y ='Y 1+'Y exp -:x;2' (36) J ('Y) = (31f'i/8'Y) (2-5/2'Y -1) + (1/2'Y2) FiG"), (37) where FJG") is defined in Eq. (30). We now put all the pieces, namely Eqs. (33), (34), (22) [to be used in (34)J, and (35), back into Eq. (32) for P p-1[(P/pkT) -lJ =€Xo(1f'/Bo)!(1/2)[1-8X oJ('Y)/31f'i]. (38) For comparison of Eq. (38) with experiment or with the results of the variational calculation, it will be convenient to introduce the concentration c in g/ml c=Mp/N o, (39) where M is the polymer molecular weight and No is Avogadro's number. Furthermore, we define an "ap parent second virial coefficient" S as S=[(P/c) -(RT/M)J(RTc)-1 j (40) the verbal description of S arises from the fact that (41) the second virial coefficient. Equations (38) and (40) give S=N€Xo(1f'/Bo)t(1/2M2)[1-8X oJ('Y)/3!]. (42) We will not compare Eq. (42) directly with experi ment, but we will compare it with the variational calcu lation for a value of X which has accidentally turned out to be close to an X used in the experimental com parison in part V. We take Xo=V2, which corresponds, according to Stockmayer and Casassa,13 to X=4 in Eq. (1). From their paper the requirement that the exact and approximate a's give the same A2 results in (Bo/B)!=0.579 for Xo= 1.4, so that the dimensionless variable v=p(1f'/B)i used in part V, is v=0.579vo. (43) (44) (45) We now compare S/ A2 for the two methods. Equa tions (41) and (42) give S/ A2={1-8X oJ('Y)/31f'!]. (46) From Eqs. (25), (29), and (37) we have tabulated in Table I, E and (S/ A2) for a range of 'Y, and from Eqs. (27) and (45), the corresponding values of v. As an incidental result, the differences between € and S/A2 give a rough measure of the validity of replacing f( R) by an average value where it multiplies OI(R) or OI'(R) in Eq. (8); if the same substitution were made for thef occurring in Eq. (32·) for P, it would turn out that S/A2=€. In Fig. 1 is a graph of S / A2 vs v from Eq. (-16), curve (3), and from the results of part V, curve (2). The latter was obtained from Table 11117 in part V at A = X 1f'i /4 = 1.8. We also present in Fig. 1 the osmotic pressure expansion broken off at the term containing the third virial coefficient. The Stockmayer-Casassa equations for A3 give, in general, or, with Eq. (45) and Xo=V2, we have for curve (1) (49) Several conclusions can be drawn from Fig. 1. (1) The linear theory gives the correct third virial coefficient. 17 To fill out the curve, the table was extended to a few smaller values of v by the procedure described at the end of part IV. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28OSMOTIC PRESSURE OF POLYMER SOLUTIONS 375 This is not surprising; it is known18 that the linear theory does in general give the correct A3, but the numerical verification is a good check on lengthy calculations. (2) The variational calculation is in quite good agreement with the linear theory in the region of small II. This is surprising, and we should understand clearly what it means. The variational calculation gives an A2 about 2% too large for A = 1.8, as is shown by comparison with the Flory-Krigbaum theory2.8 (see Table IV). The consequence of Fig. 1 is that according to Eq. (47), the variational calculation also gives A3 about 2% too large, and, in sum, A3/ A2 correctly, if we accept the A3 of Stockmayer and Casassa. (3) At a quite low concentration, the virial series broken off at As becomes a grossly inadequate description of the osmotic pressure (11= 1 corresponds to a concentration of about 2g/100 ml for the experiment considered in part V). (4) At large II, the linear theory makes S/ A2 much too small, the variational calculation makes S/ A2 slightly too small. We assert this because the variational calculation must give S( 00) correctly if the calculation has any merit at all; the value of S( 00) involves the results of the calculation only to the qualitative extent that g(R)---tl, all R, as II---too. On the other hand, A2 is made about 2% too large; hence S( ::t::J ) / A2= 1.65, according to part V, is 2% too small, and S( 'Xl) / A2= 1.5 according to the linear theory, is 10% too small. The quantitative defects of the vi rial series through A3 suggests to us that the virial series itself has a finite radius of convergence that may be exceeded experi mentally. Of course we cannot prove this assertion, because we don't know the true vi rial coefficients, but it is easily shown that the linear theory assigns a finite radius of convergence to a virial expansion. We examine, TABLE 1. Linear theory. Average .-1 of J( R) near the origin, and "apparent second virial coefficient" S tabulated against di mensionless measures of concentration 'Y and P. Potential energy parameter X =2'. 'Y p SjA2 -------- 0.1353 1.035 0.053 1.023 0.6065 1.136 0.219 1.087 1.105 1.217 0.371 1.136 2.014 1.328 0.621 1.199 3.669 1.459 1.029 1.270 6.686 1.596 1. 714 1.338 12.18 1. 719 2.899 1.394 40.45 1.887 8.764 1.463 co 2 1.5 1.6 1.5 2345678 u FIG. 1. Apparent second virial coefficient S divided by S (0) = A2, vs dimensionless measure of concentration p. X=2'. Curve (1) third virial coefficient. Curve (2) variational theory. Curve (3) linear theory. in Eq. (42), a series expansion of J( 1'). From Eq. (36), J(-y) =1' t Jx4( -'Y)n exp -(3+n)x2dx n=il = (3'Y1I'i/8) t( _'Y)n/(3+n)512• (SO) o Consequently the series diverges at l' = 1. The same radius applies to E, in Eqs. (25) and (26). A value of l' = 1 corresponds, for the example considered here (X =.4), to 11"-'0.34, as Table I shows. Nothing very remarkable occurs at 11=0.34 in fig. I, and for just this reason an attempt to fit the lower part of curve (2) by a virial series would yield virial coefficients in severe disagreement with the values actually predicted by the linear theory. Two remarks are pertinent in conclusion. First, the comments on the divergence of the virial series have relevance only for fairly good solvents. For sufficiently poor solvents the divergence will occur at a concentra tion greater than the anticipated limit of validity of the original potential, Eq. (1). Secondly, the divergence might well be an artifact of the linearization procedure; we indicate a question rather than claim an answer. III. VARIATIONAL PRINCIPLE Our reasons for discontent with the linear theory have been mentioned at the end of part I and in various places in II. In brief, the linear theory is a priori unreliable except at very low concentrations (though the suspicion is relieved by the qualitative agreement with part IV), numerically tedious, and most im portantly, because of the approximate potential we were forced to use, subjective in the relation between (Xo, Bo) and (X, B). Stockmayer and Casassa thought the approximate potential so poor for X> 4, that they abandoned its use. Here we lay the basis for our varia tional calculation, but as such an approach may have other applications, we discuss it more fully than our 18 G. S. Rushbrooke and H. I. Scoins, Phil. Mag. 42, 582 (1951). particular application might warrant. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28376 MARSHALL FIX MAN The point of departure is again Eq. (4). The first step is the removal of the differential operator v'l from the equation. To this end define Q(13) by g(13)V IV(13)=v\Q(13). (51) It is readily verified that iRIS Q(13) = ex> g(R)aV(R)/aRdR (52) suffices in the definition (51); the lower limit is arbi trary, but it proves convenient to have Q( 00) =0. Use of (51) and (52) in Eq. (4) allows VI to be factored from the integral in (4), and subsequently from the whole equation. These operations yield kT Ing(12) + V(12) +p f Q(13) g(23)dRa=K, (53) where K is a constant of integration which is found from Eq. (53) and the boundary conditions: 1 -g(R) and V (R) approach zero as R-H$;) , (54) During the variational procedure Q (13), and therefore K, will be kept constant, so it is interesting to note that K is linear in the pressure (here, osmotic pressure); from Eqs. (52) and (54), an integration by parts gives K = -(41rp/3) f RI33aQ (RIa) / a RladRI3 = -(41rp/3) f R3g(R)aV(R)/aRdR, (55) or K = 2 (P -pk T) / p, (56) by comparison of Eq. (55) with (31). There is no unique variational function for a given equation, and although variational functions can be constructed by formal procedures,19 we will simply present such a functional Wand demonstrate its suitability. Let W=kT fg(12) Ing(12)dR I+ f V(12)g(12)dR I + (p/2) f Q(13) (g(23) -1)g(12)dRadRI. (57) Possible variations og of g are subject to the constraint (58) which we impose to maintain g( 00) = 1. We assert that oW =0, together with Eq. (58), yields Eq. (53) if 18 P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Chap. 9. Q(13) is treated as known and not subject to variation. Variation of W gives oW = f[ kT Ing(l2) + V(l2) +p f Q(13)g(23)dR 3 +kT+M -K/2]og(12)dRl (59) with the aid of an interchange of the dummy variables in the double integral of (57); M is a Lagrange multi plier for Eq. (58). Now if the variations in g(12) are subject to completely arbitrary variation, it follows from oW = 0 that the coefficient of og( 12) in (59) is identically zero. Thus, Eq. (53) is recovered, after imposition of g( 00) =1 gives M= -(kT+K/2) for the Lagrange multiplier. Therefore, W is an adequate variation function. If g(R) is not subject to arbitrary variation, but has an assigned functional form with several parameters an, it is no longer true that oW =0 implies (53). Rather, by the usual application of a variation procedure, and .. oW =0 and Eq. (59) give the "best" values of the parameters from j[kT Ing(12) + V(12) +p j Q(13)g(23)dRa -K] X (ag(12)/aa n)dRI =0, (60) with one such equation for each n. Evaluation of Q( 13) with the trial g(R) in Eq. (52) completes the set of equations for the determination of all an. The resemblance of Eq. (57) to cell model calcula tions of the free energy is strong, and suggests that W might be simply related to a free energy expression under the terms of the Kirkwood superposition approxi mation. We have not been able to confirm this suspicion. Rather the treatment of Q(13), and therefore K, as constants in the variational process, seems to have the physical interpretation of a mechanical equilibrium condition. Since (1), K is the contribution of inter molecular forces to the pressure, Eq. (56), and (2), the Ihs of Eq. (53) gives an estimate of K for every value of the intermolecular separation R12, Eq. (60) determines g(12) by attempting to make the pressure the same at all points R2 when it is known that a molecule is at RI• An attempt to force a free energy out of Eq. (53) by comparison of it with Kirkwood's expressions20 for the chemical potential J.I. and g(R), which are based on a coupling parameter,20 yielded only (61) 20 See work cited in footnote reference 4, pp. 192 and 203. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28OSMOTIC PRESSURE OF POLYMER SOLUTIONS 377 where A=h(27rmkT)-l. Equation (61) is incorrect beyond the second virial coefficient, the meager result stemming from the fact that the comparison involves the superposition approximation in the first term of a virial expansion of the excess chemical potential; consequently, the coefficient of the second power of p will be incorrect. IV. APPLICATION OF VARIATIONAL PRINCIPLE In an application of Eq. (60), the choice of a trial function g(R) is governed by three considerations: the ease of the numerical work, the desired flexibility of the trial function, and our intuition regarding the expected form of the correct g(R). If X>O, the intermolecular potential Eq. (1) decreases monotonically to zero from a finite maximum XkT, as R increases. It seems un likely, then, that a calculation of the osmotic pressure would be greatly affected by the suppression of maxima and minima in g(R). To be sure, the latter behavior results even for a hard sphere potential,' which VCR) may somewhat resemble when X»l, but the "surface" of a polymer molecule remains soft, and any maxima and minima in g(R) must be more diffuse than for the hard sphere. We could reasonably begin by restricting g(R) to monotonic functions. Two further conditions on g(R) would seem advantageous: that g(R) can have the form exp( -V(R)/kT) at small concentrations, and that g(R) has the possibility of becoming identi cally unity for all R at high concentrations. A g(R) having these properties would be lng= -Kl exp( -K2KA) , where Kl and K2 are variation parameters. However, this form appeared to be too troublesome for hand calculation of the double integral in Eq. (60), and we consequently chose the qualitatively similar For the integral over Ra in Eq. (60), put N= f Q(13) (g(23) -l)dRa. (66) The integration over Ra can be effected by completion of the squares in the exponential (-N/kTX7rI) = (b+B)-la exp[ -b(1-b/b+B)R 122] -(B/b+B) (2b+B)- Ja2 exp[ -b(1-b/2b+B)R 122]. (67) For the remaining integrations in Eq. (60) we need to specify use of Eq. (63) or (64); two equations result. These are most concisely given by replacing (fJg/fJa n) in (60) by fJg/fJan~R2n exp( -bKA), (68) where n=O, 1 furnishes the two equations to be solved; see (63) and (64). Substitution of Eqs. (1), (62), (67), and (68) into Eq. (60), and integration over R12 yields f" In[l -a exp( -bR2) ]KAn+2 exp( -bKA)dR X1·3··· (2n+1)7r l + 2n+2(b+B),,+f { a1·3···(2n+1) Introduce now the reduced variables a=b/B v=p(7r/B) I, (69) (70) g(R) = 1 -a exp( -bR2). (62) as in Eq. (44), and for convenience let This form is not very good for large X and small con centrations, because of the difficulty of making g(R) stay small for sufficiently large R; but the analytical convenience of Eq. (62) outweighs this disadvantage, at least for a preliminary look at a concentration region that has not yet received much study, either experimental or theoretical. In what follows we use the exact potential of Eq. (1), and calculate the functions entering in to Eq. ( 60) . From Eq. (62) (63) A=X7ri/4. (71) Substitution of these new variables into Eq. (69) gives, upon putting n=O, 1, the final form of the variational equations for a and a in terms of A and v I2(a) +A (a/1+a)1 -Ava[(2+0)-! -a(1+0)-1(2+30)-I]=0, (72) I4(a) + (3A/2) (0/1+0)5/2 -(3Ava/2) [(1+a) (2+a)-5/2 fJg/fJa= -exp( -bKA) fJg/fJb=aKA exp( -bR2). (64) where From Eqs. (1) and (52) Q(13) =kTX[(exp -BR1S2) -(aB/b+B) Xexp-(b+B)R1n. (65) Im(a) = f"ln[1-a exp( _x2) Jxm exp( -x2)dx. (74) Equations (72) and (73) were solved for A>O by an inverted procedure. Selected values of a and 0 were This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28378 MARSHALL FIXMAN TABLE II. Variational theory. Parameters a and 0 in trial radial distribution function g(R)=I-aexp-oB.R2, for potential VCR) =kT(4Af,r·!) exp-B.R2, and p=p('1f"/B) I, with p the number density. a 0.1 0.4 0.7 1.0 0.1 0.4 0.7 1.0 0.1 0.4 0.7 1.0 A 0.0467 0.2257 0.5042 1.1599 0.0894 0.4053 0.8473 1. 7776 A 0.0462 0.2149 0.4641 1.0637 0=1 0=2 p 0.3797 0.5372 0.6365 0.9206 29.409 7.7307 4.6918 3.6491 o 0.98621 0.9209 0.8322 p=O 0.6212) A p 0=1.5 0.0659 14.590 0.3065 0.0549 0.6578 2.6170 1.4290 2.2414 0=3 0.1481 60.866 0.6486 15.558 1.3102 9.1320 2.6298 6.7028 entered, and the corresponding values of A and II computed. This procedure was dictated by the much simpler way in which A and II appear than do a and ~. The values of a and ~ were a=O.l, 0.4, 0.7, 1.0; ~=1, 1.5, 2, 3; a few miscellaneous values of a and ~ were used before the ranges useful and appropriate to A> 0 became apparent. The integrals Im(a) , Eq. (74), were computed by a four point Gauss numerical integration2I after first bringing the range of integration to O<Z<l by the substitution X= (lnZ-I)!. This procedure re sulted in Table II. We obtained the entries under 11=0 in Table II by setting 11=0 in Eqs. (72) and (73), and solving for A and ~ as functions of a. An inspection of the entries under 11=0 in Table II will immediately indicate a major problem. Because g(R) has the physical meaning of a probability, g(R) can have immediate physical significance only if g(R)?O, and necessarily then a~1. Equation (74) requires this inequality from a purely mathematical standpoint if a is real. But the apparent result of this restriction on a is that A ~ 1.064. It is not, of course, a real result even mathematically; A is an independent variable and can be assigned an arbitrarily large value. The objective, and at least formally undisputable way out of this dilemma, is to recognize that for A> 1.064 and a restricted to the real axis, our inability to find solutions of Eqs. (72) and (73) together (for 11=0) implies merely that a is no longer subject to arbitrary variation in Eq. (60). Rather a=1 for A> 1.064, Eq. (72) is rejected, as it arises from variation of a, and ~ is obtained as a function of A from Eq. (73) only, with a=1 and 11=0. If 11=0, then a can be a good variational 21 H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand Company, Inc., New York, 1943), p. 462. parameter even for A> 1.064, but for any A> 1.064, as II is decreased, II will eventually become too small for real solutions of Eqs. (72) and (73) to exist, and then we set a=l, and solve only Eq. (73) for~. If A> 1.8, the required use of Eq. (73) only, extends to a concentration so high that a great deal of over lapping of molecules occurs, and S is very close to its asymptotic value. Because Eq. (73), with a=l, is relatively easy to solve (most conveniently by obtain ing v as a function of A and ~; we record that 14(1) = -0.14502), we will not take up space with a full tabu lation of its solution. We give in Table III a tabulation of corresponding (horizontal entries) values of ~, a and II for values of 0< A < 1.8, where the entries for a< 1 were obtained by graphical interpolation of curves constructed from Table I1(A vs a, II vs a, for ~= 1, 1.5, 2,3). The entries for a= 1 were obtained from Eq. (73), as heretofore described, when the graphs indicated that no simul taneous solution of Eqs. (72) and (73) for a and II existed for the given A and ~. V. COMPUTATION OF THE OSMOTIC PRESSURE A. General For a calculation of the osmotic pressure on the basis of the variational theory, we again use Eq. (31), but it is pertinent to note, now that Table III shows g(R)~l, TABLE III. Symbols as in Table II. Ii a v o a v A=0.2 A=O.4 3 0.137 46.8 3 0.261 25.5 2 0.216 15.0 2 0.395 7.83 1.5 0.277 5.84 1.5 0.498 3.32 1.0 0.362 0.42 1.0 0.607 0.60 0.928 0.377 0 0.854 0.640 0 A=0.6 A=0.8 3 0.374 16.9 3 0.478 13.1 2 0.548 5.78 2 0.678 4.80 1.5 0.663 2.69 1.5 0.782 2.42 1.0 0.765 0.68 1.0 0.870 0.78 0.791 0.796 0 0.722 0.902 0 A=l.O A=1.2 3 0.574 10.9 3 0.658 9.6 2 0.776 4.43 2 0.858 4.17 1.5 0.871 2.30 1.5 0.939 2.26 1.0 0.950 0.86 1.0 1 0.95 0.648 0.979 0 0.575 1 0 A =1.4 A =1.8 3 0.731 8.8 3 0.845 8.0 2 0.922 3.92 2 1 3.66 1.5 0.993 2.22 1.5 1 2.39 1.0 1 1.06 1.0 1 1.21 0.522 1 0 0.450 1 0 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28OSMOTIC PRESSURE OF POLYMER SOLUTIONS 379 all R, as II-HfJ , why we did not use the alternative4 which gives more accurate virial coefficients for at least one approximate theory22 of g(R) and does not require pairwise additivity of V(Rij). (There is, of course, no reason to expect an approximate molecular theory to give completely self-consistent thermodynamic func tions.)23.24 Aside from the numerical disadvantage of Eq. (75), a numerical integration being required to obtain P, it depends most sensitively on the small difference between g(R) and unity, and thus discards the major information to come out of the variational calculation, that g~1. Substitution of Eqs. (1) and (62) into Eq. (31) gives where (77) and is a function of A =X'/I"i/4, and II. Transformation of p to the observed concentration variable c, Eq. (39), and introduction of the "apparent second virial co efficient" S, Eq. (40), gives S=[(P/c) ---(RT/M)J(RTc)-1 (7S) As we previously noted, S reduces to A2 at zero con centration. It is interesting to compare the A2 of Eq. (7S) with that of Flory and Krigbaum. 2 The factor multiplying 0---7]0) (where 7]0 is 7] at c=O), is the same for both results, but the Flory-Krigbaum result has instead of (1 -7]0) a numerically computed function F(X), the correct multiplier for the potential of Eq. (1). In Table IV (1-7]0) is compared with F(X) for a range of X, and also with a closed approximation to F(X), FOF(X), devised by Orofino and Flory,S and utilized by Stockmayer25b FOF(X) = (4/X'/I"I) In(1+X'/I"l/4) =A-1ln(1+A). (79) The range of A in Table IV is approximately that covered by polymer systems hitherto studied, and within that range the variational calculation gives fairly good values of A2, almost as good as the data, or the agreement of A2 theories with the data. At the larger values of A, the predicted A 2 are too large because 22 J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958). 23 G. S. Rushbrooke and H. 1. Scoins, Proc. Roy. Soc. (London) A216,203 (1953). 24 J. E. Mayer and G. Careri, J. Chern. Phys. 20,1001 (1952). 2i (a) W. H. Stockmayer, J. Polymer Sci. 15,595 (1955). (b) Also see a paper presented at the International Symposium on Macro molecules, Wiesbaden, October, 1959. (Private communication; to be published.) Stockmayer used Eq. (79) in his comparison with perturbation theory. TABLE IV. Proportionality factors in the second virial coefficient against potential energy parameter A =X".i/4; (1-'10) from variational calculation, F(X) from Flory and Krigbaum,' and poF(X) from Orofino and Flory. A 1-'10 F(X) poF(X) 0 0.2 0.927 0.93 0.911 0.6 0.815 0.80 0.784 1.0 0.719 0.72 0.693 1.4 0.650 0.65 0.625 1.8 0.605 0.59 0.572 2.6 0.542 0.51 0.492 3.8 0.480 0.42 0.412 5.0 0.439 0.37 0.358 the trial g(R) is unnaturally forced to rise too quickly from its minimum value at the origin. A slightly more flexible g(R) at large A, say g(R)=1---(1+dW) exp( -bR2), should be completely adequate, and even d=b, though yielding a more tedius variational calcu lation than what we have used, d=O, should better represent the physical situation. With this encouragement we now anticipate a com parison of theory and experiment. Such a comparison requires first a consistent relationship between X, B and precisely defined molecular parameters. We shall not follow the identification made by Flory and Krig baum,2 which to our mind rests on too literal an identification of the radius of the segment density distribution used to evaluate intermolecular potentials with the radius of gyration. Rather we follow Stock mayer25 and make an identification which agrees with the perturbation theory of A2 and the excluded volume effect. Without going into any but the most necessary details, we observe that the perturbation theory of A226,27 gives where fJ is a segment-segment excluded volume, n is the number of segments (S1) R02 is the mean square end-to-end distance of the random flight polymer chain, and ex is a factor which accounts for molecular expansion due to intra-chain repulsion. On the other hand, Eq. (1) gives exactly2 A2= ('/I"/B)!(N oX/2M2) (1-X21/S+···). (S2) Comparison of Eqs. (SO) and (32) permits the identi fication A =7.1SZ/a3 B=9.61/ex2R02. 26 B. Zimm, J. Chern. Phys. 14, 164 (1946). 27 A. C. Albrecht, J. Chern. Phys. 27, 1002 (1957). (S3) (S4) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28380 MARSHALL FIX MAN ~6r-------------------~(I~)-r---' 3.4 3.0 • Q '" z.a '" .... II: Z.6 6 a 10 12 14 16 Cxl02 FIG. 2. Apl?arent second virial coefficient S = [(P / c) - (RT/M](RTc)-l, with Pin g(wt)/cm', c in ~/cm3, for polyiso butylene in eyc1ohexane7; M=9Xl()4, T=303 K. Points are ex perimentaJ.7 Curve (1) Flory-Huggins theory. Curve (2) varia tional theory, a=1.18, Ro=284Xl<rs cm. Curve (3) variational theory, a= 1.21, Ro=257XlQ-s em. If a is taken to be the expansion factor appropriate to the radius of gyration of an isolated chain21),28, then (85) Eql!ation (85) permits elimination of Z from (83) A =5.63(a2 -1), (86) and so allows A and B to be expressed as functions of polymer dimensions. B. Comparison with Experiment There do not seem to be many careful determinations of the osmotic pressure in the full range of volume fractions less than about 0.15. Furthermore, only very good solvents are predicted to cause the flattening out of S(c) in this concentration range, and since this is the most unusual effect of the theory, we particularly require data on very good solvents. Because of the lack of data and also because some small quantitative, though not qualitative changes in the theory are expected when the variation of polymer dimensions with conceI;ltration is taken into account, we will confine our comparison to one experiment, that of Flory and Daoust1 on the osmotic pressure of poly isobutylene in cyclohexane, with M = 9 X 104, T = 303°K. The comparison is most simply made if we write 28 E. F. Casassa, J. Chern. Phys. 31, 800 (1959) has suggested a different choice of the numerical constant in Eq. (85) on the basis of his calculation of intramolecular repulsion in the A. theory. That is, a would not be the expansion of the radius of gyration. His suggestion is well taken, but the effect of his change would he minor here. Eq. (78) in the form S(c) = S( 00) (1-71) S( 00) = ('/r/B) iXNo/2M2. (87) (88) A rough estimate of S(O) =A2, and S( 00) is obtained from the data, Fig. 2. From Eq. (87) then follows (1-710) and entry into Table IV furnishes A. Equation (88) then yields B. With B, Eqs. (39) and (44), one obtains the proportionality constant relating c to II, and hence the complete curve of S (c), by use of Table III or its extension (see end of part IV) . In Fig. 2 are displayed the data and two variational curves Curve (2): A =2.2; a=1.18; Ro=284X1crs cm (89) Curve (3): A =2.6; a=1.21; Ro=257XlO-8 cm, (90) Also in Fig. 2 is a Flory-Huggins curve/ which describes the system very well at the higher concentrations displayed and beyond. The general form of this curve is (91) with VI the molar volume of solvent, x the ratio of the molar volume of polymer to VI, and XI a constant with respect to V2. With Vl=108 cm3/mole, and Xl =0.429,1 we obtain curve (1) (92) The values of the dimensional parameters a and Ro in curves (2) and (3) may be compared with the values determined by Fox and Flory13,29 from intrinsic viscosity measurements a= 1.30; Ro=238X1Q-S cm. (93) The agreement of the experimental data and theo retical curves (2) and (3) in Fig. 2, and the agreement of the parameters in the theoretical curves with those obtained from the fairly well-established viscosity method is good, though not perfect. Such systematic deviations as seem to occur may be rationalized or over-rationalized, as follows: (1) The molecular weight is probably closer7,8 to 8X1()4 than the considerably extrapolated M = 9X 1()4 given by Flory and Daoust. A smaller M would of course depress the experimental points primarily at small c and favor a larger A; (2) The expectation that a~l as c~oo makes it reasonable 29 T. G. Fox and P. J. Flory, J. Am. Chern. Soc. 73,1909 (1951). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28OSMOTIC PRESSURE OF POLYMER SOLUTIONS 381 that an 0: taken independent of c, 0:",1.2 in (89) or (90), should lie between unity and 0:=1.3 obtained at infinite dilution.29 These rationalizations should not be taken too seriously, but the agreement between experiment and theory and between the different computations of A2 and As seems good enough to warrant acceptance of three statements as working hypotheses: (1) The effect THE JOURNAL OF CHEMICAL PHYSICS observed by Flory and Daoust is not an experimental artifact and is worth seeking elsewhere. (2) The calcu lations presented herein give a semiquantitatively accurate description of the thermodynamic properties of chain polymer solutions up to a volume fraction of ca 0.1. (3) An adequate basis has been laid for a calcu lation of the change of 0: with c by free energy minimiza tion. VOLUME 33, NUMBER 2 AUGUST, !960 Solvent Effects on n-+rr* Transitions in Pyrazine* V. G. KRISHNA AND LIONEL GOODMAN Whitmore Chemical Laboratory, The Pennsylvania State University, University Park, Pennsylvania (Received February 5, 1960) The T->S and S->S' spectra of pyrazine and 2,5-dimethylpyrazine in hydrocarbon and EPA glasses has been measured. The S->S' and S->T spectra of the same compounds in ethanol and isopentane are also reported. When due correction is made for the effect of the rigid media, solvent shifts (from hydro carbon to EPA) of 60 cm-1 to the red in emission and the same amount to the blue in absorption are ob tained for the (H) band. The absorption maximum is shifted by approximately 1200 cm-1 to the blue and the emission maximum by about 600 cm-1 to the red. The data is explained in terms of the weak hydrogen bonding and the Franck-Condon strain resulting therefrom. It is suggested that due tothe change in the ex tension of the n orbitals in nitrogen atoms of N heterocyclics, a Franck -Condon destabilization energy results in protonic solvents. The role of the Franck-Condon principle in the n->1t* blue shift phenomenon is found to be of considerable importance in accordance with Pimentel's views. The previously reported discrepancy of 500 cm-1 in the (H) bands of the T-.S emission and S->T absorption of these compounds is entirely explained by media effects. INTRODUCTION IN a previous communication! it was noted that a discrepancy in S-+ T and T -+S 0-0 bands for pyrazine multiplicity-forbidden transitions exist. In the absence of media effects the 0-0 bands for emission and absorption should be superimposed unless the transition is orbitally forbidden. The purpose of the present investigation is to ascertain whether this dis crepancy is due to the media effects or due to the in herent nature of the transition itself. The results of this investigation throw some light on the importance of the Frank-Condon principle in solvent effects on n~* transitions. The restriction placed by the Frank-Condon principle on solvent effects has been recognized before 2.3 and extended more rigorously by Pimente14 to cases where hydrogen bonding effects the n~* spectra. However, no data on n~* transitions has been available up to now for the purpose of testing such a restriction, though its need has been recognized by Pimente1.4 The major work on * This investigation was supported by the Office of Naval Research. 1 L. Goodman and M. Kasha, J. Mol. Spectroscopy 2,58 (1958). 2 H. McConnell, J. Chern. Phys. 20, 700 (1952). 3 N. S. Bayliss and E. G. McRae, J. Phys. Chern. 58, 1002 (1954) . 4 G. C. Pimentel, J. Am. Chern. Soc. 79, 3323 (1957). solvent effects on emission, though undertaken with a different aim for '11'~* transitions has been due to Mataga.s In the following, the singlet-singlet and triplet-singlet spectra of two nitrogen heterocyclic compounds pyrazine and its 2,5-dimethyl derivative-are compared in a hydrocarbon and hydro xylic solvent and an inter pretation suggested. The multiplicity-forbidden transi tions studied here were unambiguously assigned to be n~* transitions by both solvent effects and substituent perturbation.6•7 As far as the Franck-Condon restriction is concerned, emission and absorption differ in that in the former the excited state and in the latter the ground state are in equilibrium with solvent environment; thus a com parison of the solvent shifts in emission and absorption is of importance in understanding the role of the Franck Condon principle in solvent effects. Such a study of n~* transitions is necessary as there is a question as to whether the n-7'1J'* blue-shift phenomenon can be attributed to hydrogen bonding from the solvent to the solute in the ground state involving the lone pair of 6 N. Mataga, Bull. Chern. Soc. Japan 29, 465 (1956). 6 F. Halverson and R. C. Rirt, J. Chern. Phys. 19, 711 (1951). 7 L. Goodman and R. W. Harrell, J. Chern. Phys. 30, 1131 (1959) . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.247.166.234 On: Mon, 24 Nov 2014 08:27:28

1.1715950.pdf

Apparatus for Studying Convection under the Simultaneous Action of a Magnetic Field and Rotation Yoshinari Nakagawa Citation: Review of Scientific Instruments 28, 603 (1957); doi: 10.1063/1.1715950 View online: http://dx.doi.org/10.1063/1.1715950 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/28/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dusty Plasma Structures in Stratified Discharge under Action of an Axial Magnetic Field: Mechanisms of the Rotation Inversion AIP Conf. Proc. 1397, 277 (2011); 10.1063/1.3659801 Experimental apparatus for studying rotating magnetic field current drive in plasmas Rev. Sci. Instrum. 76, 093507 (2005); 10.1063/1.2052527 Experimental and numerical study of Rayleigh-Bénard convection affected by a rotating magnetic field Phys. Fluids 11, 853 (1999); 10.1063/1.869957 Apparatus for the study of Rayleigh–Bénard convection in gases under pressure Rev. Sci. Instrum. 67, 2043 (1996); 10.1063/1.1147511 Magnetization processes in samples with modulated anisotropy under the action of nonuniform magnetic fields J. Appl. Phys. 78, 3961 (1995); 10.1063/1.359917 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53THE REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 28, NUMBER 8 AUGUST, 1957 Apparatus for Studying Convection under the Simultaneous Action of a Magnetic Field and Rotation* YOSHINARI NAKAGAWA Enrico Fermi Institute for Nuclear Studies, University of Chicago, Chicago, Illinois (Received April 8, 1957; and in final form, May 16, 1957) .\n apparatus for the study of the instability of a layer of mercury heated from below and sUbjected to the simultaneous action of a magnetic field and rotation is described. The equipment is designed to examine the mode of convection and the dependence of the critical Rayleigh number (=ag{3,d4/KV, where g denotes the acceleration due to gravity, (3, the critical adverse temperature gradient, a the depth of the layer, K the thermometric conductivity, v the kinematic viscosity, and a the coefficient of volume expansion) for the onset of instability on the values of nondimensional parameters Q,(=uH2d2/-rr2pv, where H is the strength of the magnetic field, u the electrical conductivity, and p the density of fluid) and T, (=4n2d4/1I.4V2, where n denotes the angular velocity of rotation), the features proven to be essential in such studies. By using a reconditioned cyclotron magnet as a part of the experimental arrangement, this apparatus covers wide ranges of parameters Q, (from 10 to 106) and T, (from 105 to 1010). Typical results that can he obtained by this apparatus are discussed with a few illustrative examples. 1. INTRODUCTION IK this paper we describe an apparatus which permits the study of the exact character of the onset of instability in a layer of mercury heated from below and subject to the simultaneous action of a magnetic field and rotation. It is valuable to study such convection under controlled laboratory conditions because the theoretical approaches are limited by the complexity of the problem,while the realization of such convection is to be found only in astrophysical and geophysical environments. (where g denotes the acceleration of gravity, a the depth of the layer, {3 is the linear adverse temperature gradient which is maintained and a, K, and II are the coefficients of volume expansion, thermometric conduc tivity, and kinematic viscosity, respectively), exceeds a certain determinate critical value, and further that the instability sets in as cellular convection of specified cell dimension. In the classical experimentall and theoreticaP studies, it has been established that a layer of fluid heated from below becomes unstable when the Rayleigh number ag{3d4 J?=--, KII (1) * The research reported in this paper has in part been supported by the Office of Naval Research under Contract N60ri-02056 with the Enrico Fermi Institute for Nuclear Studies, University of Chicago. 1 R. J. Schmidt and S. W. Milverton, Proc. Roy. Soc. (London) A152, 586 (1935); R. J. Schmidt and O. A. Sanders, Proc. Roy. Soc. (London) A165, 216 (1938). 2 Lord Rayleigh, Phil. Mag. (6) 32, 529 (1916); H. Jeffreys, Phil. Mag. (7),2,833 (1926); Proc. Roy. Soc. (London) AU8, 195 (1928); A. Pellew and R. V. Southwell, Proc. Roy. Soc. (London) A176, 312 (1940). Re-examining this problem under the circumstances where an electrically conducting fluid is heated from below and subject to the simultaneous action of a magnetic field and rotation, S. Chandrasekhar has found3 that the onset of convection is delayed; and that the extent of this inhibition generally depends on the value of the nondimensional parameters, 1JJ.L2 H2 cos2JJd2 Ql=. and 1('2PII (2) 4Q2 cos2t'Jd4 Tl= (3) 1('4112 where IJ, J.L, and p are the electrical conductivity, the magnetic permeability, and the density of the fluid, 3 S. Chandrasekhar, Proc. Roy. Soc. (London) A225, 173 (1954). 603 Copyrigbt © 1957 by tbe American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53604 YOSHINARI NAKAGAWA FIG. L General view of experimental setup. respectively, H is the strength of the magnetic field, Q is the angular velocity of rotation, and iJ is the in~ clination of the direction of H (respectively, 0) to the vertical. In particular, he has shown1 that in mercury, instability can, depending on the value of Ql and TI, set in either as convection and a stationary pattern of motions or as overstability and oscillations of increasing amplitude. Accordingly, the present apparatus waS designed to cover a wide range of values of Ql and T 1 and at the same time, allow the determination of the mode of convection as well as accurate values of the parameters R, Ql, and Ti• These specifications are satisfied by the apparatus to be described. It consists of three major units. The first unit, which we shall call external equipment, consists of a reconditioned magnet of a 36Hnch cyclotron of the University of Chicago and an electric motor with a variable speed transmission. The reconditioned magnet provides a uniform vertical magnetic field of a variable strength and the electric motor facilitates a wide range of variable speed of rotation. The second unit, the main body of the apparatus, is an assembly of a mercury container, a heating and a cooling system, and equip~ ment mounted on a nonmagnetic ball-bearing for measuring the temperature gradient and the tempera~ ture of mercury. When this unit is put in the magnetic field and is driven by the electric motor, the required condition of the simultaneous action of the magnetic field and rotation with heating from below is obtained. The controlling and measuring equipment stationed outside the magnetic .field comprise the last unit which provides the remote control and measurement of the strength of the magnetic field, the rate of rotation, the rate. of heating and cooling, as well as the continuous measurement of the adverse temperature gradient and the temperature of the mercury. The general view of the experimental setup with the motor and the main 4 S. Chandrasekhar, Proc. Roy. Soc. (London) A237, 476 (1956). body of the apparatus at its experimental positions is shown in Fig. 1. When the present apparatus is used with layers of mercury of a depth of 3 to 6 em, magnetic fields of SO to 10000 gauss, and rotation of 5 to 20 rpm, the ranges of the values of Ql from 10 to 106 and Tl from lOS to 1010 become available for experiments. In preliminary experiments, a layer of mercury 3 cm deep has been used and by rotating this layer of mercury with a constant angular velocity of 5 rpm the dependence of the critical Rayleigh number on the value of Ql has been studied for the value of Tl approxi~ mately 106• The results of the experiments have successfully confirmed the onset of the two different modes of convection in qualitative agreement with the theoretical predictions of S. Chandrasekhar. The details of the experimental results vvill be pubHshed elsewhere. 2. EXTERNAL EQUIPMENT Magnet Since an infinite horizontal dimension of the layer subject to a uniform magnetic field has been assumed in the theory, a large uniform external magnetic field is required. Therefore the 32t-inch cyclotron magnet of the University of Chicago with 8!-inch pole gap has been used as a part of the equipment for the present experimental arrangement. In order to provide a means of continuous adjustment of the magnetic field over a wide range, the original control system has been modified and a new control system has been added. A set of two 2S0-watt 2S00'ohm, four lOO-watt lOOO-ohm fixed resistors, and one 280- watt 1200-ohm Ohmite variable resistance with switches is placed in series in the generator field current circuit. A slight change in this resistance provides a fine adjustment of the magnet current. When the whole range of this resistance is used, the strength of the magnetic field can be varied from 3000 to 13 000 gauss. This arrangement is, however, insufficient to cover the whole aspect of the problem, because the influence of rotation is overcome by the magnetic field when the magnetic field becomes strong. A wider range of adjusta biIity of the strength of the magnetic field, especially down to weaker magnetic field, is provided by another set of resistances placed in series in the magnet power circuit. This second set of resistances consists of sixteen 660-watt 11S-volt glow coils connected in parallel; and through the combined use of these two resistances, the continuous range of control is increased from 50 to 13 000 gauss. Though the accuracy of adjustment is mostly limited by fluctuations in the resistance and in the variations of the generator, the adjustment is easily maintained within ±1% of the assigned value. A portable po tentiometer is used to measure the strength of the magnetic field in terms of the magnet current at a shunt placed in the power circuit. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53APPARATUS FOR STUDYI:\G CONVECTIO:\ 605 FIG. 2. Schematic cross section of the main body of the apparatus and the function of various units. A, base plate; B, brass rotating ring; C, mercury trough; D, stainless steel rod; E, mercury con tainer; F, stainless steel plate; G, heater; H, thermopile; I, measuring junction for the average mercury temperature; K, mercury vapor trap; R, Bakelite coupling. Nitrogen 300 V Supply D.C. Variable Speed Drive A closed type i hp 3450 rpm constant speed Graham motor with a variable speed transmission is used to drive the main body of the apparatus. This motor is located outside the magnetic field and the power is transmitted to the apparatus by means of a V belt and Bakelite pulley system (see Fig. 1). The rotational speed is set by a micrometer control attached a~ the variable speed transmission and is continuously vana?le from 0 to 200 rpm with the whole range also bemg available in the reverse direction. The motor is mounted on i-inch thick aluminum base plate with a i-inch thick felt sheet between the base plate and the motor. This assembly is then mounted on an aluminum frame which provides the working support as well as the railways for the motor. With four brass rollers attached to the bottom of the base plate, tension of the V belt is adjusted by moving the motor on the railways. Since the V belt and pulley drive absorbs vibrations of the motor transmitted to t he main body, smooth rotation is obtained in operation. 3. MAIN BODY OF THE APPARATUS The main body of the apparatus is constructed in an extremely compact manner because its dimensions a:e limited by the size of the magnetic field. The schematIC cross section of the apparatus is illustrated in Fig. 2. A l-inch thick 23X 23 inch square brass plate A is used I I L-------l I :-----l ~ : i ffiJ\1 r-------W ------+ I " I: 1/ I I ~ I I g Recorder as the base for the apparatus and the equipment to be placed within the magnetic field is mounted on this plate. . To provide for leveling and exact placmg of the plate A three equally spaced leveling screws are attached to the plate. At the bottom of this plate, four brass rollers are attached. In case of alterations or adjustments, the whole assembly is rolled out from the gap of the magnet onto the side opposite the motor. The working support and the railways for such transfers are provided by a similar aluminum frame used to mount the motor which can be seen in Fig. l. The ball-bearings of the apparatus are composed of nonmagnetic materials. Bronze balls i-inch in diameter are used with brass bases. The bottom surface of the bearings is formed by plate A on which the whole assembly of the bearings is mounted within. a brass enclosure and a rotating ring B bears agamst the bronze b~lls through two 45° contacts (see Fig. 2). Such contacts of the rotating ring B achieve a reduction in the height of the bearing assembly and at the same time provide a means for leveling and prevent the wobbling of the rotating member of the apparatus. . The eddy current induction in the rotating ring B IS minimized by its small radial dimension. Though the rotating member of the apparatus is grounded through the ring B and the bronze balls, the uniform radial distance of the contact induced no appreciable effect. Smooth rotation of the main body up to the maximum This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53606 YOSHINARI NAKAGAWA FIG. 3. Lucite top plate, mercury container, and heater. speed of 200 rpm in a magnetic field of 13000 gauss thus has been successfully maintained. To measure the angular velocity of rotation, a stainless steel rod D is mounted on a Bakelite ring placed on the top of the brass ring B. The outer end of this rod is polished and when it passes in front of the light source and photocell unit, the timing system connected to the photocell is triggered. The light source and photocell unit is made into a single assembly and it is mounted on the base plate A. A semiconductor type photocell is used in order to assure its operation in strong magnetic fields. A Pyrex glass cylinder E of lUi-inch nominal outside diameter, i-inch wall thickness, and height 10 em is used as the container for mercury; its poor thermal conductivity assures good heat insulation on the sides. The ends of the cylinder E are carefully ground because of the importance of the correct vertical placing. For convenience of assembly, two Bakelite rings are cemented on this cylinder as can be seen in Fig. 3. The bottom of this cylinder is sealed by a i-inch thick stainless steel plate F with an a ring. The surface of this plate F is carefully finished in order to eliminate the formation of convection currents due to macroscopic irregularities. To apply uniform heating at the bottom surface of the mercury, an electric heater G is placed underneath the stainless steel plate F. The electrical heater G is made of a i-inch thick transite plate; and this heater consists of B & S # 20 Nichrome resistance wire wound in a helical coil 136 inch in diameter and placed in an equally spaced double involute groove of i-inch depth. The coil is placed in such a manner that an identical length of wire is in each spiral, to avoid deformation of the magnetic field. The actual arrange ment of this heater is shown in Fig. 3 where the con tainer and the other units are shown in a semiassembled fashion. Since alternating current will induce vibrations in the heating elements, de was used. The most important variable of the experiment, the adverse temperature gradient, is measured by a nine couple copper-Constantan thermopile H immersed in the mercury. To assure that this thermopile H will not interfere with the delicate convective circulation and at the same time respond quickly to any tempera ture changes B & S # 30 wires are used. The junctions of thermopile H are soldered with Divco #422 alloy. The thermopile is wound noninductively on two Bakelite rings which are supported from the Lucite top plate in order to avoid emf induced by the magnetic field. On these Bakelite rings, each set of nine junctions is fixed at two separate levels and at each level the junctions are spaced 1 em apart along a diameter of the container E (see Fig. 3). By this arrangement, the differences in the temperature due to the relative positions of the junctions to the cells are averaged. To assure that measurements are free from the effect of boundary layers, the levels of the junctions are adjusted at least! em from the top and the bottom surfaces of the layer. The thermal emf of this thermopile has been calibrated and is 36S.3±3.2 microvolts for a tempera ture difference of 1°C between the two levels of the junctions. Since the quantities involved in the critical Rayleigh number and the nondimensional parameters Ql and Tl depend on the temperature of the fluid, a single copper-Constantan thermocouple is used to indicate the average temperature of the mercury. The measuring junction I of this thermocouple is placed outside the magnetic field in a constant temperature bath (see Fig. 2). The junction of this couple is also soldered with Divco #422 alloy. The insulation of the junction I and the thermopile H is achieved by Analac paint which was kindly provided by Anaconda Wire and Cable Company. A cooled nitrogen circulation system is used in order to remove heat at the top surface of the mercury so that the temperature of the mercury can be kept constant and at the same time a linear temperature gradient can be established through the layer as postulated in the theory of these phenomena. The cooled nitrogen enters the rotating system through a rotating coupling R, consisting of two concentric Bakelite tubes having a-ring seals in between and placed at the center of the base plate A. The nitrogen is introduced into the cylinder through a 1-inch thick Lucite plate which seals the top opening of the cylinder and provides the support to the thermocouple and thermopile. To distribute the cooled nitrogen gas evenly over the surface of the mercury, six equally spaced holes i inch in diameter are cut horizontally in a Lucite piece which is attached to the top plate. Through these holes the cooled nitrogen gas is flushed over the surface of mercury from the circumference of a thin Lucite disk (see Fig. 2). After its passage over the mercury surface, the warm nitrogen gas is released from the cylinder through the other set of six vertical holes i inch in diameter spaced equally in the top plate and is led to the vapor trap K. The mercury vaRor trap K is made of a flat brass cylindrical vessel and is mounted on a Bakelite disk This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53APPARATUS FOR STUDYING CONVECTION 607 on the edge of the ring B. It is filled with mossy zinc metal and the concentration of the mercury vapor is reduced below toxic level by thi! trap K before the gas is exhausted into the room. The assembly of the container of mercury, the heater, and the top piece is mounted on a !-inch thick Bakelite base plate which in turn is mounted on the brass ring B. At the bottom of this Bakelite plate, four concentric copper cylinders and a set of B & S # 20 copper and Constantan wires are placed. With a set of mercury filled troughs C in which these copper cylinders and the wires can rotate freely, the electrical contacts between the rotating and nonrotating members of the apparatus are maintained (see Fig. 2). The contacts between the mercury and the rest of the nonrotating components of the circuits are formed by identical metal rings such as copper or Constantan imbedded at the bottom of the troughs which are cut i-inch deep and 136-inch wide in concentric fashion in a i-inch thick Lucite disk C stationed at the center of the base plate A. No significant emf or contact noise is induced by this assembly; however, an unavoidable emf is created due to the geometrical configuration of the system. Since the contacts are maintained at different radii, when the unit is rotated with an angular velocity (Q) in a magnetic field (H), each contact cuts different amounts of magnetic flux and induces an emf, I:lE= (R1+R2) X (R1-R2)QHj4X10-2 microvolt, where Rl and R2 are the radii of the contacts and Q is the angular velocity in secI, H is the strength of the magnetic field in gauss. Since this emf depends on the radii of the contacts, the set closest to the center is used for the thermopile circuit because of less effective percentage error in the readings and the next set is used for the thermocouple circuit. The heater circuit is maintained at the outmost troughs as it is not affected by such induced emf. The induced emfs which enter the thermopile and thermo couple circuit are compensated by counter emf units at the controlling and measuring units. 4. CONTROLLING AND MEASURING EQUIPMENT With the exception of the control system of the motor, all units shown in the bottom half of Fig. 2 are assembled and placed outside the magnetic field alongside the control and measuring unit for the magnet. The rotation counter and timer system consists of a seven decimal counting unit and a lOO-kc crystal oscillator. Since the triggering circuit is connected to the light source and the photocell unit situated beside the main body of the apparatus (see Fig. 2), by the successive passage of the rod D, the time required for a single or multiple revolution of the main body is measured by this system directly to 10-5 sec. The system which controls and measures the applied rate of heating of the electric heater used to heat the layer of mercury consists of a number of rheostats connected in series in the 300-volt dc circuit with an ammeter and a voltmeter. The rate of heating is adjusted and measured to a sufficient accuracy by this system in the range from i to 2000 watts. For continuous measurement and recording of the adverse temperature gradient and the average tempera ture of mercury, Weston 1411 Inductronic dc amplifiers are used with Esterline-Angus milliammeter recorders. The range of the critical adverse temperature gradient as well as the average temperature of the mercury vary considerably with the value of Q1 and T 1; accordingly, in order to provide an appropriate amplification of the thermal emf so that the measurements can be of uniform accuracy, variable range standards are used with the amplifiers to adjust the output. In this connection, to compensate the induced emf at the mercury troughs, counter-emf units are used in front of the amplifiers, which consist of simple voltage divider networks with mercury batteries and Helipot variable resistances. Under ordinary circumstances, it is possible to change the sensitivity of the amplifier from 100 microvolt to SO millivolt for the full scale of the recorder with errors in the measurement less than 1% of the total output. The reference temperature for the measurement of the average temperature of the mercury is obtained by the thermometer placed in the constant temperature bath which is exposed to continuously stirred water enclosed in a Dewar flask. The nitrogen circulation is maintained by a small pressure head kept at the supply and the rate of the flow is measured and adjusted at the flow meter. A mixture of acetone and dry ice placed in a Dewar flask is used for cooling the nitrogen gas. To assure the efficiency of cooling, this unit is placed close to the main body of the apparatus and Bakelite tubings are used preferably for most of the nitrogen passage. 5. OPERATION The onset of convection in any volume of fluid placed in the container of this apparatus may be observed through a wide range of the non dimensional parameters Q1 and T 1. Also at any fluid depth and values of Q1 and T], the temperature gradient is easily con trolled through the amount of power supplied to the heater unit. Thus various values of the adverse tem perature gradient, namely various states of heat transfer such as conductive and convective states, are readily produced in this apparatus. Usually after the values of Q, H, and d are determined, the apparatus is set in rotation for 30 minutes before any measurement is started. Then after the fluid reaches stationary rotation, a series of measurements is performed in cluding measurements of the adverse temperature gradient, the rates of heating, the rate of cooling, the strength of the magnetic field, the angular velocity of rotation, and the average temperature of the fluid. Therefore, for a set of fixed values of Q, H, and d, a series of records of the temperature gradient and the average temperature of the fluid is obtained. Typical examples of such records, corresponding to This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53608 YOSHINARI 1\'AKAGAWA 1-8min-j FIG. 4. Typical records of the adverse temperature gradient for the case when instability sets in as overstability. Heater is turned on and off as indieated. The rates of heating applied are 60 watts (0.041 watt/cm2) for (a), 200 watts (0.136 watt/cm2) for (b) and 300 watts (0.205 watt/cm2) for (c). The values of various physical parameters are: d=3 cm; 0=5 rpm; H = 750 gauss; Ql =3.59X 102; T,= 7.72X 105• three distinct modes of heat transfer are illustrated in Figs. 4 and S. In both figures, the records (a) correspond to low rates of heating and the gradual establishment of a conductive equilibrium in the layer can be seen clearly from the gradual increase and the subsequent stationary behavior of the adverse temperature gradient after the heating is started. With increasing rate of heating, the rate of increase of the adverse temperature gradient becomes more rapid and with the onset of convection, mass motion of the warmed fluid produces a decrease in the adverse temperature gradient as it reaches the upper junctions of the thermopile; the records of (b) and (c) illustrate such cases. When a series of such records has been obtained at each experimental determination and the maximum adverse temperature gradient attained in the layer is plotted against the corresponding rates of heating, a diagram like Fig. 6 is obtained. In such a diagram, data taken from the records of the type such as (a) in Fig. 4 and 5 produce a linear segment like aa. Similar plotting of the peak adverse temperature gradient from the records such as (b) and (c) provide segments like bb and ee, respectively. Since the segment aa corresponds to the state of conductive heat transfer, the onset of convection is immediately determined in such a diagram. In the actual determination, the critical temperature gradient is determined as the intersection between the extra polated segment aa and cc after applying linear least squares approximation for each segment, following the b c ~\~-\-~_\ \ \ _\ \-~ ~ V \ ---t---t< / off 1 / J J / / I 71 I i I lon/ / / I \ \ \ \ \\\\\\\ 1 .1 _\ r'\'i\\\ \ \ \ \ ) \ ~ I off / J_ / / II / / I r--------I--/ / / / I on / / I .L J.-8min--l T 400/-Lv .i. FIG. 5. Typical records of the adverse temperature gradient for the case when instability sets in as convection. The rates of heating are the same as in Fig. 4. The values of various physical parameters are: d = 3 cm; 0 = 5 rpm; H = 1400 gauss; Q\ =1.25Xl()3; T\=7.72X105• results established empirically by the studies of Y. Nakagawa,S Y. Nakagawa and P. Frenzen,6 and Y. Nakagawa and D. Fultz.7 The segment bb is ignored as it represents a transitional stage and the data have poor reproducibility. This method of the determination of the critical temperature gradient is valid for both convective and overstable instabilities. 1.6 E 1.2 ~ 00 co. .8 .4 f3c .22 26 FIG. 6. Diagram of the adverse temperature gradient ({3) ~'S the rate of heating. The linear segments aa, bb, and cc are obtained in the conductive, transitional, and convective states of heat transfer, respectively. {3c denotes the critical adverse temperature gradient defined by aa and zc. The values of the various physical parameters are: d=3 em; 0=5 rpm; H=l000 gauss; QI =6.32XIQ2; T\=7.60X105• 5 Y. Nakagawa, Nature 175, 417 (1955); J. Sci. Earth 4, 85 (1956). 6 Y. Nakagawa and P. Frenzen, Tellus 7,1 (1955). 7 D. Fultz and Y. Nakagawa, Proc. Roy. Soc. (London) A231, 211 (1955). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53APPARATUS FOR STUDYING CONVECTIO~ 609 The onset of ove.rstable convection is illustrated in Fig. 4(b) and (c); though in both records of (b) and (c) in Figs. 4 and 5 similar breakdowns of temperature gradient-indicating the mass motion of warm fluid are obtained, distinct differences between Figs. 4 and 5 can immediately be noticed. In Fig. 4, the records show oscillatory damping of the temperature gradient after the heat supply is cut, while in Fig. 5(a) smooth decline of the temperature gradient is obtained. Further, after the first breakdown in Fig. 4, fine oscillations of the adverse temperature gradient are recorded in com parison to Fig. 5. Since the characteristic of the overstable convection is the oscillatory motion of the convection, namely the oscillatory behavior of the temperature gradient, the records in Fig. 4 are identified as the overstable convec tion by comparison with the results obtained by D. Fultz and Y. Nakagawa. The results illustrated above have been obtained in, experiments with a layer of mercury 3 cm deep, rotating 5 rpm (T= 106), when the impressed magnetic field had strengths of 750 gauss (for experiments included in Fig. 4), 1400 gauss (for experiments included in Fig. 5), and 1000 gauss (for experiments included in Fig. 6). Identical rates of heating are used in both Figs. 4 and 5 for each of the records (a), (b), and (c). It is, therefore, clear from the foregoing discussion that the present apparatus can determine the mode of convection and the type of instability, allowing the necessary measurements for the study of the dependence of the critical Rayleigh number on the nondimensional parameters (Ql as well as Tl). 6. ACKNOWLEDGMENTS The author wishes to acknowledge his indebtedness to Professor S. K. Allison and Professor S. Chan drasekhar for their advice and encouragement. He also owes thanks to Mr. K. Benford of the Electronics Shop of the Institute for technical assistance in the design and construction of the rotation time counter unit. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP: 132.204.37.217 On: Wed, 10 Dec 2014 14:00:53

1.3060459.pdf

Chain Reactions F. S. Dainton Henry Wise , Citation: Physics Today 10, 8, 24 (1957); doi: 10.1063/1.3060459 View online: http://dx.doi.org/10.1063/1.3060459 View Table of Contents: http://physicstoday.scitation.org/toc/pto/10/8 Published by the American Institute of Physics24 3 IMPORTANT McGRAW-HILL PHYSICS BOOKS STATISTICAL MECHANICS: Prin- ciples and Selected Applications By TERRELL L. HILL, University of Oregon. 432 pages, $9.00 A graduate text and reference, highly useful for physics courses. Its aim is twofold: a) to sum- marize the principles of statistical mechanics at an intermediate level, and b) to present an or- ganized and detailed account of certain important applications of statistical mechanics which are the most important part of the book. A review of quantum mechanics is included. PHYSICS FOR SCIENCE AND ENGINEERING By ROBERT L. WEBER, MARSH W. WHITE, and KENNETH V. MANNING, The Pennsylvania State University. 624 pages, $8.00 This new book gives a precise presentation of the important physical principles essential to further work in science and technology. It is the '' calculus version" of the authors' highly successful text College Physics, giving students of science and engineering insight into complex physical phe- nomena. Emphasis is on the understanding of fundamental ideas and methodology, rather than on mere memorization of facts and laws. Becent developments in modern atomic and nuclear phys- ics and concepts in quantum theory and solid state are included, along with a striking 8-page insert depicting an actual nuclear power reactor. FUNDAMENTALS OF OPTICS By F. A. JENKINS, University of California, Berke- ley, and HARVEY E. WHITE, Pittsburgh Radio Station WQED. New Third Edition. 640 pages, $8.50 A revision of a leading undergraduate text in Geometrical and Physical Optics. As before, em- phasis is on the experimental side of the subject. Each chapter is introduced by a description of the observed facts and followed by an account of the theory with stress on the understanding of physical principles rather than on a rigorous math- ematical treatment. New material includes graphi- cal methods of tracing rays through prisms and a brief introduction to concentric optics. • SEND FOR COPIES ON APPROVAL • McQ>uu»-JliU BOOK COMPANY, INC 330 W»«l 42nd Str..t N.w Y.rk 36, N. Y.scarcely (even with an extensive commentary) have given the uninstructed reader a real idea of the issues involved. And this is the more unfortunate because the general reader may be tempted to think that from these short excerpts he has mastered the whole subject. The book will therefore be more useful and more instructive to the trained scientist than to the general reader, and to him it can be recommended without reservation. This criticism applies most strongly to the final sec- tion, which deals with modern cosmology. This is, most unfortunately, a subject that the general reader has been led to believe that he can grasp without the essential background of mathematics and physics. The superficial ideas of such general readers (which are not unknown among college students) are the despair of the serious scientist. If a section is to be devoted to modern cosmology, it should, in fairness to the reader, be representative. It should, moreover, contain at least one excerpt that does not elide the mathematical difficulties. Rather than including the frank popularizations of Gamow and Hoyle, this section might have included one of Weyl's carefully reasoned expositions. And instead of ending with three simplified essays on the "perfect cosmological principle", it should have included the now classic paper by Bondi and Gold, in which the matter was first pre- sented to the scientific world. It is no kindness to the general reader to lead him to believe that he under- stands what in fact he does not. The final bibliography does indeed include some tech- nical books; brief appraisals and descriptions would have added to its value. Chain Reactions: An Introduction. By F. S. Dainton. 183 pp. (Methuen, England) John Wiley & Sons, Inc., New York, 19S6. $2.90. Reviewed by Henry Wise, Stanford Research Institute. More than twenty years have passed by since Seme- nov's book on chain reactions was published. During the intervening period a large number of chemical reactions have been studied which proceed by a chain mecha- nism. Therefore it is rather timely to come across a text which examines the underlying principles which ac- count for some of the properties of the individual re- actions involving chain centers. Chain Reactions is written for the reader who is al- ready acquainted with the general principles of reac- tion kinetics. After a very brief review of the kinetics of simple homogeneous reactions in gaseous or liquid systems, the author describes certain distinguishing fea- tures of chain reactions including the special experi- mental procedures employed in their study. Among these one finds methods for the determination of ex- plosive boundaries in gaseous systems and the initiation of chains in polymerization reactions in the condensed phase. The following chapter is devoted to an analysis of the various types of individual reactions which may occur in a chain mechanism, such as initiation, propaga- tion, branching, and termination. A few typical reac- PHYSICS TODAY A25 Tdnje cr rrererjce The chart below is a simplified repre- sentation of 11 different staff member fields at Sandia Corporation. You might call it an occupational frame of refer- ence, within which are located the many activities involved in our work—designand development of nuclear weapons for the Atomic Energy Commission. In each of the 11 fields, challenging posi- tions are available for qualified engi- neers and scientists at all degree and experience levels. Applied Research Weapon Systems Engineering Component Development Electronic Development Handling Equipment Design Standards Engineering Quality Assurance and Surveillance Environmental Testing Field Testing Manufacturing Relations Engineering Military Liaison and field EngineeringENGINEERING PHYSICS MATHEMATICS There are many other things you'll want to know about Sandia Corporation. You'll be interested in our extremely liberal employee benefits. You'll want to know about the advantages of working and living in Albuquerque, famous for its healthful year-round climate and recreational attractions. You'll want in- formation on schools, homes, and cul- tural facilities. And you'll want to know more about our work, and our back-ground as a research and development laboratory. Our illustrated brochure answers these questions, and many others. For your copy, please write to Staff Employment Section 559A. CORPORATION ALBUQUERQUE. NEW MEXICO AUGUST 195726 tions are chosen to illustrate the principles and methods involved in the various types of chain processes. The most interesting section of the book deals with the mathematical treatment of chain reactions including nonsteady state conditions. The rigorous mathematical representation may at first appear to be an exercise in the solution of complex differential equations. How- ever the labor is well justified in the application of the theoretical results to an analysis of the characteristics of spontaneously explosive reactions, such as the hy- drogen-oxygen reaction and the oxidation of carbon monoxide. The final chapter is devoted to the kinetics of an unbranched chain reaction as encountered in ad- dition polymerization of unsaturated compounds. In general this book serves the very fine purpose of examining the theoretical principles of the kinetics of chain reactions, including the interplay of mass trans- port and chemical interaction. The treatment is not di- rected at the specialist engaged in research in chemical kinetics, rather it is a brief account of a rapidly ex- panding field of scientific endeavor. The keen interest in combustion processes and polymerization reactions coupled with the availability of modem computing de- vices promises major advances in chemical kinetics dur- ing the next twenty years. An Introduction to Junction Transistor Theory. By R. D. Middlebrook. 296 pp. John Wiley & Sons, Inc., New York, 1957. $8.50. Reviewed by R. Hobart Ellis, Jr., New York City. As engineers take over radar's microwaves and the nuclear chain reaction, the physicist's frontier seems to be shifting to the study of solids. The ten-year-old transistor is the most useful contribution to come from this study. It is still so young that to most physicists it is just a tiny substitute for a vacuum tube. This book will serve as a simple path to a more subtle under- standing. For convenience we can divide transistor study into semiconductor physics, electrical action of transistors, and transistor circuitry. Dr. Middlebrook aims at the second. He offers some semiconductor theory as a foundation, discusses electrical function thoroughly, and leaves circuitry to others. In terms of minority carrier density and migration he describes transistor action for us and develops in detail the equivalent circuit on which he has worked at Stanford University. The nonpragmatic scientist, curious about the nature of things for the fun in it, may be a little unhappy at the physics in the book. The relationships among elec- tron orbitals, holes, and conduction electrons are not clearly delineated. One hard-to-take model pictures hole migration in terms of a cluster of negative mass elec- trons that moves as a unit in a direction opposite to the force of the electric field and carries the hole along in the center. But the author frequently refers his reader to Shockley's basic Electrons and Holes in Semi- conductors, in which such matters are treated exactly. He assumes the Fermi-Dirac population formula with-out derivation and discusses Brillouin zone conduction in only a cursory manner. We must compliment author and editor on the plan- ning of the book. It has been wisely said that the way to teach a subject is to describe it completely in a page, then do it over again in a complete chapter, then at book length, and so on. This book follows this plan. Quantitative descriptions follow qualitative ones, and the reader is kept constantly aware of what is ahead of him. In a few years the term "circuit design" will not im- ply only vacuum-tube circuits as it does in current book titles. People will learn to use the transistor for its unique properties as a current amplifier, and this book will help them learn. Statistische Thermodynamik. By Arnold Miinster. 852 pp. Springer-Verlag, Berlin. Germany, 1956. DM 138.00. Reviewed by T. Teichmann, Lockheed Aircraft Corporation. While statistical and mechanical thermodynamics re- main among the most basic and unifying aspects of modern science, the development of new techniques and their application to new problems lead to an inevitable change of emphasis and approach. Professor Miinster has recognized this in his work which essays to satisfy the needs of the practicing physicist and physical chem- ist while yet retaining some of the aspects of a text- book and providing the student with a thorough foun- dation. Naturally such an ambitious approach has ne- cessitated certain restrictions in topics and methods— for example, only equilibrium states of matter are con- sidered and while the basic methods are thoroughly de- scribed, many possible variants are omitted—but the book remains the most comprehensive and palatable ac- count presently available. In order to make this great mass of material more accessible to beginners, the more advanced topics for application are "starred" and can be omitted at the first reading. The book consists of four sections dealing with the foundations of statistical mechanics, the theory of gases, the theory of crystals, and the theory of liquids. It is, of course, impossible in a review of this length to list all the significant subjects described but certain fea- tures of the treatment seem of particular interest. There are included a very thorough treatment of Gibbs' method and a discussion of the ergodic problem, and an ex- tended discussion of phase transition including the new methods of Lee and Yang, a description of the general theory of condensation, and discussion of the Born- Green theory of molecular distribution function. In the section on crystals, a detailed description of the Kram- ers-Wannier theory is given and Onsager's solution of the two-dimensional Ising problem is presented in the form first given by Montroll and Newell. The implica- tion of this method for three-dimensional problems is touched on, though not as thoroughly. The electron conductivity of metals and the Nernst heat theorem are also given thorough consideration in this section. The PHYSICS TODAY

1.1740562.pdf

Thermoelectric Behavior of Nickel Oxide G. Parravano Citation: The Journal of Chemical Physics 23, 5 (1955); doi: 10.1063/1.1740562 View online: http://dx.doi.org/10.1063/1.1740562 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/23/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Anomalous magnetic behavior in nanocomposite materials of reduced graphene oxide-Ni/NiFe2O4 Appl. Phys. Lett. 105, 052412 (2014); 10.1063/1.4892476 Magnetic behavior of reduced graphene oxide/metal nanocomposites J. Appl. Phys. 113, 17B525 (2013); 10.1063/1.4799150 Assembly and measurement of a hybrid nanowire-bulk thermoelectric device Appl. Phys. Lett. 89, 233106 (2006); 10.1063/1.2400199 Thermoelectrical study of ferromagnetic nanowire structures J. Appl. Phys. 99, 08T108 (2006); 10.1063/1.2176594 Measurement of the thermoelectric power of very small samples at ambient and high pressures Rev. Sci. Instrum. 70, 3586 (1999); 10.1063/1.1149964 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:28THE JOURNAL OF CHEMICAL PHYSICS VOLUME 23, NUMBER 1 JANUARY, 1955 Thermoelectric Behavior of Nickel Oxide G. PARRAVANO* The Franklin Institute Laboratories for Research and Development, Philadelphia, Pennsylvania (Received June 7, 1954) Measurements have been made on the thermoelectric power of several nickel oxide samples containing foreign ions. Using an energy model recently propose~ for nickel ox~d.e, t~e eff~ct of the n~ture and concentra tion of additions on the Fermi level, hole concentratIOn, and mobility IS denved and dlscu.ssed. For most of the samples hole concentration can be related to chemical composition. A case of change m.valence of the addition induced upon introduction into the matrix is found. The us: of the. therm.oelectnc effect to d: termine impurity in nickel oxide is discussed. The energy scheme of mckel oXld.e which emerges from ~hls work and that of Morin is emphasized as a drastic departure from the classical model of an electncal semiconductor. INTRODUCTION A FUNDAMENTAL problem arising in connection with oxide semiconductors is concerned with the extent to which present concepts of the band theory of solids can be successfully applied to the elucidation of the electronic structure of metal oxides. In the case of nickel oxide a discrepancy between theory and ex periment has been repeatedly pointed out in the litera ture. Although a d hole conductor, nickel oxide has an electrical conductivity of the same order of magnitude as cuprous oxide, which does not have unfilled d orbitals. Different qualitative explanations had been advanced to circumvent this theoretical difficulty. Recently the problem has received new impetus from the proposal by Morini of a new energy pattern, which has quanti tatively explained the experimental data. It is well known that the thermoelectric power of electrical conductors is a truly thermodynamic quan tity, part of which is thermostatic and part kinetic. Theoretical considerations and experimental investiga tions have shown that in some instances the kinetic contribution to the total effect is negligible.2 In these cases the thermoelectric power can be considered, to a very good approximation, a thermostatic quantity. Such a fortunate situation occurs with nickel oxide.1 Therefore thermoelectric investigations on nickel oxide afford a simple, yet interesting, method of deducing thermostatic characteristics of the oxide. Furthermore, it has already been pointed out that measurements of thermoelectric power of semiconductors are not likely to be affected by the state of aggregation of the ma teria1.3 This is a considerable practical advantage when substances not in single crystal form are used. ~ With these considerations in mind, the present work was undertaken in order to gain more knowledge about the effect of the i~troduction of foreign ions into the nickel oxide lattice. This communication refers to data * Present address: Forrestal Research Center, Princeton Uni versity, Princeton, New Jersey. 1 F. J. Morin, Phys. Rev. 93, 1199 (1954). 2 C. Herring and M. H. Nichols, Revs. Modern Phys. 21, 185 (1949); F. J. Morin, Phys. Rev. 83, 1005 (1951); 93, 1195, 1199, (1954); C. A. Domenicali, Revs. Modern Phys. 26, 237 (1954). 3 H. K. Henisch, Z. physik. Chern. 198, 41 (1951). 5 on the thermoelectric power of different nickel oxide samples, from room temperature up to 900°C in ai:. It will be shown that the interpretation of the expen mental results in terms of the energy diagram proposed by :Morin leads to a quantitative fit of the data, and allows a number of conclusions as to changes of the electronic electrochemical potential and hole mobility as a function of temperature and nature and concentra tion of additions. EXPERIMENTAL Nickel oxide samples were prepared from standard ized solutions of cp nitrates, mixed in appropriate amounts, dried at 110°C, and decomposed at 400°C for four hours. The resulting oxides were thoroughly ground and fired at 900°C in air for four additi?nal hours. A pure nickel oxide sample was prepared In a similar way; it was a gray-green material, whic~ had a B.E.T. surface area of 2.2 sq mig. The chemIcal de termination of the excess oxygen4 of this sample gave a value of 7XIo-4 gO/gNiO. The main source of error in measurements of thermo electric power lies in the determination of the tempera ture difference !1T between the two ends of the sample. In order to reduce this error, the thermal contact be tween the thermocouple and semiconductor should be good and the thermocouple itself should not appreci.ably change the temperature distribution in its surroundIng.s. To meet the first requirement a large metal surface, m the form of a platinum foil, was used to contact the sample. The foil was used also as one member of a thermocouple. In this way optimum thermal contact between sample and thermocouple was assured. Further more, in order to minimize heat conduction between thermocouple junction and leads, the latter were made out of wires of as small cross section as was compatible with low electrical resistance and chemical homogeneity. The apparatus finally adopted is shown in Fig. 1. The powdered sample was placed inside a short l~ngth of Vycor tubing (7 mm i.d.), between two ~at platInum electrodes, which were cemented on lavite supports. Each electrode had a small opening in its center (not 4 W. Krauss, Z. Elektrochem. 53, 320 (1949). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:286 G. PARRAVANO lCM. FIG. 1. Thermoelectric cell. shown in Fig. 1), through which a Pt-PtRh thermo couple was placed. The tip of each of these two thermo couples was electrically insulated from the sample. A second thermocouple was welded onto each electrode on the lavite side of the platinum foil. The eight ends of the thermocouples were soldered to copper leads from the switchboard, and each junction was set in a small paraffin cup immersed in an ice water bath. The two lavite supports were kept aligned and the electrode assemblies were tightly pressed against the sample by means of two stainless steel rings, which were clamped together with two stainless steel bolts. In order to reduce thermal convection currents, the apparatus was mounted horizontally in the furnace. The temperature of the furnace was controlled by an electronic thermo regulator. Supplementary heating was provided by means of two separate coils of nichrome wire, wound around each lavite support. T 500 ~~ 450 ~3 o OI&J >I&J 400 0: ",,, ul&J I&JO f3ffi 350 llllL 300 • • • FIG. 2. Effect of AT on the thermoelectric power: o NiO, t=312°C, • NiO+Li.O, t=287°C. The temperature of each electrode, TI and T2, could be separately measured by means of the welded thermo couples. The difference in temperature, AT=T 1-T2, between the two electrodes, could be best obtained by direct measurement. For this purpose it was found that the welded thermocouples were unsuitable, because, as the resistance of the sample decreased, an appreciable amount of current would flow through the sample, thus introducing a serious error in the determination of AT. This latter therefore was measured with the aid of the central thermocouples, which were electrically insulated from the sample by means of high tempera rure cement. The insulation was continually checked 7 FIG. 3. Direct current resistivity curves for nickel oxide contining foreign ions. during an experiment. The thermoelectric voltage, developed by each sample, was measured between the two platinum wires of the welded thermocouples, with the aid of a Leeds and Northrup type K-2 poten tiometer. A high resistance mirror galvanometer was used as a null instrument. A second K-2 potentiometer was used to measure AT and the temperature of one of the two ends of the sample. In high resistance samples AT measurements were also checked by separate measurements of TI and T2• The ambient temperature was taken as the average of the temperatures of the two ends of the sample. With these precautions the de pendence of the thermoelectric power on temperature gradient was small (Fig. 2). All measurements were carried out with AT= 15-20°C. For all samples the This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:28THERMOELECTRIC BEHAVIOR OF NICKEL OXIDE 7 cold end of the specimen was found positive (hole conduction). Resistance measurements were taken with a test meter, using a calibrated rheostat. Due to the limited use of resistivity data in this work, the approxi mations involved in the determination of this parameter by means of the above apparatus, would not invalidate the general conclusions reached in this work. The re producibility of the measurements was tested by re peated experiments under similar conditions. Data agreed within ±2.S percent. EXPERIMENTAL RESULTS For all samples conductivity data could be fitted to a Boltzmann expression with more than one term (Fig. 3). However, if the degeneracy temperature is calculated with the well known formula: (1) where h is the Planck constant, k the Boltzmann con stant, p the hole concentration, and m*, the effective ~ ~ .-------------------, IJj ow >::! ><C) OW we ~ffi UlQ. 500 FIG. 4. Seebeck voltage per degree as a function of temperature. electron mass (equal to the free electron mass), it should be expected that some samples would show degenerate behavior. The expected degenerate be havior was not found experimentally, however. Its absence can be explained by assuming that Eq. (1) is not applicable to nickel oxide and/or that the effective electron mass is much larger than the free electron mass. The behavior of the conductivity and its tempera ture dependence as a function of the nature of the added ions is consistent with previous observations.' At high temperatures the conductivity of all samples tends to a common value, probably because of the approaching range of intrinsic behavior. No appreciable difference in conductivity from that of the pure oxide was found in samples containing Mg, Th, AI. It is in teresting to note that additions of Cl, which conceivably enter the lattice substitutionally, produce effects similar to the introduction of cations of charge greater than two. 5 Verwey, Haamyman, and Romeijn, Chern. Weekblad. 44, 705 (1948). w~~---------------------, ~~ ~5 ow >w ""a: ~ffi COO wa: ~~ 600 200 FIG. 5. Seebeck voltage per degree as a function of temperature. Data on the Seebeck voltage per degree for different samples are shown in Figs. 4 and S. The results for samples containing alumina and thoria were found to be similar to those for pure nickel oxide. DISCUSSION It is assumed, in accordance with the energy pattern proposed by Morin,! that nickel ions form localized levels in the energy gap between the filled and empty sp oxygen bands (Fig. 6). This assumption leads, as shown by Morin, to the conclusion that the kinetic energy term of the expression for thermoelectric power is negligible. Thus for a d hole conductor one has (2) in which Q is the thermoelectric power in volts per de gree and EF is the distance between the Fermi and the filled d levels from nickel ions (Fig. 6). Essentially this energy scheme can be considered a quantitative inter pretation of an earlier suggestion by Verwey6 on the conduction mechanism in nickel oxide. The possibility that electronic conduction in group VIII oxides involves ion pairs, like Ni+2-Ni+3, has, of course, been widely accepted in the case of magnetite. Values of EF, calcu lated according to Eq. (2) from Seebeck voltage data, are presented in Figs. 7 and 8. These plots indicate the position of the Fermi level at different temperatures and with different impurities and impurity concentra tions. For all samples EF moves upward as the tempera- WR/'#///////##//0///0 EMPTY SP BAND OF 02- - - - - - - EMPTY LEvns OF Hi+' - - - - - DONOR LEVEL ---------1-;---------- FERMI LEVEL - - - - - ACCEPTOR LEVEL --- - - - OCCUPIED d LEVELS OF HI+' W//////'////?7/,/;///ff///~ FILLED SP BAND OF 0'- FIG. 6. Energy diagram of nickel oxide (1). ----- 8 J. H. de Boer and E. J. W. Verwey, Proc. Phys. Soc. (London), 49, (extra part) (1937). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:288 G. PARRAVANO FIG. 7. Effect of temperature and foreign ions on the Fermi level. >- ~ ... UJ 0.1 FIG. 8. Effect of temperature and foreign ions on the Fermi level. ture increases. From the temperature dependency of EF it is possible to deduce that in pure nickel oxide levels at about 0.13 and 0.30 ev are present, which become ionized at 100°C and 300°C, respectively. Upon intro duction of Ag and Li the position of EF is shifted down ward by the formation of low lying acceptor levels. The case of Ag is noteworthy. Although the ionic radius of Ag+ is much larger than that of Ni+2, the data indicate that a definite interaction has occurred between silver and nickel oxides. The silver-containing sample was submitted to x-ray analysis with the hope of ob taining supporting evidence for the interaction between nickel oxide and silver oxide. The lattice constant of this sample was found slightly larger (4.1775 A) than that of pure nickel oxide (4.1767 A), but the poor resolution of the diffraction lines prevented reaching a definite conclusion. Increasing concentration of lithia in nickel oxide lowers EF and forms new levels at 0.09 and 0.05 ev. The behavior of sample NiO+Li 20 CD is anomalous at temperatures below 150°C. Additions of tungsten trioxide raise the position of EF, producing an impurity level at 0.34 ev. Thus these results parallel those obtained from measurements of electrical con ductivity, but are free from all the complicating fea tures affecting conductivity data. At temperatures be low 230°C, EF is raised by additions of nickel chloride, while chromia produces a high lying level at 0.32 ev. For all samples investigated EF was found to continue to rise with increasing temperature at high tempera tures, which confirm the conclusion that the samples have not yet entered the intrinsic region (EF(intrinsic) =0.96(1)). Hole concentration, p, can be estimated from total level density: p=N·exp( -EF/kT) (3) where N = 5.6X 1022 nickel atoms per cc of crystal. Figures 9 and 10 have been computed using Eq. (3). From these plots the effect of the nature and concentra tion of additions on hole concentration and hole ioniza tion energy can be evaluated. It is found that the con centration of holes is increased and their ionization energy correspondingly decreased by adding mono charged cations to nickel oxide, while the opposite effect occurs upon addition of cations having charges higher than two. The behavior of sample NiO+ LbO CD is again found to be anomalous. In this sample [Ni+3] =9.7X1020 ions per cc, while [Li+]=6.7X1019 ions per cc. It seems therefore that when [Li+] < [Ni+3], the addition of lithia to nickel oxide decreases the hole concentration and raises the position of EF with respect to the pure oxide. This behavior is consistent with z 0_ t=i <t" 0::" ... ~ z ~21 z o " UJ ...J 020 J: '" 2 19 FrG. 9. Hole concentration as a function of temperature and foreign ions. 20 19 eNiO vNiO+MgO DNiO+Th O2 FrG. 10. Hole concentration as a function of temperature and foreign ions. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:28THE R MOE LEe T RIC B E H A V lOR OF N I eKE LOX IDE 9 TABLE I. Additions of foreign ions to nickel oxide. Addition concentration Hole concentration (chemical) (electrical) Sample atoms per cc holes per cc NiO 9.8XlO20 NiO+Li20 CD 1.07X1()20 8.6X 1()20 NiO+Li20 (!) S.6X 1()20 6.3X1Q21 NiO+LhO (!) 1.02 X 1021 1.6X1Q22 NiO+Ag 2O 1.59 X 1021 2.SX1Q21 NiO+Cr a02 6.0X1()20 4.0X 1()20 NiO+WO a 8.0X 1()20 1.6X 1()20 NiO+Ce02 LOS X 1021 4.3X1Q2° NiO+NiCb S.2SX 1()20 4.2X 1()20 data on the lattice constant of solid solutions of NiO +Li20.7 Upon additions of lithia, the lattice constant of nickel oxide increases until [Li+]=[Ni+ 3]. This is caused by the formation of one Ni+2 for every Li+ intro duced, by the filling up of cationic vacancies. Increased amounts of lithia beyond the above value were found to decrease the lattice constant, because of the forma tion of one Ni+3 for every Li+ introduced. From Figs. 9 and 10 the concentration of holes can be estimated and compared with the chemical composition of the samples. The results are presented in the follow ing Table 1. The nickel oxide used was not spectroscopically pure. Its main impurity, sodium, was probably responsible for an impurity level at about 0.15 ev with a hole con centration of 9.8X 1020 holes per cc. The anomalous behavior of sample NiO+ LbO CD has already been dis cussed, and, with that data from Table I, the anomalous behavior can almost quantitatively be accounted for. In samples NiO+ Li20 ® and ®, the hole concen tra tion is much larger than the amount of lithium added. No explanation can be offered at present for this discrep ancy. However, it must be recalled that the treatment of the thermoelectric data followed in this work requires that QT»kT, but in samples highly doped with ac ceptors this is not true. In samples NiO+ Li20 ® and ® QT"'-'kT. For all other samples the ratio of the concentration of addition to the concentration of holes formed or destroyed in the pure oxide by the addition has been computed and is reported in the last column of Table 1. This ratio is found to be almost unity, except in the case of the sample containing ceria, but this can be explained by assuming that the solubility limit of ceria in nickel oxide is less than 2.3 percent by weight. The agreement between the concentration of addition and the concentration of holes, formed or destroyed in the pure oxide by the addition, enlarges the heuristic value of the energy model, proposed for nickel oxide inasmuch as this treatment can be used to analyze nickel oxide electrically, with regard to deviations from stoichio metric composition and with regard to impurity con tent. One more interesting fact emerges from inspection 7 L. D. Brownlee and E. W. J. Mitchell, Proc. Phys. Soc. B65, 710 (19S2). Holes found Holes destroyed Ratio (referred to pure NiO) (chemical! holes per cc electrical) 1.2X 1()20 0.89 S.3X1Q21 0.10 1.51 X 1Q22 0.068 1.52 X 1021 1.04 S.8X 1()20 1.03 8.2X 1()20 0.98 S.SX 1()20 1.82 8.2X 1()20 1.06 of Table 1. In the sample containing tungsten, this impurity was added as tungsten trioxide. However, the data show that, to a very good approximation, for every tungsten atom added one positive hole was destroyed. Therefore the hexavalent tungsten ion has been changed, upon introduction into the nickel oxide lattice, to some sort of trivalent ion. It seems, therefore, that thermo electric data on "nickel oxide" can be used to reveal the valence acquired by the foreign ion upon introduc tion into the nickel oxide lattice. The process responsible for the ionization of acceptors can be visualized as follows: Ni+3~Ni+2+p. (4) The oxygen equilibrium, viz: 4Ni+2+0~4Ni+3+2cv+2o--2 (cv=cationic vacancy), does not appreciably affect the bulk properties of the oxide up to temperatures of the order of 0.5T m(1115°K).8 At lower temperatures the number of Ni+3, ionized or not, can be considered constant and equal to the excess oxygen. The equilibrium constant of reaction (4) is K = [Ni+3]/[Ni+2][p]. Since [Ni+2] is practically con stant, from the concentration of holes and the excess oxygen it is possible to calculate a value for K. For pure nickel oxide, at 380° and 450°C, it is found that K = 7.05 and 63, respectively. From these values, LlF653= -1.5, LlF723= -3.5 kcal/mole, LlH = -14.5 kcal/ mole, LlS 66~LlS 723 = 41.4 eu. These values are highly speculative, but they seem reasonable. Hole mobility, calculated from conductivity and hole concentration, was found dependent on the type and concentration of impurity added, and the temperature. This agrees with the findings of Morin.l Cations with a charge higher than two decrease the mobility and in crease its activation energy, while the opposite effects occur with mono charged cations. This result shows that great care should be exercised in deducing changes in charge carrier concentration from conductivity data in metal oxides of the type under discussion. This, un fortunately, has not been widely done in the past, as it has been customary to assume a constant mobility value over large variations of impurity content, tem- 8 Bevan, Shelton, and Anderson, J. Chern. Soc. 1948, 1729. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:2810 G. PARRAVANO perature, and nature of the impurities. It is clear that in the interpretation of future results for this type of oxides, it will be necessary to take into account changes of electron or hole mobility with temperature and impurity content. CONCLUSION The picture of nickel oxide which emerges from the work of Morin and the work presented in this paper is a drastic departure from the classical one of a semi conductor (germanium, silicon, zinc, or cuprous oxide). Oxides containing electronic defects derived from d electrons do not seem to have a typical conduction band. Rather the conduction electrons migrate along localized levels. The formation of ion pairs for such a process depends on the energetic possibility of the lattice to deviate from the stoichiometric composition. The ex change of electrons within ion pairs is probably con strained by the presence of the large oxygen anion. This THE JOURNAL OF CHEMICAL PHYSICS constraint is reflected in the low electronic mobility of these oxides, and in the fact that it is necessary to furnish energy to "activate" the electrons, even though they have already been "separated" from the ionic cores by an ionization mechanism. Clearly, under these conditions, electrons are far from free, and such a solid should be considered to be more a semi-insulator than a semiconductor. ACKNOWLEDGMENTS I wish to thank Dr. F. J. Morin, of the Bell Telephone Laboratories, for graciously providing me with his data before publication, and Dr. C. A. Domenicali, of this Laboratory, for critical discussions and much helpful advice in the experimental part. Many fruitful discus sions on this subject with Dr. D. P. Detwiler are grate fully acknowledged. Finally I would like to thank Dr. F. C. Nix for his interest in this work. VOLUME 23. NUMBER 1 JANUARY. 1955 A Phenomenological Theory of the Soret Diffusion JAMES A. BIERLEIN Aeronautical Research Laboratory, Wright Air Development Center, Wright-Patterson Air Force Base, Ohio (Received February 22, 1954) The classical Soret diffusion problem is solved analytically for the case of unrestricted composition in a binary liquid system, taking account of the temperature-variation of the density of the mixture. Previous treatments have neglected the phenomenon of thermal dilatation entirely and have not developed a unique general solution which applies both to dilute and to nondilute mixtures. The rigorous solution derived in the present work is similar in form to de Groot's well-known equation for dilute mixtures, but contains additional parameters charncterizing the initial composition of the system and its coefficient of thermal expansion. These parameters disappear in the asymptotic approximation for a vanishing temperature gradient, but this approximation differs from that proposed by de Groot, even for dilute solutions. The asymptotic expression has practical importance for estimating the Soret coefficient and the ordinary diffusion coefficient of a system from experimental data taken during the thermodiffusional unmixing period; some examples of its application in this connection are discussed. WHEN a temperature gradient is applied to a convection-free binary liquid confined between horizontal plates, the pure Soret effect is observed; a composition gradient develops with time and asymptoti cally approaches an equilibrium state characterized by the equality of the rates of the opposing effects of ordinary diffusion and thermal diffusion. A mathe matical analysis of the transient behavior in this process was first given by de Groot,! but his treatment contains two implicit assumptions which render it highly special ized. These assumptions are that the mole fraction of one component of the mixture is essentially unity and that the density of the system is uniform throughout. The first of these conditions restricts the theory to dilute solutions, and the second (implying a vanishing 1 S. R. de Groot, Physica 9, 699 (1942). thermal expansion coefficient over a finite temperature range) is logically unsatisfying because it does not correspond to the behavior of any known liquid. Thomaes2 has reproduced de Groot's result in a slightly different form, and has also indicated the use of a chordal approximation3 to extend it to the case of systems that are not dilute; he also neglects the phenomenon of thermal expansion. Inasmuch as the theory of the Soret effect has appli cation in the interpretation of thermodiffusion experi ments,4 .• it is of interest to re-examine the problem with a view to developing a single solution which is ap- 2 G. Thomaes, Physica 17, 885 (1951). 3 R. C. Jones and W. L. Furry, Revs. Modern Phys. 18 151 (1946). ' • S. R. de Groot, J. phys. radium 8, 129 (1947). • C. C. Tanner, Trans. Faraday Soc. 49, 611 (1953). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 164.76.102.52 On: Thu, 09 Oct 2014 12:16:28

1.1722650.pdf

Measurement of Minority Carrier Lifetimes with the Surface Photovoltage E. O. Johnson Citation: Journal of Applied Physics 28, 1349 (1957); doi: 10.1063/1.1722650 View online: http://dx.doi.org/10.1063/1.1722650 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/28/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantitative analysis of copper contamination in silicon by surface photovoltage minority carrier lifetime analysis AIP Conf. Proc. 550, 215 (2001); 10.1063/1.1354400 Identification of Cr in ptype silicon using the minority carrier lifetime measurement by the surface photovoltage method Appl. Phys. Lett. 68, 3281 (1996); 10.1063/1.116574 A comparison of carrier lifetime measurements by photoconductive decay and surface photovoltage methods J. Appl. Phys. 49, 2996 (1978); 10.1063/1.325149 Measurement of Minority Carrier Lifetime in Silicon J. Appl. Phys. 27, 489 (1956); 10.1063/1.1722409 Measurement of the Lifetime of Minority Carriers in Germanium J. Appl. Phys. 26, 414 (1955); 10.1063/1.1722009 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Sun, 21 Dec 2014 23:05:59JOURNAL OF APPLIED PHYSICS VOLUME 28, NUMBER 11 NOVEMBER, 1957 Measurement of Minority Carrier Lifetimes with the Surface Photovoltage E. O. JOHNSON RCA Laboratories, Princeton, New Jersey (Received June 8, 1957) The surface photovoltage method uses the junction-like properties of a semiconductor surface as a means for studying the decay of excess carriers. No more than capacitive contact with the specimen is required to detect the surface photovoltage which, in the millivolt range, is a linear function of the excess carrier density. Theoretically, the surface method yields exactly the same carrier decay constant as the photoconductivity method when the lowest diffusion mode prevails in the specimen. This has been quantitatively confirmed for Ge: only qualitative confirmation has been made with Si. For surfaces tending toward inversion, the surface method gives larger signals than the photoconductivity method, particularly for semiconductors with low intrinsic carrier density. On the other hand, for accumu lation layer surfaces the surface method usually gives smaller output signals. 1. INTRODUCTION THE decay of excess carriers in semiconductors has been measured in a number of ways. Most commonly known, perhaps, are the Morton-Haynesl and photoconductivity decay methods.2 In the former, lifetime is deduced from the decay, with distance, of the excess minority carrier density injected by a light spot. In the latter, the lifetime is determined by the time decay of the photoconductance of a specimen that has been subjected to a brief flash of light. Both of these techniques require ohmic or rectifying contacts to the specimen. A recent publication describes a new infrared adsorption technique3 that does not require physical contact to the specimen. It is the purpose of the present paper to describe a particularly simple lifetime method requiring only capacitive contact to the specimen. This method takes advantage of the linear dependence of the surface photovoltage on the excess carrier density at low signal levels. The surface photovoltage is analogous to the junction photovoltage (photovoltaic effect) and arises from the surface space charge region which has many properties in common with the depletion region of a junction. The surface photo voltage is generated by a brief light flash and is detected by a small electrode capacitively coupled to the surface of the specimen. The signal is fed to an oscilloscope where the carrier decay constant is determined in the usual manner. 2. EXCESS CARRIER DENSITY The surface photovoltage monitors the excess carrier density adjacent to the surface. If the lowest diffusion mode prevails in the specimen, the surface density of excess carriers decays with the same time constant as carriers elsewhere in the specimen. Hence, the surface measurement should yield the same decay constant as the photoconductive decay method which monitors the integrated density distribution. This 1 L. B. Valdes, Proc. lnst. Radio Engrs. 40, 1420 (1952). 2 D. T. Stevenson and R. J. Keyes, J. Appl. Phys. 26, 190 (1955). 3 N. J. Harrick, J. Appl. Phys. 27, 1439 (1956). means that virtually all the considerations that apply to the photoconductivity method apply to the surface method. In particular, the contribution of the bulk and surface recombination to the total effective lifetime is the same in both methods. Thus, the surface method should be as capable of measuring bulk and surface recombination properties as the photoconductive decay method. The methods do differ, however, in that the surface method can be completely free of carrier drift effects due to an externally applied electric field. The surface technique can suffer from loss of signal if the surface recombination velocity, s, is high. This follows because the lowest mode excess carrier density near the surface decreases as the value of s increases. It is easy to show, however, that the loss of surface signal due to the depreciation of the carrier density at the surface will be, in all but the most extreme cases, no greater than about 50%. 3. SURFACE PHOTOVOLTAGE A picture of the energy bands at the surface of an n-type semiconductor under equilibrium conditions is shown in Fig. 1.4 For purposes of discussion the surface is shown with a negative charge in the surface states. Because of the over-all electrical neutrality require ment, this trapped charge is balanced by an equal positive charge composed of donors and holes in the surface space charge region. The equilibrium potential Yo across the space charge region can be deduced in terms of the total space charge from Poisson's equation. fi Most of the charge trapped at a semiconductor surface appears to be in "slow states" which are in, or on, the surface "oxide" layer and exchange charge with the bulk material with time constants of the order of seconds. The relatively less numerous "fast states," which have time constants in the microsecond range, do not have any effect6 on the main conclusions of the present discussion, aside from giving rise to surface 4 R. H. Kingston, J. Appl. Phys. 27, 101 (1956). • W. H. Brattain and C. G. B. Garrett, Phys. Rev. 99, 376 (1955). 6 E. O. Johnson (to be published). 1349 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Sun, 21 Dec 2014 23:05:591350 E. O. JOHNSON TRAPPED{ CHARGE IN SURFACE / SURFACE SKIN, OR "OXIDE" -+-+-.... + + + + + 1" + --DONORS FERMI ~EVEL STATES $$ SPACE CHARGE REGION ELECTRON POTENTIAL ENERGY LOISTANCE FIG. 1. Energy diagram of semiconductor surface. recombination, and so do not need to be considered in the analysis. If excess carriers are injected with light and decay at a rate that is fast compared to the slow state time constant, the charge in these states will not have time to change. Now, because over-all electrical neutrality must be maintained, the surface space charge will also remain constant. This, however, is necessarily ac companied by a flattening of the energy bands at the surface. The resulting change LlY in electrical potential across the space charge region is termed the surface photovoltage.5 This is defined by Ll Y = Y -Yo, where Y is the space charge potential when excess carriers are present. Both Y and Yo are expressed in kTle units and are negative when the energy bands bend upward, as shown in Fig. 2, and positive when the bend is downward. The surface photovoltage has almost the same functional behavior as the junction photovoltage with respect to the excess carrier density.6 It is a straight forward matter to deduce the surface photovoltage from Poisson's equation and the usual carrier statistics, if the carrier diffusion lengths are assumed long com pared to the width of the surface space charge region (",,10-5 cm). An implicit expression yielding the surface photovoltage is6 (1) where Llp is the fractional increase in hole density due to injection, Fo, is the space charge factor when no excess carriers are present, Fy is the corresponding factor when excess carriers are present, and A is the doping factor. The fractional increase in hole density is defined by Ll p = Ll pi po = Ll PiA ni; the doping factor, by A = (pol no)! = pol ni = n;j no; and the space charge factor Fy by Fy=A(e-y -1)+A-I(e Y -1)+ (A-A-I)Y. The expression for the factor F 0 is obtained by replacing Y by Yo. The excess hole density is defined by Llp = p-po, where p, po, and the other quantities used above have their usual meanings. If excess holes and electrons are light-injected at equal rates, expressions analogous to those above obtain for Lln and Lln. Equation (1), as well as those to follow, are written in terms of hole densities and are most conveniently applied to n-type material. The complimentary expres sions for p material, written in terms of electron densities, can be obtained from the appropriate relations noted above. A linear relationship between the excess hole density Llp and the photovoltage LlY, LlY Ae-Yo-A-leYo- (A-A-l) ni' (2) is obtained if th e exponential terms in Eq. (1) are linearized.7 Equation (2) is valid within a few percent if 10-1 (eYo+e-Yo-2) LlY$------ eYo-e- Yo (3) Since Yo is almost always found to exceed one kTle unit in magnitude, the linear approximation will hold at room temperature if LlY is a few millivolts, or less. In any case, the form of the photovoltage decay curve will provide a fairly good check of linearity. The more complicated case, where there is a background density of excess carriers, also gives linear results at low levels of the incremental signal. If Yo is zero, then LlY vanishes and the surface technique obviously fails. Fortunately, such a case is unlikely unless special effort is exerted to bring it about, such as by adjustment of the ambient.4 For strong inversion layer conditions, where Yo is strongly negative and the exponential terms e-Yo SPARK LIGHT SOURCE (PULSE RATE -5/5EC) t#f 0-'" AIR SPARK I GAP --L--"""", t -~----' c::::> FOCUSSING ~ __ ~ ~ENS ,-_-----, SPECIM'E'N:j~_===~r=l) PREA'-P MICA - GAIN -10 BACK _ CONTACT SURFACE PICKUP ELECTRODE TEKTRONIX TYPE 532 (AIR OR MICA DIELECTRICI FIG. 2. Apparatus for carrier decay measurements. 7 Potentials developed inside the bulk material due to the Dember effect would also be linearly related to the excess carrier density at low densities. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Sun, 21 Dec 2014 23:05:59CARRIER LIFETIMES AND SURFACE PHOTO VOLTAGE 1351 dominate, Eq. (2) reduces to l:1p no :1Y=--=-:1p. A ni n,:2 (4) Since A < 1 for n-type material, the surface signal increases linearly with the doping constant. This follows because the hole charge dominates the surface space charge when the surface has a strong inversion layer. In the n-type specimen the holes decrease inversely as the n-type doping. Thus, if very few holes are present, due to heavy doping, the injected holes will tend to have a relatively large effect on the surface space charge and hence on the surface photovoltage. Furthermore, if the doping is held constant to keep the specimen resistance constant, :1Y increases in versely as n;2. Thus, high band gap material should yield the largest signals for a given excess carrier density. Also, since ni decreases exponentially with temperature and :1Y, in volts, decreases only linearly, the surface signals should improve as the temperature is decreased. For intermediate negative values of Yo, :1Y has a value intermediate between zero and that given by Eq. (4). For accumulation layer conditions, where Yo has a large positive value and the eYo terms dominate, Eq. (4) reduces to :1p :1p :1Y=-A-=-, (5) 11; no and :1 Y decreases with the doping. This is understand able from the fact that majority carriers dominate the space charge under accumulation layer conditions. The intrinsic condition, where A= 1, is the limiting case of no doping. The same general conclusions noted above also apply to p materiaL 4. COMPARISON OF THE SURFACE PHOTOVOLTAGE AND PHOTOCONDUCTIVITY SIGNAL AMPLITUDES It is of interest to compare the size of the surface signal :1Y with the signal that would be obtained from a photoconductivity measurement on the same speci men. For the photoconductivity measurement consider a series circuit composed of the specimen, a battery of potential VB, and a signal resistor equal to the dark resistance of the specimen. The signal resistor is made equal to the dark resistance of the specimen to maximize the photoconductivity signal output. The algebra is also simplified. A simple computation shows that the signal :1 V developed across the signal resistor is given by :1V =~[l+blA:1 , VB 4 A2+b.ki P (6) for the case where V /V B«0.5, and b is the ratio of the electron to hole drift mobility. For simplicity it is assumed that the excess carrier density is uniform throughout the specimen. Equation (6) shows that the photoconductivity signal decreases with doping. This follows because the conductivity change due to excess carriers will be proportionately less if there are many majority carriers already present due to heavy doping. Comparison of Eqs. (4), (5), and (6) shows that the surface and photoconductivity methods give comparable signals for accumulation layer surfaces. However, for inversion layer surfaces the surface signal excels by the doping factor squared. For a given resistivity the majority carrier density is fixed. Hence, with high band gap material, where l1i is low, the doping factor is large. Thus, the surface technique can be very useful with high band gap material when a tendency towards inversion layer conditions is present. For example, for 7 ohm-em n-type Ge, the doping factor is 1/50 and the surface method gains by the factor 502• For n-type Si of the same resistivity it gains by the factor 108• This factor could overcome almost any conceivable loss in signal due to a high surface recombination velocity. In practice, surfaces are usually found in an intermediate state, instead of in the complete inversion that would give rise to the large gain factors noted above. Hence the gain of the surface method could be expected to be less than noted above. 5. APPARATUS The experimental arrangement is shown in Fig. 2. The excess carriers in the specimen are generated by the microsecond light flashes from the spark gap during the periodic discharges of the 2000-l'f condenser. 8 The O.S-cm air gap sparks at a repetition rate determined by the supply voltage and the capacitance and resistance of the charging circuit. For the values given in the figure the repetition rate is several per second when the supply voltage is adjusted to about six kilovolts. The excess carriers in the specimen under semi transparent electrode generate a surface photovoltage that is picked up by this electrode and passed on through the preamplifier to the oscilloscope. The electrode used in these experiments was a perforated, flat metal sheet with an optical transparency of about fifty percent. It had an effective area of about i cm2• For optimum signal output it is desirable to have the electrode area no larger than the spot of illumination. Electrode areas as small as 10-2 cm2 (the flattened end of a wire) have been used successfully. The dielectric between the electrode and the specimen can be air or mica; no significant difference has been noted between the two. The preamplifier input impedance should be high compared to that of the capacitance that couples the signal electrode to the specimen at the frequency corresponding to the inverse of the specimen lifetime. The equipment used in these experiments could handle 8 The spark gap light source used in these experiments was built by J. Gannon and A. Moore of these laboratories. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Sun, 21 Dec 2014 23:05:591352 E. O. JOHNSON photovoltage decays with time constants down to a few microseconds. The circuit return was capacitively coupled to the specimen as shown. Ohmic or rectifying contacts could also be used if desired. The salient requirement for the back contact is that it couple to a region of surface that is not exposed to excess carriers. Excess carriers near this contact would generate a voltage that might obscure or be confused with the desired signal at the photovoltage electrode. The semiconductor specimens used in these experiments were 10 to 50 mils thick and had rectangular areas ranging from about one quarter to one cm2• It was necessary to locate the specimen in a shielded box well isolated from shocks and vibrations. It is desirable to have the electrodes make rigid capacitive contact with the specimen. The additional simple circuitry for making photo conductance measurements is not shown in the figure; it is described in Sec. 4. 6. EXPERIMENTAL RESULTS The measurements were carried out primarily for a comparison of the surface and photoconductive methods. Tens of n-and p-type Ge specimens, five n-and p-type Si specimens, and one n-type InP speci men were tested. The etches CP4 and No. 59 were used in the initial surface treatment. Room air, dry oxygen, and dry ozone were used as ambients on some specimens to vary the surface potentials and also the surface recombination velocity. The carrier decay time con stants ranged from a few microseconds to hundreds of microseconds. With Ge, whenever the surface and photoconductivity techniques could be compared, they gave exactly the same carrier decay time constants. The comparisons were made by superimposing oscilloscope photographs on top of each other. No differences in the decay curves were ever detected beyond the decay time when the higher diffusion modes had disappeared. Roughly comparable signal amplitudes were obtained from Ge with both measuring techniques when the surface tendedlO toward an inversion layer. In these cases the signal amplitudes from both methods were a few millivolts. The battery potential in the photo conductivity measurements was 1.5 volts. With Ge surfaces having an accumulation layer, no measurable surface signal was obtained with the rather weak light source used. Very likely, I Yol was not sufficiently large to make Eq. (5) apply. With Si, and the one InP specimen tested, the surface signals came from a surface that tended toward in version and were, as expected, very large compared to the photoconductivity signal. In fact, the photo- 9 To be published. 10 The term "tended" is used here and afterwards to denote that the energy bands at the surface are bent in a direction to enhance the minority carrier density at the surface. In the absence of detailed surface potential measurements it is not possible to say whether or not the surface conductivity type is actually inverted. (o ) U IblQ IoIU (dlQ (elD (Ilt=] (91~ (hID ('ID IilU FIG. 3. Reproductions of oscilloscope photographs of surface and photoconductive signals. p-Type 100-cm Ge with n surface. (a) Decay of surface signal. 50Ilsec/div. (b) Tail of surface decay. 20Ilsec/div. (c) Decay of photoconductivity signal. 50llsec/div. (d) Tail of photoconductivity signal. 20 Ilsec/div. n-Type 3.50-cm Si with p surface. (e) Decay of surface signal, 100 Ilsec/div. No background illumination. (f) Decay of surface signal, 100 Ilsec/div. Background illumi nation. n-Type O.lO-cm InP with p surface. (g) Decay of surface signal, 100 Ilsec/div. No background illumination. (h) Decay of surface signal, lOOllsec/div. Background illumi natioIJ. p-Type 30-cm Ge specimen 0.14-cm thick with n surface. (i) Decay of surface signal, 50Ilsec/div. Light and electrode on same side of specimen. (j) Decay of surface signal, 50 Ilsec/ div. Light and electrode on opposite sides of specimen. conductivity signals were extremely small or not detected at all, whereas the surface signals were in the millivolt range and easily measurable. In the few cases where comparisons of decay constants could be made, the agreement between the two methods was reasonably good. There was, however, always some ambiguity from the fact that the decay constants were affected by the varying background illuminations required to reduce the nonexponential tails on the decay curves. These tails were presumably due to trap effects.u The surface technique seems to be more sensitive to these effects than the photoconductivity method. This was not studied in detail. Germanium surfaces can be pushed toward n type with wet air4 and No.5 etch.9 A surface freshly etched 11 J. R. Haynes and J. A. Hornbeck, Phys. Rev. 90, 152, (1953). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Sun, 21 Dec 2014 23:05:59CARRIER LIFETIMES AND SURFACE PHOTOVOLTAGE 1353 with CP4 also tends toward n type.6 To drive a Ge surface toward p-type dry oxygen, ozone,4 and almost any strong oxidizing agent, like quinone,12 is effective. Silicon surfaces respond roughly similarly to gaseous ambients. There is no information on the effect of No.5 etch and quinone on Si surfaces. The dichromate treatment13 of Nelson and Moore is effective in driving a Si surface toward P type. Presently, little is known about the behavior of III-V compound surfaces. Air aged Si specimens were always found to have surfaces which tended toward inversion, regardless of bulk conductivity type. Typical oscilloscope photographs of decay curves obtained from both the surface and photoconductivity methods for p-type Ge are reproduced in Figs. 3(a)-3(d). Figures 3 (a) and (b) refer to the surface method, and Figs. 3(c) and (d) to the photoconduc tivity method. The polarity of the surface signals indicates that the equilibrium value of the surface potential, Yo, was positive. That is, in the conventional band picture, the energy bands bend downward. The photovoltage curve in (a) shows the entire course of the decay. A rapid decay is seen close to t=O. This is followed by a portion of the curve that decays relatively slowly. Finally, beyond about 100 to 150 J.lsec, the curve decays exponentially with very closely the same time constant of the photoconductive curve in (c). The equality of the time constants is readily seen from a comparison of (b) and (d) where the tails of the decay curves are displayed on an expanded time scale. Presumably, the nonexponential behavior of the initial portion of the photovoltage decay curve stems from higher diffusion modes. The surface method seems to be more prone to nonexponential behavior during the initial portion of the decay than does the photo conductivity method. The curves shown in Figs. 3(e) and (f) were obtained with n-type Si where the surface tended toward an inversion layer. The curve in (e) was taken without background illumination; the curve in (f), with illumi nation. The decrease in the decay rate with illumination is quite commonly seen in Si with photoconductivity measurements. The same type of behavior was found for the InP specimen which displayed the typical surface signals shown in (g) and (h). The decay curves in Figs. 3(i) and (j) exemplify the 12 Brought to my attention by J. Hammes of the RCA Semi conductor Division. 13 H. Nelson and A. R. Moore, RCA Rev. 17, 5 (1956). utility of the surface technique. Figure 3(i) shows the carrier decay in a 3 ohm-cm p-type Ge specimen when the light flash was transmitted through the photo voltage pickup electrode in the same manner shown in Fig. 2. Figure 3(j) shows the time behavior of the excess carriers observed when the electrode and spot of illumination were on opposite sides of the 0.14-cm thick specimen. The oscilloscope trace was synchronized to the "-'l-J.lsec light pulse. The delay time, as measured by the hump in the curve occurring at 50 J.lsec, is in rough accord with the drift time to be expected from a diffusion process. The decay constants for the curves in both Figs. 3(i) and (j) are seen to be the same if only the portions of the curves beyond about 150 J.lsec are considered. The amplifier gain was doubled when the curve in Fig. 3(j) was taken. Detailed treatments of carrier drift effects have to be attended with caution. For example, in the technique described above care has to be taken to localize the region where the excess carriers are injected and, also, to distinguish between possible potential changes in the bulk and those at the surface. In addition to its use in drift studies the surface technique is well adapted for the measurement of lifetimes at different regions of a specimen. For example, the technique is particularly useful in studying effective lifetimes near ohmic contracts. 7. CONCLUSIONS A new technique for measuring the time behavior of excess carriers in a semiconductor has been described. This technique utilizes the junction-like behavior of a semiconductor surface. It requires very simple equip ment and is convenient and "clean" to use since no contacts, other than capacitive, are needed. Theo retically, the surface method yields exactly the same carrier decay constant as the photoconductivity method when the lowest diffusion mode prevails in the specimen. This has been quantitatively confirmed for Ge: only qualitative confirmation has been made with Si. The surface method seems to be more sensitive to the effects of traps and higher diffusion modes than the older method. The surface and photoconductivity techniques have comparable sensitivities to small excess carrier densities in Ge. With high band gap materials, however, the surface technique can be greatly superior in sensitivity and, thus, should be particularly useful for studies of Si and the III -V compound semiconductors. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Sun, 21 Dec 2014 23:05:59

1.1731720.pdf

Proton Magnetic Resonance Studies of Structure, Diffusion, and Resonance Shifts in Titanium Hydride B. Stalinski, C. K. Coogan, and H. S. Gutowsky Citation: J. Chem. Phys. 34, 1191 (1961); doi: 10.1063/1.1731720 View online: http://dx.doi.org/10.1063/1.1731720 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v34/i4 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 34, NUMBER 4 APRIL, 1961 Proton Magnetic Resonance Studies of Structure, Diffusion, and Resonance Shifts in Titanium Hydride* B. STALINSKI,t c. K. COOGAN,t AND H. S. GUTOWSKY Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois (Received October 27, 1960) Measurements of the proton magnetic resonance at "'26.9 Mc in titanium hydride samples, ranging in composition from TiH1.61 to TiH1.97, have been made in the temperature range from _196° to about 200°C. The second moment of the proton reso nance at the lower temperatures shows that the hydrogen atoms are randomly distributed among the lattice sites which are lo cated tetrahedrally with respect to the titanium atoms. Self diffusion of the hydrogen atoms narrows the proton resonance above room temperature. The temperature dependence of the correlation frequency for the proton motions, obtained from the linewidths, leads to diffusional activation energies which increase with hydrogen content from 9.4 kcal/g atom for TiH1.607 to 10.2 for TiH1.923. Moreover, the diffusion rate is directly proportional to the number of unfilled tetrahedral holes in the metallic lattice, which indicates that the self-diffusion takes place via a vacancy mechanism. I. INTRODUCTION THE original purpose of these experiments was to investigate the nature of the anomalies observed near room temperature in the magnetic susceptibility1.2 and in the specific heat3 of titanium hydrides ranging in composition from about TiH1.5 to TiH2• X-ray4,6 and neutron diffraction6 studies have shown that in the region of these anomalies, the titanium hydrides undergo a small, continuous tetragonal deformation typical of a second-order transition. At temperatures above the anomalies, the hydrides have a face-centered cubic arrangement of titanium atoms, surrounded by hydrogen atoms apparently in the tetrahedrally located holes. That is, when the composition approaches the stoicpiometric compound, TiH2, it achieves the CaF 2 structure. Our particular interest was the possibility that hydrogen diffusion is connected with the structural transition and with the magnetic and specific heat anomalies. Nuclear magnetic resonance methods are attractive * This work was supported in part by the Office of Naval Research. t On leave of absence from the Institute of Technology and the Institute of Physical Chemistry of the Polish Academy of Sciences, Wroclaw, Poland. t On leave of absence from the Division of Chemical Physics, C.S.I.R.O. Chemical Research Laboratories, Melbourne, Aus tralia. 1 W. Trzebiatowski and B. Stalinski, Bull. acado polon. sci. Classe III I, 131 (1953). 2 B. Stalinski, Zeszyty Nauk. Politech. Wroclaw. Chem.4, 25 (1957) . 3 B. Stalinski and Z. Bieganski, Bull. acado polon. sci. Serie sci. chim. geol. geogr. 8, 243 (1960). 4 B. Stalinski and B. Idziak (unpublished). 6 H. L. Yakel, Acta Cryst. 11, 46 (1958). 6 S. S. Sidhu, L. Heaton, and D. D. Zauberis, Acta. Cryst. 9, 607 (1956) Proton resonance shifts to higher applied magnetic fields were observed. They were measured at room temperature for all speci mens and were found to increase from 0.01 % for TiH1.607 to about 0.032% for TiH1.969. For these two extreme compositions, the temperature dependence of the shift was measured between -95°C and 190°C and was found to be similar to the bulk sus ceptibility, the shifts for TiH1.969 exhibiting an anomaly at about 13°C as does the susceptibility. These results are interpreted semiquantitatively in terms of exchange interactions which pair spins of electrons in the conduction band with those of electrons localized on the hydrogen. The results suggest that the hydrogen is held in the lattice by a combination of covalent and ionic bonding, the latter involving a net positive charge on the hydro gen. The general importance of exchange interactions in inter metallic compounds is commented upon. for investigating self-diffusion in the solid state8•7, Hydrogen diffusion is especially difficult to measure in metal lattices by classical transport experiments, because gross errors result from grain boundaries, pores, cracks, and dislocations in the metals. For example, the data reviewed by Barrer9 for hydrogen diffusion in palladium include activation energies ranging from 4.6 to 17.8 kcaljg atom. The most extensive prior study of hydrogen diffusion by proton magnetic resonance appears to be the work done on the palladium hydride system. Norberg10 found narrow proton resonance lines in a series of samples up to PdHo.8 and obtained activation energies for self diffusion by using rf pulse techniques to observe the temperature dependence of the spin-lattice relaxation time between 250° and 300oK. Similar results were reported by Torrey,!l who also investigated the proton Tl in TiH1.77. A few scattered observations have been made of the proton resonance in other solid metal hydrides. Garstens12 has looked at the proton resonance of NaH and CaH2, finding broad lines typical of rigid-lattice solids. For tantalum hydride he found13 a broad line at tempera tures below 215°K with diffusional narrowing occuring at higher temperatures. In the case of titanium hydride, Garstens12 noted a broad room-temperature, proton linewidth of about 12 gauss for three samples of composition TiRo.as, TiHo.72, 7 H. S. Gutowsky, Phys. Rev. 83,1073 (1951). 8 C. P. Slichter in Report of the Bristol Conference on Defects in Crystalline Solids (The Physical Society, London, 1955), p. 52. 9 R. M. Barrer, Diffusion in and through Solids (Cambridge University Press, New York, 1951), 2nd ed. 10 R. E. Norberg, Phys. Rev. 86,745 (1952). 11 H. C. Torrey, Nuovo cimento Supp!. to 9, Ser. 10, 95 (1958). 12 M. A. Garstens, Phys. Rev. 79, 397 (1950). 13 M. A. Garstens, Phys. Rev. 81, 288 (1951). 1191 Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1192 STALINSKI, COOGAN, AND GUTOWSKY and TiHl.b7. Similar results were obtained in the more recent room-temperature studies by Maluchkov and Finkelstein I4; they also made some unsuccessful at tempts to measure proton resonance shifts and to obtain the second moment of the proton absorption and com pare it with the value calculated for the rigid lattice. Our more detailed studies which were outlined in a preliminary report,15 failed to establish any direct connection between the hydrogen diffusion and the various anomalies which occur in the titanium hydrides. However, it proved possible to characterize the diffu sion process in very interesting detail by observing the temperature dependence of the proton linewidth over a wide range of composition and temperature. Also, we have found large, negative Knight shiftsI6 in the proton magnetic resonance position. The temperature and concentration dependences of the shift parallel the magnetic susceptibility and demonstrate that the anomalies are associated with the conduction electrons. The details of our various findings are presented here. II. EXPERIMENTAL DETAILS A. Preparation of Samples The titanium hydride samples were of the composi tion TiH1.607, TiH1.719, TiH1.776, TiH1.845, TiH1.933, and TiH1.969. They were prepared by the absorption of known volumes of high purity hydrogen in the appara tus described previouslyP The hydrogen was obtained by thermal decomposition of another sample of titan ium hydride. The titanium was produced by E. 1. du Pont de Nemours and Company. The purity was listed as better than 99.5%. Spectral analysis revealed the presence of the following impurities, in weight percent: Si'"'-'lO-3, Mg'"'-'1O- 3, Mn'"'-'lO-4, Sn'"'-'1O-2, Cu'"'-'1O-3, Fe'"'-'lO-3. Moreover, bulk magnetic susceptibility measurements gave no evidence of ferromagnetic impurities. The titanium metal powder used to prepare the samples was obtained by powdering some of the embrittled metal. The latter was made by saturating the original sponge metal with hydrogen and then degassing it by heating in a high vacuum for several hours at a temperature of about 820°C. This tempera ture should be somewhat lower than the a-fJ transition point of the metal to prevent sintering. All of the hydride samples, except TiH1.969, were synthesized by slowly cooling 2.5 to 4 g of the metal powder, in contact with the hydrogen gas, from 450°C to a temperature at which essentially all of the gas had been absorbed. The sample was then held for 20 to 40 hr at this tempera ture, which was usually 250°C or somewhat higher. 14 O. T. Maluchkov and B. N. Finkelstein, Doklady Akad. Nauk S.S.S.R. 127, 822 (1959). 16 B. Stalinski, C. K. Coogan, and H. S. Gutowsky, J. Chern. Phys. 33, 933 (1960). 16W. D. Knight, Solid State Phys. 2, 93 (1956). 17 W. Trzebiatowski, A. Sliwa, and B. Stalinski, Roczniki Chern. 28, 12 (1954). Of these five samples, all but TiH1.933 showed a dis sociation pressure of 10-2 mm Hg or less at 200°C. The TiH1.933 showed a dissociation pressure as high as 1 mm Hg at 200°C, so its hydrogenation was completed at 150°C where the pressure dropped to about 10-2 mmHg. The sample of highest hydrogen content TiH1.969 was prepared by gradually lowering the temperature from 450°C to room temperature, using an excess pressure of hydrogen. This sample was sealed under a hydrogen pressure of 431 mm Hg to prevent decompo sition in the magnetic resonance experiments carried out above room temperature. At such pressures of hydro gen, the saturation of the metal with hydrogen. is essentially complete at temperatures below 200°C. This is shown by the 5-mm drop observed in the hydrogen pressure upon cooling a sample from 300°C to room temperature. In the vacuum apparatus used, this pressure change corresponds to a composition change of TiH1.957 to TiH1.969. Microscopic examination (X 100) of the particles revealed, of course, a considerable apread in particle size, but it was apparent that no particles were larger than 10-3 cm in diameter. The samples actually used in the magnetic resonance experiments were about 1 cc in volume and, except as noted for TiH1.967, they were sealed under their equilibrium pressure of H2 in glass tubes whose outer diameter was about 9 mm. B. Apparatus and Procedure The broad-line regenerative spectrometer used in these experiments was similar to that described pre viously.I8 It was operated at a fixed radio frequency of about 26.92 Mc, and the derivative of the proton absorption was observed by automatically sweeping the field of the 6300-gauss permanent magnet. The 30-cps modulation amplitude was kept small to reduce its broadening of the absorption, and in cases where the modulation had to be large due to a low signal-to noise ratio, the appropriate corrections were made.I9 Care was taken to prevent saturation of the lines, but in some cases it appeared that these efforts were not completely successful. The volume of the sample coil was about 1.5 cc. The high conductivity of the samples led to appreciable losses in the rf coil; this required the use of higher plate voltages and greater feedback than usual in the marginal oscillator stage of the spectrom eter. Measurements below room temperature were made with a Dewar-type cryostat,18 using either liquid nitro gen or a dry-ice and alcohol slush as the refrigerant. Higher temperatures were obtained with a gas flow cryostat. The temperature of the sample was measured by means of a copper-constantan thermocouple at- 18 H. S. Gutowsky, L. H. Meyer, and R. E. McClure, Rev. Sci. Instr. 24, 644 (1953). 19 E. R. Andrew, Phys. Rev. 91, 425 (1953). Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPRO TON MAG NET I C RES 0 NAN C E IN TIT A N I U M H Y D RID E 1193 tached to the outside of the sample tube with a heat resistant glass tape. The resonance shifts were measured by comparing the magnetic field at the center of the proton absorp tion in the titanium hydrides with that in a silicone polymer, Dow-Corning Silastic Gum 400. Small pieces of the silicone gum were inserted. between the turns of the rf coil, so that the spectrum recorded was that of the hydride with a small signal from the silicone super imposed. The proton linewidth in the silicone is about 2 cps at room temperature and it is narrow enough to be a good reference over a wide temperature range. In the case of the lowest temperatures, about -95°C, at which the shift was measured, tetramethyl silane was used as the reference. It was sealed into thin glass capillaries which were inserted in the rf coil ~eside the tube containing the hydride. III. RESULTS AND DISCUSSION Our presentation of the results is divided into three main sections: the low-temperature rigid-lattice second moments, the diffusional narrowing of the absorption at higher temperatures, and the resonance shifts. One feature common to all of the experiments is that the absorption lines are simple, bell-shaped curves with no evidence of structure. Moreover, although the lines narrow at higher temperature, their shape otherwise remains essentially the same. A. Low-Temperature Second Moments Experimental Results The broad absorption lines observed at liquid nitro gen temperature do not begin to narrow appreciably until room temperature or above, depending upon hydrogen content. This suggests that the low tempera- TABLE 1. Linewidths and second moments of the proton 'magnetic resonance absorption in titanium hydride at -196°C. Observed Observed Theoretical Composition linewidtha,b second moment second momentc TiH1.607 12,1±0,1 gauss 22. O±O. 7 gauss' 21.3 gauss2 TiH1.719 12.4±0.2 23.1±0.6 22.4 TiH1.77D 12.7±0.2 24.3±0.45 23.0 TiH1.845 13.1±0.1 24.5±0.15 23.7 TiH1.933 13.5±0.15 25.7±0.55 24.6 TiH1.969 13.5±0.2 25.8±0.45 24.9 a The full width between the maximum and minimum on the derivative curve. b The errors are the standard deviations of from 5 to 10 measurements. C These values assume that the protons are distributed, at random, in the tetrahedral sites only; they were obtained by applying corrections for the tetragonal deformation and thermal expansion to second moments calculated for the fcc structure at room temperature. The latter were taken from the straight line at the bottom of Fig. 1, drawn through the points calculated from the room-temperature lattice parameters of various authors for different hydro gen con tent. 32 .... 28 "'en en ::J 0 01 ..... 24 ... z 1LI 2 20 0 2 0 z 16 0 ~ (I) 12 1.5 1.6 1.7 1.8 1.9 2.0 ATOMIC RATIO HIT! FIG. 1. The dependence of the proton second moment upon the hydrogen content of titanium hydride. The upper curve is a least-squares straight line fitted to the experimental points q,'for -196°C. The lower solid curve is a least-squares straight line fitted to secone! moments calculated for room temperature from the fcc lattice parameters of the following authors: 0 B. Stalin ski [Zeszyty Nauk. Politech. Wroclaw. Chem. 4, 25 (1957)]; () B. Stalinski and B. Idziak, uIpublished results; • H. L. Yakel [Acta Cryst. 11, 76 (1958) J. The calculations assume a random distribution of the protons among the tetrahedral sites. The dotted curve represents the conversion to -196°C of these room temperature theoretical values, by correcting for the tetragonal deformation and thermal contraction. ture corresponds to a "rigid lattice" and that some structural information might be gained from a detailed analysis of the second moments. Experimental values for the second moments were obtained by numerical integration of several derivative curves recorded at -196°C for each of the six samples. The results are summarized in Table I and show a systematic increase with hydrogen content, from 22.0 gauss2 for TiH1.607 to 25.8 gauss2 for TiH1.969. The linear extrapolation indicated in Fig. 1 gives a value of 26.3 gauss2 for the second moment of the stoichiometric compound TiH2 at -196°C. These values have been corrected for the magnetic-field modulation broadening which was of the order of 4% for all of the measure ments. The linewidths, defined as the separation in gauss between the inflection points of the absorption, show an increase from 12.1 to 13.5 gauss paralleling that in the second moment. These values agree with that of 12 gauss reported by Garstens12 for TiH1.67 at room temperature. However, our linewidths and second moments are about 20% less than the values reported by Maluchkov and Finkelstei n14 for room temperature. The discrepancies may be due, at least in part, to the fact that the signal-to-noise ratio is much better at -196°C than at room temperature. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1194 STALINSKI, COOGAN, AND GUTOWSKY Theoretical Second Moments Theoretical second moments of the proton absorp tion may be calculated from Van Vleck's results for the dipolar broadening in a rigid lattice. For a crystal powder, the relation is of the form20 AH22 = (!)I(I+1)g2J.1oW-ILrjk~, (1) 1>/0 where, for our purposes, I and g are the proton spin and magnetogyric ratio, respectively, N is the number of protons in the crystal, and rjk is the separation of protons j and k. There are similar terms20 for the dipolar broadening of the proton resonance by the Ti47 and Ti49 nuclei; however, they constitute only about 0.5% of the total. Therefore, although they were included in the numerical results given, we shall not refer to them again. In principle, the substitution into Eq. (1) of an experimentally observed second moment c~n lead to the determination of one structural parameter. How ever there is also the observed, essentially linear , dependence of the second moment on hydrogen con tent and an additional structural feature might be , inferred from it. The positions of the titanium atoms in the unit cell have been obtained quite accurately from x-ray diffraction studies.4,5 However, the loca tion of the protons is not known with any assurance.5,6 In particular, there are two types of interstitial sites which the protons might occupy in the titanium lattice. One set is located in tetrahedral positions with respect to the titanium atoms; the other, in octahedral. There is also the question as to whether or not the sites occu pied in the nonstoichiometric hydrides are located at random. The situation is favorable, inasmuch as there are two observed quantities and two structural ques tions. Nonetheless there are some complications in com paring the experimental second moments with values calculated for various possible distributions of the pro tons. The experimental values are most accurate at liquid nitrogen temperatures because of the better signal-to-noise ratio at low temperatures. Moreover, at low temperatures the residual motional narrowing of the absorption will be less. However, most of the known structural parameters and their dependence upon hydrogen content have been determined at room temperature. Also, because of the second-order phase transition, the calculation of low-temperature second moments from these data must take into account not only the volume change at lower temperature but also the slight tetragonal deformation of the unit cell. Fortunately both effects are small and can be de scribed with sufficient accuracy by a simple analytic expression so that the comparison of experimental 20 H. S. Gutowsky, G. B. Kistiakowsky, G. E. Pake, and E. M. Purcell, J. Chern. Phys. 17, 972 (1949). and theoretical second moments for different tempera tures can be readily made. The lattice constants of the tetragonal unit cell are identified as a=b¥-c, where c=a(l+A); and the lattice constant of the cubic cell, as ao. By comparing the sums of the contributions from the first and second nearest neighbors, which constitute about 90% of Li>jrii~ in Eq. (1) for the tetragonal and cubic crystals, we find that the two total lattice sums are related by the following approximate equation, a6L"'ao6(1-':2A+4A2) L. (2) I The subscripts t and c refer to the tetragonal and cubic crystals, respectively, and it is assumed that A is small compared to unity. Thermal expansion is incorporated into Eq. (2) by using the relations VI=a3(1+A) and Vc=ao3 to eliminate a6 and ao6 in Eq. (2). This gives the final result (3) which permits us to interconvert second moments at -196°C and at room temperature, providing we know the cubic lattice parameter at room temperature, the thermal expansion, and A. The x-ray studies4,5 show that the axial ratio c/ a is about 0.945 at liquid nitrogen temperature. Hence, taking A to be -0.055 and introducing the observed change in molar volume with temperature,5 Eq. (3) leads to a second moment at -196°C which is about 2% larger than at room temperature. This agrees well with the approximate 0.5-gauss2 difference actually observed and suggests that there is no significant change in the motional effects on going from room temperature to -196°C. Location of the Hydrogen Atoms Let us now consider the question of the location of the hydrogen atoms. If octahedral as well as tetrahedral sites are occupied, the observed second moment will be the following average of the second moments for protons in each of the sites, AH22 = foct (AH22) oct+ ftet (AH22) tet, ( 4) where the f's are the fractions of the protons in each type of site. We now assume a random distribution within each type of site and define the probability of an octahedral site being occupied as a and a tetrahedral site f3. Bearing in mind that there are twice as many tetrahedral sites as octahedral, this gives foct= a/(2f3+a) and ftet=2f3/(2f3+a). (5) Furthermore, in the compound TiH2-B, a and (3 are related to 8 as follows, (2(3+a) =(2-8) or 2f3=(2-8-a). (6) Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPROTON MAGNETI CRESON ANC E IN TITANIUM HYD RID E 1195 Eq. (4) require the computation of the lattice sums N-l~i>j1'i;~ of Eq. (1). However, the octahedral sites are equivalent, as are the tetrahedral. Therefore, we need only to pick a particular filled site of each kind, labeled 0 and t, and compute the sums ! ~j1'oj~ and ! E j1' Ij ~ to all other filled sites j. The assumed random distribution of filled sites makes it easier to take the sums over all sites k, weighting each by the probability of its being occupied. For the protons in the octahedral sites, this approach gives (.:1H22)oct= (~)I(I + 1) g2JLo2! E1'oj~, (7) i with ~1'oj~= ~1'ok~Pk= ~1'o!--6a+ E1'om-6,B i " I m = (115.6a+1322,B)ao~. (8) The indices land m designate all octahedral and all tetrahedral sites, respectively, surrounding the nucleus at the origin of the summation. Similarly, for the pro tons in the tetrahedral sites, one finds E1'tj-6= E1'tk~Pk= E1'IZ-6a+ E1',m-s,B i " I m = (660.5a+537.7,B)ao~. (9) The numerical values for the sums in the fcc structure are from the literature21j they may be assigned by noting that the octahedral-tetrahedral spatial relation ship is equivalent to that of Ca and F in CaF2, and that the octahedral holes and tetrahedral holes lie on a fcc and a simple cubic lattice, respectively. The appropriate substitution of Eqs. (5)-(9) in Eq. (4) gives the dependence of the second moment upon the probability a of an octahedral site being occupied, .:1H2!=[(2-20+2.92a-1.46ao-1.74a2+0.50 2) /(2-0)] X (.:1H22)tetO, (10) where (.:1H22)tetO is the second moment for the stoi chiometric hydride with only tetrahedral sites occupied. For the case 0=0, or TiH2' this becomes .:1H22 = (1+1.46a-0.87a2) (.:1H22)tetO. (11) A value of 24.6 gauss2 is obtained for (.:1H22) tetO by extrapolating the theoretical values in Fig. 1 to 0=0. This is to be compared with the experimental value of 25.7 gauss2, which is the 26.3 gauss2 observed at -196°C, corrected for thermal expansion and the lattice change. Thus the observed second moment is about 4% larger than that calculated for the fcc hydride with only tetrahedral sites occupied. It follows from Eq. (11) that a=0.025 and that the proportion of all hydrogen atoms in octahedral sites is about 1.25%. 21 H. S. Gutowsky and B. R. McGarvey, J. Chern. Phys. 20, 1472 (1952). Actually this is an upper limit to the fraction of protons occupying octahedral sites because there are several other factors which can give an increase in the second moment. One might expect the observed second moment to be low because of the residual motional narrowing effects. However, the symmetry of the lattice is such that for even moderately large amplitude oscilla tions, as many interproton distances are decreased as are increased and by about the same amount, so the net effect is very small. On the other hand, there are several possible broadening mechanisms, including spin-lattice relaxation, saturation, nonrandom distri bution of the protons, and anisotropy in the tetragonal lattice of the large proton resonance shifts. The Tl broadening could be as much as 0.3 gauss2 if the proton Tl were related to the resonance shift in the same fashion as found for pure metals21j but direct measurement of the proton T 1 gives somewhat longer timesll ("'-'0.3 sec) than predicted on this basis ("-'0.1 sec), and the proba ble broadening from this cause is correspondingly less. The Tl is long enough for saturation broadening to occur and there was some evidence for it even though efforts were made to operate the experiments at very low rf levels. In Sec. IILB., it is pointed out in the discussion of the diffusion mechanism that about 1.25% of the tetrahedral sites are inaccessible to hydrogen. If these inaccessible sites are clustered, as indeed appears to be very likely, then the second moment of TiH2--o.o25 should be (.:1H22)tetO, which would account for about 1 of the 4% discrepancy. The tetragonal deformation is about 5% at -196°C; if there were a corresponding anisotropy in the large proton shifts, it alone would account for the difference. A final argument on this point also concerns the question of random versus nonrandom distribution of the hydrogen in the lattice. In principle, the latter question can be answered by the dependence of the second moment upon hydrogen content. If the protons were distributed at random among the tetrahedral sites, one would predict, for example, that the second moment would be given by which is obtained by setting a=O in Eq. (10). At the other extreme, if the "clustering" of protons were complete so that the sample was a two-phase system of pure metal and pure stoichiometric hydride TiH2 the second moment would be (.:1H22)tetO, independent of average composition. Less extensive clustering leads to a smaller second moment, but one still larger than that for a random distribution. At first glance, Eq. (12) would seem to predict a straight line through the origin. However, (.:1H22)tetO is itself a function of 0 via the dependence of the unit cell size upon hydrogen content. The calculated values plotted in Fig. 1 include this dependence j the result over the composition range of 0=1.55 to 0=1.98 is Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1196 STALINSKI, COOGAN, AND GUTOWSKY 16 14 12 -.. .. ~ .. o' 10 :x: 00 8 :x: S 6 i 1&1 4 z ::l 2 1. 2. 3. 4. 5. 6. 5,6 TiH,.601 TiHI.719 TiH 1.775 TiH 1.845 T1HI.933 TiHI.969 FIG. 2. The observed proton line width plotted as a function of tem perature for the titanium hydride samples. Not all experimental points have been shown, to avoid confusion. The effect of decom position in sample 6 at high tem perature is shown by the dotted line. o ~ __ ~ ____ ~ ____ ~ ____ L-__ ~ ____ -L ____ ~~~~ __ ~ ____ ~ -200 -150 -100 -50 o 50 100 TEMPERATURE ,(Oe). nonetheless a good straight line, but it has a 0=0 intercept of about 7 gauss2 rather than zero. The experimental points agree quite well with this cal culated line except for the 4% displacement already noted. Also, the experimental line has a slightly steeper slope than the theoretical, and the experimental points could be fitted somewhat better by a parabolic curve, concave downwards. This sort of nonlinear behavior, if real, could result from a small tendency for cluster formation. On the other hand, if the excess broadening is from anisotropy of the proton resonance shift, the strong concentration dependence of the latter would account for the difference in slope and shape of the theoretical and experimental curves. The latter explanation is favored. 150 200 250 B. Diffusional Narrowing Experimental Results Figure 2 shows the linewidth plotted as a function of temperature for each of the six samples. The line width is defined here as the full separation between the maximum and the minimum of the first derivative of the absorption. It can be seen that the linewidth is substantially constant at low temperatures but that the absorption narrows quite rapidly with increasing temperature starting near room temperature. More over, the temperature at which the line width reaches half its maximum value depends on the hydrogen content of the sample, increasing from 66°e for TiH1.607 to 1300e for TiH1.933, as listed in Table II. In the region TABLE II. Summary of quantities obtained from the diffusional narrowing of the proton resonance observed in the nonstoichiometric hydrides TiH2--!. Ii T.a Ea Ab POC 0.393 66°C 9A±0.5 kcal 1.69XI011 sec-1 OAIXlO12 sec-1 0.281 88 9.85 2.08 0.80 0.225 86 9.85 1.64 0.80 0.155 97 10045 2.35 1. 74 0.067 130 10.2 0.56 1.23 0.031 232(?) a Tj is the temperature at the center of the linewidth change. b A is the frequency factor obtained by fitting the logve vs liT plots with the equationvc=A exp(-EaIRT). c .. o is the frequency factor corrected for the dependence oiPe upon the probability that a hydrogen has a neighboring vacant site into which it can jump; see Eq. (15). Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPROTON MAGNETIC RESONANCE IN TITANIUM HYDRIDE 1197 of rapidly changing linewidth, it was found to be important to "equilibrate" the sample for about an hour before making a measurement. Otherwise, thermal inhomogeneities could lead to appreciable errors. Decomposition of a sample at higher temperatures also could lead to spurious results. If decomposition did occur, the sample would give results expected for a sample of lower hydrogen concentration, and the ap parent linewidth change would be displaced to lower temperatures from the actual curve for the undecom posed specimen. Evidence for such decomposition was found only in the case of TiH1.969, the sample containing the largest concentration of hydrogen. For this sample, decomposition at high temperatures led to a discon tinuous decrease in linewidth. However, upon slow recooling, the sample reverted to its original composi tion, as shown by its linewidth and second moment. None of the other samples exhibited such behavior or any thermal hysteresis in the line shape, so it is believed that decomposition was important only in TiHl.969 at high temperatures. This is consistent with the condi tions under which the samples could be prepared. Activation Energy for Hydrogen Diffusion The only type of motion which can produce the observed averaging out of the dipolar broadening is self-diffusion of the hydrogen. The extent of the narrow ing depends upon the rate at which the motions occur, as shown by the analyses of BPP22 and of Kubo and Tomita.23 The rate of the motions is described by the correlation frequency lie, which may be calculated from the width 8H of the partially narrowed line by means of the following slightly modified form of the BPP equation lIe=2(2In2)1.l1l/ tan[lI'(oH/oH o) 2/2], (13) where .lll is the observed linewidth in cycles per second, and oHo is the observed rigid-lattice linewidth in gauss. This equation is applicable only to a line shape which is Gaussian in the low-temperature limit. Fortunately, the proton line shapes observed in the hydrides are very nearly Gaussian, as borne out by the fact that the ratio of oH to the root second moment, .lH2, is approximately 2. In addition, the general shape of the absorption is preserved in the narrowed lines, so that any errors resulting from the slightly non-Gaussian character should be systematic. In any event, we have used Eq. (13) to calculate the temperature dependence of lie from the observed temperature dependence of oH. These calculations were made only for linewidths for which oH/oHo lies between about 0.1 and 0.85. For narrower linewidths, inhomogeneity and Tl broadening introduces errors; while for broader lines, errors result from approxima tions in the BPP theory. 22 N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948). 23 R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954). ~ I.!I 9 7,0 6.5 6.0 5.5 5.0 4.5 I. TiH1,607 2. TiH1.719 3. TiHI.775 4. TiH1•845 5, nH1,933 I 32 40 L..._---'-__ ..1-__ ..L-__ ./..-_--i-_-J 2PO 2.25 2.75 3.00 325 3.50 '~o FIG. 3. The Arrhenius plot of the diffusional correlation fre quency Pc, for the five samples for which this plot was possible. The values for Pc were obtained from the linewidths given in Fig. 2, using Eq. (13). The least-squares straight lines are drawn through the data for each sample. The logarithm of lie has been plotted in Fig. 3 as a function of the inverse of the absolute temperature. It can be seen that these Arrhenius plots give a straight line for each of the samples. The straight lines in the figure were obtained by a least-squares fit of the data for each sample. The lines are very nearly parallel to one another. The activation energy Ea for self-diffusion of the hydrogen can be obtained from the slopes of the lines. The values range from 9.4±0.S kcaljg atom for TiH1.607 to lO.4±O.S kcaljg atom for TiH1.933. The results are listed in Table II and are plotted as a func tion of composition in Fig. 4. It appears that there is a small linear increase in Ea with hydrogen content; however, the increase is of the same order as the possible experimental error. The least-squares linear fit of the data gives an extrapolated value of 10.5 kcalj g atom for Ea in the stoichiometric hydride TiH2. These results are considered in more detail in a subsequent paper.24 It describes a very simple electrostatic model which leads to a semiquantitative explanation of the diffu sional activation energy and the dependence upon hydrogen content.24a 24 C. K. Coogan and H. S. Gutowsky, J. Chern. Phys. (to be published) . 24& W. Spalthoff has recently investigated the proton magnetic resonance in a number of hydrides including one titanium hydride, of composition TiH1.98. For this sample he reports a low-tempera ture (-190°C) linewidth of 11.8 gauss and the narrowing of the absorption at higher temperatures, centered at approx 150°C. In addition, his analysis of the linewidth change gives an activa tion energy for proton diffusion of S.8±O.8 kcal/mole. These values are all significantly below those which we report. Dis crepancies of this nature would result if his sample had suffered decomposition at the high temperatures 2S0-300°C which he used in his experiments. We wish to thank Dr. Spalthoff for mak ing his manuscript available to us prior to its future publication in Z. physik. Chern. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1198 STALINSKI, COOGAN, AND GUTOWSKY E o "0 12.0 .--------------, E ~ 811.0 ~ ~ cr 10.0 w z w z 9.0 o ~ ~ 8.0 L--_-'--_-'-_-'--_----'--_--' u ..: 1.5 1.6 1.7 1.8 1.9 2.0 ATOMIC RATIO H/Ti FIG. 4. The activation energy for hydrogen-diffusion as a function of composition. The activation energies!shown are de termined from the least-squares analysis o(~he.:data given in Fig. 3. Diffusion Mechanism The nature of the diffusion mechanism in our samples is shown with remarkable clarity and directness by the dependence of the diffusional jump frequency Vc upon the hydrogen content. This dependence is obtained in simple graphical form by taking the logvc versus 1/ T curves for the different compositions in Fig. 3 and constructing Vc versus composition isotherms from them, as shown in Fig. 5. The points plotted on this diagram are averages in that they were taken from the least squares straight lines of Fig. 3, for each specified temperature, and not directly from the experimental points because Vc was not measured for all the samples for exactly the same temperatures. It may be seen in Fig. 5 that to a very good degree of approximation, Vc is directly proportional to the stoichiometric defect 16.------------------~ 14 12 10 '" 8 '~ )( ,,>V 6 3700K 4 2 0L..e:::--'----'---'-------'------'-----' o 0.1 0.2 0.3 0.4 0.5 0.6 FIG. S. Isotherms giving the dependence of Pc upon the stoichio metric defect 0 in the compounds TiH2-<l. The data plotted here are taken from the least-squares lines of Fig. 3. 0. The latter describes the concentration of vacant tetrahedral sites. Therefore, it is clear that the diffu sion occurs via a vacancy mechanism, i.e., a hydrogen atom can diffuse only by jumping into an unoccupied, neighboring tetrahedral site, at least for the tempera tures over which oH changes rapidly. Our results are fitted approximately by a single equation of the form vc=von(0/2) exp(-Ea/RT) , (14) where 1'0 is a frequency factor, n is the number of neigh boring tetrahedral sites to which a proton can jump, and (0/2) is the probability of a tetrahedral site being vacant. Equation (14) predicts that the isotherms will be a family of straight lines going through the origin. Scrutiny of Fig. 5 shows that this is essentially so. The most striking exception is that the curves converge at the point vc=O, 0=0.025 rather than at the origin. This requires that (0/2) in Eq. (14) be replaced by (!) (0-00) where 00=0.025. Also, it may be seen that the experimental points for large 0 fall above the straight line portion of the iso therms for smaller 0, and that this is more pronounced for the lower temperature isotherms. Deviations of this sort would be expected from the linear dependence of Ea upon o. However, the deviations are actually much smaller than would be expected from this cause alone, because they are very largely compensated by the dependence of 1'0 upon o. This dependence is also linear, as shown by the values summarized in Table II. So we can write v~voO(1-Bo) and Ea=Ea°(1- Co), where 1'0° and Eao refer to the stoichiometric compound TiH2• With these revisions, Eq. (14) becomes vc=vo°(1-Bo)n(!) (0-00) exp[ -(EaO-Co)/ RT]. (15) The o-dependent term in the exponential can be approxi mated as (1+Co/RT) , for small 0 and large T, giving vc=vo°(1-BO)n(!) (0-00) (1 + Co/ RT) X exp(-Eao/RT). (16) The values of Band C/ RT are nearly equal so that (1-Bo) (1 +Co/ RT""1, at least for moderate values of 0, and our final result is vc""voOn(!) (0-00) exp( -Eao/RT) , (17) which describes the isotherms in Fig. 5 very well. The "Frequency Factor" in the Diffusion Rate Law Aspects of Eqs. (14)-(16) which merit further com ment include n, the value of 1'0 and its dependence on 0, and the significance of 00. All of these affect the fre quency factor in the rate expression for the diffusion of hydrogen. The value of n depends upon the type of path along which the hydrogen diffuses. The tetra hedral sites occupied by the hydrogen in the fcc titan- Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPROTON M A GN ETl C R ESO NAN C E IN TITANIUM H YD Rl D E 1199 ium hydride lattice are in a simple cubic arrangement with a cube edge of ao/2. It appears that the most probable diffusion path is along the body diagonals of these cubes.24 A given tetrahedral site has 8 such diagonals leading from it; however, four are blocked by the titanium atoms, so n is 4. The experimental source of the frequency factors is Fig. 3. Extrapolation to infinite temperature of the logl'c vs l/T plots gives an Arrhenius frequency factor which is the A in the equation I'c = A exp( -Ea/ RT). These values, which are listed in Table II, include the dependence of I'c upon the probable number, n(0/00)/2, of neighboring vacant sites to which the hydrogen can jump. The "true" frequency factor25 1'0 may be cal culated from A by the relation, obtained from Eq. (15) , 1'0 = A/n(!) (0-00). (18) The values which result are listed in Table II. It is found that they increase with decreasing o. If this dependence upon 0 is fitted by the linear relation 1'0= 1'0°(1-Eo), the values of 1'0 in Table II lead to a value for 1'0° of about 1.8X1012 and for E of about 2. The frequency factor is a weighted mean26,27 of the vibrational spectrum of the hydrogen atoms in the lattice. At first glance the numerical values of 1012 sec1 may seem to be low, but they are compatible with the rather flat potential function calculated24 for the hydrogen in the vicinity of the tetrahedral sites. Moreover, the steepness of the potential function and also the value of the vibrational frequency should increase with Ea; this agrees qualitatively with experi ment. The difficulty in obtaining very accurate values of 1'0 renders a more detailed discussion profitless at this time. As to the significance of 00, it does not seem probable that the passing of the isotherms in Fig. 5 through this point could arise from experimental error. Each of the isotherms is well defined and linear to a high degree in the region of interest, and all the lines intersect at the same point within very close limits. This nonzero intercept can be interpreted according to the vacancy mechanism as representing the nonavailability for the purpose of self-diffusion of a small fraction 00/2 =0.013 of the tetrahedral sites. Such an interpretation agrees with the experimental fact that it was not possible to produce titanium hydrides with more hydrogen than about TiH1.9s. The inaccessibility of this 1 % of the tetrahedral sites could arise from several causes. One possibility is suggested by the work of Bevington, 25 In this discussion, Pc should not be identified as precisely equal to the frequency with which a proton jumps from one site to another. They are related by a constant, of the order of unity, whose value depends upon the complex correlation function which describes the diffusional motions. See e.g., H. C. Torrey, Phys. Rev. 92. 962 (1953) and the work cited in footnote II. 26 S. A. Rice, Phys. Rev. 112. 804 (1958). 27 D. Lazarus, Solid State Phys. 10. 86 (1960). Martin, and Matthews,28 who found that the addition of small amounts of 0, N, and C to zirconium and titanium reduces the saturation concentration of hydrogen in the metal. The effects increase with the atomic radius of the impurity; one atom of oxygen inhibits the solu tion of one atom of hydrogen, while one atom of nitro gen inhibits 4 to 6 atoms of hydrogen. Thus, only 0.2% of the sites would need to be occupied by nitrogen to produce the observed effect, and this is not unlikely. Another possibility is that 00 represents sites which are rendered inaccessible by edge dislocations. Cottrell29 has shown, using a simple elastic stress model, that a considerable binding energy for interstitial atoms occurs on the stress-favored side of the dislocation. On the other side of the dislocation, the interstitials are squeezed out. In order to have 1 % of the tetra hedral sites inaccessible from this cause, there would need to be about 1012 dislocations/cm2, with about 20 sites affected per atom plane. These figures are reasona ble as the metal was cold-worked in the preparation of the powder and was probably not annealed to a suffi ciently high temperature subsequently to destroy the dislocations. Determination of the actual cause of 00 requires further experiments, preferably ones in which the value of 00 is changed reproducibly by cold-working and annealing of the metal or by the controlled intro duction of impurities. Relation of Diffusion to the Crystal Structure Because of the neutron diffraction 6 and proton second-moment studies, it has been assumed in this discussion of the diffusion mechanism that only tetra hedral sites are occupied by hydrogen and that this occupation is random. In addition, the diffusion iso therms themselves afford very strong evidence for the first of these points. If the octahedral sites were oc cupied to a fractional extent d, then the concentration of tetrahedral vacancies would be (d+ 0), the I'c for diffusion would be approximately proportional30 to (d+o) and the intercept of the diffusion isotherms would be at approximately o=-d. The fact that the intercept is observed at +0.025 and that there is independent support for this being equal to the con- o centration of inaccessible vacancies is strong evidence that even though the protons go through the octahedral sites, they don't sit there long. However, the diffusion data offer no conclusions on the question of random occupancy of the tetra hedral sites. If the vacancies coalesce to form vacancy pairs or groups of higher order, as do negative-ion 28 C. F. Bevington, S. L. Martin, and D. H. Matthews, Proc. 11th Int. Congress on Pure and App!. Chern. 1, 3 (1947). 29 A. H. Cottrell, Dislocations and Plastic Flow in Crystals (Oxford University Press, New York, 1953). ao There would be some deviation from a linear relationship be cause occupation of an octahedral site would partially block dif fusion to an adjacent tetrahedral site, if it were vacant; but this effect would approach zero as the tetrahedral vacancy concen tration became small. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1200 STALINSKI, COOGAN, AND GUTOWSKY vacancies in alkali halides, the probability of there being a vacancy adjacent to a particular hydrogen atom will be reduced. The net result is that "c is reduced by a factor of the form (1-k20-kg02 ••• ) where k2, kg· •• incorporate the association equilibrium con stants and give the net effect of the association on the diffusion probability. Unfortunately, this functional form is indistinguishable from the dependence of "0 upon Ea, and thus upon 0, which was discussed in con nection with Eq. (15). Finally, there does not appear to be any direct rela tion between the diffusion of the hydrogen and the various structural anomalies which occur just below room temperature. In fact, the diffusion rate is too low at these temperatures to produce any significant narrowing of the dipolar broadening, so there is little hope of detecting any anomalous change in the diffu sion rate in the vicinity of the other anomalies. C. The Proton Resonance Shift Experimental Results It was found, by the method outlined in Sec. ILB, that the proton resonance is shifted upfield in the titanium hydride samples from its position in the silicone reference. That is, the shift is in the opposite direction to that of the Knight shift found in most metals.I6 The shifts are from 0.6 to 2 gauss out of 6300, which is very large for protons. However, because of the dipolar broadening, the shifts are a modest fraction of the linewidth, so considerable pains were taken to determine whether or not the shifts were real and, once this was apparent, to obtain accurate measurements of them. Inasmuch as an external reference was used in the shift measurements, the effect of magnetic field in homogeneities was checked by changing the relative positions in the rf coil of the sample and of the silicone. This caused small changes in the apparent shift, but they were, at most, less than T\ of the observed shift. The magnetic field was swept from high to low fields, as well as from low to high, to ensure that magnetic hysteresis or lags associated with the integrating and recording system did not contribute to the apparent· shift; but no differences were found. None of the particles of titanium hydride were larger than 10-3 cm in diameter, so that one would predict31 any skin effects upon the resonance position to be less than a part in 105, which is small compared to the ob served shifts of several parts in 104• As a precaution against the possibility of larger effects being produced by aggregation of the small particles, a sample was mixed with carbon tetrachloride to break up surface contacts between particles. The same shift was ob- 31 A. C. Chapman, P. Rhodes, and E. F. Seymour, Proc. Phys. Soc. (London) B70, 345 (1957). tained. Moreover, the absorption lines did not show any evidence of the asymmetry typical of skin effects. Any correction for the bulk magnetic susceptibility is also small compared to the resonance shifts. If we assume that the sample approximates an infinite cylinder normal to the magnetic field, the measured bulk susceptibilityl leads to a correction of the order of three parts in 105. However, this is a maximum because the reference was contiguous with the sample proper. The results of all these experiments are consistent with the proton shifts being a real phenomenon, to be attributed to interactions at the atomic and nuclear levels, rather than to an instrumental or a bulk effect. Therefore, detailed measurements were made of the dependence of the shifts upon hydrogen content and temperature. Each shift reported is the mean of at least ten measurements; this number of measurements were made because the shifts are a small to moderate fraction of the line width and the signal-to-noise ratio left something to be desired. No effort was made to apply any corrections for bulk susceptibility effects because of their relative smallness and uncertainty. The proton shift was measured at room temperature for all six samples. From the results plotted in Fig. 6, it can be seen that the shift is a definite function of hydrogen content. It is approximately constant at 0.01 % between TiH1.6 and TiH1.85, but then increases sharply to 0.03% for TiH1.97. The temperature de pendence of the proton shift was measured for TiH1.607 and TiH1. 969, which are the samples at the end of the composition range investigated. The results given in Fig. 7 show that the shift in TiH1.607 is virtually inde pendent of temperature while that in TiHl. 969 shows a pronounced anomaly at about 15°C. 0.05r-----------,5 C\I 0.04F-__ ~ Q :I:J:I: : 0,03 I.L. I en ~ 0.02 fi ...J W !r 0.01 OL-__ -L __ ~L_ __ _L __ __' 1.6 1.7 1.8 1.9 2.0 ATOMIC RATIO H/Ti FIG. 6. The room-temperature shifts in the proton resonance, plotted as a function of hydrogen content. The errors shown are the standard deviations of ten or more measurements. All shifts are in the direction of higher applied magnetic fields. The curve fitted to the experimental points for TiH1.607 and TiH1.969 was calculated by means of Eq. (32). The curve at the top of the figure is the bulk susceptibility, in cgs units per gram, as re ported in the;work cited in footnote 1. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPROTON MAGNETIC RESONANCE IN TITANIUM HYDRIDE 1201 0.05 5.0 Ti HI•969 "'2 0.04 TiH 4.0 .. Ti :X:1::r: ":.. 0.03 3.0 "'2 ... " :;: t ~ (f) 2.0 w ~ !c( .... 1.0 w 0: 0.01 oL---~--~~~~=-~~~~~ -100 -50 0 +50 +100 +150 .+200 TEMPERATURE (CO) FIG. 7. The shifts in the proton resonance of TiIL.so7 and TiHl 969 plotted as a function of temperature. The errors shown are the' standard deviations of ten or ~ore meas~rements. All shifts are in the direction of higher applied magnetlc fields. The curve fitted to the experimental points for TiH1•969 at -100° and 15°C and that for TiH1.607 at room temperature were calcu lated by means of Eq. (32). The curves at the top <?f the figure are the bulk susceptibility, in cgs units per gram, as mterpolated from the results reported in the work cited in footnote 1. Comparison of the Shifts with Other Observations The most striking aspects of the proton shifts are their large magnitude and their direction. The shifts observed in the titanium hydrides are an order of magnitude larger than the total range of 3 parts in 105 for proton shifts in ordinary diamagnetic compounds. For example, we searched for a shift in solid CaH2, at room temperature, but without positive results, that is ItJ.H/H I <0.001 %. Nor, apparently, is there a report for any of the other stoichiometric hydrides of a proton shift of the magnitude found for TiH2_o• However, proton shifts have been found in two other nons to i chiometric hydrides. In the case of VHO•66 Oriani et al.32 found a relatively large upfield shift of 0.0095%, which is comparable with those reported here for the titanium hydrides of lower hydrogen content; but they also reported a negligible proton shift in TaHo.33• On the other hand, in PdHo.6, NorberglO found a small down field shift of 0.0017%, which is the direction of the normal Knight shift.16 Thus, it seems probable that large shifts are to be found only in the interstitial, metal hydrides, but not in all of them. Except for their direction, the proton shifts in the titanium and vanadium hydrides are reminiscent of the Knight shift in metals. In fact, the hydrides exhibit electrical conductivity and bulk paramagnetism similar to the pure metals. The Knight shift in metals is well knownI6 and arises from the interaction of the conduc tion electrons, near the Fermi surface of the metal, with the metal nuclei. In an externally applied magnetic field H, these unpaired electrons have a net sp~n polarization x;H which contributes a local magnetIc field at the nucleus. This local field l:!.H is proportional 82 R. A. Oriani, E. McClirnent, and J. F. Youngblood, J. Chern. Phys. 27,330 (1957). to the product of xpH and the density < ItM(O) 12)AV of the conduction electrons at nucleus M. In particular, the local field shifts the resonance position by a frac- tional amount l:!.H/H= (8'n/3)xpa( IlfM(O) 12)AV, (19) where the superscript a on Xp indicates that the Pauli or spin susceptibility is in atomic units. If the suscepti bility is given in cgs units per gram, XpG, Eq. (19) becomes where M is the atomic or molecular weight of the sample. In this expression, < ItM(O) !2)AV is unknown and cannot be calculated for most metals, and so the assumption is made that an indication of its value can be obtained from the equivalent expression for the free atom, IlfA(O) 12• There are several ways of obtain ing the latter,16 of which the most direct is from the experimental determination of the atomic hyperfine coupling constant, a(s), which is given by a(s) = (167l'/3) (fJ.r/I»)J.B IlfA(O) 12, (21) where fJ.r and I are the nuclear magnetic moment and spin, respectively, and ILB is the Bohr magneton. With the further definition of ~ as ~= < IlfM(O) !2)AV/ ItA (0) 12, (22) which is of the order of unity, we can write Eq. (20) as where gr is the nuclear magnetogyric ratio. It is to be noted that the shift results from the contact part teO) of the electronic wave function and that, therefore, only s conduction electrons make a direct contribution. The shifts predicted by this equation are opposite in sign to those observed for protons in the titanium hydrides; but the magnitudes can be comparable depending upon the values of ~ and XpG. In fact, both positive and negative proton shifts of the same magni tude as those in the titanium hydrides have been found in solid free radicals.33 These shifts are proportional to the spin susceptibility and distribution in the radical of the unpaired electron and depend upon the electron nucleus hyperfine interaction as do the Knight shifts. Therefore it is of interest to compare the proton shifts with the spin susceptibility of the titanium hydrides. Unfortunately, the latter is not directly available, as it is difficult to separate the spin and diamagnetic components from the bulk susceptibility x. However, the spin susceptibility is of the same order as the bulk susceptibility, and for the transition metals it is a pretty fair approximation to take them as equal because the 33 T. H. Brown, D. H. Anderson, and H. S. Gutowsky, J. Chern. Phys.33, 720 (1960); and earlier work cited therein. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1202 STALINSKI, COOGAN, AND GUTOWSKY effective mass m* of the electron is large.84•35 On this basis, Eq. (23) predicts that the proton shifts should be proportional to the bulk molar susceptibility, x=Mxu• However, the change in hydrogen content causes M to vary only from 49.5 to 49.9 g mole-I, which can be neglected; therefore, our comparisons are given in terms of the XU values from the literature. Figure 6 includes the dependence upon hydrogen content of XU as well as of the proton shift. It may be seen that both XU and the shift increase with increasing hydrogen content. However, XU has a small anomaly at TiH1.79, for which there may be no counterpart in the shift, but more shifts would need to be measured in the vicinity of this composition to establish the point. Probably of more importance is the fact that over the range of hydrogen content investigated, the shift increases by a factor of three while XU increases by only about 15%. In Fig. 7, the temperature dependence of XU parallels that of the shift for the two samples in vestigated. Particularly striking is the correspondence between the anomaly in the shift and in XU at 15°C for TiH1.969. This shows very clearly that the proton shifts are directly related to the magnetic susceptibility of the electrons in the hydrides. However, again the quantitative comparison is poor in that the shift at the anomaly is larger than that at -50°C by a factor of about 1.5 while XU is only about 5% larger. It seems that there is no simple, direct link between the diffusional motion of the hydrogen and the proton shifts. The sample of TiH1.969 exhibits an anomaly in the shift over a temperature range where there is negligible diffusion. On the other hand, for TiHl.607 the lie for diffusion is of the order of 10" secl at 50°C, lie changes considerably over the temperature range for which the proton shift was measured, and yet there are no anomalies in the shift. Of course, it is conceivable that there is an inverse relationship between the anomaly and the hydrogen diffusion; that is, the hydro gen diffusion disrupts the interactions producing the anomaly and reduces the magnitude of the anomaly in the samples of lower hydrogen content. Origin of the Proton Shifts From the results discussed in the foregoing section, we conclude that the proton shifts are related to some electronic interaction or property which makes an appreciable, but not dominant contribution to the bulk magnetic susceptibility of the titanium hydrides. There are several such possibilities. The diamagnetic or orbital effects of the conduction electrons produce shifts to higher fields. However, detailed calculations35 have shown that these contributions are at most of the order of a few percent of the spin effects, so it seems reasonable at this stage to consider further only the latter. 34 C. Kittel, Solid State Physics (John Wiley & Sons, Inc., New York, 1956), 2nd ed., p. 295. 36 T. P. Das and E. H. Sondheimer, Phil. Mag. 5, 529 (1960). There are three ways in which the spin polarization of the conduction electrons can lead to a shift in the proton resonance: (a) The conduction electrons themselves can have a finite density at the protons. This, however, would give rise to a Knight shift which we have already noted is opposite to the observed shifts. Therefore, any con tribution of this nature must be overcome by the mechanism(s) responsible for the negative sign of the observed proton shifts. (b) There can be a direct bonding overlap between the conduction electrons and electrons localized in 1s orbitals centered on the protons. By the Pauli exclusion principle, this would pair the spin of the 1s electron antiparallel to that of the conduction electron, and there would result a proton shift opposite to the Knight shift. This type of effect we will denote as a "direct exchange" interaction. (c) The conduction electron density on the titanium atoms can participate in an exchange interaction with the bonding electrons in atomic orbitals centered on the titanium atoms. This Hund's rule coupling tends to orient the spins of the bonding electrons parallel to those of the conduction electrons. The bonding elec trons centered on the titanium atoms transmit the spin polarization, with a change in sign, to the protons via the exchange interaction in the covalent part of the Ti-H bond. The model is similar to that for the C-H fragment,a6 where a positive spin polarization oi an odd electron in the 71' orbital on the carbon gives rise to an upfield shift of the proton resonance.33 This mechanism will be called an "indirect exchange." It is beyond the scope of this work to attempt a general a priori calculation of the various electron electron, electron-nucleus, spin-spin interactions which we have divided in the fashion just described. For example, it is very difficult to estimate the indirect exchange to better than an order of magnitude in the simpler, isolated-fragment case.36 Instead, we will present an ad hoc analysis which seeks to identify the factors responsible for the major features of the ob served proton shifts. These include the sign and magni tude of the shifts and their dependence upon tempera ture and sample composition. From a phenomenological viewpoint, we assume that the proton shift is proportional to XHa, where the latter is the spin susceptibility of electrons "on" the hydrogen atoms, expressed in gram atomic units. The effects of the direct and indirect exchange interactions between the conduction electrons, which presumably are con centrated on the titanium atoms, can be combined in the expression (24) This means that there is a probability 1/ for the spin 36 H. M. McConnell and D. B. Chesnut, J. Chern. Phys. 28, 107 (1958); and prior work cited therein. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPRO TON MAG NET I eRE SON A NeE IN TIT A N I U M H Y D RID E 1203 susceptibility XTi" of conduction electrons "on" the titanium atoms to be transmitted with a change of sign to electrons about the protons. As defined here, '1'/ includes the effects of both the degree of the spin exchange and the extent to which the orbital on the hydrogen is occupied. One might next be inclined to say that XP, the spin susceptibility of the conduction electrons, should equal XTi". However, this leads to the result that XTia is directly proportional to xu, and that the proton shift is also directly proportional to xu. But this is incompatible with the observed tempera ture dependence of the proton shifts, unless the polariza tion exchange probability '1'/ in Eq. (24) is very sensi tive to temperature. To be sure, one would expect some dependence of '1'/ upon temperature and hydrogen content; however, for simplicity we will assume it to be constant and explore the consequences. A comment on notation may be helpful here. The symbol XP has been used to designate the Pauli sus ceptibility, which is the spin susceptibility of the con duction electrons. Titanium hydride differs from metals in that according to our atomic orbital model, XP is but one of several contributions to the total molar spin susceptibility x.. Nonetheless, one would still expect the latter to be approximately equal to the bulk susceptibility in the hydride, i.e., x.=Mxu, just as xp=Mxu in the metals. The effective mass of con duction electrons is large in the transition metals, so their diamagnetic, orbital contributions to the bulk susceptibility are very smaIl34,35; and one would expect this to apply to the titanium hydride as well, because the Ti-Ti distance in the hydride is not very different from that in the metal and the conductivity is of the same order. Moreover, in the hydride, the electrons centered on the protons contribute mainly to x. be cause they are s electrons. With '1'/ constant, the temperature dependence of the proton shift can be fitted by assuming that there is a relatively large, constant component of x. which is not involved in the exchange interactions; that is (25) where xo and Xou are constants and X.,exM is the net component, in gram mole units, of the spin suscepti bility of electrons associated with the exchange inter actions. This latter quantity consists of opposing con tributions from electrons centered on titanium and hydrogen atoms. For TiHz_6, this approach leads to the relation (26) which, upon substitution of Eq. (24) and rearrange ment, becomes This can be related to the observed bulk susceptibility per gram XU via Eq. (25), giving XHa= -M(xg-xoJl) 1[1-(2-6h]' (28) Equation (28) provides a basis for calculating from the bulk susceptibility the proton shift produced by the exchange interactions. For this purpose, we modify Eq. (23) for the Knight shift by substituting XHa for Xp. This gives (AHIH)ex=a(s)xHa~H/2gIJ.l.B' (29) where ~H is the ratio of I~(O) IH2 for a ls electron on a hydrogen atom in the titanium hydride solid to that for a free hydrogen atom. In addition to the exchange inter action, there is the possibility that the conduction electrons themselves have an appreciable density at the protons and thus makes a direct Knight-type contribution to the proton shift.37 This requires a model similar to that applied with considerable success to the case of substitutional alloys,16 except that allowance must be made for the difference in conduction electron density about the hydrogen atom compared to the titanium atom. A difficulty arises, however, in deter mining the value of the spin susceptibility to be as sociated with this direct Knight shift. Probably it is as reasonable a choice as any to use XTi" for this purpose On this basis one obtains (AHIH)K=a(shTia~TiH~H/2gIJ.l.B, (30) where hiH is defined as the ratio ~TiH= < 1~(H) !2)AVI < 1~(Ti) 12)Av. (31) In both Eq. (29) and Eq. (30), a(s) is the hyperfine interaction constant for the free hydrogen atom. The Knight-type contribution can be expressed in terms of the bulk susceptibility XU by elimination of XTia with Eqs. (24) and (27). The final result is AnIH = (AHIH)ex+(AHIH)K =[a(s)~H/2gIJ.l.BJ(1-hiHITJ)xH" (l-~TiH/'I'/)M(xg-xou) =[a(s)~H/2gIJ.l.BJ 1-(2-6)71 . (32) The various experimental observations of the proton shifts are fitted at least semi quantitatively by this equation with more or less reasonable values for the four adjustable parameters ~H, ~TiH, '1'/, and Xou• Comparison of the Exchange Interaction M odd 11.!ith Experiment For the temperature dependence of the proton shift in a specific hydride, Eq. (32) reduces to the form (33) 37 This contribution was not included in the preliminary analysis reported in work cited in footnote 24. However, if it is not in cluded, an unreasonably low value of 0.125 is required for ~H in order to fit the data. Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1204 STALINSKI, COOGAN, AND GUTOWSKY where the "constant" Aa depends upon o. A value of 3.25 is obtained for Xou by fitting this equation to the experimental proton shifts and bulk susceptibilities at -100° and 15°C for TiH1.969. The solid curve drawn through the experimental shifts plotted in Fig. 7 for TiH1.969 was calculated from the bulk susceptibility at the top of the figure by means of Eq. (33) with this value of 3.25 for Xou• The parameter 'Y/ can now be evaluated via Eqs. (32) and (33) from the ratio of Aa for samples of two different compositions. In this manner a value of 0.37 is obtained for 'Y/ from the room temperature proton shifts and bulk susceptibilities of TiH1.607 and TiHl.969. The temperature dependence of the proton shifts in TiH1.607 has been calculated from the experimental bulk susceptibilities given at the top of Fig. 7, using Eq. (33) with 'Y/=0.37 and Xog= 3.25. The solid curve at the bottom of the figure shows the results. Thus far we have considered only the relative values of the proton shifts. The various constants in Eqs. (32) and (33), excepting 'Y/ and Xog but including ~H and hiH have been treated as a scale factor. The latter two remaining parameters can now be evaluated by fitting Eq. (32) to the numerical values of the shift and bulk susceptibility at a point. For consistency, we have used the values for TiH1.607 at room temperature. The result is not a unique value for both ~H and hiH but a function relating one to the other, namely, (34) By employing this condition in Eq. (32) along with 'Y/=0.37 and xou=3.25, we have calculated the depend ence of the proton shift upon sample composition from the bulk susceptibilities plotted at the top of Fig. 6. The result is the solid curve drawn at the bottom of the figure. It can be seen that this and the other calculated curves agree reasonably well with experiment. The temperature dependence of the shifts is fitted very well. However, the observed dependence upon composi tion is nearly an exponential rise as the stoichiometric compound is approached, while the calculated curve is hyperbolic. In treating the dependence upon composi tion, one might argue about the effects of the inaccessi ble sites. For instance, should not the term in the denominator of Eq. (32) be (2-0-(0) rather than (2-o)? If the former is used, one obtains a slightly different value for 'Y/; however, the agreement between calculations and experiment is not affected materially. There is some slight evidence that the positive charge on the hydrogen increases with hydrogen content24; if true, this would decrease 'Y/ and improve the fit between the calculated and experimental shifts. The values of the various parameters obtained by fitting the experimental data with Eqs. (29)-(34) appear to be feasible. Although Eq. (34) does not provide a numerical value for either hiH or ~H, an independent estimate can be made for ~H. Even here, there are a large number of factors which could affect the value, such as a van der Waals expansion of the electron orbital,38 electrostatic effects of the surround mg titanium atoms, and reduction of orbital overlap by contraction of the hydrogen orbital. Hydrogen is most closely related to the alkali metals and for Li, Na, Rb, and Cs, ~ increases in the sequence16 0.43, 0.72, 1.00, and 1.3, respectively. Therefore, a value of about 0.4 seems appropriate for ~H; and, in any case, one would not expect ~H to be greater than unity. Accordingly, hiH would be about 0.27 and no larger than 0.32. Returning to Eqs. (29) to (32), we note that this requires the Knight contribution to the proton shift to be nearly i as large as the exchange contribution. Hence, the analysis suggests that the proton shifts result from a fractional difference in the opposing effects of the Knight and exchange contributions. The value of hiH itself seems feasible. It represents the decrease of the conduction electron wave function in the vicinity of a titanium atom, and the estimated value of 0.27 is certainly less than unity as would be expected. Moreover, a value as large as 0.3 is not out of the question because each hydrogen atom is surrounded by a tetrahedral first shell of titanium neighbors for which the H-Ti internuclear separation is only "'-'1.9 A. Of course, if XP were used, instead of the smaller quan tity XTia, to calculate the Knight-type contribution, hiH would be less by a reciprocal factor. This raises the question of the significance of Xou. It represents a large component of the spin susceptibility which does not participate in the exchange interactions. This is compatible with our use of XTia in Eq. (30), which assumes that Xou does not participate in the Knight-type contribution to the proton shift, because both involve the conduction electron distribution at the "exterior" of the titanium atoms. This suggests that Xou be interpreted as a contribution to the spin suscepti bility from the titanium "cores." The outer electron configuration of titanium is 3d24s2 and it is possible that either the d or s electrons are involved preferentially in the exchange interactions. In principle, this could be checked by observing the titanium resonance shift in the hydrides. In this regard it is suggestive that Xou is very nearly the same as XU for the pure metal. In fact, the experimental results can be fitted somewhat better by using XU for the pure metal in Eqs. (32) and (33) instead of Xog = 3.25. This procedure gives a slightly lower value for 'Y/, 0.34 instead of 0.37, with a correspond ing modification in Eq. (34) for the value of hiH• The picture emerging is one in which each hydrogen atom donates some fraction of its 1s electron to the conduction band of the solid hydride. The proton shift and the change in bulk susceptibility of the hydride from that of the metal are determined largely by the electrons contributed to the conduction band and by 38 F. J. Adrian, J. Chern. Phys. 32, 972 (1960). Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsPRO TON MAG NET I C RES 0 NAN C E I N TIT A N I U M H Y D RID E 1205 the electron density remaining localized in the hydrogen Is orbital. Accordingly, the fact that TJ is less than unity results at least in part from incomplete occupancy of the hydrogen Is orbital; and it would seem that the hydrogen has a net positive charge, counter to what would be expected from electronegativity arguments. The value of 0.37 found for TJ is acceptable insofar as we can jUdge. It equals the extent to which the hydro gen Is orbital is occupied times the factor describing the extent of the spin coupling by the exchange mechan ism(s). Thus, the value of 0.37 for TJ corresponds to a range in occupation number from unity to 0.37 while the reciprocal range for the coupling factor is 0.37 to unity. In the . C-H radical fragment, for which the hydrogen Is orbital is essentially fully occupied, a factor of about 0.05 describes transmission by the indirect exchange mechanism of the spin polarization of the unpaired 7r electron to the hydrogen Is electron. This factor should be several fold larger for titanium because of its larger nuclear charge. However, it is unlikely to be as large as the required minimum value of 0.37, which suggests that the direct exchange inter action is important. An entirely different approach to the nature of the shifts can be based upon the relation between the Knight shift and the spin-lattice relaxation time in metals. Korringa39.40 has derived the following ap proximate expression for the metal nucleus, Experimental values of the proton Tl are availablell for TiH1.77 from room temperature to 400°C. The Tl value remains virtually constant over the lower 60°C of this range, decreasing from 0.39 sec at room tempera ture to 0.35 sec at 90°C. The effects of self-diffusion begin to be important at higher temperatures and the Tl decreases sharply. However, the Tl at lower temper atures is due presumably to the interactions with the conduction electrons. If Eq. (35) applies to the protons as one would expect for the model described, the room temperature shift of 0.01 % corresponds with a Tl of 0.09 sec. The factor of 4 between this value and experi ment is within the approximations. Moreover, the temperature dependence observed for Tl is in accord with Eq. (35). It is of interest that the net proton shift must be used in Eq. (35) to obtain agreement with experiment. If the Knight or the exchange contribution to the shift were used, or the sum of their absolute values, the observed TJ would be 35, 65, and 200 times longer, respectively, than that calculated with Eq. (35). 39 J. Korringa, Physica 16, 601 (1950). 40 N. Bloembergen and T. J. Rowland, Acta Met. 1, 731 (1953). A recent careful study of the shifts in NaTI, by W. D. Knight and H. E. Schone, has substantiated that the shifts are -0.16% for Na and -0.5% for TI, in qualitative agreement with Bloem bergen and Rowland (private communication); H. E. Schone, Ph.D. thesis, University of California (1961). Negative Knight Shifts in Other Systems On the basis of the very limited information available on proton shifts in other nonstoichiometric hydrides, they appear to be related to the nuclear charge Z of the metal. Starting at low Z, the room-temperature proton shifts are -0.012% for TiH1.607, -0.009 5% for VHO.66,32 0.0000% for TaHo.3332 and +0.0017% for PdHo.66.1O In view of the sensitive balance of opposing factors which govern the shifts, this apparent Z de pendence could very well be accidental. Further study of these systems is needed to check the basic features of the model proposed before one can inquire, for example, whether the differences in the proton shifts of particular hydrides result mainly from the effects of the radius ratio and electro negativity of the metal on TJ or from other differences in the nature of the conduction band. In our preliminary report/5 there was included an attempt to evaluate TJ for VHO•66 using the observed proton shift in combination with that for V51 in the hydride compared to that in the metal. It is now apparent that the situation is more complex than was then assumed. However, there are some qualitative features which illustrate what might be gained by detailed studies of the metal as well as of the proton resonance shifts. By combining Eqs. (24), (29), and (30), we can express the proton shift in vanadium hydride as /lH/H=aH~H(~yH_TJ)xva/2gH/.I.B. (36) Also, Eq. (23) gives the Knight shift for the vanadium resonance in the hydride to be ky=/lH/H =aY~Vxk.va/2gy/.l.B, (37) where Xk.ya is that portion of the Pauli spin suscepti bility which contributes to the V51 Knight shift, and Xya is that involved in the exchange interactions. A value of 0.72% is obtained for ky in VHO•66 by adding the 0.17% downfield shift observed for the V51 resonance in it compared to the metal,32 and the 0.55% Knight shift for the metal itself.16 Upon substitution of this and the other available experimental data into Eqs. (36) and (37) and dividing the latter, one finds TJ=~yH+0.094(~Y/~Ii) (Xk.ya/xya). (38) If the same values, 0.3 and 0.4, are assumed for ~y H and ~H as were employed for titanium hydride, and a value of 2.7 for ~y,16 Eq. (38) becomes (39) Even when allowance is made for the fact that 2.7 is probably too large a value for tv, Eq. (39) gives a reasonable result for TJ only if Xk.ya<Xya. That is, the spin susceptibility of conduction electrons contributing to ky is less than that of the conduction electrons participating in the exchange interactions with the hydrogen atoms. However, even though this might be Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1206 STALINSKI, COOGAN, AND GUTOWSKY explained by saying that the exchange interaction involves only the d electrons of the vanadium 3d34s outer configuration, the 0.17% V61 shift between the hydride and the metal requires that the s electrons be affected as well. More detailed analyses of such ques tions can be based upon the bulk susceptibilities. Moreover, it would be helpful if the resonance of the conduction electrons could be observed. The first report of negative Knight shifts was made by Bloembergen and Rowland,40 who observed nega tive shifts for both components in an ordered inter metallic compound NaTl. There is some doubt about the Na23 shift, which is small (-0.016%) and may be due to a chemical shift or to uncertainty about the datum for shifts. The authors tentatively suggested that the effect may be due to configurational interac tion,41 in which an inner s electron is promoted to a higher s-type orbit in the conduction band. Negative Knight shifts have also been reported for the Al27 resonance in the magnetic rare earth intermetallic compound s42 having the cubic Laves structure, for ex and (3 Mn,43 for Pt,44 for Sn and Ga in the intermetallic 41 A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. (London) A205, 135 (1951). 42 V. Jaccarino, B. T. Matthias, M. Peter, H. Suhl, and J. H. Wernick, Phys. Rev. Letters 5, 251 (1960). 43 V. Jaccarino, M. Peter, and J. H. Wernick, Phys. Rev. Letters 5, 53 (1960). 44 T. J. Rowland, J. Chern. Phys. Solids 7,95 (1958). compounds NbaX and VaX,45 and for the Pt, Ga, Sb, and Si resonances in the intermetallic vanadium com pounds VaX.46 An exchange interaction between the f electrons localized on the rare earth atom, and the conduction electrons has been proposed as the origin of the negative aluminum shifts.42 Our results for titanium hydride have been interpreted in terms of exchange interactions of somewhat different character, and it appears likely that the negative shifts in the intermetal lie compounds also involve exchange interactions. ACKNOWLEDGMENTS One of us (B.S.) would like to thank the Institute of International Education and the Ford Foundation for a Visiting Senior Scholarship which made this work possible. Another (C.K.C.) would like to thank the C.S.I.R.O. for leave of absence and to acknowledge with gratitude the support given by the International Cooperation Administration under the Visiting Scien tist Program administered by the National Academy of Sciences of the United States of America. 4l> R. G. Shulman, B. T. Wyluda, and B. T. Matthias, Phys. Rev. Letters 1, 278 (1958). 46 W. E. Blumberg, J. Eisinger, V. Jaccarino, and B. T. Mat thias, Phys. Rev. Letters 5, 149 (1960). Downloaded 23 Nov 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

1.1722230.pdf

Physical Properties of Titanium. II. The Hall Coefficient and Resistivity Georgiana W. Scovil Citation: J. Appl. Phys. 27, 1196 (1956); doi: 10.1063/1.1722230 View online: http://dx.doi.org/10.1063/1.1722230 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v27/i10 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 10 Mar 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 27. NUMBER 10 OCTOBER. 1956 Physical Properties of Titanium. II. The Hall Coefficient and Resistivity* GEORGIANA W. ScoVILt Bryn Mawr College, Bryn M awy, Pennsylvania (Received April 26, 1956) The Hall coefficient of titanium has been measured over a temperature range from 3500K to 1 tOO oK. The coefficient is -2.0X to-Um3/coulomb near room temperature, reverses sign at (675±30)OK, and increases to a value of +3.5X to-um3/coulomb at l100oK. The measurements were made on titanium samples which had a purity of 99.99%. The samples were heated by direct current, and the temperatures were determined indirectly from the resistivity measurements which were made with each Hall measurement. The resistivity is 0.48 microhm m at 350oK. It increases to a maximum of 1.76 microhm m at the crystal transi tion temperature. MEASUREMENTS of the Hall coefficient and the resistivity of a metal supply information which is useful in determining the densities and the mobilities of the charge carriers which take part in electrical conduction. This information for titanium is of special interest because titanium is a transition metal with a hexagonal structure at ordinary temperatures and a body centered cubic structure above 1155°K. Since both electrons and holes contribute to conduction in the transition group no quantitative deductions can be made from these data alone. However, it is hoped that a study of the variation of these electrical properties with temperature in the regions on both sides of the crystal transition may lead to further information on the electronic structure. The results for the low tem perature a phase are reported in this paper. The titanium samples which were used for these Hall measurements were supplied by Rem-eru Tita nium, Inc. A chemical analysis of one sample showed an over-all purity of 99.99%. The principal impurities were silicon, nitrogen, and oxygen. Although no analysis was available for the other samples, the agreement among the measurements indicates that they were of comparable purity. Each filament, 12 cm in length and about 0.3 cm in width, was cut from either 10-or 20-mil sheet titanium. It was mounted along the central axis of a chamber which could be evacuated, and it was held under tension by means of a titanium spring. The filament passed through a 0.95-cm air gap between 4A5-cm diameter soft steel pole pieces of an electromagnet which supplied magnetic fields of between 0.35 and 0040 wb/m2• Two different Hall probe arrangements were used; a three probe1 and a two probe system. In both cases the leads of 0.002-in. platinum wire were welded directly to the titanium near the center of the filament and then brought to O.057-in. Nichrome leads mounted perma nently to the chamber cover. Resistivity leads were welded to the filament at points about 3 cm each side of the Hall leads. * This paper is a portion of the thesis submitted by the author, in partial fulfillment of the requirements for the degree of Doctor of Philosophy, to the faculty of Bryn Mawr College. The samples were heated by direct current, and the temperatures were determined indirectly. Each time an equilibrium condition was reached, both the Hall voltage and the resistivity were measured. The tem peratures corresponding to the resistivities are shown in Fig. 1. This curve was determined using the equation for the power dissipated in the central section of a filament, J2pL/bd= 2 (b+d)Lft(]'(T4- To4), where I is the current, band d are the width, and the thickness of the filament, L is the distance between the resistivity leads, ft is the total emissivity, (]' is the Stefan-Boltzmann constant, T is the temperature of the filament, and To is the ambient temperature. The values used for the total emissivity were those reported by Michels and Wilford in the first paper of this series.2 Their resistivity curve for the commercial metal is shown for comparison. The lower resistivities and higher temperatures obtained with the present samples are indicative that the purity of the titanium is higher than was that of the wire available in 1948. The Hall voltages were amplified by means of a Liston-Becker, model 14, breaker type de amplifier, which under ideal conditions has a noise level which is within a factor of two of the limiting Johnson noise. The amplifier has a twenty-ohm input resistance. The amplified signal was fed to a Brown recording potentiometer. 3200 ::;1 I 160 0 /' + FILAMENT 8 >-./ I-120 / • FILAMENT 9 ;; / MICHELS & t= / WILFORD If) 80 l/) w 0: 40 20 1000 1400 TEMPERATURE OK FIG. 1. Resistivity vs temperature for titanium. t Now at Vassar College, Poughkeepsie, New York. 1 F. Kolacek, Ann. Physik 4, 1491 (1912). 2 W. C. Michels and S. Wilford, J. Appl. Phys. 20, 1223 (1949). 1196 Downloaded 10 Mar 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsHALL COEFFICIENT OF TITANIUM 1197 At each temperature the Hall voltages were deter mined as follows: The compensating circuit was ad justed to minimize the deflection obtained when the filament current was reversed in the absence of a magnetic field. With the filament current flowing in a given direction the magnetic field was established and the voltage between the probes was recorded. The magnetic field was reversed 10 to 15 times at half minute intervals. Then the filament current was re versed and the process repeated. The average change of voltage on reversal was determined as twice the Hall voltage. To check any unknown variation which might have occurred during a reading, a third set of data was recorded with the filament current again in the first direction. A known test signal was included with each recording. Measurements were generally made at in creasing temperatures, but at frequent intervals the temperature was lowered and a check measurement was taken. Figure 2 shows the variation of the Hall coeffi cient with temperature. The Hall coefficient for titanium reverses in sign from negative to positive at a temperature of (67S±30tK. This reversal occurs well below the transition tempera ture at which the crystal form changes from the hexago nal to the body-centered cubic form. Therefore it cannot be explained in terms of a gradual shift in the relative amounts of the two crystal forms present. Titanium has a hexagonal structure in the entire range in which these measurements have been made and it is probable that the coefficient of thermal ex~ pansion differs along the various crystal axes throughout this range. This would indicate a change in the relative crystal dimensions and, therefore, in the interactions of the atoms. The shapes of the energy bands may there fore be expected to change and the mobilities of the current carriers and their contribution to the Hall effect to be affected. The density of states curve for "body-centered cubic nickel" has been calculated in dependently by Slater and Koster3 and by the writer but the more difficult calculation for the hexagonal structure has not been completed. If the latter cal culation were made, it would be possible to determine whether a change in the d band with an increase in temperature was in the necessary direction to explain the observed reversal of the Hall coefficient. Figure 1 shows an extreme nonlinearity in the resistivity curve at temperatures well below the transition point. This would support the assumption of a change in the d band with temperature, for an increase in either the density of carriers or their mobilities would account for the departure from linearity. If this is the major factor influencing the conducting properties of titanium the change in the energy bands must favor an incr~ased mobility for the holes. As the temperature is increased any change in the relative densities of the holes and electrons would also 3 J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). f Z w V ~ Or---4~O~0---r--~~~+---~~-+--~1~0~00~-+ w o TEMPE.RATURE OK U_1 ..J Do FILAME.NT 5 ;i 0 F ILAM EN T 8 ::I:-2 + FILAMENT 9 FIG. 2. Hall coefficient vs temperature for titanium. lead to a change in the Hall coefficient. If the over lapping 4s band shifts its position with respect to the 3d band, the position of the Fermi level will shift and there will be a change in the fraction of the carriers in each band. If the shift favors a decrease in the density of holes there will be an increase in the hole mobility and the positive Hall coefficient would be favored. It would be of interest to make Hall measurements in the region above the transition temperature. If the ob served increase in the Hall coefficient is a result of a nonsymmetrical thermal expansion of the hexagonal structure, the coefficient would be expected to level off in the body-centered cubic region. Several effects found superimposed on the Hall effect limited the temperatures reached in this investigation. As the. temperature increased, a consistent drift ap peared m the ~ecor.ded voltages, its direction depending on both the dlrectlOn of the maanetic field and on the direction of the filament curren~. The temperature at which the drift first appeared was different for each filament, and an extrapolation back to the time of the magnetic field reversal gave good agreement with the readings which did not show the drift. A second voltage appeared superimposed on the Hall voltage. It could be separated from the Hall voltage for it did not reverse with the magnetic field. However, it did increase in size with increasing temperature and did decrease the accuracy with which the Hall voltages could be deter mined. Investigation seems to indicate that these two difficulties were due to a lack of symmetry in the ~hermal system. Plans are under way to redesign and Improve the methods used in an effort to minimize these limiting effects and make possible a determination of the Hall coefficient in the neighborhood of the crystal transition temperature. The experimental results reported in this paper show the variation with temperature of the Hall coefficient for the low temperature a phase of titanium. The coefficient is -2.0X 1O-1lni.3/coulomb just above room tempera ture. The change as the temperature is raised indicates Downloaded 10 Mar 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1198 GEORGIANA W. SCOVIL an increasing contribution to the conductivity by holes. The coefficient reverses sign at (67S±30tK and reaches 3.SX lO-nm3/coulomb at 1100oK. Several pos sible reasons for this reversal have been suggested and further work has been proposed which may lead to more definite information on the density of states of this transition metal. This work was carried out while the author held the Pennsylvania-Delaware Fellowship awarded by the American Association of University Women, and was done as part of a broader program supported by a Frederick Gardner Cottrell Grant of the Research Corporation. The author wishes to express appreciation for the support of both of these groups, and to extend thanks to Rem-Cru Titanium, Inc. who supplied the titanium samples. The encouragement and valuable criticism generously given to the author by Dr. Walter C. Michels is gratefully acknowledged. JOURNAL OF APPLIED PHYSICS VOLUME 27. NUMBER 10 OCTOBER. 1956 Viscoelastic Properties of Ice H. H. G. JELLINEK* AND R. BRILL Department of Physics, Polytechnic Institllte of Brooklyn, Brooklyn 1, New York (Received April 9, 1956) An apparatus has been constructed for the study of deformation under tension of single and polycrystalline ice. Deformations down to 10-5 cm could be measured. Deformation of single and polycrystals was investi gated as a function of time, stress, and temperature. Whereas the strain rate for polycrystaIIine ice decreases with time, that for single glacier ice increases linearly with time. The deformation for fine-grained poly crystalline ice consists of an instantaneous elastic deformation, a transient creep and a steady state creep. Deformation curves can be represented by empirical equations. The recovery curves on removal of the loads have also been investigated and the plastic flow has been deduced from the residual deformation after complete recovery. This plastic flow was found to be Newtonian within the range of stresses investigated and the viscosity coefficients can be represented by an exponential relationship as follows: 'II = 7.5· e+l6100IRT poises, where 16 100 calories is the energy of activation for the plastic flow. The total deformation can be represented satisfactorily by a large number of Voigt units representing a distribution of retardation times, in series with a MaxweII unit. The experimental results are further discussed in the light of current theories of dislocations and tentative mechanisms for the deformation of single and polycrystalline ice are proposed. VERY little quantitative and systematic informa tion on the viscoelastic properties of ice is avail able. Hessl investigated the deformation of ice bars and V. V. Lavrov2 measured the plastic flow of bars under constant load. More recently Glen3 made a study of the mechanical properties of ice cylinders, especially under compression. The present work is concerned with the strain experi enced by poly and single crystalline ice cylinders under tension. These measurements were performed over a range of stresses and temperatures. The recovery of the specimens after removal of the load was also investi gated. The analysis of the experimental results throws light on the mechanism of the deformation of ice. EXPERIMENTAL (a) Apparatus and Technique The apparatus is shown in Fig. 1. It consists of a steel frame resting on a heavy plate as shown in the figure. * Present address: S.I.P.R.E., Corps of Engineers, U. S. Army, Wilmette, Illinois. I H. Hess, Z. Gletscherkunde, 21, 1 (1940-41). 2 V. V. Lavrov, Zhur Tekhn. Fiz. 17, 1027 (1947). 3 J. W. Glen, Proc. Roy. Soc. A228, 519 (1955); with H. F. Perut .... , J. Glaciol. 2, 111 (1952); Nature 172, 721 (1953). This apparatus rests on a metal plate, which can be levelled horizontally. One of the steel columns carries a Statham strain gauge, which can be moved vertically along the steel column. There is also a gear arrangement for fine adjustment, enabling vertical movements of the order of 10-4 cm to be carried out. The other column carries a small brass platform, which can be slid ver tically along the column and screwed tight to it at any desired height. The platform itself can be raised or lowered at any fixed position at the column by means of a spring and lever device. The whole apparatus is placed in a thermostated and well insulated wooden box. A bare thermocouple inside the box showed tem perature variations not larger than 1/20°C. The output of the Statham gauge was passed via an amplifier to a Leeds and Northrup Pen Recorder. For the highest sensitivity used, one division on the recorder (total 100 divisions) corresponded to a deformation of the ice specimen of 2 x 10-0 cm. Ice blocks of about 2.5 x 2.5 x 10 cm were frozen directly on to aluminum cylinders of 2.5 cm diameter and 2.5 cm height. The surface of these cylinders onto which the ice was frozen was suitably roughened to in crease the adhesion between metal and specimen. The other surface was machined flat except for a rod of 3 cm Downloaded 10 Mar 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.1735152.pdf

Effect of Space Charge Fields on Polarization Reversal and the Generation of Barkhausen Pulses in Barium Titanate A. G. Chynoweth Citation: Journal of Applied Physics 30, 280 (1959); doi: 10.1063/1.1735152 View online: http://dx.doi.org/10.1063/1.1735152 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spatial distribution of polarization and space charge in barium strontium titanate ceramics J. Appl. Phys. 100, 054102 (2006); 10.1063/1.2336487 Effect of space charge on polarization reversal in a copolymer of vinylidene fluoride and trifluoroethylene J. Appl. Phys. 64, 2026 (1988); 10.1063/1.341733 Polarization Reversal Electroluminescence at Low Frequencies in Barium Titanate Crystals J. Appl. Phys. 37, 810 (1966); 10.1063/1.1708262 Polarization Reversal in the Barium Titanate Hysteresis Loop J. Appl. Phys. 27, 752 (1956); 10.1063/1.1722477 An Experimental Study of Polarization Effects in Barium Titanate Ceramics J. Acoust. Soc. Am. 26, 696 (1954); 10.1121/1.1907401 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 3 MARCH, 1959 Effect of Space Charge Fields on Polarization Reversal and the Generation of Barkhausen Pulses in Barium Titanate A. G. CHYNOWETH Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey (Received June 13, 1958) The rate of generation of Barkhausen pulses in barium titanate when the polarization direction is slowly reversed is profoundly influenced by the form of the voltage cycling given to the crystal. It is concluded that the rate of nucleation of new domains is determined by the field near the electrodes which, in turn, is the resultant of the applied field and a relaxing space charge field. This result follows directly if the Bark hausen pulses represent individual nucleations though this assumption is not crucial since the generation rate of the pulses parallels the total current at all stages of the polarization reversal. It is concluded also that the majority of the Barkhausen pulses occur independently of each other and of their surroundings. INTRODUCTION COMPLETE reversal of the polarization state of barium titanate crystals takes several seconds or even minutes if suitably low applied field strengths are used. During this period many individual charge pulses, called Barkhausen pulses, c-ani'le resolvedl and a study of these has-been made in the hope that it would yield information about the details of the polarization reversal process. In these investigations it was observed that the Barkhausen pulse behavior was influenced by the form of the field cycling and from the experiments described in this paper it is concluded that relaxing space charge fields were modulating the nucleation rate of new domains. The results of a study of the Bark hausen pulses themselves and their interpretation have been published elsewhere.2 EXPERIMENTAL The single crystals of barium titanate were entirely c domained with the polarization vector in the thickness direction. They were grown by the Remeika method3 from a melt containing 0.2 atomic percent of iron to counteract, somewhat, the effects of reduction at high CALIBRATING PULSE n ETC (b) FIG. 1. (a) Circuit used for production and detection of Bark hausen pulses. (b) Voltage cycle applied to crystal. I Newton, Ahearn, and McKay, Phys. Rev. 75, 103 (1949). 2 A. G. Chynoweth, Phys. Rev. 110, 1316 (1958). 3 J. P. Remeika, J. Am. Chern. Soc. 76,940 (1954). temperatures. The processed units were about 3 mm square and of the order of 10-2 cm thick, They were provided with circular evaporated platinum electrodes on opposite major faces and contact to these was made by thin strips of silver foil affixed by minute spots of air-drying silver paste. The circuit used for producing and measuring the Barkhausen pulses is shown schematically in Fig. 1 (a). To obtain reproducible data it was necessary to subject the crystal to regular voltage cycling and accordingly, the switches Sl, S2, in the circuits that controlled the voltage cycle were actuated by cams on a steadily rotating shaft. 10 0'----,3o":O--"6o---:9':-o---:
~20::-----:-:'15-:0--:-:18'::O--:-'210 TIME IN SECONDS FIG. 2. Barkhausen pulse counting rate versus time during switching for different values of the applied voltage. When both switches Sl and S2 were in position 1, a positive voltage was applied to the crystal. With the switch S3 connected to the large condenser Cl, this voltage increased slowly with the time constant RICl• With S 3 connected to the much smaller condenser C2, and with Rl=O, a steady voltage was applied to the crystal; the value of C2 was sufficient to insure that the voltage source presented a very low impedance to the crystal circuit for the frequencies equivalent to the Barkhausen pulses. At the end of the time tl, indicated in Fig. 1 (b), SlA moved to position 2, thereby dis charging condenser Cl (or C2). Shortly afterwards, S2A 280 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18SPACE CHARGE FIELD EFFECTS IN BaTiO a 281 moved to position 2, thus applying a negative voltage, V 2 to the crystal. After a time t2, switch S2.4 returned to position 1 followed by S 1.4 returning to its position 1, thus restarting the voltage cycle with the positive voltage VI. The spurious noise signals that entered the amplifiers when SlA closed to position 1 were prevented from being recorded by the scaler by switch SIB, ganged to SIA. SIB controlled a voltage applied to a gating circuit on the scaler; the time delay, a fraction of a second, between SIAl closing and the scaler being gated on was controlled by the integrating circuit R3C. and the settings of the trigger levels in the gating circuits. In all cases, this delay resulted in a negligible number of pulses being missed by the scaler. Barkhausen pulses were counted only while positive voltage was applied. At the end of 11 the scaler was gated off, thereby avoiding further spurious counts from transients. The voltage pulses arising from the Barkhausen effect were fed on to the grid of a cathode follower FIG. 3. Barkhausen pulse counting rate versus time during switching with a positive voltage immediately after having a negative voltage applied to the crystal for various periods. The value of the positive voltage was 3.0 v and it was applied for, approximately, 120 sec. through an RC coupling circuit, the constants of which were chosen so as to maintain a sufficiently high input impedance consistent with the further requirements that the circuit should have an adequate integrating time constant for the pulses and that the voltage swing at the grid due to the positive voltage should quickly die out. The resistor R completed the necessary de circuit for the crystal while its value determined the integrating time constant for the input circuit. Per manently connected into the circuit was the small calibrated condenser Ce. Voltage pulses of known height applied to this condenser produced charge pulses of known magnitude allowing the input circuit to be cali brated. The voltage pulses were led from the cathode follower to the conventional pulse amplifiers and finally through a pulse height analyzer to a scaling circuit or a ratemeter. The latter was used for the experiments described in this paper. The variation of the counting rate with time was measured when the crystal was subjected to square w ~ Cl: ~ i= Z 10 :> ° u ~ r-- If r'\ /I J h v-----. ./2 /I 71. '1/ '/ . 21 ' I r----t I' I 3 I I 6 2 I -... ~ i ---- ..... ~ '\ ~ \~ \ I ------ c-__ ---f-------- _.-- ~ V2 12 '-.::!..O TS ,~ ~ ~ :-..., "-.~ ~ " " 6 4\ 5 9" ~ '" f"... 10 2 \ o 10 15 20 25 30 35 40 45 TIME IN SECONDS FIG. 4. Barkhausen pulse counting rate versus time during switching with a positive voltage immediately after having various negative voltages applied to the crystal. The value of the positive voltage was 3.0 volts and the negative field was applied for 10 sec. voltage cycling, that is, with switch S3 connected to the smaller condenser C2. The discriminator bias level was set at about twice the noise level and the input circuit time constant was made long compared with the rise times of the majority of the pulses but short enough to avoid pulse pile-up in the amplifier at the counting rates involved. It was verified that under these condi tions, all but a small fraction of the total number of Barkhausen pulses was recorded. The integrating time constant of the ratemeter was made short compared with the rate at which the counting rate varied. The output signal from the ratemeter was recorded as a function of time on a paper recorder. RESULTS Semilogarithmic plots of counting rate versus time are given in Fig. 2 for various values of the applied field with /2= 30 sec and V 2= 24 v. For the highest values of VI used, the counting rate, R, increases im mediately to a high value followed by a decay which is exponential to within the errors of measurement. (Because of the necessarily short ratemeter time con stants, the recorder traces, in general, were rather noisy and the curves plotted in Figs. 2, 3, and 4 represent smoothed data.) At lower values of VI, R at first in creases slowly, passes through a peak, and then decays, again approximately exponentially. The initial increase in R takes place more and more slowly as VI is lowered while the time, 1M, taken for R to reach its peak in creases. Figure 3 shows a rather similar trend in the curves as the time 12 is increased, all other parameters being kept constant. A tendency towards a similar pattern occurs again in Fig. 4 as V2 is increased. These sets of curves obtained by varying Vj, V2, and t2 in turn are typical of the behavior of the many crystals tested. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18282 A. G. CHYNOWETH Curves very similar to those of Fig. 2 have been ob tained by Kibblewhite.4 In further experiments the period between the nega tive field pulse and the subsequent positive field pulse was varied. It was found that as this period was in creased, the subsequent ratemeter curves progressively changed from rapidly rising and falling to the more slowly rising and falling forms. Thus, during the off period, whatever effect the negative field had on the crystal gradually died out. By integrating under the ratemeter curves, it was established that the total number of pulses generated by the crystal during switching was unaffected by any of the field treatments to within the limits of experi mental error. This fact has been repeatedly confirmed in other measurements.2 DISCUSSION Experiments described elsewhere2 have shown that in the crystals used for these experiments, the genera tion rate of the pulses is proportional at all times to the total switching current throughout the polarization reversal. Therefore, since the current, at low fields, is controlled by the nucleation rate, so likewise will the rate of generation of the Barkhausen pulses follow the +1 + + + + ++ - + + 2// + 'Z~ (a) -t (C) 1-fJ + + ++ + + - + + + + _t (fl ,cW-t1'H <9' ___ 1.._ ],. o -t FIG. 5. piagrams illustratiI1;g !he space charge model proposed for explammg the charactenshc shapes of the counting rate curves. • A. C. Kibblewhite, Proc. lnst. Elect. Engrs. (London) I02B 59 (1955). ' nucleation rate. It has also been suggested that Bark hausen pulses represent domain nucleation events though such a direct relation is not essential to the present argument and in fact, in view of more recent studies,6 it is now known that at least some Barkhausen pulses have their origin when sideways moving domain walls approach each other closely. For present purposes, it is sufficient to suppose that domain nucleation events take place at the crystal surfaces and that nucleation will depend markedly on the field conditions near the electrodes. Thus, the pulse generation rate, which is proportional to the nucleation rate, will reflect the field conditions near the crystal surfaces. It will be assumed that, over the limited range of fields used in these experi ments, the number of domains (nucleating sites) is independent of the field, so that the number of Bark hausen pulses is proportional to the number of domains. (This assumption follows directly from the fact that the total number of pulses is independent of the field if the pulses represent either domain nucleations or domain walls colliding.) Letting N be the total number of pulses and n the number that have been generated by time t, we can then put the counting rate R= (dn/dt), proportional to the product of (N-n) and the nucleation probability. For the latter, Merz6 arrived at a function of the form exp (-a/E), where a is a constant and E is the field strength. Because, for a given field, the field factor is constant while (N-n) is continually decreasing, (dR/dt) must be negative at all times. This is clearly not the case, indicating that some additional process or factor has been overlooked. From inspection of Figs. 2, 3, and 4, it is clear that increasing either V 2 or t2 produces an effect qualitatively equivalent to reducing VI. One possible mechanism that could produce such effects would arise if the negative field gradually drives out residual domains that act as nuclei during the subsequent positive pulse. In this case, however, the subsequent ratemeter curve would be unaffected by the duration of the off-time between the negative and positive, which is contrary to experi ence. Furthermore, the constancy of N argues against this hypothesis. A more plausible hypothesis is that the magnitude and duration of the negative field can affect the switching processes when a positive field is subse quently applied through the agency of relaxing space charge fields. The presence of space charge fields would result in a field distribution in the crystal different from that produced by the applied field alone. Furthermore, if the charges are somewhat free to move, the resultant field inside the crystal will vary with time. The following model is proposed, therefore, to explain, qualitatively, the observed behavior. Consider the crystal to be initially in a space charge free or neutralized condition and suppose that there is • R. C. Miller, Phys. Rev. 111, 736 (1958). 6 W. J. Merz, Phys. Rev. 95, 690 (1954). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18SPACE CHARGE FIELD EFFECTS IN BaTiO a 283 present in the crystal a supply of carriers of both sign that are able to move under the influence of a field. (It is not essential to the argument to have both signs of carrier mobile.) An externally applied field will cause a drift of the negative charges towards the anode and of the positive charges toward the cathode. As the elec trode-crystal contacts are almost certainly imperfect there will be some pileup of carriers at the two elec trodes, as indicated in Fig. Sea), an equilibrium space charge field distribution eventually being attained as shown in S(b). The resulting space charge fields in the regions close to the electrodes will be strong and in the same direction as the applied field while in the interior of the crystal there is a much weaker space charge field opposing the applied field. Thus, the field near the electrode will increase with time in a manner suggested in Fig. S(c) where E+ is the applied field strength and E8+ is the equilibrium space charge field close to the electrode. When the applied field is reversed to E_, the space charge fields near the electrodes at first oppose the applied field. Gradually, however, they will disperse and collect in the configuration opposite to the previous one, as in Fig. S (d) with the resulting space charge field distribution represented in Fig. See). The variation of the effective field with time while E_ is applied will be as in Fig. 5 (f). Reversing the applied field again the effective field will vary with time as indicated in Fig. S(g). It will be supposed in the following that all meas urements are made while the positive field is applied and that it is maintained long enough for the space charge field to reach its saturation value E.+ On the other hand, it will not be assumed, necessarily, that the space charge saturation field Es-is attained with the negative field. Let J1. represent the change in the space charge field from its value E,+ while the negative field is applied. Then J1.=/E,+/+/Es-/ if saturation is attained in the negative direction. Let A=E++Es+. Then, at time t= 0 with the positive field applied, the effective field strength near the electrode is (A -J1.). Clearly, if the effect of the negative field on the space charge distribution is only slight, (A -J1.) > E+, and the space charge aids the applied field throughout the switching while the net field shows very little time vari ation. On the other hand, if the space charge field is modified appreciably while the negative field is applied, (A-J1.) <E+, and the space charge field at first opposes the applied field but eventually helps it. Probably, the time variation of the net field in the positive direction can be represented to fair approximation by an expres sion of the form E=A-J1. exp( -t/T), (1) where T is the relaxation time for the movement of the space charge. Thus the main effect of a mobile space charge will be to produce time variations of the net field in those regions of the crystal where nucleation is supposed to be taking place, that is, mainly near the surfaces of the 5 ,-, +' U '<,.- u ':' 10-~l~ ~ 5 a: 19 2 ~ 10-2 Z :J o U o 5 / / ~ 10-3 :J 5 / <lC 2/1 ::; 0: o z 10-0 I\. 1 :--b-. "'""" 1/ VI !\ '\ II 1\ \ 1\ 1\ -5000 1\ VOLTS/CM \ /.1-500 VOLTS/CM r- 1- -i-----t:::::. r-.. 1500 -1200 i"'-I r- ~ 1000 -.-; ,\3000 2000 I,r-j-t- t\4~ t- J'.. I\. ...... 1 .... -r--;-t--\ I o 5 10 15 20 25 30 3540 4550 55 60 65 70 TIME IN SECONOS FIG. 6. Forms of the counting rate curve for various positive fields, as obtained from theoretical analysis of the space charge model. crystal. 6 There will be much smaller effects in the interior of the crystal resulting in net fields slightly lower than the applied field. This may reduce by a proportionate amount the velocity of a domain wall. Hence, from the model presented, it is expected that the most important effect of the space charge fields will be on the nucleation rate. The expression for the counting rate can now be written, R~ (df/dt)=K(l-f) Xexp{ -a[A-J1. exp( -t/T)]-I}, (2) where K is a dimensional constant and f=n/N. Then f=O at 1=0. Solutions to this equation have been com puted for various values of the field strength param eters, A and J1.. Merz has shown7 that the value of a that is determined from switching current pulse studies depends on the crystal thickness, with a decreasing towards a constant value of about 5000 v cm-I as the thickness increases. For thin crystals it is believed that the apparently higher values of a arise because the sur face layers of the crystal do not take part in the switch ing on account of the large space charge fields located there. For the present analysis, the above value (as sumed to be the true one) was taken for a. From in spection of the counting rate curves, a reasonable value for Twas 7.5 sec. This choice is not critical since the time scale can be normalized. Figure 6 shows solutions of Eq. (2) for various values of A (in volts cm-I), a value of 500 v cm-I being chosen for J1.. Varying A corresponds to the experimental situa tion where V I is varied, and comparing Figs. 2 and 6 it is apparent that Eq. (2) represents very well the various experimental curves as long as suitable values are chosen for A. To be noted is the similarity between Figs. 2 and 6 as regards the spread in the slopes of the curves for large t and the position of the maxima. In particular, the spread in the values of VI in Fig. 2 is relatively much less than the spread in the values of A 7 W. J. Merz, J. Appl. Phys. 27, 938 (1956). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18284 A. G. CHYNOWETH .-. 10-...., -0 ~ ~ ·W 1 ~ 10-2 II <9 Z ;::: Z :::J o 5 2 U 10-3 o w N :J <i :::; II 5 2 §! 10-4 I- I / If II I ~ ~ ~o, I t--. ~ ~ ~t-... '10100 "0 5~~~S "~"I' -=-- 2~~G&- "~ i'. = 2000 VOLTS PER eM I'T-I-' o 5 10 15 20 25 30 35 40 45 50 55 TIME IN SECONDS FIG. 7. Forms of the counting rate curve for a given positive field as obtained from theoretical analysis of the space charge model for various degrees of disturbance of the space charge field while the negative field is applied. required to produce a comparable array of curves. This fact will be discussed in the following. From the above discussion of the behavior of the space charge field it is reasonable to suppose that the change, J.I., in the space charge field while the negative voltage is applied increases with both V2 and t2• How ever, it is clear that the set of curves obtained by vary ing J.I. in (2) will not resemble the experimental curves very closely. Varying /L alters the magnitude of the time-dependent factor but does not alter the eventual slope of the semilog plot of R versus t, the latter being given by exp( -a/A). The solution of (2) for various values of /L, A being put equal to 2000 v cm-1, are given in Fig. 7. It is apparent that, as in the experiments, there is an increase in tM with /L but, on the other hand, Eq. (2) leads to very little variation of the peak value of R with /L. Thus, Eq. (2) is an over-simplification when it comes to explaining the dependence of the counting rate curves V2 and t2. Better agreement could no doubt be obtained if A were made some function of /L and V 2 instead of VI alone. Furthermore, the assump tion that T is a constant is probably a poor one; cer tainly T will depend on the field and consequently, will vary with the field throughout the crystal. Also, inspec tion of Fig. 3 suggests that there may be at least two relaxation times associated with the movement of the space charge; if the series of curves for t2 ranging from 1 to 160 sec is extrapolated to t2= 0, the resulting curve shows a decay in R still slow compared with that obtained experimentally after momentary application of V 2. This shows that /L changes considerably during the first second of t2, and thereafter changes much more slowly. In spite of approximations in the theory, how ever, it is felt that the agreement with experiment is sufficient to confirm the mobile space charge field hypothesis. Further refinements could be made to the theory in principle though the solution of the modified Eq. (2) would be tedious to obtain; such a procedure is unwarranted for the present experiments. The final slope of the ratemeter curves yield values for A, the field strength in that part of the crystal where nucleation is taking place. The values of A so obtained cannot be regarded as the field right at the electrode, however, as it is likely that nucleation occurs mainly in some indistinct region towards the inner boundary of the surface space charge layer. This might account for the qualitative discrepancy between the spreads in the values of V 1 and A referred to above. It also makes any further analysis based on standard blocking-layer theories of dubious value. However, with this caution in mind, it is interesting to note that, using relations that have been derived by von Hippel et al. for the space charge behavior in alkali halide crystals,S estimates can be made of the charge concentration and carrier mobility with trapping. These are of the orders of 1014 em-a, and 10-7 cm2 secl v-1, respectively, while the corresponding Debye length is about 10"-4 em which is not very dif ferent from previous estimates of the space charge layer thickness.7,9 The mobility estimate is consistent with carrier drift with trapping at traps of about O.S to 0.7 ev in depth and this is equal to the value that has been determined for the activation energy from conductivity runslO on slightly reduced crystals of BaTiOa. Such carrier drift is similar to the drift of F centers under an applied field though color centers would not be visible to the eye at the above cencentrations and, indeed, no such color migration could be observed in these crystals. However, in some crystals known to be markedly defi cient in oxygen content, color centers have been ob served to migrate with a sharp boundary dividing the colored and uncolored regions.ll SUMMARY AND CONCLUSIONS From the experiments described in this paper it is conduded that the time variation of the Barkhausen pulse counting rate during slow switching is influenced considerably by the behavior of space charge fields in the crystal. Good qualitative agreement is obtained between the results and a theory which assumes a reasonable model for the way in which the space charge field near the electrodes varies during the voltage cycling and involves a previously established relation between the nucleation probability and the electric field strength. Consequently, the counting rate (or switching current) curves can be regarded as very sen sitive indicators of the effect of space charge fields on the switching process. In this connection it is significant to note that Kibblewhite4 found markedly different shapes of the counting rate versus time curves for barium titanate ceramics of different compositions. The results emphasize the necessity for regular voltage cycling if reproducible Barkhausen pulse data are to be obtained. 8 von Hippel, Gross, Jelatis, and Geller, Phys. Rev. 91, 568 (1953). 9 A. G. Chynoweth, Phys. Rev. 102, 705 (1956). 10 A. G. Chynoweth and W. J. Merz (unpublished material). II E. A. Wood and A. G. Chynoweth (unpublished material). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18SPACE CHARGE FIELD EFFECTS IN BaTiOa 285 Further conclusions concerning the Barkhausen pulses themselves can be drawn from the preceding experiments. First, as the probability of the occurrence of a pulse appears to be determined by Merz's nuclea tion probability expression, it is not inconsistent to relate the Barkhausen pulse itself to an individual nucleation. Second, Merz's conclusion that nuclei grow from close to the crystal surfaces is consistent with the conclusion that the occurrence of Barkhausen pulses is determined by the field near the crystal surface. Finally, the fact that, in the situation where the nucleating field is constant, the counting rate decays exponentially justifies the implicit assumption that individual nuclea tions occur independently of each other and of their surroundings. ACKNOWLEDGMENTS The author is indebted to S. Jankowski for his tech nical assistance and to Dr. R. W. Hamming and his computing staff for obtaining the solutions to Eq. (2). JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 3 MARCH, 1959 Recombination Centers on Ion-Bombarded and Vacuum Heat-Treated Germanium Surfaces SHYH WANG* AND GEORGE WALLIst Sylvania Electric Products, Inc., Woburn, Massachusetts (Received July 15, 1958) Germanium surfaces were bombarded with argon ions and then heat-treated under high vacuum. Room temperature values of surface recombination velocity, surface conductivity, dark field effect, and field effect under illumination were measured after successive heat treatments. In addition, the temperature dependence of these quantities was determined. It was confirmed that after annealing of the bombardment damage, a large number of acceptor type surface states approximately clamped the surface potential. Two types of recombination centers were identified: type 1, located near the middle of the gap and type 2, located near the valence band. The various heat treatments produced changes in the density of the type 1 centers but did not appear to effect the density of the type 2 centers. INTRODUCTION RECENTLY, a technique has been developed by Farnsworth and co-worker sI.2 to remove oxide layers from etched germanium surfaces and to produce atomically clean surfaces. The technique consists of three steps. (1) A crystal is outgassed at temperatures of 700°C or higher for several days. (2) The surface is bombarded with positive argon ions. (3) The resulting surface damage is annealed at temperatures around 500°C. From an examination of the surface by slow electron diffraction,2 it is concluded that the surface is atomically clean. Under high vacuum conditions, it is possible to keep a clean surface uncontaminated for a sufficiently long period of time so that various proper ties of the surface can be measured. Since then, many groups have used the same tech nique or a variation of the technique to study the work function,3.4 photoconductance,·-s surface con- * Present address: Department of Electrical Engineering, Uni versity of California, Berkeley 4, California. t Present address; Clevite Transistor Products, Waltham, Massachusetts. 1 Farnsworth, Schlier, George, and Burger, J. Appl. Phys. 26, 252 (1955). 2 R. E. Schlier and H. E. Farnsworth, Semiconductor Surface Physics, R. H. Kingston, editor (University of Pennsylvania Press, Philadelphia, 1957), p. 3. 3 J. A. Dillon, Jr., and H. E. Farnsworth, J. App!. Phys. 28, 174 (1957). 4 F. G. Allen, Tech. Reports 236 and 237, Cruft Laboratory, ductivity,1-9 and field effect·,7,9 of the ion-bombarded germanium surface. In the absence of electron diffrac tion evidence, it is not always obvious whether the techniques employed by various groups have indeed furnished an atomically clean surface. In particular, in much of the work step (1) has been omitted or carried out at much lower temperatures in order to avoid degradation of resistivity and lifetime. Where this has been the case, there is considerable uncertainty about the cleanliness of the surface. In spite of possible variations in the technique, it is generally agreed that the bombarded and annealed surface is strongly p type and that the energy bands at the surface are essentially fixed with respect to the Fermi level due to the existence of an enormous number of acceptor type surface states. There is also general agreement that adsorption of small amounts of oxygen makes the surface less p type and produces an increase Harvard University, Cambridge, Massachusetts, December 1955 (unpublished); F. G. Allen and A. B. Fowler, J. Phys. Chem. Solids 3, 107 (1957). 6 G. Wallis and S. Wang, Bull. Am. Phys. Soc. Ser. II, 1, 52 (1956). 6 H, H. Madden and H. E. Farnsworth, Bull. Am. Phys. Soc. Ser. II, 1, 53 (1956). 7 AutIer, McWhorter, and Gebbie, Bull. Am. Phys. Soc. Ser. II, 1, 145 (1956). 8 ]. T. Law and C. G. B. Garrett, J. Appl. Phys. 27, 656 (1956). • P. Handler, reference 2, p. 39. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 09:56:18

1.1721299.pdf

Electrical Phenomena in Adhesion. I. Electron Atmospheres in Dielectrics Selby M. Skinner, Robert L. Savage, and John E. Rutzler Jr. Citation: Journal of Applied Physics 24, 438 (1953); doi: 10.1063/1.1721299 View online: http://dx.doi.org/10.1063/1.1721299 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Period multiplication and chaotic phenomena in atmospheric dielectric-barrier glow discharges Appl. Phys. Lett. 90, 071501 (2007); 10.1063/1.2475831 The Effect of Air Ionization, Electric Fields, Atmospherics and Other Electric Phenomena on Man and Animal, by F. G. Sulman Med. Phys. 8, 723 (1981); 10.1118/1.594846 Electrical Phenomena in Adhesion: Further Comments on Electron Atmospheres J. Appl. Phys. 25, 1055 (1954); 10.1063/1.1721790 Electrical Phenomena in Adhesion. I. Electron Atmospheres in Dielectrics J. Appl. Phys. 25, 1054 (1954); 10.1063/1.1721789 The Electron Atmosphere in Dielectrics J. Appl. Phys. 25, 1053 (1954); 10.1063/1.1721788 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39438 JAMES E. MONAHAN and that the effective mean 2k will be known to better than 0.0001 in. This is an accuracy of about 0.05 per cent. Hence if all other corrections can be held to about 0.01 percent, it may be possible to determine the abso lute voltage scale at high voltages to about twice the accuracy already achieved (0.1 percent). An accuracy of 0.01 percent in analyzer length for a 40-in. radius is about 0.006 in. For the proposed value of 2k, the value of d is approximately 0.1 in. for all ratios given in Table I. Thus a length correction, r / d, less than 0.06 can be neglected. In order to obtain maximum ion current, it is necessary that b/k be greater than or equal to 2-!; and finally, as mentioned before, the radii Rand p must be of the order of the gap dimensions. With the exception of p and q, which can be evaluated approximately from Eqs. (20) and (21), the parameters listed in row 1 (Table I) are assigned arbitrarily. However, once this initial calculation is made, the behavior of the analyzer dimensions with respect to the transformation parameters described by Eqs. (22) and (23) may be used to obtain values of the analyzer ratios successively closer to any desired set. The ratios given in row 8 are within the limits of the proposed dimensions of the present analyzer. It is interesting to note that for the system described by the last line of the table (r / d = 0.03), the end cor rection neglecting the rounded corners gives tJd=0.37. This would result in an error somewhat greater than 0.05 percent in the determination of the equivalent length of the analyzer. JOURNAL OF APPLIED PHYSICS VOLUME 24. NUMBER 4 APRIL. 1953 Electrical Phenomena in Adhesion. I. Electron Atmospheres in Dielectrics* SELBY M. SKINNER, ROBERT L. SAVAGE, AND JOHN E. RUTZLER, JR. Department of Chemistry and Chemical Engineering, Case Institute of Technology, Cle'IJeland 6, Ohio (Received December 11, 1952) Rapid breaks of metal-polymer-metal adhesive specimens have shown the presence of a charge density on the metallic surface, provided the break occurs at the metal-polymer interface, and a much smaller charge if it occurs in the interior of the polymer. This is analyzed in terms of the electron atmosphere existing ex ternal to the metal in a dielectric region of low barrier. The barrier values in certain cases are otherwise known to be of appropriate order of magnitude. Measured experimental surface charge densities permit esti mation of the maximum limit of barrier height. Because of the presence of the charge distribution in the polymer, there is an electrostatic force on the adherend metal directed toward the dielectric polymer, which represents a contribution to the total adhesive forces opposing break. Such electrostatic contributions to adhesion have not previously been considered; their order of magnitude and their relation to the thickness of the adhesive are discussed. The qualitative agreement between the theory and a number of previously unexplained experimental results in the literature is shown. I. INTRODUCTION IN the course of a fundamental investigation of ad hesion and the phenomena contributing to it, it has been observed that when adhesive specimens, Fig. 1, / A-A A FIG. 1. Adhesive specimen. * With the support of the Flight Research Laboratory, Wright Air Development Center, U. S. Air Force, Contract AF33(038)- 26461. consisting of metal plates bonded by high polymer ad hesives are broken in tensile test, electrical charges are found on the two separating halves. If the opposite charges on the two halves are permitted to flow through an external circuit and the IR drop is recorded by a cathode-ray oscillograph, traces such as those in Fig. 2 are obtained. The experimental arrangements and re sults will be described in detail in the second article of this series. It is the purpose of the present paper to consider the possible origin of the charges; how the charge distribution may be related to experimental parameters, and what relation the charges may have to adhesion. Accordingly, for the present purpose, the only experimental facts resulting from this investigation which are utilized are: 1. There is a charge distribution in the polymer if the actual break occurs in a reasonably short time; the period of stressing before break may vary between wide limits, however, without alteration of the essential behavior. 2. The charge density is considerably greater if the polymer is stripped from the metal (adhesive break), [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39ELECTRICAL PHENOMENA IN ADHESION 439 FIG. 2. Oscillograph traces of potentials. than if the break occurs in the body of the polymer (cohesive break). Even in the latter case, charge den sities of low magnitude are observed. 3. The sign of the charge on each of the two halves of the specimen depends upon the metal and the polymer used in preparing the adhesive bond. The magnitude of the charge varies with experimental conditions, and breaks have been obtained in which the observed charge density has apparently been limited only by the breakdown potential of the air in the process of separat ing the plates of the specimen. 4. The phenomena in 1~3 above have been observed with a number of different adhesives, and with various metals bonded to them. Adhesion is a complex phenomenon, depending upon the simultaneous action of a number of different mecha nisms. In industrial practice, empirical methods are utilized to obtain the right amount of adhesion com bined with other desired properties of the finished materials. In the laboratory, it is found that the num ber of parameters and possible mechanisms which must be considered requires the application of painstaking surface chemistry techniques, and care in minute pro cedural detail, if the results are to be related to a single cause rather than a mixture of various causes. The fact that adhesion involves an interface between two differ ent substances introduces a degree of complexity not found in investigations of cohesive strengths. ~ot only do the two surfaces to be joined possess the forces which distinguish a surface from the interior of the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39440 SKINNER, SAVAGE, AND RUTZLER material, but also local differences in chemical composi tion, and adsorbed and other impurities affect their behavior. If nonmetals are under investigation, the use of high temperatures as an aid to obtaining clean surfaces is precluded because of the permanent changes which may be produced in the substance upon heating. In such a case, a specific preparational procedure is adopted, in order that successive specimens may have reproducible (though not necessarily specifiable) sur face characteristics. On the other hand, strong, reproducible bonds can be obtained quite regularly with suitable techniques. This would not be possible, if the practice and research in adhesives were entirely empirical, and the interface were a random mixture of unknown impurities. The adhesive bond is apparently less likely to suffer from such random factors than is the free surface before bonding.! In general, the mechanisms employed to explain the adhesion between substances have been van der Waals forces (broadly defined), or, more specifically, -dispersion (London) forces, dipole or quadrupole (Debye) forces, and induction forces. While these forces can be related to cohesion and to adhesion as well as to other behavior of materials, such as solubility, swelling, or the molecular association of certain organic substances, completely satisfactory agreement with the many phenomena of adhesion has not been obtained. This is in part due to the differing experimental conditions under which ad hesion data have been obtained, and in part to the com plex set of mutually operating phenomena involved in everyday adhesion. Experimental methods are required which identify and disentangle such factors as the role of the solvent and solvent concentration in the adhesive, the effect of the bulk elastic properties of the adhesive, the effect of surface structure of the adherend, or the extent to which the dipole concept, for example, may substitute for a direct consideration of the charge den sity distributions and potential energy in the neighbor hood of the interface. Accordingly.it becomes desirable to investigate the phenomena which occur at adhesive interfaces, and systematically to examine them for their applicability and relation to the phenomenon of ad hesion. The electric charge accompanying the breaking of the sample is one such phenomenon. II. THE EXPERIMENTAL BEHAVIOR Visualizing the adhesive specimen (Fig. 1) before and after break, and without regard to the physical nature of the adhesion before break, it is evident that any charge which may remain on the surfaces after break is directly related to a charge distribution an alogous to that in a parallel plate condenser.2 It !s possible to determine the total charge and often the 1 H. A. Perry, Jr., presented at the Symposium on Adhesion, Case Institute of Technology, April 24-5, 1952. 2 The analogy is not perfect. This will be treated in a succeeding paper. corresponding area of the plates, and thus to determine the charge density. However, adhesives do not cooper ate with the experimenter, and often the break is in the body of the adhesive, or both plates show clean metal portions and portions covered with adhesive. An independent verification of the charge density is desirable, and this can be done by suitable experimental techniques. The metal surfaces were bonded with high polymer adhesives either by prepolymerization and application, or by polymerization in place. Mter a number of pre liminary measurements, the sample was broken by a modification of the ASTM test method3; electrical con tact was made to the plates and the charge resulting from the break passed through a high resistance shunted by a cathode-ray oscilloscope. Since the plates are of small mass, the large forces applied to produce break cause rapid separation after break. This magnifies the potential difference Eo between them; at the same time, the flow through the external, measuring, circuit de creases Eo. When the separation is sufficiently rapid, an oscillograph trace with a maximum is obtained, the maximum being from ten to a thousand times Eo. As shown in another paper, analysis of the trace per mits determination of the charge density initially left on the metal surface by separation of the adhesive" Since under appropriate experimental conditions, the whole oscilloscope trace is obtained in from one to fifteen milliseconds, and the portion of the trace con cerned with the removal of the whole area of the ad hesiveS is of the order of 10"-4 second or considerably less, the experimental data are obtained before surface con tamination or other effects can produce markedly changed conditions of the surface. The existence of the charge at break can be explained qualitatively in several ways, according to the physical limitations imposed. Accepted knowledge of the solid state supplies what appears to be the most reasonable explanation. The sample consists of a dielectric bonded to metal plates on either side. The metal plates have a characteristic Fermi distribution of electrons, the dielec tric has a characteristic set of energy states modified by impurities, surface states, and proximity to the metal; between the two there is a barrier which de pends upon the specific characteristic quantities of each. Electrons or holes may be transferred between metal and dielectric. In the dielectric there will occur a distribu tion of charge dependent upon the barrier between them; if this is a negative charge, it is equivalent to an electron atmosphere. The break is made in a time of the order of, or smaller than, the relaxation time of the 3 ASTM Method of Test for Tensile Properties of Adhesives, No. D-897-49. • Surface charge density, and not the initial electrostatic poten tial between buttons is the object of the analysis. 5 The removal of the adhesive from the metal involves a succes sion of breaking of individual bonds between successive surface groups in the adhesive and metal. Obviously the time required for the rupture of an individual bond is a small fraction of the total time taken to strip the adhesive film from the metal. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39ELECTRICAL PHENOMENA IN ADHESION 441 electrons in the atmosphere which has been established in the dielectric external to the metal. Accordingly, charges remain in the dielectric and opposite charges are measured on the metal which has been pulled away from it. The sign of the charge depends upon the par ticular mechanism responsible, and upon whether the barrier is positive or negative. The break mechanically separates the metal from the dielectric against all forces acting, with, in favorable cases, negligible pre sumed disturbance of charge density.6 Thus, the surface charge observed is directly related to the volume den sity of charge in the dielectric, which in tum is directly related to what amounts to an effective work function between the two substances. The relationships will be considered below assuming the transfer of electrons,7 and the experimental results will be covered subse quently. The existence of the charge in the dielectric of the original sample before break, and the corresponding opposite charge on the metal show that part of the ad hesive force in metal-polymer adhesive pairs is electro static. Such a force has not previously been included in the analysis of adhesion. The possible significance is discussed here. III. THE PHYSICAL PICTURE Since charge was transferred, the various van der Waals mechanisms are not suitable mechanisms for the discussion. A possible interpretation is one involving chemical bonding between the metal and the polymer, such as the conversion of a double bond in a monomer unit into a single bond with the other bond attached to the metal. Under such conditions, a more stable state may result if the electron from the metal moves into the monomer structure, with an eventual increased electron density about the carbon atom which originally was double bonded. However, the picture is too simple since it implies that the energy levels in the solid are equiva lent to those in the free monomer. Looked at quantum mechanically, the electrons en tering the polymer will enter describable states, whose specification must conform to the fact that the polymer as a whole was electrically neutral before contact with . the metal. The essential difference between a metal and an insulator is that the metal has an energy band only partially filled with electrons, so that with small ap plied fields transitions to conducting states are possible, whereas in the insulator, there is a discrete energy gap 6 The measured charge density is a lower limit to the actual c~arg~ density which existed in the dielectric prior to break, SInce If local fields' on the surfaces exceed the values at which corona discharge into the air occurs, a portion of the charge is lost, l!nd since, if the total ar.ea ?f adhesive break is not so great as estimated, the charge denSity IS actually greater. Care is taken to avoid, so far as possible, such high fields. For the analysis corona discharge into air during separation wiII be left out of consideration. This will be taken up in the next paper. 7 Analogous considerations are of course possible, if the sign of the charge, or the direction of transfer is opposite to that chosen here. separating two bands, the higher of which is empty at low temperatures and the next lower filled. At ordinary temperatures this gap cannot be traversed, but at high temperatures, sufficient thermal energy is available to permit electrons to make such transitions, and, for example, salts become conducting. Considerable theoretical and experimental investiga tion has been devoted to establishing the details of the metallic state. Less has been done with nonmetals, except in the case of ionic crystals and semiconductors. Since the adhesives considered here are high polymers, much less is known about their detailed structure from the quantum-mechanical point of view. X-ray results show that some high polymers exhibit a microcrystal line structure below the second-order transition point, and that within the crystallite, there is a certain quasi-periodicity of structure due to the statistical linking of similar monomer units. On the other hand, this structure is far from the regular lattice exhibited by metals or regular crystals. Not only is the molecular weight of the linked monomer units a statis tical distribution determined by the course of the polymerization and the various chain-terminating mechanisms, but also there may exist within the mass debris of the nature of branched and cross-linked chains, and, depending upon the method of polymerization, traces of the polymerizing agent or catalyst, or some times even small concentrations of electrolyte. Insuffi cient information is available with respect to a particular sample of adhesive, no matter how carefully prepared, to permit detailed calculations of the distributions of energy levels and the degree to which they are filled, the variation of atomic charge density throughout the adhesive, or the contributions which may be expected from impurities, surface or interface states, etc. For the present purpose, it will be sufficient to treat in detail a picture representing the polymer as a region in which excess electrons Of sufficient energy can enter conduc tion levels8 and move freely with a uniform potential energy. This approximation corresponds to that made in the Drude-Sommerfeld electron theory of metals. Certain refinements of this concept will then be dis cussed. When the barrier for the passage of electrons between a metal and a contiguous material is of the order of a few tenths of an electron volt or less, an electron at mosphere of considerable magnitude may exist in the latter substance at room temperatures. Such phenomena are not normally observed in air surrounding metals, since the potential barrier for evaporation of electrons into air or vacuum is of the order of several electron volts, and only at high temperatures is the thermal energy sufficient to permit an appreciable fraction of the electrons to exist in the vapor external to the metal. 8 Excited states in the insulator which are nonconducting are not considered, since in ,such states the electron would remain in the vicinity of a particular atom, whereas the experiments show that there is often a very small but measurable charge on the surfaces resulting from break within the interior of the adhesive. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39442 SKINNER, SAVAGE, AND RUTZLER ,-- - I I XI ill I XI I 1 I K I '--'-- X2 11>2 = 2KT I .::L ~XJA T I 09 K I 2 X2 11>2 I A 1 FIG. 3. Barrier poten- ---(iI>l-t)~ I tials on contact between I a dielectric and a metal. ._- --Dielectric I - Dielectric I I I A - Dielectric 2 ----e::-B Metal I Metal 2 Metal I o d o 0 (0) Before equili brium If the contiguous surface is another metal, separation of the two permits observation of a contact potential. Similarly, if a nonmetal is brought into contact with a metal, potentials may be measured; complete investiga tion is more difficult because dielectrics are relatively nonconducting, and surface charges on insulators be have erratically. A momentary contact between an insulator and a metal is very different from an aged adhesive bond ; leaving aside chemical interaction, the former results in two interfaces with a thin film of gas between the substances. For the electron, this consti tutes an additional barrier, not existing in the intimate contact achieved at an adhesive interface; at the latter, the barrier may be very low or even negative. Certain consequences of a very low barrier are very different from those normally expected in ordinary experience. In the air, or in contact with another dielectric, an ad hesive substance, even with considerable "electron affinity," may persist in a relatively neutral state, because of the small availability of electrons. When bonded to a metal, however, it is in contact with a reservoir of electrons, whose diffusion into the adhesive takes place under conditions similar to those involved in thermionic phenomena. The charge which "evaporates" into the adhesive is chiefly retained in the region nearest to the metal, but under certain conditions appreciable charge densities may exist in the body of the adhesive. IV. THE ELECTRON ATMOSPHERE Let X be the work required to remove an electron from the lowest state in the conduction band in the insulator to infinity; the potential energy of the free electrons in the insulator. is then -x, and X may be called the "electron affinity"; similarly let cf> be the work function of the metal. When the insulator is some distance from the metal, the Fermi level of the electrons in it will be different from that in the metal. If the two substances are brought into contact, electrons flow from the higher levels in the one to the lower, unoccupied levels in the other. The exact description depends upon the nature of the barrier; here, a step function will be sufficient [Fig. 3(a)]. The flow of electrons alters the occupied Fermi levels in the two until a minimum over all energy is obtained when the Fermi levels are at the Dielectric 2 I I I Metal B d Q/ 0 (b) Space charge same height. The insulator can carry current (and there fore accumulate charge in its interior) only if electrons are in the conduction levels. The necessary energy may be obtained in several ways, one of which is tempera ture agitation. If the barrier is large, insufficient energy is gained in this way except at high temperatures; with a low barrier, an appreciable fraction will have sufficient energy to enter the conduction levels at roOIl?-tempera ture. This fraction, from the expression for the Fermi Dirac distribution, will be zero at OOK and increase with temperature. As the electrons pass into the dielectric, a space charge is formed, so that the potential seen by subsequent electrons is successively higher. At equi librium, a barrier similar to that in Fig. 3(b) will exist. Consider two metal surfaces separated a distance D in the direction of x, with two distinct dielectrics occupy ing the space between them. It is assum~d that all quantities depend only on the x coordinate. The field approaches zero and the potential energy V increases logarithmically, as x becomes large.l° For a point in 9 The following references will, in the text, be referred to by the names of the authors. R. H. Fowler, Statistical Mechanics (Cam bridge University Press, Cambridge, 1936), second edition; R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics (Mac millan and Company, Ltd., London, 1939); N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, London, 1940), especially Chapter V; F. Seitz, Modern Theory of Solids (McGraw-Hill Book Company, Inc., New York, 1940). Mott and Gurney particularly review the evidence for the existence of low or negative potential barriers at the contact between a metal and an insulator. Seitz gives a fairly detailed account of surface states. 10 The derivation is given in N. F. Mott and R. W. Gurney (reference 9), pp. 170--1; see also R. H. Fowler (reference 9), p. 367. Essentially it is: Letting Ne be the volume density of charge in the dielectric as a function of x, where e is the charge of the electron, v and D the mobility and diffusion constant of the electrons, the con dition that the electron atmosphere in either dielectric has reached equilibrium is that no current is flowing. Therefore NveF-eDdN / dx=O. (For distances so small that the electron has small prob ability of collision, this relation is not strictly true; however, to the order of approximation being used, it may be accepted.) With the Einstein relation vkT= eD, this integrates into 10g(N /No) =e/kT ./O"'Fdx, in which No is the density of electrons in the insu lator at the contact with the metal surface. Upon passing into the interior of the insulator N(x) will decrease. The positive charge per unit area on the metal will correspond to the electrons lost by that area of the metal into the interior of the dielectric. The potential energy of an electron at a distance x from the metal surface is V(x) = -e./O"'Fdx, where F is the field in the insulator; F obeys the space charge relation dF/dx=47rNe/K. Integration of the equations subject to the condition mentioned above yields Eq. (1). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39ELECTRICAL PHENOMENA IN ADHESION 443 either space, integration of the space charge equations gives v = 2kT log(ax+b), (1) where a and b are constants to be determined by the boundary conditions. Figure 3(b) pictures the energy levels; the Fermi levels are not in question in the dielectric. cp -X is the energy gap between the highest occupied levels in the metal and the lower edge of the conducting band in the dielectric. Since the work functions of the two metals may be different, and the depth of the conduction levels in the two dielectrics may also differ, the boundary conditions to be satisfied are (K being the dielectric constant), plus two additional conditions, which become obvious if the simpler case of the usual adhesive bonded speci men is considered. In this special case, there is only one dielectric, the plates are identical, and obviously the field is symmetrical about the center plane, being directed towards the two surfaces. In the present case, there is no pressing reason to expect symmetry, but, for not too great a disparity between the "emissivities" of the metal plates, there certainly is a value of x within the dielectrics for which the field changes direc tion. Calling this value a, and denoting by g the ratio of the absolute values approached by the (oppositely directed) fields from the left and the right of a, the where Region O~x~a a~x~d d~x~D x h=-; '1/ XOI (D-x) ~2=---; X02 Fora~d V -xl+2kT log(1+6) -Xl+2kTlog'7 -x2+2kT log(1+~2) N(x} NOl(l+~I)-'l N01'7-'l N 02(1 + t2)-'l K1(x-d)+K 2(d+xOl) t=-------------- (4.1) NOl and N02 are the densities at x=O, D respectively, and if a~d 2K2a=Kl(D-d+x02)+K2(d-xOl), if a~d 2Kla=Kl(D+d+x02)-K 2(d+xOl), (4.2) so that if Kl=K2 and NOl=No2, the value of a is D12. The condition determining which solution should be conditions are Two cases must ,be considered, namely, those in which a ~ d, and in which a ~ d; that is, when the change in field direction occurs in the first dielectric and when it occurs in the second. Solution is made for V and from it are obtained F and N by the relations 1 dV K dF F= ----; .r =~-. e dx 471"e dx The value of g must still be fixed by a reasonable as sumption. Since the metal surfaces may be regarded as sources of electrons diffusing outwards into the dielec tric, the location of a will depend upon the relative effectiveness of the plates in furnishing electrons, i.e., upon the ratio of the charge densities from the two. At the point at which the two charge densities from each considered separately are equal, a displacement into the dielectric in either direction will cause the contribution of the plate which is approached to be come dominant, and that from the other plate to become less. The point at which the field changes direction is therefore that at which the separate charge densities from each plate are equal. When expressed mathematically in terms of the expressions obtained for the charge densities, it is found that g can only be ± 1. The value -1 is rejected since it is equivalent to the simple case treated by Mott and Gurney. With the value g=+l, the solutions are: Region O~x~d d~x~a a~x~D used is For a~d V -xl+2kT log(l+~I) -X2+ 2kT logr -x2+2kT log(1+6) asd if Kl(D-d+x02)sK 2(d+xOl). (4.3) As shown in the schematic diagram (Fig. 3), the poten tial jump at the boundary between the dielectrics de pends only upon the dielectric constants and the differ ence of the metallic work functions, not upon the values of X within the dielectrics. The quantities xo are characteristic parameters emerging from the analysis. They have the dimensions of length, and may be regarded as scale factors affecting the mapping of the potential or field. A somewhat more intuitive picture may be obtained from the following considerations. Although the solution is not valid in the region beyond the dielectric, on the other hand, Eqs. (4) show that if it were, xo would be the distance into the metal at which the potential energy, field, and [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39444 SKINNER, SAVAGE, AND RUTZLER 3 .-JI"C. eV ~-,(·C. 3eV 12 ~ ---- I r-()olo-'ct -----..., /' - "-10 / ~ ~oao.:. -----\ , ./ \ '( --~ -r- "- ----......... \ 5 / ---'\ 4,/ ----C·1CJ""t> 0. I(T2cm ---\ 3/ V f--- "'" ',,-:,/ 0. -2 -4 ~ -6 ~ " ! -8 -10 0..1 '" l\ 1\"- "'- \ ....... \ "---o 1O-3crn -r-- ---f-.. -D 1Cr4cm 0..2 0..3 0..4 0.5 0..6 0..7 0..8 0..9 Fractional Distance from One Edge of Adhesive Sandwich .!. 0. (a) ~,·C oV .-,.0.. 3eV - - .... _--O·1CT4crn -V -----r--C·lo-'cm V r---- ---D·UT an D.~2an V r------0-10" an O .. 1(j em ------r------------ O"KT" em / ---~-.. -r-- ---- r----- 0·00-' em \ ~ 1.0. ~ VJ. v, I /' / I " / ,- 0..1 0.2 0..3 0..4 0..5 0..6 0..7 0.8 0.9 1.0. I o 0.6 ~ 1 04 & ~ 0.2 ~ F rocfionol Distance from One Edge of Adhesive Sandwich .!. 0. (b) ~ Or-~~-r--,---~-,~~--~ Fractional Oistonc:o from One Edvo 01 Adhlli" Sandwich ..!. o (e) FIG. 4. (a) Potential in a bonded adhesive specimen. (b) Charge distribution in a bonded adhesive specimen. (c) The effect of the thickness of the dielectric layer on the charge distribution within the dielectric. charge density become infinite; thus, the field (and other quantities) in the dielectric behave as if an infinite charge density existed at a distance (in non existent dielectric) of -Xo from the boundary. Within the actually existing dielectric, the Xo may be regarded as indicative of the range to which an appreciable influence of the metal penetrates [see Eq. (4.3)J, recog-nizing of course that there is no sharp cutoff of any sort within anyone dielectric. Obviously, the correct interpretation is as a scale factor; the dependence of this scale factor upon the energy levels in the dielectric and the metal is given by the definition in Eqs. (4) and by Eq. (5). In order to determine the physical magnitude of the electron atmosphere in the dielectric, it is necessary to evaluate the No's. Obviously, these depend upon the nature of the metal and the dielectric in contact. Within the metal, the electron distribution is described by Fermi-Dirac statistics. At the boundary, the condition for equilibrium is that the absolute activities of the electrons in the dielectric and the metal shall be the same. This yields,l1 for barrier heights greater than approximately 0.1 ev at ordinary room temperatures (5) To this approximation, the density corresponds to that portion of the high energy tail of the Fermi-Dirac distribution which exceeds the barrier energy. Table I gives values of No and Xo as a function of q, -x. In certain cases, for better accuracy, the calculations of the above values have utilized interpolation between the values of the Fermi-Dirac integral tabulated by Mac Dougall and Stoner.12 However, for very low barriers (q,-x=O.l or 0.05), Eq. (5) requires modification on statistical grounds. For the usual barriers, metal to air, the charge density is essentially zero, and Xo is astro nomical; (Table I, first line). In usual thermionic measurements, only the satura tion current and not No can be measured; in the non equilibrium separation (break) of an adhesive specimen (with a sufficiently rigid adhesive) the surface charge density (and thereby No) is measurable. TABLE I. Boundary volume charge densities and values of the characteristic distance Xo as a function of the barrier height and the dielectric constant. (T=300°.) Volume Barrier eharge x. (em) for height density q,-x. ev N. (em') K=1 K=4 K-IO 4.1 «10-") ( ... >10 .... ·) 2.0 6 X 10-16 2 X 109 4 X109 7 X 109 1.5 2 X1~ 1.3 X 105 2.7X105 4.3X105 1.0 4 X1()2 8.5XlO° 1.7 X 101 2.7X10' 0.8 9.1X 1()6 1.8XlO-' ·3.5XI0-' 5.6XIO-' 0.6 2.1X109 3.7X 10-3 7.4XI0-3 1.2X1()2 0.4 4.8X1012 7.7XI0-5 1.5X 10-' 2.4X10-< 0.3 2.3X 1014 1.1 X 10-5 2.2X 10-5 3.5X1O-5 0.2 1.1 X 1016 1.6Xl~ 3.2X1~ 5.1X1~ 0.1 4.9X1017 2.4XI0-7 4.8X10-7 7.6XI0-7 0.075 l.4X 10'8 1.5X1O-7 2.9X10-7 4.6X 10-7 0.05 3.5XI0 '8 0.9XI0-7 1.8X 10-7 2.9XI0-7 11 R. H. Fowler and E. A. Guggenheim, reference 9, p. 475. 12 J. MacDougall and E. C. Stoner, Trans. Roy. Soc. (London) A237, 67 (1938). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39ELECTRICAL PHENOMENA IN ADHESION 445 From the general case just derived, there may be ob tained, by specialization, the two cases of interest, namely the equilibrium distribution of electron atmos phere in (1) a metal surface with adhering dielectric layer and another metal surface at a specified distance from the free adhesive side, hereafter called the dielec tric film case; and (2) the bonded specimen. The former is obtained by letting K2= 1, and D-d have any desired value. For the latter, KI=K2, 'and if the two plates are of the same metal, N 01 = N 02. The result of letting D= d in the dielectric film case is not at all the same as the case of the bonded sample. It corre sponds rather to an unbonded nonadhesive contact, with an extremely small air film between the two. The dielectric film solution is also the equilibrium distribu-' tion, after break, and thus can be compared with the nonequilibrium existing immediately after break, cor responding to the bonded sample solution. It also represents the potential occurring at contact between a rigid nonmetallic solid and a metal.I3 The difference between the two cases is illustrated in Fig. 5. For prac tical purposes, the surface charge density on the metal in air is zero in the after break equilibrium picture; in a momentary contact, it would be a function of the time of contact, and the mobility and mean free path of the electrons in the dielectric. In the adhesive bond, on the contrary, there is an equilibrium charge density of a magnitude which is greater for smaller barrier heights. V. THE SURFACE CHARGE I, The Bonded Sample If the same metal constitutes both plates, the sample is symmetrical with respect to its center plane, which, thermodynamically, could be replaced by a nonconduct ing boundary. The charge which has diffused into the adhesive from one plate is all contained in the nearer half of the adhesive.14 The total charge per unit area of the metal surface will therefore be If and if (1= f N(x) ·e·dx=Noe . d/2 ( xod/2 ) 1 o xo+ d/2 d Xo«-, 2 d Xo»-, 2 (1= NofXr' f I (6) I' As, for example, in the measurements of R. M. Fuoss [J. Am. Chem. Soc. 59, 1703 (1937)J, in which the equivalent of a layer of air of minimal thickness was corrected for. l' This is analogous to the classiCal picture of the lines of force from one metal plate ending on charges totally within that half of the adhesive nearest to it. If the specimen is broken in a time short enough so that the equilibrium charge distribution does not change appreciably, this expression will represent the surface density of charges remaining on the metal. Within the dielectric, the charge is distributedl• as shown by N in Eq. (4). The rapid decrease of charge density from the interface into the interior is especially interesting. One of the consistent results which de manded explanation in the early experiments was the fact that a cohesive break (one occurring in the body of the adhesive rather than at the interface) showed little or no oscillograph trace with the voltage sensitivities used.16 Figure 4c shows that for surface charge densities and adhesive thicknesses of the order of those measured, the volume charge density decreases by several orders of magnitude in the first 10-6 cm. Figure 4 is drawn to show the potentials and charge densities for two different barrier values. Since in the bonded adhesive specimen with the same metal for both plates, these quantities are symmetrical about the center plane, one barrier value has been shown on the left and the other on the right. The mirror image curves FIG. 5. (a) Adhesive speci men after break. (b) Electron density distribution (if at equilibrium) in adhesive before and after break; distribution after break is that for negligible separation between adhesive and right-hand metal plate. o -I I 0-2 Zz Q -3 ~ -4 -5 rtf:::'-rll ~ HJ-I : : Direction Id ! of motion x-o 0 (a) - I. Adhesive Bonded SpecImen D'IO-'cm K'4 ()i-;r·O.leV --'2. Broken SpecImen if at equilibrium with ntlgligible sepJrfltion. Metal ~ I I I Di~leC~iC " ~Metal o 01 0.2 03 04 05 0.6 07 08 0.9 1.0 x o (b) 16 The charge density distribution is sensitive to the exact model chosen. Surface states or acceptors and donors in the dielec tric give a different volume or surface charge distribution in the dielectric. Therefore, experimental study of the distribution can contribute toward the choice between mechanisms. Qualitatively, the effects are sufficiently similar for our purpose. 18 Actually, if there is a surface charge density in a cohesive break, a very much larger potential would be expected than for the same charge density in an adhesive break, as will be shown SUbsequently. Accordingly, the experimental results show that some charge, but of a much smaller order of magnitude, exists in a cohesive break. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39446 SKI NNE R, S A V AGE, 1\ N D RUT Z L E R (including the dotted ones) represent what would be the distributions in the other half, for the one barrier. In any case, the accumulation of charge in the bonded adhesive must be a rate process, reaching eventual equilibrium during or after preparation of the sample. F or values of r/> -X of the order of 0.1 ev, the concen tration of the charge at equilibrium, in the layer near the surface in thick films permits consideration of the metal-polymer interface as a sharply defined double layer. However, the curves in Figs. 4 and 5 show that when the thickness of the adhesive film decreases, the charge spreads out through the volume, so that it must be treated as a diffuse double layer or a volume distribution. 2. Mter Break In this case, the charge density at the boundary be tween the adhering film and the metal is N 01, as before, and the dielectric constant is the same. The broken surface, however, is in contact with air. In such case, it is necessary to use the solution which considers two different dielectrics. This solution differs considerably from that just discussed. For example., at the boundary between the adhering dielectric and the air, the charge density changes discontinuously in the inverse ratio of the dielectric constants, and the potential suffers a jump of 2kT log(KI/K 2)+(r/>1-r/>2). The surface charge densities on the metal plates, when the second plate is still at essentially zero distance from the outer edge of the film which adheres to the first plate, are now: on the plate to which film still adheres 0'1= .[d Nedx+ L'" Nedx ( XOld) Kl X02 =NOle ------0'2, d+XOl K2 d+xol (7.1) and on the plate from which the film was stripped 0'2= lim [fa Nedx] (D-d)-+O D These charge distributions warrant some discussion, especially since this is the solution applicable both to the specimen after break (if it were permitted to come to equilibrium), and to the contact between a metal and a surface film on another metal. The plate on the right (Fig. 5) which, though stripped of film is still in virtual contact with the film, has a charge density which de pends upon the sign of the factor in the numerator of Eq. (7.2); thus, if K1x02 is more than K2(d+xOl), this charge is of the same sign as the space charge in the dielectric. Since in practical cases of metal in contact with air, X02 is much larger than this value, it is evident that a very different physical situation exists here than existed in the bonded sandwich. Where before, the second plate was attracted to the dielectric, now it is repelled byitY The bonded joint before break is characterized by a space charge in the dielectric and an opposite charge on the metal. If the joint is broken, immediately the sys tem is characterized by a new equilibrium condition toward which it tends, though it is far from equilibrium. Even while the separated plate is still effectively in contact with the dielectric, such equilibrium would consist of a reversal of charge so that the charge on the plate now would be of the same sign as that in the dielectric; the magnitude is (in practical cases) very 'much smaller than before. Similarly, if a dielectric film is already adhering to a metal plate, and another plate is brought up to the other side of the film, there will be a slight repulsion between the dielectric film and the new plate. Adhesion which is expected in a bonded joint, is not now to be expected, from the electrostatic forces arising from this (low temperature) thermionic mecha nism. At large separations, it may be expected that the surface charge density on the second plate will not be affected by the conditions at the other metal plate and its adhering film. For copper in air, this is of course essentially zero. In practice, other factors will enter. For example, the microscopic roughness and rigidity of solid materials will prevent close contact over a major fraction of the surface, so that the degree of electrostatic binding which is likely to exist will cause the metal to be charged in accordance with Eqs. (6) or (7), only in the vicinity of intimate contact between the two surfaces. VI. THE ELECTROSTATIC CONTRIBUTION TO ADHESION In addition to the London, Debye, and induction forces which have been utilized in the explanation of adhesion, it is evident that an electrostatic force of attraction exists between the volume charged dielectric and the oppositely charged surface. The energy supplied to break the adhesive sandwich rapidly must not only overcome any mechanical, van der Waals, dipole, etc. forces, but also produce separation of the charges. The existence of charges upon separation establishes this. What remains is to determine whether the electrostatic force is a significant contribution to the total force. 1. The Bonded Specimen Although at atomic distances it is not justifiable to consider charges smoothed out into a uniform surface, 17 Charge is transferred from the dielectric to the second plate (the one which has broken away), giving it charge of the same sign as the dielectric. If it were a neutral dielectric which furnished electrons to the metal, the approximation to the Fermi-Dirac integral, Eq. (5) (which is basic to Eq. (7.2)) would not be appro priate. However, the charge received by the second plate comes from the space charge in the dielectric, which originates in the electron sea of the first metal plate, and therefore Eq. (7.2) is valid. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39ELECTRICAL PHENOMENA IN ADHESION 447 the order of magnitude may be estimated in terms of such a picture. The classical electrostatic forces on the metal and dielectric include a tension on the metal directed toward the adhesive of magnitude f= 27ru2/ K, which here, for the bonded film, 27rNOle2[ XOI(D/2)]2 T [ D/2 ]2 (8) = =;\olkT. KI XUI+(D/2) XOI+(D/2) To a sufficient degree of accuracy, this is the electro static contribution to the force of adhesion.1s As is true for the charge density, this force is dependent upon the ratio of the thickness of the film to Xo. When Xo is much less than half the thickness of the adhesive film, the expression (8) becomesl9 f=NokT (8.1) and if the adhesive film half-thickness is much less than Xo, (8.2) Accordingly, a dependence of strength upon thickness will be observed. The magnitude of the electrostatic contribution to adhesion increases rapidly with ad hesive thickness at small thicknesses until D/2 is of the order of Xo, after which it approaches the constant value NokT, independent of how much thicker the adhesive layer is made. The greater the barrier height, the greater will be the critical adhesive film thickness, and for a given thickness, the smaller will be the electro static contribution to the adhesive force. Actually observed thickness-strength relationships show either increase, or decrease, of strength with decreasing thick ness depending upon the adhesive and the metal.2° It should be noted that for small barrier height (which is true in any case for which the electrostatic contribu tion is significant) the critical thickness is of the order of 10-6 or 10-7 cm, which is unlikely to be achieved in practice, so that a constant contribution is more likely to be observed. 18 Within the adhesive the contained charge stresses the dielec tric by the amount (K/81r-)Fl. Thus the increase in adhesion is accompanied, classically, by a decrease in cohesive tensile break ing stress. That portion of the increased stress which is analogous to a uniform (negative) pressure must be added to any external tensile stress applied to the material. For example, if the dielectric is rigid enough to be treated by the Griffith or Sack crack theory, the term Po can be replaced by (Po+KFl/87r) and the condition for spreading of the crack and eventual rupture of the material is then ( gE-y ); po= (l-cr2)c -KFl/87r in which F is the field due to the contained charge, c is one-half the length or radius, respectively, of the crack, Po is the final "pres sure" at which break occurs, u, E, and 'I, are the Poisson ratio, Young's modulus, and surface tension of the dielectric, respectively, and g=2/1I" for the Griffith crack, and 7r/2 for the Sack (penny shaped) crack, respectively. 19 See also Fowler (reference 9), p. 366. '0 G. W. Koehn, presented at the Symposium on Adhesion, Case Institute of Technology (April 24-25, 1952). The variation of the electrostatic component of the adhesion force with temperature is determined by the two quantities No and kT. A decrease of barrier (tf>-x) increases No; temperature rise increases both No and kT; thus, a fractional change in (tf>-x)/kT by change of temperature produces more change in force than does the same fractional change produced by change in tf>-x only. Of course, an increase of temperature pro duces other changes in the polymer which may override this. The temperature variation of f for large thickness is f 0:. T"12e-(<p-x) / k T and for small thickness or large barrier heights is f 0:. T3e-2(q,-x)/kT. Thus, any attempt to measure f with very thin films as a function of tem perature would give a value of tf> -X differing from that with thick films by a factor of two. In practical cases, the microsurfaces of metals and polymers are generally rough and relatively rigid. Contact between them will occur at only a fraction of the total macroscopic contact area. Charges transferred at the points of contact will largely be bound into posi tion electrostatically instead of being distributed evenly over the surface of the metal. Therefore, the charge is concentrated at points or small regions oc cupying only a fraction of the surface. Up to a certain point smoother surfaces should give greater electrostatic adhesion. The techniques used to produce suitable bonding, i.e., insuring wetting of the surface, or poly merizing in place from the monomer, will produce greater contact and therefore greater charge and conse quent greater adhesion. Tacky surfaces and consequent flexibility at the interface will permit closer contact and the acquisition of the necessary charge over a larger fraction of the interface, thus promoting adhesion. Initial rigidity at the interface is prejudicial to and liquidity is favorable to adhesion. The same is true when other mechanisms are considered.21 2. The Sample after Break For the cases dealt with here, namely metals bonded with polymers, the barrier to air renders X02 for the stripped metal so large that repulsion exists between the metal after break and the dielectric, if time is allowed for it to come to equilibrium. The actual break is so rapid that equilibrium is not achieved, and what is found is the charge distribution essentially character istic of the bonded sandwich. Therefore, the same ex pressions as before (Eqs. (4) and (6» may be accepted for the forces and charge densities.22 21 For a discussion of the experimental data and review of such other mechanisms, see the succeeding article in this series. 22 This is not true for the contact between an external metal plate brought up to an adhering film on another metal surface. When such a plate is brought up (in air) to the film, normally K1x02 will be much more than K2(d+xOl), so that there will exist a repulsion of approximately 211"u" =No2kT[1-~ (d+XOl)]. (8.3) K Kl X02 With the values of No. characteristic of even the metals with the lowest values of <Pthermionie, this force is negligible; however, it is evident that electrostatically, adhesion is not favored. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39448 SKINNER, SAVAGE, AND RUTZLER TABLE II. Values of NokT (T=3000K). NokT NokT No dyne cm-' lb in-' lO17 4Xl()3 0.06 (1019) 4XI05 6 (1()31) 4XlO7 600 However, it is of interest to examine what may occur in a very slow break. Assuming that contact is lost at a very small portion of the area by statistical fluctuations or otherwise, the charge in this region decreases, the electrostatic forces likewise decrease, and since X02 be comes very large in this small region, there is a tendency for like charge density to that in the dielectric to ac cumulate and consequently repulsion takes the place of adhesion. The repulsion increases the likelihood, under the existing tension, of neighboring areas being stripped, which then repeat the process. Accordingly, it is to be expected that a slow break will show less tensile strength, so far as the electrostatic contribution to adhesion is concerned, than will a fast break. Under practical conditions this effect may be overshadowed by other factors. VII. THE EFFECT OF INTERMEDIATE ENERGY LEVELS Actually the dielectric is more complex than the simple model of a region of uniform potential energy, -x. As in the Sommerfeld theory, the qualitative be havior is pretty well accounted for by the first approxi mation. However, when the various possible detailed structures are considered, there emerges a change in the volume electron distribution which is sensitive to the exact model. If energy levels are available below the conduction band, the electron may enter such a level and tempor arily be bound in its vicinity. Temperature agitation will eventually furnish sufficient energy, for the electron to enter the conduction band again. However, while it is in any such nonconducting level, the electron is contributing to the space charge between succeeding electrons and the interior of the dielectric. The net effect will be to cause a greater concentration of charge in the vicinity of the interface than is indicated by the previous treatment; this results in a greater field at the interface and consequent greater force. Similarly, avail able surface, or, in this case interface, states will cause an increase of charge density in the neighborhood of the interface. The method of treatment of the first type of energy level is similar to the treatment of donors and acceptors given by Mott and Gurney,9 and 'will not be sketched. For reasonable values of intermediate energy levels, the electrostatic forces at the interface are up to' one order of magnitude greater than for the simple picture, and the charge distribution is intermediate between that of the simple model, and that of interface states. An estimate of the effect of interface states may be made using the treatment of surface states by Bardeen,23 developed for another purpose. Obviously, the effect sought is that corresponding to a considerable density of surface, or rather interface states, since the opposite extreme yields the model just discussed. Accordingly, his approximate solution, V""'lol1 is appropriate. The surface state density, alone, may be estimated from his Eq. (15), page 726,23 yielding measurable values of the adhesive force as shown below. Taylor, Odell, and Fan24 have shown that grain boundaries in germanium are characterized by surface states. While actual densi ties of such states are not known for metal polymer interfaces, it is reasonable to expect that such densities will be considerably higher than at interfaces between grains of approximately the same chemical constitution. VIII. DISCUSSIO N It remains to give some attention to the orders of magnitudes involved, and to various other possible explanations. 1. Orders of Magnitude Table II gives some representative values of the electrostatic contribution to the adhesive breaking tension of bonded specimens for various barrier heights; practical adhesive film thicknesses in these cases will always be greater than Xo. Evidently on the simple model, which was used to discuss the observed charge densities occurring on break, the contribution to the adhesive tensile strength is only a small fraction of the total, except that it may be inferred that if values of No of the order of 1()21 cm-3 could be achieved, the con tribution would be significant. In the current experi ments, surface charges implying No of the order of 1019 cm-3 have been obtained, with no definite experi mental evidence of higher values, though these are not precluded.6 Such a value corresponds to about one elec tron per 1()4 atoms in the dielectric and indicates barriers of 0.1 ev or less. Higher values would corre spond to barrier heights of such sign and magnitude that an amplification of the present picture would be called for, for example, in the statistical mechanics of the equilibrium between the metal and dielectric at the interface. With reasonable values for the magnitudes involved in the surface (interface) states, electrostatic forces at the interface up to 1()3 Ibjin.2 may be computed, which is in the practical range.26 Interface states alone do not 23 J. Bardeen, Phys. Rev. 71, 717 (1947), appendix, pp. 726-7. See also F. Seitz (reference 9) pp. 320-6; J. J. Markham and P. H. Miller, Jr., Phys. Rev. 75, 959 (1949). 24 Taylor, Odell, and Fan, Phys. Rev. 88, 867 (1952). 25 If Bardeen's (EO-'Po-p)/Eo is assumed to have the value 0.1, and n the value lO14, /=21f'u,t/K is of the order of 1()3 Ib/in.2• These are rather favorable values of the parameters; however, the contribution of the volume space charge has been neglected. Ac cordingly, it is reasonable to assume that interface levels or a combination of such levels with volume space charge can, under certain circumstances, result in adhesive forces of the order of loalb for 1 square inch specimens. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39ELECTRICAL PHENOMENA IN ADHESION 449 explain the charge densities observed ~t c?hesiv~ br~, since they do not provide charge densIty m the. mtenor of the dielectric. 2. Other Mechanisms That values of cp -X in the range from 1 ev to 0.1 ev and even negative values exist in insulators is well known. That there exist charge densities in polymers bonded to metals is shown by the present experimental results. Various incidental references to what amounts to such charges may be found in the literature in fields other than adhesion. An alternative explanation of the origin of the charges, is that they result from the rupture of dipoles which have been aligned at the interface due to the forces involved in adhesion. What is referred to here is not the surface electric moment due to the double layer, but the Debye dipoles resulting from the asymmetric distribution of charge within the polymer molecules. They exist throughout the volume of the polymer, are specific for particular chemical groups within the mole cule, and are an accepted explanation of other polymer properties. However, while calculation of interactions between complete dipoles are relatively straightforward, quite arbitrary assumptions are necessary in the transi tion from dipole moments to charges resulting from their rupture. In addition, a thermodynamic treatment of such a mechanism meets the well-known difficulties arising in 'any attempt to combine electrical and me chanical quantities in a suitable thermodynamic cycle. Since charges are involved, it is more appropriate to consider the forces acting on such charges as a result of dipoles, image effects, etc., all of whic~ are summed together in the determination of the value of x. And, whether or not there are dipoles ruptured, the electron atmosphere or its equivalent will be present. Neither partial nor total dipole moments alone show any direct quantitative relation to cohesive forces,26 so that it is somewhat optimistic to expect that they will do so to adhesive tensile strengths. The possibility that the charge densities result from stripping a film of oxide from the surface of the metal has been considered. In such case, however, the charge densities found ~n the polymer faces in a cohesive break are difficult to understand unless a mechanism similar to that already treated is transferring electrons from the oxide into the polymer volume; this would qualitatively be similar to what has been discussed. The primary experimental fact would then be that the adhesion between polymer and oxide is greater than between oxide and metal. Experiments are under way to in vestigate what portion of the effects may be due to such an oxide stripping. Certain other data in the literature bear on the present explanation, and at least raise the presumption 26 N. A. deBruyne and R. Houwink, Adhesion and Adhesives (Elsevier Publishing Company, Inc., Houston, Texas, 1951), p. 16. that the flow of electrons into the polymer occurs in a manner analogous to thermionic emission. For example, Pohl27 has injected electrons into rocksalt from a pointed electrode at 550°C' this cloud is visible and can be made to move back or 'forth by appropriate fields applied through plane parallel electrodes. It is known that color centers in alkali halides are related to electron traps, which may act as donors because of the small energy gap between their levels and the conduction band. Analogous motion of electrons in the interior of a dielectric occurs for semiconductors and crystal counters. Meissner and Merrill28 have reported that the ad hesion of polystyrene to metal is small upon initial contact but increases with time of contact, and that the greater the pressure, the more rapid is the increase. It is well known that pressure sensitive adhesives show a gradual approach to final adhesive strength as a function of time. A portion of this time-dependent phenomenon would be the time required for establish ment of the charge atmosphere in the available levels in the polystyrene. Such time dependence is inherent and natural in the electron atmosphere picture but must be brought in rather artificially in other mecha nisms currently considered. Havenhill, O'Brien, and ,Rankin29 have found electro static contact potentials on various rubber and GR-S compounds after contact with clean metallic surfaces and later rapid (225 cm/sec) separation. The GR-S was negative with respect to the metal. By compound ing it with electropositive materials the contact poten tial was reduced, and greater tensile strength in co hesive break was obtained. Some samples showed greater adhesion than cohesion, adhering to the metal plunger and breaking in the body of the material. As mentioned above, volume charges within the elastomer would produce an internal repulsion resulting in de crease in measured tensile strength for cohesive break. In another set of experiments, they related the charge picked up by a metal ball rolling down a rubber or GR-S sheet to the tensile strength of the material.30 At high temperatures (see Eq. (5) above), the GR-S and rubber became more negative; the authors relate the greater decrease in tensile of GR-S over rubber to the greater "boiling off" of electrons. Milling less electronegative materials into the stock produced little effect unless a latex dispersion was used so that a relatively homo- 27 R. W. Pohl, Physik Z. 35,107 (1934). See also N. Kalabuchow, "Transference of electrons from metals to dielectrics," Z. Physik 92, 143 (1934). . .. 28 H. P. Meissner and E. W. Mernll, Am. Soc. Testmg Matenals Bull. 151, 80 (1948). 2i Havenhill, O'Brien, and Rankin, J. Appl. Phys. 15, 731 (1944) and 17, 338 (1946). 30 This technique may be regarded as similar to the break of an adhesive sandwich, since there is a rapid separation of surfaces temporarily in contact. In the succeeding article, it will be shown that the measured potential in such an experiment is proportional to the charge density which has passed through or is on a square cm of the interface, i.e., tT in Eq. (6). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39450 SKINNER, SAVAGE, AND RUTZLER geneous material resulted, at which time noticeable effect was observed. Their Figs. 8, 10, and 11 are pertinent; in particular, their variation of the "contact potential" as a function of temperature agrees with Eq. (5) if the barrier is taken as 0.088 ev for the standard, and 0.069 ev for the silica ted stock. The agreement is less good if a purely exponential function is assumed. If the electrostatic contribution to the adhesive force is of some magnitude, the recent experiments of Meissnerll are qualitatively explained by the above thermionic mechanism. Meissner measured the proper ties of unsupported thin films of various substances. His striking results included the fact that at and below approximately 0.25 micron thickness, the film of any substance measured adheres on contact with a clean metallic surface. After contact has been made to a metal adherend on one side, no adhesion is obtained upon bringing another clean metal surface up to the other side. Contact with the first metal surface would establish an atmosphere of electrons in the first por tions of the film which touched, thus drawing up the remainder of the film to the metal (if its thickness were not so great as to require forces greater than those provided by the electron atmosphere), and eventually establishing the atmosphere in the whole film, with complete adherence. The si~uation would then be as in Fig. 5; air would exist between the adhering film and any second surface of metal approaching from the other side. Reference to Eq. (8.3), shows that such a second metal surface approaching the space charge filled ad hesive film through the air would encounter a mild electrostatic repulsion, and adhesion would not result. Such phenomena would be true whether or not special techniques had been used to remove initial electrostatic charges from the films. If a barrier of not more than 0.3-0.4 ev is assumed, the critical thickness at which constant and therefore maximum (electrostatic) ad hesive force is approached is 10-' to 10-4 cm; thus, films of the thickness used by Meissner would have achieved their maximum electrostatic contribution to the adhesive force; the limitation of the universal adhesion to a thickness of 0.25 micron or less, of course, would be a matter of achieving a sufficiently thin film 31 H. P. Meissner, reported at the Symposium on Adhesion held at Case Institute of Technology, April 24-25, 1952; published in J. Appl. Phys. 23, 1170 (1952). so that the force required to produce the deformation necessary for intimate contact between film and metal over microscopic irregularities would be no greater than that achievable by the term f=NokT. IX. CONCLUSION If it is accepted that electrostatic charges received by the polymer from the metal by processes analogous to thermionic diffusion or its variants, and the resulting opposite charge on the metal, account for a portion of the adhesive forces, qualitative agreement with experi ments is obtained, and explanation of certain previously unexplained phenomena results. The significance of the electrostatic contribution can only be determined by using the correct mechanism for the states into which the charge (positive or negative) goes in the polymer. One mechanism is treated in detail and others are briefly considered here. Insufficient information is at hand to determine whether the concentrations charac teristic of the various mechanisms can reach magnitudes such that the electrostatic force becomes of an order of magnitude comparable with the total force necessary to produce break. If they can, the difficulty encountered in describing adhesive phenomena analytically using current mechanisms is understandable, since the electro static component depends upon factors such as the dielectric constant and the work function, factors which are not normally considered in fitting theory to experi mental data when using traditional mechanisms. Lack ing direct evidence, it is only possible to say that whatever effect the electrostatic terms contribute is qualitatively in detailed agreement with experiment in a number of different areas. The whole phenomenon is complex. Many things are occurring at the same time. The electrostatic con tribution is of course only a portion of the total forces acting; other forces operate, as previously considered in the literature, and calculations show that forces of the correct order of magnitude may also be obtained from them if suitable values for parameters are chosen.32 Disentanglement of the various mechanisms and achievement of a means of estimating under what conditions each is important or dominant remains a formidable task for the experimenter. 32 As, e.g., in S. J. Czyzak, Am. J. Phys. 20, 440 (1952). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.255.116 On: Wed, 24 Dec 2014 04:07:39

1.1700446.pdf

Rotational Structure of the ν3 Raman Band of Methane B. P. Stoicheff, C. Cumming, G. E. St. John, and H. L. Welsh Citation: The Journal of Chemical Physics 20, 498 (1952); doi: 10.1063/1.1700446 View online: http://dx.doi.org/10.1063/1.1700446 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/20/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extended assignments of the 3ν1+ν3 band of methane J. Chem. Phys. 102, 5126 (1995); 10.1063/1.469238 High resolution study of methane’s 3ν1+ν3 vibrational overtone band J. Chem. Phys. 100, 7916 (1994); 10.1063/1.466837 The Stark effect in methane’s 3ν1+ν3 vibrational overtone band J. Chem. Phys. 99, 1429 (1993); 10.1063/1.465388 Spectral profiles of the ν1 and ν3 bands of pressurized methane AIP Conf. Proc. 216, 363 (1990); 10.1063/1.39884 Measurement and Analysis of the ν3 Band of Methane J. Chem. Phys. 56, 5160 (1972); 10.1063/1.1677002 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21THE JOURNAL OF CHEMICAL PHVSICS VOLUME 20, NUMBER 3 MARCH, 1952 Rotational Structure of the "3 Raman Band of Methane* B. P. STOICHEFF,t c. CUMMING,t G. E. ST. JOHN,§ AND H. L. WELSH McLennan LaboraJory, University of Toronto, Toronto, Canada (Received November 12, 1951) A multiple reflection Raman tube and a high dispersion quartz spectrograph were used to investigate the fine structure of the 1'3 band of gaseous methane. Sixty-eight maxima were measured, and evidence for all but one of the 15 theoretically predicted sub-branches was found. An evaluation of the rotational constants by combination relations for Raman and infrared frequencies gives three different B values for the upper vibrational state and surprisingly, three for the ground state. A more acceptable analysis, based on a single set of rotational constants for the ground state, Bo=5.253 cm-1 and Do= 1.9x 10-4 cm-1, leads to B1 values for the various branches in the range 5.202 to 5.235 cm-1• The analysis gives 1'0= 3018.7 cm-1 and 11 =0.056. The constants obtained for the ground state are also used in an analysis of the 21'3 infrared band. INTRODUCTION EXPERIMENTAL data on the rotational structure of vibrational Raman bands of polyatomic mole cules are extremely meager; only for methanel and ethane2 has a partial resolution of such structures been obtained. Since the intensity of Raman scattering is low, the experimental difficulties associated with high resolution Raman spectroscopy are formidable and are accentuated in fine structure investigations by the necessity of working at low gas densities to prevent excessive line broadening. Nevertheless, it is important to obtain Raman spectra with an instrumental resolu tion approaching that achieved in infrared absorption, since it is certain that many problems, in particular the interaction of rotation and vibration, cannot be studied adequately by infrared investigations alone. In addi tion precise frequency data obtained from high dis persion Raman spectra of gases can be used with good effect in increasing the accuracy of molecular structure calculations. Raman investigations at high resolution have been made feasible to some degree by the development of water-cooled Hg lamps of high intensity and by in creasing the efficiency of Raman tubes for gases by multiple reflections in a system of concave mirrors.3 These methods have been used to obtain considerably greater detail of the V3 band of methane than was ob served in the early investigation of Dickinson, Dillon, and RasettLl This Raman band is one of the easiest to investigate with high resolution since its intrinsic in tensity is exceptionally high. However, it appears that with an extension of the methods used it will be pos- * A summary of some of the results of this investigation has been published as a Letter to the Editor, Phys. Rev. 84,592 (1951). t Holder of a Garnet W. McKee-Lachlan Gilchrist Post-doc torate Scholarship, School of Graduate Studies, University of Toronto, 1950-51. Now in the Division of Physics, National Re~ search Laboratories, Ottawa, Canada. t Holder of a scholarship under the Research Council of On tario, 1950-51. § Holder of a Garnet W. McKee-Lachlan Gilchrist Scholarship, 1950-51. 1 Dickinson, Dillon, and Rasetti, Phys. Rev. 34, 582 (1929). 2 c. M. Lewis and W. V. Houston, Phys. Rev. 44, 903 (1933). 3 Welsh, Cumming, and Stansbury, J. Opt. Soc. Am. 41, 712 (1951). sible to resolve the fine structure of many Raman bands of the lighter molecules. EXPERIMENTAL Since methane is transparent in the ultraviolet, the mercury line 2537 A was used for excitation. A two mirror system for multiple reflections was incorporated in the Raman tube; by this means the intensity of the scattered light entering the spectrograph was increased by a factor of nearly four.3 The tube was illuminated over 36 cm of its length by four helical lamps of fused quartz with water-cooled mercury pools. The lamps were operated at a current of eight amperes; no gain in intensity of the resonance line 2537 A is obtained at higher currents, probably because of self-absorption of the radiation in the lamp. A distressing feature of the high current quartz lamp is the rapid decrease of the ultraviolet transmissivity of the quartz; after 400 hours of operation the intensity of 2537 A radiation is reduced by a factor of at least three. The transmissivity can be restored to some extent by treating the inside of the coil with hydrofluoric acid. The spectrograph used is a two-prism quartz Littrow instrument with an j:22 lens of focal length 155 cm, giving a reciprocal linear dispersion of 27 cm-limm at 2537 A. The spectrograph and source unit are located in a thermostated room, the temperature of which is held constant to within ±0.2°C; in addition, the spec trograph is lagged with a thick wooden box. The source unit is surrounded by a plywood box from which the air is continuously removed by a fan to the outside, in order to dispose of ozone and excessive heat produced by the Hg lamps. All exposures were taken during periods of relatively constant (±2.5 mm Hg) atmos pheric pressure to obtain good definition in the spectro grams. For future work the prisms have been sur rounded by a sealed metal box in which a constant density of air is maintained. In a concurrent investigation on the pressure broaden ing of Raman lines, it was found that the half-widths of the rotational lines of the Va band of methane in crease linearly with pressure. Extrapolation of the results to zero pressure and zero slit width gave a 498 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21RAMAN BAND OF METHANE 499 TABLE 1. Frequencies in the Pa Raman band of methane. Observed frequency, Calculated frequency, cm-l Assignment cm-1 3300.5 S+(12) 3301.5 3288.8 SO(12) 3288.7 3280.9- S+(11) 3281.0 (3269.7)b SO(11) 3269.6 3260.4- S+(10) 3260.4 3250.4- SO(10) 3250.5 (3246.4) S-(10) 3239.8- S+(9) 3239.9 3231.1- SO(9) 3231.1 (3227.4) S-(9) 3219.5- S+(8) 3219.3 3211.38 SO(8) 3211.6 (3207.6) S-(8) 3198.6- S+(7) 3198.6 3192.18 S°(7) 3191.9 3177.9- S+(6) 3177.9 3171.88 SO(6) 3172.0 3157.2" S+(5) 3157.2 3152.28 SO(5) 3152.0 { RO(12) 3147.5 (3147.1) R+(11) 3145.2 {RO(l1) 3138.2 3136.5 S+(4) 3136.2 R+(10) 3135.7 3131.9-{SO(4) 3131.9 R-(11) 3131.2 3126.4- R+(9) 3125.9 3115.6-{R+(8) 3115.8 S+(3) 3115.2 3105.3- R+(7) 3105.6 3090.1- RO(6) 3090.0 3084.8- R+(5) 3084.6 3079.98 RO(5) 3080.0 3073.98 R+(4) 3073.9 (3069.8)8 RO(4) 3070.0 (3066.7) R-(4) 3067.1 (3063.8) R+(3) 3063.1 3059.9- RO(3) 3059.9 3052.1 R+(2) 3052.2 (3047.4) R-(2) 3048.0 3041.5 R+(l) 3041.3 3021.5 Q+(max) Observed frequency, cm-1 2998.5 2988.0 2976.8 2966.3 2958.4 (2954.8)- 2952.5 (2948.2) 2944.6 (2940.8)- (2937.4) 2933.3- (2922.5) (2912.0) 2906.9- 2901.2 2894.7- 2889.5- (2882.6)- (2880.9)- 2878.9- (2859.7)- 2857.18 (2838.3)8 2835.3- (2816.5)- 2813.3- (2795.0)" 2791.4- (2773.7)" 2769.28 Assignment r'(2) PO(2) P+(3) PO(3) {P+(4) PO(4) P-(4) f'(5) 00(3) PO(5) 0-(3) P-(5) P+(6) PO(6) P-(6) P+(7) {00(4) p0(7) 0-(4) P-(7) P+(8) PO(8) f 0-(5) \PO(9) PO(lO) P-(10) {00(6) 0-(6) p°(11) P-(11) PO(12) P-(12) 0°(7) 0-(7) 0°(8) 0-(8) 0°(9) 0-(9)' 00(10) 0-(10) 00(11) 0-(11) 00(12) 0-(12) Calculated frequency, cm-1 2998.7 2997.6 2988.7 2987.0 2978.6 2976.2 2974.7 2968.4 2966.1 2965.5 2965.5 2963.5 2958.2 2954.7 2952.2 2947.9 2944.9 2943.9 2943.9 2940.9 2937.6 2933.0 2922.3 2922.1 2911.2 2906.5 2902.3 2900.6 2900.3 2894.8 2889.4 2883.0 2881.0 2878.9 2859.6 2857.1 2838.2 2835.3 2816.8 2813.1 2795.4 2791.4 2774.0 2769.2 a Frequency used in the analysis. b The bracketted frequencies were measured only from microphotometer traces. value of 2.5 cm-l for the half-width of the lines. In the fine structure spectrograms a gas pressure of 60 psi and a spectral slit width of 1.5 cm-l were used. Under these conditions the half-width of unblended lines was somewhat less than 3 cm-l• For frequency calibration an Fe arc spectrum was photographed in juxtaposition with each Raman spec trum. The Raman shifts of most of the lines were deter-. mined by measuring the spectrograms with a Hilger comparator and plotting large scale dispersion curves with standard Fe lines. Each plate was measured by two observers and the results averaged. Very weak or dif fuse lines were measured from density traces recorded by a Leeds and Northrup microphotometer. Correc tions for small displacements of the Fe spectrum with respect to the Raman spectrum were made by using the known frequency of the Hg line, 2759.712A, which falls within the Raman band. Eastman 103a-0 Spec troscopic plates were used. Methane gas taken directly from commercial cylin ders fluoresced under irradiation, producing a dense continuum which obscured the Va band. The fluores cence was eliminated by passing the gas through a glass-wool filter, condensing it at liquid air temperature, and allowing it to evaporate slowly to give the required pressure in the Raman tube. RESULTS The final values of the Raman shifts given in the first column of Table I were obtained by averaging the results from four spectrograms with exposure times of 14,20,47, and 48 hr. Sixty-eight maxima were measured in the Raman pattern as compared with 14 measured by Dickinson, Dillon, and Rasetti.1 A microphotometer trace of the 14-hr spectrogram is shown in Fig. 1; beneath the spectrum is plotted the structure of the band as calculated from the results of the analysis given below. . This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21500 STOICHEFF, CUMMING, ST. JOHN, AND WELSH 27,50 0-~ II I~ 00 j-~ o. 1 I~ ~ I II 1\ I II II~ , I I I , • ib -t---lIJ---JI--t-'-,-. -t-I ----,. If-p. : __ 1-_1-1 _51-1 _4+-1 _!Jt---,Zrl pO :Q+ ~1 __ ~,_6~1--,5t~~! __ 71_2*lP+ ~ J I I o-~ --~l----!rl----'zrl---OO ' __ ~ ~*---_--.l3!,..--_--.l! __ O. V. 3ZPO I I I [ R+ I IJ. 12, 13~ 14 RO "I i~ II~ R- 12 14 • • • , ~I 5 A I 6 ~ , 7 ! I ~ , " , 28,50 em-' ~ 2 3250 , em-' 1& ~ 'l l ! 1\ 1\ I il I I " J " J I~ ,\ II I' '4 " ~ 33po • I " I~ ~ I~ I~ 2950 I~ L I I , I ~'\ljl'J I I II I, !! ! , I 4 3350 1 II~ I S· 14 50 I 5· 14 FIG. 1. Microphotorneter trace of the Va Raman band of methane. The structure of the band calculated from the results of the analysis is plotted under the trace. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21RAMAN BAND OF METHANE 501 A Raman line at dll= 3071.5 cm-I was measured by Dickinson, Dillon, and Rasetti, and also by MacWood and Urey,4 who interpreted it as the Al part of 2112, the upper state of this transition being in Fermi resonance with the upper state of III. In the high dis persion spectrograms there is no indication of any line extraneous to the rotational structure of 113 in this re gion. It is probable that the line measured by the previ ous investigators at lower dispersion is a group of un resolved lines near the maximum of the R branch. Nevertheless, it is certain that a line observed in liquid methane5,6 at dll= 3053 cm-I must be interpreted as 2112; in this case the rotational lines which are a result of anisotropic scattering are greatly broadened by the. intermolecular forces, whereas 2112 caused by isotropic scattering remains sharp. The sharp intense line corresponding to III falls in the region of the 113 band. Since this line was overexposed in the fine structure spectrograms, short exposures of the order of 15 min were made to permit an accurate measurement of the wave-number shift. The value obtained was 2916.5 cm-I; the values found by Dickin son, Dillon, and Rasetti,t and MacWood .and Urey4 were 2914.8 cm-I and 2914.2 cm-I, respectively. ANALYSIS OF THE Va RAMAN AND INFRARED BANDS The 113 band arises from the fundamental transition of one of the 2 triply degenerate (F 2) modes of vibra tion of the molecule. It was pointed out by Teller7 that Coriolis interaction between rotation and vibra tion causes a splitting of the rotational levels of the v= 1 vibrational state. The energy terms of the three rotational sublevels are given by FI+(J) = BIJ(J+ 1)+2B ltl(J+ 1)-DJ2(J+ 1)2 Fl°(J)=BIJ(J+1)-DJ2(J+1)2 (I) Fe(J) =BIJ(J+ 1)-2BltlJ -DJ2(J+ 1)2. In these expressions BI is the rotational constant and tl the vibrational angular momentum for the v= 1 state; the additional term DJ2(J+ 1)2 represents the energy caused by centrifugal distortion. The theory has been further elaborated by Shaffer, Nielsen, and Thomas,8 who have considered the Coriolis interaction with other vibrational modes, an effect first treated by Jahn,9 and also second-order interactions including centrifugal distortion and anharmonicity. The calcula tions of Shaffer, Nielsen, and Thomas show that for a simple valence force field, which is a good approxima tion for methane, the value of D is given by D= B03[(16/31112)+ (5/31122)], where III and 112 are the frequencies of the totally sym metric and doubly degenerate modes of vibration, 4 G. E. MacWood and H. C. Urey, J. Chern. Phys. 4, 402 (1936). 5 Rank, Shull, and Axford, J. Chern. Phys. 18, 116 (1950). 6 Crawford, Welsh, and Harrold, Can. J. Phys. March, 1952. 7 E. Teller, Hand-und Jahrbuch d. Chern. Physik 9, II (1934). 8 Shaffer, Nielsen, and Thomas, Phys. Rev. 56, 895 (1939). 9 H. A. Jahn, Proc. Roy. Soc. London 168, 469 and 495 (1938). respectively. Assuming Bo=5.2 cm-I, the theoretical value of D is 1.9X1O-4 cm-I• In infrared absorption the ground state combines with the V= 1 state to give 3 branches R-(J), Q0(J) , and P+(J) as observed experimentally by Cooley,1O and Nielsen and Nielsen.!1 According to the analysis of Cooley'S results by Childs,t2 the formulas (I) do not sat. isfy exactly the experimental data. In the Raman effect all transitions in accordance w\th the selection rules dJ=O, ±1, ±2, J'+J"~2 are allowed between the ground state and the Conolis sublevels of the upper state. Thus, the Raman band consists of 15 sub-branches. The complexity of the band is effectively lessened, however, by the differen relative intensities of the various sub-branches. As stated by Teller,1 the relative intensities for high J values are S+:so:S-=o-:oo:o+= 15:5: 1, R+:Ro:R-=P--:po:p+= 10: 8:3, (II) Q+:Qo:Q-=6:9:6. The highest intensities in the outer regions of the band are therefore in the S+ and 0-branches; this is mani festly the structure observed by Dickinson, Dillon, and Rasetti. In the present investigation evidence for all but one (0+) of the 15 sub-branches was found. Because of the complex structure of the band many of the maxima measured are blends of two or more lines. In the analy sis, therefore, only those lines were used which had sufficient intensity for accurate frequency measure ments, and which were shown by a preliminary analysis to be unblended. These lines are marked in Table I. Assuming rotational energy terms for the ground state of the form, Fo(J) =BoJ(J+ 1) -DJ2(J+ 1)2, and the formulas (I) for the v= 1 state, the molecular constants which can be calculated from the Raman and infrared data are Bo, BI, D, tl, and the frequency of the band origin 110. The infrared frequencies of Nielsen and Nielsen,!l corrected to cm-I in vacuum, were used. DETERMINATION OF VO, Bo-Bl, AND Bltl The values of 110, 1I0+3Bltl, and Bo-BI can be obtained directly from the following sum relations: (t){R-(J)+P+(J+ 1)} = 110-(Bo-BI)(J+1)2, (A) (!){S°(J)+O°(J+2) ) =1I0-(Bo-B1)(J2+3J+3), (B) m{S+(J)+0-(J+2)} =1I0+3B ltl-(Bo-BI)(P+3J+3). (C) It should be noted that only infrared frequencies are used in relation (A) and only Raman frequencies in 10 J. P. Cooley, Astrophys. J. 62, 73 (1925). 11 A. H. Nielsen and H. H. Nielsen, Phys. Rev. 48, 864 (1935). I. W. H. J. Childs, Proc. Roy. Soc. (London) 153, 555 (1926). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21502 STOICHEFF, CUMMING, ST. JOHN, AND WELSH '020r------,-----,---,-----,-----. 's lOI6t----j-~~__t-~:__+----+-----j <.) lOI4f-----t-----t-."< f (J) FIG. 2. Determination of po, po+3B lsl and Bo-BI from the sum relation's (A), (B), and (C). (B) and (C) and that the centrifugal distortion terms on the right-hand side cancel out, provided D has the same value in both vibrational states. When the left hand side of (A) or (B) is plotted against the appro priate function, f(J), a straight line is obtained with the intercept on the ordinate axis equal to 110 (Fig. 2). A calculation by least squares gives 110=3018.1 cm-l from (A) (ignoring the first three points) and 110= 3018.7 from (B). The small systematic difference which thus appears to exist between the Raman and infrared fre quencies is eliminated in the rest of the calculations by raising the infrared frequenciet> by 0.6 em-I. The inter cept for (C) is 1I0+3Bltl=3019.6 em-I. A departure from theory is revealed by the fact that the slopes of the three straight lines are not the same. From (A) and (C) in which lines involving FI+ and FI- states are used, the values of Bo-BI are nearly equal, but differ markedly from the value obtained from (B) in which lines involving Flo states are used. It seems advantageous, therefore, at this stage of the analysis to designate by superscripts on the B's the type of states involved in the derivation. Thus, the convergence factors are Bo+--BI+-=0.03S, from infrared data, Bo+--BI+-=0.036,} Boo-Blo=0.050, from Raman data. The value of BIt 1 can be found in two ways: (a) The difference of the intercepts for relations (B) and (C) gives directly 3B1tl=0.9 em-I. (b) The combination relation gives as the average of six values Bltl=0.29 or 3B1tl = 0.87 em-I. The two values are in fair agreement; however, the second is considered the more accurate and is used in the subsequent calculations. DETERMINATION OF THE ROTATIONAL CONSTANTS BY COMBINATION DIFFERENCES The rotational constants, Band D, for the lower and upper vibrational states are usually evaluated from appropriate difference relations. Since the infrared QO branch has not yet been resolved, Raman data alone must be used in calculating Boo and Blo; the So and 00 lines are most suitable for the purpose. The constants for P+ and P-states can be determined from combina tion differences of Raman and infrared frequencies. This procedure should be most satisfactory when s+ and 0-lines are used since these are the most accurately measurable Raman frequencies and yield the largest . combination differences with the infrared frequencies. The difference relations derived from Eqs. (I) are {1/(8J+4)}{S°(J -2)-0°(J+2)} . =Boo-DoO(2J2+2J+8), (E) {1/(8J+4) }{S°(J)-O°(J)} = BIO-DIO(2J2+2J+8), (F) {1/(6J+ 12)} {S+(J)- P+(J+3)} = Bo+-Do+(2J2+8J+ 12), (G) {1/(6J+ 12) }{R-(J)-0-(J+3)} =Bo--Do-(2J2+8J+12), (H) {1/ (6J + 6)} {S+(J) -P+(.f) -6Blt I} = BI+-DI+(2J2+4J+6), (I) {1/6J} {R-(J) -0-(J)+6B ltl} = BI--DI-(2J2+4J). (J) When the left-hand side of each relation is plotted as ordinate with f(J) as abscissa, a straight line is ob tained whose intercept and slope are Band D, re- 5~,r---,----r---.--- cm~1 ~~ ~~14 N'" 3 'in 5Z2f----+----t-'~--+---1 S.l6t---+----t----f""'--1 5.140k-----t.60.------;;,6b;-O--<1;;t40~-" f (J) FIG. 3. Determination of Boo, Doo, Blo, and Dlo from the difference relations (E) and (F). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21RAMAN BAND OF METHANE 503 spectively. In Fig. 3 the graphs for (E) and (F) are shown. The constants obtained from the six relations by a least-square analysis are Boo=5.253, Bo+=5.232, Bo-=5.233, B1o=5.200, B1+=5.202, B1-=5.197, Doo= 1.9X 10"-4, Do+= 1.5 X 10"-4, Do-= 1.0X1O--\ D1o= 1.6X 10-\ D1+= 1.0X 10-\ D1-=0.8XI0-4• The surprising feature of these results is that Boo is very different from Bo+ and Bo-although the three values for Bl are nearly equal. Also, Doo is 1.9X 10"-4, equal to the theoretical value, while all the other D values are lower. There is, however, a certain con sistency in the analysis since Boo-BID is 0.053 in reasonable agreement with the value 0.050 from (B), and Bo+ --B1+ -is 0.033 as compared with the values 0.035 and 0.036 from (A) and (C). The constants also reproduce accurately the frequencies of the lines from which they were derived. It might be expected that these values of the rotational constants would repro duce also the frequencies of the lines not used in the analysis. However, the calculated frequencies of the RO, po, S-, R+, and P-lines differ from the observed frequencies by as much as 2 em-I, that is by an amount greater than the experimental error. There is in addition a fundamental diffic.ulty which arises from the results of this analysis; each rotational level of the ground vibrational state, as well as of the upper state, appears to be split and the splitting is proportional to J(J+1). This type of perturbation might well ocCur in the upper state through a Coriolis interaction with the E part of the nearby 'V = 2 state of 112. However, according to Jahn's rule,1a the ground state (A 1) cannot be perturbed by a Coriolis interaction with any of the lower vibrational states of the mole cule. A splitting of the rotational levels in consequence of centrifugal distortion14 also seems unlikely since there is no reason why the sublevels should be desig nated as Fo+, Foo, and Fo-, as they are according to this analysis. It must therefore be concluded that the' straightforward analysis based on. combination rela tionships has yielded a result which is not understand able from the theoretical viewpoint. DETERMINATION OF Bl ASSUMING AN UNPERTURBED GROUND STATE The basic assumption in the calculation of Bo by a difference relation is that the upper rotational states are the same for the two branches involved. If the splitting of the rotational levels of the ground state as manifested by the aforementioned different values of Bo is considered an unacceptable result, it must be assumed that the sublevels of the upper state, F1+(1), F1°(1) , and F1(1), can have different values for the 13 Ii. A. Jahn, Phys. Rev. 56, 680 (1939). 14 E. B. Wilson, J. Chern. Phys. 3, 276 (1935). I , 52 o~ ~ . 0 ~ 5.18 7 5 I a 50 ~ 0 ~ 100 J(J+I) I • S" Lines 00" .. "'" ~ 150 200 FIG. 4. Determination of B1° and D1° from S· and 0° Raman lines, assuming 8°=5.253. various branches. In the analysis which follows, the ground state is assumed to be unperturbed and the constants, Bo and Do, are taken to be equal to BoO and DoD as calculated above from relation (E). It will be shown that these values lead to a consistent analysis of the Raman and infrared Va bands and also of the infrared 2)1a band. The rotational energy terms of the upper vibrational state can be calculated from the frequencies of the observed transitions and the known values of Bo, Do, and Vo. The formula for the s+ lines is quoted as an example, F1+(1)=s+(1 -2)-vo+ Bo(J -2)(1 -1) -Do(J -2)2(1 -1 )2. (III) Rotational constants for each branch are then obtained graphically from relations, derived from Eqs. (I), of the type Fl+(J) -2B1fl(J + 1) ------=B 1+-D1+J(1+1). (IV) J(J +1) The points for both the SO and the 0° branches (Fig. 4) lie on a reasonably well-defined straight line from which the values, B1o= 5.203 and D1o= 1.8X 10-4 are obtained. The convergence factor, Bo-B1o= 0.050, agrees with that calculated from relation (B). Although comparatively few RO and po frequencies are available, the points for the two branches lie on the same straight line, indicating common upper states and giving B10 = 5.217 and D1o= 1.8X 10-4. Thus, the upper states for the RO and po lines are not the same as those for the So and 0° lines. The points for the S+ and 0-lines lie on two very different straight lines (Fig. 5) giving B1+=5.206, D1+ =1.0X10"-4, and B1-=5.235, Dl=3.8X1o--4.15 The average value of the convergence factor, Bo"-Bl+-, is 0.034, in satisfactory agreement with the value 0.036 1. Although the DJ values for the S+ and 0-branches are not equal to Do, relation (C) is still valid since a computation shows that the centrifugal stretching terms effectively cancel. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21504 STOICHEFF, CUMMING, ST. JOHN, AND WELSH ~ 5.Z em 3\ \ 2 \ 5.2 / ~ ~:; 5.20 Li'::' ~:3 . ..., "") 7... 5.19 I .. • O-Lines oS· " !\ ~\ ~ ~ 1"1>-. " 518 o 50 /00 J(J./) .150 200 FIG. 5. Determination of B, and D, from S+ and 0- Raman lines, assuming Bo=5.253. from relation (C). For the weak S-and 0+ series the experimental data are insufficient to carry out an analysis. The points for both infrared branches, R-and P+, fall on the same straight line (Fig. 6) with B1+ =B1-=5.217, D1+=D 1-= 1.8X1G-4, and Bo-Bl+ = 0.036 in good agreement with the value 0.035 from relation (A). Only five lines of the R-and P+ series could be measured in the Raman spectrum since these are the weakest of the Rand P sub-branches. The aver age difference between the Raman frequencies and the corresponding infrared frequencies is 0.4 em-I. Thus, it is probable that the upper states for the R-and P+ lines are the same in Raman effect and in infrared ab sorption. For the R+ and P-series the points on the graph (Fig. 7) are more scattered, but there is no doubt that the constants for the two branches are the same, 3, ~2 em 1 ~ I " ." C '" '? 52 0-· 0- 1.n-:;' d5 ,'] N.., , 51 :J ;.: 51 51 9 B 7 • R' Infrared Lines o p' ~ ~ ~ ~ a 50 100 150 '00 250 J (J.,) FIG. 6. Determination of B, and D, from R-and P+ infrared Jines, assuming Bo=5.253. B1+=Bl-=5.234 and Dl+=D 1-=2.9XI0-4• The re sults of the analysis are summarized in Table II . When the constants obtained from the R-and P+ branches are used to calculate the infrared frequencies, the experimental values are reproduced to within 0.1 em-Ion the average. Since the analysis shows that the same rotational constants satisfy both R-and P+ branches, it should be possible to calculate Bo and Bl by combination differences of the infrared frequencies. A least-squares analysis of the data with the relation, {1/(4J+6)} {R-(J)-P+(J+2)} = Bo-BlsI-Do(2J2+6J+6) (K) gives Bo=5.249, Do= 1.7XIo-4, and B1=5.216, DI = 2.0X 10-\ which show a satisfactory agreement with the aforementioned values.I6 A further test of the con sistency of the analysis is afforded by the evaluation of Bltl from infrared data. It can be easily shown from Eqs. (I) that BlSl={Fl+(J)-FdJ)}/2(2J+l). (V) The average of ten values is BIs1=0.290±0.003 which confirms the value obtained from (D). When the constants of Table II are used to calculate the Raman frequencies, the agreement with the ex perimental values is very good as shown in Table I. The average discrepancy between observed and calcu lated frequencies for the 42 lines used in the analysis is only 0.14 em-I. The diagrammatic presentation of the spectrum under the microphotometer trace in Fig. 1 was prepared from the calculated frequencies. Relative intensities are indicated in the diagram by the width of the lines. The total intensity of a given branch line (e.g., S(6») was calculated from the formulas of Placzek and Teller17 with statistical weights due to n.uclear spin as given by Wilson.Is However, the intensity dis tribution in the components of a given branch line [e.g., S+(6), SO(6), and S-(6)] is only approximate; the matrix elements for the sublevels are not available, and it was necessary to use the relative intensities given in Eqs. (II) which are correct only for high J values. For low J values there are apparent intensity anomalies; thus, for P(2) it is easy to show from the symmetry species of the states involved that P-(2) has zero in tensity, and from the experimental results it appears that P+(2) has the greatest intensity. For the sake of completeness the weak branches, S-and 0+, have been included in the diagram although the constants for these lines could not be evaluated; average values, Bl=5.220 and DI=1.9X1G-4, were assumed. It therefore appears that a single value of Bo and the array of values of Bl, given in Table II, account for the structure of the Raman and infrared spectra. The per- 16 The lack of agreement with the theoretical formulas, found by Childs (see reference 12) for the infrared va band, appears to be the result of the use of incorrect combination relations and neglect cl~D~~ . 11 G. Placzek and E. Teller, Z. Physik 81, 209 (1933). 18 E. B. Wilson, J. Chern. Phys. 3, 276 (1935). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21RAMAN BAND OF METHANE 505 turbation apparent in the spectrum has been ascribed only to the excited vibrationai state; in this respect this analysis is more satisfactory from the theoretical viewpoint than the analysis based only on combination relationships. However, the multiplicity of Bl values is not immediately understandable. The most obvious perturbation of the rotational levels of the v= 1 state of 113 is that resulting from a Coriolis interaction with the nearby and slightly higher v= 2 state of 112. From the calculations of Jahn9 on a similar but much stronger perturbation of 114 by 112, it appears that the Fl+, Flo, and F 1-sublevels are split and also depressed from the unperturbed values. If the perturbation is small, as seems to be the case for lIa, the splitting of the sublevels would lead only to a broadening of the spectral lines, and the average depression of the sublevels would vary as J(J+l). The depression would not be the same for the three sublevels and would give in effect different values for Bl+, Blo, and Bl-. These Bl values could also vary slightly for the different changes in J, /1J = 0, ± 1, ±2, since the effective center of a given sublevel would not necessarily be the same for different transitions to that sublevel. This explanation, however, does not TABLE II. Rotational constants of the upper state of the V3 band of CH" assuming Bo=5.253 cm-I and Do= 1.9X 10-4 em-I. Branch 0-(R.E.) R+, P-(R.E.) RO, po (R.E.) R-, p+ (LR. and R.E.) 5+ (R.E.) So, 00 (R.E.) 5.235 5.234 5.217 5.217 5.206 5.203 3.8 2.9 1.8 1.8 1.0 1.8 adequately account for the distribution of Bl values in Table II. It is probable that second-order perturbations as a result of centrifugal distortion and anharmonicities, such as those considered by Shaffer, Nielsen, and Thomas, are of the same order of magnitude as the perturbation caused by Coriolis interaction with 2112. An examination of the problem from this point of view might explain the different values of Bl and D1• The average internuclear distance for the ground vibrational state ro, as calculated from the Raman value Bo=5.253, is 1.0927XIO- s cm. With the value Bo=5.249 obtained from infrared data by relation (K), ro is 1.0930 X 10-s. These values of ro are probably of the same order of accuracy. THE STRUCTURE OF THE Q BRANCH Although the individual lines of the Q branch were not resolved, the contour at high dispersion is striking; there is a sharp maximum of intensity at /111=3021.5 cm-I, a broad maximum at about /111=3017 cm-I, and then a gradual decrease in intensity towards smaller frequency shifts (Fig. 1 and on a larger scale Fig. 8). According to Eqs. (I) the structures of the three Q 5.2 em ., 3'" ~ J ° R+ Lines ° p-.. 2 0""'0 I 0 5.1 9 51 8 o 50 0,\ 0 o~ ~ 100 J(J+I) . ~ 150 zoo FIG. 7. Determination of BI and DI from R+ and P Raman lines, assuming Bo=5.253. sub-branches are given by Q+: 11= 110-(Bo-,B1)J(J+ 1)+2Bltl(J+ 1), QO: 11= 110-(Bo-B1)J(J + 1), Q-: 11= 110-(Bo-B1)J(J+ 1)-2Blt IJ. Since Bo-Bl and Bltl are positive, the QO and Q branches are degraded to lower frequency shifts, and the extent of Q-is greater than that of QO• For Q+, on the other hand, 2Bltl(J+l) is greater than (Bo-Bl) XJ(J+l) at small J values and less at large J values; hence this branch forms a band head, which corresponds to the observed maximum at /111=3021.5 cm-l. At low dispersion this head is the most prominent feature of the Q branch and has been measured by the earlier workers as "3' Actually, the band origin, as found from Spectral Slit Width FIG. 8. Microphotometer trace of the Q branch of the "3 Raman band. The structure of the branch calculated from the results of the analysis is plotted under the trace. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21506 STOICHEFF, CUMMING, ST. JOHN, AND WELSH relation (B), is at Av=3018.7 cm-t, close to the mini mum. Assuming the values Boo-B1o=0.050 and Bo+ -B1+-=0.035, the frequencies of the Q branch lines were' calculated and are plotted, with approximate intensities, under the microphotometer trace in Fig. 8. It is evident that all features of the contour are re produced. ANALYSIS OF THE 2"3 INFRARED BAND The values of the rotational constants Bo and Do obtained from the Va Raman band can be used in an analysis of the 2va infrared band, for which experi mental data are available in recent papers by Nelson, Plyler, and Benedict,19 and by McMath, Mohler, and Goldberg.20 In the following calculations the frequencies quoted by the former authors have been used. . The branches of the 2va infrared band .can be desig nated as R+(1), QO(J), and.P-(1), the upper rotational levels of which are given by equations similar to (I). Th!'! band origin Vo and the convergence factor Bo-B2 can be found from the relation (t){R+(J -2)+P-(1+1)} = vo-2B07' (Bo- B2+-)J2, (M) and also, since the Q branch lines were resolved, from the formula Q0(1) = vo-(Bo-B2°)J(J+ 1). (N) A graphical solution gives the values vo-2Bo= 5994.22 cm-I, Bo-B2+-=0.062 from (M), and vo=6004.71, Bo-B2°70.067 from (N). Assuming Bo=5.253, the 19 Nelson, Plyler, and Benedict, J. Research Nat!. Bur. Stand ards 41,651 (1948). 20 McMath, Mohler, and Goldberg, Astrophys. J.I09, 17 (1949). value of Vo from (M) is 6004.73 in good agreement with the value from (N). The difference between the values of Bo-B2° and Bo-B2+-shows that for the 2va band, there is a de parture from the simple theory similar to, but not as great as, that found in the Va band. It seems therefore appropriate to carry out an analysis of the same type as that used for Va. Assuming Bo= 5.253 and Do= 1.9 X 10-4, the upper rotational energy terms F 2+(1), F2°(1), and F2-(1) are calculated from the frequencies of the R+, QO, and P-lines, respectively, by expressions analogous to (III) above. If the B2 values for the R+ and P-branches are equal, as was found for the Va band, the value of B2s 2 can be calculated readily from an equation corresponding to (V). The average of nine values gives B2s2=0.168±0.002 cm-I. The rotational constants for the upper state, calculated by equations corresponding to (IV), are B2°=5.187, D2°=2.1XlO':"'4, and B2+-= 5.193, D2+-= 2.3X 10-4• Consistency in the analysis is shown by the agreement of the values of Bo-B2 with those obtained from relations (M) and (N). The derived constants reproduce the observed frequencies up to J = 9 with an average discrepancy of less than 0.1 cm-I• The value of S2 as calculated from B2+-S2 is 0.0324. Th~se constants for the 2va band differ slightly, but significantly, from those given by Nelson, Plyler, and Benedict. The fact that two values of B2 are found for 2va from infrared data alone shows that the rotational levels of the v = 2 vibrational state are probably per turbed in a way similar to that found for Va. However, since no Raman data· exist for 2va, the nature of the perturbation cannot be studied in greater detail. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.226.54 On: Mon, 08 Dec 2014 18:41:21

1.1700723.pdf

Viscosities of Some Lower Aliphatic Alcohols at Constant Volume A. Jobling and A. S. C. Lawrence Citation: The Journal of Chemical Physics 20, 1296 (1952); doi: 10.1063/1.1700723 View online: http://dx.doi.org/10.1063/1.1700723 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/20/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Photoionization of the Lower Aliphatic Alcohols with Mass Analysis J. Chem. Phys. 48, 5205 (1968); 10.1063/1.1668196 Further Applications of the Domain Theory of Liquid Water: I. Surface Tension of Light and Heavy Water; II. Dielectric Constant of Lower Aliphatic Alcohols J. Chem. Phys. 47, 2231 (1967); 10.1063/1.1703297 Dielectric Saturation in Aliphatic Alcohols J. Chem. Phys. 36, 2144 (1962); 10.1063/1.1732842 Study of the Mass Spectra of the Lower Aliphatic Alcohols J. Chem. Phys. 27, 613 (1957); 10.1063/1.1743799 Complex Dielectric Constants of Some Aliphatic Alcohols at 9733 mc/sec J. Chem. Phys. 21, 1898 (1953); 10.1063/1.1698700 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Tue, 23 Dec 2014 00:21:25THE JOURNAL OF CHEMICAL PHYSICS VOLUME 20, NUMBER 8 AUGUST, 1952 Viscosities of Some Lower AI~phatic Alcohols at Constant Volume A. JOBLING,* Department of Inorganic and Physical Chemistry, Imperial College of Science and Technology, London, England AND A. S. C. LAWRENCE, Chemistry Department, The University, Sheffield, England (Received February 29, 1952) From an analysis of all the published experimental data for methyl, ethyl, n-propyl, isobutyl, and n octadecyl alcohols, it is shown that, even for these compounds, at constant volume log'l is a rectilinear function of liT and the general pattern of behavior is similar to that described previously. S .. the energy of activation of viscous flow at constant volume, is plotted as a function of volume, and the curve for ethyl alcohol shows an apparently anomalous minimum. IT has been demonstrated recentlyl both by direct experiment and by analysis of published viscosity and compressibility data, that at constant volume there is a rectilinear relation between log?] and liT for a number of simple organic liquids. It has been shown that this is to be expected from a study of the fuller viscosity theories of Andrade2,3 and Eyring! and that the slope of the 10g?]-1/T isochore is a measure of Sv, the energy of activation of viscous flow at constant volume, which is a function of volume. 1'0 t c-0·0 R en ~ 1'0 i·o 60 i i i i i i , -12d'C ~ -Ioo·e , i i i 4 _80°C \ i h -60·e \ \ . iSo--40C \ 70 V Ctc) -\~ -20°c \ \ \ FIG. 1. LoglO'I ('I in poises) as a function of molecular volume, V at constant temperature (n-propyl alcohol). e, Bridgman, 24°C [Po W. Bridgman, Proc. Am. Acad. Arts Sci. 77, 115 (1949)J; 0, Bridgman, 30°C (reference 5); X, Bridgman, 75°C (reference 5); 6, atmospheric pressure, various observers. * Senior Student, Royal Commission for the Exhibition of 1851. 1 A. Jobling and A. S. C. Lawrence, Proc. Roy. Soc. (London) A206, 257 (1951). 2 E. N. daC. Andrade, Phil. Mag. 17, 497 (1934). 3 E. N. daC. Andrade, Phil. Mag. 17,698 (1934). 4 Eyring, Glasstone, and Laidler, Theory oj Rate Processes (McGraw-Hill Book Company, Inc., New York, 1941), Chapter IX. Strong evidence for the correctness of the general scheme of viscous behavior outlined in the previous paper is the fact that the behavior of those lower aliphatic alcohols for which sufficient experimental evidence is available is found to conform to a similar pattern. The log'Y/-1IT isochores are again rectilinear but activation energies calculated from them are much larger than those of the corresponding paraffins. Figure 1 shows the variation of log?] with molecular volume at constant temperature for n-propyl alcohol, a typical example. From this, the variation of vis cosity with temperature at constant volume is obtained by interpolation (Fig. 2; compare the corresponding Figs. 8 and 9 of the previous paperl). Bridgman's measurements5 of the viscosity of ethyl alcohol at high pressures are supplemented by those of Faust6 at pressures up to about 3000 atmos but the \·5 i i i i 1·0 i i i 0'$ / t c-0'0 51 en ...2 1'5 To 1·5 7,.- FIG. 2. Log1o'l ('I in poises) as a function of liT at constant molecular volume V (n-propyl alcohol). Legend as in Fig. 1. 6 P. W. Bridgman, Proc. Am. Acad. Arts Sci. 61, 57 (1926). 6 O. Faust, Z. physik. Chern. 86,479 (1914). 1296 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Tue, 23 Dec 2014 00:21:25ALCOHOL VISCOSITIES 1297 0·5' 0'0 t '·s r: !! r To To 40 45 i-I~o·c I I i I \ \ \ I ~-,oo·c \ \ \ ~ -80·C 50 55 V (<.<) -(os FIG. 3. Loglo>] (>] in poises) as a function of molecular volume V at constant temperature (ethyl alcohol). 0, Bridgman, 30°C; X, Bridgman, 75°C; e, Faust, O°C; 119, Faust, 15.1 °C; A, Faust, 30°C; ., Faust, 53.5°C; 6, various observers, atmospheric pressure and different temperatures. agreement is not very good. Bridgman7 has discussed the disparity. The two sets of observations have been combined with various atmospheric pressure measure- f ,.5' II .... .2 To t·o -'1-- FIG. 4. Loglo>] (>] in poises) as a function of liT at constant molecular volume V (ethyl alcohol). Legend as in Fig. 3. 7 P. W. Bridgman, Physics of High Pressure (Bell and Com pany, London, 1949). ~ooo 7000 t 6000 ""': -:d 5000 v '-' > lifo!) 4000 '3000 2000 0·80 o·es 0·90 0·95' ,·00 Relative volume_ FIG. 5. 6. as a function of relative volume (volume relative to that at O°C and 1 atmos pressure). e, methyl alcohol; 0, ethyl alcohol; X, n-propyl alcohol; 6, iso-butyl alcohol. ments in Figs. 3 and 4 to illustrate the lack of agree ment. The particular importance of the behavior of ethyl alcohol will he made evident later. Energies of activation of viscous flow at constant volume 6v, calculated from the isochores of Figs. 2 and 4, are plotted as functions of relative volume in Fig. 5 together with the corresponding curves for methyl and iso-butyl alcohols. The literature references to all the relevant experimental work on these four alcohols are listed in Table I. The reason for the minimum in the 6v volume curve for ethyl alcohol is not clear. It should be borne in mind that the accuracy with which 6. can be calculated is greatly reduced at the higher volumes where the isochores are short and errors in the determination of the viscosity are more significant. Bridgman5 has stated TABLE 1. Viscosities and volumes of liquids at different temperatures and pressures-collected references. Viscosity Volume Viscosity Density at high at high at low temperatures at Substance pressures pressures atmospheric pressure MeOH a, b d e,g EtOH b,c d f, g PrnOH a, b d f, g, h BuiOH b d f, g • P. W. Bridgman, Proc. Am. Acad. Arts Sci. 77, 115 (1949). b See reference 5. o See reference 7. d P. W. Bridgman, Proc. Am. Acad. Arts Sci. 49, 3 (1913). • S. Mitsukuri and T. Tonomura. Proc. Imp. Acad. Japan 3,155 (1927). f S. Mitsukuri and T. Tonomura, Proc. Imp. Acad. Japan 5, 23 (1929). g M. A. Veksler, J. Exp. Theor. Phys. USSR 9. 616 (1939). b G. Tammann and W. Hesse, Z. anorg. u. allgem. Chern. 156, 245 (1926). i W. Seitz and G. Lechner, Ann. Physik 49,93 (1916). j S. Mitsukuri and Y. Kitano. Proc. Imp. Acad. Japan 5, 21 (1929). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Tue, 23 Dec 2014 00:21:251298 A. JOBLING AND A. S. C. LAWRENCE this region, and for other liquids slight deviations of his measurements from those of other workers were noted previously. However, the depth of the minimum seems too large to be accounted for entirely by experi mental error. If the effect is genuine, it is surprising that it is not observed for methyl alcohol also, since there is some that his experimental errors are probably greatest in THE JOURNAL OF CHEMICAL PHYSICS rather uncertain evidence that a part of the 0v volume curve for water also has a positive slope. Straight lines are also obtained on plotting 10g1j against liT from the experimental data of Van Wijk, Van der Veen, ~rinkman, and Seeder8 for n-octadecyl alcohol. 8 Van Wijk, Van der Veen, Brinkman, and Seeder, Physica 7, 45 (1940). VOLUME 20. NUMBER 8 AUGCST. 19'>2 The X-Ray K Absorption Edges of Covalently Bonded Cr, Mn, Fe, and Ni* G. MITCHELLt AND W. W. BEEMAN Department of Physics, University of Wisconsin, Madison, Wisconsin (Received December 10, 1951) We discuss some recently measured absorption edges of covalent nickel complexes and edges of Cr, Mn, and Fe in similar complexes already in the literature. Low energy absorption (generally resolved as a line) is observed in all complexes where empty 4p orbital is expected from chemical or magnetic data. From the position of the line it is concluded that the metal ion is near neutral in the covalent complex. Where the 4p orbital is completely used in bonding, no low energy absorption line is observed. The method may be useful as an additional experimental check on the assignment of bonding orbitals. INTRO DUCTIO N THE elements of the first transition series and their compounds have been studied by a great many x-ray spectroscopists. We will confine the present dis cussion to the K edges1•2 which lie in the nonvacuum region at 1 or 2A. The following results will be useful. In the pure metals the absorption coefficient in creases gradually, with increasing photon energy, from a low value where no K shell ionization occurs to a maximum, some nine times greater, where K electrons are being excited into the empty 4p band of the metal. This region of the principal absorption increase has an energy extent of 15 or 20 ev in the elements under consideration. It is followed by a series of relatively small maxima and minima of the absorption coefficient which extend for 200 or 300 ev. This is the region of the Kronig structure3 with which we are not primarily concerned. A fairly standard example of a metal absorption edge in the region of present interest is that of Ni in Fig. 1. In general, no strong absorption lines are found before the main 4p absorption at 15 or 20 ev. The natural width of the K excited state and a smaller contribution from the finite resolution of the crystals used in the x-ray monochromator combine to limit the • Supported by the ONR and the Wisconsin Alumni Research Foundation. t Now with the Capehart-Farnsworth Corporation, Fort Wayne, Indiana. 1 W. W. Beeman and H. Friedman, Phys. Rev. 56, 392 (1939). 2 V. H. Sanner, thesis (Uppsala, Sweden, 1941). See also Y. Cauchois, Les Spectra des Rayons X et la Structure Electronique de la Matiere (Gauthier-Villars, Paris, 1948). 3 R. de L Kronig, Z. Physik 75, 191 (1932). resolution in these spectra to about 2 ev. However, since the absorption curve can be considered to be the sum of a large number of overlapping resonance lines of known shape, each of half-width 2 ev, analysis in terms of such lines will sometimes reveal finer detail. In particular one expects4 and findsl the initial absorp tion (near zero ev in Fig. 1) to follow an arctangent curve. The point of inflection of the arctangent gives the photon energy necessary to excite a K electron into the first empty level of the Fermi distribution. This energy is conveniently chosen as a zero for a dis cussion of absorption edge structure close to the edge since only very rarely does an absorption of lower energy occur in any of the compounds of the element. The point of inflection can be located to about one-half ev in most cases. The structure of the metal edges can be qualitatively understood 1 in terms of the electron band structure of the metal. In a few other simple cases a qualitative or semi-quantitative discussion of absorption structure within 15 or 20 ev of the edge is available. ParraH" measured the K edge of argon gas and found a number of absorption lines which he identified as 1s-mp transi tions, n=4, 5, 6, etc. Bearden and Beeman6 measured the edges of Ni++, Cu++, and Zn++ in aqueous solution and found two absorption lines whose separation agreed with the assignments ls--+ip and an unresolved ls-mp, n=5, 6, 7, etc. to the transitions. In addition the ex pected separation of the ls-+4p line of the ion from the 4 Richtmyer, Barnes, and Ramberg, Phys. Rev. 46, 843 (1934). 6 L. G. Parratt, Phys. Rev. 56, 295 (1939). 6 W. W. Beeman and J. A. Bearden, Phys. Rev. 61,455 (1942). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5 On: Tue, 23 Dec 2014 00:21:25

1.3067297.pdf

Perturbation Methods in the Quantum Mechanics of n-Electron Systems E. M. Corson Irving Kaplan , Citation: Physics Today 4, 7, 25 (1951); doi: 10.1063/1.3067297 View online: http://dx.doi.org/10.1063/1.3067297 View Table of Contents: http://physicstoday.scitation.org/toc/pto/4/7 Published by the American Institute of PhysicsAerodynamics Supersonic Aerodynamics, A Theoretical Intro- duction. By Edward R. C. Miles. 255 pp. McGraw- Hill Book Company, New York, 1950. $4.00. Foundations of Aerodynamics. By A. M. Kuethe and J. D. Schetzer. 374 pp. John Wiley and Sons, Inc., New York, 1950. $5.75. Just to be arbitrary suppose we consider the two books in the reverse order in which they appear above. The Kuethe-Schetzer book is precisely the type of book I should like to use for a group of advanced students. It seems to me that the sophistication implied in their approach is exactly what we should try to impart to all students of fluid dynamics. The almost arithmetical fluid dynamics which is frequently found in introductory courses never gives the student the perspective he needs to meet the problems in this field. From the outset Kuethe and Schetzer use the methods of vector analysis without which a really sound course in fluids is not possible. Their book covers the usual topics with nota- ble differences and additions. For example they have put in chapters on energy and wave forms, both of which deserve considerable attention (I wish the au- thors had expanded these chapters even more) and have not hesitated to introduce thermodynamics at numerous points, in several instances devoting time specifically to thermodynamics as such. Somewhat on the debit side is the lack of a more detailed discussion of viscous flow which is relegated to a relatively brief treatment. However, this is not a fault peculiar to this particular text, and actually the treatment here is better than in most similar books. Although the authors do not pretend to be writing a book in applied aerodynamics, they do include sufficient material to give the student a picture of some of the problems involved in this area. There are two excellent appendices, one on dimensional analysis and one on the Navier-Stokes and Energy Equations. Several ta- bles of coefficients and flow parameters are interspersed throughout the text, which should prove very helpful. The approach is, of course, principally analytical but particularly in the sections dealing with turbulence the authors describe some of the experimental techniques and methods used in the laboratory. Returning to Professor Miles' book we find a treat- ment that would follow rather naturally from the Kuethe-Schetzer work, although there is no intention to imply that this is anything but coincidental. The level at which Professor Miles has written his book fits in very well for anyone who has had his initial prepa- ration in aerodynamics with the Kuethe-Schetzer book;25 both use vector analysis freely and in general have the same sophisticated approach to their subjects. Pro- fessor Miles covers such topics as the potential equa- tion, linear theory of plane flow, linear potential theory in three dimensions, shock waves, and potential flow in the hodograph plane among others. Since the author is a mathematician a number of the problems take on a rather mathematical form, almost, at times, to the ex- clusion of the physical implications. It is not at all un- usual in present-day aerodynamics and plasticity to treat problems as though they were invented for the edification of mathematicians, rather than for the pur- pose of solving problems in the physical and engineer- ing world. I would guess that Professor Miles suspected this charge because in his preface he expresses the hope that readers will find his book sufficiently "practical". From the teacher's point of view the book is practical but it will have to be left to the actual designers in the field to determine how practical it may be for them. While the Kuethe-Schetzer book needs a few more problems, the Miles book has what appears to be an ample supply which will probably keep both instructor and student quite busy. From the teaching point of view, the combination of these two books would make an excellent three or four semester course starting in perhaps the senior year of college and carrying on through the first year of graduate work. There is plenty of room for expansion by the instructor within this framework and at the same time the student would be given a preparation which gives an excellent picture of modern aerodynamics and the trends it is taking. Not only has aerodynamics advanced enormously as a sci- ence in the past ten years but the art of writing text- books in this field has also marched forward. Both books should receive serious consideration in any school offering work in this area and should surely be on the bookshelf of all those interested in fluid dynamics. James Bernard Kelley Hojstra College Many Particle Systems Perturbation Methods in the Quantum Mechanics of n-Electron Systems. By E. M. Corson. 308 pp. Hafner Publishing Company, New York, 1951. $11.00. Publication of Dr. Corson's book fills an important gap in the literature of contemporary physics in that it provides a unified and comprehensive treatment of the quantum mechanics of many-particle systems. There has been a great need for a book dealing with the area between elementary quantum mechanics on the one hand and the fields of specialized applications on the other hand. Thus, there have been available many adequate books on the elements of quantum me- chanics, and a considerable number of treatises on such specialized subjects as atomic spectra, radiation theory, and solid state theory. Dr. Corson's book offers for the first time, to this reviewer's knowledge, a thorough ex- position of the theory and methods for studying the JULY 195126 properties and behavior of many-particle systems, a field which includes many of the most significant ap- plications of quantum mechanics. The scope of the book can be indicated by a brief review of its contents. In the first four chapters, the author develops the mathematical formalism which is based on the Dirac-Jordan representation theory, and discusses in a succinct, yet clear, manner the physical principles of quantum mechanics. Although these chap- ters constitute an introductory summary of the quan- tum theory of states and observables, they succeed in clarifying, at least for this reviewer, some bothersome questions. For example, the author's treatment of fixed and moving operators and representations, and of change of state in time is particularly helpful. The fifth chapter deals with the fundamentals of perturba- tion theory. The sixth chapter considers some of the properties of groups which are useful in the treatment of many-particle problems. In chapter seven, the anti- symmetry principle is studied as is the density matrix in many-electron systems. Chapters eight and nine are devoted to the methods for determining the eigen- vectors which give the best approximation to the solu- tion of the wave equation of a many-particle system (the method of the self-consistent field). The varia- tional method is treated first, with emphasis on the derivation and physical aspects of the Hartree and Fock- Dirac equations. This is followed by a discussion of the statistical or semiclassical approximation which is used to select the first approximation to the potential with which to start the iterative process of solving the Har- tree and Fock-Dirac equations. The section closes with a discussion of the Thomas-Fermi approximation. Chap- ter ten consists of a detailed treatment of the Dirac vector model including its relationship to group theory. Here the author has succeeded in avoiding an overly formal group-theoretic approach without diluting the elegance of the method. Applications to the theory of valence are considered, and the four, six, and eight elec- tron problems are discussed at considerable length. The eleventh chapter deals with second quantization. The author discusses its operational aspects, its place in the general representation theory and its relationships with the method of the self-consistent field, with the Dirac vector model and with the hole formalism. The treat- ment of this subject is excellent. The book closes with a short chapter on the S-matrix theory. The author is to be commended for undertaking the formidable task of writing a book of this scope, and for the successful achievement of his goal. His exposi- tion is clear and logical; the mathematical treatment is elegant. It is understandable that space limitations did not permit including illustrative examples in all cases where they might have been desirable. However, con- sidering the many literature references cited, the gen- eral theory and methods have been presented in a suffi- ciently detailed way so that the volume should be very useful. One must agree with the author's belief that the subject matter presented constitutes a representative cross section of applied nonrelativistic quantum me-chanics. For these reasons Dr. Corson's book is highly recommended to research physicists and to graduate students of physics. Finally, the printers, Blackie and Son Ltd., of Glas- gow, are to be congratulated on the beauty and artistry with which this volume has been made. The book's merits, both internal and external, should compensate amply for its price. Irving Kaplan Brookhaven National Laboratory Cloud Physics Cloud Physics. By D. W. Perrie. 119 pp. John Wiley and Sons, Inc., New York, 1951. $4.50. This is a book on a subject which has recently as- sumed much popularity in scientific circles, as well as among the news and magazine supplements, since it is related to "rainmaking". During the past fifty years, a number of books and monographs have appeared based on the physical and chemical properties of the atmosphere. Humphreys' Physics of the Air is an example of a book covering most of the general properties of the atmosphere. Landsberg's Atmospheric Condensation Nuclei is a monograph which deals with a highly specialized field. As a result of the many excellent basic research stud- ies in the field of meteorology which were started dur- ing World War II and have continued up to the present time, many new discoveries have been made in this important science. Cloud Physics is apparently directed toward partially filling the gap in the available printed information dealing with clouds. As such, it will be welcomed by all who are interested in the subject. A useful feature of the book is the references it con- tains to the historical development of the study of clouds. These should not be accepted as completely authoritative, however, without some additional check on the references cited. An example of a reference to the historic literature w:hich may be misleading is that of the "seeding" of clouds with dry ice which was carried out by Veraart in Holland during the 30's. A study of Veraart's experiments shows that he employed methods suggested by Gathman of Chicago in 1891 in patent applications. This involved the dumping of large quantities of dry ice into the atmosphere so that the cooling of the air would either produce cloudiness or precipitation in existent clouds. This use of dry ice to affect clouds w:as thus based on an entirely different idea than that which is used by the reviewer and his colleagues when seeding supercooled clouds with dry ice. To have any appreciable effect in the atmosphere Gathman and Veraart's method would require countless tons of dry ice. To modify supercooled clouds, a few pounds of dry ice are enough to produce a major effect in a cloud area several miles in diameter. This book is recommended primarily since it indi- cates the many gaps in our present knowledge of cloud physics and the uncertainties which exist in this impor- tant field of the physical sciences. PHYSICS TODAY

1.1721126.pdf

Reverse Characteristics of High Inverse Voltage Point Contact Germanium Rectifiers J. H. Simpson and H. L. Armstrong Citation: Journal of Applied Physics 24, 25 (1953); doi: 10.1063/1.1721126 View online: http://dx.doi.org/10.1063/1.1721126 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Schottky barrier height extraction from forward current-voltage characteristics of non-ideal diodes with high series resistance Appl. Phys. Lett. 102, 042110 (2013); 10.1063/1.4789989 Quantitative model for currentvoltage characteristics of metal point contacts on silicon rectifying junctions J. Appl. Phys. 54, 7034 (1983); 10.1063/1.331969 Forward Characteristic of Germanium Point Contact Rectifiers J. Appl. Phys. 26, 949 (1955); 10.1063/1.1722143 Evaporated Point Contact Rectifiers J. Appl. Phys. 24, 228 (1953); 10.1063/1.1721249 High Inverse Voltage Germanium Rectifiers J. Appl. Phys. 20, 804 (1949); 10.1063/1.1698530 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:35JOURNAL OF APPLIED PHYSICS VOLUME 24, NUMBER 1 JANUARY, 1953 Reverse Characteristics of High Inverse Voltage Point Contact Germanium. Rectifiers J. H. SIMPSON AND H. L. ARMSTRONG Radio and Electrical Engineering Division, National Research Council, OUawa, Canada (Received June 13, 1952) A theory of the reverse characteristic of high inverse voltage germanium rectifiers is developed, which takes account of the radial symmetry of the point contact and the presence of positive holes in the "inversion region" of the semiconductor. The field at the metal is found to consist of three components. The first com ponent varies inversely with the radius of the contact and directly with the applied voltage for larger voltages. The second component is produced by the impurity centers and varies approximately as the cube root of the voltage. The third component is produced by the positive holes in the inversion region and is approximately constant. This last component lowers the effective barrier height for rectifiers made of very pure materia!. The first is the more important of the variable components and is responsible for increases in current resulting from image force and tunnel effect at high voltages. In the former case the logarithm of the current varies as Vi and in the latter as V2 approximately. Current-voltage curves of the predicted forms have been found experimentally using short rectangular pulses varying in length from 2 to 10 J.l.sec to minimize thermal effects. Where possible, barrier heights are determined by measurements over a range of temperatures and detailed numerical comparisons of theoretical and experimental data are made. The two sets of data are in remarkably good agreement thus implying that the theoretical picture is essentially correct. 1. INTRODUCTION THE characteristics of high inverse voltage ger manium rectifiers have been discussed in con siderable detail in the literature and a summary of experimental results has been given by Benzer.1 In his discussion of the reverse characteristic, Benzer divides the current into the following components: (1) A saturation current which reaches a constant value at low voltage. (2) A component which increases linearly with voltage. (3) A component which increases more rapidly than linearly with voltage. The first two components have been discussed in some detail. Bardeen and Brattain2 suggest that the first component may be the result of the motion of holes from the semiconductor to the metal point contact and that surface effects may account for the magnitude of the current. The second component is usually considered to result from the presence of small areas of low barrier height in the contact between the metal and the semi conductor.a It has also been suggested2 that the third component may be the result of non-uniformity of the barrier. It is the purpose of this paper to investigate theoretically two other possible causes of components (2) and (3), namely the field resulting from the geometry of the contact and the presence of large numbers of positive holes in the "inversion region" of the semicon ductor immediately adjacent to the metal. Experi mental data, obtained by pulse methods to minimize thermal effects, are analysed in the light of the theo retical predictions and detailed numerical comparisons are given. 1 S. Benzer, J. App!. Phys. 20, 804 (1949). 2 J. Bardeen and W. H. Brattain, Phys. Rev. 75, 1208 (1949). 3 Johnson, Smith, and Yearian, J. App!. Phys. 21, 283 (1950). 25 2. SHAPE OF THE BARRIER WITH NO VOLTAGE APPLIED The generally accepted picture of the barrier at the surface of germanium is shown in Fig. 1, which is taken from the important paper of Bardeen and Brattain.2 The region to the left of the dotted line is an "inversion region" of the type described by Schottky and Spenke.4 It will contain many more holes per unit volume than the body of the semiconductor. The presence of these holes,-if they exist in sufficient numbers,-will have a pronounced influence on the shape of the barrier. A calculation of barrier shape taking account of this effect may be made quite readily and is carried out below. We first establish approximate values for the quanti ties shown in Fig. 1. For germanium the energy gap (ep e+ ep h) is usually taken to be 0.72 ev. A comparison of the number of electrons in the conduction band of the semiconductor with the number having energies in the same range in the metal5 gives a value of 0.28 ev for epeO, the value of epe in the body of the semiconductor, if the number of free electrons no is 4X1014/cc and the number of holes po is 1012/ cc, corresponding to a resis tivity of 4.5 ohm cm.6 Hence epllO, the value of ep" in the body of the germanium, is 0.44 ev. For ep.=:= 0.65 ev (epb=0.37 ev) the number of holes/cc at the metal, pm, is about 2X1018/CC and even for the relatively low barrier height, ep.= 0.57 ev, pm= 1017/cc. This relatively large mobile charge probably makes the barrier more uniform than would otherwise be the .case and may account, . in part, for the high inverse voltage ratings obtainable with these rectifiers. To illustrate the influence of holes on the barrier shape we consider a one-dimensional model. If we then 4 W. Schottky and E. Spenke, Wiss. Veroffent!. Siemens Werken 18, 225 (1939). 6 N. F. Mott and R. W. Gurney, FJectronic Processes in Ionic Crystals (Clarendon Press, Oxford, 1940), p. 175. 6 J. Bardeen, Bell System Tech. J. 29, 485 (1950). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:3526 J. H. SIMPSON AND H. L. ARMSTRONG z o cr ... " .... ..J .... FERMI LEVEL METAL SEMICONDUCTOR FIG. 1. Schematic energy level diagram of barrier layer at germanium surface showing inversion layer of p-type conducti vity. take the zero of energy at the bottom of the conduction band in the body of the semiconductor, we have: tPcp/dx2= (411"e2/ E) [no -Po+ChTI exp{(CPho+cp)/kT} -C.T! exp{( -cf>.o-cf»/kT}] = (411"e2/ e)[no-no exp( -cp/kT) -po+Po exp(cf>/kT)], (1) in which x is the distance measured into the semicon ductor from the metal, cp is the energy of the bottom of the conduction band, no is the number of free electrons and po the number of free holes in the bulk material. no is approximately equal to the number of donor im purities a.t ordinary temperatures. E is the dielectric con stant and T the absolute temperature. As outlined in reference 2, and C .rvChrv2(211"mk/h2) trv5X 1()1" no= C .Tt exp( -cp.o/kT) po=ChTt exp( -CPho/kT). Integration of (1) with the condition that dcf>/dx=O when cp=O gives: (dcf>/dx)2= (811"e2/ E)[(nO -po)cp-(no+po)kT +kTno exp( -cp/kT)+kTpo exp(cp/kT)], (2) dy/dx= -[(811"e2/ EkT){ (no-po)y-(no+po) +noe-II+ poell} ]!, (3) where y=cp/kT, and the minus sign must be chosen to satisfy the physical conditions. Equation (3) has been integrated numerically on the assumption that CPb= 0.40 ev (Yb= 16) to give curve A of Fig. 2. Curve B shows a "natural" barrier in which the space charge in the semi conductor is assumed to be produced by impurity centers only. The field strength near the metal is very much larger when the presence of holes in the inversion region is taken into account. Results of calculations of this type for a different value of cp b have been published recently.7 3. BARRIER FIELD FOR LARGE REVERSE VOLTAGES The magnitude of the field in the region near the metal suggests the possibility of tunnel effect or strong field emission (Schottky effect) when large reverse voltages are applied. Under this application of reverse voltages holes will migrate from the body of the ger manium to the metal. If we then take more accurate account of the geometry of the point contact than in Sec. 2, the following relations may be assumed to hold: ip= 211"r12ePOvD= 211"r[pev -eD(dp/ dr)], tPcp/dr+(2/r)(dcp/dr) = (411"e2/ E) (p+no). (4) (5) In Eq. (5) the charge of the conduction electrons in the barrier region and of the acceptors has been neglected since these quantities are both small compared to the number of holes p and the number of donors no. ip is the hole current; r1 is the radius of the barrier-semicon ductor interface; VD is the velocity of diffusion of holes from the body of the semiconductor to the barrier (see reference 2, p. 1222) ; v is the drift velocity of holes in the barrier region and D is a diffusion constant. In view of the results of Ryder and Shockley,8.9 it will be assumed that v is constant and rvl07 em/sec. To estimate the behavior of D in the barrier region is more difficult. The behavior of v implies a decrease in mobility but there is an accompanying increase in carrier temperature,9 so that perhaps the most reasonable assumption,-which will be made here,-is that D remains approximately constant. This has the advantage of simplifying the mathematics considerably. Also, in view of the short ness of the mean free path because of collisions between z 2 '" '" '" .. ... .8,----'-----,-------r---- '" .. -.. '" .. ~ o .. > o '" .6~:--___;7''----+-------+---- ,.0 "," ..... "'~ ZJ 1.L.I ~.4 BOTTOM Of CONOUCTION BAND IN SEMICONDUCTOR ~ .2r------+------+--- ~ -META -SEMICONDUCTOR o DISTANCE F ROM ME TAL (em) FERMI I..E vEL FIG. 2. One-dimensional diagram of barrier layer, A-taking account of both positive holes in the inversion layer and impurity centers, B-taking account of impurity centers only. 7 W. Bosenberg and E. Fues. Z. Naturforsch. 6a 741 (1951). 8 E. J. Ryder and W. Shockley, Phys. Rev. 81, 139 (1951). • W. Shockely, Bell. System Tech. J. 30, 990 (1951). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:35HIGH INVERSE VOLTAGE GERMANIUM RECTIFIERS 27 holes and optical vibrations of the lattice, it is assumed that (4) holds through most of the barrier. In the simple model considered here the change of hole velocity from VD to v is assumed to take place sud denly at radius rl, so that to satisfy the current contin uity relation there will also be a sudden change in the number of holes there. The hole distribution used is that given by the full line of Fig. 3 while the true dis tribution probably follows a line such as that shown dashed between A and B. To take account of this mathe matically would complicate the calculation consider ably and the error introduced by the approximation is probably quite small. The hole densities at B are related by the expression PIVI = POVD. Integrating (4) we obtain [p exp(-vr/D)Jrro = -(PoVDrND)f.r [exp( -vr/D)J(dr/r2) ro P = exp( vr / D{ Pm exp( -vro/ D) -(povDrND) lor [exp( -vr/D)J(dr/r) 1 (6) Also from (5) [r(tPq,/ dr2)+dq,/ dr J+dq,/ dr= (41Te2/ E)r(p+no), (7) Pm- <i ~ ... -' o :I: u. o a: ... m ::E ::::l Z METAL RADIUS r -- SE M ICONDUCTOR FIG. 3. Schematic diagram of positive hole distribution in barrier layer. The IuUline represents the form of distribution used in this paper. whence [r(dq,/ dr)+q, Jr1ro = (411"e2/ E{ nO(rlLr02)/2+ lorlprdr]. (8) at r=r1, dr/>/dr=O, and r/>=eV -r/>., where V is the applied voltage. At r="o, q,=r/>b, dq,/dr=F o. Substituting these conditions and combining (6) and (8) : eV -q,.-q,b-roFo= (41Te2/ E>[ nohLr02)/2+Pm exp( -vro/D) f:1 {eXP(Vr/D)}rdr -(povDrND) lor1 r{exp(vr/D)} lor{e~(-vr/D)}(dr/r)2, (E/41Te2)(eV -r/>.-roF 0) = nO(r12-r02)/2+ [(DPm/v) exp( -vro/D)][(r1- D/v) exp(vrI/D)- (ro-D/v) exp(vro/D)] -(PovDrNv{ (r1-D/v) exp(vr1/D) Ior1{exP( -vr/D)}(dr/r2)-log(r1/ ro)+(D/v)(1/ro-1/ r1) 1 (9) Once 1'1 has been determined this equation can be used to obtain the magnitude of the field Fo (in ev/cm) at the metal. To determine "1 we integrate (7) between the limits I' and 1'1 obtaining: (10) Integrating again + (povDrND) Jr1 r[exp(Vr/D)] i: [exp(-vr/D)J(dr/r)2. (e/41Te2)[rr/> Jrlro= (E/411"e2)(e V -q,.) (1'1-1'0)- nO[r12r /2-r3 /6Jr1ro-Pm[exp( -vro/ D)] i"! r1[exp(vr /D) Jr(dr)2 ro r +(povDrND) {,lfrlr[eXP(Vr/D)Jfr[exp(-vr/D)J(dr/r)2dr. (11) Jro r ro [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:3528 J. H. SIMPSON AND H. L. ARMSTRONG The integrations shown are elementary but tedious and lead to the following equation for rl, (E/41re2) (cp.-e V)ro-norl2b -ro)/2+noCrtLr03)j6-(PmD/v)[expv(rl-ro)/DJ[Crt -D/v) crt -ro-D/v)+ D2/v2J -(pmD2N) (ro-2D/v) + (povDrNv)[exp(vrl/ D) J[(rl-D/v) (rl-ro-D/v) + D2NJ i Tl[exp( -vr/D)J(dr/r2) TO The following values are chosen tentatively for the constant factors in (12): E= 16, ro= 5X 10-4 cm, no=4X1014, pm= 1017/cc, v= -107 cm/sec, po= 1012/cc, VD= -5000 cm/sec, D= 50 cm2/sec, the value applying in the body of the germanium. If we now assume that rl is about 10-3 cm,-a reasonable value in view of results obtained in the case of a "natural barrier", the relative importance of the terms of (12) may be evaluated. Taking account of significant terms only, (E/41re2)(cp. -e V)ro -nOrI2(rl -ro)/2 + no (ria -roa)/6 -PmroV/v2= O. (13) For voltages greater than 10, with which we shall be concerned, the fourth term of (13) may be neglected. The resulting equation for rl is that which would be obtained if the presence of holes in the barrier region were neglected. Thus for these higher reverse voltages, the holes in the inversion region change the barrier shape appreciably but the variation in its thickness is small. This is, of course, due to the fact that the inversion layer is very thin so that, in spite of the high density of holes, the total number of holes there is small. This result might have been expected in view of the one dimensional picture shown in Fig. 2. Dropping the fourth term in (13) and rewriting with x=rl/rO, or, following substitution of numerical values, 2x3 -3x2-[(cp.-eV)/2.0J+ 1 = 0, (15) where CPa and eV are in electron volts. This equation may be solved by the standard method (Cardan's solution) to give the following result, accur ate to within a few percent for reverse voltages greater than 10 volts, x=rl/rO=[( -2V)t+C -2V)-t+1J/2. (16) Typical values of rl (assuming '0= 5X 10-4 cm) are shown in Table 1. The validity of the approximations used in obtaining Eq. (13) is readily confirmed for these values of rl. It is now possible to determine the field at the metal from Eq. (9). Using the previously listed numerical TABLE 1. Outer radius of barrier for different applied voltages. Applied voltage V Barrier radius'l (cmXIOS) -32 -108 -256 -500 1.32 1. 79 2.28 2.77 values and rl",2X 10-3 cm, it can be readily shown that Fo= -(CPa -eV)/ro-21re2 nO(rI2-r02)/ ETo + 41re2pmD/ Ev=F1+F2+Fa, (17) the remaining terms of (9) being negligible. The terms FI, F2, and Fa are the result of the applied voltage, the impurity centers and the positive hole space charge, respectively. In this approximation Fa is constant in dependent of the applied voltage. Its magnitude is about 1()8 ev/cm for pm=1018/cc. All terms of (17) are negative since V and v are negative under reverse voltage conditions. In obtaining (17) the tendency of image force to in crease the number of positive holes near the metal has been ignored since at distances from the metal of the order of a lattice spacing the magnitude of image force becomes very uncertain. In view of other uncertainties and assumptions mentioned above the image force term may be considered to be absorbed into Fa. As far as electron flow over the barrier is concerned, however, image force plays a major role. This is considered in the following section. 4. CURRENT FLOW FOR LARGE REVERSE VOLTAGES In this section we calculate the electron flow from metal to semiconductor taking account of tunnel effect and strong field emission on the following assumptions: (a) The number of electrons flowing against the field from semiconductor to metal is negligible. This must be so since we consider only voltages greater than 10. (b) Surface flow of holes has negligible effect on the barrier shape. Such surface flow may be responsible for the saturation current as suggested by Bardeen and Brattain, but the space charge of the holes carrying such a small current will be negligible compared to that which produces Fa. (c) The field is constant over the region of the barrier for which tunnel effect and strong field emission are important, (i.e., for r -ro::; 10-jl cm). This is obviously a good approximation as far as F 1 and F 2 are concerned but needs closer consideration in the case of Fa. The field resulting from this component at distance r is actually Fa exp[v(r-ro)/DJ, as can be shown by further manipulation of the equations of Sec. 3, and this does not differ appreciably from F 3 for r -ro::; lO-jI cm. The form of the barrier is shown schematically in Fig. 4. A one-dimensional model is used since x= r -ro::; 10-jl cm. The full curved line represents the barrier when image force is taken into account. Its [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:35H I G H I N V E R S E VOLTAGE G E R M AN I U M R E C T I FIE R S 29 equation is: (18) Strictly speaking the infrared dielectric constant should be used. For germanium this is equal to E to within a few percent, however. For the reason outlined at the end of Sec. 3 the shape of the barrier between x=o and x=SXlO-8 cm probably has little significance. Except for very strong fields it may probably be assumed, however, that the macroscopic image force formula of Eq. (18) holds at Xm= (e/2) ( _FoE)-l. Hence the barrier height is about q,m= q,. -e( -Fo/ E)l. The ratio of the number of electrons of energy W transmitted through the barrier to the number incident upon it is called the transparency. It may be determined approximately by the W. K. B. method and is given byIO (19) where Xl and X2 are defined in Fig. 4. This cannot be integrated in finite terms for the value of V given by (18) and an approximate form for the barrier, OADC of Fig. 4, has been chosen. The transparency for the rectangular section OADE may be calculated exactly.IO It does not vary very much over the top quarter of the barrier and varies much more slowly with applied voltage than the transparency of the triangular section DEC. In view of 1-------- Xo =.~ x FIG. 4. Schematic energy diagram of barrier layer showing the effect of image force. The approximate form of barrier used in this paper is OADC. this our procedure will be to attempt to find, approxi mately, the way in which the number of electrons penetrating the triangular barrier varies with applied voltage and to determine numerically whether, at high field strengths, the magnitude of this component is appreciable compared with that which passes over it. The transparency of the triangular section deter mined, using (19), is T2=exp{ _2(2m/h2)ti~0-W(XO-Xm)/"'m [¢m(Xo-X)/(Xo-Xm)-WJ!dX} =exp{ -(2m/h2)![4(q,m- W)!(xo-x m)/3¢mJ}. (20) The number of electrons crossing unit area of the barrier from metal to semiconductor in unit time is, therefore, nc= (47rmkT/h3)[A i",mexp{ -W/kT-(2m/h2)![4(q,m- W)!(xo-x m)/3q,m]}dW+ J"'exp( -W/kT)dW]. (21) o "'m Here the transparency is assumed equal to unity for W> CPm and the factor A resulting from the rectangular part of the barrier is taken outside the integral, since it varies relatively slowly with W over the range of inter est. Also, since this range includes the top third of the barrier or less, the Boltzmann distribution is used. From (21) the electron current per unit area becomes i= [411"em(kT)2/h3J exp( -CPm/kT) [ "'mlkt ] X A i exp{y-a(Xo-Xm)Y!/¢m}dy+l, a= (4/3)[811"2 m(kT)3jh2J!, CPm= cp.-e(1 Fol/ E)l, y= (¢m-W)/kT. (22) 10 N. F. Mott and 1. N. Sneddon, Wave Mechanics and its Appli cations (Clarendon Press, Oxford, 1948); B. Jeffreys, Proc. Cam bridge Phil. Soc. 38, 401 (1942). The integral in (22) may be evaluated numerically or approximately by expansion about the maximum value of the function. The latter method, which is similar to the well-known method of steepest descents, leads to the following expression for the integral, 1""[ (411"iCPm IF 0 I )/3a¢.J Xexp{ (4/27) [CPmFo/ acp .]2}, (23) ~0.8X 10-51 Fo I exp[1.6X 1O-12Fo2J, (24) where Fo is expressed in ev/cm. I has values of 0.8 and 5.0 for Fo equal to lOS and SX 105 ev/cm, respectively. Since A is about 0.2, the integral term of (22) may thus be neglected for fields less than SX lOS ev/cm. For larger fields I increases very rapidly. It is fortunate that both the W.K.B. method and the expansion method em ployed in obtaining (23) become reasonably accurate for fields of the order of 1()6 ev/cm, so that (23) may be [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:3530 ]. H. SIMPSON AND H. L. ARMSTRONG used to obtain some idea of the form of the reverse characteristic if such high fields exist in the germanium. In order to determine the form of characteristic predicted by (22) we must estimate the magnitudes and variation with applied voltage of the terms of (17). For voltages greater than 10, F 1 is substantially proportional to the applied voltage, and we may put Fl=efJV/ro where fJ is a factor, discussed by Schottky,U to allow. for irregularities on the whisker surface. For lower voltages, when Fa is the dominant term of (17), fJ is probably about unity, since holes near the whisker will tend to smooth out variations in field strength over the whisker surface. Electrons will then flow into the ger manium from all parts of this surface. For voltages at which the field strength at isolated points on the whisker surface becomes greater than Fa, fJ will begin to increase and it will level off at a constant value when Fl becomes the dominant term in (17). Under this high applied voltage the effective contact area will be reduced. The term F2, calculated using (16) is plotted in Fig. 5. For voltages between 10 and 50 it may be assumed to follow the dashed straight line, so that F2= 25000+ 1500 V. The fixed term may be added to Fa. The term pro portional to V may be added to Fl, to which it compares in magnitude at low voltages. At higher voltages, how ever, when the factor fJ becomes appreciably greater than unity and the slope of Fig. 5 decreases, this com ponent is small compared to Fl and may be neglected. Summarizing, the field at the metal consists of a fixed term and a term which varies approximately di rectly with voltage. The former is probably between 1()6 and 1()6 ev/cm; the latter is V[(1/ro) + 1500] at low voltages and fJV fro at high voltages. We may now predict the general form of the reverse characteristic, correlating it with the purity of the material and the height of the barrier as determined by measurements at different temperatures. Since the predicted characteristic for I F 0 I < I Fbi will be quite different from that for IFol>IFbl, where IFbl is 40 ;-. 2 " 30 E / / /'" /" u ~ /v - ...J <I ... ... ::. ... <I 0 ...J ... ;;: 20 10 }/ V V 0 100 200 300 VOLTS 400 500 FIG. 5. Field produced at metal by impurity center charge, as a function of applied voltage. 11 W. Schottky, Z. Physik 14, 63 (1923). roughly 5 X 1 ()5 ev / cm, we consider the two regions separately. (a) Low Field Region Here the integral term of (22) may be neglected and we have iz=[411'em(kT)2/h3] exp[ -c/>.+e(IFol/E)I]/kT. (25) In the lower range of applied voltages for good rectifiers, the fixed part Fa of F 0 is considerably larger than the linear part Fl(F2 has now been absorbed into Fl and F3) and (25) may be expressed iz= [411'em(kT)2/ha][exp( -c/>mo/kT)] X[1+(eFat/2kT Et)(Fl/Fa)], (26) where Hence i varies linearly with voltage in this region. At higher voltages Fl becomes appreciably larger than Fa and a plot of logi 'liS Vi should then be linear. At still higher voltages the number of electrons crossing the barrier will become so large that their space charge will affect the shape of the barrier. The slope of the curve of logi 'liS Vi should then decrease. This will occur when i= en'll with n of order lOl6/cc. For '11= 107 em/sec and contact area 10-6 cm2 this is about 10 mao For poorer rectifiers with low barrier heights, Fa may be negligible compared with F 1 for applied voltages greater than 10. The Schottky (logi 'liS Vi) plot should then be linear down to relatively low voltages (of order 20) and a linear region, if it occurs may be merely a transition from the saturation to the Schottky region. (b) High Field Region In this case the linear part of the characteristic will still occur. If the term in (22) containing the integral is small compared with unity at low voltages, Eq. (26) will still apply for the linear component. If, however, the integral term is large compared with unity, the current in (26) will be multiplied by an extra factor. Using (22) and (23) with Fl«Fa we obtain ih= iz(411'IA/3) (QFa) [1 + Fl/Fa][exp(4Q2FN27)], (27) where Q= (3c/>m/4c/>.e)[h2/8rm(kT)3]i, (28) A is the factor, previously defined, for the rectangular part of the barrier. The slope of the linear section will thus be greater than that of a good unit showing Schottky effect only . For larger applied voltages, when Fl exceeds Fa, the exponential term of (23) will then vary much more rapidly than the linear term of (23) or the Schottky term [exp( -c/>m/kT)] of (22). Hence i=JTl exp[( -c/>m/kT)+(BV2/J'S)], (29) where J and B are constants. Thus a plot of 10g(iT-l) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:35HIGH INVERSE VOLTAGE GERMANIUM RECTIFIERS 31 against V2 should be linear. The variation of logi with temperature for constant applied voltage will depend upon the relative magnitudes of the terms in the square bracket. In practical cases the first term will dominate as the reverse current would otherwise be excessive. Hence the variation of logi with temperature should be similar to that given by (25). Some idea of the relation between the values of <Pm and the lengths of the linear sections in the two cases may also be obtained. Tunnel effect will probably playa part only in those units having high thin barriers and small currents. Hence the fixed field F 3 will be larger for these units than for those showing Schottky effect alone and their linear component will exist over a wider volt age range. Thus, if the linear part of the characteristic can be ascribed to the mechanism mentioned, its length should be greatest for the best rectifiers. The extent to which characteristics obtained experi mentally confirm these predictions is considered in the following section. 5. EXPERIMENTAL RESULTS In this section results obtained on a dozen commercial rectifiers, tested at room temperature are summarized. Characteristics of three units were taken at several different temperatures to determine activation energies. Most of the units were IN34's or IN39's made by two manufacturers, but isolated examples of other types (e.g., IN63's) were also tested. All units tested fitted into one of the classifications described below and the characteristics are believed to be intrinsic properties of high inverse voltage germanium rectifiers and not the result of a particular manufacturing technique. The theoretical developments of preceding sections take no account of thermal effects at the point contact and these effects must, if possible, be eliminated by the experimental method. In an attempt to do this, the applied voltage was applied in short pulses, the duration of which could be varied from 2 to 10 ILsec, at the rate of one per second. Current and voltage were measured oscillographically. The results obtained at the two ex tremes of pulse length did not differ and no detectable rise in current occurred during the time of application of each pulse, even at the highest voltages. Thus, heat ing effects, if they occur, seem to be relatively small. The linear component which appears in dc results occurs also under pulse conditions in many cases and has a slope of about 0.3 lLa/volt for the best rectifiers tested. This compares with values of 0.5 /l3./volt or higher ob tained under steady current conditions. The difference is apparently due to thermal effects. A slight initial drift under dc conditions is in fact discernible at the low value of 5 volts. For some good 'rectifiers at room tem peratures this linear region may extend from 10 to 50 volts or somewhat higher, while for others, almost as good, it may be quite short or missing entirely. For poorer rectifiers there is usually no linear section under 1000 .00 .0 0 o .. 0.0 . o I RE VERS! vOLTAGE , V-APPLIED , ,i,REVEFISE CURREtliT Imol / / / , .----, / V // . / /1 25 V / .0 .00 / .5 . 200 25 vl'Z 20 . . . 300 400 SOD 600 V FIG. 6. Schottky plot of reverse current, under pulse conditions, of a good commercial rectifier (lN39) showing the decrease in slope owing to the charge of the conduction band electrons. pulse conditions although thermal effects may make part of the characteristic essentially linear under dc condi tions. Above 50 volts the slope increases. Defining i. as the zero voltage intercept, which should be approximately equal to the saturation current, we may plot log(i-i.) against Vi. A straight line results over a considerable range for many units. A typical example is shown in Fig. 6. The linear part of the Schottky plot commences at from 50 to 100 volts. The slope of the curve begins to decrease when the current has reached a value of 10 or 20 rna, as predicted. It will be noted that under these pulse conditions, the rectifier operates satisfactorily at much higher voltages than its dc turnover voltage. This is characteristic of all the units tested and apparently confirms the suggestion12 that breakdown in these de vices is the result of heating. Plots similar to Fig. 6 have been made for different temperatures for 1N39's and IN34's which showed pro nounced Schottky effect. Some typical preliminary re sults for a 1N39 are replotted in Fig. 7. These give activation energies of 0.19 ev and 0.30 ev at 196 and 64 volts, respectively. Using (25), with Fo={3V/ro the height of the barrier with no voltage applied may be calculated. This is cf>. of Fig. 1 and its value is 0.45 ev approximately. This is so low that the calculated value of F3, the fixed field, is negligible. Corroborative evi dence is the fact that the ordinary reverse character istic has no appreciable linear section. Data obtained for 1N34's are qualitatively similar to the above but the barrier height is lower. One unit of this type gave 0.28 ev for the zero voltage barrier height. At 150v the barrier height is 0.1 ev. Whether there exist units having appreciable linear sections at low voltage and Schottky regions at high voltage has not yet been ascertained. Many character- 12 L. P. Hunter, Phys. Rev. 81, 151 (1951). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:351. H. SIMPSON AND H. L. ARMSTRONG 1000 r----r---.".---,----,------r---, 100 k--cc----t---t----+---t-----r---j " 3- '" 3_0 3.1 3.2 3_3 3.· IOOOIr FIG. 7. Activation energy plots for a good rectifier (lN39) showing strong field emission (Schottky effect). istics which appear linear over short ranges of voltage may merely be the result of the transition from the sat uration part to the Schottky part of the characteristic. Thus, (26) has been neither verified nor disproved ex perimentally. Three units have been found in which the linear part of the characteristic extends from about 20 to 80 volts as shown for one unit in Fig. 8. The slope of the linear part at room temperature is about t ~a/volt. The slopes of the Schottky plots in these cases are constant from 60 to 100 volts approximately but then increase as shown, for one unit, in curve A of Fig. 9. The logarithms of these higher values are plotted against V2 in curve C and indicate that the increase of current in this region is probably caused by tunnel effect. Plots of logi against V and V3 (curves B and D) confirm that the V2 curve gives the closest approach to a straight line. In view of the above it should probably be assumed that the linear part of the SChottky plot in Fig. 9 has no special significance and merely occurs during the transi tion from a linear i-V characteristic to a "tunnel effect" curve. The "Schottky region" does, in fact, occur at a lower voltage than it would if the unit showed a true Schottky effect. The range of voltage in the above measurements was limited in order to improve sensitivity at lower voltages. In order to ascertain whether or not the expected decrease in slope of the tunnel effect curve occurred at larger currents, the apparatus was altered temporarily and curve E of Fig. 9 taken on one of the units. It shows the predicted drop at about 6 rna. The form of (22) with (23) substituted for the integral is so complicated that only the order of the slope of a (logi vs V2) plot can be predicted. If the temperature variation of the exponential term of (27) is ignored, how ever, the barrier height can be obtained by plotting log [slope linear partlTJ against 1jT, as can be readily seen from (26). (It is shown in the following section that the above procedure is probably acceptable). One unit unit analyzed in this manner had a barrier height of 0.45 ev approximately. This quantity is cf>. -e( 1 F 31 I E)! and is thus different from the barrier height determined from a Schottky plot. A plot of 10g(iT-!) vs (liT) gives a straight line, within the experimental error, but a (1/T3) plot gives a result practically as good. Thus, the statement made in connection with (29) cannot be confirmed by a plot of this type using the data at present available. The mag nitudes of the factors in the square bracket of (29) can, however, be determined and the statement confirmed in this manner. This is done in the following section. It must be emphasized that all of the above results are of a preliminary nature and that more accurate measurements over wider ranges of voltage and temper ature should be made to confirm them completely. Also, in order to eliminate thermal effects positively, single pulse methods are desirable. The general trend of the experimental curves described in this section do seem to confirm the theoretical predictions fairly well, however. It remains to determine whether or not numer ical values of slopes and magnitudes involved are of the correct order and consistent among themselves. 6. NUMERICAL VALUES (a) Units Showing Schottky Effect We first consider the magnitude of the slope of the Schottky plot in Fig. 6. Using Eq. (25) with Fl =ej3Vlro substituted for Fo, the slope of Fig. 6 becomes 0.435 ({3/ro)! (300el E)t/kT. The measured value is 0.225. Putting ro=SXlO-4 we obtain {3~8, which is quite a reasonable value. Using the same value of ro, values of j3 ranging from 5 to 8 have been found for IN39's and from 5 to 12 for IN34's. The number of tests made has not been large enough to determine whether whisker radius and roughness are factors in determining differences between IN39's and IN34's, 400r----.-----r----,-----r------r----~ 3OOj---I- " 200 j---I---+-:::7'~-+---t- --+----1 .3- '"I 100 i------r---t-----:::J;,..-""--l-- o 20 40 60 VOLTS eo 23'C 100 FIG. 8. Reverse characteristics, under pulse conditions, for a good rectifier having an appreciable linear section. 120 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:35HIGH INVERSE VOLTAGE GERMANIUM RECTIFIERS 33 but, in view of the differences in barrier height men tioned in Sec. 5, this seems unlikely. The magnitude of the current for the unit of Fig. 7, calculated for a contact area of 10-6 cm2 using (25) and the experimental values of barrier height, is a factor 10 or so larger than that obtained experimentally. Pre sumably, a factor of this order should be allowed to account for the fact that the experimentally measured current does not come from all parts of the whisker, so that the two results agree. (b) Units Showing Tunnel Effect The theoretical value of the slope of the (lni vs P) curve is, using (24), 1.6X 1O-12(/3/ro)2. The experimental value obtained from curve C of Fig. 9 is 0.5X 10-4. For ro= 5X 10-4 this leads to /3= 3, a satisfactory value. Using the experimental value above for the slope the exponential term in (23) or (24) is found to be exp(2.3) at 200 volts. The variation of this term over the range of temperatures used is fairly small so that the pro cedure above in which the temperature variation of the second term of (29) was ignored is probably justified. The fact that curve C of Fig. 9 becomes linear be tween 150 and 200 volts indicates that the fixed field Fa is probably between 100(/3/ro) and 150(/3/ro) or be tween 6Xloo and 9Xloo ev/cm. The reduction in the barrier owing to image force, with no voltage applied, is e(1300Fal/E)t or 0.09 ev, approximately. There is a further effective reduction of between kT and 2kT because of tunnel effect before the e~perimental value of 0.45 ev is obtained. Hence the "true" barrier height is of the order of 0.6 ev. Using the data given at the be ginning of Sec. 2, the value of pm, the number of holes at the metal is found to be about 4XI017/cc. Using the relation for Fa given in (17) with the previously men tioned values for the various quantities gives Fa= 2X loo ev / cm. The discrepancy between this value for Fa and that estimated from the experimental data is not un reasonable and may be readily accounted for by varia tions of D or v from the values applying in the german ium body or, perhaps, by inaccuracies resulting from some of the approximations made. The slope of the linear part of the tunnel effect characteristic may be estimated very approximately using (26), (27), and (28). It is about 0.1 ~a/volt, which is in satisfactory agreement with the experimental value of 0.3 ~a/volt. 7. DISCUSSION In view of the results of this paper, we may make the following statements concerning the reverse character istic of high inverse voltage rectifiers: (a) Data taken under dc conditions cannot be inter preted theoretically, except possibly for voltages less than 5, unless account is taken of thermal effects. A calculation in which this were done would be of doubtful value as rather drastic assumptions regarding dissipa-10,000 .-----,-,---,-r--r-,,-:;-:[;:""" 1.000 1----1-+-/-=-+-v.~--¥t---r- ~ 100 I-".i~~---j...t!!--+---:;~+"-'-'-Tt----jr- .~ I ." 100 10 I 200 15 20 V 1/2 10 v2 x 10-4 V ANO v3 X 10-5 I • 300 FIG. 9. Various plots for rectifiers showing tunnel effect. A B, C, D--Plots of log(i-i,), for first unit, against Vi, V, P, and V~, respectively. E-Plot of }og(i-i,) against P for sec~md unit showing decrease in slope owmg to the charge of conductron band electrons. tion of heat would be involved. The easier and more fruitful approach is to make the experimental measure ments under pulse conditions. Such data are susceptible to theoretical interpretation. (b) For most ordinary rectifiers and for some very good ones, the increase in current under high reverse voltages may be ascribed to lowering of the barrier caused by image force. The field responsible for this lower ing results chiefly from the geometry of the system. The differences between ordinary rectifiers and very good ones in this group are apparently the result of differences in barrier height which in turn are probably the result of differences in purity of the germanium, although surface states may also playa part. (c) For some very good rectifiers the increase of current seems to be cheifly the result of lowering of the barrier caused by tunnel effect, and Schottky effect is almost completely masked. The main field in this case is also that due to the geometry of the point and thus varies with applied voltage, but a large fixed field is apparently also present because of the existence of large numbers of positive holes in the inversion region near the point. The linear region which occurs in the pulse characteristic of a unit of this type is largely owing to the fact that this fixed field is much larger than the variable field at low voltages. There is a possibility that a fairly large fixed field may exist for some of the better units described in paragraph (b) above. If so, these units should also have linear sections in their character istics. This has not yet been thoroughly investigated experimentally. The true barrier height seems to be greatest for those units showing tunnel effect as expected. The difference between true and "effective" barrier heights becomes greater as the purity of the germanium, and the number of positive holes near the metal, increases. Thus, the [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:3534 J. H. SIMPSON AND H. L. ARMSTRONG increase in effective barrier height with purity of the material becomes small when the number of holes near the metal becomes large. There is some improvement in the reverse pulse characteristic, however, because the mobile charge causes a decrease in field strength at irregularities on the whisker surface and increases the length of the linear section at the expense of the more rapidly rising part. If the whisker point could be made very smooth, operation at very high voltages under pulse conditions would probably be improved by in creasing its radius. This would, of course, increase the current at low voltages because of the increased area but the very rapidly rising tunnel effect portion would commence at a higher voltage. The same remarks prob ably do not apply under dc conditions, however, as in this case it is important to keep the current at lower voltages as small as possible, so that a very small whisker radius is desirable. The agreement between experimental results and theoretical predictions described above is quite remark able and leads us to believe that the basic assumptions made are essentially correct. There are, however, several factors which may modify the conclusions reached in certain cases. The most important of these is the exist ence of areas of low barrier height which has been as sumed in the multicontact theory of Johnson et al.3 Such areas may be important at low voltage especially in poorer units and may account for the low barrier heights of some 1N34's. For good units at the voltages considered here, however, it seems doubtful that they are important, in view of the similarity of results on different units and the agreement with theory. A second factor which may be important is the trap ping of holes near the metal. This has the effect of decreasing the magnitude of the hole velocity and of in creasing the fixed field term of Eq. (17). Further experi mental data are required before the importance of this factor can be assessed. It seems possible to explain the results so far obtained without resorting to this mech anism. A lowering or change in thickness of the barrier of the type described here will be produced by an increase in the number of holes near the metal, resulting from an injected hole current, so that one of the mechanisms mentioned may play a part in transistor operation. Since the collector barrier in a transistor is probably lower than that of a crystal rectifier, Schottky effect should be the dominant factor. Further discussion of this subject is beyond the scope of this paper, however. In conclusion, it may be observed that the experi mental results described herein are in remarkably good agreement with the theoretical predictions. While part of this agreement may be fortuitous, there seems to be little doubt that the basic concepts discussed are correct. In fact the .results obtained seem to confirm exception ally well the ideas concerning metal-semiconductor barrier layers that haye developed during the past few years. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 136.165.238.131 On: Tue, 23 Dec 2014 17:16:35

1.1747293.pdf

Classification of Spectra of CataCondensed Hydrocarbons John R. Platt Citation: J. Chem. Phys. 17, 484 (1949); doi: 10.1063/1.1747293 View online: http://dx.doi.org/10.1063/1.1747293 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v17/i5 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions484 JOHN R. PLATT same symmetry class. Each computed level properly refers to the center of gravity of a singlet and triplet of the same kind. The energies are expressed in units of {3 (Coulson's 'Y), where {3, the resonance or bond integral corrected jor overlap,' must be determined empirically. For accurate comparison with experiment, we must know the triplet levels. In naphthalene, one of these, 3La, is known (Fig. 2), and the other, 3Lb, can be estimated from the esti mated singlet-triplet separation in benzene16 and from the position of 1Lb• In azulene, it seems not unreasonable to assume the same singlet-triplet separations as for the corresponding levels in naphthalene. The computed values for the two lowest states are compared with the observed values in Table III. With {3=22,200 cm-1 in naphthalene, both centers of gravity are predicted within 100 cm-1. With the same value of {3 in azulene, errors of 4500 cm-1 are found, but a lower value of {3, 18,500 cm-I, will fit the data within 200 cm-1. The comparison of the computations on the upper states is quite crude since only the upper singlets are known in either molecule. The computed centers of gravity are on Fig. 2, using a {3 of about 23,000 cm-1 for both' molecules. (This figure is taken from reference 12, where this value of (3 was used in another comparison.) For each state, the deviations of computed levels from observed sing lets are in the same direction and of about the same size in both molecules, if the assignments given are correct. The method of making these assignments 16 C. C. J. Roothaan and R. S. Mulliken, J. Chern. Phys. 16, 118 (1948). THE JOURNAL OF CHEMICAL PHYSICS is discussed in detail elsewhere.12 This qualitative agreement tends to support them. No intensity computations were made here. I t should be noted that the values for {3 for azulene and naphthalene obtained from these spec troscopic considerations are not in agreement with the corresponding {3's obtained from thermochemi cal (e.g., heats of combustion) measurements of resonance energies. For example, the {3spect'S for azulene, benzene, naphthalene, and anthracene are respectively 18,500 em-I, 20,600 cm-I, 22,200 cm-I, and 24,300 cm-1. The corresponding {3's, calculated from heat-of-combustion resonance energies,16 are (10,000 cm-1?), 13,400 cm-I, 14,000 cm-I, and 14,000 cm-1. No explanation of these discrepancies is offered, and their investigation is desirable. Here we have not compared the observed energy levels in azulene with the HLSP computations of Sklar.5•13 Although his predicted frequencies for the lowest singlet-singlet transitions for a number of compounds agree well with the observed position of the first absorption bands, the spread in his excited electronic levels is much too large in azulene as well as in benzene and other molecules, and the sequence of his predicted levels is wrong, for ex ample, in benzene. He attributes the unduly large spread to his neglect of ionic structures, which are known to be essential for describing excited states by the HLSP method.l3 The LCAO description which Mayer and Sklar applied later to benzene17 is much more satisfactory in predicting excited states,16 and a modification of the latter approach has been used here for that reason. 17 M.G. MayerandA. L.Sklar, J. Chern. Phys. 6, 64S (1938). VOLUME 17, NUMBER 5 MAY. 1949 Classification of Spectra of Cata-Condensed Hydrocarbons JOHN R. PLATT Physics Department, University of Chicago, Chicago, Illinois (Received November 1, 1948) The classification of 1I'-orbitals in a cata-condensed aromatic system is like that of the orbitals of a free electron traveling in a one-dimensional loop of constant potential around the perimeter. To take into account electron interaction, certain quantities corresponding to angular momenta may be added or subtracted. Introduction of the cross-links in the molecule removes the degeneracy. The first excited configuration in such systems gives two low frequency singlet weak absorption bands and two higher singlet strong dipole absorption bands. Selection and polarization rules are given. The levels are identified from the spectra and some of their properties are determined. An explanation is given of the regularities found by Klevens and Platt. A systematic nomenclature is given. The results agree qualitatively with LCAO theory, can be applied easily to unsymmetrical molecules, and can possibly be extended to other types of ring systems. I. THE FREE-ELECTRON MODEL THE aromatic spectra which were extended into the vacuum ultraviolet in the preceding paper1 (hereafter called I) show many empirical resemblances and regularities. In benzene, the LCAO (linear-combination-of-atomic-orbitals) mo- 1 H. B. Klevens and J. R. Platt, J. Chern. Phys. 17, 470 (1949). Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsCLASSIFICATION OF SPECTRA 485 lecular orbital theory satisfactorily explains the excited levels.2•3 In the linear condensed-ring sys tems, or polyacenes, only the centers of gravity of singlet-triplet pairs have been calculated;4 and in tensity predictions are too large, especially for the low-frequency bands,! probably because electron interactions have not been included. The LCAO theory has not yet been applied to account for the regularities in the spectra of the unsymmetrical condensed ring systems. (Complete and up-to-date bibliographies of other theoretical and experimental work on aromatics will be found in I and in refer ences 2, 4, 7, 12, 13, 16, 19,20,25, and 27.) Another molecular orbital approach-the free electron orbital method-will give less accurate but more rapid results, if simplifying assumptions are made. It can describe electron interactions in a simple way. Without introducing parameters to be determined empirically, it will give the approximate height of the excited states. It gives correct polariza tions, accounts for the general weakness of the low frequency bands, and can easily be extended to unsymmetrical molecules. It explains the regu larities observed in I, and leads to a useful new classification for the electronic states. Empirically similar bands in different compounds are given the same symbol, instead of different symbols as they are in theories which emphasize symmetry con siderations. The method assumes that the 1r-electrons of a planar conjugated system are free to move along the bonds throughout the system under a potential field which is, in first approximation, constant. This description was used to explain the diamagnetism of aromatic molecules by Pauling5 and Lonsdale. 6 Schmidt7 generalized it further by thinking of the conjugated system as simply a large flat box con taining a Fermi gas of unsaturation electrons (1r-electrons) in analogy to the two-dimensional metal model of graphite. He discussed excited states and spectra, but without any systematic correlation with the known levels. The idea of almost-free motion along the bonds has always lain in the back ground of molecular orbital discussion. 8-11 Lennard J ones9 coined the term "mobile electrons" to indi cate this motion of the 1r-electrons throughout the whole molecule. Hiickel8 used complex molecular 2 C. C. J. Roothaan and R. S. Mulliken, J. Chern. Phys. 16, 118 (1948). 3 M. G. Mayer and A. L. Sklar, J. Chern. Phys. 6, 645 (1938). 4 C. A. Coulson, Proc. Phys. Soc. 60, 257 (1948). 5 L. Pauling, J. Chern. Phys. 4, 673 (1936). 6 K. Lonsdale, Proc. Roy. Soc. IS9, 149 (1937). 7 O. Schmidt, Zeits. f. phys. Chern. 47B, 1 (1940), and previ- ous papers. 8 E. Huckel, Zeits. f. physik. 70, 204 (1931). 9 J. E. Lennard-Jones, Proc. Roy. Soc. ISBA, 280 (1937). 10 R. S. Mulliken, J. Chern. Phys. 7, 369 (1939). 11 C. A. Coulson and G. S. Rushbrooke, Proc. Carnb. Phil. Soc. 36. 193 (1940). orbital eigenfunctions in treating benzene and re lated problems. These represent electron waves traveling around the benzene ring. He also used "orbital ring quantum numbers" describing angular momentum, as we shall do here, and showed how they add and subtract to give a "total ring quantum number" which characterizes the state of the system. However, the LCAO method with real co~fficients has usually been regarded as the best starting point in molecular orbital theory. Bayliss!2 has recently returned to the idea of the Fermi gas in one dimension to explain the spectra of polyene chains. Kuhn!3 has announced that a modification will predict the strong allowed frequency in poly acenes; he makes no mention of the other transi tions classified in the present paper. In the present approach, it is assumed that the 1r-orbi tals retain their main 1r-electron property, i.e., a node in the'molecular plane. This quantiza tion perpendicular to the plane of the molecule is assumed to be independent of the quantization in the plane. The closed-shell IT or single-bond elec trons are assumed to be more tightly and locally bound than the 1r-electrons and cannot be treated by a free-electron approximation. They will not be further discussed here. The cata-condensed ring systems, whose general formula is C4n+2H2nH, include some of the most important aromatics and carcinogens. In them no carbon atom belongs to more than two rings and every carbon is on the periphery of the conjugated system. This makes possible a further simplification. Postulate: The classification of 1r-orbitals in cata condensed systems is like that of the orbitals of a free electron traveling in a one-dimensional loop of constant potential around the perimeter. The use of the perimeter is a convenience in visualization because of the importance of angular momentum in electron interactions and in selection rules. This postulate amounts to asserting that the wave equation for 1r-electrons is approximately separable in three coordinates, one along the perimeter, one per pendicular to the plane of the molecule, and one perpendicular to these two at the perimeter, and, further, that the main difference of the orbitals frorn each other is with respect to quantization in one of these coordinates only, i.e., along the perimeter. To find the perimeter-free-electron orbitals and energies, the perimeter may first be distorted into a circle of the same length. The orbitals are those of a plane rotator. The energies are then E=q2h2/2mI2= 1,21O,000q2/12, where q is an integer, 0, 1, 2, ... ; h is Planck's constant; m is the mass of the electron, I is the 12 N. S. Bayliss, J. Chern. Phys. 16, 287 (1948). 13 H. Kuhn, J. Chern. Phys. 16. 840 (1948). Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions486 JOHN R. PLATT FIG. 1. One-electron sta tes ; and shells. length of the perimeter (in angstroms), and E is the energy (in cm-I) measured upward from the constant potential. The levels are quadratically spaced as shown in Fig. 1 and are all doubly de generate except the lowest, since electrons that have a finite velocity may travel either clockwise or counterclockwise around the loop. The orbital ring quantum number g, which measures angular momentum, determines the num ber of nodes of the wave function and the selection rules. It may be thought of as a vector perpendicular to the plane of the molecule. In the zig-zag perim eter, angular momentum is no longer a consta.nt of the motion, because of the constraints, but lmear momentum is. When the periodic potential due to the atoms is introduced, g no longer describes accurately any momentum. But it remains a good quantum number because. it stilI determines the number of nodes around the perimeter. In many-electron systems, the vector q's of the different electrons may be added and subtracted algebraically and not vectoriaIIy,. since they ~re restricted to one dimension-to gIve a total nng quantum number Q for the system.I4 Th~s proce~ure gives a qualitative model for electron mteractlOns similar to the vector model for representing these interactions in a diatomic molecule. II. NAMES AND CHARACTERISTICS OF STATES Shells, Configurations, and States In cata-condensed systems of n rings, the 2 (2n+ 1) carbons each bring one 7r-electron to the system. These will fill the successive shells as shown in Fig. 1. The highest filled shell will ~e fo~ q = n, and we wiII designate the 4 electrons m thIS shell as I-electrons, those in the next lowest shell as e-electrons, and so on. The first empty shell will be the g-shell, with g = n+ 1, the next, the h-shell, a~d so on. Since the "optical" electrons are those m the last one or two filled shells, it will be convenient to have a notation which is the same for these top shells in molecules of the same general type even when the ring quantum numbers are different. 14 I am indebted to Prof. Mulliken for suggesting the names, "orbital ring" and "total ring quantum number" for q and Q. Huckel (see reference 8) used the symbols, k and K, for these quantities. In the states of interest, the total momentum number, Q, may take on the values, 0, 1, 2, .... Such states we may designate by A, B, C, .... We will see that it may also take on the values· .. , 2n, 2n+1, 2n+2,.·· .. These states we may designate ... , K, L, M, ... , the letters being kept the same regardless of the value of n. In the perimeter-constant-potential approxima tion all of the states are doubly degenerate except the A state since the total momentum may be in either direction. Introduction of the crosslinks and of the periodic potential due to the atoms alo~g the perimeter removes the degeneracy except m some states of highly symmetric molecules such as benzene and triphenylene. Because of its removal, the states will be split. The two components may be given subscripts a and b, whose significance we will examine later. The most important spectra will be electron-hole spectra, produced like those of the rare-gas atoms by the excitation of one electron from a filled shell to an empty one. Spectra produced by the excitation of two electrons will be weak, as in atoms, and will require higher excitation energies. The excited el:c tron interacts with the remaining tenants of Its former shell, or, more simply, with the hole it leaves behind. Thus, when an electron goes from an I to a g orbital the new system has states with the I and g ring q~antum numbers added or subtracted, i.e., with Q = (n+ 1) ±n = 1 or (2n+ 1). These are Band L states. Removing the degeneracy, they become Ba, Bb, La, Lb. The electron and hole spins may be anti parallel or parallel, giving two sets of excited states, singlets and triplets. This makes 8 type.s of states for this j3g configuration. All singly eXClted configurations will have 8 states.* The Pauli prin ciple makes the ground state singlet and single .. The lower configurations, and the states to whIch they give rise, are given in Table I. . The configurations have been grouped accordmg to their arithmetic sum of momentum numbers, ~q. 6l:q (from ground state) 3 2 o TABLE I. Configuration {' .... .f3i · ... e'f4h · . d'e4f4g { ... Ph · . e~f4g · .. .f3g · .. .f4 States 1.3Da,b 1,'Na,b 1,'Da,b 1,'La,b I, 'Da,b I, 'Ja,b 1,'Ca,b 1,'Ma,b 1,3Ca,b 1,3Ka,b 1,3Ba.b 1,3L •. b IA * Note added in proof; This assumes ~hat degem;rate stat~s are counted twice. There is also the obvIOUS exceptIOn of eXCI tation from q=O states as in the e3f4g configuration of benzene. Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsCLASSIFICATION OF SPECTRA 487 2345234 log ~ 2345234 60000 40000 , "- L. " 20000 'A " " 0 "-.... " .... .... 'L. 0:) 0:::0 OXOO:XX:O ft --:L 2"'+30 1120+30 11211-.& -ta-311 J. 'L. I .. . .. Iforb] (allowedl 'A LCAO obs obs LCAO OJ <0 's • ...... .: 'C. " ...... - 'a. 'L, ,,- / GZ"'I u-/ :;r-- 'L. / 'L. RZ·'3 "'1. / ,/ ,/ '''' .,.. .., (forll) 'A LCAO obI ObW 0:::0 FIG. 2. Electronic energy levels of polyacenes and isomers. Comparison of LCAO center-of-gravity predictions with observed singlets in naphthalene, azulene, anthracene. Note: Lowest vibrational level shown. Remaining structure of each electronic band omitted for clearness. Length of horizontal lines indicates logEmax of transition from ground state on scale at top of the figure. Lines for triplets drawn to left to indicate logEmn is less than 2. In a large system with almost equally spaced one electron states, the configuration energies will be grouped the same way. In smaller molecules with increasing spacing near the j-shell, the e--'>g absorp tion bands will lie at lower frequencies than the j--'>h bands. The first ultraviolet bands will therefore be of j--'>g type and. the next of e--'>g type. Such groupings may be seen in the energy levels of Fig. 2. Higher bands become weak, and only the Rydberg series bands, which fall outside this 1r-electron classification, can be seen at very short wave lengths. Empirically, the order of states of a given con figuration is determined by the Hund rule, which holds for molecules as well as atoms.15,a Triplets lie below singlets; and within the singlet group, or the triplet group, states of high Q lie below low Q states. Thus in the states in Fig. 2, which have been identified by this scheme as described below, one 3L always lies below any lL; and the lL's always below the lB's. Selection and Intensity Rules The selection rules are as follows. 1. Singlet-triplet strongly jorbidden. This is the familiar rule in light atoms and in molecules con taining light atoms. Triplet states in aromatics are known only from phosphorescence and from absorp tion with very long paths. l6·G. Herzberg, Molecular Spectra and Molecular Structure. I. Diatomic Molecules (Prentice-Hall, New York, 1939), (a) p. 360, (b) pp. 136, 264. 2. ~Q must be odd in centrally symmetric systems. This will be justified under Rule 3. ~Q= 1 is always allowed and strong-the dipole transition when the perimeter is a circle. The higher values of ~Q would be forbidden multi pole transitions if the perimeter were a circle. They would be allowed but weak, as in polyenes,1O,12 if it were a double straight line. They will therefore be generally weak, but their exact strength will depend on the molecular shape. Very high ~Q values, near 2n, will be practi cally forbidden, but may become stronger through vibrational interaction, like the forbidden bands in benzene. Thus, in the observed spectra (see I), the lA -lB transitions have oscillator strengths from O.S to 3.0. The lA -lLb transitions in benzene as well as in the other molecules have oscillator strengths near 0.002; and the lA -lLa, near 0.1. 3. ~'l:,q must be odd in centrally symmetric systems. This is equivalent to Rule 2 in the molecules we are discussing. In centrally symmetric systems, the states may be divided into "even" and "odd," or "g" and "u," depending on whether the electronic wave function is symmetric or antisymmetric with respect to reflection in the center of symmetry. Even states have 'l:,q even; odd states have 'l:,q odd. The proof of Rule 3 is therefore the same as the proof of the selection rule evenf-todd, even~ I~even, odd~l~odd; the procedure for this proof is like that indicated for diatomic molecules by Herzberg.15h Vibrations destroy the symmetry, and in several Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions488 JOHN R. PLATT symmetric polyacenes, what seem to be even-even tr~ns~tions .(1:4 _ICb) have been found (see Fig. 2) wlth mtenslties 1/5 to 1/2 as strong as in the un symmetric isomers. Large values of A};q become more and more improbable by the Franck-Condon principle, because they begin to involve large changes in electron velocity. By correspondence-principle, AIq gives the order of the harmonic in the analogous classical flat metal oscillator. A};q = 1 corresponds to the fundamentals, A};q = 2 to the first harmonics, and so on. The Polarization Diagram At this point we examine the meaning of the subscripts a and b, which are related to the polariza tion of absorption bands. If the perimeter were a circle, the states ABC , , , . .. would correspond to };, A, IT, ... of a plane rotator configuration. A transition A~B corre sponds to };~IT, and so on. The possible polariza tions of such a transition are the same as those of the corresponding one-electron transition, (f~7r. They are determined by the location of the nodes of a 7r-orbital. This leads to a convenient diagram for deter mining polarization of absorption from the ground state. Draw the perimeter line. Indicate the atom positions, and mark where the line is cut by planes of symmetry. Expand the line to a circle. For the transition between the ground state and a state of momentum number Q, draw the nodes for a free electron standing-wave orbital of momentum num ber q = Q. Since the orbitals are doubly degenerate, there will be two independent sets of these nodes. Each set will have 2Q nodal cuts across the perim eter. The nodes of one set will lie at antinodes of the other set. Where there is a symmetry axis, they are easy to construct. In one set the nodes will lie on the axis, in the other set the antinodes. We will call that set, a, which proves to have a nodal line cutting crosslinks (see Fig. 3) when the perim eter is returned to its normal shape; b, that set in which the crosslinks are uncut. The polarizations of transitions from the ground state will be the same as if we located + and - point charges at alternate antinodes, shrank the perimeter back to its normal shape, and determined the direction of the resulting electric moment. If we normalize the sum of all + charges to +t, and the sum of all -charges to -t, the size of the electric moment in this diagram will give a rough qualitative idea of the transition moment integral, or effective dipole length Z (Q in the usual notation). From this we can estimate the oscillator strength,16 f= 1.08XlO-6vGZ2cm/N 16 R. S. Mulliken and C. A. Rieke, Rep. Prog Phys VIII 231 (1941). . . , FIG. 3. Polarization diagrams for low absorption bands of representative molecules. where v is the frequency of the absorption band (in wave numbers) ; where G is 1 for a non-degenerate state, 2 for a doubly-degenerate state' and where Z is the dipole length (in A). ' This method of estimating Z and f amounts to replacing ClJFod'lr exe) in the transition moment in tegral by a real charge density, p, which has the same symmetry properties and which represents the corresponding transition in a one-electron system. A normalization of p to +! and -! would corre spond to approximating the vi's by two normalized orthogonal point functions which each take th~ absolute value l/(n)t at n points in the molecule. However, since the effective charges are not so localized, but are distributed, the effective dipole lengths are shorter by about a factor of 2, and this can be taken into account by normalizing the p's to + t and. -t as indicated above. Thus, for a long polyene chain transition with one node we would have effective charges of +t and -t ;t each end of the chain, effective dipole length t the chain length. Bayliss gives this dipole length from the transition moment integral as 0.21 times the chain length.12 Similarly in benzene, the one-electron transition moment integral for an f~g transition i~ a cir~ular ring i.s t the diameter, and the polariza tlOn dlagram usmg the rules above would give about the same value for the effective dipole length. Perhaps it should be emphasized that such a polarization diagram is a device for visualization and not a classical model. It is not suited for accu =ate inte~sity predictions, since they require proper mtroductlOn of the total wave functions in the ~ra~sition moment integral; but it will give polar lzatlOns correctly. Several of these diagrams are Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsCLASSIFICATION OF SPECTRA 489 shown in Fig. 3 for some important transitions in anthracene, phenanthrene, and azulene. The two highly allowed IA -IB transitions are always polarized along mutually perpendicular axes in the molecule. It is these two transitions which go over by the correspondence principle into the classical fundamental electrical oscillations of a fiat metal plate, which were used as analogies by Lewis and Calvin.11· 18 Classically, i' would vary in versely with Z in the above formulas, and the intensity of each of these transitions would be proportional to the molecular diameter, suitably averaged, parallel to the transition axis.lO Empiri cally, according to the assignment of Fig. 2, the longitudinal transition, IA -IBb, lies at the longer wave-length, as does its classical counterpart. The IA _IG bands, as shown by Rule 2 and as confirmed in Fig. 3, will be forbidden in centrally symmetric systems. The polarization diagram for the lA _1Gb band in long, non-centrally-symmetric systems will presumably be cut by roughly parallel nodal lines transverse to the length, as IA -IBb was, and as shown in Fig. 3. If the system has no center of symmetry, but has an axis of symmetry, as phe nanthrene and azulene do, the transition, IA _1Gb, will then be allowed and will have a moment parallel to the symmetry axis. In all long bent systems, this moment will probably be roughly transverse to the long dimension, as in phenan threne. The moment of IA -IGa will be perpendicu lar to the symmetry axis, if any, or roughly longitudinal in long bent systems. The lA -IL bands will have the following special property. In their polarization diagram, the number of nodes on the perimeter is 2Q=2(2n+l), which is just the number of carbons on the perimeter, so the nodal separation is just the mean carbon carbon distance. In the diagram for IA -ILa, every peripheral bond will be cut by a node and we must put the effective charges on the atoms. For IA -ILb, the atoms are at nodes and the effective charges go on the bonds. It is seen from Fig. 3 that in these transitions the elementary dipoles are nearly can celed within each ring, regardless of gross molecular shape, so the total intensity will be very small. Other transitions of high AQ are weak for the same reason. For hexagonal ring systems, when some bonds are just parallel or perpendicular to a symmetry axis, the IA -ILa moment will be parallel to these bonds, the IA -ILb moment perpendicular. In azu lene, which has an odd number of carbons in each ring, the a and b naming is ambiguous by the· rule given above, but we can continue to draw the effective charges at the atoms in IA -ILa• As may 17 G. N. Lewis and M. Calvin, Chern. Rev. 25, 273 (1939). 18 G. N. Lewis and J. Bigeleisen, J. Am. Chern. Soc. 65, 520, 2102, 2107 (1943). be seen in Fig. 3, this interchanges the polarizations in the two L bands from their directions in the isomer, naphthalene (whose polarization diagrams are like those of anthracene). In the classical macroscopic fiat plate analogy, the intensity of these very high multi pole lA -IL transitions must approach zero, as they would correspond to the oscillations of a line of microscopic dipole charges around the edge of the plate. They should therefore not correspond to the Lewis and Ccrlvin fundamental vibrations; Coulson4 empha sized that this correspondence did not hold for the lowest frequency bands, which are the ones identified here with lA -IL. Height of Levels The average height of a configuration should be given by the energy of the one-electron jump from the ground configuration. It is easily shown that this would predict an average frequency of absorp tion almost inversely proportional to the perimeter length. Actually, the frequency of the strongest band varies about inversely as the 0.3-0.5 power of the length (see I). The same difficulty, that the predicted frequency varies too rapidly with length, is found in the free-electron theory of polyenesl2 and in the LCAO theory of polyeneslO and of polyacenes.4 Addition of the periodic potential in the free-electron model may improve the predic tionsP The arrangement of le\'els of a given configura tion is given by the H und rule, as noted above. The particular advantage of the free-electron model is that it may be extended easily to unsym metrical cata-condensed systems. For all systems of a given number of rings, the perimeter is the same, the number of crosslinks is the same and the area is the same. Therefore the location of the levels should be approximately the same in isomers. This is seen in the spectra of I for isomers having the usual 6-carbon rings. The levels of azulene, with a 5-carbon and a 7-carbon ring are systematically lower than those of its 6-carbon isomer, naphthalene. Electron Density The possible importance of high local charge density in determining reactivity and potency in carcinogenic hydrocarbons such as benzanthracene, dibenzanthracene, and their derivatives, has been emphasized by Schmidt,6 and quantitative pre dictions of density variations have been made by Daudel and Pullman and others.19 The free-electron model with suitable boundary conditions will give alternating charge densities in successive bonds near the end of a polyene chain, though Bayliss 19 For bibliography, see P. Daudel and R. Daudel, J. Chern. Phys. 16, 639 (1948). Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions490 JOHN R. PLATT did not point this out. However, the perimeter model gives uniform charge density for condensed systems, if they are unsubstituted. Introduction of crosslinks will produce local density variations of Schmidt's type. Substitution on the perimeter will introduce a perturbing potential, and cause mixing of the wave functions and further variation of charge density. In this way, the free-electron model could probably be extended to account for directing effects in benzene rings and for similar effects in the carcinogens. I t is unfortunate that the terminology proposed here uses two letters, A and B, in different senses from their use in group theory. But spectroscopy has taken its toll throughout the alphabet and there seems to be no other series of consecutive letters with fewer conflicts. Especially, it seemed wise to avoid the established angular momentum letters in either English or Greek, since the case here is so different. It is hoped that the different subscripts will help to avoid confusion with the group theory designations. III. CORRELATION WITH SPECTRA In I the levels of 17 cata-condensed systems were identified according to the scheme just outlined. Levels of a few important molecules are given again in Fig. 2. Justification of this system of naming must be in two parts: (a) justification of giving the same names at all to levels of different compounds; and (b) justification of the particular names chosen, since they now imply definite proper ties. Subsequently some general support from LCAO theory, and some ambiguous identifications, will be discussed. (a) Empirical Identity of Levels in Different Compounds The following general results are derived from I; details may be found there. 1. Levels of a given type have unambiguous idetttify ing characteristics, which do not change from com pound to compound. The most useful characteristics are intensity and vibrational sharpness of the transition from the ground state to the level in question. There is almost no mixing or confusion of properties. 2. Levels of a given type move in a systematic way from compound to compound. All lie on smooth and almost parallel curves as a function of length. They generally lie near the same energy in isomers. 3a. Each empirical type occurs once and only once in each compound. 3b. The number of lower levels remains constant from compound to compound up to any particular upper level as far as the spectra have been carried. There are a few reasonable exceptions where de generacies occur or where a weak band is expected to be overlapped by a stronger band and its upper state cannot be located accurately. Levels that resemble each other so much in different compounds must have some important properties in common. The same name should be given to all of them to indicate these properties. Actually, according to any present theories, in cluding the free-electron method used for the interpretation here, we should expect more change than there is in the band characteristics, such as the "forbiddenness" of the IA -ILb bands, in going from benzene to the unsymmetrical 4-ring systems, and we should expect more mixing and loss of identity of band types than there is in the spectrum of any of these larger systems. Nevertheless, the empirical evidence for distinctness and preservation of identity of band types seems fairly convincing. We can proceed only by taking that evidence at face value in justifying the names and predicting the properties of the different types of states. (b) Justification of Names There is a band in benzene at 39,000 em-I. LCAO theory shows the upper state is a singlet, probably of symmetry B2u,3 though B1u is not ruled out.2 Assuming B2u is correct, the polarization dia gram must have nodal lines through the atoms. This kind of state we have named ILb• If anything like it exists in the larger molecules, it will also be ILb• There is just one similar band found in every aromatic molecule, except where it is ex pected to be hidden by a stronger one. It always has nearly the same intensity, logE"V2.5, and the same kind of sharp vibrational structure running to about 6 bands, as in benzene. The upper states of these bands, when plotted against the number n of rings, lie on a smooth curve starting with the 39,000 cm-1 state of benzene. We therefore call them ILb, and hope that we have guessed correctly the com mon property. Similarly, by LCAO theory, the benzene 48,000 cm-1 state is IBI .. ,3 though IB2u is not ruled out.2 Assuming the former, our diagram must have nodes between atoms. This state lies on a smooth curve with similar levels in the longer molecules. All transitions to these levels have 5 or 6 diffuse bands with logE"V3.8. They are therefore lLa, according to this scheme. The scheme predicts transverse Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsCLASSIFICATION OF SPECTRA 491 polarization for lA -lL" in polyacenes (Fig. 3). This has been found in naphthalene and anthra cene. 4. 20 This correct prediction is an argument for the correctness of our assumption that lLb has symmetry lB2u in benzene and that lL" has sym metry lBlu. These should be together in benzene and split in the other molecules studied in I. The benzene 55,000 cm-l transition is the strongest in benzene. By LCAO it is allowed and of type lAlg_lE lu, which corresponds to our definition of lA -lB. It lies on a smooth curve with the strongest transitions in the longer compounds, when plotted against length. With increasing length, these bands have an increasingly sharp first peak followed by two or three weaker diffuse bands. The intensity of transi tions on this curve increases with molecular length as expected. In naphthalene and anthracene these transitions are longitudinally polarized. 4, 20 They are at the same wave-length as the strong bands of polyenes of the same molecular length (see I). Therefore they have all the properties expected of the 10ngitudinailA -lBb bands. In each of the compounds except benzene another strong band lies higher. Its intensity does not in crease much with molecular length. It can therefore be the transverse lA -lBa, which should have this property. lCb,lC a A band appears between the two lB bands in every compound but benzene. (A band which may be similar appears above the B band in unsym metrical alkylbenzenes.) It is doubtful in naphtha lene, but clear in anthracene, about as strong as lA -lLa• Its intensity increases in the non-centrally symmetric isomers azulene and phenanthrene, be coming about as strong as lA -lBa. Therefore it is possibly forbidden in centrally symmetric mole cules. Among the low transitions, this behavior is expected of lA _lC and lA -lK bands. The intensity when it is strong is more like the dipole lA -lB than the multipole lA -lL so it is probably lA _lC. Since no similar band appears lower, but one appears higher in the 4-ring systems, the band between the two 1 B bands can be assigned to lA _lCb and the higher one to lA -lCa. Presumably the lA -lK bands lie lower but are weaker than lA _lC (and also forbidden in centrally symmetric molecules), and are concealed by the strong lA -lBb bands or by lA -lLa• 3La The phosphorescent level has been assigned to 3Blu (our 3La) in benzene3 but 3B2u(3Lb) has not 20 R. N. Jones, Chern. Rev. 41,353 (1947). been ruled out.2 In the whole group of molecules, its separation from lLa varies smoothly with length, from 19,000 cm-l in benzene to 12,000 cm-l in three-and four-ring systems, and is constant among isomers even when both levels move far. On the other hand, its separation from lLb varies widely, especially among isomers (cf. anthracene-phenan threne in Fig. 2). Since the singlet-triplet separation should be roughly constant, this phosphorescent level then must be of the same type as lLa, and so is called 3La• In turn, this is an argument for the assignment of 3Blu to the phosphorescent level in benzene, if we accept the reasoning above as establishing the lLa level as lBlu in benzene. 2l (c) Identifications and LCAO Predictions Coulson4 calculated, by the LCAO method with out neglect of overlap integrals, the center of gravity positions and the polarizations of singlet triplet pairs in polyacenes. If the center of gravity of the lowest transverse pair of bands, as given by the present assignments, is close to his lowest pre dicted transverse center; and if the observed lowest longitudinal center is close to his predicted lowest longitudinal, then his calculations give a general support to these assignments. This is found to be the case, as was shown in I. Coulson's ignorance of the triplet states and of lLb caused confusion in his original comparison with the data. The upper triplets are unknown, so Coulson's calculations must there be compared with the singlets, and they show as good agreement as they do with the lower singlets. (See Fig. 2.) Thus his predicted total spread of the states corresponding to our f~g system is about 20,000 cm-l in naphtha lene, 30,000 cm-l in anthracene where the total spread of the singlets, according to our identifica tions, is about 27,000 cm-l in both cases. In his upper two singlet transitions, the longitudinal one should be at lower frequencies. This agrees with the present identifications of lBb and lB". His predictions for the third forbidden transition of e~g type in naphthalene and anthr~cene also lie within about 5000 cm-l of the level we have 21 H. Shull at the University of California has also shown from analysis of the phosphorescent band in benzene that it is made allowed by the same types of vibrations as those which would make the lA I. -IBlu band allowed. (Private communica tion.) The two other independent pieces of evidence in the present paper which show its empirical relation to this singlet band are: (a) the almost constant separation of the phos phorescent band in benzene and the larger molecules from IA-IL., which we have concluded is probably lAl._IB lu in benzene; and (b) the good agreement between the center of gravity of this pair of bands in the different molecules and the LCAO calculations-any other pairing would destroy the agreement. Nevertheless, R. S. Mulliken points out that we cannot conclude with certainty that the phosphorescent state in benzene should be labelled 3Blu until the theoretical spectro scopic combining properties of corresponding singlet and triplet states have been established. Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions492 JOHN R. PLATT named correspondingly, 1Gb• (See Table I.) The two lower forbidden transitions, our IA -IK bands, should also be weaker according to the present arguments, and have not been found. Mann's LCAO calculations on the azulene levels22 also support the identifications, as shown in Fig. 2. The predicted centers of gravity deviate from the singlets in the same direction and about the same amount as the corresponding LCAO predictions deviate in naphthalene. A mistake in identification would have been revealed by large changes in these deviations. Because of the reversal of polarizations from naphthalene to azulene, Mann's results also confirm the correctness of the nodal properties assumed here in the polarization diagrams for the L bands. In azulene, IA -ILb should be transverse, IA -ILa longitudinal, as indicated by the diagram in Fig. 3. These bands are identified at 14,000 em-I and 28,0'00 em-I, respectively, by the usual empirical resemblances. Now the LCAO calculation also gives the low singlet-triplet pair as transverse, the next pair as longitudinal. The separation of the pairs is great enough that the sequence of the singlet levels could not reasonably be interchanged from the LCAO-predicted sequence of the pairs, if we re member how accurate the LCAO energy predictions are in the polyacenes (I). Therefore, if the LCAO has predicted these polarizations correctly, then the properties assumed here for the L states also predict them correctly. The failure of the LCAO predictions of intensities does not much weaken the im portance of the good agreement on energies, because intensities are known to be generally much more sensitive to the accuracy of the wave functions than the energies are. ( d) Possible Misidentifications There are three bands described in I in which there is some question about the identifications according to the present scheme. The weak band called 1Gb in naphthalene, and presumed to be forbidden, seems to be still weaker or has moved in acenaphthene, which is essentially an alkyl-substituted naphthalene. One would expect it to be somewhat stronger. On the other hand, the difference in strength is not much larger than the experimental error. The band called IA _1Gb in naphthacene, and presumed to be forbidden, becomes stronger in bent 4-ring systems, as it should. However, the band we have called IA _1Gb is also stronger in chrysene. This molecule again has a center of symmetry, so that this band should again be forbidden. If there is not a mistake in identification, perhaps some 22 D. Mann, J. R. Platt, and H. B. Klevens, J. Chern. Phvs. 17. 481 (1949). . interaction with allowed transitions nearby will account for the larger intensity. On almost any theory, the IB state in triphenyl ene, which has D3h symmetry, should be single and doubly degenerate. However, two strong bands have been found with about the usual IBb-IBa separation, about 10,000 em-I. The higher band must be of some type other than IBa, but it is not certain what. These points deserve further study. It does not seem likely that they will invalidate the classifica tion, since the latter is supported by nearly a hundred other identifications in which the band properties and their variations among isomers are correctly predicted. IV. COMPARISON WITH THE LCAO METHOD: BENZENE The free-electron method will also be justified by its close basic correspondence with LCAO method in the construction of orbitals and states, in cases where the latter method has been worked out. The comparison will be made for two such cases: ben zene and anthracene. In benzene, one free electron in the loop of perimeter l = 8.4A would have energies E= 17,000q2 em-I. No arbitrary constants are involved in this for mula. These one-electron states are plotted in the left half of Fig. 4. The quadratic spacing with interval ratios 1: 3: 5 agrees qualitatively with LCAO calculations which include overlap integrals. The latter give the in terval ratios 1: 4: 5, as shown on the right in Fig. 4.23 If the LCAO bond integral, -(3, is set at 28,000 em-I, the energies agree within about 10 percent with the free-electron energies. However, a smaller value of (3, about 20,000 em-I, fits the observed st<!-tes better.2 The LCAO theory predicts only four one-electron states, with the top one single, where free-electron model predicts an infinity of higher states, all doubly degenerate. The difference is due to the restriction of LCAO to combinations of atomic p-orbitals. The additional free-electron states corre spond to combinations of d-andf-orbitals and so on. The difference is unimportant in practice since the observed spectra involve only the lowest excited states. On the free-electron model, the classification of the lower states in the six-7r-electron benzene prob lem is as given in Table I I. Excitation of an electron from the f-shell to the g-shellieads to states with Q=2-1 and Q=2+1, according as the electron and hole q-vectors are 23 C. c. J. Roothaan, private communication. Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsCLASSIFICATION OF SPECTRA 493 added or subtracted. These have angular mo mentum 1 (B-state) and 3 (L-state) and would correspond to 11-and ~-states, in diatomic notation, with respect to the 6-fold axis. Because of the high symmetry the B-state is not split, but the L-state is so this configuration gives three singlet and three triplet states. Four of these six excited states are known as shown in Fig. 2. The group-theory classification is fairly well established2•3 except for the triplet whose identity seemed to be established by the arguments of the last section. The position of the other levels has been estimated.2 The forbidden lA -IL bands are supposed to become somewhat allowed through vibrational interactions of the proper symmetry. In the strong allowed transition, the polarization diagram method gives an effective dipole length, Z, about 1/4 of the greatest diameter, or 0.7A. This gives J=0.59. This is in fair agreement with the experimental value, 0.79 for the integrated J-+g transition intensity.24 Because of the crudeness of the method, only the relative intensity estimate as compared to anthracene (below) is of significance. The relative agreement is fairly good. The LCAO method predicts J=2.35.16 When this is corrected by a standard empirical factor of 0.3, it is in fair agreement with the observed value. The center of gravity of the levels of the f3g system in benzene is estimated 2 from the spectrum to lie at about 44,000 cm-1 which may be compared with the free-electron prediction of 51,000 cm-I. V. COMPARISON WITH THE LCAO METHOD: ANTHRACENE For anthracene, with one free electron on an ideal perimeter loop of length 1 = 19.6A, the energies will be E = 3150q2 cm-I. Again this quadratic spacing agrees well qualita tively with the increasing spacing found in Coulson's LCAO calculations4 (Fig. 5). TABLE II. Configuration Free-electron states Group theory notation fB 'B lEI" 3EI" .. .f3g IL 3La IBI" 3BI" IL: 'Lb lB," 'B2" .... j< IA IAI• Coulson's orbitals are divided into two groups: without (P, Q) and with (R, S) a longitudinal nodal line; and each of these into two sub-groups: without (P, R) and with (Q, S) a transverse nodal 24 J. R. Platt and H. B. Kleve~s, Chern. Rev. 41. 301 (1948). E em" 180 110000 80000 q h 3-__ f3 ___ ·4.0 9 2 o o o '1.33 40000 ;; ----. 80 ~ ____ -..!.ll- Free-electron lCAO FIG. 4. Comparison of free-electron and LCAO states and shells: benzene. line across the center of the molecule. The free electron q = 0 orbital with no nodes corresponds to his Pl. The free-electron degenerate q= 1 with two nodes on the perimeter corresponds to his two non degenerate orbitals, QI with nodes at the sides, and RI with nodes at the ends of the perimeter. Simi larly q = 2 wi th 4 nodes on the perimeter corresponds to his P2 and SI; and so on. The number of nodes for corresponding levels is the same in the two systems, and the only difference is in the removal of the free-electron degeneracy. The surprising thing is not that the degeneracy is removed, but that the splits are not wider. Thus in naphthalene and anthracene, the LCAO levels are arranged in order of increasing free-electron q; and crossovers occur first in naphthacene. (Coulson's levels for anthracene are accidentally degenerate, as shown in Fig. 5.) By choosing Coulson's LCAO factor 'Y = 23,000 cm-I, the energy predictions of the two schemes agree as shown in Fig. 5. All but two of the 7 excited free-electron states lie between their split LCAO counterparts. The maximum separation of any free-electron state from the center of gravity of its split counterparts is 7000 em-I. The lower states of anthracene would then have the correspondence in the two systems shown in Table III. The identification of the ol;>served states by this system in Fig. 2 differs from Coulson's identifica- Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions494 JOHN R. PLATT E 160000 120000 80000 40000 h g /--- 5.-4.0 6---( 5 ---------P·R;2.21 U .L Q3s.-1.33 4 0 p. -46 g g: 3 . R. 37 8 2 ___ --=:::::::::?=.!~= 0'5, .78 ______ ~ P'R l03 _..:::::::=....-_ Q,'1.33 ee _---..IL..lB 1 Free-electron LCAO FIG. 5. Comparison of free-electron and LCAO states and shells: anthracene. tion. Coulson knew only the two singlet bands 26,000 cm-l (experimentally transverse) and 39,000 cm-l (experimentally longitudinal), and naturally identified the latter with his longitudinal N -V2• The identifications of Fig. 2, setting 39,000 cm-l as his R2-S2 state and the hidden 28,000 cm-l as V2, are in much better agreement with his energy pre dictions, as was shown in I. If this is correct, his intensity calculations on N -Vl are about ten times too high, and those on N -V2 are st;veral hundred times too high.l,4 Con figurations involving his P 3 state actually make only weak transitions to ground. The qualitative success of the present model would imply that electron in teractions, and the mixing of transverse R2 -P 3 with transverse Q2-S3, and of longitudinal Q2-P3 with longitudinal R2-S3, may have to be considered in order to get the correct intensities by LCAO theory. The free-electron prediction of about 22,000 cm-l for the center of gravity of the f3g states (Fig. 5) is to be compared with its experimental location near 30,000 cm-l• The frequencies predicted by either method become too low in long chains, as remarked above. For intensities, the effective dipole lengths may be estimated from the polarization diagram, giving the values shown in Table IV. The lA -lBa transi tion is across the molecule and its dipole length is correspondingly near that of benzene, about 1/4 the width, or 0.7A. The lA -lBb transition is along the molecule and its dipole length is taken as 1/4 the chain length. The lA -lLa band has a polarization diagram with nodes between the atoms, as seen in Fig. 3. If we think of the end rings as having balanced charges, the remaining center dipole with 1/7 of the charge will give an effective transverse dipole length of about Ht·2.8A) or about O.IA. The lA -lLb has a polarization diagram with nodes at the atoms, as seen in Fig. 3. This would give a zero moment if all the effective charges were equal and the rings were perfectly hexagonal, but will give a small longitudinal moment otherwise. The known polarizations agree with these assign ments, and the sequence of intensities is correctly predicted. More accurate calculations do not seem justified without introducing the periodic potential and connecting links, and the vibrations, especially in computing the multipole bands. It is strange that the benzene lA -lL transitions should have the same intensities as their counter parts in larger, and even in unsymmetrical, mole cules. In the latter there are many mechanisms variations in bond distance and electron density, the crosslinks, and so on-by which even multipole transitions should gain intensity and "become allowed," just as Coulson expected for the L bands. It is similarly strange that these benzene bands have much the same diffuseness or sharpness as in the larger molecules. It is usually supposed that lA -lLa is moderately strong and diffuse in benzene, and lA -lLb is weak and sharp, because the former is nearer the allowed transition and more perturbed, or possibly because of predissociation in the former. But lA -lLa gets far away from strong allowed transitions in the other molecules and remains moderately strong and diffuse, while lA -lLb comes between it and the strong allowed transitions in pentacene, yet remains weak and sharp. It seems improbable that predissociation would occur just TABLE III. Anthracene: names of states and polarizations of absorption bands abbreviated. Free- electron Coulson Polarization states notation Symmetry of transition Configuration (Singlets and triplets) species to ground . . d'e'f'g D • Q1-S2 B2a trans Db R1-S2 B1a long Jb Q1-P, B1. long J. R1-P, or V. B'a trans .. . e'i'g c • P2-S2 B'g forb Cb 51-52 A1g forb Kb p,-p, or V. A'g forb K. Sl-Pa or Va B,g forb . . . . pg B • Q,-S, B,. trans B. R,-S, B,. long L. Q2-P,orV2 B1a long L. R,-P,or V, B,. trans ....... .f' 'A N A'g Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsCLASSIFICATION OF SPECTRA 495 at the lLa level in all these molecules, especially since this level moves more widely than any other. Evidently there is more in common in corresponding bands than current theory will allow. This is useful for identification but puzzling. Possibly all these sources of extra intensity in the larger molecules are negligible, like the ones used in the rough calculations of Table IV, which pre dicted too low L-band intensities. If this were so, the only important contribution to intensity in these bands would always be the result of vibrations, just as in benzene. Then, since vibration amplitudes and charge densities must be much the same in com pounds with similar force constants and masses and 1I"-electron densities, we would have an explanation of the constancy of the L-band properties. In benzene and anthracene, we see that the free electron model does not contradict the LCAO method in principle at any point. The only practical divergence is on predicted L-band intensities, but this may disappear when electrori interaction is included in the LCAO treatment. No doubt by introducing the periodic potential and making vector-model-type estimates of the size of the electron interactions in the free-electron scheme, we could improve the quantitative predictions and determine the sequence and location of the excited states. This might mean a considerable increase in the ease and accuracy with which we can describe the spectra of unsymmetrical systems. VI .. SPECTRA OF OTHER TYPES OF MOLECULES The free-electron model can be adapted to classi fying other conjugated ring and ring-chain spectra. The kind of classifications required may vary with the type of topological connectedness of the con jugated system. In simple polyene chains, for example, the levels of the first excited configuration reduce to our lBb and aBb, with selection rules and intensities depending only on the bending of the chain and its dimensions. For ring-chain systems and peri-condensed sys tems (in which some carbon atoms belong to three rings), study of the spectra25•26 shows that in many cases the lowest bands still seem to be of Band L types. However, there is more mixing of characters, both vibrational and intensity, just as is expected in the larger cata-systems (Fig. 8 of I). Identifica tion of types is correspondingly less certain in many of the spectra. We can account for the existence in these mole cules of L states and other states giving weak multipole absorption bands, by means of a theorem of Coulson and Rushbrooke.lO This applies to sys- 26 E. C1ar, Aromatische Kohlenwasserstoffe (Springer-Verlag, Berlin, 1941). ' 26 H. B. Klevens and J. R. Platt, unpublished results. TABLE IV. Estimated oscillator strengths in anthracene. fest fobs Polar- (free- Transition ization Z elect) (from I) lA-lB. trans .7A 54,000 0.30 0.65 lA-lBb long 1.8A 39,000 1.4 2.28 lA-lL. trans .1A 26,000 .003 .10 lA-lLb long .OOA (28,000) .0000 (.002) (phenan threne) tems containing rings with only an even number of carbon atoms. By LCAO theory with neglect of overlap, the theorem says that the occupied and unoccupied one-electron states may be classed in pairs with equal and opposite binding energy. Call the members of any pair, a and a'. If we translate this into the language of classical interacting oscil lators, state a corresponds to a particular phase difference between adjacent oscillators. Call this difference (11"/2) -a. Then state a' corresponds to phase difference (1I"/2)+a. Similarly, translating into free-electron terms, the phase difference in the Schrodinger wave between the two ends of a bond will be (11"/2) -a in state a, and (1I"/2)+a in state a'. When an electron jumps from state a to a', the electron and hole momenta in the two-dimensional network may be added or subtracted. This gives two kinds of states, with total momenta corre sponding to a phase difference between adjacent carbons of either 2a or 11". On the polarization dia gram, an absorption band to the state with 2a has only a few nodal lines across the molecule, i.e., it is a dipole or a low "multipole." An absorption band diagram to the state 11" will have nodal lines either in the middle of every bond, in which case it is like lA -ILa; or at the ends of bonds, in which case it is like lA -ILb. Thus it seems reasonable that the first few singlet bands of the more complicated systems can again be called lA -ILb, lLa, lBb, lBa. If mixing of char acters is strong, a more exact treatment, or perhaps a dIfferent kind of approximation, will be required. Probably the first four distinctive bands in the spectrum of pyrene25 are of these four types. In aniline and styrene there are also four prominent bands.26 In diphenyl,26 only three are seen, but the fourth weak one reappears in fluorene, which is a bent alkyl-substituted diphenyl so this weak band may simply be hidden in diphenyl itself. I must express my especial debt to Herzfeld27 for his emphasis on the importance of the nodes, and to Mulliken, Longuet-Higgins, and many other colleagues for helpful criticisms and discussion. Note added in proof: Several very recent applica. tions of the free-electron method are described in a footnote at the end of I. 27 K. F. Herzfeld, Chern. Rev. 41, 233 (1947). Downloaded 18 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

1.1698794.pdf

Nuclear Magnetic Resonance in Metals. II. Temperature Dependence of the Resonance Shifts B. R. McGarvey and H. S. Gutowsky Citation: The Journal of Chemical Physics 21, 2114 (1953); doi: 10.1063/1.1698794 View online: http://dx.doi.org/10.1063/1.1698794 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/21/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Temperature dependence of polymer-gel dosimeter nuclear magnetic resonance response Med. Phys. 28, 2370 (2001); 10.1118/1.1410124 Temperature dependence of 195Pt nuclear resonance chemical shifts J. Chem. Phys. 61, 2985 (1974); 10.1063/1.1682446 Nuclear magnetic resonance study of the transition metal monoborides. II. Nuclear electric quadrupole and magnetic shift parameters of the metal nuclei in VB, CoB, and NbB J. Chem. Phys. 60, 2310 (1974); 10.1063/1.1681364 Nuclear Magnetic Resonance in Molten Salts. II. Chemical Shifts in Thallium Halide—Alkali Halide Mixtures J. Chem. Phys. 42, 631 (1965); 10.1063/1.1695983 Temperature and Pressure Dependence of the Co59 Nuclear Resonance Chemical Shift J. Chem. Phys. 39, 3349 (1963); 10.1063/1.1734200 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:142114 MIZUSHIMA, OHNO, AND OHNO changing the value of the intermolecular potential, his result may contain a numerical error. Miyak09 used the same potential for all kinds of collisions as in Boo(') (T), and obtains the value of B(T) for the case of para :ortho= 1: 3. As -114A3 at 2SoK and -128A3 at 200K this is larger than the result for pure para hydrogen, while the experimental data shown in Table VI seem to indicate that B(T) of pure para hydrogen is larger than that of ordinary hydrogen. This contradiction will be improved, if the dependence of the intermolecular potential on the rotational quantum THE JOURNAL OF CHEMICAL PHYSICS number is taken into account as pointed out III Sec tion V.22 ACKNOWLEDGMENTS The authors wish to thank Professor M. Kotani and the members of his laboratory for their useful sugges tions and economical help. They also wish to thank Misses N. Mizushima and N. Sugawara for performing numerical calculations, and Mr. Grayson for assistance with English. 22 In the theory of viscosity also, the effect of statistics is known to have the wrong sign (E. W. Becker and V. Stell, Phys. Rev. 87, 525 (1952». VOLUME 21, NUMBER 12 DECEMBER, 1953 Nuclear Magnetic Resonance in Metals. II. Temperature Dependence of the Resonance Shifts* B. R. McGARVEyt AND H. S. GUTOWSKY Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois (Received July 16, 1953) The temperature dependence of the nuclear resonance shift in metals has been investigated in lithium, sodium, rubidium, cesium, and gallium. The resonance shifts were found to change by no more than 5 to 6 percent over temperature ranges of 200°, including the melting point. For sodium, the observed temperature dependence of the resonance shift is correlated directly with the volume dependence predicted theoretically for. the mass magnetic susceptibility due to the conduction electrons. In the other metals, effects appear which are related apparently to the volume dependence of the wave functions for electrons at the top of the conduction band. The room temperature resona.nce shifts of Sn1l7 and SnU9 and the temperature dependence of the Rb87 line width are also reported for the metals. The resonance shift of tin is 0.705XID-2. The estima tion of activation energies for self-diffusion from the temperature dependence of the line widths is discussed. INTRODUCTION A PREVIOUS paper! was concerned primarily with the magnetic resonance absorption line shapes and widths and their temperature dependence in several metals. In most of the metals the resonance line was broadened significanbly by the interaction of the nu clear spins with the conduction electrons, the interac tion also determining the short spin-lattice relaxation time. In the case of lithium and sodium, magnetic dipolar broadening was important in the solid at tem peratures below about 2S0oK, with self-diffusion nar rowing the resonance line at higher temperatures. The main part of this paper reports measurements of the temperature dependence of the resonance shifts in lithium, sodium, rubidium, cesium, and gallium. The resonance shifts in metals were first observed by Knight2 and have been attributed 3 to the magnetic interaction between the nuclei and the conduction * Supported in part by the U. S. Office of Naval Research. t U. S. Atomic Energy Commission Predoctoral Fellow. Present address: Department of Chemistry and Chemical Engineering, University of California, Berkeley 4, California. 1 H. S. Gutowsky and B. R. McGarvey, J. Chern. Phys. 20, 1472 (1952). 2 W. D. Knight, Phys. Rev. 76, 1259 (1949). 3 Townes, Herring, and Knight, Phys. Rev. 77, 852 (1950). electrons. The resonance shifts have been expressed by Townes, Herring, and Knight as (1) where tJ.H = H c-H; H co the higher field, is the reso nance field, at a given frequency, for a compound of the metal and H is the resonance field for the metal; M is the weight in grams of an atom of the metal; Xm is the contribution of the conduction electron spins to the mass susceptibility of the metal, and < I ~k(O) 12)F is the value for the square of the conduction electrons' wave functions at the nucleus, averaged over the top of the Fermi band. The resonance shifts have been measured for quite a few metals!-6 at room temperature, but, except for a preliminary reportT on Na23, no investiga tion of the effect of temperature upon the resonance shifts seems to have been published. In principle, the effect of temperature upon the reso nance shifts may be obtained by considering in Eq. (1) the two terms which are responsible for the shift and 4 D. F. Abell and W. D. Knight, Phys. Rev. 85, 762(A) (1952). • W. D. Knight, Phys. Rev. 85, 762(A) (1952). 6 H. E. Walchli and H. W. Morgan, Phys. Rev. 87, 541 (1952). 7 H. S. Gutowsky, Phys. Rev. 83, 1073 (1951). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:14NUCLEAR MAGNETIC RESONANCE SHIFT IN METALS 2115 its magnitude. To a first approximation, both Xm and (l1fk(O) 12)F are independent of temperature. There is possible a slight dependence upon atomic volume of each of these terms, which could give rise to an indirect dependence of the shift upon temperature. The small changes observed in the shift with temperature are evidently the net result of the volume dependence. The interpretation is complicated by there being two terms, neither of which is known independently. In addition to the temperature dependence of the resonance shifts in several metals, we report the reso nance line widths and shifts observed for Sn117 and SnU9 at room temperature, as well as the temperature dependence of the Rb87 line width. The line widths in tin are of interest in regard to possible mechanisms for the broadening of the resonance lines in metals. All of the other metals observed have nuclear spins greater than! so that quadrupole coupling of some sort might contribute to the line widths. But the spins of Sn117 and SnU9 are! and quadrupole effects cannot exist. Nonetheless, the observed line widths in tin exceed the predicted dipolar broadening, proving the importance of other than quadrupole effects. EXPERIMENTAL Most of the apparatus and procedures have been de scribed.t·8 All of the measurements were made with the larger of our permanent magnets, at a field of 6365±2 gauss. The sample of tin was a fine powder with an oxide coating providing sufficient insulation for use without further treatment. The other metal samples were those used in 1. The room temperature line widths and resonance shifts in tin and the temperature dependence of the Rb87 line width were measured with a super-regenera tive spectrometer.8 This system can be operated at high rf levels with a decent noise figure and is useful for samples such as the metals which have a short spin lattice relaxation time T1• The two tin isotopes with magnetic moments are of low abundance (Sn1l7 -7 .54 percent, Snu9-8.62 percent) and their resonances could not be detected in either the metal or salt with any of our apparatus other than the super-regenerative spectrometer. The resonance shifts for Snm and SnU9 were obtained by measuring the resonance frequencies, at constant magnetic field, of the metal and the salt. The metal and the salt samples were placed alternately in the same rf coil and the oscillator tuned to give a maximum output from the narrow-band amplifier.8 The system detects the dispersion rather than the absorption observed with most setups, so the maximum in the derivative of the resonance line is its center. The resonance frequencies were then measured with a BC-221 Signal Corps fre quency meter set by reference to WWV. Care was exer- 8 Gutowsky, Meyer, and McClure, Rev. Sci. Instr. 24, 644 (1953). cised to insure that the carrier frequency was measured and not one of the side bands. In I, the line width oH was taken to be the separa tion of the two peaks of the derivative of the resonance absorption line. However for rubidium the absorption line became too weak for accurate measurement below room temperature, because the line broadened. So the derivative of the Rb87 resonance dispersion was plotted several times at each temperature with the super regenerative spectrometer, and the width of the center peak at half-maximum intensity was determined. In the next section these values are related to the customary absorption line widths. The Rb86 resonance was not observed ·because its low resonance frequency offsets its high isotopic abundance to give a weaker signal than Rb87• The constructional details have been given else where8 for the cryostat used in these line-width measure ments and in the temperature dependence of the reso nance shifts. The temperature dependence of the resonance shifts was observed with a dual rf bridge system activated by the same signal generator. Two identical metal samples were used; one was placed in an rf probe at room tem perature while the other was in the cryostat, the two samples as close together as possible in the magnet gap. A switch enabled the output of either bridge to be con nected to the detection system. The resonances from the cryostat and the reference were observed alternately and the difference in applied field required for each was measured as a function of temperature. For lithium, sodium, and gallium the absorption lines were displayed on an oscilloscope, while for rubidium and cesium the derivative of the absorption line was observed with the narrow-band amplifier and the line center taken as the point midway between the minimum and maximum. The measurements are reported in terms of the quan tity o(AH/H) which is defined as follows: o(AH) = H26°C-H T, H H260C (2) where H 25'C is the applied field for the resonance in the sample at 25°C and HT is the resonance field for the sample at temperature T, the radio-frequency remaining constant. The temperature of the sample in the cryostat was measured with a copper constantan thermocouple immersed directly in the sample. RESULTS AND DISCUSSION Line Width and Resonance Shift in Tin The resonance shift was measured at room tempera ture with respect to a saturated aqueous solution of SnCb, which was 0.2 molar in MnCb. The paramagnetic Mn+2 ions were added to reduce T 1 so the super regenerative system would not saturate the resonance. The resonance in the solution required the narrow-band amplifier for detection even though the line was narrow. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:142116 B. R. McGARVEY AND H. S. GUTOWSKY I.~r---'-------r------'----' :z: '" 1.0 o.~ D IA.P. o.oL-.,..!IDl-:..o-----l-.m-:c~~h~-.t-ur-.-·-K------,3-hoO.-------' FIG. 1. Temperature dependence of the magn~tic .reso~ance ab sorption line width for Rb87 in the metal. aH IS given In gauss. The points plotted as squares were obtained from the resonance dispersion, as described in the text. The results are: for Snl17, (~H/Hc)=0.701XIO--2,and for Snus (~H/Hc)=0.709XIO--2. The error is estimated to be ~ 1 percent of the measured value, and within this error the resonance shifts are the same for the two iso topes. The metal sample was the (3 form, i.e., white tin, as confirmed by x-ray analysis.s The derivative of the resonance dispersion in the metal was plotted for both isotopes at room tempera ture giving curves similar in general appearance to that'shown elsewhere8 for eu63• However, the discus sion in I is based on the second moment and width of the resonance absorption. When the dipolar broadening is small, the absorption has the Lorentzian shape f(~H)=4Tl[1+16TTl2(~H)2J-1, (3) where ~H is the magnetic field separation from the line center. The absorption and dispersion are related by the expressionlo 1 f+OO f(~H)d(~H) P(~Hl)=- ,(4) '/r --'fJ ~H-~HI where P(~Hl) is the dispersion shape function. The width of the dispersion derivative at half-maximum, 8Himax, is given via Eqs. (3) and (4) in terms of the width of the absorption derivative 8H by the ex- pression (5) The theoretical dipolar second moments ~H22 of the absorption lines for both Snl17 and SnUB are about 0.06 gauss.2 The experimental 8H values for Snl17 and Sn1l9 obtained through Eq. (5) are 2.5 and 3.0 gauss with uncertainties of ±0.5 gauss. The relation between 8H and ~ 22 ·depends upon line shape; in the extreme case of a rectangular shape, IJH = 2VJ~H2 so the maximum IJH possible in tin from dipolar broadening is 0.9 gauss, with 0.5 gauss a more likely value. Even allowing for the g We are indebted to Walter E. Thatcher of the department's x-ray laboratory for the analysis. . 10 G. E. Pake and E. M. Purcell, Phys. Rev. 74,1184 (1948). error introduced by assuming a Lorentzian shape in computing 8H from the observed IJHlmax, the resulting values are nearly double the maximum dipolar width. TI values basedl upon the observed resonance shift are 1.4XIQ-4 and 1.3XIQ-4 sec. If only TI broadening were present, the line widths computed from the equa tionl T 1= 1/2VJ'/r8H would be 0.4 and 0.5 gauss. In the other metals the observed nondipolar broadening of the resonance exceeds that predicted from the resonance shift by a factor of about two. Adding the Tl and di polar broadening, the line widths should be in the range of 1 to 2 gauss, which, in view of the indirect procedure used, agrees reasonably well with experiment. More over, white tin is tetragonal and anisotropy in the resonance shift with crystal orientation might also contribute to the line width. Temperature Dependence of the Rb87 Line Width Absorption line widths of Rb87 were measured at several temperatures above 3000K and the results re ported in I. The dispersion line widths have since been measured at lower temperatures with the super-regen erative spectrometer. The absorption lines were nearly Lorentzian in shape, 1 so IJH values can be estimated with Eq. (5) from the observed IJH!max data. Both sets of data are given in Fig. 1; the 8H values computed from the dispersion curves are indicated by squares while the directly observed 8H values are plotted as circles. The data join very well, indicating the general validity of the proced\lre. The rubidium resonance line is broad in the liquid like that of cesiuml and is similar to lithiuml and so dium7 in having a transition below the melting point. But at lower temperatures it does not decrease like cesium. The width at the higher temperatures and in the liquid state is due mainly to the short spin-lattice relaxation time TI as is the case in cesium, while the greater width at the lower temperatures must be due to the contribution of both dipolar and Tl broadening. The dipolar contribution to the line width should be _._" -EQUATION e 4. .. _ .......... ---EOUATION 10 >00 "0 4 TEMPERATURE -K FIG. 2. Temperature dependence of the resonance shift for Nail. The broken lines represent the theoretical predictions of Eqs. (9) and (10) for this metal. The solid circles are data reported pre viously in reference 7. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:14NUCLEAR MAGNETIC RESONANCE SHIFT IN METALS 2117 approximately 0.5 gauss in the rigid lattice, and this is about the magnitude of the change in line width during the transition which occurs below the melting point. Therefore, this transition is evidently associated with self-diffusion of the metal atoms as in sodium and li~ium. The small discontinuous change at the melting pomtll may be the effect upon the dipolar broadening or upon Tl of a cooperative change in the self-diffusion rate. The Tl broadening should decrease at low tem peratures as it did in cesium because of the increase in T 1; however, any line narrowing of this sort in rubidium occurs below the 900K limit of our observations. The combination of dipolar and Tl broadening in rubidium prevents any very good estimate from the line width transition of the activation energy for self-diffu sion such as made for lithium! and sodium.7 The T! broadening is negligible for the latter two metals but is significant in the heavier elements. Moreover, it seems likely that self-diffusion would produce time-dependent field gradients in the metals. If the self-diffusion were fast enough, the gradients would average substantially to zero, but at slower rates there might be some broad ening of the resonance line by quadrupole interactions which could also contribute to TI. Seymour!2 has re~ ported a line-width transition in aluminum, centered at 330°C, but did not obtain very reasonable values for the diffusion processes. The TI, and perhaps quadrupole effects as well, may cause the discrepancies. Overhauser!3 has suggested that the method described by Wert and Marx!4 could be used to estimate activa tion energies for self-diffusion from the temperature TM, at which the resonance line begins to narrow, rather than from its change with temperature. The method assumes a constant factor Vo in the equation v= voe-EalRT 10 -10 r-----------~.----------,-----~/~----. -(QUAT ION 9 ----- - -EQUATION 10 o 00 ~>- ~ ) t M.P. 25'0 00 TEMPERATURE 0" FIG. 3. Temperature dependence of the resonance shift for Rb8? The broken lines represent the theoretical predictions of Eqs. (9) and (10) for this metal. 11 The resonance shift increases upon melting so the line width change cannot result from an increase in T, as suggested in r. 12 E. F. W. Seymour, Proc. Phys. Soc. (London) A66, 85 (1953). 13 A. L. Overhauser, private communication. We wish to thank ~r. O~erh.aus~r for reading the manuscript of our paper and dISCUSSing It wIth us. 14 C. Wert and J. Marx, Acta MetanurgicaJ,~113 (1953). 100r--,------r-------.--~ so -SOL----"l;;,------ ....... 20b.O----------3..!O~O _-.J Temper.ture ~K FIG. 4. Temperature dependence of the resonance shift for CSl33 in the metal. and also requires in this case that there be the same critical frequency of self-diffusion when the resonance line starts narrowing in the different metals. Under these conditions we have Ea=ATM, where A can be evaluated from a known Ea and T M. Using Nachtrieb's directly measuredI6 Ea of 10.5 kcal for sodium and the line width7 TM of 160oK, A is 65.5 cal deg-I. Differences in the critical diffusion frequency are corrected by adding the term RTM In (VNa/VM) to ATM; VNa and VM are the frequency widths of the resonance lines in sodium and in the metal, at TM• In this way, activation energies of self-diffusion of 10.3, 13.9, 14.8, and 29.8 kcal are estimated for CSI33 LF, Rb87, and A127, respectively, from data in thi~ article and in I. The 13.9 kcal for lithium is to be com pared with the 9.8-kcal value found in I from the change in line width with temperature. The 29.8 kcal for aluminum is in better agreement with the expected value12 of 33 kcal than is the 21-kcal value obtained by Seymour. Temperature Dependence of the Resonance Shifts The observed resonance shifts are given as a function of temperature in Figs. 2, 3,4, and 5 for sodium, rubid- r M.P. o -. 300 350 400 TEMPERATURE, -K . FIG. 5. Temperature dependence of the resonance shift for Gall !n the metal. These da~a were ob~aIned with the setup described In refer~nc~ 7. The pOint at 298 K was obtained in the super cooled lIqUId; the resonance has not been detected in the solid. 16 Nachtrieb, Catalano, and Weil, J. Chern. Phys. 20, 1185 (1952). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:142118 B. R. McGARVEY AND H. S. GUTOWSKY ium, cesium, and gallium, respectively. In the case of lithium, no measurable difference was found between room temperature and 200°C, outside the experimental error of ±2X 10--6, and this includes any possible change occurring at 186°C, the melting point. Each point on the sodium, cesium, and gallium plots is the average of several separate measurements, while for rubidium each measurement is plotted. The rubidium data are given in this manner to demonstrate more graphically the apparent realness of the peak in the curve in the region of 254 OK. The broadness of the rubidium resonance below 2200K and of the sodium resonance 7 below 2000K precluded measurements of significant accuracy; however, an observation of rubid ium at 800K showed that its resonance shift below 2200K follows the general trend indicated by the solid line in Fig. 3. The gallium resonance could not be found in the solid below the melting point, 303°K, probably because of quadrupole broadening in the tetragonal lattice. The earlier measurements for sodium above room temperature are plotted as solid circles in Fig. 2, and they agree well with the present data. The previous results at lower temperatures7 were inaccurate because of an incomplete correction for changes in the field at the sample caused by the materials of the cryostat itself. The proton resonance in hexane was used to determine the correction; however, possible differences in the glass sample containers were overlooked. In these experiments the latter were found to be important below room temperature. The corrections are a signifi cant fraction of the observed effects only for the sodium measurements below room temperature. From the data given, the resonance shifts are seen to depend only slightly upon the temperature; the maxi mum change over the temperature ranges investigated is about 5 or 6 percent of the total shift. The smallness of the temperature dependence accounts for the ap parent absence of an effect in lithium; in it the resonance shift is so small that the expected change with tempera ture is within the experimental error. It was pointed out in the Introduction that the observed temperature effects express the net influence of the atomic volume upon the two terms, Xm and (I ~k(O) 12)F, in Eq. (1). Of the theoretical work on metals, none seems to give any firm indication of how the latter term varies as the lattice constant is varied. Fortunately, more can be said concerning the possible volume dependence of the spin paramagnetic susceptibility Xm. One would expect Xm to increase with an increase in volume. As the volume in which the conduction elec trons are confined increases, the separation of the different energy levels decreases; this increases the energy-level density and gives a corresponding increase in Xm. In this manner the Pauli theory16 for the para magnetic susceptibility of metals predicts that Xm 18 W. Pauli, Z. Physik 41, 81 (1927). should be proportional to VI. In the preliminary ac coune of temperature effects for Na23, the Pauli theory was found to predict a rather smaller change in reso nance shift than that observed. It was suggested that chemical shifts of the resonance in different com pounds of the metal could introduce some uncertainty in the actual value of the conduction electron shift, which might account for part of the discrepancy. Another pertinent factor, which has been noted inde pendently by Overhauser,13 is that the Pauli theory assumes the conduction electrons are free and non interacting. Sampson and Seitz17 have refined the Pauli theory for the alkali metals by taking into account the exchange and correlation energies of the electrons. For Xm they obtain the expression Xm=MJL02/201, where JLo is the Bohr magneton and 1 2.32X 10-20 a= (h2/6m*) (9/3271'2)1 r/ r, (6) (2r.+6.75X 10-8) +1.46XlO-2°r8 • (7) (r8+2.70XlO-8)2 m* is the effective electron mass for the metal, and '. is the radius of the atomic sphere for the metal and is found from the equation (8) where V is the volume of the metal and N is the number of atoms contained in that volume. If the term (I ~k(O) 12)F were independent of volume, the change in resonance shift would be given from the Sampson-Seitz theory by the expression This function is plotted for Na23 in Fig. 2 and for Rb87 in Fig. 3. For comparison the expression which is what the simple Pauli theory would predict, is also plotted for both Na23 and Rb87. Similar plots are not given for CS133 and Ga7l since it is apparent from Figs. 4 and 5 that neither Eq. (9) or Eq. (10) would give a temperature dependence for the shift which agrees with experiment. As can be seen in Fig. 2, Eq. (9) gives quite well the observed temperature dependence of the Na23 resonance shift. In making this plot, the coefficients of volume 17 J. B. Sampson and F. Seitz, Phys. Rev. 58, 633 (1940). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:14NUCLEAR MAGNETIC RESONANCE SHIFT IN METALS 2119 expansion used were Hackspill'sl8 for temperatures above O°C and were Dewar'sl9 for temperatures below O°C. The value used for the percentage volume change at the melting point was that of Losana.2o There ap pears to be a great deal of uncertainty about the latter quantity, with reported valuesI8.2Q-23 varying between the 1.5 percent of HackspilP8 and the 2.53 percent of Endo.22 For all of the various values) Eq. (9) agrees somewhat better with experiment than Eq. (10), but Losana's value of 2.17 percent gives with Eq. 9 almost the exact change at the melting point. The divergence between experiment and Eq. (9) at lower temperatures could be due to Dewar's datal9 being somewhat in accurate. For the other metals the situation is more complex. The Rb87 shift increases with temperature but not nearly as much as predicted by either Eq. (9) or (10), while the CSl33 and Ga71 shifts actually decrease with increasing temperature. The thermal expansion data used in plotting Eqs. (9) and (10) for Rb87 were those reported by Hackspill,18 and the volume change at the melting point was that reported by Losana,2o which agrees well with that of other observers.19.22.23 The theory of Sampson and Seitz for Xm should be particularly good for sodium, so that agreement be tween Eq. (9) and the experimental data furnishes good evidence that in this metal the term < 1 Y,k(O) 12)F is inde pendent of volume. Or alternately, as we shall see later, there is an indirect argument that Eq. (9) is a better representation for Xm of sodium than is the free electron theory. The exact volume dependence of Xm for the other metals will probably not be given by either Eq. (9) or (10), but it is very unlikely that Xm would decrease with increasing volume. Thus it is clear for cesium and gallium that the decrease in resonance shift with in creasing temperature must come about from a decrease in <1Y,k(O) 12)F. The dependence of the latter term upon the volume is a rather difficult question. For one thing, those wave functions which have been computed for metals are poorest in the region of interest to us, that is, at the top of the conduction band. If an electron were perfectly free, the square of its wave function at every point would be inversely proportional to the volume, from 18 L. Hackspill, Ann. Chim. 28, 633 (1913). 19 J. Dewar, Proc. Roy. Soc. (London) A70, 237 (1902). 20 L. Losana, Gazz. Chim. Ital. 65, 851 (1935). 21 A. Bernini and C. Cantoni, Nuovo Chimento 8,241 (1914). 22 H. Endo, J. lnst. Metals 30, 121 (1923). 23 E. Hagen, Ann. Physik 19, 537 (1883). the normalization condition. But the effect of normal ization on the volume dependence of the wave function at a nucleus is not necessarily the same. An increase in the volume over which the wave function is normal ized will always trend to decrease the value of the wave function at most points, but since the nature of the wave function may also change with volume, the change in value of the wave function at a specific point is uncertain. Assuming a free election model, we have the VI de pendence of Xm from the Pauli theory and the V-I effect of normalization upon <I Y,k(O) 12)F. These combine to give a net v-t equation for the resonance shift. This agrees qualitatively with the experimental results for Ga 71 and CS133; however, the predicted decrease in the resonance shift with increasing temperature is about half that actually observed for these two metals. The behavior of the rubidium shift suggests a decrease in <1Y,k(O)12)F with increasing volume but not nearly a V-I dependence, since the experimental data do not fall too far below the curves plotted from Eqs. (9) and (10). To summarize, it appears that <1Y,k(O) 12)p decreases relatively little for sodium, if at all, with increasing volume, more for rubidium, and so much for cesium that the increase in Xm is overcompensated and a net decrease in the resonance shift results. The systematic trend with nuclear charge is probably significant. At least it suggests that any change in < 1 Y,k(O) 12)F with in creasing volume for sodium is probably a decrease rather than an increase. Therefore, the better fit of the data by Eq. (9) implies that the Sampson and Seitz theory is really a better expression for Xm than is the free-electron theory. At present no explanation can be given the peak in the rubidium resonance shift occurring at 254°K. No transitions have been reported at this temperature for any of the other properties of rubidium metal, such as structure, heat capacity, or electrical resistance. ACKNOWLEDGMENT The continuing interest of Dr. C. P. Slichter in our research on metals has been very helpful to us. The direct observation of nuclear relaxation phenomena in metals by his research group has been valuable in our choice of experiments and in the interpretation of our results. Also, we wish to thank Mr. R. E. McClure for his assistance with the measurements on Ga71• This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.63.180.147 On: Tue, 12 Aug 2014 17:10:14

1.1698725.pdf

Paramagnetic Resonance Absorption in Solutions of K and Na in Liquid NH3 Clyde A. Hutchison Jr. and Ricardo C. Pastor Citation: The Journal of Chemical Physics 21, 1959 (1953); doi: 10.1063/1.1698725 View online: http://dx.doi.org/10.1063/1.1698725 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/21/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ultrasonic absorption in liquid Na–K and Na–Rb alloys J. Chem. Phys. 61, 92 (1974); 10.1063/1.1681675 Velocity of Sound in Na–NH3 Solutions J. Chem. Phys. 54, 2279 (1971); 10.1063/1.1675166 Electron Paramagnetic Resonance Study of Solutions of Europium in Liquid Ammonia J. Chem. Phys. 44, 2954 (1966); 10.1063/1.1727162 Paramagnetic Resonance of Color Centers in NaNO3 J. Chem. Phys. 23, 1967 (1955); 10.1063/1.1740630 Paramagnetic Resonance Absorption: Hyperfine Structure in Dilute Solutions of Hydrazyl Compounds J. Chem. Phys. 21, 761 (1953); 10.1063/1.1699023 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER II NOVEMBER, 1953 Paramagnetic Resonance Absorption in Solutions of K and Na in Liquid NHa* CLYDE A. HUTCHISON, JR. AND RICARDO C. PASTOR Institute for Nuclear Studies and Department of Chemistry, University of Chicago, Chicago, Illinois (Received August 2, 1953) Paramagnetic resonance absorption in solutions of K and Na in liquid NHa has been investigated The resonance has been studied at a frequency of approximately 7.0.106 cycle sec! and a magnetic field ~f 2.5 gauss. The measurements have been made at 240oK, 274°K, and 298°K and over a concentration range from 0.004 M to 0.7 M. The intensities and susceptibilities, spectroscopic splitting factors widths shapes and extents of saturation of the resonances have been determined. The paramagnetic partd of the ~agneti~ suscept!b!l!t~es in static ~elds ~ave been calculated from the rf data and have been compared with the susceptlb!htl~s measured III stat.lc ~elds by other investigators. The relatIOns of the trapped electron model of s~ch solutIOns to these quantItatlve measurements on the paramagnetic resonance absorption have been conSidered. INTRODUCTION AS. a result of .extensive in.vestigations1 ?f the phys Ical properties of solutIOns of alkalI metals in liquid NHa, it is clear that the metal is, in these systems dissociated into positive metal ions and electrons, the latter being trapped in the solvent. One of the important lines of evidence leading toward the development of a satisfactory model of these solutions has resulted from the study of their magnetic susceptibilities in static fields. These susceptibilities have been investigated by Freed and Sugarman2 for the case of K and by Huster3 for the case of Na. Measurements were made at 2400K and 2200K in the case of K and at 238°K and 198°K in the case of Na. In the concentrated solutions and down to concentrations of a few tenths molar, the static magnetic susceptibility is quite small, of the order found in the metal. As the concentration is decreased, the molar susceptibility increases. It approaches in dilute solutions something in the vicinity of the value of the susceptibility of an Avogadro's number of free electron spins. The susceptibilities increase with increas ing temperature at all concentrations which were investigated. (In the extremely dilute solutions the susceptibilities presumably obey Curie's law with variation of temperature.) It is apparent from these measurements that increasing the concentration or lowering the temperature results in a pairing of electron spins. This is qualitatively the behavior to be expected of a metal. But not only is the behavior of the magnetic susceptibility not quantitatively that to be expected for a metal, but also the electrical conductivity at the concentrations for which the susceptibilities were * This work w~s assisted by the U. S. Office of Naval Research. 1 For summanes of the results of these investigations with reference to suitable models of such systems and for references to the literature see: (a) C. A Hutchison, Jr , and R. C. Pastor, Revs. Mod. Phys. 25, 285 (1953); (b) C. A. Hutchison, Jr., J. Phys Chern. 57, 546 (1953)· (c) J. Kaplan and C. Kittel, J. Chern. Phys. 21, 1429 (1953)· (d) W. N. Lipscomb, J. Chern. Phys. 21, 52 (1953); , (e) T. L. Hill, J. Chern. Phys. 16, 394 (1948)· (f) W. Bingel, Ann. Physik 12, 57 (1953). ' 2 S. Freed and N. Sugarman, J. Chern. Phys. 11,354 (1943). 3 E. Huster, Ann. Physik 33,477 (1938). measured corresponds to electron mobilities at least 100 times smaller than those of electrons in a metallic conduction band. It was proposed by Freed and Sugar man2 that the electrons were trapped by the solvent and that there were pairwise interactions between these trapped electrons similar to those which lead to the F' centers in crystals in which two electrons are trapped in a single vacancy.4 These pairs would be expected to be diamagnetic and would thus lead to lowered suscepti bilities. This model was elaborated by Ogg5 who postu lated the existence of cavities in the liquid which served to trap single electrons and pairs of electrons. He cal c~lated the radius of these cavities to be lOA. The large SIze of these traps has been the subject of both experi mental and theoretical controversy. More recent detailed calculations by Lipscomb1(d) have shown that the cavities which form the traps would be expected to have sizes about the same as that of an NHs molecule and that such a model is in agreement with the experi~ mental data on the volumes of the solutions. Kaplan and KitteP(c) have considered the cavity trapping model in some detail and have shown that it successfully correlates much of the experimental knowledge of these solutions. Hutchison and Pastor6 first reported the observation of paramagnetic resonance absorption in solutions of K in liquid NHs at microwave frequencies. Observations of the resonance have also been made by Garstens and Ryan7 and by Levinthal, Rogers, and Ogg.8 This resonance was found to be extremely sharp and to have the smallest width of any known paramagnetic reso nance absorption. In fact the width, 0.1 gauss, initially reported by Hutchison and PastorS proved to be caused largely by field inhomogeneities. In view of the intrinsically interesting features of this paramagnetic resonance absorption and because of the importance for an understanding of these solutions of 4 F. Seitz, Revs. Mod. Phys 18,384 (1946). 6 R A. Ogg, Jr., J. Chern. Phys. 14, 295 (1946),14,114 (1946); J Am. Chern Soc. 68, 155 (1946); Phys. Rev. 69, 668 (1946) 6 C. A. Hutchison, Jr. and R. C. Pastor, Phys. Rev. 81, 282 (1951) . 7 M A. Garstens and A. H Ryan, Phys. Rev. 81, 888 (1951) 8 Levinthal, Rogers, and Ogg, Phys. Rev. 83, 182 (1951). 1959 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:381960 C. A. HUTCHISON, JR., AND R. C. PASTOR the detailed magnetic information which such experi ments yield, the investigations were extended over a wide range of temperatures and concentrations. Preliminary results of such experiments with K in liquid NHa were reported in two symposia held re cently.!(a) , !(b) This paper gives a detailed report and discussion of the data on intensities and susceptibilities, spectroscopic splitting factors, widths, shapes, and saturation, for the paramagnetic resonance absorption in solutions of K and N a in liquid NHa, and preliminary results on K in NDa. EXPERIMENTAL PROCEDURES AND APPARATUSES. PREPARATIONS AND ANALYSES OF SAMPLES The solutions of alkali metals in liquid NHa were prepared by distillation, first of alkali metal and then of NHa into Pyrex capsules which were subsequently sealed. The samples were prepared in an evacuated sys tem. The storage vessel for the NHa was made of steel, and it was joined to the Pyrex preparation system proper by means of a copper-to-glass seal. The capsules, 1 cm in diameter and 3 cm in length, were attached to the lower ends of vertical tubes 1 cm in diameter by means of constrictions which facilitated the sealing. Just above every constriction was a side arm, 0.5 cm in diameter and 10 cm in length leading to a bulb in which approximately 0.5 g of metal was placed before the evacuation of the system. The side arms were con structed with five right angle bends. The system was evacuated, the alkali metal was fused and heated to remove occluded gases, and the vacuum system was baked by heating nearly to the softening point. Tank NHa had originally been placed in the steel storage cylinder with a large q~ntity of sodium metal. The hydrogen which was evolved had been released from the cylinder. NH3 from this cylinder was now condensed in a glass bulb containing alkali metal in the preparation system. This was the final step of purification of the solvent. The metal in the bulbs at the end of the bent side arms was then heated and moved along the walls to form mirrors just above the constrictions. The side arms were removed. The walls of the tubes on which the mirrors were located were cooled with a swab which had been dipped in a dry ice-acetone mixture, and the proper amounts of metal were washed down into the capsules. Additional NH3 was distilled into the capsules until the desired levels of liquid were obtained. The contents of the capsules were frozen in liquid N 2 and sealed off from the vacuum system. The capsules were stored at dry ice temperature when they were not being used for measure ments. The con tents of the capsules were anal yzed chemically. The capsules were weighed and placed one at a time in a stainless steel bomb which contained 25 cma of H20. The bomb was placed in hot water causing the capsules to explode. The contents of the bomb were filtered. The residue of broken glass was dried, ignited, and weighed. The filtrate was boiled down to such a small volume that all NH3 was removed. An excess of standard HCI solution was added quantitatively, and the solution was titrated at about 98°C to 100°C with a standard solution of NaOH, using methyl red as the indicator. The standard HCI was usually 0.005 M. Blank analyses were made which indicated the presence of about 5.0.'10-7 mole of alkali, and this was deducted from the determined numbers of moles in the samples. It is believed that the errors were largely those of the titra tion, and that they were equal in all cases to about 0.1 cm3 of 0.005 M HCI or 5.0.10-7 mole. This would be an error of about 10 percent for a 0.005 M solution of alkali metal in liquid NHa or an error of 0.5 percent for a 0.100 M solution. Analyses of known solutions of NaOH mixed with NHa carried through all steps of the analytical procedure except the breaking of the capsules confirmed this viewpoint. The errors introduced by the procedure of breaking the glass capsules were probably much smaller than the errors of titration. The tris-p-nitrophenylmethyl samples, used as stand ards, were made by the reaction between tris-p-nitro phenylmethyl sodium and tris-p-nitrophenylmethyl bromide in an atmosphere of dry N 2. The former com pound was obtained by filtering the dark blue product which resulted on treating a pyridine solution of tris-p nitrophenylmethane with sodium ethoxide dissolved in alcohol. The latter compound was the residue left on refluxing tris-p-nitrophenylmethane with an excess of Br2 for 24 hr and then evaporating the Br2 in a vacuum. The residue was recrystallized from chloroform-ether. The method used was a modification of the procedure of Ziegler and Boye.9 The free radical content of a given sample was determined from the volume uptake of O2 by a weighed portion, 1 mole of O2 being equivalent to 2 moles of tris-p-nitrophenylmethyl. MEASUREMENT AND RECORDING OF THE ABSORPTIONS a. The Static Field Since the resonance was extremely narrow, the width at the points of maximum slope being as small as 20.10-3- gauss, it was necessary to have a static magnetic field that was homogeneous to better than 1.10-3 gauss over the volume of the sample. The resonance was first observed at 23700.106 cycle sec! the associated field strength being approximately 8500 gauss. A field of this value with sufficient homogeneity was not available. The width of the resonance being essentially independ ent of the field strength the investigations were carried out at approximately 7.106 cycle sec!. The associated magnetic field was approximately 2.5 gauss, and it was not difficult to obtain such a field with sufficient homogeneity. Moreover because of the narrowness, the intensity was sufficiently great to permit the use of these very low field strengths. (In the case of large fields and frequencies difficulties were also encountered 9 K. Ziegler and E. Boye, Ann. 458, 248 (1927). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38MAG NET I eRE SON A NeE INS 0 L UTI 0 N S 0 F KIN L I QUI D N H 3 1961 because of the relatively large conductivity of the solutions.) The static field was supplied by a solenoid of circular cross section 96.16 cm long and 12.591±0.002 cm in outside diameter. The solenoid was wound with 33.225 ±0.045 turns per cm as determined by means of a travelling microscope and scale. (33.225 is SO times the average of the reciprocals of the lengths of SO SO-tum sections of the solenoid and 0.045 is SO times the mean deviation from the mean of these reciprocals.) The current for the solenoid came from an electronically regulated supply. A motor driven potentiometer in the supply circuit provided a very slow and nearly linear variation of current with time. The voltage across a resistor in series with the solenoid was measured and recorded for 5 sec at intervals of 30 sec by a recording potentiometer throughout each determination of a magnetic resonance curve. The recording potentiometer was calibrated by comparison with another potentiom eter, and the resistor was calibrated by comparison with a resistor which had been calibrated by the National Bureau of Standards. The recorder chart was calibrated in terms of the output of the other potentiometer immediately after each scanning of the magnetic resonance signal. For K or Na in liquid NHa the current through the solenoid was scanned from 0.054 to 0.070 amp or a resultant field of 2.0 to 2.8 gauss and for tris-p-nitrophenylmethyl, 0.000 to 0.120 amp or -0.6 to 5.2 gauss. The solenoid was mounted so that it could be rotated both horizontally and vertically and main tained in a desired position. The axis of the solenoid was aligned parallel to the magnetic field of the laboratory by means of the resonance signal from a solution of K in NHa. The direction of the solenoid was varied until the difference between the currents at constant fre quency required to produce maximum resonance ab sorption with (a) the angle between the solenoid field and lab field less than 7r/2, and (b) this angle greater than 7r/2 was a maximum. b. The Modulating Field The static field was modulated by means of an auxili ary coil. The auxiliary coil was a short solenoid (2 cm in length) with a single layer of 12 turns. It was located just inside and at the midpoint of the main solenoid. The £enter of the rf coil was located at the center of the auxiliary modulating coil. The auxiliary coil was trans former coupled to a power amplifier whose input signal was a SO-cycle seci sine wave from an audio oscillator. The potential across a resistor in series with the modu lating coil was measured with an ac voltmeter which had been calibrated against an electrodynamometer. The field modulation Hw equivalent to a given voltmeter reading was determined by employing the magnetic resonance of K in NHa. The change in voltage across the calibrated resistor in series with the solenoid which was required to move the resonance frpm one extreme of the modulation to the other at fixed frequency was determined. This together with the geometrical con stants of the solenoid and the calibration of the resistor in series with the solenoid was sufficient to determine the amplitude of the modulation field as a function of the reading on the voltmeter which was across the resistor in series with the auxiliary coil. The range of peak to peak modulation for solutions of alkali metals in liquid NHa was from 0.0025 to 0.0218 gauss at 298°K, 0.0059 to 0.0238 gauss at 274°K, and 0.0198 to 0.0594 gauss at 240oK. c. The Radio-frequency Field The radio-frequency field was supplied by a coil which together with a . variable condenser formed the grid tank circuit of a regenerative oscillator-detector. The coil consisted of 30 turns of number 20 Formex coated copper wire, was 3.0 cm long and 1.27 cm in outside diameter. The capsules containing the solutions were fitted inside this coil. The coil was mounted with its midpoint at the center of the static field solenoid and with its axis perpendicular to that of the solenoid. It was connected by a coaxial line to the remainder of the circuit outside the' solenoid. It was completely sur rounded by a copper box which served as a shield. The frequency of the radio-frequency field was in the range 6.0.106 to 8.0.106 cycle seci and was usually 7.0.106 cycle seci. The voltage across the coil was controlled by varying the regeneration and was measured con tinuously during measurements of magnetic resonance by means of an electronic voltmeter. The loaded Q of the coil and the shunt capacitance C at resonance at the working frequency were determined with a Q meter. The frequency of oscillation was determined with a heterdyne-type frequency meter and the measurements of frequency were found to be accurate to 1 part in 20000. All measurements of magnetic resonances with K or Na in liquid NHa were made with 0.05 rms volt across the coil. At circuit resonance, for a frequency of 7.0.106 cycle seci and a capacitance of 1.43.10-10 f, this potential corresponds to 0.0051 gauss at the center of the coil, and the rf field should be proportional to the coil voltage. At room temperature, for a frequency of 7.0.106 cycle secI, the Q of the coil plus sample varied from 90 to 120 for the range of concentration (0.65 M to 0.004 M) of K in liquid NHa investigated, and the capacitance needed for circuit resonance was about 1.43 .1O-1Of. The Q's obtained using the low-temperature apparatus were about 20 percent lower. With solutions of Na in liquid NHa the Q's were about 10 percent lower at room temperature than those obtained with equiva lent concentrations of solutions of K in liquid NHa. d. The Detecting and Recording System The detected SO-cycle seci audio signal from the oscillator-detector circuit passed through three tuned This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:381962 C. A. HUTCHISON, JR., AND R. C. PASTOR stages of amplification followed by a phase-sensitive detector. The reference signal for the phase-sensitive detector was supplied from the generator for the auxili ary modulating coil. The rectified output of the ampli fication system was fed to a recording potentiometer. By means of an automatic switch the same potentiom eter was used for recording both the magnetic signal and the current through the solenoid. A Watkins PoundlO calibrator was used for determining signal intensities. The regenerative oscillator-detector-ampli fier-recorder system was used to compare the magnetic signal with the calibrator signal produced when a 50- cycle sec! signal of known amplitude was placed on the grid of the calibrator tube at known fixed plate current. The calibrator signal was supplied from the modulation generator and measured with an electronic voltmeter which had been calibrated against an electrodynamom eter. The recorder chart was calibrated in terms of calibrator gird voltage immediately after each scanning of the magnetic resonance signal. In experiments in which the Watkins-PoundlO radio-frequency spectrom eter was substituted for the regenerative oscillator detector, the magnetic signals were not detectably different in shape from those obt.ained with the re generative oscillator-detector. The regenerative circuit was used because of its much greater signal to noise ratio. For all measurements the amplitude of the modu lation of the static field of the solenoid was sufficiently small that the output of the phase-sensitive detector was proportional to the first derivative of the magnetic signal with field strength. The ratio of (a) af calibrator volts necessary to simulate the magnetic derivative signal to (b) the field modulation amplitude, was equal to the slope of the curve of hypothetical dc calibrator grid voltage versus field strength, i.e., to the slope of the curve of magnetic signal versus field strength. THE DATA AND THEIR TREATMENT The data consisted of (a) curves drawn by the recording potentiometer showing the variation of the magnetic derivative signal and of the current through the solenoid with time; (b) meter readings showing the rf coil voltage and frequency, field modulation ampli tude, calibrator ac grid voltage, Q of the rf coil and capacitance across the coil; (c) thermometer and ther mocouple readings; (d) results of chemical analyses of the solutions and standards; (e) data obtained during the calibration of the recording potentiometer, meters, resistors, etc. The magnetic derivative signal and the solenoid current signal were both recorded along the width of the chart of the recording potentiometer. The length of the chart was proportional to time. At the conclusion of each recording of the magnetic derivative signal, 10 G. D. Watkins and R. V. Pound, Phys. Rev. 82, 343 (1951). records of a series of calibrator signals and records of a series of known voltages from another potentiometer were made on the same chart paper. The calibrator signals were produced by known ac grid voltages. Hence, the number of calibrator ac grid volts per nominal in. of chart width was determined. The cali bration against the other potentiometer gave the num ber of volts per nominal in. of width of chart and this combined with (a) the measured slope of the current signal versus length of chart, (b) the calibration of the resistor in series with the solenoid (the voltage across which gave the current signal), and (c) the geometrical constants of the solenoid, determined the number of gauss variation of field at the center of the solenoid per nominal in. of length of chart. The width of the chart, thus, could be converted to a calibrator ac grid volt scale and the length to a magnetic field scale. The length of the chart was divided into small intervals convenient for numerical integration. The areas of these strips were converted to volt gauss by use of the calibration factors measured as just described. SumMations of the volt gauss, corresponding to these strips, up to a given field strength gave a quantity which when divided by the constant field modulation amplitude was the number of hypothetical calibrator dc grid volts that would simu late the magnetic signal (not derivative signal) at that given field strength. A plot of these first integrals up to given field strengths versus the field strengths gave the magnetic resonance absorption line. A second numerical integration gave the areas under these curves of cali brator dc grid volts versus field strength. These second integral areas were proportional, in the absence of magnetic saturation, to numbers of absorbers in the samples being investigated, other things such as sizes and positions of samples, frequency, etc., being kept constant. This may be seen as follows: The calibrator was coupled to the rf coil in such a manner that if the rf resistance of the coil is represented by <R, the voltage placed on the grid of the calibrator was proportional to '~<R/ <R2 where ~<R was the change in <R produced by this voltage. It is also true that if the ~<R was produced by a magnetic resonance absorption, then for a coil with large Q, ~<R/ <R2= -~C/Qrnag where ~ is the ratio of (a) the field energy stored within the boundary of the sample to (b) the total energy stored in the field, and Qrnag is the ratio of (a) the energy stored in unit volume of the field in the sample to (b) the magnetic energy absorbed per unit volume and per radian in the sample. Now since Qmag-! is equal to 471' times x", the imaginary component of the susceptibility, then ~<R/<R2 was equal to -471'~wCxl/. The area obtained as the second integral Jooo (calibrator dc grid Voltage) dH, described above was proportional to Jooo (~<R/ <R2)dH. (The integral Jooo is written assuming that the only contribution to spin resonance absorption was in the region investigated.) Therefore, from the immediately preceding discussion it This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38MAG NET I eRE SON A NeE INS 0 L UTI 0 N S 0 F KIN L I QUI D N H 3 1963 is seen that 1 ~ i~ -f (calibrator dc grid voltage) dHo: X" (H)dH. ~wC 0 0 The experiment was carried out at a constant frequency. For a sharp symmetrical absorption the integral, fooox" (H)dH, for the present experiments may be equated to the corresponding integral, Hm8:.:fooo X (X" (v)/v)dv, for an experiment in which the field is kept constant and the frequency is varied. (Hmax is the value of H in the constant v experiment at which the absorption is a maximum.) But fooo(x"(v)/v)dv is, according to the Kronig-Kramersll relation, just X the susceptibility in a static field (magnetic moment per unit volume divided by field strength). Therefore it is seen that g~ i~ X 0: -- (calibrator dc grid voltage) dH. ~w2Ch 0 It should be noted that this X is just the paramagnetic part of the susceptibility. X is proportional to the num ber of absorbers per unit volume. It is therefore clear that the numbers of absorbers per unit volume in the samples were proportional to (~W2C)-1 times the results of the second numerical integrations of the recorder charts described above. The second integrals for the samples of alkali metal in liquid NH3 were compared with those for the standard substance, tris-p-nitro phenylmethyl, and the proportionality constant was thereby eliminated. Since only the part of the imaginary susceptibility X" caused by the flipping of spins in the static field was observed, it is clear that the static susceptibilities calculated from the rf data were just the paramagnetic parts of the susceptibilities and did not depend upon any diamagnetic effects of the systems. ~ was not determined but instead a quantity R was measured. R was defined as the ratio of the magnetic energy stored within the boundaries of the sample to the energy that would have been stored if the energy density were everywhere the same as at the center of the coil. It will be seen that the previous relation may then be transformed so as to read 1 ioo xmo:-- (calibrator dc grid voltage) dH, mRw2C 0 in which m is the number of moles in the sample and Xm is the molar susceptibility. Calling the .right-hand side of this proportionality A/mRw2C, the static para magnetic susceptibilities of the samples of alkali metals in liquid NH3 were calculated by means of the relation, X = N AvfJ2(_A_) (_A_) -1. f m kT at mRw2C mRw2C I • ---- 11 H. A. Kramers, Atti congr. intern. fisica, Como, 2, 545 (1927); R. de L. Kronig. J. Opt. Soc. Am. 12, 547 (1926). N Av[f2 / kT at is the magnetic susceptibility of the standard substance at the temperature at which its curve of calibrator ac grid voltage versus field was recorded and the subscript st everywhere refers to the standard substance. CALF, the integral from zero up to maximum absorption was actually employed for the comparisons rather than A.) f is a factor which corrected for the saturation. f was determined by measuring the maxi mum deflections of the phase-sensitive detector as a function of rf coil voltage at various concentrations of K in liquid NH3. Since the width and shape did not vary in the region 0.05 volt (operating voltage) and less, the second integrals were proportional to these maximum deflections. f is the ratio of the area extrapolated to zero voltage to the area at 0.05 volt. The f values deter mined at room temperature for K were used in the treatment of the data for both K and Na at all tempera tures. A curve drawn through the measured values of f versus concentration was employed to interpolate the values of f for given concentrations. The experimental data entering into the calculation of Xm and the values obtained for Xm are listed in Tables I and II. (The data in these tables were obtained from a series of measurements employing techniques which were considerably improved over those used in the measurements referred to as Series I and II in the symposia mentioned above. None of the rf data of these Series I and II is included in Figs. 1 to 4. Figs. 5 and 6 give results of all series of runs.) In Figs. 1 and 2 the static molar susceptibilities Xm of K and Na in liquid NH3 calculated from the rf data as described above are plotted against the concentration at the three different temperatures. The curves through the points were visually fitted to them, and these curves were all drawn so as to pass through the points, NAvf12/kT, at zero concentration which are shown as horizontal bars on ordinate axes. In Figs. 3 and 4 these visually fitted curves are given, without the experimental points, together with curves that were visually fitted to the data of Freed and Sugarman2 and of Huster obtained from measurements made in static fields. The values of X were obtained by mUltiplying 10-3 . Xm by the concentrations M. The lengths of recorder charts from the maxima of the magnetic derivative signal to the minima were meas sured. These lengths when mUltiplied by the factor equal to the number of gauss variation of static field per unit length of chart gave the widths AHM s between the points of maximum and minimum slopes of the magnetic resonance curves. In Fig. 5 these widths are plotted as a function of concentration. To avoid con fusion caused by overlapping of curves and points three separate graphs are given corresponding to the three temperatures. In Fig. 6 just the visually fitted curves are presented without the experimental points and all with the same ordinate scale. Data on widths for two concen trations of solutions of K in ND3 are given in Table III. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:381964 C. A. HUTCHISON, JR. , AND R. C. PASTOR TABLE I. Experimental data for K in liquid NHa• 2 3 4 5 6 8 9 10 11 12 Amount Molar Alkali Tempera- Geomet- Capaci- Modula- Width suscep- Suseep- Serial Sample Concen- Metal ture Cel-trieal factor tanee tion amph- maXlmum Low field tlbihty tlb!lity No. No. tratlOn ,10' sius '2 .1012 tude slope half area ,10' '10' M m'lO' 2R C H .. I1HMB ALF x,,'lO' X'lO' mole I-I mole I-I mole deg farad gauss gauss volt gauss em' 1 K141 0.00402 0.3358 -33.2 1.82 124.6 0.0317 0.057 0.00979 981 3.94 2 K149 0.0138 1.183 -33.2 1.85 124.1 0.0475' 0.059 0.0210 595 8.21 3 KI43 0.0195 1.589 -33.2 1.85 124.5 0.0317 0.056 0.0275 574 11.2 4 Kl48 0.0267 2.306 -33.2 1.87 124.2 0.0594 0.062 0.0276 395 10.5 5 K142 0.0337 2326 -33.2 1.88 124.6 0.0317 0055 0.0226 315 10.6 6 K145 0.0361 2.797 -33.2 1.86 124.7 0.0317 0.055 00298 349 12.6 7 K150 0.0793 6.686 -33.2 1.87 124.3 0.0317 0.055 0.0363 176 14.0 8 KISS 0.155 14.00 -33.2 1.86 124.1 0.0317 0.052 0.0576 131 20.3 9 Kl44 0.177 1642 -33.2 1.86 124.3 0.0317 0.049 0.0587 113 20.0 10 K151 0.203 19.98 -33.2 1.82 124.2 0.0317 0.049 0.0715 115 23.3 11 Kl46 0.219 16.78 -33.2 187 124.7 0.0198 0.044 0.0632 117 25.6 12 K147 0.312 27.57 -33.2 1.83 124.4 0.0198 0.046 0.0770 86 26.8 13 K153 0.419 33.63 -33.2 1.90 124.2 0.0317 0.051 0.0569 48 20.1 14 K156 0.728 62.01 -33.2 1.90 124.8 0.0396 0.125 0.1089 50 36.4 15 K157 1.551 130.7 -33.2 1.88 16 K141 0.00376 0.3358 0.8 1.79 124.3 0.0238 0.040 0.01489 1549 5.82 17 K149 0.0129 1.183 0.8 1.81 124.3 0.0198 0.036 0.0337 978 12.6 18 Kl43 0.0182 1.589 0.8 1.81 124.3 0.0238 0.036 0.0399 869 15.8 19 K148 0.0250 2.306 0.7 1.85 123.8 0.0158 0.032 0.0517 763 19.1 20 K142 0.0315 2.326 0.8 1.87 124.2 0.0238 0.039 0.0479 687 21.6 21 K145 0.0338 2.797 0.8 1.83 123.9 0.0158 0.031 0.0615 750 25.4 22 K150 0.0742 6.686 1.2 1.85 124.2 0.00792 0.030 00826 406 30.1 23 KISS 0.145 14.00 0.8 1.83 124.2 0.00594 0.030 0.1191 277 40.2 24 KI44 0.166 16.42 08 1.83 123.7 0.00594 0.030 0.1275 256 42.5 25 K151 0.190 19.98 0.7 1.79 124.2 0.00792 0.030 0.1311 217 41.2 26 K146 0.204 16.78 0.8 1.85 124.7 0.00792 0.032 0.1123 217 44.3 27 K147 0.291 27.57 0.8 1.81 123.8 0.01190 0.039 0.1456 170 49.5 28 K153 0.392 33.63 0.7 1.87 124.2 0.00594 0.039 0.1462 129 50.6 29 K156 0.681 62.01 0.8 1.87 122.8 0.0792 0.190 0.2572 121 82.4 30 K157 1.451 130.7 0.8 1.87 31 K141 0.00356 0.3358 25.5 175 142.9 0.02176 0.024 0.01426 1301 4.63 32 K149 0.0123 1.183 24.6 1.79 142.4 0.01474 0.D25 0.0415 1054 13.0 33 Kl43 0.0172 1.589 25.2 1.79 142.6 0.01123 0.022 0.0490 919 15.8 34 Kl48 0.0237 2.306 25.2 1.83 142.4 0.00878 0.023 0.0606 767 18.2 35 K142 0.0299 2.326 25.2 1.84 142.7 0.01123 0.025 0.0616 766 22.9 36 K145 0.0320 2.797 25.0 1.81 142.5 0.00632 0.020 0.0738 778 24.9 37 K150 0.0703 6.686 24.2 1.83 142.4 0.00562 0.023 0.1144 496 34.9 38 KISS 0.137 14.00 24.2 1.81 141.2 0.00316 0.D25 0.1750 363 49.7 39 Kl44 0157 16.42 25.0 1.81 142.3 0.00421 0.027 0.2209 385 60.4 40 K151 0.180 19.98 25.3 1.75 142.3 0.00246 0.026 0.2280 336 605 41 K146 0.193 16.78 25.2 1.83 142.0 0.00351 0.032 0.2091 349 67.4 42 K147 0.276 27.57 25.2 1.77 141.3 0.00702 0.046 0.2487 255 70.4 43 K153 0.371 33.63 25.0 1.86 142.3 0.0176 0.061 0.2497 193 71.6 44 K156 0.645 62.01 26.5 1.86 45 K157 1.375 130.7 25.0 1.84 Graphs of the first integrals of the recorder curves sample gy to that of another g", was then given by which give the shapes of the resonance absorption lines gy/gz= (VI/[ IEII+ IE,::I]) y (VI/[ IEII+ lET ~I]):: have been given by Hutchison and Pastor.l(a) Ratios of the spectroscopic splitting factors g for different samples were determined by means of an experimental procedure which involved reversal of the in which the subscripts f and r denote the forward and direction of the current through the solenoid, thereby reverse directions of the current in the solenoid. From cancelling the effect of the earth's field. With the 5 to 8 pairs E" E, were measured for each of a number solenoid's axis aligned parallel or antiparallel to the of samples. The results are summarized in Table IV earth's field, the output of the phase-sensitive detector along with the results on other substances and at other was observed as the current through the solenoid was frequencies. varied. The value of the voltage across the resistor in The magnetic saturation was investigated in a series series with the solenoid at which this output passed of measurements in which the maximum magnetic through zero on its way from maximum to minimum derivative signal with variation of field was examined derivative signal was called E. The ratio of g for one as a function of the voltage on the rf coil. The results of This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38MAGNETIC RESONANCE IN SOLUTIONS OF K IN LIQUID NHa 1965 TABLE II. Experimental data for Na in liquid NHa• 2 3 4 5 6 Amount Geomet- alkali Tempera- trical Serial Sample Concen- metal ture Cel- factor No. No. tratlon ·10' SIUS ·2 M m·l0· 2R mole I-I mole deg 1 Na135 0.0558 4.801 -33.1 1.91 2 Na138 0.0563 5.273 -33.2 1.90 3 Na137 00784 8.134 -33.2 1.87 4 Na136 0.210 19.45 -33.2 1.89 5 Na124 0.347 35.65 -33.2 1.84 6 Na139 0.982 104.2 -33.2 1.90 7 Na135 0.0521 4.801 0.6 1.89 8 Na138 0.0526 5.273 1.0 1.86 9 Na137 0.0732 8.134 0.8 1.84 10 Na136 0.1962 19.45 08 1.87 11 Na124 0.324 35.65 0.7 181 12 Na139 0.919 104 2 0.8 1.86 13 Na135 00495 4.801 24.1 1.88 14 Na138 0.0499 5.273 24.2 1.86 15 Na137 0.0696 8.134 23.8 1.82 16 Na136 01864 19.45 23.8 1.85 17 Na124 0.3080 35.65 24.3 1.78 18 Na139 0.873 104.2 24.0 1.86 these measurements at room temperature for several concentrations of K in liquid NHa are summarized in Table V along with data on the width of the resonance as a function of coil voltage. In order to calculate T1, the spin-lattice relaxation time, the maximum magnetic derivative signal at zero rf field was calculated by extrapolation of the measure ments at finite fields. The ratio maximum derivative signal Dex=------------------ maximum derivative signal at zero rf voltage was equated to (1+r2HrrT1T2)-i according to Eq. (17) of Bloembergen, Purcell, and PoundP T 1 was then calculated from the relation in which Hrf2 was obtained from coil geometry and circuit constants and T2 is equal to 2/3lr!:.HMS. !:.HMS is the width at point of maximum slope. 1000 y Calculated StatiC Pararnagnehc '" '" Suscephblhhes of K and No 11 LKluid NH3 ~ 800 -\ from Resonance Absorptions .. '( \! ~ ~ ap~~ l6OOY$'~" KmL">*I%. ___ --.. ---A -A ~ ~" ,,"-Y "'"l">*INH3 -y _ -. :'400 -; ~+ <::""""t~-t----t--+-----I--l ', .... ~ ""'-! I'~-I>~ 200 ~~---""i------ ... -A...!-o .... A I ' .. - WOO Q200 Q300 0IlOO Q50Q o.soo 0.700 o.aao M.moJ rl FIG. 1. 12 Bloembergen, Purcell, and Pound, Phys. Rev. 73, 679 (1948). 8 9 10 11 12 Molar Capaci- Modula- Width suscep~ Suscep- tance han amph- maX1mum Low field tlbihty tlbllity ·10" tude slope half area ·10' ·10' C H .. tJ.HMS ALF Xm"lO' X·lO• volt mole I-I farad gauss gauss gauss em' 124.3 0.0198 0051 0.0355 215 12.0 124.3 00198 0.056 0.0393 217 12.2 124.3 0.0198 0052 0.0448 163 12.8 124.3 0.0198 0.043 0.0763 111 23.3 124.4 0.0139 0.043 0.0994 77 26.7 124.2 0.00594 0.030 00876 539 28.1 124.6 0.00594 0.033 0.1086 615 32.3 123.9 0.00792 0.031 0.1242 463 33.9 1240 0.00594 0030 0.1856 276 54.2 124.1 0.00594 0.037 0.1922 154 49.9 127.8 0.00351 0.Q25 0.1196 718 35.5 128.2 0.00351 0.Q25 0.1722 947 47.3 128.0 0.00351 0.023 0.2248 816 56.8 127.1 0.00351 0.Q28 0.370 539 100.0 128.6 0.00351 0.032 0.354 278 85.6 The T l'S so calculated varied from 2 to 5 times the inverse line width time T2• The errors in the determinations of the integrals of the recorder curves were essentially all in the intensity determinations the measurement of the field making a negligible contribution. The measurements of the integrals for the solutions of K in NHa were most precise in two intermediate concentration ranges from 0.06 M to 0.1 M and from 0.2 M to 0.3 M where the errors were probably of the order of 3 percent or less. The errors in the determinations of relative amounts of tris-p-nitro phenylmethyl in various independently chemically analyzed samples was about 2 percent. Three factors were responsible for loss of precision in the low, middle, and high ranges of concentration of K in NHa. These were (a) the fact that X decreased very rapidly in the more dilute solutions although Xm was increasing rapidly and this resulted in very weak signals; (b) at the high concentrations although X was large, the width was also large and this lowered all the intensities 1600 150 140 1300 1200 ~ A I Calculated Static F'<lramagnellc SusceptlbtlrtleS - of.K and No 11 LIquid NH3 from Resonance AbsarptlOllS 11100 T 1000 i900 ~ 80 -;"7 '~H: -'-.-~-... -''':::-, K 1'1 liquid NH3-Set'It~m ---A ---~ --It. _ ~ "'.L,q""NH3 _.y -----l< c-'~~~ I r--A~~-;"'.L 't I f--' ,. I --';-.:::::.~ o ~ I I -~--L 't ----·1-= 00 ~ :.:",.;;;:..:..;-::--! 1 600 500 400 300 200 100 a -'",;:,,:~ ~ r-A ~~~~~~_L " -------::::::.--~ ~ T '=T'-------...., :--00<. I 0.010 0.000 0D30 0D40 a.oro Q.06() ().()7Q 0Il80 Q090 0100 M;N r' FIG. 2. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:381966 C. A. HUTCHISON, JR., AND R. C. PASTOR IIOOr---r--,----::-:----:-:-::--::------:---. 1000 \ Calculated Static Paramagnetic 900 -\ Susceptibilities of K and No In liqt.id NH3 ~ from Resonance Absorptions 800 -~ Room Ice BPLlqLIId Temp PoInt NH3 '"'l-700 - KI1LlquldNH3.Serlesllo-- -__ _ 13 600 -~'" No In LIQUJd NH3 ------ ~ 500 _\~, ~ Susceptlbilifles Measured 11 Static Fl8lds CD-400 -_\*-~":.~~~ - -K Freed and Sugorrrm -33·C Q 300 \d~~~'1'-d,",,----- No....... -35'C '" \ '<::----. ----200 ~ T--",-",~=::~-=--~-+--+--1 100 ,~-------~- __ ...l.=-=~=-_=_~~_=_~2:... o o 0.600 0700 o.eoa FIG. 3. and gave small derivative signals; (c) at the middle concentrations the magnetic derivative signals were very intense, and in order to avoid overloading the calibrator, the derivative signals had to be decreased by means of reducing the field modulation amplitudes and ~~o---:-.--~---:-~-~--------, Calculated StQtlC Paramagnetic Susceptibilities of K and Na In Liquid NH3 from Resonance Absorptions 500f----+~ 400f----+-_+.:> 3001---t---+---f roor--1--+-~-~~ IOOr--1--+-~-+--+--~-r-~~=F==~ oL-~n-~~~,,~~~-n,~~~nf.~~o-~ FIG. 4. this led to errors in the determination of the modu lation amplitudes. The errors in the low, middle, and high ranges were probably about 5 percent or 6 percent except in the very dilute region where the errors of chemical analysis predominated. //"'-- 0 // o~ .-/ -0 I'" • , , OJB OJ. OJ' 0-Width of Resonance ot MaXimum Slope vs Con~w~nCEFOff BPLKU)N-i3-~/ 0 0 I OJOO ~ !; 0.100 li ODB 0 KW LIQUID Mi3.SERlES 1--... .. D __ .t. .. m--A ---A Na"LIQUD~ ¥ ){ OD' 0 o P1""*"-_ ... ;",,,, ~---"",-A-Al'o----- . 1 0.02 0 / o.oao 1/ I -0 ;/A' l'. 0.04 J...-r ...... II li o -If!-~ "t aD20 ,~~ I' cwo Q200 Q.3OO 0000 FIG. 5. , -r,' 1 I "I ---;/ , I I I I I / / / / ./ 0.000 DISCUSSION. SUSCEPTIBILITIES Figures 1-4 summarize the calculations of the para magnetic susceptibilities in static fields from the rf data and compare the values so obtained with the suscepti bilities measured in static fields by other workers.2, 3 The essential features of these curves are the following. (a) At all temperatures and for both K and Na the static molar magnetic susceptibilities calculated from the rf data are relatively small at the higher concentrations and increase with decreasing concentration until they reach something near the value, NAv{32/kT in the very dilute solutions. In this respect the behavior is similar to that observed in static fields.2, a (b) At all concentra tions which were examined the molar magnetic sus ceptibilities increase with increasing temperatures. Although presumably at the extremely low concentra tions the reverse would be true the lowest concentra tions which were employed were still above the cross over point. In this respect the behavior is similar to that observed in static fields.2, a In the present experiments the range of temperatures is larger than was the case for the static field measurements. (c) At the higher con centrations, for the temperature at which both rf and static field data are available (bp of NHa), the curves for the two approach a small approximately constant difference, the quantities obtained from the rf measure ments being larger. In the dilute solutions the smooth curves which have been drawn visually through the points show the static field data for K slightly higher and the static field data for Na slightly lower than the values from the present research. (The rf curves for K and Na are indistinguishable at this temperature.) However, this happens only below 0.065 molar, and at these low concentrations the possibilities of analytical errors affecting all the curves (rf and dc) become much more important, and the errors in the recorder curves become larger because of the weak signals. The differ ence between the rf curve for N a and K and the dc curve for Na is just about constant over almost the entire concentration range. (d) The rf curves for K and Na are identical at the bp of NHa. At 274°K, the Na curve is slightly higher than the K curve, and at the highest temperature 298°K the Na curve is very much higher than the K curve throughout the range from a few thousandths to a few tenths molar. The attempts to correlate the optical, chemical, electrical, thermodynamic, and magnetic properties of the solutions of alkali metals in liquid NHa by means of a model capable of quantitative analysis have been ·numerous. One model which has received considerable attention in recent years has consisted in supposing that the metal is dissociated into positive metal ions and electrons and that the electrons are trapped in cavities in the solvent. Moreover, it has been supposed that the electrons may be trapped singly or in pairs in these cavities, there being a reversible dissociation of the pairs on diluting the solutions. The pairs would pre- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38MAG NET I eRE SON A NeE INS 0 L UTI 0 N S 0 F KIN L I QUI D N H 3 1967 TABLE III. Experimental data for K in liquid ND3• 2 3 4 5 6 Amount Geomet- alkali Tempera- trical Senal Sample Concen- metal ture Cel-factor No No. tration ·10' sius 2 M m·IO· 2R mole I-I mole deg 1 K161 00329 2.05 29.5 2 K162 0.0648 2.15 292 sumably be in singlet states and therefore not contribute to the paramagnetism of the solutions. Such a model has its origins in the work of Kraus, was explicitly discussed by Freed and Sugarman2 and was elaborated 'by Ogg." Detailed discussions of the cavity trapping model and calculations based on it have been presented by HillI(e) and quite recently by Lipscomb. I (d) Most recently, Kaplan and Kittel1(c) have discussed the model giving particular attention to the spin resonance results. One of the questions on which light is shed by results of the paramagnetic resonance absorption experiments is that concerning the diamagnetism of these solutions. The diamagnetism of the electrons in the cavities is an effect which is inherent in the cavity model of the paramagnetic absorbers and one which cannot be measured in static field experiments or rf experiments TABLE IV. Spectroscopic splitting factors· 2 3 4 Concen- tratIOn 23600 '10' 14 10' 7·10' mole I-I cycle sec-1 cycle sec-1 cycle sec-1 gH/gT 099973 099788 ±O 00012 KK/gH o 99882 ±0.00007 gK/gT 00036 0.99906±0 00012 KK/gT o 99855 ±O 00010 gK/KT o 157 o 99909 ±O 00006 gK/gT 0180 0.99904±0 00010 gK/gT 0276 o 99906 ±O 00006 gK relT 2.0008 gK relH 2.0013 gK relP 20012 gH relP 20037-20041- gT relP 2.0039 & Depending upon onentatIOn of crystals. Column Entnes 1 These are the symbols for the ratIOs of g factors and for relatlve g·s. The symbols have the following meanings gH g factor of hydrazyl. gT g factor of tris-p-nitrophenylmethyl EK g factor of K In liqUid NH, gK "IY where Y IS H (hydrazyl) or T (tns-p-nitrophenylmethyl) (KK/gy) ·20037. (See references 13,6,14.) gx "IP where X IS H (hydrazyl), T (tns-p-mtrophenylmethyl), or K (K In liqUid NH.) ~'3 0420 '10-'. (See references IS, 16) IIproton (Proton and electron resonances were both measured in the same field.) 2 These are the concentrations of solutions of K In liquid NH,; umt mole I-I. 3 These are the values of the correspondmg entnes in column 1 measured at a frequency of 23 600 '106 cycle sec-I. 4 These are the values of the correspondmg entnes in column 1 measured at a frequency of 14 '10' cycle sec-I. S These are the values of the correspondmg entries In column 1 measured at a frequency of 7 '10' cycle sec-I. ----- 13 Holden, Kittel, Merritt, and Yager, Phys. Rev. 77,147 (1950). 14 Hutchison, Pastor, and Kowalsky, J. Chern. Phys. 20, 524 (1952). 16 H. Taub and P. Kusch, Phys. Rev. 75, 1481 (1949). 16 J .. Schwinger, Phys. Rev. 73, 416 (1948). 8 Q 10 11 12 Molar CapacI- Modula- Width suscep- Suscep- tanC'e tlOn amph- lTlaximUm Low field tlbillty tlbillty .1012 tude slope half area 10' ·10' C H", tJ.HMS ALP volt Xm 106 x·IO' mole 1-1 farad gauss gauss gauss cm' 0.0176 0039 0.0176 0.035 taken by themselves. In the static field experiments, only the sum of the diamagnetic and paramagnetic susceptibilities is measured. The paramagnetic reso nance experiment, as explained previously, gives just the paramagnetic part of the susceptibility. Comparison of the latter with the static measurements gives the diamagnetism associated directly with the paramag netic species. On the basis of the cavity trapping model one may then form some opinion concerning the radii of the cavities which contain the electrons. At the highest concentration for which there are available data from both the rf and static field experi ments ("-'0.5 M) the value of Xm from the former is 55.10-6 and from the latter is 29· HI-6. At this con centration the pairing of the electrons is nearly com plete, the rf data showing that the pairs are only 0.035 dissociated. (The temperature in question is the bp of NHs.) We may, therefore, cakulate the mean-square radius for the electron's distribution in the cavities in which the pairs are trapped assuming that the diamagnetism of two electrons in a cavity of given size is just twice that of a single electron in a cavity of the same size. Qualitative calculations have shown that a quite large correlation between the positions of the two electrons has only a small effect on the diamagnetism and the assumption is therefore probably quite good. Since e2NAV/6mc2= 2.829·1OIO(r2)Av we see that the difference in susceptibilities 26.10-6 corresponds to (r2)A.=3.0A. If now it is postulated that the cavities are spherical boxes and the electrons have vanishing probabilities of occurrence outside the boxes, it is found that the boxes must have a radius of 5.7A. On the other hand, if one imagines the situation which must actually exist in which the wave functions pene- 0.100 Q200 Q.3OO O~ O!!OO 0,600 0.7'00 o.eoo FIG. 6. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:381968 C. A. HUTCHISON, JR., AND R. C. PASTOR TABLE V. Experimental values of ~aturation factor! and width at maximum slope AHMs as a functIOn of concentratIOn at room temperature for K in liquid NIIa. 1 2 3 Sample Concen4 No tration rf Field mole 1-1 gauss K57 0.0135 (0.000) 0.010 0.021 K53 0.0138 0037 0.048 0080 K73 "-'0.02 (0.000) 0.005 0.010 0021 0.031 0.041 0.062 K88 0.152 (0.000) 0.004 0.010 0021 0.041 K166 0.380 (0.000) 0.005 0.010 0021 0.041 0062 0.072 0.082 • By extrapolatlon. 4 Max. Defi. Mod Amp. volt gauss-l (43.5)- 33.6 25.5 18.2 148 5.42 (42.6)- 37.9 33.7 24.8 18.4 13.2 40 (178.0)- 166.0 147.0 1160 68.8 (55.7)- 54.8 53.7 50.5 38.9 28.4 224 18.1 5 AHMS gauss 0.022 0.027 0.D28 0.034 0.048 0.021 0.021 0.024 0.031 0.033 0.044 0031 0.033 0034 0.039 0.058 0.058 0.059 0065 0.073 0078 0.084 6 f 1127 1.124 1.094 1.016 trate the dielectric to some distance it will be seen that the cavities would be somewhat smaller than this. The experimental data on volumes of dilute solutions of Na in liquid NH3 show an increase in volume of 71A3 per dissolved Na.l(d) If this volume is increased by amounts necessary to account for the difference in volumes of Na+ and Na and also to take account of the electrostriction caused by the Na+ the volume to be accounted for by cavities in the NH3 is 143A3 according to Lipscomb.I(d) This leads to 3.2A as the experimental cavity radius of the singly trapped electrons. If one minimizes the sum of the electron energy and dielectric polarization energy of a single electron trapped in a box in a dielectric he finds h2/2me2 or 10.3A as the radius of the cavity with a single electron, and the size of the cavities containing two such electrons would be con siderably larger. However, Lipscombl(d) has carried out a detailed calculation of the size of the cavity containing a single electron including the effects of ~lectronic polarization of the dielectric, electrostriction energy, and surface energy of the cavity and gets about 4.8A as the radius. He did not calculate the size of the cavities containing two electrons; this problem has been discussed by Hilll(e) but without consideration of all the effects which entered into Lipscomb's calculation. In principle it is of course possible from the assump tion of a specific model in which single electrons are trapped in cavities of one size and pairs are trapped in cavities of another size, to determine the sizes of both by means of a comparison of the static and rf data. The slope of the plot of (a) the difference between the molar susceptibility calculated from rf data and the static molar susceptibility versus (b) the ratio of the rf molar susceptibility to N A,IN kT is just N A,e2 /6mc2 times the difference between the mean-square radii of the two different cavities. Unfortunately, however, the static field data in the region from 0.1 M to 0.5 M are not sufficiently numerous to fix the curve of susceptibility versus concentration with sufficient precision for any meaningful comparison with the rf data. It is clear that if the single traps and double traps are of the same size, then the difference between the susceptibilities calcu lated from rf data and the static molar susceptibilities should be constant over the entire range of concentra tions which have been investigated. This is just about the case for the difference between the rf data on Na or K and the static data on Na, the latter being avail able over a much wider range of concentrations than are the static K data. However the Na static data are apparently less precise than the corresponding data onK. It may be said that the value for the size of the cavities in which pairs are trapped, (r2)A,i= 3.0A, as determined from comparison of the rf and static field susceptibility data are in reasonably good agreement with Lipscomb's calculated value 4.8A lId) for the radius of the cavities containing single electrons assuming no penetration of the dielectric or with the expe~imental value 3.2A for the radius of these cavities as determined from measurements of volume changes. The question of the manner in which the suscepti bility should vary with concentration is one which has been examined in detail by Hill I (e) and very recently by Kaplan and KitteLl (c) Hilll(e) has considered the transition from trapped pairs to trapped single electrons to be a simple dissociation equilibrium and has given both a classical and a quantum-mechanical discussion of the manner in which the extent of dissociation and hence the magnetic susceptibility should vary with concentration. Hill I (e) evaluated the interaction energy for two electrons in a cavity using first order perturbation theory, and was led to the conclusion that the energy of trapped pairs was greater than the energy of trapped single electrons. He pointed out, however, that treating the species involved as ideal classical gases the extent of dissociation would still increase with temperature. It should also be remarked that he neglected the surface energy and electro striction effects which were considered by Lipscomb.l(d) He then made a calculation using Bose-Einstein statistics for the pairs and Fermi-Dirac statistics for the single electrons and found that at a temperature equal to 220 oK, a concentration equal to 0.1 M and a ratio of the energy of trapped pair to twice th~t of a singly trapped electron equal to 4, that 0.28 of the trapped electrons are singly trapped and the remainder are held as pairs. The rf value of xmkT / N A,{12 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38MAG NET I eRE SON A NeE INS 0 L UTI 0 N S 0 F KIN L I QUI D N H 3 1969 TABLE VI. Experimental values of degree of dissociation a; a times the square root of concentration, aMi; and one half the square root of the equilibrium constant for the reaction e2-= 2e-, aMt/ (l-a)i, as functions of concentration and temperature. 2 3 4 5 6 7 8 a~t 10 11 a aMi (I-a)i Concen- tration 2400K 274°K 298°K 2400K 274°K 298°K HooK 274°K 298°K M W mole molei molei molei molei mole· mole· I-I 14 14 H H H H ev 0.002 0.822 0.953 0.968 0.0367 0.0426 0.0433 0.0970 0.195 0.243 0.101 0.006 0.616 0.858 0.905 0.0477 0.0665 0.0701 0.0760 0.176 0.228 0120 0.012 0.455 0.735 0.816 0.0498 0.0805 0.0893 0.0675 0.156 0.208 0.122 0.020 0.331 0620 0.709 0.0469 0.Q28 0.255 0.541 0.630 0.0426 0.036 0.205 0.483 0.573 0.0388 0.045 0.168 0.431 0.517 0.0357 0.060 0.134 0.364 0.446 0.0327 0.080 0.112 0.305 0.390 0.0317 0.100 0.0979 0.258 0.359 0.120 0.0883 0.229 0.328 0.0306 0.160 0.0768 0.186 0.281 0.0307 0.200 0.0697 0.155 0.246 0.0312 0.300 0.0544 0.113 0.188 0.0298 0.400 0.0435 0.0928 0.157 0.0275 0.500 0.0352 0.0848 0.143 0.0249 is (extrapolating the data to 2200K from the plots of dln[aMi/(1-a)!J/dt versus l/T mentioned below) found to be 0.063 which is considerably lower than Hill's estimate. On the other hap.d, Hill finds that if the energy ratio be taken as 1/1.4 instead of 4 the fraction of the electrons that are singly trapped is 0.14 which is considerably closer to the observed value. At 3000K for the first choice of the energy ratio Hill gets 0.36 as the fraction dissociated and the rf experiments give 0.36 for this fraction. The smaller value of the energy ratio would give a much smaller fraction. For a solution with concentration 0.2 M but with other conditions the same as in the first case Hill gets 0.16 dissociation and the experimental results are about 0.05. The classical calculation for the first set of conditions gives 0.42 as the fraction of electrons singly trapped which is in much worse disagreement with the rf data than is the other calculation. The value 1/1.4 for the energy ratio used above corresponds to 0.14 ev for the difference between! the energy of a trapped pair and the energy of a singly trapped electron. This is in quite good agree ment with the value of this energy calculated from the temperature dependence of the rf data as discussed below. Kaplan and Kitte}lcc) have proposed a detailed model for which minimization of the free energy leads to the expression [1+a2(e2w/kT -1) J/ (l-a) = 2N/n, (1) in which a is the ratio of Xm as determined in the rf experiments to the free electron spin susceptibility, namely NA,(32/kT; 2W is the difference in energy between a pair of trapped electrons and two singly trapped electrons; N is the total available number of trapping sites, and n is the total number of trapped electrons, i.e., the total number of dissolved alkali 0.0877 0.1003 0.0573 0.142 0.186 0.129 0.0905 0.106 0.0494 0.134 0.174 0.138 0.0915 0.109 0.0435 0.127 0.166 0.147 0.0913 0.110 0.0391 0.121 0.158 0.154 0.0891 0.109 0.0351 0.112 0.147 0.158 0.0862 0.110 0.0337 0.103 0.141 0.156 0.0793 0.114 0.0320 0.0903 0.139 0.167 0.0743 0.113 0.0319 0.0824 0.133 0.153 0.0693 0.110 0.0324 0.0754 0.127 0.146 0.0621 0.103 0.0306 0.0659 0.115 0.140 0.0587 0.100 0.0281 0.0617 0.108 0.142 0.0600 0.101 0.0254 ' 0.0627 0.109 0.137 metal atoms. Since at all concentrations investigated in the present rf work e2w/kT»1 we may write (2) From the lowest concentrations up to somewhat higher than 0.5 M it is true that a2e2w/kT»1 and with error no greater than 10 percent over this range one may write (3) This is just the expression, as far as concentration dependence is concerned, that would result from the consideration of a simple dissociation equilibrium between doubly and singly trapped electrons, i.e., a2M/(l-a) = constant in which M is the concentration and a is the fraction dissociated which in the present case on the basis of the cavity trapping model is just XmkT/NA,(32. The con stancy of this product is examined in Table VI. Kaplan and Kitte}lcc) have in their paper chosen to discuss their model in terms of a still further approximation a«l which is seen from Table VI to be good to within 10 percent only down to 0.1 M. The constancy of aM! is examined in Table VI. It will be seen that this last approximation gives considerably better agreement than does the better approximation mentioned above. It is to be noted that at the bp of NH3 the static magnetic susceptibilities calculated from rf data are indistinguishable for K and Na within the errors of the experiments in the range of concentration investi gated. This fact is, of course, in agreement with the predictions of all of the models that have been discussed inasmuch as they assume the metal to be completely dissociated into positive ions and trapped electrons and the magnitude of the paramagnetism is determined This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:381970 C. A. HUTCHISON, JR., AND R. C. PASTOR only by the extent of the electron pairing. Thus, the positive ions play no role in determining the magneti zation. It will be seen, however, from Tables I and II and Figs. 1 and 2. that as the temperature is raised, a difference between the Na and K appears which increases with temperatun;. At 0.100 M the suscepti bility of Na is about 15 percent greater than that of K at 274°K and about SO percent at 298°K. Apparently some effect of the alkali metal ions on the relative numbers of singly and doubly trapped electrons must be taken into consideration. Whether Debye-Huckel type interactions would be sufficient to account for these pronounced effects or whether it would be necessary to introduce the idea of electrons trapped on positive metal ions to form metal atoms, and in equi librium with singly and doubly trapped electrons, is not clear. The variation of the susceptibility with temperature is of interest in the examination of any of the models. The HilP(e) relation requires that a2Mj d In- dT=d lnKjdT= IjT(!+2WjkT), l-a whereas Kaplan and Kittel! (c) speak only of the ex ponential term in T and one would write in this case for the approximation (3) the expression a2Mj d In- dT= 2W jkP. l-a Plots of 10g[aMlj (l-a)lTiJ versus 1/T and 10g[aMlj (l-a)lJ versus IjT gave only slightly curved lines over the range of three temperatures and the slopes were about the same in either case and varied very little with concentration. The values of W calculated from the plots of 10g[aMlj (l-a)lJ versus 1/T are given in Table VI. The values for! the dissociation energy of pairs to form singly trapped electrons according to these plots range from 0.10 ev at the lowest con centrations to a maximum of about 0.17 ev at about O.lM and then drop to slightly lower values at the higher concentrations. The plots of 10g[aMlj (l-a)lTIJ versus IjT gave values of W which were essentially the same as those in Table VI. SPECTROSCOPIC SPLITTING FACTORS Three important observations concerning the spectro scopic splitting factors or g's for K are that (a) they are independent of concentration in the vicinity of 7.106 cycle sec!; (b) they are the same at the fre quencies 23 500.106 cycle sec! and 7· 106 cycle sec! if the g's of the standard comparison substances are not different at these two frequencies; and (c) they have the value 2.0012±0.0002. The fact that they are independent of concentration over a rather wide range is indicative of the fact that the electron is bound in the same manner at all concentrations. The value of the g is about 0.0011 lower than the free electron value 2.0023. As has been indicated by Kaplan and Kittel!(o) this is compatible with binding on the protons of the NH3 molecules surrounding the cavity in analogy with the binding that has been found by magnetic resonance experiment to exist for the electrons trapped in vacan cies in alkali halide crystals to form F centers.l7·!8 The Bloch-Siegert!)! effect which results in a shift of the center of resonance by the factor 1-[(2H1)2/16H 02J= 1-1.6'10-7 is too small to be of importance in this connection. WIDTHS One of the most striking features of this resonance is its extreme sharpness. There is evidently an extra ordinarily large narrowing associated with exchange effects and with the great mobility of the electrons in these solutions. Abrahams and KittePO find that for nl spins per cm3 distributed randomly on a simple cubic lattice the dipole-dipole width should be about nl·10-19 gauss. This would give as the width 4.3 gauss at 0.5 M, 2.16 gauss at 0.1 M and 0.84 gauss at 0.02 M which makes it clear that the narrowing is very large. Kaplan and KitteP(o) discuss the effect of the rapid diffusion of the singly trapped electrons on the dipole-dipole width using the methods of Bloembergen, Purcell, and Pound!2 and find that below 0.1 M the diffusion narrowed dipole-dipole broadening is very small compared with the observed widths, but that such broadening might be responsible for the increased widths at the higher concentrations. It has been seen that the spin-lattice relaxation time Tl is of such a magnitude that it is not more than 2 to 5 times greater than the inverse line width time T2• Kaplan and KitteP(c) have considered the problem of the broadening caused by nuclear hyperfine structure which has proved to be very important in the case of the electrons trapped in vacancies in the alkali halides.2! Motional narrowing must be considered here also and the conclusion reached is that the broadening for electrons trapped on the protons surrounding the cavity might be expected to be of order 10 gauss for a rigid structure. However, Kaplan and Kittell(c) estimate that the rotational relaxation fluctuations might narrow the line to 0.03 gauss at the bp of NH3. This is not too far from the observed widths. They also estimate that the characteristic frequency We of the fluctuations is about 1011 radian sec1. Since in the present experi ments W is approximately 4·107 radian sect, it is clear that WTe«1. Bloembergen, Pound, and Purcell12 have 17 A. H. Kahn and C. Kittel, Phys. Rev. 89, 315 (1953). 18 C A. Hutchison, Jr. and G. A. Noble, Phys Rev. 87, 1125 (1952). . 19 F. Bloch and A. Siegert, Phys. Rev. 57, 522 (19401. 20 C. Kittel and E. Abrahams, Phys. Rev. 90, 171 (1953). 21 Kip, Kittel, Levy, and Portis, Phys Rev. 91, 1066 (1953). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.59.171.71 On: Thu, 11 Dec 2014 21:53:38MAGNETIC RESONANCE IN SOLUTIONS OF K IN LIQUID NHs 1971 shown that when this condition obtains one would expect that TI':::::.T 2• This is seen to be in agreement with the present experimental observation that for K in liquid NHa Tl is not much larger than T2• It should be pointed out that the variation of the width at infinite dilution would be expected to vary with temperature inversely as the ratio of the tempera ture to the viscosity. This relation will be seen to be nearly quantitatively obeyed for the three temperatures which were investigated. It should also be remarked that the rapid rise of the widths at the higher concentrations may perhaps be associated with the increasing electrical conductivity. Actually the very rapid rise begins at each temperature at just the concentration at which the skin depth is about 1 cm and this is the order of the linear dimensions of the samples. The preliminary data on solutions of K in NDa gave at two concentrations widths which are considerably greater than those for K in NHa. Two effects at least are to be considered. One is that the deuteron nuclear magnetic moment is much smaller than that of the proton, and we might therefore expect somewhat sharper absorptions because of residual effects of the nuclear hyperfine structure which are not washed out by the motional processes mentioned above. The other is that the zero-point vibrational frequency of the deuterons is very much less than that of the protons and that the inversion frequency of NDa is very much less than that of NHa and these effects might result in considerably less motional narrowing and thus lead to broader lines. Actually the interpretation of the experimental results is very much beclouded by the fact that the viscosity of NDa is unknown and in particular the viscosities of solutions of K in NDa must be measured before an interpretation of the results can be given. In any event there is probably no marked narrowing of the line at comparable viscosities and the two effects on the width probably cancel each other to a considerable extent. The change of width with rf coil voltage begins only at voltages considerably higher than that employed in the determination of the rf susceptibilities. In the case of a resonance as sharp as that of the alkali metals in liquid NHa one must consider the Rabi width associated with the flipping of the spin by the perturbing rf magnetic field. For very small rf fields such as were used in these experiments the Rabi width is just V1. rms rf field strength or for the conditions of the present experiments 0.147·rms coil voltage. The operating coil voltage was 0.05 volt so that the Rabi width was about 0.006 gauss. This is not less by a very great factor than the width of the narrowest resonances, namely 0.020 gauss. SHAPES The shapes of the resonances are very close to Lorentz ians and deviate greatly from the Gaussians.1(a) For absorptions down to 0.2 to 0.1 of the maximum the Lorentzian curve is followed closely and then at lower absorptions the experimental curve drops appreciably below the Lorentzian. This is in agreement with the theory of Anderson and Weiss. 22 SATURATION The spin-lattice relaxation times determined by the method of Bloembergen, Purcell, and Pound12 from the variation of the output voltage of the phase-sensitive detector with rf coil voltage are several times as large as the times corresponding to the widths of the absorp tion lines. The fact that these determinations give relaxation times of the same order of magnitude as the inverse line width times indicates that the spin-lattice relaxation may make a considerable contribution to the observed widths. No calculations of spin-lattice relaxa tion times on the basis of the cavity trapping model have been made. ACKNOWLEDGMENTS Most of the electronic equipment employed in these experiments was designed and constructed by Clarence Arnow. Edward Bartal constructed the solenoid and its mounting and did other necessary machine work. Jack Boardman carried ou t a large part of the numerical integrations and other calculations. Arthur Kowalsky synthesized the tris-p-nitrophenylmethyl and analyzed it. Their help and assistance is gratefully acknowledged. 22 P. W. Anderson and P. R. Weiss, Revs Mod. Phys. 25, 269 (1953). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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1.1748246.pdf

Near Infrared Pleochroism. II. The 0.8–2.5 Region of Some Linear Polymers L. Glatt and J. W. Ellis Citation: The Journal of Chemical Physics 19, 449 (1951); doi: 10.1063/1.1748246 View online: http://dx.doi.org/10.1063/1.1748246 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/19/4?ver=pdfcov Published by the AIP Publishing Advertisement: This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57THE JOURNAL OF CHEMICAL PHYSICS VOLUME 19. NUMBER 4 APRIL. 1951 N ear Infrared Pleochroism. II. The 0.8 -2.St' Region of Some Linear Polymers L. GLATT*t AND J. W. ELLIS Department of Physics, University of California, Los Angeles, California (Received July 21, 1949) Polarized r~diation is. used to study. the. pleochrois~ of. aligned specimens of polythene, Nylon, polyvinyl alc~hol, polYVinyl chionde, and polyvInylidene chionde, In the 0.8-2.5" overtone and combination region. Assignments of observed absorption bands to various vibration modes of (coupled) >CHz, N -H, and C=O groups are attempted. The symmetry of an infinite planar-zigzag >CH2 chain is discussed as an approximation to ~hat of p~lythene an~, t~ a lesser ext~nt, Nylon. Some limitations in the interpretation of observe~ pleoc~r~lsm .and In the application of. selectio~ rules are discussed. A pleochroism study, with unpolanzed radiation, IS made of the molecular alignment In oriented "Parowax." A study is made also of the rupture of the intermolecular N -H· . ·0 bonds in molten and premolten Nylon. I. INTRODUCTION INFRARED absorption is, for a given vibration mode, most intense when the exciting E-vector is parallel to the direction (for degenerate modes, directions) of the oscillating electric moment increment characteristic of that mode. The infrared pleochroism of a specimen, or variation of absorption intensity with respect to the E-vector, can often yield considerable information as to the vibration modes which are responsible for the vari ous observed absorption bands and also concerning the orientation of the molecules in the specimen. This is, naturally, true only for a crystalline or ordered sub stance in which, moreover, the molecules have some element of symmetry. Several investigations early in the century used polarized infrared radiation to study crystals of in organic salts. Schaefer and Matossi summarized, in 1930, the results of investigations on carbonates, nitrates, and other ionic crystals.! Ellis and Bath, in this laboratory, were perhaps the first to use infrared H /' (l c /f': --+-)I He/' : ~" t Z FIG. 1. Section of a > CH2 chain. * Most of this material was taken from a dissertation presented in partial satisfaction of the requirements for a Ph.D. degree. t Now at the Cryogenic Laboratory, Ohio State University. 1 C. Schaefer and F. Matossi, Das Ultrarote SPektrum (Verlag. Julius Springer, Berlin, Germany, 1930). pleochroism for the correlation of the vibration spectra of organic substances with their structures.2.3 Several such studies have been conducted recently, in England for the most part. Reference will be made later to these contemporary investigations. The present report is concerned with some linear polymers that were semi-aligned by mechanically stretching or rolling the specimens. Polythene, "Paro wax," 6.10 polyamide (a "Nylon"), polyvinyl alcohol, polyvinyl chloride, and polyvinylidene chloride are the substances discussed in this paper. Later papers will deal with single-crystal as well as with polymer speci mens. Whenever possible, the authors have attempted the assignment of observed absorption bands to the various vibration modes of (coupled) > CH2, 0-H, N-H, and C=O groups. In making these assignments, it was attempted to correlate the observed pleochroism with available data on molecular orientation and with symmetry selection rules. Figure 1 shows a diagram of a normal saturated > CH2 chain. The carbon atoms form a planar zigzag. The planes of the CH2 groups, shown in projection, are normal to the chain axis. The four single bonds about each carbon atom are distributed essentially at the tetrahedral angle. Schematic diagrams for the three fundamental modes of a free CH2 group (not to scale) are shown in Fig. 2. Similar figures are shown for the three restricted rotation modes that result when the > CH2 group is attached to the rest of the molecule. Spectrograms of all the specimens were made on a quartz prism recording spectrograph, which was used in the overtone and combination band region between 0.6 and 2.7 p..4 The fundamental X - H bond stretching region in between roughly 2.7 p. and 4.0p.. A Glan Thompson type calcite polarizer was used with this instrument. It is not possible, unfortunately, to send plane polarized radiation through a non-isottopic medium with the E-vector vibrating in any arbitrarily chosen direction. The cross section of the index ellipsoid parallel 2 J. W. Ellis and J. Bath, J. Chern. Phys. 6, 221 (1938); 7, 862 (1938). 3 J. W. Ellis and J. Bath, J. Am. Chern. Soc. 62, 2859 (1940). 4 J. W. Ellis, Rev. Sci. Instr. 4, 123 (1933). 449 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57450 L. GLATT AND J. W. ELLIS .~ (0) )/ •. .<f>. (b) [, c: ~H I (e) ]/a (d) (e) r 'y' z (0) t. x-t (h) FIG. 2. The three vibration and the three restricted rotation modes of a >CH2 oscillator. to the transmitting surface is, in general, an ellipse, the major and minor axes of which are, for normal incidence, the only two vibration directions for which plane polarized radiation can be transmitted as such. If a vibration direction for the incident beam other than one of the above two mutually perpendicular ones is chosen, then elliptically polarized light is transmitted through the specimen. This limitation upon the use of polarized radiation has not always been appreciated. The converging beam of the quartz spectrograph has a semi-angle of about 10°. This causes most of the light to impinge upon the specimen at slightly oblique incidence. The alignment between specimen and polarizer is seldom reliable to better than 3°. These deviations from the ideal condition cause the E-vector within the speci men to vibrate in a compact bundle with slightly varying azimuths and hence tend to diminish the variation in absorption intensity with respect to polarization direction. II. PREPARATION AND CRYSTAL NATURE OF THE SPECIMENS The specimens were mostly in the form of aligned multilayers imbedded in CCh to minimize scattering loss. The total thickness of the various specimens ranged from 0.04 to 0.5 mm. The two polarization directions used were those perpendicular and parallel, respectively, to the direction in which the films had be~n stretched or rolled. Untreated polymer films exist in a composite phase in which both crystalline and amorphous regions are dis tributed at random.5o 6 Treatments such as stretching, rolling, pressing, and annealing tend to increase greatly the ratio of ordered to amorphous regions. A small percentage of amorphous regions remain even in fully stretched or rolled specimens. Short side groups, if present, tend to disrupt the ordered structure in their vicinity. Longer side chains line up with the parent chain." 1. Polythene and "Parowaxll7; -(CH2)n- Oriented specimens of polythene were graciously supplied by Dr. Wilfried Heller, now at Wayne Uni versity, and his former co-worker at the University of Chicago, Dr. Hans Oppenheimer. These specimens were plastic films which had been stretched until the align ment of the molecular chain axes was as closely parallel to the stretch direction as possible. 8 Unoriented specimens of "Parowax" were sliced froJ!! a block. Semi-crystalline specimens were prepared by slow cooling of a melt on a hot water surface. To mini mize disorientation caused by the heat of the beam, these latter specimens were placed at the exit slit of the instrument. The crystal structure of polythene has been de termined by C. W. Bunn9 and by A. Charlesby.lO Two chain molecules pass through a unit cell, which contains four CH2 units. The CH2 planes are all parallel to the c(OOl) plane, but the symmetry axes of the CH2 triangles that belong to different chains are not parallel to each other. The planes of the zigzag carbon backbones are all parallel to the c-axis, but those of the two adjacent chains passing through the same unit cell form angles of ",82° with each other. The symmetry axes of the CH2 triangles lie in the carbon planes and are thus also inclined at ",82° to those of adjacent chains. In the stretched specimens the various microcrystals have their c-axes aligned with the direction of stretch, but have their a-axes (and b-axes) randomly distributed in the plane normal to this direction. Charlesby found in his specimens an average angle of 9° between the c-axes of the microcrystals and the stretch direction. 2. NyJonl1; -[NH-(CH2)6-NH-CO (CH2)s -COJn- A sheet of 6.10 polyamide 0.095 to 0.12 mm thick was very kindly supplied by Mr. R. B. Aken of DuPont. Fully stretched or rolled films were approximately 0.045 mm thick. Strips of rolled or pressed Nylon tennis string • E. M. Frith and R. F. Tuckett, Trans. Faraday Soc. 40, 251 (1944). 6 S. D. Gehman, Chern. Revs. 26, 203 (1940). 7 L. Glatt and J. W. Ellis, J. Chern. Phys. IS, 884 (1947). 8 W. Heller, Phys. Rev. 69, 53 (1946). 9 C. W. Bunn, Trans. Faraday Soc. 35,482 (1939). 10 A. Charlesby, Proc. Phys. Soc. (London) 57, 496 (1945). 11 L. Glatt and J. W. Ellis, J. Chern. Phys. 16, 551 (1948). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57INFRARED ,PLEOCHROISM 451 were also examined. Nylon tennis string, as sold com mercially, is pre-stretched. Some unstretched tennis string, desirable for testing the orientation effects of pressing, was also supplied by Mr. Aken. Spectrograms were also made of molten Nylon specimens. A heated absorption cell12 was used. The crystal structure of two polyamides, 66 and 6.10, have been fully determined by C. W. Bunn and E. V. Garner.J3 There is only one chain molecule passing through each unit cell. In a Nylon crystalloid, the molecular backbone planes are thus all parallel to one another. The chain axes are parallel to the crystallo graphic c-axis. The oxygen atoms of one molecule lie on the same level as the NH units of the next; the mole cules are linked into parallel sheets by intermolecular N -H· . ·0 bonds which extend perpendicularly to the c-axis. Excellent structure diagrams are given by Bunn and Garner.13 3. Polyvinyl Alcohol; -(CH 2-CHOH)n- Dr. Maurice L. Huggins of the Eastman Kodak Company supplied us with clear films of polyvinyl alcohol. To orient the films, they were either rolled be tween heated rollers or stretched under the heat of a commercial infrared lamp. Spectrograms were made for specimens oriented by each method. An x-ray study of the structure of this substance has been made by R. C. L. Mooney.14 Some features of her synthesized model have been objected to by Bunn and Peiser,t5 and by M. L. Huggins (in a correspondence with the authors). It seems probable, however, that Mooney was correct in concluding that the molecular chains are in pairs linked together by hydrogen bonds, with the C-C bonds in each molecular zigzag and the 0-H· . ·0 bonds between the two molecules both tending to be in the (101) plane. The molecular chains have their axes parallel to the crystallographic b-axis. It is likely that there are departures of considerable magnitude from this orientation resulting chiefly from a randomness relative to the plane of the carbon zigzag of the oxygen atoms in each chain ;15 there are two alterna tive positions for each 0-H group. This randomness was not considered by Mooney in her structure analysis. Strips of polyvinyl alcohol that are rolled between heated rollers become doubly oriented with the (101) planes preferentially in or near the plane of the strip and with the b-axes in the direction of roll. In stretched specimens the double chains lie along the stretch direc tion but in random orientation about it. Dr. Huggins was so gracious as to communicate the above informa tion to the authors. 12 A. E. Richards and H. W. Thompson, Trans. Faraday Soc. 41, 185 (1945). 13 C. W. Bunn and E. V. Garner, Proc. Roy. Soc. (London) A189, 39 (1947). 14 R. C. L. Mooney, J. Am. Chern. Soc. 63, 2828 (1941). 1& C. W. Bunn and H. S. Peiser, Nature 159, 161 (1947). 4. Polyvinyl Chloride; -(CH2 -CHCI-CH2- CCIH) n -and Polyvinylidene Chloride; -(CH2 -CCI2)n- Both of these substances, in the form of amorphous sheets, were obtained from Dr. M. L. Huggins. Speci mens were oriented by warming somewhat with a heat lamp and then stretching. The long chain molecules of the vinyl-polymer have their carbon backbones in a planar zigzag; the fiber repeat distance is, however, 5.0A, approximately twice that for polythene and polyvinyl alcohol. This has been interpreted as implying that the chlorine atoms are alternately placed, first 011 one side and then on the other with respect to the plane of the zigzag.16 The crystal structure of polyvinylidene chloride has been investigated by R. C. Reinhardt;17 his results indi cate that the carbon backbones are neither planar zigzags nor uniform spirals. He postulates a sort of serpentine configuration of the C-atoms with the C-C-C bond angles at 122° rather than the usual 109°-112°. This sort of a configuration would cause the CH2-(and the CCb-) planes to have an appreciable inclination toward the chain axis, and thus considerably diminish the pleochroism in the CH2 bands. It is there fore significant that the> CH2 bands in polyvinylidene chloride exhibit considerably less pleochroism than do those of any of the other, planar zigzag, polymers in vestigated (see Fig. 9), this, in spite of the fact that oriented polyvinylidene chloride specimens have a very high degree of crystallinity and very little branching in the chainsP . III. SELECTION RULES AND COUPLING FOR AN EXTENDED INFINITE > CH2 CHAIN Consider an infinite normal > CH2 chain with the planar zigzag carbon backbone in the yz plane (see Figs. 1 and 2). The full spatial symmetry of such a chain forms a "strip-group" of infinite order whose factor group is isomorphic to the finite point group D2h. There is point symmetry C2v(z) about a z-axis through each carbon atom. The normal modes of a > CH2 chain containing n links18 can be subdivided into nine sets of n-fold multiplets; one multiplet set corresponding to each of the three true vibration, three restricted rotation, and three restricted translation modes characteristic of a > CH2 vibrator. The multiplet component of a given set differ in the phase relations between the various> CH2 links; their frequencies can be expected to be closely packed or overlapping. It is evident, even without resort to group theory, that the net dipole moment increment of the chain can be expected to be negligible for most of these multiplet components. 18 M. L. Huggins, J. Chern. Phys. 13, 37 (1945). 17 R. C. Reinhardt, Ind. Eng. Chern. 35, 422 (1943). 18 Whitcomb, Nielsen, and Thomas, J. Chern. Phys. 8, 143 (1940). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57452 L. GLATT AND J. W. ELLIS (a) Y.so (b) ~If / "-7 ,/ (c) }lao (d) Va1T FIG. 3. 0-and ... -coupled C-H bond stretching modes in a > CH2 chain. Winston and Halford19 have analyzed the classifica tion of motions of a crystal and the selection rules for spectra according to space symmetry. Their analysis shows, impl~citly, that each of the nine sets of n-fold multiplets contains either one normal mode only or none at all with a non-zero dipole moment increment. The sets containing an active fundamental can be identified by analyzing modes of one> CH2 group under its local point symmetry C2.(z). The active member of the corresponding set is the mode that belongs to the totally symmetric representation of the (invariant) translation sub-group of the strip-group. This means that for infrared active fundamentals there must be zero TABLE I. Selection rules and band assignments frequencies of polythene bands. Fundamentals Freq. C,,(s) Mode (em-I) A2; f Bl; M. Va r B2; M. w 2853 1460 (1300) 2925 725 1310 (or 1375) Mode 2va 21'. 1'.+0 v.+w v.+t va+v, va+o v,+w va+t Binary combinations Computed Observed Band freq. freq. num~ (em-I) (em-I) bers 5790 ?{5782 2 5650 5782 2 4250 4247 5 4180 or 4250 4100 5720 5671 3 4325 4322 4 4110 4216 6 (or 4180) 4180 19 H. Winston and R. S. Halford, J. Chern. Phys. 7, 607-616 (949). phase difference between the motions of all > CH2 units that are interchangeable by pure translations. Adjacent > CH2 units are oppositely directed and cannot be interchanged by pure translations. It is evident that for infrared activity, the dipole moment increments of adjacent units must reinforce rather than negate one another. Of the modes shown in Fig. 3, for example, Jln and Jla" are infrared active while JI.o and Jlao are inactive. This would also follow from an analysis under the D2h factor group representations. The analysis of combination (and overtone) modes of the CH2 chain is considerably complicated by the C-C coupling of the chain units. Whereas the selection rules for fundamentals permit infrared activity by, at most, one component from each of the nine n-fold multiplets, the space symmetry selection rules permit activity, for any binary multiplet, by essentially n among the n2 pairs of components that can be formed. Thus, no composite transition corresponding to an arbitrary pair (or triplet, etc.) of multiplet sets is strictly forbidden by symmetry considerations.19 . Although it can be expected that most of these above :~ + , ' UJ,oo Uy I , I.T~ tsr.!' FIG. 4. Spectrograms with polarized radiation of stretchecl polythene film: (a) E " stretch direction; and (b) E ..1 stretch direction. modes will have very weak absorption, if n is large and they lie in a narrow frequency range, they mayaccumu late to produce considerable absorption. Thus, even neglecting anharmonicity, the center of the envelope for a doubly excited mode need not fall at the sum of the two active components. If the active fundamentals were both the lowest members of their respective sets, then the active combinations could well appear atfrequencies exceeding their sum. If the active fundamentals were the highest frequencies of their respective multiplet sets, the converse could be true. It has been suggested by a reviewer ofthis paper that the hypothesis outlined above may account for the apparently anomalous shifts from predicted positions, assuming normal anharmonicities, of some of the observed absorption bands discussed later in this paper. The space symmetry representations of the n per mitted binary transitions, for each pair of n-fold· multiplets, contain all factor group, and thus also all activity, representations of the space group. It is to be expected, therefore, that there will be a diminution of the pleochroism as well as possible apparent violations This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57INFRARED PLEOCHROISM 453 of the factor group and "site" symmetry selection rules. It is nevertheless proposed, in line with arguments advanced by Winston and Halford, to use as a suitable approximation the selection rules of the "site" sym metry, C2v(z), of the > CH2 units. The C2.(z) selection rules will be applied, therefore, to the observed > CH2 type bands of polythene and the other polymers. It is realized, of course, that the above discussion for an infinite extended > CH2 chain is only approximately applicable, even for polythene. Aside from the finite length of the polymer chains, there is some branching, and also some units other than > CH2 exist to cause complications. There is the possibility, moreover, that coupling by the crystal field between adjacent chains cannot be neglected safely in compari son to the coupling by the C-C bonds within a chain. The symmetry species of some of the more important > CH2 oscillator type modes are cataloged in Table I. These symmetry species were derived from the character FIG. 5. Spectrograms with unpolarized radiation of "Parowax:" (a) specimen with molecules randomly oriented [(a'l wide slits]; and (b') specimen prepared from a melt on the surface of hot water [(b) wide slits]. tables of the irreducible representations for the C2v(z) point group.20 The symbols for the dipole moment com ponents are listed adjacently to those of the irreducible representations to which they belong, f meaning that infrared activity is forbidden by the "site" symmetry (but not strictly forbidden by the full spatial sym metry). The wave number frequencies given for the fundamental modes (see Fig. 2) are consensus values from the literature for polythene. The alternate 1375 cm-1 frequency for the > CH2 "wagging" mode w is more usually assigned to the symmetrical deformation mode of the methyl units which are said to be present in small quantities on the side chains of polythene.21. 22 This 1395 cm-1 band shows, however, the y-type activity ex pected of a > CH2 w mode and has been so assigned by 20 J. E. Rosenthal and G. M. Murphy, Revs. Modern Phys. 8, 317 (1936). 21 H. W. Thompson and P. Tarkington, Proc. Roy. Soc. (London) 184A, 3 (1945). 22 Elliott, Ambrose, and Temple, J. Chern. Phys. 16,875 (1948). I l38~ I /.e~ FIG. 6. Spectrograms of stretched Nylon (6.10): (a) un polarized radiation; (b) E II stretch direction; and (c) E .1 stretch direction. at least one investigator, T. Simanouti,23 who gives arguments for questioning the methyl assignment. The predicted frequencies given in Table I for the binary combination and overtone modes were computed by adding the observed frequencies of the appropriate pairs of fundamentals and subtracting approximately 50 or 60 cm-1 as an anharmonicity term. It was pointed out previously that the n-fold multiplicity of permitted components in a composite binary transition can cause a significant shift from frequencies computed in the above fashion. A discussion of the proposed band assignments given in the last column of Table I, together with some possible alternative assignments, is given in the follow ing sections of this paper. The arbitrary band numbers listed after the observed (polythene) frequencies are those assigned to the absorption bands in Figs. 4-9. IV. RESULTS AND THEIR INTERPRETATIONS This section of the report shows reproductions of spectrograms made on the quartz-prism instrument. All of the spectrograms show several atmospheric water vapor bands, and the band at 2.20}.l caused by the quartz prisms. The numbered bands are those caused by j l.foB"., I 1.11".. FIG. 7. Spectrograms with un~olarized light of Nylon (6.10): (a) molten specimen kept at 300 C; and (b) solid specimen kept at 200°C. 23 T. Simanouti, J. Chern. Phys. 17, 734--7 (1949). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57454 L. GLATT AND J. W. ELLIS I 1.38"., I I.By. -----a ~----b ~A'-' -"=:;;:::==:=::; c T 'IS? I Z.29-I 2.S~ FIG. 8. Spectrograms with polarized. radiation of vacuum desiccator-dried polyvinyl alcohol: (a) rolled specimen; E II roll direction; (b) rolled specimen; E 1. roll direction; (c) stretched specimen; E II stretch direction; and (d) stretched specimen; E 1. stretch direction. absorption in the specimens. The positions and relative intensities of the numbered bands are listed in tables. Owing to causes which will be discussed later, it was often quite difficult to determine these relative in tensities with 1\.ny degree of accuracy. The comparative intensities of widely separated bands are accurate to only one significant figure. The relative intensities listed were, in general, averaged from' several spectrograms rather than taken from the particular spectrogram shown in the figures. 1. Bands in the 1.7V (5900 em-I) and 2.3v (4300 em-I) Regions Designations on the spectrograms reproduced in Figs. 4-9 make it evident which of the absorption bands of the other polymers correspond to those of polythene. The positions of these> CH2 type bands are tabulated in Table II; Table III gives their. relative intensities. I /.38"., I /.8 ''''' FIG. 9. Spectrograms with polarized radiation of stretched polyvinyl chloride and polyvinylidene chloride: (a) polyvinyl chloride; E II stretch direction [(a') wide slits]; (b) polyvinyl chloride; E 1. stretch direction reb') wide slits]; (c) polyvinylidene chloride; E 1. stretch direction [(e') wide slitS]j' (d) polyvinylidene chloride; E II stretch direction [(d') wide slits . The values for polyvinylidene chloride are tabulated separately in Table IV. a. Pleochroism in Nylon Bunn and Garner13 outline procedures for obtaining doubly oriented strips of Nylon film with the backbone planes almost parallel to the surface. A specimen of this sort would be expected to exhibit a considerably reduced absorption for "x-active" and a somewhat enhanced absorption for "z-active" modes. It will be recalled that the x-axis was chosen normal to the plane of the carbon backbone, and the z-axis parallel to the symmetry axes of the CH2 triangles. Nylon (6.10 polyamide) specimens initially in the form of unoriented films, stretched fibers, and unstretched fibers were variously treated, after the fashions outlined by Bunn and Garner, in an attempt to obtain double orientation. Unfortunately, only a limited degree of double orientation was achieved. There was very little significant contrast between the spectrograms of the singly oriented stretched specimens and those of any of TABLE II. Positions of the >CH2 bands. Band Polythene Nylon Polyvinyl ale. Polyvinylenl. No. freq. (em-I) freq. (em-I) freq. (em-I) freq. (em-I) 1 8237 8292 8319 8333 2 5782 5804 5838 5828 3' 5740 3 5671 5695 5688 5708 4' 4360 4 4322 4342 4329 4329 5 4247 4257 4255 4279 6 4216 4219 7 4184 4202 4202 4211 8 4130 4161 9 4095 4100 10 4018 4010 4010 4075. the specimens in which it had been attempted to produce double orientation. Some of the pressed fiber specimens did, however, appear to show a slight decrease in the intensity of band 3 relative to band 2 (see Fig. 6). This intensity change, which was in the opposite direction from that expected on the basis of the relative positions of the bands, may have been only apparent. It was difficult to estimate the correct background from which to measure the intensity of the bands involved. b. Pleochroism in Polyvinyl Alcohol More definite evidence qLn be had by comparing the spectrograms of stretched, with those of rolled polyvinyl alcohol. It was remarked earlier that the rolled speci mens are believed to have the zigzag carbon backbones, and thus the CH2 z-direction preferentially located in or near the plane of the sheet. It is evident from Fig. 8 and Table III that the ratio of the intensity of band 3 to that of band 2 is definitely diminished in rolled as compared to the stretched specimens. The intensity ratio of band 5 to band 4 is, on the other hand, slightly This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57INFRARED PLEOCHROISM 455 increased. This last is, however, somewhat unreliable. The energy background in the 2.3JL region changes too abruptly and is pulled down too severely by the overlying 0-H· . ·0 absorption to allow accurate de terminations of relative intensities. These same handi caps also restrict the accuracy of the relative intensity determinations in the 1.7JL region but not to so great an extent. The relative intensities listed in Table III are fairly reliable for the comparison of bands in the near vicinity of each other. For widely separated bands the listed intensities serve merely as orders of magnitude. This is particularly true of the polyvinyl alcohol bands. c. Assignments The above discussed pleochroism observed in the polyvinyl alcohol bands gives an admittedly flimsy basis for the assignment of the 1. 7 JL and the 2.3 JL doublets to vibration modes. The interpretation is further com plicated by the existence of the carbinol C-H units. The pleochroism evidence is the weakest for the 2.3JL doublet; it appears safe, however, to assign band 4 to the "x-active" va+o type modes and band 5 to the "z-active" v,+o type modes (see Table I). These as signments are in good agreement with positions, separa tions, and relative intensities of the observed bands. Four possible assignments, none of them completely satisfactory, are shown below for the 1.7 JL doublet. (1) (2) (3) (4) Band 2 Band 3 (apparently "z-active") 2v. type; "z-active" 2va type; "z-active" 2va type; "z-active" va+ v. type; "x-active" (apparently "x-active") va+v. type; "x-active" va+ v, type; "x-active" 2v. type; "z-active" 2v, type; "z-active" None of the above assignments is completely for bidden by the C2v(z) selection rules. The last two are, however, in disagreement with the evidence from the pleochroism observed in polyvinyl alcohol (and perhaps in Nylon). The third of the assignments is the only one for which the observed band pos}tions are in good agreement with the frequencies computed from the active components of the fundamental multiplets. This is evident from Table I. It is difficult, however, to reconcile so strong an activity as that observed in band 2 with what would be expected from a "z-active" mode com pounded out of "x-active" and inactive multiplet com ponents. This latter criticism also holds for the second assignment, which, however, is in agreement with the observed pleochroism. The first assignment is in agreement with the ob served pleochroism and does not incur the difficulties mentioned above. It is therefore somewhat hesitantly favored by the authors even though it gives the poorest agreement between observed and computed frequencies. A justification, in terms of multiplet structure, has already been advanced for the assumption of such large frequency shifts. In addition, there is a precedent, from TABLE III. Relative intensities of the >CH 2 bands. Stretched Stretched Stretched Rolled Stretched Band polythene Nylon pvl. ale. pvl. a1c. pvl. chI. No. .L II .L II .L II .L II 1 0.6 0.2 0.7 0.4 O.S 0.3 O.S 0.3 2 4 1.9 3.2 1.3 3 1.2 3.6 1.5 3' 3 1.7 1 2.2 1.2 1.7 0.7 1.2 0.9 4' 4 10* 6 10* 7 10* 6 10* 7 5 8 5 10-7-9-5 10-6 6 0 3 0 1.7 7 n 0.5 6 0 4 0 5 0 8 9 5+ 0.5 10 2 0 5 5 5 5 * Intensity arbitrarily set at 10 . .L E-vector perpendicular to stretch (or roll) direction. II E-vector parallel to stretch (or roll) direction. .L II 0.3 0.2 4 2.5 0.05 0.05 1.6 1.3 0.1 0.2 10* 7 4 3.5 6 5 5 4 4 3 2 1 the spectra of water vapor, for the assumption of an extraordinarily large negative anharmonicity term for the va+v. type combinations.24 The authors have also found a like phenomenon in the > CH2 bands of many of the organic crystals that will be discussed in a later paper. There is precedent also for the possible assump tion of a positive anharmonicity term for the 2v. type mode. In the v. type vibration of methane, CH., a positive anharmonicity term is associated with the repulsive forces between the H-atoms which approach one another as they move on the surface of a hypo thetical sphere.25 In 2v, type modes of> CH2 oscillators, the two H-atoms approach one another as the C-H lengths contract, and a repulsive force probably is present. d. Polyvinyl Chloride and Polyvinylidene Chloride. The absorption bands in these two polymers will not be discussed in detail. The 1. 7 JL region in the former is very similar to those in polythene, Nylon, and polyvinyl alcohol. There is no distinct band that can be attributed to the C-H bond in the CHCI units; this is similar to what was found in polyvinyl alcohol. The bands in the region beyond 2.2JL are too broad to be conveniently classified with those of polythene and the other polymers. The 1.7 JL region in polyvinylidene chloride is markedly different from that of the other polymers. This difference TABLE IV. The polyvinylldene chloride bands. Band Freq. Relative intensity No. (cm-') .L II 1 8439 0.4 0.4 2 5924 1.4 1.2 3 5787 4 3 4 5624 9 8 5 4370 10* 8+ 6 4272 8 10- 7 4172 4 6 8 4125 3 5 24 G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand Company, Inc., New York, 1945), p. 282. 25 D. M. Dennison, Revs. Modern Phys. 12, 207 (1940). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57456 L. GLATT AND J. W. ELLIS TABLE V. The perturbed N-H···O and C=O bands in stretched Nylon (6.10). Band Frequency Nos. (10-' 'cm-l) I' 6.789 2' 6.636 3' 6.523 3'/ 6.757 4' 6.373 5' 6.223 6' 5.13 7' 4.975 8' 4.883 9' 4.70 10' 4.591 Relative intensities* .L 0.01 0.01 1.4 0 0.7 0.4 2 2.3 " 0.02 0.02 0.5 0 0.4 0.4 o 5 o 0.03 0.04 0 Assignments Perturbed 2v(N-H) Free 2v(N -H) ?Perturbed 2v(N-H) v+/l(O-H) of bound H20 3v(C=0) vH(N-H) * The listed intensities are relative to the arbritrary value 10 for the CH, band number 4. is very probably caused by the difference in skeletal structure.16 Band 2 (Fig. 9, c and d), which is definitely absent from the spectrograms of the other polymers, may be caused by the 2va type mode. Bands 3 and 4 are quite broad and probably complex. This is also true of the longer wavelength bands. A comparison of the spectrograms of polyvinylidene chloride with those of the other spectrograms shows that there is a greatly reduced pleochroism, particularly in the 1.7 p. region. It has been pointed out in an earlier section of this paper that this decrease in pleochroism is a logical result of the altered structure of the carbon backbone. The authors wish to suggest that pleochroism studies of this sort might prove to be a useful implement to x-ray and electron diffraction methods in investi gating the structure of long chain compounds. It may be that the pleochroism in some of the longer wavelength, skeletal type modes would give a fairly sensitive indi cation as to whether the backbone structure was a planar zigzag, a uniform spiral, etc. 2. Unusual Pleochroism in the 2.372- (4216 em-I) Band The weak band 6, at 4216 cm-l in polythene, is excited only by the E-component vibrating parallel to the molecular chain axes. The pleochroism exhibited by this "y-active" band is thus not only the most pronounced but also is in the direction opposite to that of any of the other> CH2 bands. Figure 6 shows that this "y-active" band also appears, considerably diminished, at 4219 cm-l in oriented Nylon specimens. It is too weak to be evident in the unpolarized spectrograms of Nylon, and is completely absent from the spectrograms made of polyvinyl alcohol and of polyvinyl chloride. Since band 6 appears distinctly for polythene speci-mens as thin as 50p., the mode responsible for it very probably involves displacements within the CH2 units. Table I shows that possible assignments of this band are to the "y-active" v.+w or va+t type modes. The first of these assignments is the more attractive. The w .. mode is essentially a coupled motion of the entire chain in which all the hydrogens vibrate against all the carbons in a direction parallel to the chain axis. The oscillations could be expected to be restricted severely by the presence of the alternate CHX links in vinyl polymers, and to a lesser extent by the -CO-NH -links in Nylon. The above might also hold true for certain combi nations of wand v type modes, particularly for v.o+w ... This would explain the decreased intensity of band 6 in Nylon and its failure to appear in the vinyl polymers. The previously discussed "multiplet structure" of com posite transitions can again be called upon to justify the fact that the observed frequency is about 36 or 106 cm-I higher than that computed from the active fundamentals, depending upon the choice between the two w fre quencies listed in Table I. Band 6 could possibly be assigned to the va+t type modes. One would, however, expect this type of mode to be active only very weakly, if at all, since none of the component Va or t type fundamentals belongs to a "y-active" representation. It is, in fact, questionable whether the fundamentals of the> CH2 "twisting" type modes have been observed at all in the infrared. 3. Additional > CH2 Bands Band 1, which occurs at 8237 cm-I in polythene, is very likely caused by some sort of a 3v type mode. A comparison of the position of this band with that of the fundamental Va and v. modes shows that a rather large (negative) anharmonicity term occurs. Bands 7, 8, 9, and 10 may result from combinations of the> CH2 type modes with some frequencies of the carbon backbone. There is a slight possibility that some of these bands, possibly 8 and 9, which are apparently absent from Nylon and the vinyl polymers, originate in the methyl side groups. 4. "Parowax" The findings for "Parowax" are an example of a pleochroism study made with unpolarized radiation. The spectrograms made for amorphous "Parowax" specimens (Fig. 5 j a, a/) are essentially identical with those of unoriented polythene. The unpolarized light spectrograms made for the crystalline specimens give convincing proof of an excellent alignment of the chain axes normal to the surface of the films.7. 26 As is shown in Fig. S, band b', the "parallel" band 6 at 4216 cm-l is completely missing and the intensity ratios and general appearance of the observed bands are unmistakably like those of the spectrograms made for oriented 36 J. J. Trillat and T. V. Hirsch, Compt. rend. 195, 215 (1932). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57SOLID-STATE LUMINESCENCE 457 polythene when the E-vector was directed perpen dicularly to the chain axes. 5. N :-H···O and C=O Absorption in Nylon The perturbed N-H···O and C=O bands in Nylon were discussed in an earlier preliminary report ;11 so also was the change in structure of certain of these bands in molten specimens and in specimens kept at 200°C, below the 260° melting point. The preliminary report did not, however, show spectrograms. The N -H· . ·0 and C= 0 bands are indicated in the spectrogram of Fig. 6 by primed numbers. The positions and relative THE JOURNAL OF CHEMICAL PHYSICS intensities of these bands are tabulated in Table V. This table also gives assignments for some of the bands. Figure 7 shows spectrograms of molten and of 200°C specimens. Band 3'f, which appears at 6757 cm-1 in these spectrograms, is caused by the free 2v(N -H) bond stretching mode. The N -H· . ·0 bond ruptures in the molten specimens, and apparently to some extent in the hot unmolten specimens.27 The perturbed 0-H· . ·0 bands in polyvinyl alcohol were discussed in a preceding report.28 27 Fuller, Baker, and Pope, J. Am. Chern. Soc. 62, 3275 (1940). 28 Glatt, Webber, Seaman, and Ellis, J. Chern. Phys. 18, 413 (1950). VOLUME 19. NUMBER 4 APRIL. 1951 An Absolute Theory of Solid-State Luminescence* FERD E. WILLIAMS General Electric Research Laboratory, Schenectady, New York (Received December 26, 1950) The absorption and emission spectra of the thallium-activated potassium chloride phosphor at various temperatures has been computed theoretically. An ionic model is used. The radial charge densities of free Tl+ in the ground 'So state and in the excited 3P,0 state are evaluated using the Sommerfeld modification of the Fermi-Thomas method for the core and the Hartree self-consistent field method for the two outershell electrons. From these wave functions and from the known ionic radius, polarizability, and re pulsion energy cohstant p for the ground state, these parameters are evaluated for the Tl+ in the excited state interacting with Cl-. The variation of repulsion energy with interatomic distance a is shown to be equal to the variation of S2/a with a, where S is an overlap integral. The Tl+ in the 'So and the 3P,0 states are sub stituted in dilute concentrations for the K+ in KCl, and the change I. INTRODUCTION THE quantitative interpretations of solid-state luminescent phenomena have been phenomeno logical. Fundamental interpretations based on the elec tronic theory of solids haye produced only qualitative results. Detailed atomistic explanations frequently have been based on a configuration coordinate model,l Diagrams of potential energy versus a position co ordinate have been applied to problems of molecular spectra2 and rates of chemical reactions.3 In these applications, the position coordinate can be precisely specified as an interatomic distance. To describe the atomic rearrangements involved in electronic processes in solids r~quires, in principle, energy contours in con- * Presented October 10, 1950 at the National Academy Meeting in Schenectady. See Science 112, 428 (1950) and Phys. Rev. 80, 306 (1950). , F. E. Williams, J. Opt. Soc. Am. 39, 648 (1949). 2 A. G. Gaydon, Dissociation Energies (Dover Publications, New York, 1950). 3 Glasstone, Laidler, and Eyring, Theory of Rate Processes (McGraw-Hill Book Company, Inc., New York, 1941). in total energy of the sy'stem is calculated as a function of the change in the Tl+ nearest Cl-distance Aa with the condition that the remainder of the lattice rearranges to minimize the total en ergy. Madelung, exchange repulsion, van der waals, ion-dipole and coulomb overlap interactions are included. The absorption spectrum is computed by recognizing that the various atomic con figurations of the system in the ground state have probabilities in accord with a Boltzmann function. The emission spectrum is similarly determined by summing over configurations of the sys tem in the excited state. The computed spectra at various tem peratures are found to be in good agreement with experiment. In addition, new insight is obtained on the detailed mechanism of solid-state luminescence. figuration space having three times as many coordinates as there are particles involved. In describing the transi tions and rearrangements occurring during lumines cence, a useful simplification consists of defining a "center" as the excited atom plus the group of associ ated atoms participating in the rearrangements and' then vaguely describing the coordinates of the center by a single average configuration coordinate. Seitz,4 Gurney and Mott,6 and Pringsheim6 have used such potential energy diagrams to illustrate qualitatively various effects in the luminescence of solids, and Williams and Eyring? have correlated phenomeno logically various diverse properties of luminescent solids with calculations based on a simple configuration coordinate model. The use of a model involving potential energy plotted against an unprecise coordinate to describe the lu- 4 F. Seitz, J. Chem. Phys. 6, 150 and 454 (1938). 6 R. W. Gurney and N. F. Mott, Trans. Faraday Soc. 35, 69 (1939). 6 P. Pringsheim, Revs. Modern Phys. 14, 132 (1942). 7 F. E. Wil1iam~ ~nd H. Eyring, J. Chern. Phys. 15, 289 (1947). This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 137.99.26.43 On: Wed, 23 Oct 2013 22:16:57

1.1747622.pdf

The Vibrational Spectra of Molecules and Complex Ions in Crystals III. Ammonium Chloride and DeuteroAmmonium Chloride E. L. Wagner and D. F. Hornig Citation: The Journal of Chemical Physics 18, 296 (1950); doi: 10.1063/1.1747622 View online: http://dx.doi.org/10.1063/1.1747622 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/18/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Analysis of the heat capacity results of deuteroammonium and ammonium solids J. Chem. Phys. 89, 2324 (1988); 10.1063/1.455075 Vibrational Spectra of Molecules and Complex Ions in Crystals. VIII. The Infrared Spectrum of a Mixed Crystal of 1DeuteroAmmonia and 2DeuteroAmmonia J. Chem. Phys. 23, 1053 (1955); 10.1063/1.1742190 Vibrational Spectra of Molecules and Complex Ions in Crystals VII. The Raman Spectrum of Crystalline Ammonia and 3DeuteroAmmonia J. Chem. Phys. 22, 1926 (1954); 10.1063/1.1739942 The Vibrational Spectra of Molecules and Complex Ions in Crystals. V. Ammonia and DeuteroAmmonia J. Chem. Phys. 19, 594 (1951); 10.1063/1.1748298 The Vibrational Spectra of Molecules and Complex Ions in Crystals IV. Ammonium Bromide and Deutero Ammonium Bromide J. Chem. Phys. 18, 305 (1950); 10.1063/1.1747623 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12296 E. L. WAGNER AND D. F. HORNIG cm-1 to the red (solvent not specined). Sheppard8 states, however, that while the shorter wave-length bands are displaced to the red (solvent unspecined), the longer wave-length bands move to the violet in solution. * From Eqs. (10)-(12), one would expect AI' to be about 1/10,000 of AI' for the (V, N) transition. When the electrostatic displacement becomes so small, other effects such as that of the solvent cagel2,13 become predominant, and the quantitative evaluation of AI' becomes impossible. For the well-known visible continua of bromine and iodine we have j=0.002926 and j=0.015,21 respectively. If a is chosen as 0.6A to conform to the results for the (V, N) transition in Table I, Eq. (10) predicts AI'= 780 cm-1 for bromine in n-hexane. With the actual value of a= 2.5A, the predicted AI' is about 10 cm-1• The experi mental value is 100 cm-t,26 while for most solvents, particularly if they are associated, the displacements * Added in proof: K. Lauer and R. Oda (Berichte, 69, 851 (1936)) give data for the 2600A benzene system in 11 solvents. The displacements to the red vary from 100 to 400 cm-l and are qualitatively in agreement with the refractive index law. 2& Bayliss, Cole, and Green, Australian J. Sci. Research, Series A 1, 472 (1948). THE JOURNAL OF CHEMICAL PHYSICS AI' are to the violet ranging to 1400 crrrl.26 Similar experimental results obtain for iodine.27 These are again cases where relatively small electrostatic dis placements are superimposed by other influences such as the solvent cage effectl2, 13, 26 which causes displace ments to the violet. Other authors27 have ascribed the violet shift of the iodine continuum in associated solvents to solvate formation. Whether the true expla nation be solvation or caging, these and the electrostatic polarization effect are of the same order of magnitude in weak bands, and the observed displacements must be regarded as their resultant. Subject to the complications arising out of the superposition of the other effects particularly in weak bands, the qualitative dependence of the electro!)tatic polarization effect on j, a (or a), and n seems to be jus tined by the experimental results. The quantitative agreement is good for the (V, N) transitions in isoprene and benzene, although the data do not allow one to decide which of Eq. (10)-(12) is the best. The cases of the ultraviolet absorption of bromine and iodine are much less satisfactory from the quantitative viewpoint. 27 O. J. Walker, Trans. Faraday Soc. 31, 1432 (1935). VOLUME 18, NUMBER 3 MARCH. 1950 The Vibrational Spectra of Molecules and Complex Ions in Crystals III. Ammonium Chloride and Deutero-Ammonium Chloride* t E. L. WAGNERt AND D. P. HORNIG Metcalf Research Laboratory, Brown University, Providence, Rhode Island (Received July 11, 1949) The infra-red spectra of thin non-scattering films of NH,Cl and ND,Cl were obtained at 28°, -78° and -190°C. A convenient low temperature transmission type cell usable for such films is described. No indication of fine structure due to free rotation of the NH,+ ions was found. Instead, evidence is presented for the existence, both above and below the X-point, of a torsional lattice mode involving the NH,+ ions. The limiting frequencies of the torsional oscillations were observed at about 390 and 280 cm-I for NH,CI and ND,CI, respectively. These values agree quite well with the frequencies calculated on the basis of a purely electro static potential function. The spectra of the low temperature modifications indicate strongly that the structures belong to the 1. INTRODUCTION A LL of the simple ammonium salts have second order phase transitions in the vicinity of -30° to -60°C, but the nature of these transitions has never * Based in part on a thesis submitted by Edward L. Wagner in partial fulfillment of the requirements for the Ph.D. degree at Brown University (1948). t This work was supported in part by ONR under Contract N60ri-88, T.O. 1. t Tennessee Eastman Corporation Fellow, Brown University (1946-48). Present address: Department of Chemistry, State College of Washington, Pullman, Washington. space group Ttll in which the NH4+ ion symmetry is Ttl. Of the eight observed bands, two are assigned to the triply degenerate fundamentals 1'3 and V4, one to the overtone 2v" one to the com bination V2+V, which resonates strongly with Va, one to the com bination of the totally symmetric mode, VI, with the limiting lattice frequency, JI&, and two to the combinations involving the lattice torsional mode, V6, i.e., V,+V6 and V2+"6. The spectra of the room temperature modifications are consistent with a struc ture in which the NH4+ ion tetrahedra are randomly distributed between the two possible equilibrium orientations in each unit cell. The X-point transformations are probably simple order disorder transitions between the two modifications. been clarined. Paulingl has advanced the hypothesis that these transitions mark the onset of essentially free rotation, while Frenkel,2 on the other hand, has sug gested that such transformations are order-disorder transitions in the orientations. In the case of NH4Cl a considerable amount of ex perimental data is available. Lawson3 has demonstrated that the evidence obtained from the measurement of 1 L. Pauling, Phys. Rev. 36, 430 (1930). 2 J. Frenkel, Acta Physicochimica 3,23 (1935). 3 A. W. Lawson, Phys. Rev. 57, 417 (1940). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12AMMONIUM CHLORIDE SPECTRA 297 the heat capacity at constant volume (Cv) in the vicinity of the ~-point is strongly opposed to any rotation, or even near rotation, in either phase. His data seem in controvertible. Earlier spectroscopic evidence which appeared to show rotational structure on at least one of the vibrational bands4•5 has been shown to be defi nitely incorrect,S although recently it has been argued that Raman polarization measurements indicate rota tion about a single threefold axis.7 The x-ray evidence of Smits and MacGillivry8 and Ketelaar9 has confirmed that the N and the CI atoms are arranged in a CsCllattice in both phases although the lattice expands by one percent as the temperature is raised through the transition at -30.5°C. It has been shown that below the transition point NH4CI exhibits a piezoelectric effectI°.ll which vanishes in the room temperature phase. Similarly, the Raman spectro graphic investigations of Menzies and Mills12 show a distinct lattice line at 183 cm-1 in the low temperature modification (Phase III)~ which is absent in the room temperature modification (Phase II).~ However, Krish nanl3 has found that this line (as well as several others) persists in Phase II although with much less intensity. None of these data are opposed to the structure of Phase III which is consistent with the x-ray evidence, namely a CsCI lattice with space group symmetry T i, but they offer no definite evidence regarding the nature of the phase transition. In the first paper of this series14 it was shown that the fundamental vibrations of the crystal which may be active in the infra-red or Raman spectrum are those which are allowed under the crystallographic point group. The selection rules for the fundamentals of the NH4+ ion may also be obtained from the local symmetry of the ion in the crystal. Since the local symmetry of the ion is T d if the space group is T dl, both of these sta te ments lead to the conclusion that only those funda mental vibrations allowed under the tetrahedral group, T d, should be active in either spectrum if the symmetry of the low temperature phase is indeed T i. Since all of the fundamentals are then allowed in the Raman spectrum, it does not afford a critical test. However, 4 R. Pohlman, Zeits. f. Physik 79, 394 (1932). 6 C. Beck, J. Chern. Phys. 12, 71 (1944). 6 E. L. Wagner and D. F. Hornig, J. Chern. Phys. 17, 105 (1949). 7 L. Couture and J. P. Mathieu, Proc. Ind. Acad. Sci. 28A, 401 (1948). 8 A. Smits and C. MacGillivry, Zeits. f. physik. Chemie A166, 97 (1933). 9 J. A. Ketelaar, Nature 134, 250 (1934). 10 A. Hettich and A. Schleede, Zeits. f. Physik 50, 249 (1928); Zeits. f. physik. Chernie A168, 353 (1934). 11 S. Bahrs and J. Engl, Zeits. f. Physik 105, 470 (1937). 12 A. C. Menzies and H. R. Mills, Proc. Roy. Soc. London 148A, 407 (1935). ~ This is the designation used by A. Eucken, Zeits. f. Elektro chemie 45, 126 (1939). In this system the phase (or modification) just below the melting point is designated by the number I and successive phases at lower temperatures are designated by suc cessively increasing Roman numerals. In ~CI Phase I has the NaCI structure which changes to Phase II at 184.3°C. 13 R. S. Krishnan, Proc. Ind. Acad. Sci. A26, 432 (1947). 14 D. F. Hornig, J. Chern. Phys. 16, 1063 (1948). only the two triply degenerate internal modes and one triply degenerate limiting lattice mode are allowed in the infra-red spectrum. No such definite predictions can be made regarding Phase II. It seems clear from free energy considerations that the transition must consist of some sort of dis ordering process, in which case two possible effects can appear in the spectra: (1) Selection rules may be re laxed and the transitions not allowed in Phase III may become active in Phase II, and (2) degeneracies may be lifted and degenerate vibrations in Phase III may be spread out into narrow bands in Phase II. The infra-red spectrum of NH4Cl has been previously studies by several investigators and Pohlman4 has most recently studies both phases. The spectra obtained have been used to argue either that the ammonium ion does not possess tetrahedral symmetry, or that the simple selec~ion rules previously mentioned are incorrect.lI;' 16 In order to throw further light on these questions the infra-red spectra of both NH4Cl and ND4CI were studied in Phase III at -190°C and -78°C and in Phase II at 28°C on films essentially free of scattering and with considerably higher resolution than pre viously used. It appears that the difficulty was due to a consistent misinterpretation of the spectrum of the NH4+ ion. Since a torsional lattice vibration is observed in both phases, there can be no question of free rota tion in either phase, and it appears that the ~-point transition is of the type suggested by Frenkel2 and con sists of a change from relative order to relative disorder of the equilibrium orientations of the NH4+ ions. A detailed theory of the Bragg-Williams type based on this idea has been developed for NH4CI by Nagamiya.n U. EXPERIMENTAL METHODS AND APPARATUS The ammonium salts employed in these investigations were reagent grade chemicals. Thin films of these salts on rocksalt plates were used in obtaining the spectra. The films were prepared by subliming the ammonium or deutero-ammonium halide through a loose glass wool plug onto a polished rocksalt plate which had been cleaned with the high frequency discharge of a vacuum leak tester. The sublimations were performed in a closed glass system under a pressure of about lo--2-mm Hg at 110-130aC. Under such conditions the films ob tained were quite clear. They exhibited only very little scattering in either visible or infra-red radiation pro vided that the thicknesses were less than about 0.6 micron. The majority of the films investigated had thicknesses, as estimated by the interference colors of reflected white light, of about 0.1 to 1.0 micron. The deutero-ammonium halides were prepared by means of a simple exchange reactionl8 in which the 16 R. Ananthakrishnan, Proc. Ind. Acad. Sci. SA, 76 (1937). 16 L. Couture, J. Chern. Phys. 15, 153 (1947). 17 T. Nagarniya, Proc. Phys. Math. Soc. Japan 24, 137 (1942). 18 K. F. Bonhoeffer and G. W. Brown, Zeits. f. physik. Chemie B23, 171 (1933). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12298 E. L. WAGNER AND D. F. HORNIG corresponding ammonium salt was dissolved in the minimum amount of 99.6 percent D20 at approximately 70°C and the exchange water evaporated off in a vacuum at room temperature. This process was repeated four or five times with fresh batches of D20. A conversion to 99.5 percent deuterated ammonium salt has been claimed for this procedure.19 Our samples contained considerably more than 0.5 percent hydrogen at the time the spectra were obtained, possibly acquired in handling because of the somewhat more hygroscopic nature of the deuterated compounds. The halide films were maintained at the appropriate temperature by means of the low temperature absorp tion cell illustrated in Fig. 1. This is a direct trans mission type cell which can conveniently be inserted THERMOWELL COOLANT RESERVOIR HOUSEKEEPER COPPER-GLASS SEA-:::L...r---.<;:=, ROCKSALT F Il.M-SUPPORT W '''~. ,~ GR OUND OINT I! It COPPER COOl.ING-BLOCK PlUG ( ROCKSAl. T WIND~ ~ .' )) :: '\ \\ VACUUM JACKET FIG. 1. Infra-red absorption cell for use at low temperatures. into the beam of almost any infra-red spectrometer without changing the optical path. The cell was com posed of a copper cooling block, a coolant reservoir, and an enveloping jacket. The cooling block consisted of a piece of solid copper through which a rectangular hole had been machined in such a way that a shoulder existed in the center of the block. The rocksalt plate on which the ammonium halide film had been sublimed rested against this shoulder and was held in position by means of an annular copper plug. A thin layer of vacuum grease between this shoulder and the rocksaIt plate was found necessary to obtain adequate thermal contact. The temp~rature of the cooling block was measured by 19 Clusius, Kruis, and Schanzer, Zeits. f. anorg. allgem. Chemie 236,24 (1938). means of a fine thermocouple inserted into a thermo well which was embedded in the block, and the tem perature of the rocksalt plate was measured by another fine thermocouple whose junction was embedded into a hole drilled halfway through the salt plate. The glass coolant reservoir was attached to the cooling block by means of a copper to glass Housekeeper seal which gave an extremely satisfactory vacuum-tight joint. The rock salt or silver chloride windows were sealed onto the flat ground ends of the glass jacket with clear Glyptal lacquer. Prior to cooling, the assembled cell was evacuated to a pressure of about lo-4-mm Hg. Under such condi tions it was found that the cooling block quickly at tained the refrigerant temperature. However, at liquid nitrogen temperatures the rocksalt plate supporting the ammonium halide film was S° hotter than the block. At -78°C no temperature difference between the block and the plate could be detected. In the radiation beam of the spectrometer the temperature of the rocksalt plate did not increase. Under the normal operating conditions of this cell less than 200 ml of liquid nitrogen were required to maintain the temperature of the rock salt plate at -190°C for one hour. When the cooling block was maintained at liquid nitrogen temperature the exterior windows did not cool perceptibly. The spectra were taken with a modified Perkin Elmer Spectrometer Model12B. The modification con sisted of replacing the usual system for focusing the ra diation source onto the entrance slit with a double beam arrangement for shifting the beam alternately through the sample and through a blank cell. With this instru ment alternate point by point recordings of the trans missions of the sample and a blank were obtained. In addition, the instrument recorded the zero or base line positions after each transmission point. This has the advantage of essentially eliminating errors in transmis sion due to base line drift, a factor which may become significant in low temperature work. The optical paths of the spectrometer were com pletely enclosed. Prior to recording a spectrum the housing was thoroughly flushed out with dry nitrogen gas until the CO2 band at 4.26tL and the 2.67 tL. H20 band were hardly discernible. Under such conditions, the maximum absorption in the 6.3tL water vapor band was less than 10 percent. This procedure was adopted when it was ascertained that the point by point method of recording spectra leads to the appearance of spurious peaks in the plotted spectrum. This results from the paucity of points along the steep slopes of deep at mospheric absorption bands. These false peaks do not necessarily coincide with those of the atmospheric bands, but may be situated along one or both of the slopes depending upon their relative steepness. Calcium fluoride and rocksalt prisms were utilized in these investigations. The slit widths employed were such that the theoretical widths of the frequency bands resolved were (for the fluorite prism) 9 cm-l at 3tL, This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12AMMONIUM CHLORIDE SPECTRA 3.3 cm-1 at 7 p., (and for the rocksalt prism) 4.6 cm-I at 8p. and 2.7 cm-1 at lOp.. Ill. EXPERIMENTAL RESULTS I The spectra obtained for NH4CI are shown in Fig. 2. In these curves the noise level has not been smoothed out nor have they been corrected for scattering. All the data of these spectra pertain to a single film whose thickness, as estimated from the color of reflected light, was about 0.2 micron. In addition to the records re produced here, spectra were also obtained for eight other NH4CI films, ranging in thickness from 0.1 to 1.0 micron. No difference between the various spectra could be detected except that in the thickest film the frequencies of maXImum absorption seemed to be shifted to higher frequencies by 5 to 10 em-I. Since it has been shown that NH4CI smokes with particle sizes about equal to the thickness of our films have the bulk crystal structure,20 the cause of the apparent frequency shift is probably not due to a change of crystal struc ture with thickness. The corresponding spectra of ND4Cl are given in Fig. 3. Three different films of ND4CI were studied. Unfortunately, the films of this salt were always con taminated with some NDaH+ and ND2H2+ ions so that some of the observed peaks must be ascribed to these contaminations. This was done by analogy to the corresponding deutero-methanes, all of which have been studied,21 and by comparison with the Raman spectra of the various deuterated ammonium chlorides.22 The observed frequencies of maximum absorption, the relative integrated peak intensities, and our assign ments are given in Tables I and II. In the case of over lapping bands the intensities were partitioned in what seemed to be the most reasonable way but, particularly in the case of the 3.2p. region in NH4CI, this partitioning was undoubtedly inaccurate. The relative intensities of the bands are believed to be accurate to about 15 percent. The most striking difference between the spectra of NH4CI and ND4CI is the contrast between the very strong single peak at 2336 cm-1 in ND4CI and the strong doublet at 3080 cm-1 in the corresponding NH4CI spectra. Relative to the sharp peaks at 1067 cm-1 in ND4CI and 1403 cm-1 in NH4CI, it is found that the total intensity of the doublet at 3080 cm-1 in NH4Cl is about equal to the intensity of the strong single peak at 2336 cm-1 in ND4Cl. This suggests strongly that the doubling is due to Fermi resonance between the strong fundamental, va, and an almost coincident combination level. The temperature behavior of the bands at 1760 and 2000-2100 cm-l in the NH4Cl spectrum is very interest- 20 J. Trillat and A. Laloeuf, Comptes Rendus 227, 67 (1948). 21 See G. Herzberg, Infra-Red and Raman SPectra of Polyatomic Molecules (D. Van Nostrand Company, Inc., New York 1945), p.309. 22 R. Ananthakrishnan, Proc. Ind. Acad. Sci. SA, 175 (1937). 1-=_:;.;-;;:;.-:.=====---- + :;l q.. ., . ., on • oS ., N .. I ~- ~-.. ! , OJ !II o o N ", .. 8 o il '" 299 cJ ~ - This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12300 E. L. WAGNER AND D. F. HORNIG PERCENT TRAw.PnSS~ ing. They change from broad symmetrical bands at ~ g i 0 0 .. OJ 2«> .. .. room temperature to curiously shaped multiple peak ::I. q § bands at -190°C. There is no evidence of discontinui-g- ties in their behavior at the X-point. The details of the .... ~-.,.. .,; shape of the 1760-cm-1 band, determined on a thicker 5-g film, have been published in an earlier communication,6 eli ,i}-and those of the 200o-2100-cm-1 band are given in iii 0 Fig. 4. It is interesting to note that these are the only 0 !\! ::I. two bands in the spectrum which are significantly 0- iii 0 shifted in the spectra of the heavier halides of am- ~ ::I. monium·11 "'-,.; One of the most salient features of the NH4Cl spec- 0 .... 0 .,.. ::I. ~ trum is the complete disappearance at low temperature 0-,.; of the subsidiary peak on the high frequency side of the 0 0 1403-cm-1 band. The intensity of this peak, which ~-!!l .; U amounts to 10 percent of the main peak at room tem- 0 0 perature, drops quite abruptly at the X-point but ap- 0 0 !!! ~ parently it does not completely disappear at this tem-:l, 0- 0 d perature. .; 0 t:: >-< All of the lines in the spectrum show a notable >-< >-< '" sharpening as the temperature is lowered and the weak 0 '" U ::I. 0 oj peak in the center of the NH4CI doublet at 3080 cm-1 !!! ..<:: .. on-~ c 'i':? ,..; is only resolved at the lowest temperature. In general, z -0 ~ 0 0 ~ the intensities of the fundamentals decrease by about 0 oj .. !!! !!! U , 30 percent between temperature and liquid 0 room ::E V) ::I. 0 '" .-..: nitrogen temperature, while the intensities of com-q-8 a: N on N '" ..... bination and overtone bands are relatively less affected Q. oj II> >-< by this temperature change. 0 '" >-< 0 ~ '" OJ ;a til oj ..<:: IV. THE AMMONIUM ION FUNDAMENTALS ~ ~ 8 .5 The fundamental ion frequencies of the NH4+ ion ::I. N 0 "l-N .. may be expected to lie slightly higher than those of ~ ci .... Z methane. It seems apparent, therefore, that the low .,.. ---.. .... frequency bands at 1067 and 1403 cm-1 in the two 0 /"'-'-'-'-'---;'-0 E chloride salts may be assigned with confidence to the ~ ::l .!:: N U triply degenerate bending mode, 1'4, of the tetrahedral ! '" c. model. Indeed, the ratio of these frequencies in ND4+ ::I. § til 0- -0 and NH4+, 0.7605, is very close to the corresponding .. OJ '" .... ~ ratio between CD4 and CH4, namely, 0.7622. The width 8 .t: ~ at half-height of this line in both salts is only 6 cm-1 at l6 >-< ...; -190°C. The sharpness of this degenerate vibration 8 '-' therefore affords strong evidence that the ion sym-... ~ OJ metry is genuinely tetrahedral at low temperatures. 2 Similarly, the very strong peak at 2336 cm-1 in the OJ spectrum of ND4CI is certainly the triply degenerate "- stretching frequency, Va. The corresponding frequency "'-~ § in NH4Cl must lie approximately at the 3080-cm-1 OJ doublet mean. If it is assumed that the combination 0 level "borrows" all of its intensity from the funda- ~ mental (certainly a good assumption since Va is by far ..... 8 the strongest band in the spectrum), a more accurate ..,. ill estimate of the unperturbed frequencies of the two i levels may be obtained from their relative intensities. ::I. A first-order perturbation calculation yields j;J- § o=S(R-l)/(R+1), ~ ~ ~ ~ ~ 9 ~ ~ ~ !I 0'" 01 , , , I '1-I , I I I , II The data for the bromides and iodides will be presented at a e01 later date in this Journal. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12AMMONIUM CHLORIDE SPECTRA 301 TABLE 1. Observed frequencies, relative intensities, assignments, and inferred frequencies for NH.Cl and ND.Cl in Phase III. NH,Cl Inferred Obs. freq. Obs. freq. Assign- freq. at -78°C at -190°C ment (em-I) (em-I) (em-q "1+111 3231 3228 (3) 3223 (6) 2)1,+", 3219 .. (F.) [30861t 3126 (34) 3126 (41) 3090 (I) 3089 (5) 1'2+". [30861t 3047 (38) 3050 (42) .. (A.) (3048)§ (3048)§ 2., 2806 2833 (6) 2828 (8) .2+11, 2074 2018tt(1) 200S-2100tt(2) "4+"6 1788 (3) 1817}(4) 1794 .. (E) 1683 (1712)§ .,(F.) 1447 (I) 1402 (10) 1403 (10) .,(F,) 391 .,(F.) (l83)§ (183)§ * ND.H+ and ND.H,+. t Computed unperturbed trequencies. tt Broad band. ND,Cl Inferred Ob •. freq. freq. at -190°C (em-I) (em-I) 3129" 3089" 3059" 2331 2315 [232S1t 2336 (94) 2292 2279" [22721t 2260 (16) 2246" 2224" 2134 2142 (12) 1486 1420" 1405" 1360} 1348 (4) 1205 1132* 1067 (10) 281 § From Raman spectrum (see references 12 and 13). where h is the separation of the unperturbed levels, Sis the observed separation of the peaks and R is the ratio of the observed intensities of the two branches of the resonating doublet. In this way the frequencies written in brackets in the inferred spectrum columns of Tables I and II were obtained. The Teller-Redlich frequency product ratio for the two F2 species is then vaDV4D --=0.5725. vaHV4H For comparison, the harmonic value of this ratio is 0.5528, while the corresponding observed ratio for CD4 and CH4 is 0.570. It is interesting that the fre quencies of the two F2 modes are essentially identical in the spectra of NH4CI, NH4Br and NH4I. The band at 1794 cm-I in NH4CI has sometimes been assignedl3 to the doubly degenerate bending vibration, V2. However, the corresponding band in ND4CI, which is plainly identified by its temperature behavior, oc curs at 1348 em-I. The frequency ratio is then 0.7514, whereas if it were actually V2 the ratio would be closer to 0.707. The observed value of this ratio cited21 for methane is 0.6906.** Thus, it seems unquestionable that the 1794-cm-1 band in NH4Cl cannot be the doubly degenerate vibration, V2. If we assume that the level which resonates with Va in NH4CI is V2+V4 (the only possible binary combination), and make a corre sponding assignment to the peak at 2260 cm-l in ** It should be noted that the ",-modes have never actually been observed in CH. and CD,. TABLE II. Observed frequencies, relative intensities; assignments,. and inferred frequencies for NH,Cl and ND,Cl in Phase II. NH,CI Inferred Obs. freC!. Assign- freq. at 28°C ment (em-I) (em-') "'+". 3209 3200 "3(F2) [3097Jt 3138 (39) "2+'" [3085Jt 3044 (31) "I(AI) (3041)§ 2", 2806 2810 (6) V2+". 2041 2000tt(1) v.+". 1762 (3) "2(E) 1682 (1710)§ J/,(F2) 1445 (1) 1403 (10) ".(F,) 359 ".(Fo) (168)§ * ND,H+ and ND,H,+. t Computed unperturbed frequencies. tt Broad band. § From Raman spectrum. ND,CI Inferred Ob •. freq. freq. at 28°C (em-I) (em-I) 3121* [2337Jt 2350 (84) [2265Jt 2252 (13) (2214)§ 2132 2129 (12) 1469 1413" 1335 (3) 1200 (1215)§ 1126* 1090}(10) 1066 269 ND4Cl, we obtain the frequencies 1683 and 1205 cm-l for the values of V2 in NH4Cl and ND4Cl, respectively. The frequency ratio is then 0.7159 and the isotope check is good. A further check on this frequency, al though a rough one, is supplied by the combination band at 2000-2100 cm-l which we shall discuss in more detail later. The corresponding room temperature Raman lines occur at approximately 1710 and 1215 cm-l• These values are probably more correct than the frequencies we have calculated. V. COMBINATION AND OVERTONE BANDS The selection rules for fundamental vibrations are exceedingly strict in that spectral activity is restricted to those fundamentals which are totally symmetric with respect to translation. This translational restric tion is very much relaxed in the case of combinations and overtones since the only requirement is that 1ihe combining modes have the same wave number vector," and it is not necessary that, in the individual normal vibrations, the vibrations of all cells be in phase. The resultant level, however, is totally symmetric with respect to translation. In the case of molecular modes, the frequencies will not in general be strongly affected by the phase shifts between neighboring molecules, since the interactions are usually weak. Thus all fre quencies within the branch lie in a narrow region. Con sequently, the entire band of allowed frequencies com prising the overtone or combination should lie close to the value computed in terms of the totally symmetric fundamentals. However, the resulting band is a dis tribution function and may therefore show structure,. 23 M. Born and M. Bradburn, Proc. Roy. Soc. London 188A, 161 (1947). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12302 E. L. WAGNER AND D. F. HORNIG including subsidiary peaks. The temperature behavior of many of the combination bands indicates that they may well be ternary bands where the third component is the Debye frequency distribution. No calculations on the detailed envelopes of such bands have yet been made. The first overtone of 114 is observed as a band with a maximum at 2828 cm-I in NH4CI and 2142 cm-I in ND4Cl. In each case the frequency of maximum absorp tion is slightly greater than 2114 when 114 is taken as the limiting frequency which is itself allowed, If the fre quency in the 114-branch increases in going from the completely in-phase to the completely out-of-phase motion, we expect the center of the overtone, 2114, to be greater than twice the frequency of the in-phase motion (which we observe) and also to have a width approxi mately equal to twice the width of the 114-branch. The shape of the band changes appreciably with tempera ture, chiefly in that the low frequency side of the band tends to vanish as the temperature is reduced. The same phenomenon has been observed in the Raman spectrumY; It is interesting that the intensity of this overtone is just a? great or greater than that of the fundamental. This may be cau:;ed by the fact that the number of transitions entering an overtone is enor mously greater than in the case of a fundamental, so that although individual transitions may be exceedingly weak the integrated intensity may become quite considerable. The combination 114+ 112 has been discussed previously. Because it is so closely coupled to the intense funda mental 113, it is impossible to say anything about its envelope or intensity in NH4Cl. However, if the 114- distribution increases we would expect the 112-dis tribution to decrease in going from in-phase to out of-phase motion since the in-phase 112-motion has the same phase relation between adjacent N - H bonds as the out-of-phase 114-motion and vice versa. In this case the 114+112-band would be relatively narrow since the width should be approximately equal to the difference between the two distributions. This is actually the case as may be seen from the ND4CI spectrum where the combination level does not resonate seriously with "3. The bands observed at 1794 cm-I, 2020 cm-1 and 3223 cm-I cannot be accounted for in terms of intra molecular frequencies. The latter can, however, be interpreted as a combination between the totally sym metric mode of the NH4+ ion (which has been assigned to 3048 cm-1 on the basis of the Raman spectruml2) a.nd the optical branch of the lattice spectrum whose greatest density of frequencies is in the vicinity of the Reststrahlen frequency, 183 cm-I. The other two bands must involve still a different characteristic lattice frequency. VI. TORSIONAL LATTICE VIBRATIONS In addition to the vibrations of a CsCl type space lattice, the NH4CI crystal lattice spectrum must also contain three branches arising from the coupled tor sional oscillations of the NH4+ ion. The limiting fre quency of this type of motion, which we shall designate 116, is of symmetry species FI so that it may be active in the Raman spectrum as a fundamental but is forbidden in the infra-red spectrum. In order to consider combina tion bands which may involve these modes, an estimate of the frequency of torsional oscillation would be useful. The purely electrostatic potential energy in the vicinity of a threefold axis in a CsCI structure is given by v=~{ ~ ~ ~ [{ (l-a)+a( ~+ ~) r +{ (m-a)-a( ¢-~) r+{ Cn-a)l2rt -~ ~~ [{ (~-a )+a( ¢+ ~) r +{ (~-a)-a( ¢-~) r +{ (: -a) rrl (1) where a is the lattice constant, a is the x, y, or z coordi nate of a point on the threefold axis (the origin is taken. at the center of the cube of eight chlorine atoms), ¢ is the angle (small) about the z axis by which the point in question is obtained from the point a, I, m, n are in tegers, and u, v, ware odd integers only. If as a rough model of the charge distribution on an NH4+ ion we assume a charge fe on each hydrogen atom, where e is the electronic charge, we find for the force constant of the torsional oscillation about the z axis + (m-a)2+ (n-a)2} -0/2 -Cl+m)l (l-a)2+(m-a)2+(n-a)21- iJ -L L L [3a(~_~)2j (~_a)2 "vw 2212 +(;-a Y+(: -a Yf-5/2 -(;+~){ (~-a r+(~-a Y +(: -a y}-i]}. (2) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12AMMONIUM CHLORIDE SPECTRA 303 0.0'.------------------,100 +~ g-O.I ..J .. __ -,,---"'------ 90 ~ ... _'" ):10 27°0 ----- 80 Z -190"0 --- ~ FREQUENCY IN WAVES PER eM FIG. 4. 4.9 micron band of NILCl in Phase II at 27°C and Phase ITI at -190°C. iii 70 !1! ~ Using the value j=0.2 which was estimated by Pauling,24 the values a=3.86A and rNH= 1.0A, we then calculate for the torsional oscillation frequency, by considering the contributions of 180 surrounding ions, the value li6= 262 cm-l. It is interesting to note that considering only the 14 nearest neighbor ions this frequency comes out to be 265 cm-I. If instead we use the values suggested by Nagamiya,17 j=0.277 and rNH= l.13A we obtain li6=360 em-I. These estimates neglect entirely the effects of repulsion (which would lower the frequency), of the polarizability of the halogen and NH4+ ions (which would raise the frequency), and of multipolar terms due t~ the non spherical charge distribution in the NH4+ IOns other than the one considered (which would be small). Since the frequency of oscillation is determined almost entirely by the nearest neighbor halide ions, the fre quency should not depend greatly on t~e relative phase of the torsional motion of the NH,+ IOns. Hence the distribution of frequencies in this branch should cover only a narrow range compared, for example, to the acoustic branches of the lattice spectrum. If the band at 1794 cm-I in the low temperature ammonium chloride spectrum, to which no combina tion of internal modes can be assigned, is identified with the combination li4+ li6, the torsional oscillation frequency is approximately 391 cm-I. A corresponding assignment to the band at 1348 cm-l in the deuterated salt yields 281 cm-l. The ratio of the~ frequencie.s, 0.718, is in satisfactory agreement WIth the ratIO predicted by the Teller-Redlich rule, 0.707. Using our previously derived value of approximately 1680 cm-l for li2, this same type of assignment accounts for the presence of the band at 2000-2100 em-I as li2+ li6. Although this band also has a sharp low fre quency edge at low temperature, it is quite broad and apparently possesses several maxima. Its detailed structure obviously requires further investigation. Nevertheless, a qualitative estimate of the width .and shape of this as well as the li4+li6-band ca~ be ?btamed in the following way. If we assume that m gomg from 24 L. Pauling, Nature of the Chemical Bond (Cornell University Press, Ithaca 1944), p. 72. the in-phase to the out-of-phase motion the frequency of the torsional lattice mode, li6, decreases, we would expect the li'+li6-band to be narrow since we found previously that the overtone indicates that the li,-dis tribution increases. This is indeed the case. Also, since it has been shown earlier that the li2-distribution prob ably decreases, the li2+ li6-band may be expected to have a width approximately equal to the sum of the two distributions and to be distributed to lower frequencies than the value predicted from the in-phase modes. This is consistent with the observed spectrum. Furthermore, this one frequency, 390 cm-I, can ac count qualitatively for a series of heretofore inexplicable Raman frequencies observed by Krishnan;13 e.g., that at 560 cm-I can be assigned to the limiting frequency li6+ li6, where li6 is the Reststrahlen frequency; that at 760 cm-l can be assigned to the first overtone 2li6, and the diffuse bands at 1065-1145 cm-I and 1280-1335 cm-I in the room temperature spectrum may be assigned to the difference bands li, -li6 and li2-li6, re spectively. Finally, this frequency, on the basis of Eq. (2), should be sensitive to the lattice dimensions, and should therefore be lower in NH,Br and NH,I. It is found that the bands corresponding to the 1794 Cm-I and 2000-2100 cm-I frequencies in NH4CI are the only ones in the spectrum which are significantly shifted, in NH4Br both bands occur at frequencies , about 60 cm-l lower and in NH41 they occur about 110 cm-l lower. It seems inescapable, therefore, that the torsional frequency occurs at about 390 cm-I in Phase III of NH4Cl. If this conclusion is correct, this vibra tion occurs at about 360 cm-l in Phase II. Consequently, there can be no question of free rptation in this phase in Pauling's sense.1 In order to interpret the envelope of the combina tions involving li6 in detail it will be necessary to obtain the complete frequency distributions for the li6 branch and for the molecular branches as well as an intensity estimate for all the combinations between branches. vn. THE RAMAN SPECTRUM The Raman spectrum of NH4Cl has been studied by Krishnanl3 over the same temperature range as in this work. Although Krishnan has listed a very large num ber of lines, his microphotometer tracings ar~ in general very similar to the infra-red spectra pubhshed here. In particular, Krishnan has found li4 to be very sharp, but in his case'it exhibits a double peak at low tempera tures. It is not clear why this should be so, but it should be noted, first, that li, hiLS never been observed in the Raman spectrum of methane and, second, that the lines obtained by Krishnan are of relatively low intensity. The 3100 cm-I region of the Raman spectrum is even more complicated than in the infra-red spectrum, since in addition to the strong resonance doublet it contains the even more intense totally symmetric fundamental, lil. The Raman spectrum in the vicinity of 1790 em-I and 2000-2100 cm-I parallels the spectrum This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12304 E. L. WAG N ERA N D D. F. HO R N I G reported here. Menzies and Millsl2 also observed a Raman line at 1712 cm-1 in the spectrum at -150°C which may be the doubly degenerate vibration, 1'2, which we have placed at approximately 1680 em-I. However, this fundamental is also unobserved in the Raman spectrum of methane and, since its motion is quite similar to that of 114, it is surprising that its in tensity should be so much greater than that of 1'4 in the ammonium halides. Krishnan has also observed a series of frequencies between 3200 cm-1 and 3500 em-I, all of which can be qualitatively accounted for in a variety of ways using the assignments obtained here. Polarization measurements have not been obtained in Phase III, but Couture and Mathieu7 have de termined the polarization of all the main Raman lines at room temperature. In every case the reported polariza tion is consistent with the assignments obtained here on the basis of a tetrahedral model. Polarization meas urements in Phase III as well as low temperature Raman spectra of the deuterated compounds would be of value in these interpretations. VIll. THE NATURE OF THE A-POINT TRANSITION The simplest crystal structure of NH4Cl consistent with the x-ray data is that of space group Ti. In this structure the ammonium ion tetrahedra are identically situated in every unit cell and the hydrogen atoms lie on the threefold axes. It is in terms of this structure that it has been possible to interpret the low tempera ture (Phase III) spectrum of NH4CI and ND4Cl. With this symmetry the crystal may exhibit a piezoelectric effect, which has been found by Hettich,IO may show a first-order Raman lattice spectrum, as has been ob served by Menzies' and Mills12 and by Krishnan,13 and should be optically isotropic, as is the case.lO In order to interpret the phase transition, a similar knowledge of the structure of the room temperature modification (Phase II) is necessary. Now it is clear for two reasons that the ammonium ions in this phase are not rotating. First, the narrow line due to the fundamental 1'4 is characteristic of a pure vibrational transition and shows no evidence of rotational structure, particularly on its low frequency side. (Rotational lines might be expected to be spaced at about 15 cm-l intervals.) Second, the identification of a torsional oscillation frequency at a frequency much higher than rotational frequencies removes all of the degrees of freedom which might be available to rotation. However, it is apparent that there are two equivalent equilibrium orientations for the ammonium ion in a single unit cell of a body-centered structure. In both of these the 'hydrogen atoms lie on the threefold axes. If, therefore, the structure of Phase II is taken as one in which am monium ions are randomly distributed between these two possible orientations, the infra-red spectrum can be satisfactorily interpreted. Couture and Mathieu7 have argued that a disordered phase should yield a depolarization ratio p=6j7 rather than the observed ratios which are characteristic of cubic crystals. This argument would be valid for completely random orientations such as would be ob tained if there were free rotation, but for the model suggested above the depolarization ratios would be only slightly perturbed from those of an ordered cubic crystal, since none of the axes of the tetrahedra are shifted with respect to the crystal lattice. In our Phase II structure the crystal symmetry is lowered and, consequently, the symmetry of the crystal line field about any ammonium ion is no longer genuinely tetrahedral. A variety of symmetrically non-equivalent configurations of nearest neighbor ammonium ions are possible. Some of these configurations slightly displace the chloride ions, and this, in turn, probably gives rise to the main perturbation on the NH4+ ion. The most obvious effect to expect is the destruction of the de generacy of the triply degenerate vibrations, and, indeed, the most striking change of the Phase III to Phase II transformation is the shoulder which develops on the high frequency side of 1'4. This is illustrated by the spectra in Fig. 2. It is interesting to compare this behavior with that of NH4Br and NH4I, which in Phase III do not possess tetrahedral NH4+ ion symmetry.tt In these compounds there is a sharp second peak which merges with the main peak as the temperature is raised through the transition point and the line as sumes a shape which is almost identical with that of the chloride. The fact that the infra-red spectra of the three halides in Phase II are essentially identical is consistent with the idea that all three have the structure for Phase II discussed above. A more detailed discussion of this transition will be given at a later date. However, it is reasonable to expect that, as in disordered alloy structures, the near neighbor structures are chiefly those of the most stable crystalline forms although over any extended volume of the crystal there is com plete randomness. In this case the Phase II structure might be expected to approximate locally a mixture of NH4CI and NH4Br structures, which is consistent with the observed spectrum. A further consequence of such a randomly oriented structure would be that the lattice vibrations of the crystal should be spread out by the destruction of much of the lattice symmetry. This has been qualita tively observed by Krishnan in the Raman spectrum and may be the reason for the very considerable broad ening of the bands at 1794 cm-1 and 2000-2100 cm-1 which involve the torsional lattice vibration. tt The x-ray symmetry of NILBr and NILI in Phase III is D4h7 so that the NIL+ ion symmetry is Yd. In Phase II all three halides have the CsCI structure. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 213.100.235.35 On: Tue, 13 May 2014 17:27:12

1.1747022.pdf

Contributions to the Theoretical Treatment of Ammonium. I J. I. Horváth Citation: The Journal of Chemical Physics 16, 851 (1948); doi: 10.1063/1.1747022 View online: http://dx.doi.org/10.1063/1.1747022 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/16/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theoretical study of proton transfer in ammonium nitrate clusters J. Chem. Phys. 117, 2599 (2002); 10.1063/1.1489995 Interpretation of the low temperature phase transitions in ammonium and potassium chromic alums in terms of vibrational and static crystal distortion contributions to the spin Hamiltonian zerofield splitting parameter D. III J. Chem. Phys. 64, 4504 (1976); 10.1063/1.432078 Approximate Treatment of the Viscosity of Idealized Liquids. I. The Collisional Contribution J. Chem. Phys. 22, 1728 (1954); 10.1063/1.1739885 Errata: Contributions to the Theoretical Treatment of Ammonium. II J. Chem. Phys. 19, 978 (1951); 10.1063/1.1748428 Contributions to the Theoretical Treatment of Ammonium. II J. Chem. Phys. 16, 857 (1948); 10.1063/1.1747023 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52THE JOURNAL OF CHEMICAL PHYSICS VOLUME 16, NUMBER. 9 SEPTEMBER, 1948 Contributions to the Theoretical Treatment of Ammonium. I. J. I. HORv ATH* Institute of Physical Chemistry of the Technical College of Budapest, Hungary (Received December 31, 1946) In this paper we determine the electron distribution, the binding energy, and the ionic radius of the positive ammonium molecule ion. The general idea of our method is that the molecule is divided by a spherical surface which contains the protons and it is supposed, in our first approximation, that the charge of protons is distributed uniformly on this surface. Now, we we have inside our sphere a nitrogen nucleus, the charge of which is over compensated by ten electrons and so a N3-ion is formed. The whole formation can be regarded from the outside of the sphere as similar to the Na+ ion, because the charge of the four protons has been added to the charge of the N nucleus. In a second approximation we pay attention to the fact that the protons are not exactly uniformly distributed on the spherical surface but on the points of a tetrahedron. We have taken into consideration the inhomogeneous field of protons by using the perturbation calculation. The ionic radius of our molecule ion is determined as usual in the statistical theory of atoms. Finally, we check our result with a cycle process. We do not use semi-empirical parameters. IT is interesting that there exist radicals com posed of non-metallic atoms which have the same properties as metals. The most common example is ammonium, which forms salts similar to those of alkali metals. It is obvious that the four protons of the radical penetrate the electron cloud-of the nitrogen and it may be presumed that the molecule has, in addition to the closed shell of the positive ammonium radical, one s-electron, just as the alkali metals have. The theoretical treatment of ammonium is an especially interesting problem, because pure ammonium is experimentally unknown. Con sequently, its constants have not been estab lished. We cannot ignore the exceedingly re markable fact that, on the one hand, ammonium (just as alkali metals do) forms an amalgam, which, though sufficiently liable to decay, can be prepared under high pressure and has the same properties as the amalgams of alkali metals; on the other hand, up till now. pure ammonium metal has not been produced. In the first part of our paper we shall determine the electron distribution, binding energy, nitro gen-proton distance, and ionic radius of the positive ammonium ion; in the second part the wave function of the valence electron, ionization energy of ammonium, and the eigenfrequencies of NH4+ will be determined. I. In the case of the positive ammonium ion, nitrogen is linked with four hydrogens, while one electron is given to the anion. The four protons are on the surface of a sphere, which has a radius R, and the molecule ion has tetrahedral symmetry. The theoretical treatment of molecules of such * At prest!nt at the Institute of Physics, Med. Faculty, )niversity of Debrecen, Debrecen, Hungary. type would be possible generally on grounds of 851 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52852 J. 1. HORVATH an ionic model,! supposing that in the center would be a negative ion with a fourfold charge surrounded by four protons. According to this premise, the binding energy would be derived from the Coulomb and polarization energy. For this, however, the polarizability of the ion should be known veryexactIy. This is possible in the case of the halogen atoms (for this reason the ionic model is a very good start for the discussion of hydrochloric acid), but in the case of higher negative ions the values of polarizability are practically completely unreliable. Now, in the case of ammonium it must be taken into con sideration that the center is a negative ion with triple charge which has to bind four protons. This structure could not be explained electro statically. We have used Neugebauer's method2 for our calculation. Neugebauer has calculated the methane molecule with similar assumptions. The general idea of this method is that the molecule is divided by the surface of a sphere, which con tains the protons and has a radius R. Further more, it is supposed, in the first approximation, that the charge of the protons is distributed uniformly on this surface. Now we have inside our sphere a nitrogen nucleus with a sevenfold charge, which is overcompensated by the ten electrons to a N3-ion. This is surrounded by the four P.f0tons, the charge of which is handled in the first approximation as uniformly distributed on the spherical surface. The whole structure can also be regarded, from the outside of the sphere, as similar to the Na+ ion, because the charge of the four protons has been added to the charge of the nitrogen nucleus. In a second approxima tion we pay attention to the fact that this assumption is rough and the protons are really on the points of a tetrahedron. We have taken into consideration the inhomogeneous field of the protons in terms of perturbation calculation, where the energy is produced because the in homogeneous distribution is small enough for the application of this method, as will be shown m the calculation. I J. H. De Boer, Electron Emission and Absorption Phenomena (Cambridge University Press, Teddington, England, 1935), p. 31. 2 Th. Neugebauer, Zeits. f. Physik 98, 638 (1936); Mat. Term. Tud. Ert. Budapest 36, 450 (1937); 57, 182 (1938). This model forms a transition between the ionic and the covalent interpretation of the chemical bond. It approaches the facts very well and points out the indubitably right view that the chemical bond is brought about by the valence electrons, which are on a common energy level. We do not use Slater and Pauling's theory of valence forces,s because the relations of sta bility are given automatically in case of tetra hedral symmetry. Let p(r) be the density distribution of valence electrons which we wish to normalize so that the number of valence electrons may be correctly given far from the nucleus, that is,4 J p(r)dv=8, (1) where dv is the element of volume and the integral, now and always, if not otherwise stated, shall be extended over the whole space. As, on the one hand, the maximum of the density dis tribution of electrons on the level Is is low inside our tetrahedron, so, on the other hand, the valency electrons bring about the chemical bond. Therefore, we regard the charge of the Is elec trons as being united with the nucleus. We have, in this way, neglected the effect of two electrons only and the calculation will be much shorter. However, we mention that this approximation is not to be used in the case of a central atom of higher atomic number. When determining energy terms we take the mutual effects into consideration as follows: the mutual effect among the protons, between the protons and the nitrogen nucleus, between the protons and electrons, between the nucleus and electrons, and, finally, must take into considera tion the mutual effect among electrons. Electrostatically the mutual effect of protons is given, based on a simple geometrical con sideration, by (2) where a = 109°28' is the tetrahedron angle. As we imagine the electrons on. the level Is to be united with the nucleus, the repulsion potential of the nucleus is 4(SjR). But the elec- 3 J. C. Slater, Phys. Rev. 38, 1109 (1931); L. Pauling, Phys. Rev. 37, 1185 (1931). 4 We use Hartree's atomic units. See D. R. Hartree, Proc. Camb. Phil. Soc. 24, 91 (1927-28). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52AMMONIUM. I. 853 trons, which are inside our sphere, shield this effect of the nucleus with one attracting term in the form -(4/R)fp(r)dv (where the integral must be extended over the inside of the sphere). Therefore, the mutual effect between the protons and the nucleus as well as between the protons and the electrons inside our sphere is summed up in the following term: The mutual effect between the protons and the electrons outside our sphere is given simply by where the integral must be extended now outside the sphere. The energy term, which corresponds to the attracting effect of the nucleus after uniting the ls electrons with the nucleus, is given by (5) The greatest mutual effect between the elec trons is that which follows in consequence of electrostatic forces. At a distance r from the nucleus, the potential energy is plainly -f[p(r)/lr-r/IJdv. Therefore, the next en ergy, term is E6=t f f [p(r)p(r/)/ I r-r/l Jdv'dv. (6) This term contains the mutual effect of each electron on itself, which we do not correct here, but shall take into consideration later. As a result of Pauli's exclusion principles we must pay attention to Fermi's kinetic zero-point energy, which is Then we ought to introduce the exchange ener gies between the electrons with parallel spin and with an antiparallel one. The first is given by6 6 W. Pauli, Zeits. f. Physik 41, 81 (1927). 6 P. A. M. Dirac, Proc. Camb. Phil. Soc. 26, 376 (1930); F. Bloch, Zeits. f. Physik 57, 545 (1927). Geiger-Scheel, Handbuch der Physik (1933), Vol. 24, p. 485. H. Jensen, Zeits. f. Physik 89,713 (1934); 93,232 (1935); L. Brillouin, J. de phys. et rad. 5,185 (1934). TABLE I. The values of A (R) and C(R). R 1.5 1.6 1.7 1.8 1.9 2.0 A(R) 4.58870 6.78759 9.99601 14.62811 18.93651 29.89032 C(R). 1.42999 1.32085 1.21955 1.13909 1.09742 1.04180 E7= -(3/4) (3/7I')1/3f p4/3(r)dv. (8) This term also contains the mutual effect of electrons on itself. As is shown by the statistical theory of the atoms,7 this compensates for our former error. We write the second exchange energy, called by Wigner correlation energyS and employed by Gombas9 in the statistical theory of atoms in the form Es= -f [ap4/3(r)/(pI/3(r)+b)Jdv, (9) where a=O.05647 and b=O.1216.10 Finally we pay attention to the polarization energy which has its foundation in the inhomo geneity of the proton tetrahedron. The same thing happens as in the theory of crystals.u We write the formula of polarization energy in the form where in the denominator there is one energy, instead of the usual difference, one mean fre quency, which we calculate so that tRe polariza bility should be given correctly. In the numerator the elements of the matrix are given by H12(SS) = f ifi*v2ifidv, and Hl(SS) = f ifi*vifidv, (11) (12) 7 A. Sommerfeld, Atombau und Spektrallinien, II (F. Vieweg & Sohn, Braunschweig, 1939), second edition, pp. 700-702. 8 E. Wigner, Phys. Rev. 46, 1002 (1934). 9 P. GombolS, Zeits. f. Physik 121, 523 (1943). 10 In connection with the two latter energy terms I have to mention that in this case they mean only a lesser kind of correction and do not considerably influence the place of the minimum of the energy. But in the case of a molecule having a central atom with higher atomic number (e.g., SiH4, see in Nature (in press)) they are more important factors. 11 P. Gombols and Th. Neugebauer; Zeits. f. Physik 92, 375 (1934); Th. Neugebauer, ibid. 95, 717 (1935) and reference2, This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52854 J. I. HORVATH TABLE II. The energy terms and the total energy as a function of R. R E. -E, -E, -E. E. E. -E, -E. -E. -E 1.6 2.44956 2.57690 4.36332 36.77885 22.28027 11.24672 3.99087 0.35492 0.11477 12.2308 1.6 2.29646 3.15352 3.87304 36.19630 22.07200 10.07200 3.83082 0.35411 0.08664 12.5788 1.7 2.14138 3.43661 3.26172 35.08775 21.53023 9.76558 3.70924 0.35402 0.07089 12.4304 1.8 2.04130 3.65737 2.74001 34.13760 20.91274 9.24820 3.57544 0.35391 0.05937 12.3146 1.9 1.93385 4.13677 2.00165 33.61115 20.52973 9.18355 3.53073 0.35202 0.05111 12.0630 2.0 1.83717 4.22810 1.63172 32.86035 20.03026 8.90180 3.42544 0.34824 0.04417 11.7879 where 1/1 is the wave function of the molecule-ion, v is the perturbation energy which we will now form into-a series of Legendre functions in the usual way: 4 4 V= LVi=L:{[1/(R2+r2-2Rr COS~i)!]- (1/R) 1 i=l i=I 4 00 =L L Pk(Cos~i)(rk/Rk+I), r<R, (13) i=l k=l and 4 4 v= LVi= L {[1/(R2+r2-2R+cos~i);]- (1/ R) 1 i-I i=1 4 '" = L L Pk(COS~i)(Rk/rk+l) i=1 k=1 4 + L[(1/r) -(1/R)], r>R. (14) i=l In the last term 1/R is constant. Consequently, we can now neglect it and as we have supposed, 1/r is united with the charge of the nucleus; therefore, we can write (14): 4 00 V=L L Pk(COS~i)(Rk/rk+I), r>R. (15) i=1 k=! In our approximation we confine ourselves to the first term. Equations (2) to (lO) give the energy which is necessary to carry the four protons and the eight electrons from our molecule-ion into infinity. II. Our calculations are as follows. As wave func tion we have functions of N3-inside our sphere and the wave functions of Na+ outside our sphere ,which we write with the help of Morse, Young, and Haurwitz's formula.12 They are 12 P. M. Morse, L. A. Young, and E. S. Haurwitz, PhI'S. Rev. 48, 948 (1934). given by 1/IN'-1s: 9.78447·e-6,7r, 2s: 1.20799(r· e-l.67Sr -1.29134· e-6,14166r), {V'1 cos~ (16) 2P: 0.74559·r·e-1.2884Ir. sin~·eil" sin~'e-il" and 1/IN.,+ 1s: 19.74699·e-IO,7r, 2s: 6. 92899(r· e-3,34375r -0.61030. e-IO,03J25r), lV'1'cos~ 2p: 8.45082· r' e-3,39163r. sin~' eil" ' sin~·e-i .. (17) where these functions are orthonormalized per definitionem. After neglecting 1s electrons we introduce the electron density defined by lC(R) {21 if;N3-, 2812 +61if;N3-.2pI2l, r~R, p(r) = . (18) C(R)A (R) {21if!Na+. 2.12 +61if;Na+.2pI2l, r;;;R, where the constants C(R) and A (R) are deter mined so that the wave functions of N3-and Na+ may be equal at place R and that this may satisfy the conditions (1), consequently 21if!N'-. 2.12+61 if;N'-, 21' 12 and =A (R) {21 if;Na+, 2812+61 if;Na+, 2p 121 C(R) { iR47rr2[21 if;N'-, 2.12+611/1N'-, 21' 12]dr +A (R) f'" 47rr2[21 if;Na+. 2.12 R (19) +61 if;Na+, 21' 12]dr} = 8. (20) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52AMMONIUM. I. 8SS The derivatives of the wave functions are not equal in the place R, but that does not matter because it is possible to prove that this does not playa part in the calculation of the energy, as the protons deform the electron cloud in the place R independently of this, and this effect reduces our error. We sum up the values of A (R) and C(R) as functions of R in Table I. We remark that in case of a central atom which has more than one closed shell, we must normalize the wave functions of different shells separately, as can be understood without any difficulty. We have calcu1ated the integrals with the help of Simpson's formulal3 where p is an even integer number and Yi is the value of this function Y = f(x) in the ith point. The error of this approximation is (p 'w2/180)f(4)(~), a~~~a+pw. (22) It is very difficult to determine the integral (6). By means of one semiclassical argument by Neugebauer2 we have written this double integral "in the form E5=t f"'{(1/rf) fT '471'r2p(r)dr o 0 + j,"'47rr2(p(r)/r)dr }47rrf2p(rf)drf, (23) which can be calculated by a machine without great difficulty. The appropriate term of the correlation energy has pier) in its denominator. Consequently, the calculation of the integral would be very lengthy, if carried out exactly, but as we shall make no greater mistake than 1 percent if we multiply b also by the normalizing factor, we use this simplification. \ In connection with the polarization energy we make the following remark. It is well-known that (24) 13 H. Geiger and K. Scheel, Handbuch der Physik (Berlin, 1928), Vol. 3, p. 626. if k= 1, 2 and r ~P3(COSt?i) sint?dt?dcp "f2"{ 4 }2 Jo 0 ,-I =(2/7)271'(4+12(11/17)1. (2S) Furthermore, the second term vanishes because j"Pk(COSt?) sint?dt?=O if k=l, 2, 3. (26) o Therefore, the polarization energy is -(l/h;;) . (1/7H 4+ 12(11/17) 1 R X (1/ R8) f 471'r2p(r)r6dr, r~R, (27) o "inside our sphere and -(1/h;;) . (1/7)( 4+ 12(11/17) 1 XR6f"'47rr~(p(r)/r8)dr, r;;;.R, (28) R outside our sphere. We shall determine the average frequency at present unknown-by obtaining the known a-polarizability of Na+ rightly based on the formula a=[2·P2(SS)/h;;], (29) where 00 +1 2" P2(SS) = f J f r2 cos2t?p(r)drd(cost?)dcp. (30) o -I 0 The polarizability of Na+ can be seen in well- tr~H+8e-+cr -E-~~;+~N+~N+~N / ",,+.fs,.t3JN NI(CI- N+3H+H+Cr -Eg t ltDJU NHj:t Nt/-{-tHtfftcr +QNH,Clf 'I +DHt f~+2I{tlC{z Nt~tl(+CI- iD.vtlDc~t ! -£11(( N+2H,,+CI--N+H,,+'r(+ CIt e tJH, FIG.!. Cycle for computation of the NH.+ energy. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52856 J. I. HORVATH known experimental tables,14 but in this case we introduce with hv one semi-empirical con stant. We can avoid this by using the theoretical value of Gomba,s.lo This value is not very good (agreeing only in magnitude with the experi mental value), but can be used because this energy term is only 7 percent of the total energy, and therefore is unimportant. So we find the total energy as the function of R. Now, based on the "variation principle" we determine its minimum, in this way obtaining the stable state. The value of the energy terms and of the total energy as a function of R is summed up in Table II. We can see that the minimum is at R= 1.6aH, which is 0.845A. The calculated value of energy is 12.56788e 2/aH = 340.34 ev. III. It is not easy to check our result because we have no experimental data. Therefore, we use the following method. In the statistical theory of the atom the density distribution of electrons does not vanish on the limit of the atom and the ion, but it has there a definite value, which determines also the radius of the ion.16 In case of a model, which is supplemented by the correlation energy, this value is, according to Gombas,to _ p(ro) =0.003074. (31) In our case this takes place at 3.1aH, which is 1.63A. Goldschmidt's crystallographic value17 of this is 1.43A, so our value is higher by 11.3 percent. This is natural because it is well known that in crystals the ions are very compressed in consequence of electrostatic forces. Therefore, the crystallographic radii are always smaller. We check our value of the energy with the help of an extended cycle-process of Born and Haber18 as is shown in Fig. 1. As we have calculated the energy which is necessary for us to remove the four protons and -eight electrons from our molecule ion into infinity, we start with a fivefold ionized nitrogen 14 K. Fajans and G. Joos, Zeits. f. Physik 23, 1 (1924), «=0.196.10-24 cm3• 16 P. Gomba.s, Zeits. f. Physik 122, 497 (1944), '" = 0.850 .10-24 cm3• 16 H. Jensen, Zeits. f. Physik 93,232 (1935). 17 V. M. Goldschmidt, Chern. Berichte 60, 1263 (1927). 18 M. Born and C. Lande, Verh. d. D. Phys. Ges. 20, 210 (1918); F. Haber, ibid. 21, 750 (1919). atom, four protons and eight electrons, and also a chlorine ion. We unit the electrons with the nitrogen atom and three protons, in this way freeing the ionization energy of electrons (Jl,N, J2,N, Ja,N, J4,N, J5,N, 3JH). We unite two hy drogen atoms into a hydrogen molecule, and in the next step a hydrogen atom and a proton into a hydrogen molecule ion, thus gaining their dis sociation energy (DH2, DH2+)' Now from the negative chlorine ion we take away an electron and add this to the hydrogen molecule ion. For this we must invest the energy Eaff corresponding to the electron affinity of chlorine, but this sets free the ionization energy of the hydrogen molecule (JH2). We change the nitrogen and the chlorine atom into 'molecules, then we win their dissociation energy (DN2, DCl2). When we unite these molecules into ammonium chloride crystal the heat of formation (QNH4C1) is set free. We supply to the ammonium chloride its lattice energy (Eg) because it dissolves to positive am monium and negative chlorine ions. Finally we supply ammonium with our calculated energy and are back at our starting point again; we finish our cycle process. The difference with zero is our calculation error. Our experimental data are as follows: The ionization energies of nitrogen are 14.48, 29.47, 47.40, 72.04, 97.43,19 JH= 13.53,19 DH2 =4.72 (the zero-point energy here is 0.27),20 DH2+ = 2.78 (the zero-point energy is 0.14),20 Eaff=3.75,21 JH2=15.235,22 DN2=11.9,23 DCI2'=2.46, QNH4C1 = 3.28,24 every datum is to be taken in ev. For the lattice energy of ammonium chloride we have no experimental data. In order to determine this we use the Madelung's formula2D Eg=545(d/M)i, (32) (d is the density of halogene and M is the molecular weight) which is sufficient in case of a crystal of the rock salt type. This data is also written in ev. In our cycle we must pay attention to the 19 I. Naray-Szab6, Kristalykemia (Mernoki Tovablkepz8 Intezet, Budapest, 1944). 20 O. W. Richardson, Proc. Roy. Soc. 152, 466 (1929). 21 F. A. Henglein, Zeits. f. anorg. Chemie 123, 159 (1922). 22 O. W. Richardson and P. M. Davison, Proc. Roy. Soc. 123,466 (1929). 23 R. T. Birge and H. Sponer, Phys. Rev. 28, 477 (1926). 24 V. F. Thomsen, J. prakt. Chern. (2) 21, 477 (1880). 26 A. Eucken, Lehrb. d. phys. Chem. (Akademische Verlagsgesellschaft M, B. H., Leipzig, 1934), p. 370. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52AMMONIUM. II. 857 following corrections. The zero-point kinetic energy of nuclei always appears in 'molecules. Therefore, the calculated dissociation energy has a smaller negative value than the experimental data. In some cases we pay attention to this. In our cycle a positive energy term corresponds to the vibration, which comes from the nine degrees of freedom of our molecule ion. But unfor tunately, we have no experimental data for this. This does not cause any great mistake because this error does not exceed 1-2 percent as shown in other molecules. The energy, which is calculated from our cycle is 423.57 ev, or expressed in atomic units 11.94944e2/aH while our result was 12.56788 e2/aH. The difference is 5.17 percent, which is satisfactory, if we pay attention to the error beyond our control. THE JOURNAL OF CHEMICAL PHYSICS VOLUME 16, NUMBER 9 SEPTEMBER, 1948 Contributions to the Theoretical Treatment of Ammonium. II. J. I. HORVATH Institute of Physical Chemistry of the Technical College of Budapest, Hungary (Received December 31, 1946) In this second part of our paper we have determined the ionization energy of the ammonium molecule and the eigenfrequencies of the positive ammonium molecule ion. Because we know the electron distribution of the positive radical, we can determine the eigenfunction of the valence electron of ammonium in the same way as is done in cases of alkali metals with larger atomic number. In Section I of Part II we review this method. In Section II we determine the eigenfrequencies of the ammonium molecule ion by Neugebauer's method. IN Part I of this paper! we have determined the electron distribution of the positive am monium ion. The substance of our method was that the molecule is divided by the sphere surface which contains the protons and it is supposed, in our first approximation, that the charge of the protons is distributed uniformly on this surface. Inside our sphere we have a nitrogen nucleus with sevenfold charge, which is over compensated by the ten electrons and so a N3- ion is formed; outside, as the charge of the pro tons has been added to the charge of the nucleus, an ion similar to Na+ comes into existence. In a second approximation we have paid attention to the fact that the protons are not really on the sphere surface but on the points of a tetra hedron. We have taken into consideration the inhomogeneous field of the protons based on per turbation calculation. In this way we have also determined the binding energy of our molecule ion, the distance 1 From the references of Part I T. Neugebauer's works should be emphasized: Zeits. f. Physik 98, 638 (1936); Math. Term. Tud. Ert. Budapest 56, 450 (1937); 57, 182 (1938). between nitrogen nucleus and protons and, finally, its ionic radius. In every case the dif ference between our result and our check was small enough. Now we complete our molecule ion with one electron into an ammonium molecule, and later, in. Section If, we determine~the eigenfrequencies of NH4+. I. Since, in respect to its chemical properties, the ammonium molecule is quite similar to the alkali metals, we may presume that, in addition to the closed shell of the positive ammonium molecule ion, the ammonium molecule has one s electron, just as the alkali metals have. Since the electron distribution of the positive radical is known, we can determine the eigenfunction of the valence electron of ammonium in the same way as in the cases of alkali metals with higher atomic number. We ought to determine the eigenfunction of the valence electron by solving SchrOdinger's equation, but it is well known that it is impos- This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.33.16.124 On: Mon, 01 Dec 2014 13:16:52

1.1723858.pdf

The Magnetic Susceptibilities of Metals Dissolved in Liquid Ammonia Simon Freed and Nathan Sugarman Citation: The Journal of Chemical Physics 11, 354 (1943); doi: 10.1063/1.1723858 View online: http://dx.doi.org/10.1063/1.1723858 View Table of Contents: http://aip.scitation.org/toc/jcp/11/8 Published by the American Institute of Physics354 S. FREED AND N. SUGARMAN for similar peaks found in the case of liquid phosphorus which also has a tetrahedral mole cule. The peak near 6.4A represents the average distance between the centers of adjacent mole cules, whereas the subsidiary peaks between about 3.5A and SA represent frequently re curring distances between nearest approaching atoms in different molecules. The presence of peaks implies slight preferred orientations, proba bly due to valence saturation within the mole cule, and a consequent slight repulsion between nearest chlorine atoms in different molecules. In order to use the areas under reconstructed isolated peaks for determining the number of nearest-neighbor molecules, the areas under the subsidiary peaks should be included, as well as that under the large 6.4A peak, since these areas arise from parts of neighboring molecules. Fur thermore, it seems clear that other and probably less well-defined peaks at distances greater than THE JOURNAL OF CHEMICAL PHYSICS 6.4A should exist, to correspond to distances be tween atoms on opposite sides of adjacent mole cules. Hence, the way in which the curve to represent the nearest neighboring molecules should be drawn in at distances greater than 6.4A is not well enough determined at present to obtain any reliable molecular coordination num ber. It is only possible to set a lower limit to this number. Following the contour of the distri bution curves from about 3.2A up to about 6.8A, and then sloping off along the dotted line from there on, gives an area corresponding to a minimum number of nearest molecules. On the questionable use of the same type of area equa tion as was used for atomic scatterers, this leads to a coordination number of about five from the curves of Fig. 3. It is believed, however, that this number should be considerably higher in accord with the above interpretation of the significance of peak areas. VOLUME II, NUMBER 8 AUGUST, 1943 The Magnetic Susceptibilities of Metals Dissolved in Liquid Ammonia SIMON FREED AND NATHAN SUGARMAN C. H. Jones Laboratory, Unive.rsity of Chicago, Chicago, Illinois (Received April 30, 1943) Here are recorded data on the magnetic susceptibilities of potassium and of cesium dissolved in liquid ammonia at 2400K and 220oK. Also included are a few data on calcium and barium. These solutions throughout the entire range of concentrations were regarded as representing an electron gas. Such systems would permit the distributions of the magnetic moments of the electrons to be followed con tinuously from the degenerate region of the Fermi Dirac statistics to those distributions where the quantum and classical statistics are indistinguishable. The general features of the magnetic behavior of a free electron gas METALS dissolved in liquid ammonia have the property, primary for the present in vestigation, of dissociating into electrons and positive metal ions.1 In dilute solution, the conduction is electrolytic with the electrons as the negative ions carrying 1 This classical work of Kraus is summarized by him in J. Frank. lnst. 212, 537 (1931). Also in C. A. Kraus, The Properties of Electrically Conducting Systems (Chemical Catalog Company, New York, 1922). were recognized, overlaid, however, by interactions charac teristic of the environment of the electrons in the solutions. In the light of the magnetic susceptibilities as well as of other properties, a description is proposed for the structure of the solutions with special reference to the conduction electrons. Barium was found to dissociate into two electrons per gram atom and it is inferred that calcium dissociated likewise, but much greater dilutions would be required for a satisfactory demonstration. A method of considerable sensi tivity and accuracy has been developed for measuring the magnetic susceptibilities of substances at low temperatures. most of the current. The conductivity changes continuously with concentration and becomes metallic in the strong solutions. When saturated, a solution of sodium possesses about twice the conductivity of mercury at room temperature. Along with this behavior go changes in color which is a pale transparent blue in great dilution, deepens into a dense dark blue when more metal is dissolved, and acquires in concentrated solu-MAGNETIC SUSCEPTIBILITIES IN AMMONIA 355 tions a metallic bronze-like luster giving the solution every appearance of being a liquid metal. When the solven t is evaporated, the original metal is regained. In the case of lithium and also of calcium the metallic crystals which first deposit from the solution contain ammonia bound stoichiometrically, Li(NH3)4 and Ca(NH3)6. The former even at low temperatures dissociates into ammonia gas and the metal powder. These phenomena speak strongly for regarding the solutions as a liquid metal whose concentra tion can be varied continuously over a great range. Like metals, the solutions may presumably be conceived as an electron gas which in better approximation is to be counted as subject to the microscopic fields in the solution. The magnetic measurements were undertaken in order to follow the distribution in spin or magnetic moment among the electrons as a function of concentration and temperature. At high electron concentration, we would expect according to Pauli's well-known work2 almost all the electrons to have paired-off their spins, the degenerate state of the Fermi-Dirac statistics, whereas in very dilute solutions, the elementary magnets would be independent of each other and conform to those distributions of the Fermi-Dirac statistics which are indistinguishable from the distributions of the classical statistics. To achieve a comparable variation in spin with the high electron densities prevailing in metals, it would be necessary according to theory to raise the temperature above the so-called degeneracy tem perature, the order of 20,000°. By diluting the electrons of the metal, in our experiments with liquid ammonia, the degeneracy temperature would be greatly reduced, even considerably below the temperature where magnetic measure ments are possible. The degeneracy temperature is given by h2 (3N)~ To= 2mk 87r V ' where h is Planck's constant, m is the mass of the electron, k is Boltzmann's constant, N is the number of electrons in the volume V. 2 W. Pauli, Zeits. f. Physik 41, 81 (1927). In a previous work,3 it was shown that in its general features, the course of the variation of the susceptibilities was consistent with this outline. When sodium was at a concentration O.SM, its atomic susceptibility was of the same order of magnitude as that of the bulk metal. The de generacy temperature for this concentration was about 2000°K, ten times higher than the temper ature of the substance 2300K and, hence, the electrons were degenerately distributed. At o FIG. 1. Apparatus for measuring magnetic susceptibilities of substances at low temperature. 0.002M with the degeneracy temperature at about SocK, the atomic susceptibility had in creased about one hundred fold and had attained values expected of an electron gas according to classical statistics. These results were confirmed in the more ex tensive investigation of Huster and Vogt.4,5 However, the data lacked the precision required for interpretation, especially in very dilute solu tions where the actual forces to be measured were as. Freed and H. G. Thode, Nature 134,774 (1934). 4 E. Huster and E. Vogt, Physik. Zeits. 38, 1004 (1937). 5 E. Huster, Ann. d. Physik 33,477 (1938). 356 S. FREED AND N. SUGARMAN FIG. 2. Apparatus for preparing solutions of metals in liquid ammonia. extremely small. Since the atomic susceptibility of sodium proved to have less than the theoretical value, the measurements had to be pushed to extreme dilutions in order to discover the limiting susceptibility. For this reason, we turned to solu tions of potassi~m whose electrons would sup posedly be "freer" than those of sodium; the conductivities at the same concentrations were known to be consistently higher. The suscepti bilities were actually found greater and a satis factory precision was obtained in the measure ments. Also included in this communication are susceptibilities of solutions of cesium, of barium, and of calcium. EXPERIMENTAL Apparatus A refinemen t6 of the Gouy method of meas uring magnetic susceptibilities was made avail able for substances at low temperatures without appreciable decrease in accuracy. A vertical tube divided into equal sections by a horizontal partition is suspended from the arm of a microbalance so that the partition comes into the center of the pole gap of a magnet. One-half of the tube is filled with the solution and the other with the pure solvent. Under these conditions the susceptibility of the dissolved substance per unit volume in dilute solution is given directly by 2F K = Ksolution -Ksolvent A (HI2_H22) , 6 S. Freed and C. Kasper, Phys. Rev. 36, t002 (1930). where K is the specific volume susceptibility of the solution minus that of the solvent, F is the vertical force exerted on the balanc~ arm, A is the cross-sectional area of the tube, HI is the field strength at the partition, while H2 is that at the ends of the tube. In the more concentrated solu tions, account must be taken of the fact that the solvent material in unit volume of the solution is not identical in amount with that in unit volume of pure solvent. This correction is usually small and altogether negligible in the more dilute solutions. To extend the usefulness of this procedure for substances at low temperatures, the method con sisted in suspending the glass tube G (Fig. la) in an atmosphere of stagnant hydrogen inside a tube H which was kept at the low temperature by the surrounding refrigerant liquid. A stream of dry hydrogen entered the stationary tube H at the upper end J and flowed to the outside atmosphere at such a rate as to keep the air with its moisture from descending during the experiment and af.fecting the weight of the tube G. Bames with small openings could be mounted horizontally on the ends of the tube K which was perforated to allow easy entry of the hydrogen from the tubes J. The refrigerant, liquid ammonia, was kept in the Dewar vessel F. Its cap, not shown, was provided with connections to a manometer and pump for reducing the pressure and consequently the temperature of the refrigerant. The tubes T, T contained single junction thermocouples which abutted into H. The temperatures at the different levels of the hydrogen atmosphere as read by the thermocouples rarely varied by more'than one half degree even under unfavorable conditions. B was a capillary tube for the passage of air into the liquid ammonia to lessen the bumping of the liquid while it was being pumped. The Dewar vessel F had a circular cross section with exception of the region C (Fig. Ib). This fi tted in to the pole gap and was therefore designed to separate the pole pieces as little as possible and yet accommodate the tubes T and B. The cylin drical tubing had been here enlarged on one side by means of a carbon form pressing out the glass while it was heated and softened without affecting the diameter of the circular portion. G contained a trap at the lower end as shown MAGNETIC SUSCEPTIBILITIES IN AMMONIA 357 (Fig. la). By this means the bubble of ammonia gas occupying the space left between the liquid and seal would not rise to the partition when the tube was inverted to its normal position for weighing. The liquid in G extended, then, in an unbroken column except for the horizontal glas~ partition. Calibration and Measurement The Gouy method for determining susceptibilities re quires three measurements of force, a blank run on the container, a calibration with a substance whose absolute susceptibility is known, and finally the measurement of the material of interest. Both portions of C were filled with liquid ammonia for the blank run. Ideally under these conditions the force would be zero but since the system was not perfectly symmetrical small correction forces were to be expected. The dissymetry due to the temperature gradient was so small that it was ignored. The extreme variation in the forces was as a rule not more than ;:1::0.003 mg from day to day. Pure liquid ammonia served for the calibration. Its susceptibility had been determined within an accuracy of one to two percent by Huster. The relative susceptibilities of the solute as measured by the present method were practically independent of the value accepted for the susceptibility of liquid ammonia. The absolute value was, however, determined by it. For the calibration, the upper part of the tubeC was filled with liquid ammonia, the lower portion was left evacuated and the net force noted on the balance at definite field strengths, i.e., definite currents in the electromagnet. The computations were made here as well as in all the measurements at 15.0 amp. with the corresponding field at about 15,000 gauss. The value at 15.0 was taken from a graph of forces against current. Although the individual points, in the measurements of the solutions, differed from the average by as much as 0.005 mg, it was found that in a twenty-four hour period, this average did not change by more than ±0.002 mg. This variation together with that of the blank produced a probable error of ±0.005 mg in the net force. The tube factor defined as the ratio of the susceptibility to force changed less than 1 percent over the temperature range -33°C to -53°C. At lower temperatures, the weighings were somewhat less accurate. When the bath was pumped to obtain lower temperatures, the ammonia bumped. This was almost entirely eliminated by aspirating air through the capillary tube B. To obtain a uniform temperature throughout the refrigerating liquid it was found necessary to heat the bath slightly throughout its length, a function performed with bare platinum resistance wire. In this manner, the tempera ture of the bath could be controlled to better than a degree with a temperature difference of less than one-half degree between the top and bottom. Preparation of Solutions Potassium and Cesium The solutions were prepared in an.all glass vacuum line system adapted by H. G. Thode and R. P. Metcalf of this laboratory from the procedures developed by Kraus and by Gibson and Phipps.7 The metal was introduced into D at the point B (Fig. 2a) by distillation. A length of capillary had been filled with the metal and a section of the appro priate length was broken off for each concentration. The desired amount, within ten percent, was slowly distilled into D by the use of free flame. Ammonia which had been twice dried over potassium was condensed into D. The condensation was complete after there was enough solution in D above the upper siphon so that C could be filled and an aliquot portion of the remainder could be siphoned into T. The solution was then stirred by creating a slight vacuum and so bringing about vigorous boiling. The solution was then successively siphoned into C, T, and S. C was sealed off at E and the solutions in T, S, and D were kept for analysis. Calcium and Barium Because of the high melting points of these metals, another method was devised for introducing them into D. TABLE I. Susceptibilities per gramatom XA' Force Free electron Moles (corrected) K solxl5f NH3 gas liter (mg) xA XIO' xA XI0' Potassium 2400K 0.00341 0.123 0.00432 1268 1530 0.00384 0.127 0.00452 1180 1525 0.00406 0.152 0.00505 1240 1522 0.00481 0.160 0.00570 1180 1520 0.00812 0.222 0.00790 974 1505 0.00960 0.246 0.00819 853 1480 0.0318 0.359 0.01280 402 1060 0.482 1.126 0.0390 29.9 310 Potassium 2200K 0.00354 0.078 0.00286 809 1630 0.00422 0.100 0.00332 790 1620 0.00501 0.118 0.00417 834 1600 0.00844 0.116 0.00411 488 1550 0.0331 0.213 0.00765 232 1220 0.500 0.558 @.0194 -7.7 310 Cesium 2400K 0.00415 0.139 0.00470 1130 1500 0.00582 0.214 0.00724 1245 1480 0.00690 0.207 0.00700 1013 1460 Cesium 2200K 0.00432 0.096 0.00325 755 1620 0.00605 0.183 0.00618 1020 1580 0.00718 0.139 0.00472 657 1560 Calcium 2400K 0.00244 0.077 0.00274 1140 3060 0.0100 0.145 0.00531 938 2840 Calcium 2200K 0.00~5 0.069 0.00245 978 3300 0.01 0.079 0.00283 271 3040 Barium 2400K 0.00106 0.066 0.00235 2280 3400 7 G. E. Gibson and T. E. Phipps, J. Am. Chern. Soc. 48, 312 (1926). 358 s. FREED AND N. SUGARMAN 1600 i(Oi~~2) \ 1400 \ \ 1200 'O~ .!..(o.l1Z) JI< l 2,40 i= 1000 :; ;;; 800 ~ ... ~ 600 ::> '" ~ 400 ~ 4: 200 POTASSIUM A 240'K V 220'K CESIUM <2SI 240'K ~ 220'K 0.005 0.010 0.015 0020 0.025 0.030 0.035 0.OiOm'O.48t 0.~85 0.490 0.495 05~OO MOLES/LITER FIG. 3. Atomic magnetic susceptibility as a function of concentration. The apparatus represented in Fig. 2b was sealed onto the vacuum line; F was joined to F on D (Fig. 2a). After the system had been thoroughly pumped, flamed, and flushed with gaseous and liquid ammonia, a piece of the metal was put into B and the system highly evacuated. Ammonia was then condensed on the metal so that the saturated solution which formed ran down the spiral capillary tubing C and through the filter plate D, a 01 Selas clay filter. This solution was siphoned into D through the stopcock F. Analyses of Concentrations The volumes of the solutions contained in traps S, D, and T were determined by distilling the ammonia into three measuring cells of the type described by Kraus.8 After the ammonia had been distilled from S, D, and T, the residue was dissolved in water, boiled to remove ammonia and titrated with O.OIN HC!. The concentrations of the solu tions in D were usually about 5 percent higher than those in T and S which agreed with each other within better than 1 percent. The higher concentration in D was doubtless due almost entirely to some undissolved metal which had con densed during the distillation considerably above the solu tion line. Because of good agreement in the analyses of the solutions in T and S and because siphonings were rapid, the concentration was taken as the average in T and S. In a few instances, D was averaged in if S0me mishap occurred to one of the other. Previous work9-u has shown that the densities of the dilute solutions were practically identical with that of pure ammonia. Only in the most concentrated solution of potassium is the difference appreciable and the correction was applied. The actual density was obtained by interpo lation from the data of Johnson and Meyer. The experimental errors were incurred almost solely through the errors in weighing; the errors in analyses were relatively small. 8 C. A. Kraus, J. Am. Chern. Soc. 43, 749 (1921). 9 C. A. Kraus, E. S. Carney, and W. C. Johnson, J. Am. Chern. Soc. 49, 2206 (1927). 10 w. c. Johnson and A. W. Meyer, J. Am. Chern. Soc. 54, 3621 (1932). u E. Huster, Ann. d. Physik 33, 477 (1938). The errors in the analyses of potassium and of cesium amounted to less than 1 percent, in calcium 2 percent, in barium 3.5 percent. EXPERIMENTAL RESULTS AND DISCUSSION OF DATA The data for the four metals measured are gathered in Table I and are also presented in the graph (Fig. 3). The following features of the data support the idea held at the beginning that the solutions may be taken to represent an electron gas. (1). In great dilution, the atomic susceptibility approximates the value which would arise from independent elementary magnets possessing one half unit of spin. (2). The concentrated solutions which obvi· ously possess strongly metallic properties furnish atomic susceptibilities in rough agreement with the presence of an electron gas free to move throughout the volume. For example, Huster found that a saturated solution of sodium had an atomic susceptibility of about 80 X 10-6 whereas a free electron gas in the same volume would have about 40X10-6• (3). With increase in atomic volume, the atomic susceptibility increased. At 240°, the concentration where the steep increase sets in occurred at about O.OtM. The ideal theory has 0.02M as the concentration where the degenerate gas begins to pass into the classical distributions at 2400K. With increasing temperature, Fig. 3 indicates, the difference between the correspond ing concentrations diminishes. It is evident that the model of a free electron MAGNETIC SUSCEPTIBILITIES IN AMMONIA 359 gas must be extended to include interaction with the environment. Free electrons would because of their motion in a magnetic field show a dia magnetism12 cancelling one third of the spin magnetism. The data establish that such a cancellation does not occur; the two-third value of the spin paramagnetism indicated by the lower horizontal line on the left of the graph is exceeded and the susceptibilities continue toward the full value due to spin, the upper horizontal line. Peierlsla has raised the question whether Landau's conclusions are valid for a condensed system where the average frequency of collisions between the electrons and their environment is high. The actual susceptibilities are in general lower than computed14 for a free electron gas (last column of Table I) and even more drastic de parture from such a model comes to light in the temperature dependence of the susceptibilities which decrease rather than increase with de creasing temperature. To account for such a behavior, the idea naturally suggests itself that electrons are re moved by the positive ions to form diamagnetic potassium molecules 2K++2E--p.K 2. It should be noted that the formation of potas sium atoms or of molecular ions would instead increase the susceptibilities. Huster had assumed the presence of diatomic molecules ill solutions of sodium. A computation shows, however, that the concentration of the molecules needed for agree ment with the susceptibilities appears in conflict with the molecular weight of dissolved sodium as revealed by Kraus'ls measurements of vapor pressures. There are other factors too which indi cate that the hypothesis of dissolved diatomic molecules is rather forced. The very dilute solu tions and the very concen tra ted solu tions both are in agreement with a system consisting only of electrons and separate ions. The existence of the metals such as Li(NHa)4, Ca(NHa)6' Sr(NHa)6, Ba(NHah containing the same number of am monia molecules as completely surround the 12 L. Landau, Zeits. f. Physik 64, 629 (1930). 13 R. Peierls, Zeits. f. Physik 80, 763 (1933). 14 N. F. Mott, Proc. Carnb. Phil. Soc. 32,108 (1936). 15 C. A. Kraus, J. Am. Chern. Soc. 30, 1197 (1908). positive ions in their salts indicates strongly that the ions are separated from each other in these metals by ammonia. We shall take the single point of view that throughout the entire range of concentrations all these solutions consist of posi tive ions and electrons in ammonia and that no appreciable concentration of diatomic linkage exists between metal atoms or ions. We shall then inquire into the disposition of the conduction electrons to which the lower susceptibilities may be ascribed. The specific volume susceptibility of a de generate gas is given by16,17 K = 2p.2(dZjdE)E=Eo, where K is the susceptibility per unit volume, p. is the Bohr magneton, and (dZjdE)E=Eo is the number of energy levels per unit energy range at the top of the Fermi distribution. The energy of interaction of the magnetic moments of the electrons with the external field serves to uncouple the pairs of electrons in the filled cells and send some into empty ones so that the spins are in line. The effectiveness of this process depends upon the number of cells (or levels) which are available to the energy of interaction. Therefore, the more widely separate are the levels, the lower is the susceptibility. It has been shown by Bardeenl8 that when resonance energies are included the resulting decrease in the density of levels lowers the magni tude otherwise computed for the heat capacity of metals. Magnetic susceptibilities are affected in the same general way. This resonance binding may, we propose, be represented as follows: solvated ion solvent where the ammonia near the K+ corresponds to 16 W. Pauli, Zeits. f. Physik 41, 81 (1927). 17W. F. Mott and H. Jones, Theory of Properties of Metals and Alloys (The Clarendon Press, Oxford, 1936), p. 184. 18 J. Bardeen, Phys. Rev. 50, 1098 (1936). See, e.g., F. Seitz, Modern Theory of Solids (New York, McGraw-Hili, 1940), p. 421. 360 S. FREED AND N. SUGARMAN one of the molecules of solvation and the am monia structure facing it represents the solvent, in general, but may also on occasion stand for ammonia of solvation of another potassium ion. Structures with the electron near one of the hydrogen atoms may be imagined stabilized by the many equivalent or nearly equivalent struc tures which the situation may assume. The mobility of the electron in electrolysis may be viewed as the passage of the electron from one hydrogen atom to another with some differenti ation between the ammonia molecules of solva tion and of solvent. Also contributing to the decrease in suscepti bility and especially to its decrease with the temperature is the interaction of pairs of electrons which have some resemblance to pI centers known in systems composed of a metal such as potassium dissolved in a crystal as potassium chloride. In a site where a negative ion is missing, two electrons can lodge with some stability,19 This pair is probably diamagnetic. In addition to the interaction having analogy with pI centers, the electron pairs are further stabilized through resonance in the same way as the unpaired elec trons. In our present schematization another electron would be attached, let us say, to one of the hydrogen atoms on the ammonia molecule to the right. The interaction of these electrons to form a diamagnetic pair would then account for the decrease in paramagnetism. Here again the electrical conductivity would be ascribed to the quantum mechanical passage through barriers somewhat as Farkas20 assumed in an over simplified model. In this way there appears no contradiction with the molecular weights as derived from vapor pressure measurements. 19 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (The Clarendon Press, Oxford, 1940), p. 128. 20 L. Farkas, Zeits. f. physik. Chemie A161, 355 (1932). With further increase III electron concen tra tion, the increased interaction may be viewed as an increase in the Fermi zero-point energy which operates to oppose the local trapping within barriers. Such a process is consistent with the gentle increase in susceptibilities at still higher concentrations which Huster found. The lower susceptibilities of sodium compared with potassium (or of calcium compared with barium) may be linked chiefly to the relative sizes of these ions. The smaller ion induces a greater polarization in the ammonia of solvation with more positive charge localized on the hydro gen atoms. Relative to these hydrogen atoms, the exchange energies are increased with consequent decrease in the density of energy levels. Asso ciated with this influence lowering the suscepti bilities there is another aspect-the deepening of the energy barriers would favor an increased stability and concen tration of the electron pairs. * The fact that the atomic susceptibility of barium is definitely greater than that arising from one electron per gram atom is sufficient evidence that the ion dissociates into two elec trons per gram atom which interact of course, with their environment. It appears highly prob able that calcium dissociates in the same way but, as would be expected, the interaction is greater. We wish to express here our gratitude for much assistance to Dr. RichardP. Metcalf and to Dr. Warren C. Johnson whose knowledge of the properties of liquid ammonia systems was always at our disposal. * In a structureless medium, one expects on purely electrostatic grounds that the pairing of electrons would also take place to a greater degree with smaller ions. The Debye-Hiickel ion atmosphere is here replaced by the electron atmosphere and with increased electrostatic potential from the smaller ion, there would be a greater probability of having more than one electron at a given distance from the center of this ion.

1.1697596.pdf

Physics in 1946 Philip Morrison Citation: Journal of Applied Physics 18, 133 (1947); doi: 10.1063/1.1697596 View online: http://dx.doi.org/10.1063/1.1697596 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/18/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Making Physics: A Biography of Brookhaven National Laboratory, 1946–1972 Phys. Today 53, 55 (2000); 10.1063/1.882943 Making Physics: A Biography of Brookhaven National Laboratory 1946–1972, by Robert P. Crease Phys. Teach. 37, 503 (1999); 10.1119/1.880383 Report of the Director for 1946 Rev. Sci. Instrum. 18, 444 (1947); 10.1063/1.1740970 Duane Roller, Recipient of the 1946 Oersted Medal for Notable Contributions to the Teaching of Physics Am. J. Phys. 15, 176 (1947); 10.1119/1.1990920 Proceedings of the American Association of Physics Teachers: The St. Louis Meeting, June 20 and 21, 1946 Am. J. Phys. 14, 341 (1946); 10.1119/1.1990862 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51Journal of Applied Physics Volume 18, Number 2 February, 1947 Physics in 1946 By PHILIP MORRISON Cornell University, Ithaca, New York NINETEEN forty-six was the first year of peace. Throughout the world, and espe cially in the United States, the generally difficult problems of reconversion began in earnest. Re conversion meant much for physics and physi cists. The strange symbols of wartime work the SCR-S84 radars, the HVAR rockets, the VT fuzes, the compounds of tuballoy-began to lose their compelling interest and their jealously guarded secrecy. The great" amateur" wartime laboratories, where academic physicists and their industrial and engineering colleagues had worked so hard and so successfully on war program, be gan to dwindle. The publication of volumes of reports began. And physicists went back, a little rustily, to the problems of the days before the war. Most of those problems were still there, for not much fundamental progress had been made during the war years. But gone was the reluc tance to do big things, gone the sometimes valuable, sometimes hampering isolation of the research worker. Physics, especially nuclear physics and its related frontier fields, had grown up. It was pretty well organized, in the wake of the OSRD and the Manhattan Project. The work of the year very much reflected the problems of the physicist. Much work was be gun, but not very much completed. People had plans, often great and exciting ones, but still only plans. Some wits began to talk of the latest "Physical PREview." Teaching loads were heavy, and the flood of students, good and serious and deserving students, drew many research workers away fn?m problems they had almost begun to remember. New laboratories had sprung up, like Oak Ridge and the Argonne, and in the first postwar year began to make their mark on public research. Most striking of all was the essential disap pearance of the peacetime international com munity of physics. The United States, tired enough by years. of war bu t still incredibly rich by the standards of Europe and Asia, was pre eminently the home of physics. Our once good communication with the rest of world had atrophied with the secrecy and the contingencies of wartime. By the end of the year, only a begin ning had been made towards a return of the days when physics was truly international. Few physi cists hoped more wistfully for anything than for the full restoration of the ways of peace in travel, in publications, and in the spirit of a world-wide SCIence. The present account is for all these reasons a fragmentary account, mainly of "work in prog ress," and mainly of work here in America. But it is work of the highest promise for physics, work carried on with the highest hope that in 1947 and the years ahead the best of the old spirit will come to employ the great new tools which are the legacy of war. 133 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51SUPERCONDUCTMTY One of the most spectacular results of the year was in the tradition of the world before the war. Professor Richard A. Ogg, working essentially alone in the Chemistry Department of Stanford. University, discovered a substance which is electrically superconducting at liquid-air tem peratures and even above.1 This statement has little in common with the dramatic discoveries of new particles, but this work may be as funda mental a step towards the understanding of the strange phenomenon of superconductivity as was the finding of the neutron to nuclear physics. For decades it has been known that some metals and alloys cooled to very low tempera tures lost all electrical resistivity. Resistances of many ohms dropped by twelve or fourteen orders of magnitude when the transition temperature was reached. Strange electrical and thermal ex periments could be performed with such extra ordinary solids. By moving a ring of the material out of a magnetic field in which it was cooled, a current was induced. This current died away witp a time constant given by the inductance and the extremely small resistance of the cold metal. It is told that the Leyden laboratory workers delighted to prepare such persistent currents, and to send the loop, still cold in its Dewar, to a distant place where the lecturer could demonstrate that the ~urrent was still flowing, hours or even days after it had been started up! Such conductors demonstrated Lenz's law with a vengeance: their low resistivities meant that the induced currents cancelled the external fields to a T, and magnetic fields thus cannot be set up within superconductors. They are almost perfectly diamagnetic. The large scale theory of such phenomena is fairly clear, but it is quite unsure what is the electronic and atomic mechanism by which the state of superconductivity is caused. Only one thing seemed essential: extremely low tempera tures. No superconductor was known which showed the effect above ten degrees absolute. Ogg had long been interested in a famous and strange set of substances, the solutions of the alkali metals in liquid ammonia. For twenty years these materials have provided work for 1 R. A. Ogg, Phys. Rev. 69, 243 (1946). 134 physical chemists concerned with the nature of ionization in solution. The example which proved so interesting was the dilute solution of sodium in ammonia, in concentrations of twenty or thirty grams of metal per liter of solvent. That solutions of this kind are sufficiently strange in their other properties was well known. They look like metals, with a quickly-frozen solution having a metallic steely blue luster. At -50°C or so, the homogeneous solution separates in two differently-colored phases, bronze and blue, if slowly cooled. It is not hard to prepare the ma terial. In one method, an annular glass trap is employed, attached to a vacuum line.2 Water and air must of course be kept out. The sodium metal is placed in the system, and anhydrous ammonia brought into contact with the metal. The system is kept cool, below the boiling point of the ammonia, at -34°C, and the solution forms. To demonstrate the superconductivity it is enough to place the liquid solution in its vessel inside a solenoid coil, and to freeze it in liquid air within a matter of ten seconds or so. When the frozen ring is removed from the liquid air, it is placed near a flip coil, and the magnetic field of the ring detected by a galvanometer deflection. A magnetic field will repel the ring strongly, be cause of the very large negative susceptibility of the substance. Such tests seem to prove the superconducting nature of the substance. It is not quite so simple as that, as usual. Out of a few score preparations, only a few are suc cessful. Apparently the large volume change on freezing causes the solid to crack seriously, and the resistance is the resistance of the cracks and not of the solid itself. One must be patient to find the effect. Two attempts3,4 to confirm Ogg's work failed to do so, probably for this reason, but it has been confirmed by at least one other worker. Why the interest? One more superconductor for the handbooks does not sound exciting. But there is a vast difference. This material is a super conductor not at 5 or 10 degrees absolute tem perature, but at more than 90oK. Recall that thermodynamically the gap between these tem- 2 J. W. Hodgins, Phys. Rev. 70, 568 (1946). 3 Boorse, Cook, Pontius, and Zemansky, Phys. Rev. 70, 92 (1946). 4 Daunt, Desirant, Mendelssohn, and Birch, Phys. Rev. 70, 219 (1946). . . JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51peratures is much greater than the gap between the temperature of liquid air and room tempera ture. Consider, for instance, the amount of work required to remove a given amount of heat energy, say by the'use of a mechanical refrigera tor. If the region to be cooled is at liquid-air temperature and the condenser of the refrigera tor at room temperature, less than one-third as much work is required as if the heat had to be moved from a region at lOoK to a condenser cooled by liquid air. Thus Professor Ogg's solutions hold the suggestion that a supercon ductor can perhaps be made at room tempera ture. One cannot predict such a result, of course, but it would be a foolhardy statement now to deny its possibility. A little speculation on what would happen to laboratory apparatus and in deed to all electrical devices if fairly large cur rent density could be achieved without appreci able ohmic loss is worth while. Even if no such near-fantastic result ever ensues, the first super conductor at temperatures so "high" as those of liquid air is sure to prove a challenge and a stimulus to the theory of this strange phenom enon. Ogg predicted this property of the sodium ammonia system from his own theory5 of the structure of this unusual substance. His theory is formally strange and unconventional to most physicists, and is unconvincing to this author at least. It involves the notion of paired electrons which move in cavities in the solvent, after escaping from the valence bonds of the alkali metal. These paired electrons have no resultant spin and satisfy a condition years ago proposed to explain superconductivity as an effect bound to occur with particles which have no spin. The stability of these arrangements and the order-of magnitude of concentrations and temperatures do not appear evident from Ogg's so far pre liminary theory. It is to be hoped that the next year will bring wider interest in this remarkable problem both from the experimental and the theoretical side. There is hardly a laboratory which cannot make a superconductor now! Liquid hydrogen generators and the rest of the special techniques of cryogeny are no longer needed. 5 R. A. Ogg, Phys. Rev. 69, 668 (1946). VOLUME 18, FEBRUARY, 1947 THE PUSH TOWARD HIGH ENERGY The one most characteristic feature of nuclear physics is the high energy per atom involved in the reactions. Where chemical reactions release a few electron volts for each atom involved, the reactions of nuclear physics release or require millions. The very fact of such high specific energies makes it possible to' do experiments in which only individual atoms take part, observ ability following only from the highly energetic nature of the motions and transformations of the reactants. Nuclear physics-with of course the spectacular and decisive exception of the chain reaction of uranium--does not often work with macroscopic amounts of matter. But to give even single atoms the millions of electron volts they need often requires large scale apparatus. The engineering which is so typical of such laboratories today began in the first efforts to concentrate a few million volts in ions of hydrogen and heavy hydrogen. Since Rutherford's Cavendish days, this has been done by simply setting up a few million volts potential difference in the laboratory (with a,c., as in a surge transformer set, with electrostaticaIIy generated d.c., as in the familiar Van de Graaff machines) and letting the ions .fall through the potential difference in a vacuum tube. Always the demand was for higher energy. The first energy barrier· which it was sought to overcome was the electrostatic barrier, the energy required to bring one positively-charged projectile very near the positively-charged nucleus. One wanted the charges near enough so that the nuclear matter could touch and "stick," to produce new radioactive nuclei and to initiate nuclear reactions. The size of this barrier is easy to compute: the energy of repulsion is Zze2/r, where ze is the charge of the projectile (generally e or 2e for protons, deuterons, or alphas), Ze the charge of the target nucleus, ranging to 94e for the heaviest known, and the distance r is the nuclear radius, never much larger than 10-12 cm. The answer turns out to be some ten million electron volts needed to cross the barrier for the most difficult cases. N"o one has yet established a potential difference above six million or so. But the desire to produce particles with such energies has led to ingenious solutions. 135 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51The most famous of these is the cyclotron, in which a modest radiofrequency potential differ ence of a hundred or two kilovolts is applied again and again to the moving particle which is bent into a spiral path by a large magnetic field. Successive passages across the r-f drop occur uniformly spaced in time and, "re.sonant" with the r-f frequency, lead to high final energies. The Crocker cyclotron at Berkeley, the largest machine of its kind, is capable of making forty million-volt alpha-particles, which cross any nuclear barrier with energy to spare, and make possible many complicated reactions. Why go further? The answer is, of course, that there are still more remote objectives. The probing of nuclear matter by bringing the" test charge" which is. the rapidly moving projectile closer and closer to the nucleus has given us information about distances of the order of 10-12 cm or somgwhat less; we would like to look at matter even more intimately. The closer the impact, the more energy transferred, the finer the detail of ex ploration. We learned many years ago that there were particles of very short life, which we can not find in our laboratories, but which stream in to the earth in the cosmic rays. These objects, known now as mesotrons, because of their mass of 200 electron masses (intermediate between that of electron and proton), are believed on quite general grounds to be associated with those extraordinary attractive forces which make nuclear matter sticky, which bind together the neutrons and protons of the nucleus. The argument for the connection is essentially this: The electromagnetic forces are by now familiar. They are long range forces, falling off as the inverse square of the distance between two inter acting particles. Now, these forces, which propa gate, of course, with the finite velocity of light can be thought of as caused by the transfer of quanta between the moving charges. This is quite evident for the transverse electromagnetic waves, but it is true at least formally for the static Coulomb field as well. From this point of view electric charges are surrounded by quanta. If there is energy available, these quanta may be set free to go as far as they will. Otherwise they must be emitted, last a very short while, and then be reabsorbed by the particle which 136 emitted them or by another particle nearby. By analogy the same mechanism is used to account for nuclear forces by the exchange of mesons, It turns out that the Compton wave-length, A = h/mc, of the particles transferred determines the effective range of the forces. Thus, for quanta, the rest mass is zero, and the" Compton wave length" infinite, and the range infinite. The force decreases only for geometrical reasons, and is simply inverse-square. But the nuclear forces act only at short range, about 2 X 10-13 cm. The rest mass corresponding to such a range is about two hundred electron masses, just that observed for the mesons of cosmic rays. The inference is strong that around nuclear particles there is a transient cloud of mesons, and that if one supplies enough energy by collision or even by energetic "light" quanta, such mesons can be set free. The study of this mechanism would correspond for the nuclear forces to the study of Maxwell's equation, and could lead to the understanding of the nuclear forces in detail. It must be said that since the first suggestion of Yukawa in 1936 that such mesons might exist, reinforced by their discovery in the cosmic rays two years later, the best theorists have worn thin their patience on this theory. No consistent description has yet been given of the properties of nuclear forces. All the more has this challenged experimenters to make mesons in the laboratory and there to study them in the number and the detail which is impossible while their source is still only the cosmic ray. How much energy does one need? No one knows. There are clues. Certainly at least 100 Mev is needed, just for the rest energy, known to be 200 times the 0.51 Mev which is the electron rest energy. Perhaps they can be made only in positive and negative pairs, as are electrons in the field of a nucleus. Then at least 200 Mev is needed. More will be required to make them in some quantity, for one must do more than merely tickle the threshold of the reaction. So a popular target for the ingenious builders of machines has been 300 Mev. There are some who think that many mesons must be made at once rea,lly to study the details of their creation; these p~~si mists (or optimists) are planning in billions of volts, but still for some years in the future! The first postwar year saw active construction JOURNAL OF APPLmD PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51on large machines and smaller prototypes, all shooting at the 300-Mev target, or thereabouts.6 Most direct attack was that planned long ago at Berkeley, where a giant cyclotron had been built. This seventeen-foot magnet was built in 1940, and used during the war in the experiments which led to the construction of the great electromag netic separation plant at Oak Ridge. By simply extending the cyclotron design to real California scale, 200-Mev deuterons were expected, some what short of the popular goal, the more because such particles directed at nuclear particles have an effective velocity not much more than half of that in the laboratory system of reference. The transferrable energy is given by the velocity with which the particles are brought together computed in the frame of reference in which their center-of gravity is at rest. Even this energy is most difficult to get with a cyclotron. In the cyclotron, the particles must all spiral out in step, at constant phase, returning in each circuit to the region between the D's, where the high r-f voltage is applied. The familiar reson ance condition is that 27rf = eH/mc = "', where e and m are the charge and relativistic mass of the . I' (rest energy + kinetic energy) partlc e, l.e., m = , c2 H the constant magnetic field, f the frequency of the r-f oscillator and '" the angular velocity of the particle. This implies that the particles at all radii have the same resonant frequency, though some go fast and others slow. This is the key to the success of the machine. But the mass value, hence the resonant frequency. is not, in fact, constant, if the relativistic varia tion of mass with velocity be considered For 200-Mev deuterons this is not a small effect. The hope was, however, that by running up the highest possible r-f voltage, several million volts, some particles could be gotten out in spite of their having fallen a little out of phase on each step, for the number of times they need circle in the machine. is only a hundred. The giant cyclo tron has indeed been operating at Berkeley since November 1, producing its deuterons as planned, but the principle of operation is quite changed, and the deuterons make not a hundred circles in their spiral path, but nearer to ten thousand. 6 L. Schiff, Rev. Sci. Inst. 17,6 (1946). VOLUME 18, FEBRUARY, 1947 And the r-f oscillator is not a giant device pro ducing thousan.ds of kilowatts but an oscillator which would be considered small even for an ordinary-sized cyclotron. The story of this change is also a story of 1946. It began, of course, much earlier. The start was probably the development of the betatron in which the accelerating voltage is not applied by any electrodes, but is simply the induced e.m.£. caused by the changing magnetic flux in the a.c. magnet, acting on the electrons in their vacuum. doughnut as if they were the secondary turns of a transformer. The total induced voltage in present design is only some seventy-five volt" per turn, but the electrons come out with a hundred million volts in the General Electric betatron. These particles have circled the primary about a million times, traveling some thing like a thousand miles in the vacuum tube. The key to this machine is evidently the stability of the electron orbits. so that a chance disturb ance from stray fields or by collision with a gas molecule does not throw too many electrons against the glass walls of the vacuum tube. This is a good machine for energies up to several hundred million volts. The 100-Mev General Electric model produced the first man-made radiation of such high energy, but, in spite of hopes and even the illusion of success, probably has made no mesons. Professor Kerst is now en gaged in constructing a 300-Mev model, of quite advanced design, at the University of Illinois in Urbana. In 1945 Professor E. M. McMillan, then at Los Alamos, of the University of California. proposed a new type of acceleration. Actually, the same proposal had been published months earlier and completely independently by the Soviet physicist V. Veksler. The McMillan Veksler idea7 stems in a way from the demon stration in the betatron that it was not foolhardy to plan for very long paths in the vacuum tube, if the conditions of stability are properly ful filled by the design of the machine. The fact that one could count on geometrical stability had been proved in the betatron. McMillan began to think of the use of radiofrequency accelerating electrodes, as in the cyclotron, with stability in 7 E. McMillan, Phys. Rev. 68, 143 (1945); V. Veksler, J. Phys. (USSR) 9, 153 (1945). 137 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 1. The first synchrotron to operate in America. This device is installed at the General Electric Research Laboratory at Schenectady, and is under the supervision of Dr. H. C. Pollock. The vacuum tube is visible between the magnet poles. time, or, as he calls it, phase stability. With such a condition, the resonance condition of the cyclotron might be far from satisfied by all particles, and yet eventually after many turns, the particles might gain a large energy. For definiteness, think of a particle in the ... _'t..------. __ ---. __ I I I I I 2 34 ~t FIG. 2. An oscillogram showing operation of the 75-Mev GE synchrotron. The trace is cut off every 100 microsec. (1) marks the time of electron injection. At this time the machine is operating as a betatron. At (2) the r-f voltage is turned on. The electrons have by now about two-Mev energy. At (3) the r-f is turned off and at (4) is a signal from a photo-multiplier tube placed in the x-ray beam. For this trace the energy was about 20 Mev. 138 cyclotron at perfect resonance. Each time it reaches the gap between the accelerating elec trodes the r-f voltage has reached its maximum. If the particle happened to arrive a little early, the r-f voltage has not yet climbed to its maximum and the particle gets a little less energy than it should have gotten. It will then make the next turn in the field with a somewhat too high angu lar velocity (the equivalent mass is too small) and arrive still earlier next time. Such a particle lacks phase stability and will gradually faIl more and more out of step. It will never gain much energy. Now consider a very different case, that of a particle which arrives when the r-f voltage drop across the gap between electrodes-the accelerating field-has fallen to zero. This particle is just 180 electrical degrees out of phase with the r-f voltage. I t will be in resonance, but it will gain no energy in passing the gap. There are two such nodes in the complete cycle. Let us consider the one in which the voltage across the gap is changing from being slightly accelerating through zero to be slightly decelerat ing. (The other node will not provide stability.) Now suppose the out-of-phase particle arrives just a bit early. It will feel a small amount of an accelerating electric field. It will gain energy, its equivalent mass will increase, its angular velocity go down, it will begin to lag, and next time it will be more nearly 180 degrees out of phase again. If it came a bit late, it would cross a small decelerating potential drop, and be reduced in energy. The mass would go down, the angular velocity go up, and it would come more closely to the 180 degree phase. A particle exactly out-of- FIG. 3. Another oscillogram of synchrotron operation. The length of the sweep is 1000 microseconds. The trace begins at the time the r-f was turned off, and the electrons soon begin to strike the internal target. The irregular signal shown is taken from the output of a photo-multiplier tube placed in the emergent x-ray beam. Note the complex structure of the beam. This effect is still unclear. The machine is operating between 60 and 70 Mev. JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51~ 5YNCH ROTRON FIG. 4. The synchrotron being built at Berkeley for 300-Mev electrons by the group under Professor E. M. McMillan. The flux bars re ferred to bars of iron which saturate at a modest field. Until they saturate they strengthen the central field of the machine so that the electrons will accelerate un der betatron conditions. \Vhen they saturate the electrons are moving at a modest energy, and the syn chrotron r-f voltage is turned on. WT" MNA'l "TO ~OW ''''''TlAIOIt.. COIL, POL.f..,~ VAC.uUM C.}.1AMf)(Jl. phase then will have phase stability; it will tend to maintain this condition both at low and at relativistic velocities. Now suppose either the magnetic field or the alternating frequency be slowly increased, changing a little, but only a little, during the time the particle is making its turn in the magnetic field. The stability in phase will act to insure that the particle increases its mass to keep resonant with the increased fre quency. The particle will cross the gap just enough before the time when the electric field is zero so that it gains energy sufficient to keep it near resonance. As long as the frequency or magnetic field changes slowly, the particle will oscillate in phase back and forth across the 180 degree time, gaining on the average just enough energy to be resonant under the new conditions. Thus after many turns, picking up a very small amount of the available accelerating energy on each turn, the particle can reach high energy. The frequency may be varied with constant magnetic field, and the particle will spiral out as its energy increases. Or the r-f frequency and orbit radius may stay constant or nearly so and the magnetic field increased. Or combined changes may be made. In all such devices the particles will not be accelerated in a steady stream as in a cyclotron, but in pulses, repeating as rapidly as the frequency is varied or the magnetic field made to increase. VOLUME 18, FEBRUARY, 1947 These devices, of which there is clearly a large family, are generically called "synchrotrons." The name is based on the analogy between the motion of the particles and that of the rotor of a synchronous motor. The rotor spins at exactly the synchronous speed .with no load. But loading the shaft does not change the speed. The phase slips behind far enough so that the field differ ences will supply the needed energy to the load. In the same way the particles in the synchrotron slip out of exact phasing, "hunting" in fact for the node, but gaining energy just enough to compensate for the changing frequency or magnetic field. 8 The first American synchrotron completed is a seventy-five million volt electron machine, Fig. 1. The pulses of fast electrons from this machine are shown in Fig. 2. A synchrotron was made even earlier in Britain9 by placing a small electrode for the r-f voltage inside a betatron doughnut. The output of the machine went from four millions to more than double that, since the saturation of the central part of the magnet core in the betatron limits the field at the orbit to less than half of practical design saturation values. The stability of orbits in the betatron demands that the central flux change be larger by a 8 D. Bohm and L. Foldy, Phys. Rev. 70, 249 (1946); H. C. Pollock, Phys. Rev. 69, 125 (1946). 9 Goward and Barnes, Nature 158, 413 (1946). 139 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:515VNC.HROTRON VACUUM CHAMl>ER A~EM illY t;AC.H POLL P,f.(.L CDMRl5EO 0'-FORT.,.. ~C.T()II.~. ~ $ ... TO" .......... >JATEI> OF .014 I"''''' T .. ,(.,,-$PE<;.j",,- STKL 4"nT ~~NJ'EU Wl1"I1 ,....uL ... T' ... " <-, VA<'VWM-PR.oOF 'R~l)Ul(~ fI,IOIVIO\J"'<, $£"TO"'$ AU ",",PA,QAYtt:l W'i~ ji J~ ~&~R... '-A$1<.,5."T6. LDWl:.R. rolE. INNER GRID srUf> STt.M DRIVlfJG _ STE.M FIG. 5. The vacuum as sembly for the Berkeley synchrotron. Note the r-f electrode "dee." It is con· structed as a grid instead of as a solid electrode to reduce eddy-current losses and fields which would be set up by the strong a.c. magnetic field. The r·f energy is supplied by a pulsed water-cooled oscil lator at about 48 megacycles. The stubs are provided to adjust the position of the nodes on this rather high impedance structure. OOT"E.I\. (,R>D 110'.:>£0 -n:> O\J'l'E1l: VA". c.HAMeu WA.u... 1\1 A:>5C>J\1!IIx. nRJ'-ltt-lG " $Tt1& $T£M~ ~"'fI.O Tl) o.>n\ot. OF ~u.... ,"",0 ~c..~D T .. 'W WAu.. -ro 'IJJ.J~ 6ft10 <:If OEE. definite factor than that at the orbit. Synchrotron stability does not require any such condition, for the voltage gain comes from the r-f field imposed by electrodes, aI,.ld not from the transformer effect except incidentally. The GE machine actually begins the acceleration of its electrons as a betatron; then, when they have reached an appropriate starting energy, the r-f is turned on. The giant cyclotron is working now accelerat ing heavy particles not as a conventional cyclo tron, but as a synchrotron, or as it is sometimes called, a synchro-cyclotron. Frequency modula tion of the r-f supply accomplished by a rapidly rotating condenser in the plate tank of the r-f FIG. 6. Photograph of a model of the Berkeley synchro tron showing the driving stem and stubs. 140 oscillator brings out the high energy beam, after thousands of turns, with a modest amount of r-f power. The synchrotron principle maintains resonance acceleration in spite of the relativistic variation of mass.10 The synchro-cyclotron ac celerates deuterons, which are caused so far to strike an internal beryllium target, producing a beam of very fast neutrons. The first operation of the machine was on schedule, November 1, 1946, and experiments with the new fast neu trons are already in progress. No mesons have FIG. 7. The other side of the synchrotron model, showing the vacuum pumps in their pit, and the large vacuum manifold. 10 Richardson, MacKenzie, Lofgren, and Wright, Phys. Rev. 69,669 (1946). JOURNAL OF ApPLmD PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 8. A general view of the giant Berkeley cyclotron, now operating for 200-j\Iev deuterons as a synchro-cyclotron. Note the large radial crane of thirty-ton capacity overhead, the magnet with its 184" pole pieces (there is a man visible against the right leg of the magnet yoke), the two 32" oil diffusion pumps used for evacuating the vacuum "can," whose thick iron walls are visible between the pole pieces, and the rails, to the left of the pumps, used for removing the ion source. The tem porary concrete block shielding wall and a cloud-chamber set-up are seen at the right. (The photo graphs of this machine were furnished through the kindness of Professor R. L. Thornton, in charge of the giant cyclotron group at Berkeley.) yet been seen; but important results on the nature of nuclear forces seem sure already (December IS, 1946). It is planned to bring the deuteron beam itself out of the machine in the near future. Michigan,!1 Massachusetts Institute of Tech nology, Cornell, the General Electric Laboratory, Berkeley, and no doubt other laboratories are now engaged in design and construction of synchrotron-type electron accelerators of the 300-Mev size, in a variety of ingenious exempli fications of the same principle. The mechanical and electrical engineering problems involved in such work are not small, and many a design variant may have its own special advantages. General problems like starting the particles in their orbits, removing the particles from the machine after acceleration, detection methods specially suited to these pulsed sources, and 11 H. R. Crane, Phys. Rev. 69, 542 (1946); Phys. Rev. 70, 800 A (1946); Pollock et al, Phys. Rev. 70, 798 (A) (1946). VOLUME 18, FEBRUARY, 1947 many others may be expected to build up a whole science of very high energy physics in the next year or two. One simple and elegant achievement of the year in this field was the first successful removal of a high energy electron beam from the Urbana betatron, at about 20 Mev.12 The device which did the work is simply a piece of iron shaped like a U-channel, mounted in the vacuum doughnut in the right position. The electrons enter the U-shaped slot, where the iron walls shield them from most of the magnetic field, and shoot out in the field-free space in a straight line to a target or a thin window outside the machine. The success of this device should make possible many experiments with fast electrons, up till now never available in a well-directed beam. The idea of bending the fast particles in circu lar orbits and causing them to retrace their steps is evidently economical. But if the magnet grows 12 Skaggs, Almy, Kerst, and Lanzi, Phys. Rev. 70, 95 (1946). 141 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 9. A view of the big cyclotron from the side oppusite to that shown in Fig. 8. In the right foreground. on the truck which moves on rails in the floor. are mounted the round vacuum housing for the rotating condenser, its associated vacuum pump, behind them the oscillator housing. The vacuum chamber is here shown clearly in operating position in the field. To the left may be seen the horizontal rod, which extends into the vacuum chamber holding the in ternal beryllium probe against which the beam is made to strike. Neutrons were produced at 100 Mev or more on N"ovember 1, 1946. to be very large the economy of the scheme begins to seem less obvious. Perhaps it is better to let the particles fly in a straight line down a long tube. If one can set up a certain electric field strength, the cost of the linear accelerator goes up about as its length (the parts are simply re peatedly placed end-to-end); but if a large magnet needs to be scaled up with the magnetic field kept constant, the volume, weight, and hence the major part of the cost will go up as the cube of the radius. But the energy of particles in their equilibrium orbits goes up only as the radius or, if they are still moving much more slowly than light, as the square of the radius. Thus for sufficiently high energy a linear accelerator will be cheaper than a scaled-up magnetic device. The idea is an old and unsuccessful one; but the familiarity with microwave techniques gained in wartime makes it appear practical once more. Professor Alvarez at Berkeley is now engaged in building a large linear accelerator for protons. Protons enter the machine with a few million 142 volts energy, from a one-step accelerator, in this case a Van de Graaff machine. The vacuum tube down which they fly is one long resonator, made of many resonant cavities placed back-to-back and so driven by individual but phased micro wave oscillators at 150 cm that the moving pro tons enter each cavity in phase with an accelerat ing electric field. By the time they cross the cavity, one cycle has elapsed, the field behind has dropped to zero, and the next cavity is beginning to acquire an accelerating electric field. There is one difficulty with this simple picture. The protons do not move with uniform velocity, but constantly increase in velocity. If the cavities are driven, as they must be, by an oscillation of one frequency, some trick must be used to keep the protons in step. This is done by adjusting the repeat length of the resonant cavities l so that ll(f = Ale where A is the wave-length of the driv ing oscillator and rIle the velocity of the particle relative to that of light. If the outside cavity diameter is held fixed, the cavities must be JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51modified in shape to keep them resonant, though of different lengths. In transmission line language the line is a loaded line. A short coaxial cylinder is placed inside the outer cavity cylinder, and the protons move down the axis of the whole cavity. Adjusting the diameter of these central tubes keeps resonance. Each cavity will during half the cycle be decelerating: the electric field will point the wrong way. During this time the proton must be inside the central tubes, and shielded from the wrongly-directed field. The tubes are hence called drift tubes, because within them the protons" drift" under no field.J3 To maintain phase stability and geometrical focus ing, it is necessary to place conductors athwart the drift tubes, through which the protons must pass. Thin beryllium foils have been developed for this work. In trials more than a million volts per foot has been developed in a cavity. As yet no coupled cavities have been tried for accelera tion, !:lut the first forty-foot unit is soon to be tried out. It is expected that it will be delivering protons at more than thirty-two million volts by February, 1947. New high power radar oscillators FIG. 10. A close-up of the rotating vacuum condenser whose housing is seen in Fig. 9. The many teeth which produce the varying capacity are plainly visible. The oscillator frequency, which is in the neighborhood of 10 Mc, is varied about 20 percent at a modulation rate of a few hundred times per second by means of this device. 13 Oppenheimer, Johnston, and Richman, Phys. Rev. 70, 447 A (1946); L Alvarez, Phys. Rev. 70, 799 A (1946). VOLUME 18, FEBRUARY, 1947 FIG. 11. The forty-foot linear accelerator at Berkeley. Note the large tank of the horizontal pressure Van de Graaff machine at the far right. This is the ion source for the machine. The outer steel tank for the linear accelerator tube itself is conspicious, flanked by the thirty-odd war surplus l.S-m radar oscillators on either side. Protons in the thirty-to-forty-million volt range are expected by February, 1947. New high power magnetrons are under construction to replace the many oscillators shown here. Only a few of the new magnetrons would be needed for a forty-foot section. (Photo from Professor L. W. Alvarez, heading the linear accelerator group.) have been developed for the job, using the pulsed magnetron principle. Here again the output beam will be pulsed at a rather slow audiofre quency, for the sake of magnetron cooling and performance. A similar linear accelerator but without the drift tubes and without phase stability can be built for particles moving near the speed of light.14 High energy electrons are planned for at M.LT. and at Purdue by groups working on the design of such a device. If mesons are not made in 1947, it will not be for want of effort! It is to be hoped that enough leisure will be left to plan the experiments which these machines will make possible. THE LEGACY OF THE WAR The year was marked by the widespread if delayed publication of results of the key war projects in physics, and above all, by the return to their old laboratories of hundreds of war-ex perienced physicists, brimful of information about what had been done, and confident in their understanding of whole fields of technique which had been vague general possibilities in t940. 14 J. Slater, Phys. Rev. 70, 749 A (1946). 143 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:511. Radar To this author one of the most fascinating p'art~ of micro~ave radar has been the duplica tiOn m the radiO spectrum of the familiar results of physical optics. The" optics" of microwaves mirrors, ~~atings, diffraction at openings-are by now famlhar to everyone. Not so familiar is the microwave lens.16 An optical lens works, of course, because the phase velocity of the electro magnetic wave is reduced by the coherent scat tering of the bound electrons in the dielectric. Shaping the lens is a matter of adjusting the delay introduced along each ray path to produce a net wave front with the form desired usuallv changing the radius of curvature while I~aving ~ spher~cal incoming wave still approximately sphencal. No one has made bound electrical oscillators on a larger scale for microwave optical glass: But the fact has been used that the prop agatiOn of a wave in a wave guide, a region bounded by conducting walls, proceeds with an increased phase velocity, greater than that of light in free space. (If the reader is worried about relativity, he should review the distinction 'be tween wave and group velocity.) By spacing a set of copper plates, their planes parallel to the direction of propagation of the wave, a half wave-length apart, a change in the phase velocity of the microwave can be secured. Shaping these plates in forms not unlike those of ordinary optics (b.ut with a refractive index always less than one) microwave lenses can be made which quite suc cessfully act as condensing lens, and even pro duce a fairly sharp focus. They are of course good only for one wave-length. "Chromatic aberra tion" is large! Quite the opposite point of view has been used with good effect in recently-reported experi ments using microwave radar technique. These experiments were really experiments in measur ing the absorption spectrum of water vapor molecules in the region of wave-lengths from 7 mm to about 1. 7 cm. An absorption line was found at 1.34 cm and its shape measured as a function of the pressure. The same line could be predicted roughly from the older work on the rotational fine structure in the infra-red region around 20 microns wave-length. Absorption ~ 16 W. E. Kock, Proc. I. R. E. 34, 828 (1946). 144 of course a quantum phenomenon and it is satisfying to see that the analysis: of the in strument-the "spectroscope"-employed came very naturally if one remembered the quantum nature of microwaves,16 The instrument consisted of an eight-foot cubical box of copper, arranged to be filled with moist air. Through the box were strung, at random, the detectors, in this case thermo couples .. The box was coupled to a pulsed magne tron OSCillator-several were used to get points a.t several wave-lengths-and the' space-and ttme.-averaged energy density measured as pro portt.onal to the thermoelectric e.m.f. This energy denSity for a constant excitation is compared with and without the water vapor. The oscillator pumps a fixed number of quanta into the box each second. These quanta are eventually ab sorbed, either by the walls of the box, the thermo couples, solder, air, and fingerprints of any actual ~xperiment, o~ by the molecules of water vapor m the box. Smce the thermocouple reading is proportional to the density of quanta built up m the box, all that is required is to compare the rate of loss due to the water vapor with some known absolute rate of loss of quanta, and the absorption of water vapor is known for that particular wave-length. But how to get an absolute loss rate? It is difficult to compute such a quantity for walls of a copper cube, the more because of the actual complex nature of the experimental geometry. But the quantum idea gives a direct method. It is necessary only to make a hole in the side of the box. All the quanta that strike the hole must leave the box. This number can be calculated just as the number of molecules leaving an orifice to form a molecular beam can be calculated. The expression is just the familiar kinetic theory formula: tM": 'amber of quanta leaving through a hole of ar per second is given by lnc, where n is the nmff 'er of quanta per ,unit volume and c, of cours¢i!i-their mean velocity-the velocity of light. The whole method of procedure was checked both by ex periment and by the more complicated exact calculations of wave theory. Randomness had to be guarante~d. No stand ing wave pattern could be allowed. This was as- 16 W. E. Lamb, Phys. Rev. 70, 309 (1946); Becker and Autier, Phys. Rev. 70, 300 (1946). JO~RNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 12. A face of the shielding wall of the graphite moderated pile at the Clinton Laboratories at Oak Ridge. The operation shown is that of exposing a sample to a strong neutron flux, for the production of a radio isotope. The sample in an aluminum holder is being inserted into a hole in a graphite block can be pushed back along a s!ot into the interior of the pile. The slot is then filled with more graphite blocks to complete the structure, and the shield hole is then plugged. Note the many removable "stringers" of graphite for which shield openings are pro vided. sured by the slow rotation of some copper bladed fans within the enclosure, which changed the geometry and made it sure that a random set of normal modes was excited. Diffraction effects were small, because the hole was made large compared to the wave-length. Apertures up to several hundred square centimeters in area were used. The whole experiment not only gave re sults and pioneered a method sure to be useful for the study of molecular structure, but formed a beautiful illustration of the unity of our dual picture of radiation. A number of other gas absorption studies have been made at microwave frequencies, using more conventional techniques, like attenuation in a wave guide, or using the atmospheric propaga tion itself.'7 2. Atomic Energy The chain reaction of uranium and its con sequences are too well known and too extensive to give any general account here. The slow but steady publication of the results of this project will eventually form a sizeable branch of nuclear physics, as they now do in secrecy. The great 17 Beringer, Phys. Rev. 70, 53 (1946); Dicke et al., Phys. Rev. 70, 340 (1946); Townes, Phys. Rev. 70, 665 (1946). VOLUME 18, FEBRUARY, 1947 FIG. 13. A face of the "hot lab" at the Clinton Labora tories. This is a concrete cubicle fitted to permit the re mote operation of chemical plant adequate f?r. the in organic separations needed !o produce pure radIO Isot?peS from fission product. The air and st~am valve~ and h!les, the thermocouple potentiometer, plalll.ly seen III !he nght foreground are typical for any chemical operatlO!l' The periscopes through which the operators are ~atchlllg the reaction behind the concrete wall are the eVidence of the special problems met in the handling of kilogram equiv alents of radium. continuing laboratories at Chicago, Oak Ridge, Los Alamos, and Berkeley, as well as the new regional laboratory at Brookhaven, Long Island, and the power development laboratory at Sche nectady will become an integral part of American physics. Some topics selected out of the many possibilities from the Manhattan project wi'll be sketched here. a. Isotopes Some hundreds of separate lots of artificial radioactive isotopes have been sold to research workers already by the Isotope Section, Research Division, Manhattan Engineer District, PO Box E, Oak Ridge, Tennessee. This activity will surely continue and expand under the newly formed Atomic Energy Commission, which took over control of the great project from the Army on January 1, 1947. The long-lived soft beta emitter, CI4, made in the high neutron flux of the Clinton graphite pile by the reaction N14(n,p)C14 which goes well with thermal neutrons, is the material of greatest interest as a tracer in bi ological studies. The two-week beta-emitter, P32, is also in demand, chiefly for its therapeutic value in some cases of leukemia and related dis eases. It is made by neutron capture in the 145 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 14. A microphotograph of the first sample of the new element americium (Z-95) ever isolated. The lower part of the photograph shows the eye of an ordinary needle to fix scale. Ab<?ve it is a smal! glass capillary, which is the test tube for this sort of chemistry. In the capillary can be seen the curved meniscus of the supernatant solution. At the i;>ottom tip of the inside of the glass capillary is the ~raYlsh precipitate of americium hydroxide. (Photo furn Ished by Professor G. T. Seaborg, discoverer of Am.) normal phosphorus isotope, P31. A third popular product is 1131, which is a fission product, pro duced in the exposed uranium metal slugs of the pile. These three are only examples, but they indicate the three types of reactions which go with ease in the reacting pile. The technique of separating a particular radioactive isotope out of the complicated mixture which may be formed by bombardment is well shown in Figs. 12 and 13. It always involves remotely controlled chemical operations, with all personnel pro tected from the radiation emitted by thousands or tens of thousands of curies of activity, The handling of large amounts of radioactive sub stances is now a well-developed branch of engineering. b. The Transuranic Elements The element of highest atomic number and highest atomic weight which is found in nature is of course uranium. But the periodic table now goes well beyond that. In the year 1946 the dis closure and naming of the transuranics had pro- 146 gressed through neptunium, number 93, and plutonium, 94, to americium, 95, and curium, 96.18 In 1942 plutonium was produced only by cyclotron bombardment beginning with natural uranium. It was obtained in microgram amounts, and the extraordinary techniques developed by the radio chemists to handle such ultra-micro chemistry have been admired widely by now. The elements at the top of the table are still made in microgram amounts by cyclotron bombardment. But the starting point may now be the man-made element plutonium, available in kilogram lots! It is interesting that the chem istry of the transuranics, especially of trivalent curium and americium, seems to prove that a new rare earth like series of the periodic table begins with actinium, atomic number 90, and that successive elements essentially are made by adding electrons, not to the outermost 7 s shell, but to fill the 5f shell. This is in close analogy to the rare earths, which may be called the lan thanides, for at lanthanum the 4f shell begins to be filled. The transuranics we should call acti nides. It is appropriate here to say that identity and name are now suggested, though not yet officially, for all the elements of the table up to curium, 96.18 The blanks are to be filled in this way, all by known radioactive isotopes of the element in question: Element 43 will be named by Professor Segre of Berkeley, probably to be called tech nesium, from techne-art, to indicate that it is artificial only. I t has no stable isotopes. The previous identification of a stable 43, called masurium, is certainly in error. Element 61 is now known to be a rare earth fission product. No name has been suggested as yet. Element 85, an unstable halogen, will be called by the beauti ful name astasine, from a-not, and stasis-stand ing still. Element 87 is known from radioactive work only, not chemically, and on these not quite complete grounds may be named francium, since it was discovered in this sense by a French radio chemist, and is believed to be a homologue of cesium.I9 18 Seaborg, Science (Oct. 25, 1946). 19 See, for example: Seaborg and Segre, Phys. Rev. 55, 808 (1939) for element 43, Corson, MacKenzie, and Segre, Phys. Rev. 57, 459, 1087 (1940) for element 85, Perey, J. de phys. et rad. 10,453 (1939) for element 87. Private com munications from E. Segre, D. Corson, C. Coryell, and A. Turkevitch. JOURNAL OF ApPLffiD PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51c. The Pile as an Instrument The chain-reacting pile forms an excellent device for the determination of the probability of neutron absorption, especially for thermal neutron absorption, by any material. The sub stance is placed into the reacting structure at a point where the neutron density is fairly high. At such a place, evidently, the absorption of neutrons will have an important effect on the criticality of the whole structure. Since the time required for the neutron density to double in a barely-critical pile is approximately proportional to the difference between the actual reproduction factor of the structure and unity, careful meas urements of pile level changes with and without samples introduced provide a very sensitive way of measuring neutron absorption. Macroscopic samples are used, of course, and the cross section per atom can be measured at least relatively with high accuracy. This method has been applied to many elements-all that could be tried-and even, for engineering purposes, to manufactured materials. By oscillating the sample to and fro, from one point of the pile to another where the neutron density was quite different, sinusoidal change in neutron level of the whole pile is pro duced.20 This allows the elimination of slow drifts and the amplification of the effects by the use of resonant galvanometer systems. In the hands of the group at Oak Ridge this has become a method suitable for both precision measurement of ab sorption cross sections, in favorable cases, and in the detection of very small cross sections. A general review of all the neutron cross sections over the whole periodic table is in preparation based on the extensive project work. The pile work was mainly done at the Argonne Labora tory in Chicago and the Clinton Laboratory at Oak Ridge; very beautiful work in the region of fast neutrons was done mainly at Los Alamos; and neutron spectroscopy, with cyclotron veloc ity selectors or with crystals (see below) at all these places and by sub-contract at Cornell and Columbia. Much of this work is in the press for release; some has appeared this past year.21 20 E. O. Wollan, in press. 21 For example: Bailey et al., Phys. Rev. 70, 583 (1946) Barschall, Battat, and Bright, Phys. Rev. 70, 458 (1946) Rainwater and Havens, Phys. Rev. 70, 136, 154 (1946) H. H. Goldsmith, a survey article in press. VOLUME 18, FEBRUARY, 1947 d. The Pile as a Source of Neutrons A well-collimated beam of thermal neutrons defined by cadmium slits only millimeters wide and meters apart was long a dream of neutron physicists. When the intensity in addition leaves little to be desired, real progress has been made. This was made clear in 1946 by the experiments reported from the Manhattan Project-still only fragmentarily-in which the beam of neu trons from a pile was put to work. The Bragg crystal diffraction of neutrons is not a new effect. It has been somewhat obscurely demonstrated in several laboratories. A neutron moving with the energy corresponding to thermal equilibrium in the lattice of the block of graphite in which it has been slowed has a wave-length of a few angstroms. This is just right to give strong low order diffraction maxima from natural crystal gratings. The effect has been put to work in the construction of crystal spectrometers,22 using not x-ray tubes, sources of the diffracted waves, brass slits, and photographic plates for detectors, but cadmium slits defining a sharp beam of thermal neutrons f[(?m the pile, and boron-filed ionization chambers as detectors, The apparatus has had many uses. At Chicag022 the neutrons have been used to investigate the scattering of neutrons from crystalline compounds. The neutron is scattered with different phase and amplitude by different nuclei. The intensity of scattering FIG. 15. Another section of the face of the Clinton pile. This suggests at least the kind of geometry applicable to the production of a strong collimated neutron beam. A beam is here emerging from a small hole in the pile face (the opening marked 20) and being caught in the large lead brick housing in the center of the photo. 22 Abstracts by Fermi, Zinn, Sturm, Turkel, and L. Marshall, Phys. Rev. 70, 103 (1946); Borst et ai., Phys. Rev. 70,557 (1946). 147 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 16. J\ German-built V-2 rocket being made ready for flight at the proving grounds in the White Sands region of New Mexico. Note the truck at the left evidently supply ing the liquid oxygen fuel to the rocket. determines the probability of the process, or the scattering cross section. This has long been in vestigated. But the interference of the scattered amplitudes from the planes of differing nuclei in crystals, or even in molecules, gives a way to measure the hitherto unmeasured phase shifts in the scattered neutron wave. They show up in the relative intensities of the different orders of diffraction from various crystal planes, for ex ample. This effect has no parallel in x-ray dif fraction, where the wave is electromagnetic and its phase shift uniform. When this complication is unravelled, use of the neutron diffraction as a supplement to x-ray crystal analysis is likely to interest the workers in this field. Neutrons feel the effect of hydrogen and especially deuterium atoms much more than do x-rays, which are capable of interaction only with the electron cloud. This may lead to the study of the hydrogen atom location in some compounds with the aid of neutrons. The neutrons which have been thermalized leak out of the pile with a velocity distribution just that of a molecular beam of hydrogen leaking out of a chamber at a little above room temperature. Collimated and di rected suitably against a crystal of salt or calcite 148 or what you will, a spectrum of the neutrons will be spread out in angle, according to the well-known Bragg law, n"A = dsinll. Here A is the de Broglie wave-length, "A = h:mv, and thus the distribution in angle is a distribution in velocity. Putting absorbers in the diffracted beam at any angle will serve to produce the attenuation for the energy of neutrons present at that angle of deflection. This is a physically monochromatic beam of neutrons, not simply a device for selecting the effects of a particular velocity group, like all the familiar time-modulated schemes. Energy resolution and range of usable energies is about as good as the best time-modu lation schemes, and-apart from the pile-very much simpler. This scheme has been responsible already for the mapping of several resonance peaks in neutron absorption, for example those of In, Rh, Sm, Eu, and Gd. FIG. 17. A war-head of the V-2 fitted with research apparatus, in this case the electronic circuits of a cosmic ray telescope whose Geiger-Muller tubes "look" through the openings in the war-head. This installation was made by the Applied Physics Laboratory of Johns Hopkins, at Silver Springs, Maryland. JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51The strong beam has given us one result of decisive importance, a precision re-measurement of the magnetic moment of the neutron.23 At the Argonne Laboratory, the heavy-water moderated pile, which produces a strong thermal beam, was used as the source for an experiment measuring the magnetic moment of the neutron by polariz ing and analyzing the neutrons with ferromag nets through which they passed. The magnetic moment was obtained by observing the resonant frequency at which the polarized beam was partly depolarized by induced transitions in a magnetic field, coming from the Larmor preces sion of the neutron magnetic moment. (This is the exact experiment of Alvarez and Bloch, an analog to those of Rabi with molecular and atomic beams.) The proton and deuteron mo ments were measured as calibrating data in the same magnetic field, using the method of nu clear induction in one form (see below). The result was given a conservatively estimated pre cision of one part in fifteen hundred. To this accuracy the magnetic moment of the deutreron is shown now to be just that of the free neutron plus that of the free proton plus a small calculable contribution arising from the non-spherically symmetric motion of the proton in the deuteron, the known slightly cigar-shaped distribution of charge. No effect of the binding on the nuclear particles themselves is found, to a rather high accuracy. This point is a long debated one in the theory of the lightest nuclei, and sets a necessary condition to be met by any meson theory of the future. 3. Rockets The great technical development of the Axis Powers during the war was certainly the V-2 rocket. For the first time it was possible to launch a projectile of some size into the region beyond the atmosphere. This wartime feat again has meaning for physics. Last year two dozen rockets were launched in New Mexico, most of them reaching altitudes of fifty miles or more. The flight lasts for several minutes, and up to a ton of apparatus can be carried aloft. In coopera tion with the ordnance and industrial teams en gaged in stud)ling the rockets as weapons, 23 Arnold and Roberts, Phys. Rev. 70, 766 (1946); d. Alvarez and Bloch, Phys. Rev. 57, 111 (1940). VOLUME 18, FEBRUARY, 1947 1-= lL ~ W 0 ::> I-5 <:t 3150,000 300,000 200,000 "0,000 100,000 J -"\, -200 5£. 1505£(; .... :.· )' t , I. ., '\ :~ 100 SEC. 1 ;.~ 250 SEC. -.-300 SEC. ::-. 50 SEC. !5O,OOO o J o ii' J iii i J iii i I 50,000 100,000 I!5O,OOO HORIZONTAL RANGE, FT. FIG. 18. The graph of the spectacular trajectory of a successful V-2 flight, The high altitude data were obtained by radar tracking. Note that about four minutes is passed in regions beyond the reach of balloons or planes, above 100,000 feet. Not long, but something! several laboratories have taken advantage of the chance to study the region beyond the atmos phere. Here are the cosmic-ray primary particles, not yet complicated by cloudbursts of secondary particles which they cause on striking the atmos phere. Here may be measured the spectrum of the sun and of the stars, not through the dark glass of the air, but as they come through empty space. Both spectra and cosmic-ray measure ments have been made with interesting results.24 Too little has yet been done to draw any valid conclusions, but it is certain that the exploration 24 GoHan, Krame, and Perlow, Phys. Rev. 70, 776 (1946); private communication from Applied Physics Laboratory, Silver Springs, Maryland. 149 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51of space beyond the air will have real importance for more than one.field of physics. THE MASS OF THE MESON For the years since its discovery, the meson has been examined in the cosmic ray in an effort to measure all its properties. Most fundamental, after its charge, is perhaps its rest mass. If the cosmic-ray meson is that predicted in connection with nuclear force theories, its mass should be somewhere between 150 and 250 times that of the electron. Up until 1946 about twenty-five mesons had been observed with sufficient ac curacy to make possible some measurement of their mass. The spread in values was great, and it was by no means sure that the meson had only one definite mass. Perhaps the meson was not a unique particle but a whole family. . An excellent experiment just reported25 seems to have shown that the ordinary meson which makes up the penetrating component of cosmic rays at sea level has one mass, about 202 ± 5 times the electron rest mass. This measurement was made by a very careful and successful application of the usual technique. The method of measuring the mass is ordinarily this: The momentum of the meson can be meas ured by measuring the curvature of its path as made visible in a cloud chamber across which there is applied a known constant magnetic field. This measurement is made, di«icult»)' the accidental deflections of the track C of droplets caused by the irregular and turbulent motion of air in the cloud chamber. Then the ionization which the particle produces is measured, by counting the droplets left in the chamber gas, or by seeing what thickness of matter it will penetrate before it comes to rest. This gives the rate at which the particle loses energy by inter action with the electrons of matter, and this is known as a function of the velbcity of the particle. The' scheme then is to measure the curvature of the track and on the same particle to determine what thickness of lead it will penetrate. The experiment here reported divided up these ,measurements between two cloud chambers. The upper chamber was free of any obstacles, and carefully controlled in temperature. Track curva- 26 W. B. Fretter, Phys. Rev. 70, 625 (1946). 150 ture could be measured with minimum difficulty from turbulence. A second chamber Illounted two feet below the first and in the same plane was expanded simultaneously, and photos were taken of both chambers. The lower chamber was crossed by eight half-inch sheets of lead. By observing in which plate the track appeared to end the range in lead could be gotten to fair ac curacy, and the velocity calculated. The particles observed could be assigned a momentum with a spread due to the error in curvature measure ment, and a range with the error coming from the finite plate thickness and other sources. The particles then could each be assigned a mass and a definite error. The spread of values was from 142 to 264 electron masses. Statistical analysis of the data gives the result that a unique mass of 202 m is quite consistent with the data. The observations agree among themselves just as well as can be expected from their individual estimated errors. These mesons have the single mass 202 m. NUCLEAR INDUCTION For some years the magnetic moment ,which is intrinsic to nuclear particles and to their motion within the nucleus has been an important object of study. This is a very small magnetic moment indeed, roughly two thousand times less than that associated with a single atom. First evidence for and some measurements of these tiny mag netic dipoles were spectroscopic. They interacted with the current respesented by the orbital motion of the electrons around the nucleus in the atom, and the different orientations the nuclear magnetic moment assumed in the magnetic field resulting from the electrons' motion gave rise to atomic energy levels. These levels are very close together: the energy difference is very small, and the difference in frequency and hence wave length of the emitted spectral lines very small indeed. Lines which originate in this way are said to be lines of the hyperfine structure of spectra. About ten years ago another and more elegant method was devised, which reached its present form in the work of Professor Rabi and his co-workers at Columbia. Here the nuclei are examined as they stream in molecular beams. Such beams are made to pass through strong in-. homogeneous magnetic fields. The nuclear mag- JOURNAL OF APPLmD PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51FIG. 19. The receiver-transmitter coils first used to meas ure the nuclear magnetic moment by nuclear induction. The four heavy turns are the transmitter coil, and the more finely-wound receiver coil is seen within, its axis at right angles to the transmitter coil. Within the receiver coil is placed the sample encased in a half-inch spherical glass bulb. The whole arrangement is shielded heavily. Openings in the shield allow the installation of a rotating copper paddle for adjusting stray fields from without. (Photo graph by Professor Felix Bloch of Stanford University, where this work was done.) netic dipoles feel a force which will depend upon their quantized orientation in space, and the molecular beam will split into several compon ents. These components are each deflected by a different amount, and the deflection may be measured. The deflections one gets with any realizable magnetic fields are small, of the order of hundredths of a millimeter. The method could not by the nature of things be very precise. An elegant adaptation of the molecular beam tech nique was made by Rabi. He used'two equal but oppositely directed inhomogeneous fields. The beam passes through the first and is split into its components. Then the beam is reunited by passage through the second, and strikes a de tector. But between the analyzer and the second field there is placed a third region. Here a uniform magnetic field is produced, and also a small radiofrequency magnetic field. The large uni form field has of course no effect on the nuclear VOLUME 18, FEBRUARY, 1947 FIG. 20. Oscilloscope traces of the nuclear induction signal. The vertical deflection is proportional to the precessing magnetic moment (or its component in the direction of the axis of the receiver coil), the horizontal deflection to the applied d.c. magnetic field, which has a small, 60-cycle variation in magnitude. The traces shown are those of the proton signal from a water sample. In the top trace the applied field is above the resonance field on the average. The d.c field was lowered in about a second's time to a value below resonance. The proton signal de creased slowly. The next three traces were taken at suc cessive five-second intervals after the reduction of the d.c. field. Note that the signal slowly reverses to a below-re sonance condition, though no external change is now going on. This time lag, or "memory," is the consequence of the time it takes for the small nuclear magnets to realign themselves into thermal equilibrium with the new applied field. The time is clearly a few seconds in the case of water. (The double trace is a result of stray 6O-cycle pick-up, which separated the forth-and back sweep in the camera exposure.) dipoles. But the radiofrequency "quanta" may be absorbed whenever their frequency reaches a value such that the quantum energy hv equals the energy difference between one orientation of the nuclear magnet in the uniform field and an other one. Classically one writes hv = JLII, where II is the uniform field strength and JL the mag netic moment of the nucleus. Fields around a thousand gauss correspond to resonant fre quencies of a few megacycles for'typical nuclei: Now the nucleus which" absorbs" the quantum is flopped over to a new orientation, cannot be refocused in the second field, and never reaches the detector. By measuring the detector response as frequency or magnetic field are varied, the shapes of the lines corresponding to resonance 151 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 152.2.176.242 On: Mon, 01 Dec 2014 02:12:51with the nuclear magnetic moment can be found. This leads to precision comparisons of nuclear moments, depending only upon frequency meas urements, always easy to do well. No nuclear quantities are known with the precision with f:'-,#",j<' , which nuclear magnetic moments can be found in this way. But the apparatus is delicate and the skill and patience required great. Nor can every substance be gotten into a molecular beam and detected. It was exciting, then, when not only one but two investigators suggested methods by which the same phenomenon-the flip of the nuclear magnets when they feel an oscillating field of the . right frequency-could be detected with no fancy molecular beam techniques. For the first time the macroscopic effect of the nuclear mag nets on the magnetic polarization of bulk ma terial was taken into account. Think of a sample of material placed in a magnetic field. The nu clear magnets will line up in the magnetic field to get into the state of minimum potential energy. But of course the thermal agitation of the molecules will act to disorient the magnets. The net result will be given by th,e usual Curie Langevin law: the resultant nuclear moment pe(.. cc at thermal equilibrium will be(~:pproximateIY.l np.(JJHlkT), where n is the number of nuclei per cc with magnetic moment p. per nucleus, H is the applied magnetic field, and kT is Boltzmann's constant times temperature. This is a small con tribution to the paramagnetism at best. For protons in water in even pretty high fields it is almost unobservable. What is measured is not the d.c. effect, but a resonant effect at radio frequency. In one experiment, a resonant cavity was made for about 30 mc.26 It was filled with paraffin, and was placed in a strong magnetic field, at about 7000 gauss. A weak r-f magnetic field in the cavity was kept perpendicular to the d.c. field. The d.c. field was slowly varied until a sharp resonance absorption was observed. The absorption of r-f energy had changed the Q of the cavity, reduced its output, and affected a detector which had been balanced out off re sonance. The same advantages can be claimed for this method as for the molecular beam. An even simpler technique was applied some- 18 Purcell, Torrey, and Pound, Phys. Rev. 69, 37 (1946). 152 what later. Here the effect observed is the pre cession of the resultant nuclear magnetic moment as resonance is approached.27 The observation is made, not of the reaction upon the driving circuit, but of the e.m.f. induced directly by the preces sing nuclear magnets in a pick-up coil in which all e.m f. had been balanced out off resonance. The nuclear-induced precessing field has a com ponent perpendicular both to the constant field and to the r-f field. It is this component which is observed. No more is needed than a radio oscil lator, a lecture table magnet, a transmitter receiver coil arrangement, and appropriate radiofrequency amplifiers and oscillograph cir cuits. The coils of Fig. 19 are the heart of the apparatus. It should be noted that the methods of nuclear induction, as they are called, depend upon the nuclear magnets reaching thermal equilibrium, with their moments odented not at random, but with the equilibrium resultant value. It is just the macroscopic want of cancellation which makes the whole effect. But the mechanism by which the nuclei come to thermal equilibrium is complex and little-known. If it took weeks for equilibrium to arise, the experiment would be very difficult. The time taken will be a function of the chemistry of the compound and of many atomic features. A whole new subject in atomic physics is opened up by this technique, as well as a simple new supplement to the existing study of magnetic moments. The use of the technique for isotope analysis without any destruction of the sample, isotope analysis by radio, so to speak, may prove of great importance in tracer work with stable isotopes. The whole subject is a good example of how new ideas may arise in fields believed already carried to their highest develop ment. The world of physics is surely infinite. This sketchy review of the first year of peace is full of promise. The promise will be fulfilled only if physicists can share with all men the pros pect which carries all our hopes, the prospect of the many years of peace that lie ahead. I am glad to acknowledge the kind cooperation of all the busy men who answered letters and supplied photographs to make this account pbssib1e. 27 F. Bloch, ·Phys. Rev. 70, 460 (1946) (theory); Bloch, Hansen, and Packard, Phys. Rev. 70, 474 (1946) (exp't). JOURNAL OF APPLmD PHYSICS [This article is copyrighted as indicated in the article. 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1.1712751.pdf

Electrical Conductivity of Metals John Bardeen Citation: Journal of Applied Physics 11, 88 (1940); doi: 10.1063/1.1712751 View online: http://dx.doi.org/10.1063/1.1712751 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/11/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The electrical conductivity of microcellular metals J. Appl. Phys. 100, 044912 (2006); 10.1063/1.2335672 Electrical conduction in metals Phys. Today 31, 44 (1978); 10.1063/1.2994869 Condensation and Electrical Conduction in Metallic Vapors Phys. Fluids 10, 2179 (1967); 10.1063/1.1762015 A Note on the Paper ``Electrical Conduction Between Metallic Microparticles'' J. Appl. Phys. 37, 4590 (1966); 10.1063/1.1708093 Electrical Conduction between Metallic Microparticles J. Appl. Phys. 37, 1594 (1966); 10.1063/1.1708572 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14Ele.ctrical Condllctivity of Metals JOHN BARDEEN Department of Physics, University of Minnesota, Minneapolis, Minnesota I. Introduction THE classical theories of the electrical con ductivity of metals as developed mainly by Drude and Lorentz, while succeseful in some re spects, most notably in regard to the Weide mann-Franz law, encountered serious difficulties. These difficulties have been almost completely removed by modern quantum theory as applied to the problem by Sommerfeld, Houston, Bloch, Mott, and others. Before discussing the modern theories, it will be instructive to give a review of some of the salient experimental facts which an adequate theory of conductivity must explain, and also to give a brief discussion of the older theories, as they contain many elements of truth. In fact, the basic concepts are largely un changed, even though the formal theory is quite different in appearance. The fundamental idea of the theories of Drude and Lorentz, and of all later theories, is that the current is carried by electrons which have become detached from the atoms which make up the metal and may move about more or less freely. The most direct experimental proof of this fact was not obtained until some years after the theory was first proposed. Experiments of Tolman and various collaborators indicate that the current is carried by particles of negative charge and with a mass close to that of an elec tron. The earliest is that of Tolman and Stewart.l A coil of wire rotating at high speeds is connected to a sensitive ballistic galvanometer. When the coil is suddenly stopped from rotating, an im pulse of current is sent through the galvanometer. The kinetic energy which the electrons acquired when the coil was rotating is dissipated by the resistance. From the velocity of the coil and the total charge carried by the impulse of current, together with a knowledge of the resistance and dimensions of the coil, one can find the ratio of the charge to the mass of the particles that carry the current. Tolman and Stewart find for 88 this ratio, elm, the following values: for Cu for Ag for Al 1.60X107 e.m.u. 1.485 X 107 e.m.u. 1.54X107 e.m.u. These values are rather close to the value for perfectly free electrons,2 1. 77 X 107 e.m. u. Later experiments made with the use of oscillating cylinders gave similar results. These experiments indicate that the current is carried by electrons which move through the crystal lattice. II. Summary of Experimental Facts A. CONDUCTORS AND INSULATORS Perhaps the most important thing an adequate theory of conductivity must explain is the re markable difference in conductivity between metals and insulators. The resistivity of metals if of the order of 10-5 ohm cm, of insulators, the order of 1012 ohm cm. There is no satisfactory explanation on any classical basis. B. TEMPERATURE DEPENDENCE At high temperatures, the resistance of a pure metal is roughly proportional to the absolute temperature. It decreases rapidly as the tem perature is lowered, and at very low temperatures is proportional to P. On the other hand, the resistance of insulators increases as the tempera ture is decreased. This is a characteristic property which distinguishes metals and nonmetals. Gruneisen3 has shown that the resistance of most metals can be given by a universal function of the temperature: R=Ref(TIG). (1) Here e is a characteristic temperature for the particular metal, which is generally rather close to the Debye characteristic temperature for specific heats. Experimental points for a number of metals are shown in Fig. 1. JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14There is thus indicated a close connection between the conductivity of pure metals and the thermal motion of the atoms which is responsible for the specific heat. W. Wein has suggested that the conductivity is proportional to the mean square amplitude of the motion of the ions. The resistance would then be pro portional to the absolute temperature, T, at high temperatures, and to T4 at very low temperatures. The theoretical curve shown in Fig. 1, which was used by Gruneisen, is based on an approximate theoretical expression due to Bloch, which will be discussed more at length later on. According to this expression the re sistance should be proportional to rs instead of T4 at low temperatures, and is in better agreement with experiment. Both theory and experiment indicate that one of the most important factors in the resistance of a pure metal is the mean square amplitude of motion of the atoms of the crystal. C. DEPENDENCE ON PRESSURE The resistance of most metals decreases with increasing pressure. Fig. 2, shows the resistance 0.1 .. AU e-175 • NA e-202 • eu e-333 .. ilL e·396 ~ NI e-472 Q.2 0:4 TIe FIG. 1. Temperature variation of resistance of various metals. The curve is a plot of the Bloch-Gruneisen function (Eq. (28». Data from values quoted by Meissner (d. Bibliography). VOLUME 11, FEBRUARY, 1940 SA cs ----------PB RB ~:::----=======:::--==:::::=- NA 10.000 2.0,000 '0.000 PRESSURE KOICM' FIG. 2. Pressure variation of resistance of various metals. Data from Bridgman (reference 4). as a function of pressure for a number of the softer metals in the pressure range extending to about 30,000 atmospheres. These curves have been obtained recently by Bridgman.4 The re sistances of a few of the metals (Li, Sr, Ca) increase with increasing pressure; some (Rb, Cs, Ba) show a reversal, decreasing at low pressures to a minimum and then increasing as the pressure is further increased. Most of the anomalous metals are shown in the figure; the normal behavior is a gradual decrease with pressure . Gruneisen has explained this normal decrease as due to the decrease in the thermal motion of the atoms as the pressure is increased and the atoms are bound with greater forces to their positions of equilibrium. D. MATTHIESSEN'S RULE Any actual metal contains a certain amount of impurities and as the temperature is decreased to the absolute zero, the resistance does not go to zero, but approaches a constant value which depends on the amount of impurity. The purer the metal, the smaller is the residual resistance. Matthiessen has pointed out that the resistance is the sum of two terms, R=Ro+R T, (2) 89 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14a constant term (independent of temperature), which is proportional to the amount of impurity, and a temperature dependent term, characteristic of the pure metal, which goes to zero as the temperature approaches the absolute zero. This is known as Matthiessen's rule, and is in approxi mate agreement with experiment for the increase in resistance of a metal due to a small concentra tion of a second metal in solid solution. Electrons may be scattered by impurity atoms in the lattice, giving rise to an added resistance. For small concentrations, the resistances caused by impurities and by thermal motion are additive. In finding the resistance of a pure metal, a correction is generally made for the residual resistance. 20 "'0 . 1& N ~ 0 . :I % 0 ,.. I-:; i= co iii I&J II: o 20 40 60 80 100 Y. " VOLUME FIG. 3. Resistivity of heterogeneous mixtures. Data from values quoted by Meissner. E. RESISTANCE OF ALLOYS If an alloy is made up of a mixture of micro crystals of individual pure metals, the resistance is roughly the average resistance of the com ponents. Fig. 3 illustrates the curves for the resistances of the Pb-Sn, Pb-Cd, Zn-Cd, and Zn-Sn series as a function of concentration. These metals are mutually insoluble in one another. If, on the other hand, there is a solid solution, the resistance is much greater than that of either of the components. Results for the Ag-Au alloys, which form a solid solution for all con centrations, are given in Fig. 4. The resistance is 90 a maximum at about 50 percent concentration. In this alloy there is no superlattice; the Ag and Au atoms are distributed at random over the lattice points. Let us now see what happens when an ordered structure is formed. Fig. 5 illustrates the Cu-Au system, which has been studied extensively by a large number of workers. The quenched alloy, in which there is no super lattice, has a resistance curve of the same general shape as that of the Ag-Au system. If the alloy is annealed, a superlattice is formed in the neighborhood of 25 percent Cu, 75 percent Au, and of 50 percent Cu, 50 percent Au. The Cu and Au atoms then take up more or less regular ordered positions in the lattice. It is seen that the resistance drops markedly and approaches that of a pure metal. We will consider this system in more detail later on; we now merely want to emphasize the fact that an ordering of the atoms decreases the resistance. The facts enumerated above indicate that resistance is due to irregularities in the lattice, and that these may be caused by either (1) thermal motion of the atoms; (2) impurities, or, in alloys, from a random solid solution. A further cause is the disorder existing in a liquid or amorphous solid. These are but a few of the facts which a complete theory of conductivity must explain. The theory should give the absolute values of the conductivities of the different pure metals and their dependence on temperature and pressure. Why does copper have a low resistance and iron a comparatively high resistance? III. Early Theories The early theories of Drude and Lorentz5 did not attempt to give an explanation of these facts in any detail; they merely attempted to give the mechanism of conductivity. It will be instructive to consider the elementary theory of Drude as it is very simple and its fundamental notions are preserved in the later theories. The essential idea is that the current is carried by electrons which may move about more or less freely, but are subject to collisions with the crystal lattice. The mean time between collisions will be called 2T (T is the time of relaxation). We suppose that on the average the momentum of the electron is destroyed at each collision. In JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14spite of the large interaction between electrons, we suppose, for simplicity, that the electrons move freely between collisions so that the equation of motion is mdvx/dt= -eF, in which -e and m are the charge and mass of the electron, and F is the field strength, which is assumed to act in the x direction. The mean drift velocity in the direction of the field is therefore Vd= ~. (-eF/m)· 2r= -eFr/m. (4) If there are N electrons per unit volume, the current will be J = -Neva, and the conductivity (5) Drude made the admittedly simplified assump tion that all electrons move with the same mean velocity, U, and that this velocity could be obtained from the equipartition law: (6) where K is Boltzmann's constant. The mean free path I is defined by: 1=2rU. (7) • o II~----~----~----~~--~ • z o al z o > t: > Ii iii III --...... "" "- / , / , / OOK' / ~ , , , E O~----~------~-----+----~ o 21 10 AS ATOMIO % 11 10096 AU FIG. 4. Resistivity of Ag-Au alloys. Curve for OOK extrapo lated from results of Clay, quoted by Meissner. In terms of the mean free path, u=Ne21/2mU. (8) This formula gives about the right order of magnitude for the conductivity at room tem perature if N is of the order of magnitude of the VOLUME 11, FEBRUARY, 1940 11r---;---~----+----1 21 10 11 ATOMIC % 100" AU FIG. 5. Resistivity of Cu-Au alloys. (a) Quenched from 650°C. (b) Annealed at 200°C.* number of atoms per unit volume and I of the interatomic distance. However, in order to account for the temperature variation, it is necessary to assume that I increases very rapidly as the temperature is decreased. This peculiar variation was difficult to explain. Furthermore, one would expect that as the pressure is increased and the mean distance between atoms is de creased, 1 will decrease, and consequently the conductivity will decrease. As we have seen, this is contrary to the normal behavior. Perhaps the most serious difficulty was in connection with the specific heat. Classically the electrons should contribute 3R/2 times the number of free electrons per atom to the molar heat capacity. However, most of the heat capacity of metals is accounted for by the thermal motion of the atoms. Any contribution from the electrons must be very much less than R. As is probably well known to most readers, this difficulty was removed by Sommer feld by the application of Fermi-Dirac statistics to the electrons. More refined calculations by Lorentz, based on the same physical assumptions, merely served to emphasize the difficulties inherent in the Drude theory. * Data from Johansson and Linde, Ann. d. Physik 25, 1 (1936). 91 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14The great success of the theory was its explana tion of the Wiedemann-Franz law which states that the ratio of the thermal to the electrical conductivity is proportional to the absolute temperature. On the same assumptions that were used in the theory of the electrical Drude found for the thermal conductivity: (9) so that (10) The constant of proportionality, which involves only the fundamental physical constants, is in approximate agreement with the experimental values. In the classical theories there was no way to calculate the absolute value of the conductivity of a metal, nor to explain the differences in the conductivities of different metals. IV. The Hall Effect One of the most important phenomena in the historical development of the subject is the Hall effect. An electric current flows in a plate normal to which there is a magnetic field. -eF • • 6 • ~ • • • • • • • c5 • ~ • • • • • • • • • • FIG. 6. Illustrating transfer of charge by associated electrons, according to Hall. A potential difference is then set up across the plate in a direction transverse to the direction of the current. A very rough explanation is that the paths of the electrons are bent by the magnetic field. In order that there be no com ponent of current in the transverse direction, a potential gradient must be set up. The sign of the gradient depends on the sign of the charge 92 which carries the current. For most metals, the sign is such as would be expected for electrons (negative charge), but some metals show an anomalous sign (Zn, Cd). This fact gave rise to the dual conduction theories, which were de veloped mainly by Hall.6 He assumed that the current is carried not only by "free" electrons but also by "associated" electrons. An atom in the metal may become ionized by giving rise to a "free" electron. The ion is then left positively charged. A bound electron from a neighboring atom may be transferred to this ion. In this way the position of the positively charged ion may move through the metal (d. Fig. 6). A current arising from such a transfer of charge would give rise to an anomalous Hall coefficient. We will see later on that the modern theory gives some justification for this point of view. V. The Sommerfeld-Bloch Theory The remainder of the article will be devoted mainly to modern theories of electrical con ductivity and their applications to various problems. Only a very brief discussion of the basic concepts of the general quantum theory of metals, which we shall need for the discussion of conductivity, will be given here. For further details, the reader is referred to the excellent series of articles by Seitz and Johnson~hich have appeared in this Journal, or to one of the recent books on the subject (d. Bibliography). According to the modern picture, the valence electrons (i.e., the electrons outside of closed shells) become detached from the atoms which make up the metal and are free to move through out the crystal lattice. These are the conduction electrons, which not only give the metallic bond, but also play the major role in the various electric and magnetic properties. Our discussion will be confined to these electrons; the ion cores will usually be replaced by an effective potential field in which the conduction electrons move. This picture is not greatly different from that of Drude and Lorentz. \Vhat is responsible for recent progress is the application of quantum rather than classical mechanics to the problem. The two principles of quantum theory most important for the new development are (1) the wave property of electrons, and (2) the Pauli exclusion principle. JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14A. ELECTRON WAVES Associated with each electron is a wave which is propagated in the direction of motion of the electron. For free electrons (i.e., electrons subject to no forces) the wave-length, X, is given by the de Broglie relation: X=hlmv, (11) where h is Planck's constant, m is the mass of the electron, and v is the velocity. This relation has been verified experimentally by Davisson and Germer, Thomson, and others. If the electrons, instead of being free, are considered to be moving in the periodic potential field of the crystal lattice, the relation between wave length and velocity is no longer given by (11) but by a somewhat more complicated equation which will be given later (Eq. (17)). Instead of using the wave-length or velocity to define the state of an electron, it is more convenient to use the wave vector, k, whose direction is the direction of propagation, and which has a magni tude 27r IX. The energy of the electron, E(k), will be a function of the vector k. For free electrons (12) The latter equality is a consequence of (11) as may easily be verified. Again, the general ex pression for the energy of an electron moving in a periodic field is more complicated. The Pauli exclusion principle, as applied to the present situation, states that at most only two electrons can be in the same state, or; in other words, have the same wave-length and direction of propagation. The factor two arises from the two possible values of the electron spin. In an infinite crystal, the possible wave-lengths are continuously distributed, but in any finite crystal, they are limited to a discrete set of values. As long as the wave-length is small compared with the dimensions of the crystal, the number of states between two wave-lengths X and X+~X is proportional to the volume of the crystal and is independent of its shape. Thus the number of electrons per unit volume which can have wave-lengths in the interval X, X+~X, is independent of the size or shape of the crystal, as of course it must be, if the properties of the crystal are independent of size and shape. VOLUME 11, FEBRUARY, 1940 B. SPECIFIC HEAT OF ELECTRONS One consequence of the exclusion principle is that even at the absolute zero of temperature the energies of the different conduction electrons are spread over a wide band which may be several electron volts in width. Fig. 7 shows the average number of electrons in the different states at T=OoK, and also at some higher temperature. At T=OoK, all states with energies below a certain maximum, Emax, are occupied; ~Or-----------------~ 0.5 T=T, o FIG. 7. The Fermi distribution function, giving the probability that an electronic state is occupied at a given temperature. those above, unoccupied. At higher tempera tures, a few electrons in states near the top of the filled band become excited to states of higher energies, and a distribution somewhat like that shown in Fig. 7 results. The fraction of electrons which are excited is of the order KT I Emax, and the average energy of excitation of those elec trons which are excited to higher states is of the order KT. Thus the average thermal energy. per electron is of order (KT)2jE max, and if there is about one free electron per atom, the specific heat is of order (KTIEmax)R where R is the gas constant. Since at room temperature, KT j Emax is less than 0.01 for most metals, the heat capacity of the electrons is negligible compared with that of the thermal vibrations of the lattice (,,-,3R). It is only at very low (liquid He) temperatures that the electronic specific heat can be observed. As Sommerfeld first demonstrated, an applica tion of the Pauli exclusion principle and Fermi Dirac statistics gives an explanation of the specific heat difficulty which was inherent in the theories of Drude and Lorentz. C. SOMMERFELD THEORY OF CONDUCTIVITY In the theory of Sommerfeld, as in the theory of Drude, it is assumed that each electron moves 93 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14freely between collisions, and so is accelerated by the field. The field acting on a given electron due to the ions and due to the remaining electrons is neglected. Drude's formula (8) for the con ductivity will therefore apply. The maip. differ ence is that the mean free path l=2rU (13) will be much larger, since U, the mean velocity of an electron, will be much larger than that obtained from the equipartition theorem (kinetic energy=!KT). 1£ one assumes about one free electron per atom, it turns out that the mean free path at room temperature is of the order of 100 interatomic distances instead of, as in the Drude theory, the order of the distance between neighboring atoms. The long mean free path is a consequence of the wave property of the electrons. While it is difficult to see how a particle could move very far through a crystal lattice without being scattered, it is not difficult to see that a wave can be propagated without serious attenuation. The situation is similar to the propagation of a light wave, or perhaps better, an x-ray, through a crystal: the wavelets scattered from each atom interfere constructively so as to continuously build up the wave front. An estimate of the mean free path on this basis was first given by Houston. D. THE BLOCH THEORY The concept of electrons moving in an electro static field having the period of the crystal lattice was introduced by Bloch. Just as for the case of perfectly free electrons, it is possible to assign to each electron wave a definite wave length and direction of propagation. Instead of being plane, the waves are modulated by a function having the period of the lattice. The wave function is of the form -.f;k(X, y, z) = Uk(x, y, z) exp [i(kxx+klly+k.z)]. (14) The vector k, with components (kx, ky, k.) is the propagation vector defined above. The second factor, exp [i(kxx+kyy+k.z)] is the wave func tion of a plane wave; it is multiplied by a factor Uk(X, y, z) which is periodic with the period of the lattice. The energy, E(k) of an electron in 94 the state k depends on the particular field III which the electron moves. E. VELOCITY OF THE ELECTRONS To obtain the velocity of an electron in the state k, one may use the expression for the group velocity of any waves. For the case of waves moving in one dimension, this expression is: v=dvld(l/A). (15) In our case, the frequency v is given by Elh, and since k = 27r lA, v = (27r Ih)(dEldk). (16) In the general case of three-dimensional motion, the expression for, say, the x component of the velocity is vx= (27rlh) (iJEliJkx). (17) The current contributed by the electron is The total current can be obtained by summing the contribution from each electron over all occupied states. If more electrons are traveling in one direction than the opposite, there will be a net component of current in this direction. Since the states we have been considering are stationary states, this current will not diminish in time. Thus a perfect periodic lattice will have no resistance. Resistance is due to irregu larities introduced by thermal motion, or by the presence of foreign atoms, which destroy the periodicity of the lattice. This is just what the experiments seem to demand. F. ENERGY STATES It is convenient to think of each electronic state as represented by a point in a three dimensional k space. The coordinates of the point are the components (kx, kll, kz) of the wave vector k. This k space is exactly similar to the reciprocal lattice space, which has been of so much use in the theory of x-ray diffraction. Points representing allowed states are uniformly distributed in k space. The number in the element dkx, dky, dk. is (V /87r3)dkxdkydk. where V is the volume of the crystal. The energy of an electron is not in general a continuous function of the wave vector. From JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14the theory of the motion of an electron in a periodic potential field, it is found that the whole of k space is divided up into regions or zones in each of which the energy is a continuous func tion of k. Across the boundary between two zones, the energy jumps discontinuously from one value to another. Each zone corresponds to a band of allowed energies which may be sepa rated by a gap of forbidden energies from the next higher band. On the other hand, the energy bands may overlap, the lowest level of the upper band being lower than the highest level of the lower band. The zones or energy bands may be correlated with the states of electrons in the free atom. One may imagine that the lattice constant of the crystal is continuously increased until the atoms are so far apart that there is little interaction between them. The energy bands will become narrower and narrower, until finally they go over to the discrete levels of the free atom. Thus one speaks of the s band of a monovalent metal which arises from the s level of the valence elec tron of the free atom. The d levels of the transi tion elements break up into several distinct d bands in the metal. There can be two s electrons in the free atom, and the total number of states in the s band is such that it will accommodate two electrons per atom. Similarly, the total number of states in the d bands corresponds to ten electrons per atom. G. METALS AND INSULATORS If every allowed state in a band is occupied by an electron, the total current is zero, because for every electron traveling in one direction there will be another going in the opposite direction with the same velocity. If a band is only partially full, more electrons may be going in one direction than the opposite, giving a resultant current. The first case is characteristic of insulators, the second of metals. In an insulator any band which contains any electrons at all is full, and there is a gap of forbidden energies to the higher, un occupied bands. An insulator must have an even number of valence electrons per unit cell. A metal contains bands in which only a fraction of the states are occupied. The divalent metals (Be, Ca, etc.) have just enough electrons to fill the lowest VOLUME 11, FEBRUARY, 1940 (s) band, i.e., two per atom. Since they are good conductors, it must be concluded that a higher band overlaps the lowest, so that there are electrons in two different bands, each par tially full. H. ACCELERATTON OF ELECTRONS The configuration of electrons in a metal may be described by giving the distribution of over the occupied states in k space. If an electric field acts on the metal, the distribution will no longer be symmetric about the origin, but will be dis placed in the direction of the field, giving a resultant current. The new distribution results from an equilibrium between transitions among the states due to acceleration by the field and those due to the scattering which gives rise to the resistance. The classical expression for the acceleration of an electron by an electric field F is m(dv/dt) = -eF. (19) If we use the relation mv=hk/27r, appropriate for free electrons, it is found that the rate of change of the wave vector k with time is dk/dt= -27reF/h. (20) What effect will the periodic field of the crystal lattice have on the acceleration? It is not difficult to show that (20) will still be valid, even though Eq. (19) can no longer be used, and even though the velocity is no longer proportional to k. The energy of an electron as a function of k along the direction of the applied field is illus trated schematically in Fig. 8a. The wave vector, k, according to Eq. (20) increases uniformly with time, so that a point which represents the state of an electron will move to the right with con stant velocity. We have seen (Eq. 18) that the velocity of the electron itself is proportional to the slope, dE/dk. The velocity of an electron whose state is represented by the point AJ where the curvature is positive, increases with time, since dE/dk increases as k increases. On the other hand, the velocity of an electron whose state is represented by A2 decreases with time, since dE/dk is decreasing in this region. The electron then acts like a particle with negative mass. When the state reaches the point Aa, the velocity is zero. At this point there is a very 9S [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14small probability that the electron will jump to the next higher band. Normally, the electron will reappear at A3' (which really represents the same state as A3) and then retrace its course over the lower band. The electron suffers Bragg reflection at A 3, and starts moving in the opposite direction. 8a. -eF )II k ... FIG. 8. (a) Motion of electron in a band under the influence of an applied electric field. (b) Motion of positive hole in band. Let us now suppose that all the states are occupied except one, which is represented by the point A 1 in Fig. 8b. If all the states were occu pied, the resultant current would be zero; the total current is therefore the negative of the current carried by an electron in the state AI. This is equal to the current carried by an electron with velocity equal and opposite to that at Ai, i.e., just the current of an electron in the state A 1'. Under the influence of an electric field, F, the whole distribution of electrons, and therefore also the position of the unoccupied state or "hole" will move to the right with uniform velocity. Correspondingly, the point AI' will move to the left in the direction in which a positively charged particle would move. If, as shown in the figure, the point A/ is in the region of negative curvature, the velocity and current will increase with time. Thus the "hole" in the otherwise filled band acts like a particle of positive charge and positive mass in the region 96 of negative curvature, and like a particle of positive charge and negative mass in the region of positive curvature near the bottom of the band. We shall see (Section V) that the above considerations may be used to give an explana tion of the anomalous sign of the Hall coefficient. In a normal metal, all the states with energies below some maximum energy, Emax, will be occupied, and those with energies greater than Emax will be unoccupied. (We neglect, for the moment, the small dispersion due to thermal agitation.) All these electrons will be accelerated by the applied field, so that the whole distribu tion in k space will move in the direction of the field. At first the density of electrons is changed only in those states which lie near the surface of the Fermi distribution, i.e., those with energies near Emax. A plot of the density of the occupied states in k space for a normal metal, and also after an electric field has been applied for a short time, is shown in Fig. 9. Let J be the total current, as obtained by summing the contribution from each electron, jk, (Eq. (18» over all occupied states. We may define the effective number of free electrons per unit volume by means of the equationS dJ/dt=e2 FNeff/m. (21) In case the lattice field vanishes, so that the electrons actually are free, Neff is equal to N, the actual number of electrons per unit volume. In general Neff may be either greater or smaller than N. The expression for Neff is particularly simple if the surfaces of constant energy are spheres, so that E depends on the magnitude but not the direction of k. In this case 47r2mN( 1 dE) Ncff=----- h2 k dk E=Ern•x• (22) Thus Neff is large for a wide band and small for a narrow band. It is a measure of the relative ease with which an electron may travel from one atom to another in the crystal. If a zone is com pletely occupied by electrons, Neff =0, since dE/dk vanishes at the surface of the zone. 1. CONDUCTIVITY In a metal with finite resistance, the dis tribution in k space will shift in the direction of JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14the applied field until an equilibrium has been reached when the effect of the field will be just balanced by collision processes. The equilibrium distribution will be just that which would be obtained if the electrons had been accelerated for a time t = T in the absence of resistance. Here T is called the time of relaxation (d. Section III). For fields of ordinary magnitude, the equilibrium distribution will differ but slightly from the normal distribution in zero field. The conductivity may be expressed in the Drude-Lorentz form, we merely need to insert Neff, as defined by Eq. (21) in place of N, (23) The factors which determine T will be discussed later; we first wish to give an explanation of the anomalous sign of the Hall coefficient exhibited by some metals. J. HALL COEFFICIENT If the conduction band is only partially full, so that the states near the top of the Fermi distribution lie in the region where the curvature (d2E/dk2) is positive, the electrons behave normally; i.e., as particles of negative charge and positive mass. The sign of the Hall coefficient will be that which is expected for electrons. Let us now suppose that the band is nearly full, so that the electrons near the top of the Fermi distribution (which are those important for con ductivity) have states in the region of negative curvature. It is then better to think of conduction by the unoccupied states or "holes" near the top of the band. These behave (d. Section V, H) as particles of positive charge and positive mass. They give rise to a Hall coefficient of anomalous sign. In many metals, electrons occupy two unfilled bands, in one of which conduction is by "holes" and in the other by "ordinary" electrons. The sign will then depend on which of these gives the predominant effect. Conduction by "holes" is the quantum-mechanical analog' of conduction by associated electrons, as visualized by Hall (Section IV). ' The magnitude of the Hall coefficient can be worked out, and the correct order of magnitude is found. Numerical agreement is obtained for the alkali metals. VOLUME 11, FEBRUARY, 1940 -eF ~ a. FIG. 9. Shift of the electron distribution due to an a pplied field, VI. Resistance of Pure Metals A. HIGH TEMPERATURES The resistance of pure metals is due to the thermal motion of the atoms. A time of relaxa tion, T, may be defined for collisions due to thermal motion only at high temperatures. At low temperatures, the quantization of the differ ent modes of vibration of the lattice is important. At each collision a vibrational quantum of energy, hll (II is the vibrational frequency) is either taken from or given to the lattice, so that the collisions are inelastic. If the energy of the quantum is small compared with KT, as it will be if the temperature is well above the Debye characteristic temperature for the metal, it is possible to neglect this quantization, and to treat the vibrations classically. It then turns out that the probability of a collision of an electron with the lattice, and consequently 1/ T, is proportional to the mean square amplitude of motion of the ions in confirmation of Wien's hypothesis. The mean square amplitude is pro portional to the absolute temperature, and is inversely proportional to the mass of the atom, M, and to the square of the vibrational fre quency, II, (24) Thus the resistance should be proportional to the absolute temperature, in agreement with experiment. The vibrational frequency is pro portional to the characteristic temperature, 8, so that 1/T,,-,(x2)ave"-'T/M8Z. (25) A second factor of importance in (l/T) is the density of states in energy near the top of the Fermi distribution. A large density means that there are a large number of states into which the electrons can be scattered, and so a large 97 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14value for (l/T); The scattering probability is directly proportional to the density. The density is, of course, high for a narrow band and small for a wide band. O.80~-----:5:700:------=IO'::OO:-------C'50~O·C TEMPERATURE FIG. 10. Plot of the relative resistance of various metals divided by the absolute temperature, T, illustrating devia tions from proportionality of resistance with T at high temperatures. Data from values quoted by Gruneisen and by Meissner (ef. Bibliography). Deviations from the proportionality of re sistance with absolute temperature at very high temperatures ("-'1000°C) have been explained by Mott as due to the change in e as a result of thermal expansion. As the metal expands, e de creases, so that R/T increases as the tempera ture rises. This is the normal behavior, but for some metals, in particular for those in the transition group, R/T decreases with increasing temperature. In this group of metals there is a narrow d band, and at high temperatures the density of states changes appreciably in the range of energies of width KT at the top of the Fermi distribution. The effective density de creases as T increases, which explains the decrease in R/T observed for these metals. Fig. 10 gives a plot of R/T as a function of T for various metals. B. Low TEMPERATURES In order to discuss the temperature variation of resistance at low temperatures, it will be necessary to investigate the interactions of the electrons with the lattice vibrations in somewhat more detail. Following Debye, we may analyze the vibrations into a system of independent waves. The very long waves are just the ordinary sound waves; the shortest waves correspond to adjacent atoms vibrating in opposite directions. Each wave is described by the propagation vector q. The direction of q gives the direction of propagation of the wave, and its magnitude is 98 27r/>', where>. is the wave-length. There are three independent vibrations for each q., corre sponding to the transverse and longitudinal sound waves. If the velocity of the wave is c, the frequency is /I =c/X =cq/27r (26) and the energy of the vibrational quantum is hv=hcq/27r. In each collision of an electron with the lattice a vibrational quantum is either emitted from or absorbed by the lattice. The selection rules for the transition of an electron from the state k to the state k' are as follows k'=k+q k'=k-q (emission) (absorption). (27) The vector relations are illustrated in Fig. 11. The angle, IJ, between k and k' is the angle through which the electron is deflected by the collision. In addition to these rules, there is the requirement of conservation of energy. Thus, for absorption, the energy of the electron in the final state, k', must be equal to the energy of the electron in the initial state, minus the energy of the vibrational quantum. At low temperatures, the only vibrations which will be excited are those with low energies, and these have long wave-lengths and therefore small values of q. The low resistance is due not only to the fact that the amplitudes of vibration are small, but also to the fact that the electrons are deflected through small angles (d. Fig. 11). A quantum can be absorbed by the lattice even though no quanta are originally present. Thus it would appear that a metal has resistance at the absolute zero of temperature. However, a quan tum can be absorbed only if the electron can lose an equivalent amount of energy. The average excitation ene"rgy of the electrons near the top of the Fermi distribution is of the order KT. An electron cannot lose more than this amount, because all the lower states are occupied. Bloch's derivation of the formula for the tem perature dependence of resistance involved a number of assumptions and approximations, among which are: (1) Debye theory for the thermal vibrations. (2) Thermal equilibrium of vibrations. (3) E(k) a function of 1 k I. In addi tion, certain assumptions about the form of the JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14interaction between the electrons and lattice TABLE I. Pressure coefficient of resistivity of various metals. vibrations were made. The function giving the distribution of electrons in k space, which was used in calculating the conductivity, is only an approximate solution of the Boltzmann equation. The formula derived states that the resistance is proportional t09 This formula was used by Gruneisen, and is plotted in Fig. 1, together with the experimental points for a number of metals. The check is extremely good, especially when one takes into account the approximate nature of the theo retical derivation. In this respect the theory is somewhat analogous to the Debye theory of specific heats. The formula shows that the resistance at very low temperatures should be proportional to Ta. This law can be derived independently, and is on a firmer theoretical foundation. Most metals seem to follow this law fairly well,1° but there are some devia~ions. In particular, the resistance of some of the transition metals at extremely low temperatures « lOOK) seem to follow a P law. Baberll finds a term proportional to P in the resistance arising from collisions between elec trons. The effect is never large, but should be most pronounced in metals having a very narrow conduction band, such as the d band of the transition metals. FIG. 11. Illustrating the vector relations for the scattering of an electron wave by a lattice wave. C. PRESSURE VARIATION OF RESISTANCE We have seen that the conductivity may be expressed in the form (23) and that (1/1') is proportional to (1) the mean VOLUME ll, FEBRUARY, 1940 -(d LOG Rldp)1012 (2,BV ole.) 10" METAL (c.g.s.) (c.g.s.) Li -4.0 21. Na 73. 40. Mg 5.9 9. Al 4.8 6. K 190. 91. Ca -8.9 Fe 2.7 2.0 Co 1.1 2.0 Ni 2.1 2.0 Cu 2.3 3.0 Rb 200. 120. Sr -47. Mo 1.5 1.1 Ag 4.0 4.8 Cs 220. 157. Ta 1.7 1.8 Pt 2.1 1.9 Au 3.4 3.3 Pb 15.4 12.5 square amplitude of the thermal motion of the atoms, which, according to the Debye theory, is proportional to T / Me2, and (2) the density of the electronic states in energy at the top of the Fermi distribution, which is inversely propor tional to dE/dk. As we have mentioned earlier, the normal decrease of resistance with increasing pressure is accounted for by the increase in e with pressure, which results from the stronger binding forces as the atoms are brought closer together. The relative change in resistance with pressure due to the change in e is d log R/dp= -2d log e/dp. (29) The dependence of e on pressure can be esti mated from Gruneisen's formula12 {3Vo/KCv= -d log e/d log V, where {3 is the volume coefficient of expansion, K is the compressibility,* Vo is the volume occu pied by one gram of the substance, and Cv is the specific heat. Since the compressibility K=-dlogV/dp, d log e/dp={3V o/Cv• (30) Values of -2d log e/dp as calculated from (30) are compared with the observed values of d log R/dp for a number of metals in Table I. * We have also used K to denote the heat conductivity. There should be no confusion. 99 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:140.7 0.6 a 0~1~0-----1~4----~18~--~2~2----~2~8----~3~O 1.1 -NA a 1.024 28 32 38 44 ATOMIC VOLUME [0 MI"I Olt4] FIG. 12. Plot of a (Eq. (31» for sodium and lithium as a function of volume. The agreement, at least as to order of magnitude, is in general good, but there are some notable exceptions, e.g., Li, Ca and Sr, for which the resistance increases with pressure. Since e cer tainly increases with pressure, these anomalous values must be attributed to electronic factors. We will first discuss the monovalent metals and then give a possible explanation, due to Mott, of the anomalous behavior of some of the divalent metals. A plot of the variation of the resistance of the alkali metals with pressure, as observed by Bridgman, is shown in Fig. 2. Na behaves nor mally, the resistance decreasing with increase of pressure. The resistance of Li increases with pressure, while K, Pb, and Cs show a minimum, the resistance at first decreases and then, with further increase in pressure, the curve reverses and the resistance increases. To a first approximation the electrons 111 a monovalent metal may be treated as free elec trons. The energy is then E = ~mv2 = h2k2/87r2m. A better approximation for E is R = ah2k2/87r2m, ( 31) where a is the effective number of free electrons per atom. Values of a which have been computed 100 for Li and Na from fundamental principles14 are plotted as a function of the atomic volume in Fig. 12. It is found that a is about 0.65 for lithium and decreases as the volume decreases. It is nearly equal to unity for sodium and increases slightly as the volume decreases. To make a very rough calculation of the variation of resistance with pressure, we may assume that the charac teristic temperature, e, is inversely proportional to the square root of the compressibility, K (Einstein's formula). Since T is proportional not only to 82, but also to dE/dk, or a, we find that the resistance is proportional to (32) or The subscript zero refers to the values at zero pressure. Values of RjJ/ Ro for sodium as com puted from Eq. (33) are compared with the ex perimental values in Fig. 13. The agreement is fairly good. Similar calculations can be made for lithium. The decrease in a with pressure would tend to 1,0 o.s Q. R/Ro [EXP) h. O</<fJ,/rK/a':Io 0.8 0.4 b O~L--- ____ ~ ________ ~ ________ ~ ___ o 10 PRESSURE 20 30 lI(G/cM2l(10-1) FIG. 13. Comparison of experimental and theoretical values of the relative change of resistance of sodium with pressure. JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14make the resistance increase. Frank13 has at tributed the observed rise in resistance to a decrease in Neff. However, the decrease in ex shown in Fig. 12 is not sufficiently great to counter-balance the effect of the increase in El, so that Eq. (33) gives a net decrease in re sistance, in contradiction with experiment. Of course, the factors given in Eq. (32) are not the only ones which affect the resistance. Another important one is the term which represents the interaction between the electrons and lattice waves. Unpublished calculations by I. Weiner on the basis of the author's theory17 indicate that the further terms have little effect on the pressure variation of resistance of sodium. They act in such a direction as to increase the resistance of lithium at high pressures, but the effect is not very large, so that there ·is still a small net decrease in the calculated resistance as the pres sure is increased. There is some evidence that the conduction electrons in lithium lie in two overlapping bands, and it may be necessary to take this into account to obtain complete agree ment between theory and experiment. A discussion of the conductivity of divalent metals, where two Brillouin zones are of impor tance, has been given by MotUS An excellent qualitative discussion of his general method is given in the introduction to the second of the papers listed.16 If there are two overlapping energy bands, which we may designate by a and b, the conductivity is Here Ta and Tb are the times of relaxation of the electrons in bands a and b, respectively, and ex and {3 are the corresponding effective members of free electrons per atom. Mott shows that Ta and Tb will be of the same order of magnitude, but that ex and {3 may be very different. If there is a low density of states in band a and a high density in band b, then ex will be very much larger than {3, and most of the current will be carried by elec trons in band a. The resistance will then be due largely to transitions which send electrons from band a to band b, and the times of relaxation, Ta and Tb, will be inversely proportional to the density of states at the top of the Fermi distri bution in band b. If, as the pressure is increased, the overlap is increased so that the density of VOLUME 11, FEBRUARY, 1940 states at the top of the Fermi is increased, the resistance will increase, since Ta and Tb will . decrease. This will occur if either a or b has lower energy. A schematic diagram of two su~h over lapping bands is shown in Fig. 14. Manning and Krutter17 have made approxi mate calculations of the energy bands of Ca. They find a dense d band lying above and over lapping the normal s band (which contains nearly two electrons per atom). They find that the overlap increases with pressure, so that the resistance should increase with pressure, in qualitative agreement with experiment. ~ o > ~ CD Z iii C ENERGY .... ENERGy ..... FIG. 14. Occupied electronic states in two overlapping bands at (a) zero pressure, (b) some high pressure (sche matic). D. ABSOLUTE VALUE OF THE CONDUCTIVITY We have seen that one of the important factors which determine the resistance of a metal is the square of the amplitude of vibration of the atoms. Due to this calise alone, the conductivity would be proportional to MEl2, where El is the Debye characteristic temperature and M is the atomic weight. In Fig. 15 we give a plot of II / MEl2, which should depend on purely elec tronic factors, as a function of the atomic number of the element. The most striking features are 101 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14"' !?I~ ~ . 11 v1 " ': < ~ \ IJ ATOMIC NUMBER FIG. 15. Values of u/ M02 plotted as a function ofthe atomic number. There is a gap for the rare earth metals. the relatively high values for the monovalent metals, and the low values lor the transition elements. Note, for example, the large difference in the values of IT / M82 for the neighboring pairs (Ni, Cu), (Pd, Ag) and (Pt, Au). In the transition metals there is an incomplete d band which is overlapped by an s band which is similar to the s bands of the monovalent metals. (Cf. Fig. 16.) The current is carried mainly by electrons in the s band but the large resistance is due to scattering from the s band to the narrow d band. The prob ability of transitions of this type is large because of the high density of states in the d band. Such transitions cannot occur in the monovalent metals, because the d band is fully occupied. The comparatively high resistances of the divalent metals is probably due to the small effective number of free electrons for these metals. This is true to a much greater extent for such semi-metallic elements as Bi, Sb, and As. A calculation of the absolute value of the conductivity of a metal from fundamental prin ciples is very difficult because little is known about either the electronic wave functions or the vibrational spectrum of most metals. Such calcu lations have been attempted only for the mono valent metals. To illustrate the degree of agree ment that is obtained, results of a calculation by the author18 are given in Table II. This calcula tion was based on the following assumptions: (1) The wave functions of the electrons were assumed to be nearly the same as those for free electrons throughout the major part of the volume. (2) The Debye theory was used for the lattice vibrations. (3) The perturbation potential, 102 which gives the scattering of electrons by the lattice vibrations, consisted of two parts: (a) the change in the potential of the ions, and (b) the change in the potential of the self-consistent field of the valence electrons when the ions are moved from their equilibrium positions by the lattice waves. The agreement is fairly good for Na and K (the metals for which the assumptions are prob ably best justified), but the calculated conduc tivities of the remaining monovalent metals are too large. E. FERROMAGNETIC METALS The resistance of the ferromagnetic metals rises rapidly as the Curie temperature is ap proached. There is a discontinuity in the slope of the resistance-temperature curve at the Curie point, and above the curve is much like that of a II. o ) I- iii z 11.1 o 3d BAND 4. BAND [IO,NI] 1'll,cU] ENERGY .. FIG. 16. Schematic picture of wide s band overlapping narrow d band. normal metal. The curve for nickel, which has been investigated more extensively than any other ferromagnetic metal, is shown in Fig. 17. Gerlach19 has suggested that the resistance in the neighborhood of the Curie point can be given as JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14the sum of a normal resistance, Rn, which is given by the Bloch-Gruneisen function, and a second term, RF, which is a function of the spontaneous magnetization: R=Rn+RI<,=Rn+C(102-J2). (35) Here 10 is the intensity of the spontaneous mag netization at the absolute zero, 1 the actual mag netization at the temperature T, and C is a con~tant independent of temperature. The subject has been treated theoretically by Mott.20 The resistance of nickel is due largely to transitions which take electrons from the s to the d band. Each state in the d band can ac commodate two electrons, one of each spin. At low temperatures, when the sp~ntaneous mag netization reaches the saturation value, all the states in the d band are occupied by electrons whose spins are parallel to the direction of the magnetization. The remaining electrons have spins antiparallel but there are not enough to completely fill the band. The chance that an electron changes its spin during a collision is rather small, so that only half of the electrons in the s band, namely those with anti parallel spin, can make transitions to the d band. At higher temperatures, when the magnetization is not complete, electrons of either spin can make these transitions, so the resistance rises. The curves a and c in Fig. 17 are suggested resistance curves for paramagnetic nickel and for nickel mag netized to saturation. The three points shown give the resistances as computed by Mott for the actual magnetizations at the given temperatures. The calculations are rather complicated, and do not yield the simple form suggested by Gerlach. The agreement with the experimental curve is fairly good. The resistance decreases when the specimen is magnetized by an external magnetic field. It is TABLE I I. Electrical conductivity of monovalent metals. T=O°C (ohm-1 cm-1 X 10-4). META1. EXPERIMENTA1. CALCULATED Li -11.8 28 Na 23.4 23 K 16.4 20 Rb 8.6 33 Cs 5.3 22 Cu 64. 174 Ag 66. 143 Au 49. 142 VOLUME 11, FEBRUARY, 1940 not as yet certain whether this effect can be quantitatively explained as simply due to the change in magnetization. FIG. 17. Resistivity of nickel as a function of tempera ture. (a) Calculated for hypothetical paramagnetic nickel. (b) Experimental curve. (c) Calculated for magnetization at OOK. (0) Calculated for observed magnetizations at given temperatures. (After Mott, reference 19.) VII. Conductivity of Alloys A. DILUTE SOLUTIONS We have already discussed Matthiessen's rule (Section II, D) which gives the increase in re sistance due to a small concentration of a foreign metal in solid solution. This rule receives a simple explanation according to the quantum theory of metals. A perfect periodic lattice has no resistance. In a pure metal, resistance is due to thermal motion, which destroys the perio dicity of the lattice. In a solid solution electron waves may also be scattered from the foreign atoms present. For small concentrations, the resistances due to these two causes are additive. Matthiessen's rule will apply if the temperature dependent part, resulting from the thermal vibra tions of the lattice, is independent of the con centration. The increase in resistance due to the dissolved metal is usually very large. At room tempera ture, the resistivity of a metal may be doubled 103 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14by the presence of one atomic percent of im purity. Resistance measurements are often used as a test of the purity of a specimen. The effects of different solute metals may vary over a considerable range, and for the most part have not been amenable to theoretical treatment. Some interesting relationships have, however, been pointed out by Norbury.21 By comparing the increase in resistance caused by one atomic percent of different metals dissolved in a common solvent, he showed a marked dependence on the valencies of the solvent and solute metals. In general, the greater the difference in valencies (or in the horizontal position in the periodic table), the greater is the additional resistance. The effect is most pronounced for metals dissolved in Cu, Ag, and Au. In later work, Linde22 has found that, for these metals, the increase in resistance varies approximately with the square of the difference in valencies. If Z + 1 is the number of electrons outside of a closed d shell (so that Z=O for Cu, Z=l for Zn, Z=2 for Ga, etc.) the additional resistance varies with Z2, as shown in Fig. 18. A simple explanation has been given by Mott.23 All electrons outside of the closed d shells, from the dissolved as well as from the solvent metals, go to make up the conduction electrons which are free to travel throughout the metal. The core of a dissolved atom has an excess charge over a solvent atom of amount Ze. Elec trons are scattered by the field of this excess charge. It follows from the Rutherford law that the scattering is proportional to the square of the charge (Ze) and it is this factor which makes the additional resistance proportional to Z2. Quan titative calculations, taking into account the .;;' ~IO :i o I :I x ,g AS screening of the excess charge by the conduction electrons, give the correct order of magnitude for the effect. Mott24 has also investigated the additional re sistance due to one atomic percent of metal A dissolved in metal B as compared with that of one atomic percent of B dissolved in A, and found that under certain conditions, these should be approximately equal. B. RANDOM SOLID SOLUTIONS In the present article we will be concerned only with the conductivity of a single homogeneous phase. (Many alloys consist of a mechanical mixture of two phases; in this .. ase the conduc tivity is roughly the weighted average of the conductivities of the individual phases.) We first limit the discussion to the case of a disordered alloy; i.e., we assume that the atoms of the metals making up the alloy are distributed at random over the lattice points. Ordered alloys will be discussed in the next section. The resistance of an alloy may' be expressed as the sum of two terms R=Ro+RT, (36) a temperature dependent term, RT, which results from the thermal motion of the atoms, and is much the same as the resistance of a pure metal, and a second term, Ro, which is the resistance at the' absolute zero of temperature. The second part represents the resistance caused by the fact that the lattice is not perfectly periodic due to the random distribution of the different atoms which make up the alloy . ...... ID~IO " a 0 I ~ AU x au 0 -S a::: FIG. 18. Increase in resistance due to one atomic percent of various metals dissolved in Cu, Ag, and Au. The abscissa are proportional to the squares of the difference in valency between the solvent and solute metals. Similar results are obtained if the solute metals come from other 'rows of the periodic table. Data from Linde (reference 21). 104 JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14A theory of the resistance of binary mixtures on the basis of the Bloch theory has been given by Nordheim. He finds that if x is the concen tration of atom A, so that (l-x) is that of atom B, the resistance, Ro, is proportional to x(1-x). A plot of Ro as a function of the concentration x is a simple bell-:-;haped curve with a maximum at a concentration of 50 percent. As we have seen (Fig. 4) this is just the type of curve which is found experimentally for the Ag-Au alloys. Similar curves are found for other binary alloys, the component metals of which are mutually soluble in all proportions, and which exist in but a single phase, e.g., K-Rb; Pt-Pd; In-Pb. (Cf. Fig. 1 Q.) A somewhat different type of curve is found for alloys of the noble metals with the transition metals, as illustrated in Fig. 20 for the case of alloys of Cu, Ag, and Au with Pd. The maximum resistance is shifted toward the Pd end of the curve. The explanation of Mott is 'briefly as follows. As we have already mentioned, the rela tively high resistance of the transition metals, including Pd, is due to the presence of an in complete d shell, so that transitions from the s band, which carries most of the current, to the ... o ~30 I :IE :x: o 20 o 20 IN-PS 40 80 80 100 ATOM I 0 " FIG. 19. Resistivity of K-Rb, Pt-Pd, and In-Pb alloys as a function of the atomic concentration. T= 25°C for all curves. Data from International Critical Tables. d band are possible. In the alloys under consider ation the d band i:; incomplete for all concentra tions of Pd above about 40 percent. The ab normally high values of the resistances of the alloys above this concentration are due to s-d transitions. VOLUME 11, FEBRUARY, .1940 ... o >< :I 40 ~ 80 :I :x: o 100 ATOMIO % PO FIG. 20. Resistivity of Cu-Pd, Ag-Pd, and Au-Pd alloys as a function of the atomic percent Pd. Cu-Pd and Ag-Pd from Svensson,* and Au-Pd from Geibel (International Critical Tables). In Fig. 21 we give a plot of the resistivity of the Cu-Ni alloys at several temperatures, ac cording to the measurements of Svensson. These alloys are ferromagnetic for concentrations of nickel above about 40 atomic percent. Constan tan, with the approximate composition CuNi, is a member of' this series. Svensson attempts to correlate his results by expressing the resistance as the sum of three terms: (1) RT, resulting from thermal motion, (2) Ro, which gives the resistance due to the random distribution of the different types of atoms, and (3) RF, a ferromagnetic part which is a function of the magnetization. A qualitative theoretical discussion has been given by Mott.20a C. ORDER-DISORDER ALLOYS25 We have shown (Fig. 5) the resistance concentration diagram for the copper-gold alloys. The quenched alloys give a simple bell-shaped curve similar to that of the silver-gold alloys. If the alloys are annealed at temperatures below 400°C, the resistance drops sharply in the neigh borhood of the compositions CuAu and CusAu. On the basis of chemical and resistivity measure ments, Tammann suggested that the atoms take up more or less regular ordered positions at these concentrations. The ordered structure was * Ann. d. Physik 14, 699 (1932). 105 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14'" o ,. 50 2 040 I 2 % o >... > 30 ~ 20 UJ 1&1 II: CU ATONIC % HI FIG. 21. Resistivity of Cu-Ni alloys. At 0, 250 and 500°C from Svensson.* At -273°C from Krupkowski and de Haas.t later definitely proved for the CuaAu alloy by analysis of the "superstructure" lines in the x-ray pattern by Johansson and Linde.26 Since that time, many other ordered alloys have been found. Resistivity measurements have played an important role in both their discovery and in vestigation, supplementing x-ray and specific heat data. The copper-gold alloys have perhaps been investigated more thoroughly than any other system, and much of our discussion will be confined to CuaAu as a typical example. The face-centered lattice of the copper-gold system can be considered to be a superposition of four interpenetrating simple cubic lattices. At high temperatures, the atoms are distributed at random over the lattice points, but somewhat below 400°C the gold atoms of CuaAu tend to congregate on but one of these simple cubic lattices, forming an ordered structure. At the composition CuAu, below a certain critical tem perature, the copper and gold atoms segregate on alternate planes. The spacing between these planes changes, and the structure changes from cubic to tetragonal. • Ann. d. Physik 25, 263 (1936). t Comm. Leiden No. 194, 1 (1928). 106 The equilibrium degree of order depends on the temperature. The distribution is random above the critical temperature, Te, and order gradually takes place as the temperature falls below T". A number of different definitions of the degree of order existing in a crystal have been suggested, perhaps the best known being that of Bragg and \ViIliams.27 Let us designate the two types of atoms in the alloy by A and B. When perfect order exists, the,A atoms will occupy a subsidiary lattice, and positions on this lattice will be called a-sites. Similarly, the B atoms will occupy f3-sites. In the partially ordered crystal, let the probability that an A atom be on an a-site be PA. Finally let rA. be the value of PA for a completely random distribution (e.g., r A = i for Cu3Au, if A refers to gold). The degree of (Ionr, distance) order is defined as : P.4 -rA S=---, l-rA (37) so that S= 1 for perfect order, and S=O for a random alloy. This definition has been criticized in that no account is taken of any ordering of neighbors that may exist, and attempts to take the local ordering into account have been made by Bethe,28 Peierls,29 Kirkwood,30 and others. The I~r-------------~~---- __ ffi 0.5 Q II: o o 0.26 0.76 1.00 FIG. 22. Theoretical curves of long distance order, S, as a function of TITe for an alloy of the type. A.B .. (a) Bragg-Williams. (b) PeierIs (for face-centered cubiC lattice). most suitable definition of order for considera tions of resistivity measurements is not known. The two critical distances are the electronic wave-length ('" lOA) and the mean free path JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14('" lOOA); the amount of order existing in re gions of each of these dimensions probably plays a role. Theoretical curves of Bragg and Williams and of Peierls giving the degree of order, S, as a function of temperature for an alloy of the type A J3 are shown in Fig. 22. The long distance order changes abruptly at the critical tempera ture, and there is a latent heat at the transition point. This is contrary to the behavior of alloys of the type AB, for which there is no latent heat, and the degree of order starts in at zero at the transition point and gradually rises as the tem perature falls. . In Fig. 23 we show the resistivity-temperature curves for Cua.Au as observed by Sykes and Evans.a! Curve a is for the equilibrium state, which was obtained after long annealing. In curve b, the cooling rate was 30°C per hour. Curve c is extrapolated from above the critical temperature to the value obtained for an alloy quenched from 450°C, and represents the re sistance curve for the completely random alloy. Due to the relatively slow rate at which different atoms in the crystal interchange places, the rate of approach to the equilibrium configuration is slow except at temperatures close to the transi tion point. At room temperature, the rate is so slow that the alloy will exist indefinitely in a metastable state. Measurements of the resis tivity as a function of temperature for different rates of cooling or heating have been found ex tremely useful in determining the rates of ap proach to equilibrium. Unfortunately, however, resistivity measure ments cannot be used for exact quantitative work because the resistivity is not simply related to the degree of order. Following the general theory we have discussed earlier, we should ex pect that the resistance is the sum of two terms, a temperature dependent term, RT, which arises from the thermal motion of the atoms, and a second term, Ro, giving the resistance due to the disordered arrangement of the atoms which will be dependent on the temperature only indirectly through its dependence on order. Purely for simplicity, Bragg and Williams assumed that the second term is a linear function of the degree of order, S, vanishing when the order is perfect. As may be seen from a comparison of Figs. 22 VOLUME 11, FEBRUARY, 1940 and 23, there is some justification for this as sumption. The subject has been treated theo retically by Muto,32 who extended Nordheim's treatment of random alloys to cover the order disorder alloys. He finds a quadratic dependence of Ro on S. The calculation assumed no correla tions between the types of atoms on neighboring sites, and so neglected any local ordering that may be present. 14 '" ~ 12 " 2 o I 2 5 10 >-!: 8 > j::: III (is ~ 6 TEMPERATURE FIG. 23. Resistivity as a function of temperature for Cu.Au. (a) Quenched; (b) Cooled at a rate of 30°C/hr. (c) Slow cooling, in equilibrium above 350°C. The path OA is discussed in the text. Data from Sykes and Evans (reference 30). In a very interesting series of papers, Sykes and J ones33 have investigated the formation of nuclei and their effect on the resistivity of CuaAu. The quenched alloy is heated rapidly to about 350°C, and the resistivity rises to the point 0 (Fig. 23), corresponding to a random alloy. Ordering has not had time to set In. If the alloy is kept at this temperature, the resistance gradually drops along the line OA until finally it reaches the equilibrium value. In each crystal, the gold atoms may segregate on anyone of the four simple cubic lattices which make up the face centered I:'tructure. Nuclei may start on any of these at random and grow until they meet one another. The crystal then consists of a large number of small regions in each of which the structure is ordered, but there is a discontinuity or jump in phase at the boundaries. Some of these 107 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14FIG. 24. Resistivity as a function of the apparent size of nuclei. Annealing temperatures: (a) 376°C; (b) 346°C· (c) 298°C. Data from Jones and Sykes (reference 22). ' regions will gradually grow at the expense of the others, so that the nuclei increase in size. As this happens, the resistance drops. The average size of the nuclei during different stages of this process were estimated by Sykes and J ones from the widths of the superstructure lines in ·the x-ray powder diagram. Different times of anneal were taken, corresponding to about equal changes in the resistance, and then the alloys were quenched in water. In this way a whole series of metastable states, corresponding to different stages in the growth of the nuclei, were obtained for the study of the resistivity and its dependence on the size of the nuclei. In Fig. 24 we give a plot of the resistivity as a function of the apparent nuclei size for three different temperatures of anneal as ohtained by Jones and Sykes. The true size of the nuclei is estimated to be about one-half of the apparent size, f, as determined by x-ray methods. The differences between the three curves are due to the differences in the degree of order in the in dividual nuclei at the various annealing tem peratures. The curves become straight lines for values of (1M below about 10-2 (f in A). In this region the nuclei size is large compared with the width of the boundaries, and the change in resistance can be considered to be due to the reflection of electrons from the boundaries. Tak ing the effective number of free electrons per atom equal to unity, Jones and Sykes estimate that the reflection coefficient at a boundary is 108 about one-tenth. Actually, it may be somewhat greater than this value. There is an anomaly in the specific heat of the quenched alloy as it is heated between 100° and 200°C, long before there is any appreciable effect on the electrical resistance. This must represent a very early stage in the formation of nuclei. Probably a large amount of local ordering takes place which involves comparatively large energy changes, but which is confined to such small re gions that it has little influence on the resistance. The resistivity of an ordered alloy is changed markedly by cold work. In Fig. 25 we give the results of Dahl on the effect of Rlastic deforma tion on the resistivity of a CU3Au alloy. With sufficient deformation, the resistivity of the annealed alloy approaches that of the quenched alloy. As this occurs, the x-ray superstructure lines disappear, indicating the return of a random structure. A large change of resistance on cold work is often taken to be an indication (although not a certain one) of an ordered alloy. As a second example of the effect of order on resistivity, we show in Fig. 26 the resistance of J3-brass (CuZn) as a function of temperature. The o mo~ REDUCTION IN CROSS-SECTIONAL AREA PERCENT FIG. 25. Effect of plastic deformation on the resistance of CU3Au alloy. Data from O. Dah\.* structure is body-centered cubic, and in the ordered state the Cu and Zn atoms tend to segregate on the two interpenetrating simple cubic lattices which make up the structure. Ac- * Zeits. f. Metallkunde 28, 133 (1936). JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14cording to the theory of Muto,32 the resistance due to disorder, Ro, should be proportional to (1-.)'2) where S is the degree of order (Eq. (37)). Plots of 1-S2 as a function of temperature ac cording to the theories of Bragg and Williams and of Bethe are shown below the resistivity curve. D. AGE-HARDENING ALLoys34 When a supersaturated alloy is quenched from a high temperature, one or more of the phases may begin to precipitate out of the solid solution, either at room temperature or after annealing. This causes a gradual change in the physical properties of the alloy. The hardness usually increases with time to a maximum, after which there is a gradual decrease. The electric and magnetic properties also change during the hardening process. The resistance sometimes de creases, but often there is an initial increase, followed by a decrease. The maximum in re sistance usually comes before the maximum in hardness. The resistance will depend, of course, on the distribution of atoms in the lattice. A random distribution yields a high resistance, and a con glomerate of crystals of individual pure metals or compounds a comparatively low resistance (d. Figs. 3 and 4). Thus one might expect the resistance to decrease as a metal precipitates from solid solution. However, as Mott35 has pointed out, the random distribution may not be the one which gives the highest resistance. If the atoms are distributed in very small clusters, such that the average size of a cluster is of the order of magnitude of an electron wave-length ('" lOA) they may be more effective in scattering the waves than if the clusters are larger or smaller. The maximum in resistance may therefore corre spond to a very early stage in the precipitation. VIII. The Weidemann-Franz Law It is not our purpose to discm;s heat conduc tivity, or such subjects as thermoelectricity and the various galvano- and thermomagnetic effects in metals, even though they are very closely related to our present study. We must, however, make some mention of the Weidemann-Franz law. As is well known, heat may be conducted in a metal either by the conduction electrons or VOLUME 11, FEBRUARY, 1940 through the lattice itself, by means of the thermal vibrations. In insulating crystals, only the latter is effective. In metals, conduction by electrons is usually much the greater of the two. The Weidemann-Franz law can be derived under quite general considerations on the basis of the Bloch theory, and states that the ratio of the heat conductivity, K, to the electrical con ductivity, (T, is 18r---~----~----~---'-----r----' "0 o ~:f 2: 1 o 100 200 aoo 400 soo 600·0 TEMPERATURE FIG. 26. Top. Resistance of !3-brass (composition 51.25 atomic percent Cu, 48.75 percent Zn) as a function of temperature. Data from W. Webb.* Bottom: Plot of I-52, where S is the degree of long distance order, according to the theories of Bragg and Williams and of Bethe. The numerical factor is slightly larger than that obtained from the Drude theory (7r2/3 in place of 3) and agrees somewhat better with the experi mental values. The derivation applies only to conduction by electrons, and so neglects any contribution to the heat conductivity by the lattice vibrations. It is valid whether the re sistance is due to disorder (as in alloys) or is due to thermal motion of the atoms, but in the latter case only for temperatures above the Debye characteristic temperature. Furthermore, it is supposed that KT is small compared with the width of the conduction band, so that deviations may be expected, for example, in the transition metals at high temperatures. * Phys. Rev. 55, 297 (1939). 109 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14IX. Conclusion In this brief survey, no attempt has been made at completeness. We have merely attempted to give illustrative examples of some of the various types of problems connected with the conduction of electricity in metals and alloys, and, wherever possible, to correlate them on the basis of the Bloch theory. Among the subjects upon which we have not touched are (1) dependence of re sistance on crystallographic directions, (2) re sistance of liquid metals, (3) the effect of poly morphic transitions, (4) the influence of an external magnetic field on the resistance, (5) the temperature coefficient of the resistance of alloys, (6) the structure sensitive properties, such as the effect of cold work, and (7) the resistance of thin films. With a deeper understanding of the funda mental causes of resistance in metals and alloys, resistivity measurements promise to play an increasing role in the study of other physical. properties. This is especially true of problems connected with the arrangement of different types of atoms in an alloy. There is a great need for further study, of both an experimental and a theoretical nature, and if this article provides any stimulus for further activity in this direction, it will have served its purpose. Before concluding, it might be well to point out some phenomena for which the Bloch theory has not as yet provided even a qualitative under standing. The best known of these is, of course, superconductivity. A second is the discovery of De Haas and co-workers35 at Leiden that there is a minimum in the resistance-temperature curve of gold at liquid helium temperatures. The in crease in resistance becomes very rapid at ex tremely low temperatures « 1 OK). The curve suggests that the resistance may become infinite at the absolute zero. The position of the minimum depends on the purity of the gold, shifting to lower temperatures as the amount of impurity is reduced. No explanation of this phenomenon has been given. A third example is provided by the existence of a class of semi-conductors with in completely filled d bands.36 We have mentioned (Section V, G) that a necessary condition for an insulator is that there be an even number of valence electrons per unit cell; if the number is odd, one would expect a metal. Substances like MnO, CoO, Mn304, etc., violate this rule since they are not metallic, but are semi-conductors. The Bloch theory is based on the assumption that each electron moves independently in a periodic potential field. The electrostatic inter action of the electrons is neglected, except insofar as it is taken into account through the potential of the average space charge of the electrons. The theory is extremely valuahle because it provides a simple physical picture which gives explana tions (in many cases, quantitative) of a very wide range of phenomena. It should be recognized, however, that it may be necessary in some cases to go beyond the Bloch picture to explain things which depend to a large extent on electron inter action, or on the cooperative action of a large number of electrons. Bibliography (a) References to Books The references given below refer only to the few specific examples of various phenomena which have been discussed in the text. A fairly complete summary of the experimental literature extending up to about the year 1934 is given by W. Meissner, Handbuch der Experimentalphysik, Vol. XI/2. Other sources of experimental data are: 1. G. Gruneisen, Handbuch der Physik, 24. 2. G. Borelius, Handbuch der Metallphysik, 1/1 (Leipzig, 1935). 110 3. W. Hume-Rothery, The Metallic State (Oxford, 1931). The following books and articles devoted mainly to the theory have been found useful: 1. A. Sommerfeld and H. Bethe, Handbuch der Physik 24/2 (1933). 2. N. F. Mott and H. Jones, Theory of the Properties of Metals and Alloys (Oxford, 1936). 3. A. H. Wilson, Theory of Metals (Cambridge, 1936). 4. H. Frohlich, Elektromentheorie der MetaUe (Berlin, 1936). 5. A. H. Wilson, Semi-Conductors and Metals (Cambridge, 1939). JOURNAL OF APPLmn PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14(b) References to Journal Articles 1. R. D. Tolman and T. D. Stewart, Phys. Rev. 8, 97 (1916); 9, 164 (1917). 2. It has been shown by Darwin that the experiment should give the free electron value of elm; the "effective" mass does not enter. C. G. Darwin, Proc. Roy. Soc. A154, 61 (1936). 3. E. Gruneisen, Verh. d. D. Phys. Ges. 15, 186 (1913); Leipziger Vortrage, 46 (1930); Ann d. Physik 16, 530 (1933). 4. P. W. Bridgman, Proc. Am. Acad. 72, 157 (1938). 5. P. Drude, Ann. d. Physik I, 566 (1900); 3, 370, 869 (1900); 7, 687 (1902); 14,936 (1904). H. A. Lorentz, Electromentheorie der Metalle (Leipzig, 1909). 6. The theory is summarized in a recent book: E. H. Hall, A Dual Theory of Conduction in Metals (Cambridge, 1938). 7. F. Seitz and R. P. Johnson, J. App. Phys. 8, 84, 186, 246 (1937). 8. Cf. Mott and Jones, p. 96. 9. F. Bloch, Zeits. f. Physik 59, 208 (1930). See also Gruneisen, reference 3. A simplified derivation has been given by F. Sauter, Naturwiss. 7, 109 (1930). 10. W. J. de Haas and G. J. Van den Berg, K. Onnes Lab. Leiden Comm. Nos. 241-252, Suppt. No. 82A (1936), have measured the resistivities of a number of metals between 10K and 200 K. The results can be expressed in the form CTb with b usually between 4 and 5, but for Pt, b = 2. 11. W. G. Baber, Proc. Roy. Soc. A158, 383 (1937). 12. See, for example, J. K. Roberts, Heat and Thermo- dynamics (Blackie, 1933), p. 437. 13. N. H. Frank, Phys. Rev. 47, 282 (1935). 14. J. Bardeen, J. Chem. Phys. 6, 369 (1938). 15. N. F. Mott, Proc. Phys. Soc. Lond. 46, 680 (1934); Proc. Roy. Soc. A153, 699 (1936). See also the theoretical discussion by A. H. Wilson, Proc. Roy. Soc. A167, 580 (1938). 16. Cf. also Mott and Jones, p. 265. VOLUME 11. FEBRUARY, 1940 17. M. F. Manning and H. M. Krutter, Phys. Rev. 51,761 (1937). 18. J. Bardeen, Phys. Rev. 52, 688 (1937). Similar calcula tions have been made by E. L. Peterson and L. W. Nordheim, Phys. Rev. 51, 355 (1937). 19. W. Gerlach, Zeits. f. Physik 59, 847 (1930). 20. (a) N. F. Mott, Proc. Roy. Soc. A153, 699 (1936); (b) A156, 368 (1936). See also A. H. Wilson, refer ence 15. 21. A. L. Norbury, Trans. Faraday Soc. 16, 570 (1921). 22. J. O. Linde, Ann. d. Physik 10, 52 (1931); 14, 353 (1932); 15, 219 (1932). 23. N. F. Mott, Proc. Camb. Phil. Soc. 32, 281 (1936). 24. N. F. Mott, Proc. Phys. Soc. Lond. 46, 680 (1934); also reference 22. 25. For a general review of this subject, see F. C. Nix and W. Shockley, Rev. Mod. Phys. 10, 1 (1938). See also G. Borelius, Proc. Phys. Soc. 49E, 77 (1937). 26. C. H. Johansson and J. O. Linde, Ann. d. Physik 78, 439 (1925). 27. W. L. Bragg and E. J. Williams, Proc. Roy. Soc. A145, 699 (1934); A151, 540 (1935). 28. H. A. Bethe, Proc. Roy. Soc. AlSO, 552 (1935). 29. R. Peierls, Proc. Roy. Soc. A154, 207 (1936). 30. J. G. Kirkwood, J. Chem. Phys. 6, 70 (1938). 31. C. Sykes and H. Evans, J. lnst. Metals 58,255 (1936). 32. T. Muto, Sci. Papers lnst Phys. Chem. Res. 30, 99 (1936); 31, 153 (1937). 33. C. Sykes and F. W. Jones, Proc. Roy. Soc. A157, 213 (1936); A166, 376 (1938). 34. For a brief review of this subject, see C. H. Desch, Proc. Phys. Soc. Lond. 49E, 103 (1937). 35. Discussion following a paper by M. L. V. Gayler, ]. lnst. Metals 60, 55 (1937). 36. W. ]. de Haas, H. B. G. Casimir and G. J. Van den Berg, Physica 5, 225 (1938), where references to earlier work may be found. 37. ]. H. de Boer'and E. J. W. Berwey, Proc. Phys. Soc. Lond. 49E, 59 (1937). 111 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.209.6.50 On: Fri, 19 Dec 2014 04:06:14

1.1714797.pdf

Physics in 1941 Thomas H. Osgood Citation: Journal of Applied Physics 13, 3 (1942); doi: 10.1063/1.1714797 View online: http://dx.doi.org/10.1063/1.1714797 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/13/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Arthur Compton’s 1941 Report on explosive fission of U-235: A look at the physics Am. J. Phys. 75, 1065 (2007); 10.1119/1.2785193 Palma, Antonio•1941–1984 J. Acoust. Soc. Am. 77, 1273 (1985); 10.1121/1.392155 Henry Crew, Recipient of the 1941 Oersted Medal for Notable Contributions to the Teaching of Physics Am. J. Phys. 10, 28 (1942); 10.1119/1.1990311 Frances Gertrude Wick, 1875–1941 Am. J. Phys. 9, 382 (1941); 10.1119/1.1991725 Proceedings of the American Association of Physics Teachers: Pasadena Meeting, June 17, 1941 Am. J. Phys. 9, 245 (1941); 10.1119/1.1991694 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16Physics in 1941 By THOMAS H. OSGOOD Michigan State College, East Lansing, Michigan I As through the streets at eve we went It might be half past ten- Oh! we fell out, my friend and I About the cube of (x+y) And made it up again. And blessings on the fallings out Between two learned men Who fight on points which neither knows, \Vho talk but never come to blows, And make it up again! SO we used to sing as undergraduates, happily unaware of the significance of the topics on which our professors were wont to disagree. In passing, one heaves an academic sigh of regret that all differences of opinion, however mo mentous, cannot be debated and adjusted by the amicable and civilized methods which character ize scientific controversy. A case in point which has recently added new interest to the study of x-rays and crystals is the occurrence of unex pectedly strong reflections in directions which are not accounted for by elementary theory. These anomalous reflections are found both when the Laue method and when the Bragg method of x-ray diffraction are used. The actual observa tion of spots and streaks which do not fit into a Laue pattern is a matter of many years' standing, but it has been customary to pass them off rather lightly in some quarters as due to irregularities in the mosaic structure or in the surface structures of individual crystals. Only recently have attempts been made to devise experiments which accentuate the intensity of these anomalous reflections. As soon as this was done, it became apparent that the spots were not due to accidental variations from one crystal to the next, but that they represented some special scattering property of crystals in general. I t is not possible to give here any logical sketch of the theories involved, but a few facts will indicate the present situation. VOLUME 13, JANUARY, 1942 When a monochromatic beam of x-rays falls on a crystal face, the crystal acts merely as an absorber and scatterer unless the Bragg relation nX = 2d sin (J is satisfied. In the latter case, of course, a very strong reflection, or scattering in a preferred direction, takes place. But no beam of radiation is perfectly monochromatic, x-rays scarcely ever come in parallel beams, and crystals are not perfect. For these three reasons the x-radiation reflected by a crystal at the Bragg angle can never be in the form of an infinitely narrow pencil, but possesses a finite width. Under ordinary experimental conditions, reflec tion is easily detected half a degree from the Bragg angle even in the case of so-called perfect crystals like diamond or calcite. If a calci te cleavage face be lightly ground, so as to increase the total reflected power, the range of reflection increases about lO-fold on account of the randomness introduced into the orientation of the surface particles.1 It is therefore necessary to use specially good cleavage surfaces in the investigation of anomalous reflections near the Bragg angle. An experiment of Siegel's2 will illustrate the occurrence of such reflections. A KCI crystal was found which had so perfect a cleavage face that a rotation of the crystal through 6 minutes of arc in either direction from the Bragg angle . for the copper Kala2 radiation was almost enough to suppress the reflection of the character istic copper lines. The crystal was mounted in a Bragg spectrometer in the usual way; the graduated circles were accurate to 1 minute or better, and the customary ionization chamber was replaced by a photographic plate. The crystal was then set so that it was, let us say, 20 minutes away from the Bragg angle, and a long exposure, perhaps 20 hours, was made. The Bragg reflection was, of course, recorded on the photographic plate, but only with reasonable intensity in spite of the long exposure, because of the slight offsetting of the crystal. Very close 3 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16to the Bragg spot in this exposure appeared another, rather more fuz?y, but of comparable intensity. Referring to Fig. 1, suppose that radiation from a copper target were incident along BC on the crystal, and that the KOi rays would then be reflected along CB'. Then B CB' FIG. 1. Reflection of monochromatic x-rays from a crystal face A CA', showing the direction of the diffuse reflection CD in relation to the Bragg reflection CE'. represents the ordinary Bragg reflection. With no other geometry could the reflected intensity be greater. The angles BCA, B'CA' are, of course, equal. Now suppose that the crystal is offset a little, and that the incident radiation falls on the crystal along RC. Some faint reflec tion still occurs, along CR', with the angles RCA and R'CA' still equal. But another reflection occurs along CD, causing a fuzzy spot to appear rather close to R', but always between R' and ]3'. I t is this diffuse reflection over which there has been so much argument-not about its occur rence, but about its interpretation. A pictorial idea of the phenomenon can be obtained from Fig. 2, which was taken with an arrangement like that of Fig. 1. The spots CL, Oi, in any strip of Fig. 2 are found in the directions R', D of Fig. 1. Diffuse spots are also shown in an even more striking manner by two equal-scale photo-. graphs (Figs. 3 and 4) reprinted from a paper by Gregg and Gingrich.3 One print was made, as in an ordinary Laue photograph, with continuous radiation; the other with monochromatic radia tion. In the latter case the Laue spots are suppressed, and the diffuse spots, caused by monochromatic radiation, stand out prominently against the background. In discussing their own experiments on the subject, Raman and Nilakantan4 state: "We presented evidence proving conclusively that the lattice planes in a crystal give a second kind of geometric reflection of x-rays which we 4 designated as the modified or quantum reflection, to distinguish it from the unmodified or classical reflection discovered by Laue and his collabo rators in 1912. The process which results in the modified reflection was clearly established by the experimental results in the cases studied by us. In the language of classical optics, a modified reflection by the lattice planes results from the dynamic variation of their structure amplitude consequent on an oscillation, relative to each other, of the interpenetrating lattices in the structure of the crystal. In the act of such reflection, the primary x-ray frequency is altered by the addition or subtraction of one or another of the characteristic infra-red frequencies of the crystal, In the language of quantum mechanics, the modified reflection represents an inelastic collision of the photon with the crystal in which the two exchange energy." Zachariasen, however, has been developing the theory of the diffuse scattering of x-rays by crystals which accounts generally for the radial streaks frequently found on Laue photographs. He finds theoretically that under appropriate conditions, the intensity along these streaks may increase and decrease so fast that a streak is recorded merely as a spot, the position of the spot being calculable for a particular wave-length in terms of the Bragg angle and the glancing angle of incidence. He shows too, from a re measurement of some of Raman and Nilakantan's photographs with rocksalt and diamond crystals, that the positions of the spots found by the Indian authors are very satisfactorily accounted for by his own theory. He even goes so far as to say that "there seems to be no acceptable basis for the assertion of Raman and Nilakantan that the theory of diffuse scattering is incapable of giving the correct positions of the diffuse maxima. There is thus no experimental justifica tion for the statement that the effect described by Raman and Nilakantan (observed earlier by others) is not a diffuse scattering phenomenon." In their latest contribution to the discussion, Raman and Nilakantan energetically contradict the suggestion that the phenomenon dealt with by them is explicable as "diffuse maxima in the scattering of x-rays by elastic waves of thermal origin," and to support their contention, consider the following three points. First, the intensity JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16of the scattering of x-rays by elastic waves of thermal origin is theoretically proportional to the total number of atoms effective in scattering, and the effects due to this type of scattering would therefore be very small unless large volumes of a crystal were involved. But their effect can be recorded under the same experi mental conditions as the Bragg reflection, and with narrow beams and thin crystals. Second, according to Raman and ~ilakantan's interpreta tion of Zachariasen's theory, the scattering coincides with the Bragg direction, and in such circumstances, becomes indistinguishable from it. At all other angles, they maintain, the maximum should be smeared out, and no longer appear as one sharp spot. Nevertheless, the anomalous spots which they themselves observe are still fairly sharp even when the crystal is set as much as 10° from the position appropriate for classical scattering. Third, since the diffuse scattering depends on the thermal vibrations of atoms, the scattered intensity should be sensitive to changes in the temperature of the crystal. 2 3 FIG. 2. Three examples of diffuse reflection of x-rays. Copper K radiation with a weak background of general radiation was incident at glancing angles of about 36°, 31°, and 28° on a 100 cleavage face of rocksalt in strips numbered 1, 2, 3, respectively. The diffuse 400 reflections of the Ka-and K(3-rays are indicated by a and (3. The 400 Laue spots are labeled CL. The narrow lines on the ex treme left are fiducial marks at the positions of the 200 Bragg reflection of the Ka line. (Courtesy of G. E. M. ]auncey.) However, their observations with ·diamond crystals at ordinary and at liquid-air tempera tures show very little dependence of the in tensi ty upon temperature. This report may well be contrasted with that of Baltzer,5 who finds, m the case of rocksalt, a strong dependence of the diffuse spot intensity on temperature. VOLUME 13, JANUARY, 1942 In the discussion of these experiments and theories, J auncey and others bring out some significant points. ".According to Zachariasen the clastic waves are a consequence of the thermal agitation of the atoms in the crystal as in Oebye's theory of the specific heats of solids. According to Raman and Nath the clastic waves are excited by the incident x-rays by means of a quantum process," but it seems to be agreed by others that these investigators offer no indis putable proof of the change in frequency which is a basic part of their theory. An approximate equation which gives the positions of the modified reflections, and which fits experimental results well for rocksalt, for diamond, and for sylvine "can be obtained by at least two-if not three-different theories," and therefore cannot be used, unless in a more exact form, to force a decision. Kot only will it be necessary to give more~study to the intensities as well as to the positions of the spots before a choice between the theories can be made, but the theories themselves will have to undergo some reinterpretation. In a recent letter to Nature, K. Lonsdale6 gets right to the heart of the matter. According to her, the reason why many different assump tions (the existence of small groups of atoms, of waves of the Raman type, of elastic heat waves) give in the various theories the same formula for the displacement of the anomalous spot from the center of the Bragg reflection, is that in the derivation of the formula it is assumed that the spreading of the intensity of reflecting power around each reciprocal lattice point is inde pendent of direction. The simple formula usually quoted is just a geometrical way of expressing the fact that near the reciprocal lattice points the distribution is spherical. Actually, at greater distances the distribution is far from spherical, though the actual shape is not known. Even Raman and Nath's more general formula is simply a way of expressing a geometrical relationship. The interpretation of the observa tions according to one theory or the other is therefore a matter of choice, for it cannot prove that anyone is to be preferred. \Vhat is needed is more information concerning the effects of elasticity, crystal perfection, temperature, etc., on the positions of the anomalous spots; and 5 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16some increase in accuracy, for at the moment of writing the agreement of calculated with ob served positions is termed excellent when no difference greater than 5 or 10 minutes of arc is found. II New ways have been found of accomplishing the fission of heavy atoms. From the University of Rochester, Dessauer and Hafner7 report the fission of uranium and of thorium by protons. In their principal experiments, they allowed a beam of 6.9-Mev protons from a cyclotron, shown in Fig. 5, to fall on a 10-mil sheet of FIG. 3. Laue pattern of rocksalt, for comparison with Fig. 4. Some diffuse scattering is visible in the form of streaks, due to the multiplicity of wave-lengths in the radiation. (Courtesy of N. S. Gingrich.) thorium, which was too thick to be penetrated either by the proton beam or by fission frag ments. If protons caused fission, therefore, only the front face of this 10-mil thorium sheet could release fission fragments. It was necessary, however, to anticipate the possibility that the impinging protons might cause a proton-neutron reaction in the heavy metal. The neutrons thus created would, of course, be able themselves to cause fission. However, since neutrons can pene trate even such a heavy metal as thorium with some ease, any neutron-induced fission would take place almost anywhere in the 10-mil thorium sheet. As a result, if the fission in this experiment was a secondary process due to the incidental creation of neutrons, then fission products could be collected not only from the front of the 6 thorium sheet, but also from the back. Dessauer and Hafner found that no fission fragments emerged from the back of the 10-mil thorium sheet, but from the front there came recoiling fragments which could be collected on a suitable aluminum foil placed a short distance away. As might be expected, the decay curve of the recoil fragments caught on the aluminum foil was such as to indicate the presence of many different periods. The threshold energy for thorium fission by protons was found to be at 5.8 Mev. Since the specific charge of alpha-particles is only half that of protons it would not be reason able to expect the former to be able to enter the nuclei of heavy atoms unless their energies were considerably in excess of the threshold quoted above for protons. The potential barrier of a uranium atom, for an incident alpha particle, is estimated to be nearly 30 Mev, an energy which happens to be within the range of the 60-inch cyclotron of the Crocker Radiation Laboratory. Using this instrument, Fermi and Segrc8 bombarded ammonium uranate with 32- 1Iev alpha-particles, finding among the products several tell-tale isotopes of iodine and tellurium which definitely establish the fission of uranium by this bombarding agent. Xeutrons, deuterons, gamma-rays, protons, alpha-particles-an impressive list containing all the simple massive particles-are now known to be effective in splitting heavy elements. The probabilities of the separate processes do not differ by more than a few powers of ten, as Table I shows. The detection. of the smaller cross sections listed is almost at the limit of present-day technique. The fission fragments of uranium (235) under impact of slow neutrons include both stable and unstable isotopes of the clements numbered 35 to 43 and 51 to 57, inclusive. Those which are unstable decay as beta-rayers, thus constituting a number of radioactive series. Decay periods have been measured with some accuracy, and in about half of these cases, chemical analyses and information from the field of artificial radio activity have led to the identification of the nature and the isotopic number of the element responsible for a particular rate of decay. Until JOuRNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16the work of Anderson, Fermi, and von Grosse9 was published, however, only the roughest qualitative estimates were available concerning the proportions of fission fragments which followed eg.ch of these radioactive series. These TABLE I. Experimental cross sections for fission. Energy Particle Mev Neutron Thermal Neutron 0.2 Neutron 0.5 Neutron 1-10 Neutron >10 Proton 6 Deuteron 8 Deuteron 9 Alpha 32 Photon 6.3 Photon 17 U em' 3 X 10-24 1 X 10-25 6X 10-26 5XlQ-25 7X 10-25 > 1 X 1Q-28? 4X 10-27 2.2XlQ-26 3.5X 10-27 3 X 10-27? Th cm2 1 X 10-25 1.4 X 10-25 1 X 1Q-28? 3XlQ-27 1.5 X 10-26 1.7 X 10-27 1.5 X 1Q-27? investigators separated one convenient member from each of a number of these known radio active series, and compared the {3-ray counts from a known quantity of each. The probability of the occurrence, or branching ratio, of a particular radioactive series from the slow neutron fission of uranium could then be found. As an indication of the results, it will be recalled that the following three sequences, among many more, occur in the decay of the fission products: 51Sb127 -(80 hr.)----752Te127 -(10 hr.) ----7531127 (stable), 52Te131- (25 min.)----753 P31-(8 days) ----754Xe131 (stable), 54Xe139- (30 sec.)----755CS139- (6 min.) ----756Ba139- (86 min.)----757La139 (stable). The branching ratios turned out to be 0.18, 1.6, and 6.4 percent, respectively. All told, these three workers investigated 9 of the 12 series known among the heavy products of fission, and one among the light products. The sum of the branching ratios of all 12 of the heavy product series should be 100 percent. The actual sum of the 9 which were examined comes to only about 50, so the remaining 50 must be accounted for ·either by the 3 not studied, or else by other series not yet discovered. \\Then neutrons of energy less than 10 Mev are used to bombard uranium, the resulting fission fragments all have atomic numbers between 35 and 57, but none occur with atomic numbers 44 VOLUME 13, JANUARY, 1942 to 50, incl usive. This unsymmetrical fission will undoubtedly have important theoretical implica tions, particularly in conjunction with the recent experimental findings of Nishina, Yasaki, Ki mura, and Ikawa.l° These authors find that with increased neutron energy, the fission may take place in a more symmetrical manner, though the proportion of symmetrical fragments is relatively small even with the 17 -Mev neutrons from the Li(d, n)Be reaction. Among the elements now found in the 44-50 gap (and these have been checked by Segre and Seaborgll) are 46Pd111 and 46Pdl12 which decay as beta-rayers through 47Ag111 and 47Ag112 to stable 48Cdlll and 48Cd1l2, respec tively. In addition, the Japanese workers found Ru (44) and Rh (55) fragments after bombarding pure uranium oxide with more than 100 micro ampere hours of fast neutrons. To achieve this identification it was necessary first to separate rhodium (as a metal) and ruthenium (as a sulphide) from the sample of uranium oxide. Then each of these was carefully purified to be sure that none of the more abundant fission products such as molybdenum, palladium, silver, cadmium, antimony, tellurium, iodine, cesium, barium, and lanthanum remained. When this was done, the rhodium fraction showed one 34-hour period, while the ruthenium showed two, of 4 and 60 hours. Since Nishina and his col leagues also found an indium isotope, the only element of atomic number 35 to 57, inclusive, not yet reported among the fission fragments is Sn (50). FIG. 4. Companion pattern to Fig. 3. Monochromatic Mo Ka-radiation was used in place of polychromatic radiation. The spots here are due to diffuse scattering of the monochromatic radiation. They are not Laue spots. (Courtesy of N. S. Gingrich.) 7 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16I t has been known for more than two years that the fission fragments of uranium fall into two groups, as judged by their ability to ionize gases. Corresponding to this method of grouping, the fragments were found to have ranges in air of 1.S and 2.2 cm. The researches which have been conducted under Bohr's guidance at Copenhagen show that a similar distinction between the light and heavy fragments can be made from cloud chamber,observations. It might be thought that there should be no ambiguity in deciding from cloud-chamber pictures, such as B¢ggild, Brostr¢m, and Lauritsen12 have made, which tracks belonged to one group and which to the other. But there are two difficulties which prevent such a direct identification. First, though the most commonly occurring fragments have masses in the ratio of about 3 : 2, yet the mean ranges in air of these particles appear to be more nearly in the ratio 4 : 3. The inherent stopping power of a thin layer of uranium, though amounting to less than ten percent of the range, nevertheless tends to make the grouping less obvious. Second, in each group the straggling, which may be described as the incidental variation from the mean range, is large, much larger than for alpha-particles. In classifying the fragments, therefore, the actual length of a track is of little value and some other criterion must be used. It turns out that the number of 8 branches on the tracks, in a specified section of each, is a reliable guide in separating the groups, the heavier fragment having a shorter track and more branches than the lighter. The figures for the ranges derived from the cloud-chamber observations reduced to standard air by the rules which are applicable to alpha-particles are 19 and 2S mm (in argon) and 23 and 30 mm (in helium). Setting beside these the 1S-and 22-mm ranges quoted above, it is clear that there are still some discrepancies to be explained between the relative stopping powers of different gases for the fission fragments, although the main features13 of the slowing down process of these heavy particles have been well established. The act of fission by neutrons is presumed to proceed along the following lines. A neutron is temporarily captured by the heavy nucleus, raising it to an excited state. Then, one of two FIG. 5. 7-Mev protons emerg ing from the Rochester cyclotron. A short distance before the end of its range in air the beam is par tially intercepted by a fluorescent screen. (CourtesyofG. Dessauer.) things may happen; either a neutron is emitted, or the nucleus divides. \Vith variation in the incident neutron energy, and therefore in the state of excitation, the relative probabilities of these two happenings will change. In particular, if a large incident neutron energy elevates the U239 nucleus to a high state of excitation, neutron emission will occur, but it may also happen that the remaining U238 nucleus is still in such a high state of excitation that it will itself undergo fission. ~ow the higher the state of excitation of JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16U238, the more likely is fission to occur in com parison with neutron emission. Therefore, if we admit the possibility of fission of both U239 and U238, an increase in the cross section for fission may well be expected with increasing neutron energies. The available experimental evidence bears only indirectly on the details of the fission process, but the scheme outlined hereI4 accounts well for the observation of Ageno, Amaldi, and othersI5 that the cross section for fission in uranium by neutron impact remains practically constant for neutron energies from about 1 Mev to 10 Mev, but that it increases considerably for neutrons with still higher energies. A sum mary of these results is found in Table I. III Carbon has two stable isotopes, CI2 and C13, whose' abundancesI6 are 98.9 and 1.1 percent. Using these figures we find that CI2 is about 90.0 times as abundant as C13. A mass-spec trometer study of the relative abundances of C12 and CI3 recently completed by Murphey and NierI7 discloses however, that the CI2jCI3 ratio varies from 89.2 to 93.1 depending on the source of the carbon. There is no doubt that limestones, whatever their ages, give low values; that carbon from plants is unusuaIly rich in C12; and that in meteoric carbon intermediate values occur. But it is too early yet to hazard an explanation of the significance of the variations. Of the radioactive isotopes of carbon, the most useful as far as applications to other fields of science are concerned is Cl4, an unstable atom which decays by emitting beta-particlesl8 of energy about 145 kev. It can be manufactured with deuterons according to the reaction Cl3(d, p )Cl4 or by the action of slow neutrons on nitrogen, Nl4(n, p)Cl4. The beta-particles, having a range of nearly 20 mgjcm2 in aluminum, equivalent perhaps to 0.2 mm of plant tissue, are energetic enough to be detected by Geiger counters even if produced in moderately thick specimens. The period is so long that no appreci able decay has been detected in a preparation 9 months old. Ruben and Kamen estimate the half life to be between 103 and 105 years. Hence this 04 isotope should prove very useful for tracer work in living organisms. All nuclei absorb neutrons, but with a facility VOLUME 13, JANUARY, 1942 which depends markedly on the velocity of the neutrons. In generctl, slow neutrons are absorbed more readily than fast neutrons. There are also particular narrow ranges of energy throughout which, for a given nucleus, the probability of neutron absorption is extremely high. When this occurs, the capture is described as a reso nance process. A beam of fast neutrons, reason ably homogeneous as regards velocity, can now be created without great difficulty from one of a number of nuclear reactions such as Li(d, n)Be. Of course, neutrons can be passed through hydrogen-containing materials and the emerging neutrons, called thermal neutrons, having suf fered numerous collisions with particles of the same mass as their own, are slow moving ones. But the very process of slowing them down spreads their velocities into a Maxwellian distribution such as is found in the molecules of a gas. The production of a homogeneous beam of neutrons of very low energy (say 0.1 electron volt) therefore requires the use of some kind of velocity spectrometer, so that a beam with velocities in a narrow range may be selected. Methods of achieving the desired result by modulating a high voltage source of ions have already been referred to in this series of articles, and convenient references to the original work may be found in a paper by Baker and Bacher.19 These authors used beams of slow neutrons of well-defined energy to investigate resonance processes. Cadmium, for example, has for several years been known to absorb thermal neutrons very easily. Baker and Bacher show that this strong absorption is due to a resonance phe nomenon with a maximum probability of ab sorption at 0.14 ev. The position which they find for this maximum in cross section is much more precisely determined, but of the same order of magnitude as that given by other earlier investi gators. Cadmium has eight stable isotopes altogether, of which only two have relative abundances less than 5 percent. It seems probable that the resonance absorption of 0.14- ev neutrons is due either to Cd111 or to Cd1l3, which are present in ordinary cadmium to the extent of only 13.0 and 12.3 percent, respectively. The cross section at resonance turns out to be 4.2 X 10-20 cm2, an extraordinarily high value for such a process, as a glance at Table I will show. 9 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16IV At last a gratifying semblance of orderliness is apparent in the deductions which can be made from the various cosmic-ray experiments that have been performed during the past decade. Many of the older tentative hypotheses con cerning the nature of the primary radiation can now be abandoned in favor of one which provides a logical background for the interpretation of numerous counter experiments. The present concept of the incoming primary radiation and of its subsequent behavior in passing through the atmosphere arises directly from the experi ments of Schein, Jesse, and Wollan,20 and of Swann21 and his collaborators, though it should not be forgotten that the most recent work has been guided by the experience of a host of earlier investigators. An admirably clear state ment of the present situation has been made by Swann, who emphasizes the following points. (1) "There is only one type of primary radiation, a charged particle radiation-probably protons --comprising particles of heavy mass." This is an assumption without which many contradic tions appear in the interpretation of the counter experiments. (2) "By processes at present unknown, the primary radiation gives birth, probably indirectly, in the upper atmosphere, to mesotrons." (3) "Those mesotrons which are born approximately at rest will have such short lives22 [and low speedsJ* that they will disinte grate before they have travelled more than 300 meters. They will, in fact, disintegrate in the stratosphere, and in so disintegrating, will give rise to electrons which, on account of the disintegration occurring from mesotrons at rest, will emerge on the average equally in all direc tions. (4) The mesotrons formed with higher energy will disintegrate at lower altitudes, because of their longer lives [and rapid motion],* and because they disintegrate at high energy, will give rise to electrons which possess on the average a forward component at these lower alti tudes." We shall try to sketch the most straigh tforward experimental evidence which is available in support of these statements. Near sea level, * Phrase in brackets added by the present reviewer. 10 counter-tube "telescopes" show the intensity of cosmic-ray particles of 'all kinds arriving verti cally at any particular locality to be much greater than that of cosmic rays arriving from near the horizon. At higher altitudes, however, the maximum of intensity in the direction of the zenith is much less pronounced. In other words, the directions of travel of these cosmic-ray particles are more random at high altitudes than at low, where a strong concentration in the vertical direction is observed. The statements (3) and (4) above are consistent with these experimental findings. If there were any appreci able proportion of electrons in the primary incoming cosmic rays, an intensity maximum should occur in the zenith direction up to the highest altitudes at which observations have been made. In the absence of such an effect, the most obvious deductions are that there are no such incoming electrons, and that cosmic-ray electrons are produced in the atmosphere, in agreement with point (2) above. Fairly direct evidence for the existence of penetrating massive primary particles is offered by observations noted below. At the moment, the production of mesotrons by these primaries is merely an inference made necessary by the knowledge that mesotrons have but a transient existence. The formation of mesotrons by protons has not yet been observed in the laboratory. Perhaps the new Berkeley cyclotron will provide protons with energies in the range where the process occurs with a reasonably high probability. Conclusions of essentially the same nature have been reached by Schein, Jesse, and Wollan from their studies of the vertical intensity and the production of mesotrons at high altitudes. Their experiments measured the intensity of the hard component up to heights at which the barometric pressure was only 2 or 3 cm of mercury. With different arrangements of coun ters, like that shown in Fig. 6, these penetrating particles were recorded only after they had passed through thicknesses of lead varying from 4 to 18 cm. Strangely enough, the intensity was not appreciably affected by variation in the thickness of lead and the intensity curve rose steadily as a function of altitude, showing no signs of a maximum followed by a rapid fall near the top of the atmosphere, as earlier experiments JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16seemed to show. "Because of the constancy of the penetrating power of the particles we measure," say these three authors, "and because they are not shower producing, we conclude that there are no electrons of energies between 109 and 1012 ev present at the highest altitudes reached. Since the energy required for electrons to penetrate the earth's magnetic field of 510 geomagnetic latitude is about 3 X 109 ev, and since our measurements were carried out to within the first radiation unitt from the top of FIG. 6. Automatic cosmic-ray counting apparatus carried to very high altitudes by balloons in the experiments of Schein, Jesse, and vVollan. The two vertical rows of per forated cylinders on the left are covers of amplifying tubes. To the right of these are seen the shielded ends of six counter tubes. The striped bundle at the lower right is a package of dry batteries. Just above them is an air-tight metal cylinder enclosing the high voltage supply. At the lower left is a metal drum containing a camera for record ing cosmic-ray intensity, barometric pressure, and tem perature. The whole frame is covered during a flight with Cellophane and silver paper to keep the temperature inside reasonably constant. (Courtesy of \V. P. Jesse.) t One radiation unit is the average distance in matter which an electron goes before losing half of its energy through radiative processes. This distance is about 0.4 cm in lead or 275 meters in standard air. VOLUME 13, JANUARY, 1942 the atmosphere, it seems difficult to assume the presence of electrons (E < 1012 ev) in the primary cosmic radiation, and hence they must be replaced by some penetrating type of charged particles. The mesotrons themselves cannot be the primaries because of their spontaneous disintegration. Hence it is probable that the incoming cosmic radiation consists of protons." In support of these views they quote further evidence which shows that mesotrons are pro duced in multiples mainly by ionizing non shower-producing particles (mesotrons can also be produced by non-ionizing radiation); and that the number of penetrating particles ob served in their counter experiments near the top of the atmosphere is approximately the same as the number of primary particles deduced from ionization chamber measurements. The' problem of the nature of cosmic rays is, however, still far from solution. A note by Cocconi23 points out that the rather simple picture which has just been given neglects some important points. Chief among these is that it fails .to account for the numbers of electrons which are found at high altitudes or even at sea level by at least an order of magnitude. It may be necessary then to assume that the primary protons, in addition to producing mesotrons singly or in bunches,24 create also the photons and electrons which produce large showers and a considerable part of the electron component. If this hypothesis be correct, the next step will be to find under what circumstances the various processes are most likely to occur. v Some comment on the lifetime of the mesotron is desirable, since this concept plays an important part in Swann's argument as stated above. Yukawa had employed such a particle in theoretical work a few years before it was found experimentally, and from the beginning postu lated that it must be unstable. After its recogni tion as an important component of cosmic rays several experimental estimates of its lifetime were made, the values given in 1939 ranging from 1.7 to about 3 microseconds. Mesotrons do not all have the same penetrating power, and therefore may be assumed to possess a range of 11 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16velocities, and the very fact of their motion will make their lifetimes appear different from those of mesotrons at rest. Rossi and HaI122 point out that if to be the lifetime of a mesotron measured in a frame of reference in which the mesotron is at rest, and t the lifetime measured in a frame of reference in which the mesotron is moving at {3 times the speed of light, then t=to(1-{32)-~, following one of the simplest rules of the principle of special relativity. If now L be the average distance traveled by mesotrons of velocity {3c from their creation to disintegration points, then L = c{3t = pto/ p. where p. is the rest mass and p 'the momentum of a mesotron. The probability of decay per centimeter path, 11L, is, therefore, inversely proportional to the momentum. Hence, theory predicts that fast moving mesotrons have a longer expectation of life than slow ~oving ones. The test of this variation in lifetime depends on the so-called "absorption anomaly." To explain the principle of this test, let us consider mesotrons moving vertically downwards over Colorado. At Echo Lake (alt. 3240 m) an observer might find how much the mesotron intensity would be reduced by passing through 200 g/cm2 of iron. At Denver, nearby (alt. 1616 m), another observer might find how much the mesotron intensity had been reduced after passing through the 1624 vertical meters of air lying between the two stations mentioned. The thickness, 200g/cm2, of iron is chosen because the difference between the average barometer readings at Echo Lake and Denver is 108 mm of mercury, or 147 g/cm2 of air. Using the best available theories, it can be shown that if mesotrons were stable, losing energy by collision processes only, then 147 g/cm2 of air is equivalent to 200 g/cm2 of iron. Should an experiment reveal that 200 g/cm2 of iron is not equivalent as an absorber to the layer of air between Echo Lake and Denver, then the discrepancy must be attributed to the decay of mesotrons. If the observations could be made on mesotrons of different momenta it would be possible to test the equation L=pto/p.. This is just what Rossi and Hall22 have tried to do. The only feasible way of distinguishing be tween mesotrons of different momenta is to see 12 if they can or cannot pass through specified thicknesses of material. Whether or not lead is actually used for the purpose, it is customary to express the ranges of mesotrons in terms of their penetration through this metal. By using appropriate thicknesses of iron and lead around and between their counter tubes, Rossi and Hall were able to investigate separately two groups of mesotrons: those with ranges between 196 FIG. 7. Equipment at Echo Lake under six tons of iron plates used as an absorber. (Courtesy of B. Rossi.) and 311 g/cm2 of lead, and those with ranges greater than 311 g/cm2 of lead. The type of experiment to be done determines roughly what these thicknesses should be, though their precise values have no particular significance. l\Iost people think of Colorado as a vacation land, but the handling of such thick absorbers, covering counters from 25 to 60 cm long, at the altitude of Echo Lake could scarcely be a restful contribu tion to a holiday (Fig. 7). For both groups of mesotrons it was found that the apparent absorption caused by the 147 g/cm2 of tenuous air between Denver and Echo Lake exceeded that caused by the compact 200 g/cm2 of iron at Echo Lake. This is merely an additional proof of what was already known, viz., that in passing JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16through an absorber a beam of mesotrons is reduced in intensity both by absorption and by loss of mesotrons due to decay. The latter process, though negligible in dense absorbers, makes an important contribution to the absorp tion in media of low density where the mesotrons travel long distances. The crux of the experiment, however, lies in the observation that for the slower moving mesotrons the excess absorption was 2.5 times as great as for the faster moving mesotrons. Now allowance must be made in this factor of 2.5 for the longer time required by the slower moving mesotrons to traverse the 1624 vertical meters of air between Echo Lake and Denver. An application of the Bethe-Bloch formula concerning the slowing down of moving particles led Rossi and Hall to the conclusion that the momenta of their two groups of mesotrons were roughly in the ratio 3 : 2. Such a ratio of mo menta (and therefore roughly of transit times) is not nearly enough to account for the whole factor 2.5 quoted above, so that the conclusion seems inevitable that the slow-moving mesotrons disintegrate in a shorter time than fast-moving mesotrons. Definite as this result appears to be, the experiment on which it is based will have to be refined considerably before it can be used to test quantitatively the theoretical dependence of a mesotron's distance of travel upon its momentum. The division of the cosmic-ray mesotrons into only two momentum groups can hardly be called resolving them into a momentum-spectrum, but it will stand as a notable achievement of 1941. Although the disintegration of mesotrons into electrons and neutrinos is an essential part of the description of the general behavior of cosmic rays as summarized above, yet the justification for this point of view has, up to the present time, been found in the simplification it intro duces into the general picture, rather than in the weigh t of direct experimental evidence which can be used to support it. The direct evidence, indeed, consists of a few photographs of cloud tracks, but much indirect evidence is available from experiments on the anomalous absorption of mesotrons in air and in solids. New direct evidence, such as is contained in a series of VOLUME 13, JANUARY, 1942 papers by Rasetti,25 is therefore of great im portance. Rasetti's experiments were performed with rather elaborate groupings of Geiger-Muller coincidence and anticoincidence counters. One group of counters recorded the absorption of mesotrons in aluminum or in iron, and another group was arranged to detect the delayed emission of other particles, presumably the disintegration electrons from the decaying meso tr:ons. This second group of counters could be made insensitive for various prearranged times, of the order of 10-6 sec., immediately after the tripping of the first bank of counters by a flying mesotron. At the end of these prearranged times, the second group of counters became, and remained, sensitive. A series of observations, with different delay times, thus furnished the distribution in time of the secondary particles. From it, Rasetti calculated the mean life of the disintegrating mesotrons as 1.5 X 10-6 second, a result in good agreement with other determi nations carried out by different methods. On the further details of this decay process we can do no better than quote Rasetti's own words. "The present experiments," he says, "seem to indicate a number of disintegrating electrons per mesotron definitely smaller than unity .... The results, however, are in agreement with the assumption that only half of the mesotrons undergo free decay. Since the analysis of meso tron tracks in a magnetic field has shown that there are about as many positive as negative mesotrons, or a small excess of positive, the result found is what should be expected if only mesotrons of one sign (positive) undergo free decay. Actually, if, according to the calculations of Tomonga and Araki, reactions with nuclear particles are much more probable than spon taneous disintegration for negative mesotrons, then we should only record an electron for each positive mesotron absorbed. The nuclear reac tions produced by negative mesotrons will probably lead to excited states of nuclei and eventually give rise to electrons through pro cesses of ,B-decay. It is exceedingly unlikely, however, that such particles could be emitted with sufficient energy and within a sufficiently short time to be registered in the present experiments." 13 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16VI "In the past, the acceleration of electrons to very high voltage has required the generation of the full voltage and the application of that voltage to an accelerating tube containing the electron beam. No convenient method f<?r re peated acceleration through a small potential has been available for electrons, although the method has been highly successful in the cyclo tron for the heavier positive ions at velocities much less than the velocity of light." So Kerst26 begins his paper on "The acceleration of electrons by magnetic induction." The emphasis here is on the word electrons. To bring out its signifi cance we must consider the repeated acceleration of charged particles as it takes place in the cyclotron dees. The operation of a cyclotron depends essentially on the fact that the time, t, taken by a particle of mass m and charge e, moving with velocity v, to trace a semicircle of radius r in a plane perpendicular to a uniform magnetic field H is given by t= (m/e) (7r/H). This expression is independent of v and of r; therefore in the same field all particles with the same m/e will describe semicircles, no matter what their radii have to be, in the same time, provided m/e does not vary with v. Since the charge of a particle may be considered constant, only the variation of m with v concerns us. For a 2-Mev alpha-particle, the mass differs only by about 0.05 percent from the rest mass, so the time taken to cover the largest semicircle in the cyclotron will differ from .the time required to cover the smallest semicircle by about this amount. The mass of a 2-Mev electron, however, is about five times its rest mass. Hence, the time required by such a light particle to cover the largest semicircle in a cyclotron would be about five times as long as the time required to describe the smallest semicircle. Hence the various por tions of an electron beam of high energy would be hopelessly out of phase in cyclotron dees. When the magnetic flux in a certain space is changing, an e.m.f. is set up in a loop of wire surrounding the space. The electric force is present, indeed, whether the wire is there or not. Suggestions and attempts to use this principle in accelerating electrons have been made from time to time during the last 15 years, but it is only 14 eighteen months since they were incorporated in a successful working apparatus built at the University of Illinois. Guided by the experience gained there, Kerst has just completed the new accelerator, shown in Fig. 8, in the Research Laboratories of the General Electric Company. We can give here but the barest outline of the apparatus, the data referring to the Illinois machine. A glass tube perhaps 5 cm across, in the shape of a ring about 20 cm in diameter (i.e., shaped like an American doughnut) sur rounds the closely spaced pole pieces of a lami nated magnet excited by a 600-cycle current. Electrons, liberated from a filament near the outer wall of the evacuated tube are accelerated in gradually contracting orbits to a circle of predetermined radius within the tube until they strike a target. The magnetic field between the magnet poles is not meant, at any instant, to be uniform. It must vary radially in such a way that electrons which happen to be circling a little too far out or a little too far in are brought quickly back to the proper orbit. To find out how the ring of moving electrons is finally made to deviate enough from its circular path to strike a target, the reader must turn to the original paper. We can merely indicate that it is accomplished by having certain parts of the poles reach saturation before others. The design of the magnet controls the success or failure of the instrumen t. During each revolution the electrons are accelerated by an amount equal to the instan taneous e.mJ. which would be induced in a wire placed at the position of the orbit. The accelera tion is completed during the first quarter-cycle, while the field is increasing, but the electron speeds are so high that perhaps 100,000 revolu tions are possible in this time, before the electrons strike the target. During the third quarter-cycle, acceleration of the electrons proceeds in the rev~rse direction, so that the target, bombarded from both sides, will emit x-rays intermittently 1200 times per second, but with a marked spatial asymmetry. Currents to the target in the Illinois instrument are about one-thirtieth micro ampere at 2.3 Mev. The induction accelerator is a promising source of high energy photons; and when some mechanical difficulties have been overcome, the electron beam will be brought out JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16of the accelerating chamber to provide a strong source of electrons for nuclear investigations, and it may soon be possible to duplicate some low energy cosmic-ray phenomena under con trolled conditions in the laboratory. In strong contrast to the modest dimensions of this new induction accelerator, Fig. 9 shows a recent photograph of the new giant cyclotron now under construction in California. This picture was taken on October 3, 1941. Half of the steel frame of the walls of the 24-sided building which will cover the cyclotron is already in place. When finished, it will be 160 ft. in diameter. The magnet itself consists of 3700 tons of 2-inch steel plates bolted together into a rec tangular frame, one side of which is below ground level. One bolt is allowed for every 10 ft.2 of surface. To guard against damage by earthquake the plates are also welded together at the edges. Each of the 92 plates in a horizontal member is 52 ft. long and 6.25 ft. deep, with a weight of 13.5 tons. This leaves a rectangular opening 45.3 ft. wide and 17.5 ft. high in the middle of which the pole pieces will be placed. Most of the lower pole piece is already in posi tion ; when the upper one has been assembled and hung in place, an air gap of depth 40 inches and diameter 184 inches will remain to accom modate the accelerating chamber containing the dees, To excite the magnet, nventy-four "pancakes" wound from copper strip 4 X 1: inch will be used, twelve around the upper core and twelve around the lower. These windings will be cooled by circulating oil. will be needed to complete the construction. The equipment is designed so that at the start 70-Mev deuterons should be produced, and it is expected that this energy can be raised to 100 Mev. At this latter energy the magnetic field required will be 10,000 oersteds, and the oscillator providing the dee voltage will operate at a wave-length of 40 meters. The instrument is being set up on the Berkeley campus of the University of California, at an altitude of 881 ft. above sea level. Those who are familiar with this campus may be able, from the picture, to verify that it is about 100 yards behind the Big C. I t is expected that ap proximately two more years FIG. 8. The new induction accelerator constructed in the Research Laboratories of the General Electric Company. 13-Mev electrons have already been produced by it, and higher energies are expected. (Courtesy of D. W. Kerst.) VOLUME 13, JANUARY, 1942 15 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16VII Those who have seen Darrow's racy review27 of the present knowledge of Helium, the Super fluid will be interested in two short papers by Kapitza and Landau28 concerning the thermo dynamic properties of this remarkable substance. It will be remembered that liquid helium can exist in two different forms, depending on the temperature, which are known as He I and He II. Of these, the latter is the colder, the transition temperature or A-point between the two forms having been established with some accuracy at 2.19°K. Chief among He II's re markable properties are its extremely low viscosity, lower than that of hydrogen gas if measured by conventional methods, and its ability to transport heat at a prodigious rate. The reasons for this abnormal "conductivity" are not entirely clear since they depend on the theoretical concept of the liquid's structure. It is certain, however, that next to none of the heat transferred is carried by the classical process of conduction. When a temperature gradient is maintained across a volume of He II, it seems that a strange counter current is set up, the liquid behaving as though it consisted of two separate components. There is a current of "normal" He II which carries heat from the hot region to the cold region, and a counter 16 current of "superfluid" liquid directed from the cold to the hot part. This differs from the ordinary mechanism of convection in that the return current of superfluid liquid carries no heat, and travels without friction. Thus, if two vessels of liquid helium II at slightly different temperatures are connected by a narrow tube, the pressures applied at the ends of the tube being the same, heat will be "conducted" from the warmer to the colder, through the liquid in the tube, nearly as fast as the warmer liquid could run away unimpeded through the tube if FIG. 9. The giant cyclotron now under construction on the University of California campus at Berkeley. The size of the mag net can be judged by the figures of two men working under and near it. (Courtesy of D. Cooksey.) the colder vessel were removed. If such a fantastic behavior seems to the reader to be beyond the bounds of reason, we hasten to explain that in order to bring out the differences between He II and ordinary liquids we have perhaps been guilty of an over-bold simplification of the process. Nevertheless, although the presence of a counter-current is, in the present review, based on a few qualitative remarks in Kapitza's and Landau's papers, yet its existence is in agreement with the observations of Daunt and Mendelssohn on the passage of He II through a porous plug, and with those of Allen and Jones29 on the fountain effect. Landau's concept of the nature of He II, as far as can be gathered from his brief paper in The Physical Review, is that it is a quantum liquid in which there can be no continuous JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16transition between the states of potential motion and vortex motion, but only a discrete finite jump. The identification of the superfluid with the lower state of potential motion, and of the normal He II with vortex motion is then a natural development. Not until the liquid is in the higher vortex state can there be any possi bility of dissipation of energy, and the motion of the superfluid is therefore thermodynamically reversible since there is no entropy change associated with its passage from one place to another. The quantization of the liquid is in terms of longitudinal compressional waves and must therefore be a kind of macroscopic quanti zation, quite different from any atomic or molecular quantization. At temperatures above the X-point, of course, the superfluid state of He is not present (or should one say, not occupied?). But it is only fair to state that the two types of liquid appear first in the mathematical formula tion of the problem. How much their physical counterparts have independent existence is a matter for further experiment. VIII The electrical failure, and the causes of elec trical failure in crystals, glasses and other insulators are matters of some importance in the field of applied physics, but it is only recently that experimental results on electrical breakdown have been consistent enough to warrant their use as a test of present-day theory. Two papers by von Hippel and Maurer,30 and von Hippel and Lee31 give a readable summary of the present situation. \Vhen a crystal such as KBr is broken down by a gradually increased electric field it is found that the voltage at which failure occurs depends upon the temperature of the specimen. From about 3 X 105 volts/cm at -200°C to about 8Xl05 volts/cm at +60°C the breakdown field increases more or less smoothly. But above this higher temperature the field necessary to cause breakdown grows smaller, drppping to 5 X 105 volts/cm at 200°C, On the hypothesis that breakdown is due to a cumulative ionization-by-collision process within the crystal, a qualitative explanation of the behavior of KBr in the lower temperature range can be given in terms of the motion of electrons VOLUME 13, JANUARY, 1942 through the lattice. When the crystal is cold the small thermal energy of vibration keeps the lattice points relatively undisturbed, so that electrons pass easily; but with increasing temper ature the agitation of the lattice grows more violent, and electrons, formerly nearly free to run through the periodic field, are now much more likely to be scattered from their original direction. The scattering thus introduces a more frequent "stop and go" motion among the electrons and higher applied fields are required to cause ionization by collision. I t seems necessary, however, if the same kind of theoretical explanation is to be used for the higher temperatures as for· the lower, to assume further that the number of secondary electrons needed to constitute a breakdown current can be formed at lower fields, if the temperature is higher. Another way of saying this is that the number of free secondary electrons per cm path produced by collision must be an increasing function of the crystal's temperature. Here, already, the fundamental ideas of Townsend's theory of ionization by collision in gases are being extended to situations they were not de signed to cover. Some objections may be raised against such an extrapolation, but nonetheless it is profitable to pursue this line of thought as far as possible. Von Hippel and Lee suggest that in addition to simple ionization by collision, two other effects should be taken into account. These are, first, the possible capture by atoms of slow electrons, just after their liberation, to make negative ions. This would diminish enormously the mobility of the charges so captured, making them no longer effective in liberating new free electrons by collision. There is fairly good evidence from absorption spectra that such a process occurs, but it will be prevented from gaining control by the re-liberation of the elec trons from the negative ions by thermal vibra tions of the crystal. Briefly, then, Townsend's theory of cumulative ionization gives a fair explanation of the increase of breakdown voltage with increasing temperature, as indeed, Froh lich32 and others have shown; but in order to account for the subsequent decrease in break down voltage at somewhat higher temperatures, auxiliary assumptions along the lines suggested by von Hippel and Lee are required. This more 17 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16elaborate theory does not yet appear to have been worked out in any detail. Quite a different situation is found in studying the electrical breakdown of a permanently disordered substance like glass. The most interesting case which von Hippel mentions is that of silica glass, whose behavior he contrasts with that of a single crystal of quartz. The latter, in the range -80°C to +60°C, shows a variation in breakdown voltage continually increasing with temperature, the magnitude depending slightly on the direction of the crystal axes; but the amorphous silica glass, over the same temperature range, shows an ever-quick ening diminution in breakdown voltage remi niscent of the behavior of a crystal in which disorder of thermal origin is already well ad vanced. The orderly arrangement of a crystal la ttice is therefore characterized by a breakdown voltage with a positive temperature coefficient, the amorphous state by a negative coefficient. IX A few years ago the magnetic moments of some nuclei were measured experimentally by an ingenious method developed by Rabi, Millman, Kusch, and Zacharias.33 For a' new type of apparatus the experiments possessed a highly creditable precision, the error being of the order of a few tenths of one percent. The LF nucleus, for example, was found to have a magnetic moment of 3.250 nuclear magnetons. The method used by these four workers involved a combina tion of magnetic fields, inhomogeneous, homo geneous, and oscillating, the accuracy of the final result depending on the precision with which the value of the steady field was known, which had to be measured in more or less standard fashion by flip-coils. It turned out that the lines in the radiofrequency spectrum could be located with an accuracy far in excess of that available for measuring the magnetic field. Therefore, to make the fullest use of this pre cision another way of measuring the field had to be developed. When an atom emits radiation while it is in a magnetic field, the spectrum shows a Zeeman pattern, a splitting of normally single lines into various components whose arrangement can be 18 calculated if the strength and direction of the field are known. In atomic spectra there are some lines in whose Zeeman patterns the separation of the component lines depends strongly on the strength of the field used; and there are other lines whose Zeeman pattern separations are quite insensitive to slight changes in the field. Millman and Kusch34 reversed the usual pro cedure of determining a Zeeman pattern from the field and instead proceeded to measure magnetic field strengths from the separation of the Zeeman components. That is, Zeeman patterns were used to calibrate the magnetic field. This amounted essentially to comparing the magnetic moment of an electron in an atom, responsible for the Zeeman effect, with the magnetic moment of a nucleus. The latest values of the moment of the Li7 nucleus and of the proton, to quote only two examples, are 3.2532, and 2.7896. The latter value is measured, of course, from observations on the behavior of molecules like N aOH and KOH, which contain a proton, and derives its importance from the fact that it enables. a determination to be made of the moment of the neutron, by subtracting the proton's moment from that of the deuteron. The neutron's moment thus found can then be compared with the value obtained for free neutrons. The ques tion has been raised by Rarita and Schwinger, however, whether a simple subtraction of the two moments really gives the moment of the neutron. They suggest a rather more complicated way of calculating which leads to a slightly different result. The figures are: -1.933, by simple subtraction; -1.911, by Rarita and Schwinger's calculation; and -1.935, experi mentaJ.35 At the moment of writing, the accuracy is overwhelmingly in favor of Millman and Kusch's indirect method, and, therefore, to quote these authors, "it is highly desirable that the moment of the free neutron be measured with a precision comparable to that obtained for the difference" between the moments of the deuteron and proton. Another topic of perennial interest in the field of optics is the velocity of light. It has long been regarded as one of the fundamental constants of nature which plays an increasingly important part in atomic physics, yet its absolute value is JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16not known to better than one part in 20,000. Suggestions have been made, on the basis of the experimental results of the last three quarters of a century, that this velocity may be a slowly varying periodic function of time, but determi nations made before 1900 possess so large a probable error that these suggestions can hardly be taken seriously. Prior to 1936, all the meas urements which ha:ve any claim to reliability involved the visual measurement of the displace ment of a spot of light, or the manual control of the return of a spot of light to a prearranged position. Most physicists will therefore read with interest a paper by W. C. Anderson36 in which he summarizes his experimental values of the velocity of light determined by a purely automatic method. Personal errors in this experiment could come in only in the measuremen t of the films on which the records were made. into voltages which are amplified and recorded. It can be shown that the resultant voltage is dependent upon the phase relation of the orig-inal lig-ht modulations and this voltage will be either a maximum when the beams are .in phase, or a minimum when one beam is an odd number of half-cycles behind the other. By noting the path difference for a given minimum position, the velocity of light is readily computed by the relation: c=2fs/n, where n=the number of half cycles phase difference, s = the optical path difference between the two light beams, f = the The method can hardly be described more briefly than in Anderson's own words. "A light beam is passed through a modulator where it is made to vary sinus oidally in intensity about some steady value. From the modulator the beam passes through a half-silvered mirror, a portion being re flected from the surface over to a movable mirror. From this mirror the beam is re turned, passing through the half-silvered mirror to a photoelectric cell. The other portion of the original beam transmitted by the half silvered mirror passes over a much longer path and is re turned along the same path, being reflected this time from the half-silvered mirror over to the same photoelectric cell. A tuned circuit converts these photoelectric currents FIG. 10. The Smith-Putnam Wind Turbine on Grandpa's Knob in the Green Mountains, Vermont. The dimensions of the structure may be estimated from the figure of a man near the wind-indicating instruments at the left center of the picture. (Courtesy of J. B. Wilbur.) VOLUME 13, JANUARY, 1942 19 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16frequency of modulation of the beams, and c = the velocity of light." In this velocity determination, just as in the majority of earlier experiments, the directly measured velocity was not the velocity in free space, but rather the velocity in air. Therefore the measured values had to be corrected, as the phrase goes, to vacuum, using the known refractive index of air. Furthermore, the directly measured velocity was the group velocity whose magnitude depends upon the relative speeds of disturbances of different wave-lengths in the medium. A satisfactory reduction to vacuum, in which there is no dispersion, thus requires a knowledge and use of the dispersion caused by air. Anderson seems to have paid more attention to this group velocity correction than his predecessors did, justifying his procedure by the statement that in some cases the correction amounted to as much as 7 km/sec. The final result of his experiments gave the' velocity as 299,776±14 km/sec. In 1935 Michelson, Pease, and Pearson reported the velocity to be 299,774 km/sec., but the closeness of this to Anderson's value must be regarded as somewhat fortuitous, since the 1935 experiments revealed a monthly periodic variation of velocity with time, of the order of 20 km/sec., an effect which, if it be real, has thus far received no satisfactory explanation. x The winds and the tides, the most restless of the elements, are nevertheless among the most difficult to harness on a commercial scale. For some time now an interesting experiment in applied physics, a serious large-scale attempt to derive useful electric power from the wind, has been under way on a hill in Castleton, Vermont. This is the Smith-Putnam Wind turbine on Grandpa's Knob in the Green Mountains. Although no technical details are available for publication, we believe that a professional eye will discover much that is scientifically inter esting in the picture reproduced in Fig. 10. We have no information as to its performance, but it is said to have been constructed to feed into the regular electrical power network in Vermont. Teachers of physics may like to use it in problems on angular motion in mechanics; and their 20 students may be surprised to find that if the speed of the turbine were 18 r.p.m., and if it were developing 1500 hp, then the main shaft would need to have about the same diameter as the propellor shaft of a 20,000 ton ship, and the wing tip would be traveling at 100 mi./hr. It is obviously unnecessary to review in any detail the many important papers or groups of papers which have appeared during the last year in the Journal of Applied Physics. In the tenth of a series of articles on lubrication contributed by the Gulf Research and Development Com pany, Morgan, Muskat, and Reed37 study the curious stick-slip process which occurs when one solid slides over another. This phenomenon, as recorded in these reviews some years ago, was first reported by Bowden and Leben, who found that the process of slipping was an intermittent one, and that during the slip-stick cycle, con siderable local heating occurred. The explanatory mechanism suggested by Bowden and Leben, and the observations which they made have been examined and tested in the current paper by Morgan, Muskat, and ~eed. Not only is the problem an interesting one for experimental investigation, but there is ample scope for new theoretical work by reason of the compressional waves which must inevitably be set up by such a sequence of rapid jerks. Historians of the future, we believe, will choose the cyclotron and the electron microscope as the most notable new instruments of the age. Both are remarkable because they contribute to several fields of science. By its massive design and large dimensions, the cyclotron has un doubtedly captured popular imagination; but from a strictly technical point of view it might be argued that the electron microscope is a tool of wider application. Physicists, biologists, engi neers, chemists, bacteriologists-all have prob lems whose solution is, in a short time, almost assured. Papers will be found in earlier issues of this Journal describing how the instrument has been adapted to study thicker specimens than before ;38 to observe surface structure ;39 and to measure the thicknesses of tiny objects.4o By increasing the attainable resolving power of scientific instruments by more than an order of magnitude it has indeed become the herald of a new era. JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 199.212.67.168 On: Tue, 23 Dec 2014 01:46:16Under appropriate circumstances its performance transcends that of the best modern optical microscope by as much as the first optical microscope added to human vision. A poe't, perhaps, untrammeled by the traditional meticu lousness of the language of technical papers, could do justice to its possibilities. And while we wait for a poet, let us at least rejoice, in our inarticulate way, that we live when our science, too, is alive. The writer is indebted to several of his colleagues at Michigan State College for their helpful criticism of parts of the manuscript; and to those gentlemen who, in addition to providing illustrations for thi~ article, have aided him by friendly correspondence. BIBLIOGRAPHY (1) P. Kirkpatrick, Phys. Rev. 59, 452 (1941). (2) S. Siegel, Phys. Rev. 59, 371 (1941). (3) R. Q. Gregg and N. S. Gingrich, Phys. Rev. 59, 619 (1941). Many other pictures can be found in Proc. Roy. Soc. A179, 8 (1941). (4) Nearly all the work dealing with this point can be traced from a few papers: C. V. Raman and P. Nilakantan, Phys. Rev. 60, 63 (1941); G. E. M. Jauncey and O. J. Baltzer, Phys. Rev. 59, 699 (1941); and W. H. Zachariasen, Phys. Rev. 59, 766 (1941); 59, 860 (1941). (5) O. J. Baltzer, Phys. Rev. 60, 460 (1941). (6) K. Lonsdale, Nature 147, 481 (1941). See also G. D. Preston, Nature, 147, 467 (1941). (7) G. Dessauer and E. M. Hafner, Phys. Rev. 59, 840 (1941). (8) E. Fermi and E. Segre, Phys. Rev. 59, 680 \1941). (9) H. L. Anderson, E. Fermi, and A. von Grosse, Phys. Rev. 59, 52 (1941). (10) Y. Nishina, T. Yasaki, K. Kimura, and M. Ikawa, Phys. Rev. 59, 323 (1941). See also reference 15. (11) E. Segre and G. T. Seaborg, Phys. Rev. 59, 212 (1941). (12) J. K. B~ggild, K. J. Brostr~m, and T. Lauritsen, Phys. Rev. 59, 275 (1941). (13) N. Bohr, Phys. Rev. 59, 270 (1941); W. E. Lamb, Jr., Phys. Rev. 58, 696 (1940). (14) N. Bohr, Phys. Rev. 58, 864 (1940). (15) M. Ageno, E. Amaldi, D. BocciareIli, B. N. Cacia puoti, and G. C. Trabacchi, Phys. Rev. 60, 67 \1941). See also reference 10. (16) J. J. Livingood and G. T. Seaborg, Rev. Mod. Phys. 12, 30 (1940). (17) B. F. Murphey and A. O. Nier, Phys. Rev. 59,771 (1941). , (18) S. Ruben and M. D. Kamen, Phys. Rev. 59, 349 (1941). VOLUME 13, JANUARY, 1942 (19) C. P. Baker and R. F. Bacher, Phys. Rev. 59, 332 (1941). (20) M. Schein, W. P. Jesse, and E. O. Wollan, Phys. Rev. 59, 615 (1941). (21) W. F. G. Swann, Phys. Rev. 59, 770 (1941). \22) B. Rossi and D. B. Hall, Phys. Rev. 59, 223 (1941). (23) G. Cocconi, Phys. Rev. 60, 532 (1941). (24) E. O. WoIlan, Phys. Rev. 60, 532 (1941). (25) F. Rasetti, Phys. Rev. 60, 198 (1941). (26) D. W. Kerst, Phys. Rev. 60, 47 (1941); see also D. W. Kerst and R. Serber, Phys. Rev. 60, 53 (1941). (27) K. K. Darrow, Rev. Mod. Phys. 12, 257 (1940). (28) P. L. Kapitza, Phys. Rev. 60, 354 (1941); I". Landau, Phys. Rev. 60, 356 (1941). (29) J. F. Allen and H. Jones, Nature, 141,243 (1938). (30) A. von Hippel and R. J. Maurer, Phys. Rev. 59, 820 (1941). (31) A. von Hippel and G. M. Lee, Phys. Rev. 59, 824 (1941). (32) See M. F. Manning and M. E. Bell, Rev. Mod. Phys. 12, 215 (1940). (33) I. I. Rabi, S. Millman, P. Kusch, and J. R. Zacharias, Phys. Rev. 55, 526 (1939). (34) S. Millman and P. Kusch, Phys. Rev. 60, 91 (1941). (35) L. W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940). (36) W. C. Anderson, J. Opt. Soc. Am. 31,187 (1941). (37) F. Morgan, M. Muskat, and D. W. Reed, J. App. Phys. 12, 743 (1941). (38) V. K. Zworykin, J. Hillier, and A. W. Vance, J. App. Phys. 12, 738 (1941). (39) L. Marton and L. I. Schiff, J. App. Phys. 12, 759 (1941). (40) V. K. Zworykin and E. G. Ramberg, J. App. Phys. 12, 692 (1941). 21 [This article is copyrighted as indicated in the article. 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1.1714934.pdf

Thermionic Emission from an OxideCoated Cathode H. Y. Fan Citation: Journal of Applied Physics 14, 552 (1943); doi: 10.1063/1.1714934 View online: http://dx.doi.org/10.1063/1.1714934 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/14/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in OxideCoated Cathodes Phys. Today 7, 30 (1954); 10.1063/1.3061648 Thermionic Emission from OxideCoated Tungsten Filaments J. Appl. Phys. 24, 49 (1953); 10.1063/1.1721131 Thermionic Emission and Electrical Conductivity of OxideCoated Cathodes J. Appl. Phys. 23, 599 (1952); 10.1063/1.1702257 Effect of Coating Composition of OxideCoated Cathodes on Electron Emission J. Appl. Phys. 21, 1115 (1950); 10.1063/1.1699552 Pulse Emission Decay Phenomenon in OxideCoated Cathodes J. Appl. Phys. 20, 415 (1949); 10.1063/1.1698387 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37Thermionic Emission from an Oxide-Coated Cathode H. Y. FAN Radio Research Laboratory, National Tsing Hua University, Kunming, China (Received May 18, 1943) The properties of thermionic emission from a cathode coated with barium oxide are studied. The cathode is indirectly heated and its temperature is measured by a thermocouple. It is found that the emitted electrons have a Maxwellian distribution corresponding closely to the temperature of the cathode. The emission is studied with accelerating voltages up to 1300 volts. The current rises more steeply than predicted by Schottky's theory but begins to bend at the upper end. The variation of the work function and the factor A with the state of the cathode is studied by glowing the cathode at different temperatures. It is found that both the work func tion and the factor A vary. No appreciable decay of emission with time is obserVed. The effect of drawing emission on the work function and the factors A is small. THERMIONIC emission from oxide-coated cathodes has been studied extensively. Many results are, however, contradictory and the properties of such emitters are not yet thor oughly clarified. We present in this paper the results of experiments made on a cathode coated with barium oxide. Emissions in retarding and accelerating fields were studied and the variations of emission constants with the state of the cathode were investigated. The experiments were made at low temperatures, so that the state of the cathode was not affected by the measurement itself. EXPERIMENTAL ARRANGEMENT For measurements of emission in a low field it is necessary to eliminate the potential drop along the cathode. In the case of oxide-coated cathodes the indirectly heated type is best suited for this purpose. The cathode used' consists of a nickel alloy tubing, 2 mm in diameter and 4 cm long, with a loop of tungsten wire, embedded in alumina, serving as the heater. The coating is barium oxide, applied in the form of barium carbonate with amyl acetate and a trace of collodion as binder. The collector is a nickel cylinder, 1.29 cm in diameter and 1 cm long, with a nickel guard ring at each end. The guard rings are of the same diameter and 1.S em long each. They serve to eliminate the end effect of the applied field and the effect of the non-uniformity of the cathode temperature at the two ends. The cathode temperature is usually measured 552 by one of three methods: (1) direct optical pyrometer measurement; (2) measurement of the heating power input; (3) measurement of the core resistance. The first method is not applicable at low temperatures. The second method involves determination of the total emis sivity of the coating surface, which varies with temperature. Besides, in the case of indirectly heated cathodes it is difficult to determine ac curately how much of the total heating power is dissipated through the central portion of the cathode where the temperature is different from the two ends. In using the third method there is also the difficulty of non-uniform temperature of the cathode. In the case of filaments potential leads could be used to measure the resistance of the central portion. When the cathode is in the form of a tubing as in our case, this method is not applicable. We measure the temperature of the cathode by means of a thermocouple. A tungsten wire is spot-welded to the inside of the cathode near the center. The wire is held under tension, so that it does not touch the cathode at other places except the welded spot. The temperature of the core is measured by the thermal e.m.f. of this joint. This method is not applicable to filaments because of the fact that the wire welded to a filament disturbs its temperature. The core of our cathode is tubing 2 mils in thickness and the tungsten wire used is 3 mils in diameter. It is thought that the cathode should be massive enough for the disturbing effect of the tungsten wire to be negligible. To check this assumption JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:3750 0 L. / V 0 V 0 L V o / II 10 10 30 J. T/'~rmQ/ emf (t.neJt,n caModeJ Nil/volts FIG. 1. Calibration of tungsten-cathode thermocouple. a tube is made with two wires spot-welded to the cathode, one on the inside and one on the outside, at two different points on the same circum ference. The outer wire should cool the welded joint whereas the inner wire, being nearer the heater, should heat the joint. For a wide range of the heater current the thermal e.m.£. of the two joints showed no difference. We feel, there fore, justified in using this method. The thermal e.m.£. of the tungsten-core couple is calibrated against a Chromel-Alumel thermocouple. For the calibration the two thermocouples are placed in an evacuated quartz tubing which is heated to different temperatures by an electric heater wound on the outside. To prevent any tem perature difference the joints of the two thermo couples are spot-welded together and their thermal e.m.f.'s are taken simultaneously. Figure 1 shows the calibration curve. The thermocouple measures the temperature of the core. The tem perature of the oxide surface should be lower. However, according to Moore and Allisonl the 1 G. E. Moore and H. W. Allison, J. App. Phys. 12,431 (1941). VOLUME 14, OCTOBER, 1943 temperature drop across the coating (thickness less than 100J..!) cannot be more than a few degrees. Furthermore; in such a composite cathode the emission could not be a function of the oxide surface alone but should depend upon the temperature of the bulk of the oxide also. In view of these considerations we shall take the thermal e.m.f. as measuring the "cathode tem perature.' , The structure of the tube is shown in Fig. 2. To seal to the glass envelope the thermocouple wires have to be welded to lead-in wires. The joints between these wires are removed far from the rest of the tube to prevent them from heating up. Figure 2 shows also the electric circuit used for measurement. The emission current flows through a high resistance in the grid circuit of the FPS4 tube. The potential drop across the resistance is balanced by the potentiometer Pl' There is, then, no potential difference between the collector and the guard rings during the measurement. The collector current is measured by the voltage supplied by the potentiometer Pl. The current is measured down to 10-12 ampere. The thermal e.m.f. indicating the cathode tem perature can be measured with an accuracy of 0.02 mv, which corresponds to less than 1°C. The accuracy of the temperature measurements depends, however, upon the accuracy of the calibration of the Chromel-Alumel thermocouple, for which the table given by Hoskins Manu facturing Company in the Handbook oj Chemis- FIG. 2. ~easure ment circuit. try and Physics is used. This table differs slightly from the table given in the International Critical Tables. 553 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37ELECTRON EMISSION IN A RETARDING FIELb If the conduction electrons in the cathode have a Fermi-Dirac or Maxwellian distribution and if the surface potential barrier allows all electrons with velocities (normal to the surface) above a certain value to pass through and reflects all electrons with velocities less than this value, then the emitted electrons should have a Maxwellian energy distribution corresponding to the tem perature of the cathode. Such results have been obtained for pure tungsten and platinum cathodes. For composite emitters, however, surface reflection may vary with the energy of the electrons in a more complicated manner. Indeed, Nottingham2 has shown that electrons emitted from thoriated tungsten do not follow the Maxwellian distribution curve corresponding to the temperature of the cathodes and that at low retarding fields the emission current has a much smaller slope. Nottingham explains this as due to surface reflection of the type R:=: e-E./c• In the case of oxide-coated cathodes Koller3 and Rothe4 found that, although the energy dis tribution of the emitted electrons was Ma.x wellian, it corresponded to a temperature much higher than the observed cathode temperature. This, in fact, is similar to Nottingham's result for thoriated tungsten. On the other hand, Davisson5 and Demski6 found that the electron temperature calculated from the energy distri bution curve agreed c1o~ely with the cathode temperature. The tube used by Rothe did not have suitable structure necessary for such measurement. The experiments of Davisson and Koller were not reported in detail. Demski measured the cathode temperature by an optical pyrometer and worked in the range 1110 to 14SQ°K. The electron temperature checked within a few percent with the temperature of the cathode, the difference becoming larger the higher the temperatures (13.3 percent at 14S00K). He explains this as due to variation of cathode activity with measurement. To avoid such trouble it is best to work at low temperatures. 'W. B. Nottingham, Phys. Rev. 49, 78 (1936). 3 R. Koller, Phys. Rev. 25, 671 (1925). 4 H. Rothe, Zeits. f. Physik 36,737 (1926). 6 L. H. Germer, Phys. Rev. 25,795 (1925). 6 A. Demski, Physik. Zeits. 30, 291 (1929). 554 This is possible with our method of measuring cathode temperature. Schottky7 has shown that if the emitted elec trons have Maxwellian energy distribution, then the relation between the current and the retard ing potential is given by Xd[(eV)iJ+f'" ceVlkTd[(eV)iJ}. (1) kT (peV/kT)i kT In our case p!=[R2j(R2_r2)]i= 1.011. Setting this factor equal to one, the second term on the left-hand side is the probability integral, the value of which can be taken from tables. If the ratio Rjr is very large, the integral in the first term could be set equal to one. We have Rjr= 6.43; such a' simplification would not be a good approximation. We calculate log i/io for different values of e VjkT, evaluating the integral by numerical integration. From such data theo retical curves for different values of T can be plotted. Figure 3 shows the experimental points and the corresponding theoretical curves for different temperatures of the cathode in the range 582 to 9S1°K. For curves 3,4, S, and 6, the saturation current was not measured. In fitting the experimental points to the theoretical curves we used the contact potential found from curves 1 and 2 to fix the relative horizontal posi tions, but the relative vertical positions were not fixed, the saturation currents not being known. The relative vertical positions of the experimental points and the theoretical curves shown for these curves are arbitrary. Each point was taken twice, on increasing and on decreasing voltage. In most cases the two readings differed so little from each other that, to avoid confusion, they are not shown separately. We see from Fig. 3 that in cases 1 and 2 the experimental points fall very closely on the theoretical curves. In cases 3,4, 5, and 6 we can at least say that the experimental points lie very nearly paralIel to the theoretical curves. Actu- 1 W. Schottky, Ann. d. Physik 44, 1011 (1914). JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37I / ! I I / i i v 1/ I II 1/ I / I : V V v v 17 I / J I. I l7 17' )1 11 j IJ I ---FIG. 3. Emission in retarding fields. I l2.~ /-..~ 1/', / T't,aq'Jo; l!,':'~~n"IC 7 / ~-+---- I V / r [7 l7 i ~ :/ I~ ! ." ." -j.O ·os ally, in cases 2, 3, 4, and 5 the experimental points have a slightly steeper slope giving the energy distribution of the emitted electrons a temperature lower than that of the cathode. The difference between the two temperatures is, however, less than 5 percent in the worst case. We conclude that the emitted electrons have Maxwellian distribution corresponding to the cathode temperature within experimental error. ELECTRON EMISSION IN AN ACCELERATING FmLD Assuming mirror image force between the emitter and the electrons Schottky8 derived the following relation between the emission current and the accelerating field E (2) The logarithm of i plotted against yE should be a straight line with the slope e'/kT. Experimen tally this relation does not hold at low fields, since the emission current rises much faster with in creasing field. The curve becomes a straight line with the predicted slope only at sufficiently high fields. For composite surfaces, such as thorium on tungsten, the departure from Schottky line depends upon the degree of activation, being larger for partially activated states. To explain such phenomena, the patch theory has been 8 W. Schottky, Physik. Zeits. 15, 872 (1914). VOLUME 14, OCTOBER, 1943 / v V r{".SU"K / I /' 1/ J 1/ . / ., , .. .. coltettar volta.ge: volts proposed. It takes into account the surface in homogeneity of the emitter due to different emission properties of the various crystal facets in the case of clean metal cathodes, or due to different amounts of adsorbed active material at various parts of the surface in the case of composite cathodes. Becker9 showed, by using a simplified model, that with the proper choice of the size of the areas of different activities and the degree of difference in activity a curve could be obtained to fit the experimental points. His calculatioIts show that the smaller the size of the areas the higher must be the field for the current to approach the Schottky line and the larger the difference in work function between the various areas, the larger is the departure and the slower is the approach to the Schottky line. Electron optical pictures10 of emission from oxide-coated cathodes actually show strongly varying inten sity. Thus one should expect departures from the Schottky line on the strength of the patch theory alone. Figure 4 shows the experimental curves taken at various temperatures. There is some irregu larity at small fields, the cause of which is not known. Above E= 150 volt/cm the curves are straight within experimental error. The slopes are, however, over three times higher than the values predicted by Schottky's theory. The range of the accelerating voltage was then extended to 9 J. A. Becker, Rev. Mod. Phys. 7, 95 (1935). 10 Benjamin, Huck, and Jenkins, Proc. Phys. Soc. SO, 345 (1938). 555 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:371300 volts. The results are shown in Fig. S. The curves are straight up to E=2500 volt/em, the ~ o.s o.lt ~ 1---+--+---t----;;:::>!II!or----t-~ ~. S ~'o!:---!---+---+. --+s---fl.,,:----;-l,t vVz votts'" FIG. 4. Emission" in accelerating fields. slopes are given in Table I, together with the corresponding values according to Schottky's equation. The slopes of the experimental curves are much higher than predicted by Schottky's TABLE I. Temperature Theoretical slope Experimental,slope Ratio of the OK loglO/voltl IOglO/volt< slopes 693 757 827 0.00635 0.00581 0.00536 0.0242 0.0207 0.0196 3.81 3.56 3.65 theory, but the ratios of two corresponding values are about the same for the three different tem peratures. At the higher end of the voltage range the curves begin to bend with decreasing slope. Unfortunately, the insulation of the tube does not permit carrying the measurement to much higher voltages to see whether the curves will approach the Schottky line. In this connection, indirectly heated cathodes are not the most suitable type to use for investigation. On account of the large diameter of the cathode, very high voltages must be used to obtain large fields. According to Benjamin, Huck, and J enkins10 the size of the areas with different activities cor responds to the size of carbonate particles from which the oxide coating is prepared. The particle size is usually of the order of 10-3 or 10-4 cm. It 556 is interesting to note that this is the right order of magnitude required by Becker's patch theory. VARIATION OF EMISSION CONSTANTS The oxide coating of a thermionic cathode is an impurity semi-conductor with excess Ba, Sr, or Ca atoms as impurity supplying electrons to the conduction band. Fowlerll has derived, on the basis of the theory of semi-conductors, the following equation for the emission current (8k5?rm)l i=De . n!T5/4E!(llEl+llE.)-llE./kT hI where n is the concentration of impurity atoms in the oxide, f1El is the energy gap between the bottom of the conduction band and the top of the next lower band (2), f1E2 is the energy dif ference between the energy level of the impurity atoms and the top of band (2), f1E", is the energy difference between the top of band (2) and the energy of an electron at rest outside the oxide, and D is the transmission coefficient of the oxide surface. In contrast to the emission from clean metals, A I is not a universal constant hut depends upon n. In fact, A for oxide-coated cathodes is very small compared with values for clean metals. This is not entirely due to D which may not be so greatly different from one. Thus :! 0 , S v . 0 ~V t---. vr V· ~ V ~ 1" ", /" /. V ~ V ~ l\1·1f / 5!~ / 1~"4, / II 1/ , " " " " " " .. VV2 volts'l.t FIG. 5. Emission in accelerating fields. 11 R. H. Fowler, Statistical Mechanics (Macmillan Com pany, New York, 1936), second edition, p. 401. JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37A should vary with the state of the oxide, both on account of its variation with n and on account of the variation of D with the surface condition. The factor .:lE",-H.:lEl+.:lE2) is equivalent to the work function. Of the three factors, .:lEl and .:lE2 are approximately fixed for a given oxide, whereas t>.E", may vary with the surface condition such as the amount of adsorption of barium or strontium and oxygen. Thus the work function should also vary with the state of ac tivation. In this respect it is interesting to consider the point of view adopted by de Boer.12 He considers the emission not as the passing of conduction electrons through the surface but as thermal ionization of the individual atoms ad sorbed on the surface. According to this picture, the electrons inside the oxide pass from one barium atom to another, each time transforming these atoms into ions, until they reach the atoms adsorbed on the surface, where they are emitted by thermal ionization. So far as electron con duction inside the coating is concerned such a picture is merely an equivalent way of looking at the matter, but the factor A will depend not only upon the concentration of barium or stron tium atoms inside the oxide due to its effect on electron conduction, but also upon the number of atoms adsorbed on the surface, beiRg directly proportional to it. What is more important is that the work function is now the energy required for the thermal ionization of the adsorbed atoms. According to de Boer, when the adsorption is low (less than a full layer), the effect of the adsorbed atoms on the ionization energy of each other is negligible; consequently the work function of the oxide should be independent of the state of activation and should remain constant. It seems to us that even with this picture the work func tion should not remain constant, since the amount of adsorption certainly affects among other things, the difference between the energy levels inside and outside the oxide, on account of the double layer effect. The energy levels of the adsorbed atoms are more or less fixed with respect to those of the oxide. With variation of the amount of adsorption, the energy required to 12 J. H. de Boer, Electron Emission and Adsorption Phenomena (Macmillan Company, New York, 1935). VOLUME 14, OCTOBER, 1943 release an electron from an adsorbed atom and emit to the outside space should not remain unaffected. From the above considerations we expect that both A and the work function will vary with the state of activation of the oxide. The experimental results conflict with each other. Some claim that the work function remains constant while A varies;13 other results are just the opposite ;14 still others state that both A and the work function vary.IS It seems, however, that most of these results are not reliable enough to be conclusive, because the temperature range used has been too short to give reliable Richard son plots or the temperature has been too high to prevent a change of the state of the oxide during measurement. . To determine the emission constants one has to measure the zero field emission. On account of the space charge there is no sharp breaking point where the curve goes through zero field. We find, however, that in the region of low accelerating voltage the current rises very slowly. The fact that our cathode has a large diameter and that the field increases rather slowly with the voltage is helpful in this respect. Negligible error is made if we take the current at 0.1 or 0.2 volt on the accelerating side, instead of at zero field. This is evident from Fig. 6, curve (a). We FIG. 6. Effect of po tential difference be tween collector and guard rings. • • 4 0 " \ 0. ....... /" '( • o.s 10 1.0 V volts have observed a peculiar phenomenon: Some times the curve log i versus V shows a hump near zero field. The hump appears every time after glowing the cathode at a sufficiently high 13W. Espe, Wiss. Ver. Siemens-Konz. 5, No.3, 29, 46 (1927). H. Kniepkamp and C. Nebel, Wiss. Ver. Siemens Konz, 11, 75 (1932). 14 W. Heinze and S. Wagener, Zeits. f. Physik 110, 164 (1938). 15 F. Dete1s, Zeits. f. Hochfrequenz. 30, 10, 52 (1927); W. S. Huxford, Phys. Rev. 38,379 (1931). 557 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37temperature and disappears after a large current is drawn at sufficiently high temperature. We conclude that the hump is due to a contact potential difference between the collector and the guard rings. Although these are made of the same material, because of the difference in tem perature between the central portion and the two ends of the cathode the rate of evaporation of barium and oxygen and their deposition on the collector and the guard rings is different, causing a potential difference. Curves (c) and (d), taken at the same time, are for the guard rings 0.1 volt positive with respect to the col lector, in the case of curve (d), and 0.1 volt negative in the case of curve (c), and bear evi dence to this explanation. Curve (c) has just the same shape as curve (b). The fact that only curves with a hump like curve (c), but none with poor saturation like curve (d), were ob served, indicates that the collector is sometimes positive, but never negative, with respect to the guard rings. The magnitude of the hump of curve (c) far exceeds the worst case observed. Fortunately, with an accelerating voltage of over one volt, the two curves (c) and (d) approach the same value, which differs but little from the zero field current. It should be mentioned that the measurement of emission in retarding and accelerating fields was made with the tube in the condition showing no hump. The contact potential between the collector and the cathode, as determined from the bending position of the log i versus V curve, varied from 0.1 to 0.7 volt. Had the surface condition of the collector remained unchanged, this variation of the contact potential difference should corre spond to the change in the work function of the cathode. But the latter, as obtained from the Richardson plot, showed no correlation with the shift of the bending position of the log i versus V curve. Furthermore, since the collector is made of nickel, its work function should be around 5 ev, at the same time the work function of the cathode varied from 1.5 to 1.8 ev. The contact potential difference should thus be several volts. The fact that it was less than one volt, and varied with no apparent correlation with the work function of the cathode, shows that the surface condition of the collector was not that 558 of clean nickel and that it varied with the treat ment of the tube. Since barium as well as oxygen may be evolved from the cathode and deposited on the collector, such phenomena are not sur prising. We found that, in general, after glowing the cathode at high temperature the bend in the curve log i versus V shifted to higher accelerating voltage, whereas drawing large emission had the opposite effect. For the de termination of emission constants, low cathode temperatures were used, 480° < T < 7200K, so that no change of contact potential difference took place during each set of measurement. Figure 7 shows a set of log i versus V curves N + 4 /' I , / V / I V V ,/ i// ~I V ~ / / / L( I 0 0.< a, 02 0' 04 V volts FIG. 7. Emission at different temperature. 05 and Fig. 8 shows the Richardson plots for various states of activation. Since the current i is not sensitive to the power of Tin Eq. (3), we used P instead of P/4 to reduce Eq. (3) to the form of Richardson equation for comparison with the data of others. Figure 8 shows that the points fall very closely on straight lines. The current measurement is accurate to within 2 or 3 percent. Any small deviation of the points must be due to inaccuracy in temperature measurement. The largest deviation of a point from the straight line corresponds to about 5°C. The experimental JOURNAL OF ApPLffiD PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37-, :S i 1\ ~ \ , , I 0 <4 I~ 1\ \ [\ \ :~. 1\ \ ~ \ \ f\ \ ",,\ \ \ ,\ \ !\ ~ \ " . " " ., ,.0 FIG. 8. Richardson plot. points in the Richardson plots cover a range of more than 4 for loglo i. The upper limit is im posed by the precaution against a change of the state of the cathode due to high temperature and large emission, whereas the lower limit is set by the tube leakage. Table II gives the emission TABLE II. Cathode glowing Temperature Duration Work function No. oK min. ev IOglOA 1 1390 7 1.691 -0.21 2 1350 4 1.571 -0.25 3 1200 30 1.520 -1.02 4 1270 10 1.571 -0.43 5 1320 7 1.689 -0.15 6 1300 10 1.590 -0.56 7 1100 30 1.521 -0.97 8 1320 4 1.641 -0.14 9 1320 7 1.696 -0.33 10 1120 60 1.551 -0.65 11 1200 30 1.560 -0.45 12 1150 45 1.503 -1.06 13 1390 7 1.830 +0.34 14 1140 30 1.605 -0.79 VOLUME 14, OCTOBER, 1943 constants determined after various treatments of the tube. The numbers of the Richardson plots in Fig. 8 correspond to the numbers in this table. An examination of the table shows that high temperature flashing increases both the work function and A, which come back to lower values after glowing the cathode at a lower tem perature. Our cathode, being indirectly heated, has a large heat capacity, which makes its tem perature variation slow. After flashing it may reactivate to some extent as the temperature drops slowly, so that we could not vary its state of activation over a wider range. The lowest value observed for the work function is about 1.5 ev. After flashing the cathode at high tem peratures several times successively we could no longer bring the work function to this value: It was then about 1.6 volt. This is probably due to poising by oxygen liberated at high tempera tures. Figure 9 shows the relation between the work function and log A. Poor scattering of the points may be due to the inaccurate determina tion of the latter. ~5 0 <C -<>5 l.------/ . l---.::.-' ~ ~ ~ 0 00 -S 1,0 I 'I, . t6 1.1 Work function .I,dron volts FIG. 9. Relation between work function and log A. EFFECT OF DRAWING EMISSION I t has been reported in several papers that when emission is drawn from an oxide-coated cathode the emission current decays with time, approaching a final value considerably smaller than'the initial current, sometimes less by a factor of ten or more. In the range 6500K < T <1000oK, the greater part of the decay takes place in the first few minutes, and the lower the temperature the slower the approach to the 559 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37final value. All these measurements were made on cathodes coated with a mixture of barium and strontium oxides.is In our case the measurements made for the Richardson plots were carried up to 7000K and in the measurements with accelerating fields the temperatures used were as high as 8S0oK. The emission current was steady and showed no sign of decay. To make sure of this fact four curves were taken at 790oK, 880oK, 9S0oK, and 990oK, with a constant voltage of 22.5 volts. The cathode was first glowed at the test temperature for some time. Then the collector voltage was applied by dosing a switch and the emission current was taken at different time intervals. The curves are shown in Fig. 10. Curve 4 shows no change of °O~-+--~,.r-~,,--~,,~,--~,--~ $t(01'l4$ mirUlteo" Tim~ FIG. 10. Variation of emission with time. emission with time, whereas curves 1, 2, and 3 show a slight drop within the first few seconds. However, in these cases the cathode temperature also dropped slightly as the voltage was applied. This was indicated by a deflection of gal vanometer G1 shown in Fig. 2. The galvanometer reading came back to the balanced position again when the collector voltage was removed. The slight drop in emission was thus apparently associated with the drop in cathode temperature due to emission. The emission current has also the effect of heating the cathode by going through the oxide. Figure 11 shows the cathode temperature and 16 J. A. Becker, Phys. Rev. 34, 1323 (1929); J. A. Becker and R. W. Sears, Phys. Rev. 38, 2193 (1931); H. Kniepkamp and C. Nebel, reference 13; J. P. Blewett, Phys. Rev. 55, 713 (1939). 560 ,. ,..---,-----,----,---, /. f---+----+-_-+ __ -j(lfO s ,JOti Time minutu FIG. 11. Variation of emission and cathode temperature with time. the emission current at 1125°K with an applied voltage V = 11 0 volts. Upon applying the col lector voltage the cathode temperature fell below the original value, then rose above it. Apparently the cooling was due to emission and the heating was due to conduction of the emission current through the oxide. The variation in emission was evidently to a great extent due to the variation of the cathode temperature. I t seems that, at temperatures and emission densities corresponding to the curves in Fig. 10, the emission of current has very little effect on the state of the oxide. In cases such as shown in Fig. 11 it is difficul t to tell, due to the disturbing effect of the large variation of cathode tem perature. Table III gives the emission constants TABLE III. Cathode Tube treatment tempera- Emission drawing Work ture Voltage Current Duration function No. OK volt rna min. ev log,. A 1 1065 120 1.592 -0.73 2 1065 20 2.5 120 1.592 -0.73 3 1120 120 1.648 -0.42 4 1120 60 13 90 1.620 -0.57 5 1120 110 30 45 1.608 -0.60 7 1065 180 1.598 -0.84 8 1065 110 8 60 1.590 -1.06 determined after various treatments of the tube. When emission was drawn the collector voltage was removed after the heating current, otherwise the effect of drawing emission might be reduced due to the slow cooling of the cathode. It is seen that the effect of drawing emission is small, both log A and the work function are slightly reduced. JOURNAL OF APPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 155.33.16.124 On: Fri, 28 Nov 2014 12:11:37

1.1714951.pdf

Physics in 1942 Thomas H. Osgood Citation: Journal of Applied Physics 14, 53 (1943); doi: 10.1063/1.1714951 View online: http://dx.doi.org/10.1063/1.1714951 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/14/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A Lecture on Bomb Physics: February 1942 Phys. Today 48, 27 (1995); 10.1063/1.881468 Nobel Lectures, Physics, 1942–1962 Am. J. Phys. 33, 419 (1965); 10.1119/1.1971617 American Institute of Physics Report for 1942 Rev. Sci. Instrum. 14, 148 (1943); 10.1063/1.1770147 The Status of Courses in Physics and of Physics Departments in Institutions of Higher Education—October, 1942 Am. J. Phys. 11, 78 (1943); 10.1119/1.1990447 Proceedings of the American Association of Physics Teachers: Meeting at the Pennsylvania State College, June 25–26, 1942 Am. J. Phys. 10, 209 (1942); 10.1119/1.1990383 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19Journal of Applied Physics Volume 14, Number 2 February, 1943 Physics in 1942 By THOMAS H. OSGOOD Michigan State College, East Lansing, Michigan I. CRYSTALS AND PHOTONS N EARLY twenty years ago, alpha-particles from natural radioactive sources and protons resulting from the disintegration of light elements were counted by scintillations, lumi nescent points of light which are visible under a microscope when such particles impinge on a screen of zinc sulfide. As gracious expressions of professional courtesy, workers in the field of radioactivity would sometimes exchange speci mens of the zinc sulfide which they used in their experiments. In 1924, the late Lord Rutherford received such a gift from a friend in Germany. Upon examining the powder, he found it too coarse for his liking, and therefore directed his laboratory assistant, Crowe, to grind it to a finer size in a mortar before sprinkling it on the castor-oil smeared glass plate which served as a screen. When this was done, it was a source of keen disappointment to Rutherford and all who participated in his experiments to find that the new screen was quite unsatisfactory, in as much as only a small fraction of the incident alpha particles and protons produced observable scin tillations. After some correspondence, a new sample' of zinc sulfide was received from the same source, with a suggestion that the material be used in its coarser form. This time it proved to be thoroughly satisfactory; but there remained S3 the unanswered question, why did the grinding destroy the luminescent properties of the crystals? In the intervening years, experimental and theoretical work on photo-conductivity and related topics have added enormously to our understanding of the behavior of atoms, ions and electrons in crystals, and make it possible to give a fairly satisfactory answer to the question now. Complete summaries of recent work, and of the present knowledge of solids, are available in two books by Mott and Gurneyl and by Seitz,2 as well as in a series of papers by many authors, stemming from the early work of Poh!. A qualitative answer to the twenty-year-old question will give at least an inkling of the concepts which enter into one aspect of the modern theory of solids, though it can give little idea of the scope of the theory as a whole. First it is necessary to mention that studies of the photo-conductivity of crystals have led to a clear understanding of the relations between crystals and photons, and have provided a good explanation of the coloration and darkening of crystals under the influence of light-matters which are of extraordinary importance in the theory of the photographic· process. One of the most significant observations is that crystals of [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19the alkali halides, for example, when heated in an atmosphere of their proper alkali vapor, can absorb an excess of these alkali atoms, even in chemically distinguishable amounts. The ab sorption is not a surface effect, so there must be within the crystals some sorts of vacancies. known as trapping centers where these excess atoms can reside. N ow all specimens of zinc sulfide are not fluorescent, nor is all zinc sulfide photo-con ducting. To prepare the fluorescent variety, the ordinary sulphide is heated alone, or with a small amount of another heavy sulfide, such as copper, manganese, or silver. This treatment not only renders the crystals fluorescent, but at the same time causes them to become photo-con ducting. It appears, then, that there is a close connection between the two effects. As the heat treatment proceeds, the zinc sulfide decomposes slightly, and a preferential evaporation of sulphur atoms takes place from the surface. An ab normally large number of zinc atoms is thus left behind in the solid, and metallic atoms of the accompanying heavy sulfides may also infil trate to the crystal. These excess foreign or zinc atoms diffuse throughout the crystal, perhaps as positive ions, but more likely as neutral atoms. If some are in the form of ions, there must be a corresponding number of free electrons wander ing, or trapped, in the lattice, for the crystal is electrically neutral. Be that as it may, the luminescence of the crystal under the influence of ultraviolet light or of flying particles is def initely conditioned by the presence of these excess zinc atoms or ions, which may be raised by a local stimulus to excited levels. During the return to their normal levels, the light which constitutes fluorescence or a scintiIIation IS emitted. However, the internal economy of such a luminescent crystal is sensitive to external dis turbances. Even the mechanical forces involved in crushing the crystals in a mortar, or the heating which may be a secondary effect accom panying rupture, may be enough to cause the excess zinc atoms or ions in the finely powdered crystal to form small aggregates of zinc' metal, which will be definitely anchored at one spot in the crystal. Indeed, the process is probably closely akin to that whereby a photographic 54 film, upon exposure to light, or upon being pinched, undergoes local electrolysis, forming free bromine and numerous tiny bunches of free silver atoms which are the black grains in the developed emulsion. Once this has happened in zinc sulfide, the excess solitary metallic atoms are no more, and scintillations will be produced but rarely by impinging alpha-particles or protons. II. THERMAL DIFFUSIO N Different kinds of molecules can usually be separated from a mixture by combinations of physical and chemical means. With the dis covery of isotopes, and with the recognition of their importance in the study of artificial radio activity, the need has grown more urgent for a method of separation involving physical means alone. While the problem has become more pressing, it has also become more difficult, because the only differences between isotopes are differences of mass amounting usually to not more than a few percent. Maxwell's demon could do the trick, but something much more practical is required. A mass spectrograph can also achieve a complete separation, but with a yield only about one-ten-millionth of what is desired. Several new methods have been tried in the last twenty years, many achieving partial success.3 Perhaps the most promising at the moment is that involving thermal diffusion. This is nqt the same effect as the ordinary diffusion process which is described in elementary accounts of the kinetic theory of gases, and which occurs and is manifest in a gas mixture maintained at a uniform temperature. By virtue of the process of ordinary diffusion a local concentration of a foreign gas, introduced artificially into a gas-filled chamber, will distribute itself uniformly within the limits of statistical error throughout the whole volume, merely because all the molecules are moving. On the other hand, thermal diffusion shows itself in a mixture of different kinds of particles which is kept at a non-uniform tem perature. For the sake of simplicity, let the gas mixture under consideration be confined between two vertical parallel walls Hand C,' a short distance apart. Let H be heated and C cooled. Then thermal diffusion creates a concentration gradient perpendicular to Hand C, the lighter JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19particles being the more numerous near the wall H. Strong convection currents then come into play, carrying the abnormal concentration of light particles upward and the abnormal con centration of heavy particles downward. Thus, after a time, the upper part of the enclosure will be over-populated with lighter particles, the lower part with heavier particles, whence these "fractions" can be drawn off by appropriate means. Unfortunately, this argument amounts to nothing more than a statement of the facts of the case; as an explanation it appears to be entirely illusory, for nearly all writers on the subject are emphatic in their statements that an explanation of the phenomenon cannot yet be expressed in simple terms. In estimating the success with which the separation of a binary mixture of molecules of dissimilar masses may be carried out it is customary to calculate4 the value of a thermal diffusion factor a, which connects the coefficient of ordinary diffusion with the coefficient of thermal diffusion and with the concentrations of the two types of particles whose separation is desired. The expression for IX in terms of the fundamental characteristics of the mixture involves, in addition to the masses of the constituents, a function which depends on the law of force which is assumed to be operative between the particles. If a spherically sym metrical power law of repulsion is considered, such that the force between a pair of particles is inversely proportional to the nth power of their distance apart, then IX is found to contain a factor (n-5). Hence it follows that for n=5, that is, for what are known as Maxwellian par ticles, the thermal diffusion factor a is equal to zero. Therefore, for such particles, no separation by thermal diffusion is possible. It also follows that the concentration gradient between the walls Hand C mentioned above may be reversed in direction, depending on whether n is greater than 5 or less than 5. The simplest available derivation of the general shape of the function representing the diffusion factor is to be found in a note by Frankel,5 who employs dimensional analysis for the purpose. The factor has also been interpreted in a somewhat different fashion by Grew,6 whose experiments show the reversal of VOLUME 14, FEBRUARY, 1943 sign in the separation of mixtures of varying proportions of ammonia and neon. It is, of course, certain that the forces between . molecules are not in general spherically sym metrical nor describable by an inverse power law. Nevertheless, the closeness with which such a law approaches the truth may be gauged by the accuracy with which it can be used to predict a thermal diffusion factor which can be compared with an experimental value. In the case of neon, whose force field should, by any elementary argument, be spherically symmetrical, the agree ment between theory and experiment is only moderately satisfactory. In the case of methane, whose force field is known not to possess the required symmetry, the agreement is poor. A further difficulty arises because the simplf' assumption of a spherically symmetrical field of force leads to a value of a which is independent of temperature. Experimental results show that this is not even approximately true. In spite of the many defects of the theory, the process of thermal diffusion is in practice an effective separator of isotopes. Clusius and Dickel, whose work is mainly responsible for the renewed attention which is now being paid to this subject, used a multiple-tube unit with an effective length of 36 meters, and obtained from ordinary hydrogen chloride light and heavy samples of chlorine which were about 99.5 per cent pure. Since the method is applicable also to liquids, it will no doubt be used by those who seek to develop practical sources of atomic energy in an effort to separate the priceless 235 isotope from ordinary uranium. 1lI. THE NUCLEUS The exploration of the atomic nucleus is not exploration in the true sense of the word. No one can look into a nucleus to see what is there. Rather, thc problem is one of deriving a plausible, but in the first place tentative, structure, and then seeing how it withstands various tests that can be applied to it. The process might be likened, indeed, to the task of a geographer of the middle ages who attempted to describe a foreign country, its people, and its physical features, solely on the basis of what goods were carried to it and from it in ships. 55 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19There arc two parts in the problem of nuclear structure: First, what is a nucleus composed of; second, how is it held together? The answer to the first question is now fairly definite, but the second requires much more research. The forces which are familiar in the laboratory, and upon which the operation of most laboratory instru ments depends, forces like those described by Coulomb's law, are still in control when atomic distances of the order of 10-8 or 10-9 cm are involved. But at distances only one-ten thou sandth as great it is not surprising that new types of forces appear. These new forces are what hold the components of the nucleus together. They come into play only when the components approach very close to one another. In technical language, we are interested in the proton-proton and proton-neutron interaction. The most ob vious way of finding out about such interactions is to let protons be scattered by other protons or neutrons by protons and so on. By varying the speed of the incident particles the closeness of approach can be varied, and the short-range forces can be found as functions of the distance of separation. The investigation of the scattering of protons by protons has been going on for several years, but only in a few places because of the complexity of the apparatus required to S6 FIG. 1. The vertical acceler ating tube, guarded by external metal rings, of the voltage quad rupling outfit used by Ragan, Kanne, and Taschek at the IT niversity of \Visconsin. In the background is a bank of con densers and the power supply unit for the arc. (Courtesy of R. F. Taschek.) provide the necessary stream of fast-moving particles. The general nature of the proton proton forces is known up to incident proton energies of about 2 Mev. A great deal of a more detailed nature remains to be discovered. Ex periments of half a dozen years ago showed that the scattering of protons by hydrogen, that is by other protons, was far from what could be expected from a simple Coulomb law of force. The scattering was such as might occur if the repulsive Colomb force, operative at compar atively large distances, changed rather suddenly into a strong attractive force, when the distance between the particles was of the order of 10-13 cm. Now there must be some critical distance (for some particular velocity of the incident protons) at which these two types of force com pete, as it were, for control of the scattering process. Should an incident particle pass just outside this critical distance, it will be scattered normally by the Coulomb field; but should it pass just inside, it will be whipped around the target nucleus and will suffer a deflection which differs considerably from that which occurred in the former case. There will therefore be a scarcity of scattered particles at some definite angIe with respect to the incident beam, for some particular energy of incidence, which must, however, be JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19large enough to bring true nuclear forces into play. This minimum in the scattering curve has been shown to lie at 45° for 400-kev protons, but until Ragan, Kanne, and Taschek7 studied the region in detail the available information was only approximate. The space-variation of the short-range nuclear forces of which we have spoken cannot, it is found, be described by a simple power law such as r-n• The variation may not even be con tinuous. Probably the most successful and com pact way of representing these forces is to specify what kind of a potential weIl they are equivalent to. For example, Breit8 and his co-workers in 1939 published an analysis of proton scattering up to that date, and concluded that the results could be represented over a range of 2 Mev by supposing that each proton behaved as a square "potential well" superposed upon an existing Coulomb field. Unfortunately, the relevant data were not able to give an unambiguous estimate of the shape of the weIl; a considerable latitude had to be permitted. Now the new observations of Ragan, Kanne, and Taschek remove the am biguity. With a high voltage generator at the University of Wisconsin they studied the scat tering of protons by hydrogen in the region 200 to 300 kev. A view of the accelerating tube of this outfit, which has already been described in the literature, is shown in Fig. 1, and the FIG. 2. The spider-like unit on the right is one of the scattering chambers used by Ragan, Kanne, and Taschek. By changing the de tector from one arm to another, protons can be observed scattered at various angles from the primarY beam, which comes down the large inclined tube behind the chamber. Part of the pumping system is seen at the left. (Courtesy of R. F. Taschek.) VOLUME 14, FEBRUARY, 1943 pumping and detecting systems, situated a short distance below the accelerating tube, are seen in Fig. 2. The reasons for choosing this range of energy for particular study are as follows. On either side of the minimum referred to above, that is, about 300 kev, or about 500 kev, the scattering is par ticularly sensitive to the shape of the potential weIl which is chosen to represent the proton. Hence, an accurate determination of the amount of scattering in one or both of these neighbor hoods wiIl give information from which a correct choice can be made of the well parameters. ";t the time of their theoretical work, Breit, Thaxton, and Eisenbud could not decide whether a well of width 2e2jmc2 and 10.5 Mev deep, or a weIl three-quarters as wide and nearly twice as deep represented the results the better. Both gave the same scattering at voltages around 2 Mev, and the scattering at lower voltages was not known precisely enough to make a distinction between the two weIl-forms possible. The new work from the University of 'Wisconsin, even though difficult to perform with the necessary accuracy because of the low penetrating and ionizing power of the low-energy protons, decides conclusively in favor of the potential weIl 10.5 Mev deep and 2e2jmc2 across. This is, however, not the only type of potential well which can be made to fit the observed data. Others have been 57 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19tried, with moderate success. But the problem does not end here. With a knowledge of proton neutron interaction as well, binding energies, and then masses can be calculated for such particles as He3 and He4• If the assumed potential well for the proton leads to the wrong result for these masses, then the constants describing the poten tial well must be further modified, or a new shape must be tried. The first attempts are naturally made with the simplest forms of well which offer some prospect of success. The interaction between pairs of elementary particles, such as proton-proton or proton neutron, are known to be of very nearly the same magnitude. Nevertheless, the theoretical bases on which the detailed calculations of nuclear structure rest amount to little more than a series of assumptions, in need of being tested. If an experimental result is in agreement with anyone set of assumptions, the validity of the others may be ruled out, but this agreement, in itself, does not imply the absolute correctness of the hypotheses with which the agreement is found. Experimental work thus proceeds asymptotically, as it were, towards a complete understanding of these short-range nuclear forces. The time is not ripe for a critical experiment which can solve the problem completely, but during the last year several contributions have been made towards the solution of certain aspects of the problem. For example, Carro1l9 has studied the interaction of slow neutrons with protons, using as target material the hydrogen nuclei in a series of hydro carbons. This has a bearing on the prediction made by Fermi that the cross section under such conditions is influenced by molecular binding. As typical results, Carroll found the cross section of the proton to slow neutrons to be 32 X 10-24 cm2 for hydrogen gas, and 49 X 10-24 cm2 for gaseous butane, thus showing that the effective target area per proton increases with increasing complexity of the bombarded molecule. In a paper published a few months ago, TatePO re ported an investigation of the protons scattered by high energy neutrons. For experimental con venience, of course, the neutrons were flung at the more or less stationary protons, some of which recoiled under the influence of close col lisions, and were detected in ionization chambers. He found that the scattering, referred to the 58 center of gravity of the system of particles, was isotropic, within the ten percent accuracy of his experiments. Such a result is interesting because of its relation to the earlier theoretical work of Rarita and Schwinger,u These two authors worked out the consequences to be expected in proton-neutron scattering on the basis of three different assumptions. One of these assumptions, which predicted asymmetrical scattering, now appears to be ruled out. When theory and experiment concern them selves with scattering by very complicated nu clei, a much more general point of view has to be taken. No longer can the details of the process be considered separately; a statistical or thermo dynamic treatment is required, in which the heavy scattering nucleus is considered to possess some of the attributes of a drop of liquid. Such a nuclear model has already had some success in the interpretation of the results of nuclear fission. When it is applied to the scattering of neutrons by lead, an energy spectrum of the inelastically scattered neutrons can be predicted. Several years ago, Weisskopf predicted that fast neutrons, all of one velocity, incident upon a heavy element would be scattered inelastically in such a way that the neutrons, after scattering, should show a Maxwellian distribution of velocities. This point has been tested by Dunlap and Little12 using neutrons from the D - D reaction with an energy about 2.5 Mev. The distribution of scattered neutrons turns out not to be Maxwellian, but to have a much greater proportion of high energy particles than a clas sical distribution would have. The reason for expecting a Maxwellian distribution is that the moving neutron is presumed to be captured temporarily by the scattering nucleus, then ejected like a molecule evaporating from a warm drop of liquid. It seems more likely now that the re-emission of a neutron takes place, or may take place, before "thermal" equilibrium has been attained in the nucleus, and that the scattered neutron ~merges from the temporary nucleus at some point where a fortuitous concentration of kinetic energy still exists. This mechanism would account for scattered neutron energies in excess of those corresponding to a calculated equi librium temperature of the nucleus. Such a definite, mechanical picture of the nucleus must, JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19however, be developed further with caution, and only insofar as additional experimental checks are available. IV. THE NEUTRINO Until about ten years ago it was generally believed that a nucleus was built up of protons, or groups of protons and negative electrons. The emission of beta-particles and of alpha-particles by naturally radioactive nuclei with energies greater than could be derived from the extra nuclear structure was considered almost sufficient evidence of their presence there. But with the discovery of the neutron there came a change in point of view, which threw into strong relief the difficulties attending the assumption that elec trons existed in the nucleus. These difficulties were concerned with such things as nuclear spin, nuclear magnetic moments, potential barriers, and the distribution in space of the electron matter waves. Protons and neutrons are now believed to make up the nucleus. What then of the negative electrons which are, without doubt, emitted by radioactive material? The modern view is that such disintegration electrons are created at the moment of their emission, just as a photon can be created and emitted by the extra-nuclear structure when a change of total energy occurs. Thus some of the difficulties of the continuous beta-ray spectrum are solved, but not all of them. Here we are not concerned with the line spectrum of beta-rays, which arises through the action of nuclear gamma-rays on outer electrons. The beta particles which appear to come from the nucleus have a wide energy spectrum, from zero to a definite upper limit, which is different for dif ferent members of a radioactive series. Thus there is a fixed change in the energy of the original nucleus, but a varying amount appears as energy of the beta-ray. In the face of the enormous mass of evidence which elsewhere supports the conservation laws of energy and of momentum, it seems unreasonable to throw them aside for this one case. Another solution of the difficulty is needed. Fermi, in 1934, assumed that another undetected particle, t~e neutrino, took part in the disintegration, carrying away a varying amount of energy so that the desired balance could be maintained. VOLUME 14, FEBRUARY, 1943 In one sense this is a very logical solution, but since it puts the burden on something which cannot complain, it can hardly be called a satis factory one; for the properties which the neutrino must possess to function as it is supposed to are nearly all negative~no mass, no charge, no magnetic moment, spin one-half. The problem is therefore to detect the neutrino to see if it carries away about the amount of energy expected. The usual physical methods are useless, so that secondary effects must be used. Several attempts have been made to demonstrate the neutrino's existence by measuring the recoil momen ta of other particles which take part in the disintegration process. Any unexpected momentum they possess must have come from the neutrino's recoil. The results of such experi ments have not been entirely convincing, but, as the years pass, and data accumulate, the exist ence of the neutrino becomes more and more certain. A new investigation of this matter by Allen13 is probably less clouded by ambiguitites of interpretation than any previous work. To simplify the problem as much as possible, it is desirable to deal with a simple nuclear reac tion. At least three entities must be involved, the nucleus, the beta-particle, and the neutrino. Therefore, to measure the momentum of the neutrino, it is necessary to measure the sum of the momenta of a nucleus and of a disintegration electron. The experimental technique would be simpler if the momentum of only one particle had to be observed. Kan Chang WangH has pointed out that such a situation can be realized when a radioactive nucleus moves one step lower in the atomic sequence by absorbing a K electron into the nucleus. Be7 is an unstable nucleus of this type, with a half-life of 43 days. If its decay is traced from counts registered by thin-walled Geiger-Muller tubes, it is found, by interposing suitable absorbing screens, that only gamma-rays are emitted, not accompanied by electrons. The reactions involving Be7 appear to be Be7+ek~Li7+17+1 Mev, Be7+ek~Li7*+?]+O.SS Mev, Lj7*~Lj7+hv+O.4S Mev, S9 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19though recent evidence from other studies suggests that the numerical parts of these equa tions may be in error by ten percent. In one case, an intermediate excited nucleus Li7* is formed, but in any event, a neutrino, denoted by 1/ is emitted. The detection of the neutrino therefore involves the observation of the Li7 recoil nucleus. Allen impregnated a platinum foil with active Be7, platinum being used on account of its high work function. By a preliminary heating, Be atoms were encouraged to diffuse to the surface of the foil, whence the resulting Li7 atoms could recoil with as little energy loss as possible. Because of the relative values of the work func tions, Li7 atoms resulting from the disintegration of Be were given off as positive ions, but other atoms evaporated in the neutral state. By virtue of their charge, the Li7 ions were easy to handle. They emerged from the treated platinum foil with a range of velocities, depending upon the depth at which they originated. Some of these recoiling nuclei were caught by the first electrode of an electron multiplier tube, which increased the current to such an extent that it could be handled readily with vacuum tubes. Such current was, of course, propor tional to the number of Li7 ions striking the first electrode of the multiplier. These ions, on their way to this electrode, were controlled by a retarding potential, and their number was measured as a function of the retarding potential until that potential was large enough to prevent any of them from being detected. In this way Allen found that the maximum energy of the recoiling Li7 nuclei was about 45 electron volts. If there were no neutrino at all, then the energy of these nuclei would be that derived from the recoil of the gamma-ray of 0.45 Mev, or perhaps from the 1.0-Mev gamma-ray. The latter possi. bility was ruled out by the results of a supple. mentary experiment with coincidence counters which showed that no gamma-ray of this higher energy was observed at the instant when any Li7 ion was being detected. The O.4S-Mev gamma-ray could give the recoiling nucleus only 15.6 electron volts of energy. This is so far below the figure of 4S electron volts found by Allen that it is safe to conclude that the difference (about 30 electron volts) was given to the recoiling nucleus by the neutrino. 60 Theoretical considerations show that the amount of excess energy, 30 electron volts, is only slightly dependent on the mass of the neu trino. The usual assumption is that the neutrino's mass is zero; but if it were as large as 0.2 of the electron's mass, the excess energy would be diminished by only one electron volt. In this respect Allen's experiment is inconclusive, but it has increased enormously the probability that the neutrino really exists. V. COSMIC RAYS I t was remarked a year ago that a gratifying semblance of orderliness had been brought into the study of cosmic rays by the hypothesis that practically all of the primary incoming radiation consisted of fast moving particles, probably protons. Although this conception is definite and clear cut, yet there has not been time for it to make its mark in simplifying the interpretation of the behavior of cosmic rays subsequent to their entry into the atmosphere. The nature of the primary cosmic rays on the one hand and their behavior in passing through matter on the other, therefore remain as two distinct problems. A helpful summary of the processes which occur during the interaction of cosmic rays with matter has recently been given by Rossi and Greisen15 in the Reviews of Modern Physics. They summarize the phenomena which occur in a qualitative way and also treat most of them mathematically. Some of these effects involve nuclear transformations; others do not involve nuclear change at all. The nuclear processes are comparatively rare and do not contribute sig nificantly to those effects which are observed by counter trains, effects such as the absorption and scattering of mesotrons, and the production of showers by electrons and photons. The non nuclear effects are of two kinds, collision processes and radiation processes. In the former of these the primary particle affects atomic electrons directly. The atom may be raised to an excited state, and slow-moving electrons may be ejected. In the latter, the primary particle is accelerated by the field of the nucleus and in doing so gives out radiation. For some particles one of these two effects predominates; for mesotrons and protons of moderate energy, for example, the JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19energy lost in passing through the atmosphere is due almost exclusively to collision processes. Photons are emitted by charged particles only when they interact closely with the electric field of the nucleus. It may seem to be a step backward to state in 1942 that according to the classical electromagnetic theory a charged particle emits radiation whenever it undergoes acceleration, and to base on it parts of the theory of the passage of cosmic rays through matter. But it enables the role played by the controlling variables to be estimated in a qualitative way. For example, the radiation loss will be a rapidly increasing function of the atomic number because the ac celeration which a particle experiences will depend on the charge of the nucleus. By the same token, the radiation loss suffered by elec trons will be much larger than that of heavier particles like mesotrons or protons because the latter will not suffer nearly such severe accelera tions in a nuclear field as will the lighter electrons. The energy lost by radiation processes is carried away by photons having a spectrum of very wide limits, but in anyone act a large per centage of the available energy is usually trans formed. Hence, in strong contrast to the behavior of mesotrons and protons, whose energy is frittered away gradually, very fast electrons in passing through matter surrender only trivial amounts of their energy to low-energy processes, while changing a great deal into energetic quanta. These high energy quanta may now produce positive and negative pairs or else may produce new fast-moving electrons by Compton recoils. Thus a new supply of fast-moving light particles is created. The give-and-take continues; these fast-moving electrons again lose energy by radiation and the resulting photons again produce fast-moving particles. The whole process is sum marized in the phrase "cascade shower." It is obvious from the description given that a shower may begin either with a high energy photon or a high energy electron. It will be clear that if cosmic rays go through a very thin layer of material, the chance of a well-developed shower being formed is rather slight. On the other hand, if the rays enter extremely thick material, many of the ~howers which are produced will be absorbed in the VOLUME 14, FEBRUARY, 1943 material itself. There must exist a certain thick ness of material, different, of course, for different materials, at which the showers are most fully developed. Transition curves, which show the growth of the ionization due to these showers, indicate that the thickness at which full develop ment occurs is not quite the same for photons as for electrons. Since the penetrating ability of photons and of electrons is not the same, they will, in the first instance, begin to produce showers at different depths, but after the cascade process has been established there should be no essential difference in the transition curves for the two types of primary agent. Theory predicts that on the average, the photon-induced process should lag behind the electron-induced process by about the distance in which a photon has a twenty-five percent chance of being absorbed. The theory, which is quantitative, and which is much more specific than this qualitative state ment implies, predicts the distance to be 0.3 cm in lead. Nereson16 finds an experimental value very close to this figure. The secondary processes accompanying the absorption of cosmic rays are very complicated. One method of attack on the problem is simply to find out with the help of suitable coincidence counters what different events occur simul taneously, with the presumption that such simul taneous events may be causally related. When simultaneous events have been recognized, then the conditions under which they are observed can be made increasingly stringent, and the number of variables controlling their appearance can be reduced to a reasonable number. Although no one experiment can be considered a crucial one, yet the data obtained from a well-chosen series may be expected, upon analysis, to indicate the key events which occur. Along these general lines, Korff and Clarke,17 in the high Rockies, have searched for a con nection between the occurrence of showers and the production of neutrons. They find that coin cidences between these events occur for showers generated in a variety of substances. The rate of production of neutrons is small, being of the order of 10-3 or 10-4 neutron per g per sec. at the summit of Mt. Evans, an altitude where cosmic rays are much more intense than at sea level. 61 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19FIG. 3. View near the top of Mt. Evans, Colorado. The simple prismatic building is the cosmic-ray laboratory operated jointly by the Cniversity of Denver, the University of Chicago, and Massachusetts Institute of Technology. (Courtesy of S. A. Korff.) The site of their experiments is shown in Fig. 3. In the middle distance is a hut which appears to be, and is, nearly all roof. This is the cosmic-ray laboratory maintained jointly by the University of Denver, the University of Chicago, and Mas sachusetts Institute of Technology. In the same laboratory, Bostick18 used a cloud chamber to study the non-electronic particles which accompany showers. He found heavy particles, which he interpreted as slow mesotrons or slow protons, in about 6 percent of his pic tures; and in about 1 percent of the total expansions, there were pairs of penetrating par ticles, which must obviously have been produced nearby, and could not be primary particles coming from outer space or from the fringes of the earth's atmosphere. In closely related experi ments, Auger19 also found some low energy mesotrons accompanying the soft component. The origin of the soft component of the rays is still a matter of debate. I t is known to consist 62 of electrons, which may arise from a variety of secondary processes, and may even include the last remnants of primary electrons, much slowed down by their journey through the atmosphere. Another possibility is that practically all the soft component can be accounted for by the electrons which are one of the decay products of mesotrons. This last hypothesis has been de veloped mathematically by Rossi and Greisen,20 who were able to compare their theoretical results with the experimental values found by Greisen21 for the intensity of the soft component as a function of altitude and of zenith angle. It will hardly surprise the reader to learn that the comparison led to the conclusion that there was something correct and something incorrect about each of the hypotheses. l<.ossi and Greisen sum marize this phase of their work in the statemen t that electrons arising from the action of meso trons, by decay or otherwise, should show a much less rapid variation of intensity, both with alti- JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19tude and with zenith angle, than is actually observed. The source of this fast-varying elec tronic component in soft cosmic rays is therefore probably to be found in secondary processes accompanying the absorption of fast primary electrons. At sea level, however, and also in the very high atmosphere, the numbers of electrons arising from mesotron decay appear to be nearly adequate to account for the whole intensity of the soft component. The facts that mesotrons are produced in the high levels of the atmosphere, and decay on their subsequent downward flight have been among the most helpful basic concepts in the correct inter pretation of cosmic-ray phenomena. Going one step farther, Auger and Daudin22 have inquired whether or not other parts of the atmosphere may not also make contributions to the mesotron population. Their experiments showed that some particles emitted under thick layers of lead have the same penetrating power as low energy meso trons. This result would indicate that at least some such particles are normally produced in air at low altitudes. In a further search for quantitative data, Rossi23 and his friends have continued their ex periments to determine the lifetime of the mesotron. Since the lifetime of a mesotron varies with its momentum, being longer for energetic particles, it is necessary to specify its rest lifetime, as it were. Most values taken from the literature in the last two or three years run from one to three microseconds. In the recent experi ments of Rossi, the value comes out at 2.8±0.03 microseconds. Turning now to the problem of the primary radiation, Millikan, Neher, and Pickering24 have recently coordinated several results of a world wide cosmic-ray survey (Figs. 4 and 5) to provide a working hypothesis concerning the distribu tion-in-energy of the rays as they enter the earth's atmosphere. They are very careful, however not to commit themselves as to the actual nature of the incoming radiation, beyond admitting that 60 percent of it may consist of charged particles, and it is to be regretted that they are yet unable to point out the features of their theory which are, and those which are not, in agreement with the proton hypothesis. The general argument which these writers VOLUME 14, FEBRUARY, 1943 follow is this. They point out that all competent authorities agree that the only acceptable origin of stellar energy is the transformation of mass into other forms of energy, following the well known Einstein relationmc2=E. Stellar energy is apparently adequately provided for by as suming, inside stars, the synthesis of simple atoms into more complex ones, but such a process does not give particle or photon energies as large as are observed in cosmic-ray experi ments. It is therefore very satisfactory to know by direct observation that cosmic rays do not come from the stars but from a much more uniform distribution of sources throughout the nearby uniyerse. Millikan, Neher, and Pickering assume that in interstellar space complete atomic annihilation is going on. The energy thus released is adequate to account for the observed cosmic ray energy, and is much larger than that avail able from the synthesis of heavy atoms from simpler ones. The spectrum of the radiation thus generated in space will be governed by the total energies of the atoms which are most frequently found there, and here astrophysical evidence shows that the commonest atoms are hydrogen, helium, carbon, nitrogen, oxygen, and silicon. These occur in the approximate ratios 100, 10, 1, 1, 1, 1, respectively. Taking each of these atoms in turn, the three authors show that the magnetic fields of the sun and earth would debar radiation due to the annihilation of hydrogen from reaching the earth's surface, but that radi ation from the other five annihilation processes would be able to reach the earth in certain localities. The less energetic rays will be able to reach low altitudes in the earth's atmosphere only at high magnetic latitudes, while the most energetic ones will penetrate to sea level at the geomagnetic equator. The five species of atoms, helium, carbon, oxygen, nitrogen, and silicon give rise to radiations of energy approximately 2 billion electron volts, 6 Bev, 7 Bev, 8 Bev, and 14 Bev, respectively. In proceeding northward or southward at a fixed altitude from the mag netic equator, the cosmic-ray intensity, due only to the 14-Bev component at the equator, will be enhanced by the appearance and detection of the less energetic components, each coming in at a fairly definite magnetic latitude. The 8-Bev radiation begins to be detected, for instance, at 63 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19a magnetic latitude of 33° or 34°, but is not found between this and the equator. Hence the over-all result is a series of steps or plateaus in the cosmic-ray intensity curve. The experimental evidence is not nearly as straightforward as these simple statements imply, but from an analysis of data from many parts of the world Millikan, Neher, and Pickering make out a fairly reason able case for their hypothesis. The relative abundances of interstellar atoms should provide a check on the relative intensities of these five energy bands. At the moment, it is safe to say that the predicted energy in each band is not in contradiction to the observed energy, but quan titative estimates are still so rough that it is too early for a test of the whole hypothesis on this basis. What the general reader would welcome would be an interpretation of the findings of other schools of cosmic-ray research in terms of this five-band theory. VI. X-RAYS Considerable differences of opinion still exist concerning the diffuse reflections of x-rays from crystals. It will be remembered that a paper by Zachariasen25 a few years ago revived an interest in the occurrence of faint spots in diffraction patterns obtained either by the Laue or the Bragg method. Since that time the pages of several journals have been enlivened by spirited discussions as to the real cause of these anomalous spots. A very extreme point of view is held by FIG. 4. Launching pilot balloons to carry an electroscope to near the limit of the atmosphere at Bismark, North Dakota. Readers who are not familiar with western states will find intersting extremes of terrain in Figs. 3 and 4. (Courtesy of R. A. Millikan.) 64 JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19Raman and his school in India, which is at variance with views now current in Great Britain and the United States of America. A clear sum mary of the situation has been given by Preston,26 who is quoted directly in the following account. The progress made in the investigation of this interesting subject in India has recently been reported in the Proceedings of the Indian Academy of Science of October, 1941, which is wholly devoted to a "Symposium of Papers on the Quantum Theory of X-Ray Reflection and the Raman Reflection of X-Rays in Crystals." This description of the subject matter reflects very clearly the difference in the interpretation of the phenomena which exists between French, inner ican, and British physicists on one hand, and the group of Indian physicists at Bangalore working under the leadership of Sir C. V. Raman on the other. \Vhen a single crystal of, say, aluminum, rocksalt, or sylvine is mounted in a beam of x-rays which traverse the crystal in a direction parallel to a cubic axis, a Laue photograph is obtained if the beam consists of white radiation. In addition to the Laue spots, there appears on well-exposed plates a number of streaks, running through the Laue spots, which apparently should have no business there. \Vhen the composite x-ray beam is replaced by a monochromatic beam, the usual theory of crystal diffraction tells us that no reflection should be observed unless the crystal is oriented in a particular way. In general, the condition for reflection will not be satisfied, except by a lucky accident, and a blank photograph should result. However, in fact, reflections are observed. They are faint, but have the symmetry proper to the crystal axis in which the incident x-ray beam lies. It is this background of non-Laue diffuse reflections that is attracting the attention of x-ray crystal lographers. To what is it due? There appear to be several possible answers to this question. The most comprehensive is perhaps to say that the diffuse background arises as a direct result of departures from geometrical per fection in the crystal architecture. A periodic flaw, a regular precipitation of an impurity, and the temperature vibrations of the atoms are all possible and probable causes. It is the task of the experimenter to devise means of identifying VOLUME 14, FEBRUARY, 1943 FIG. 5. Two young Indian gentlemen watch while \V. H. Pickering receives cosmic-ray signals coming from a transmitter high in the atmosphere. The scene is the top of a building at Agra, India. (Courtesy of R. A. Millikan.) the differen t causes so that use may be made of the machinery provided by the mathematician to enlarge our knowledge of the solid state. Those features of the background reflections which occur in crystals of pure substances, so that flaws and chemical segregation are excluded from con sideration, are evidently of great importance. There is general agreement on all sides that this background is due to movement of the atoms or molecules of the crystal from the positions of perfect geometrical alignment. The differences of interpretation arise in assigning the cause of the movement, and in the mechanism of x-ray scattering. In Great Britain and the United States, the whole effect is ascribed to the static geometry of the crystal, so the diffuse background is a picture of the dynamic vibrations of the crystal. The undisturbed crystal can be regarded as a medium in which the density varies periodi cally in space, and the effect of the temperature vibrations is to superimpose a spectrum of much longer elastic waves on the natural periodicity of 65 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19the crystal. The whole crystal then diffracts as a unit, and the resulting pattern consists of the usual Laue reflections plus a background due to the motion of the atoms. The point of view adopted by the Bangalore group of physicists is rather different. To begin with, the quantum aspect of the interaction of x-rays with the atoms of the crystal is em phasized. The radiation scattered from an oscil lating atom is stated to consist of three com ponents the amplitudes of which depend on the amplitude of oscillation of the atom, and which differ in frequency. The first component, which gives rise to the normal reflections, has the fre quency /I of the primary x-ray beam, while the other two have frequencies /I±/I', where /I' is the frequency of atomic oscillation. The frequency change is unfortunately too small to be measured directly. This description of the mode of forma tion of the anomalous spots can be shown to lead to certain contradictions. For example, it can be used to show that the diffuse spots must be in part controlled by the elastic properties of the crystal; but such a dependence is essentially denied by Raman's primary assumptions. Preston concl udes, perhaps wi th undue cau tion, by saying that the quantum hypothesis as set out by Ramah is unlikely to be reconciled with the elastic wave explanation unless one or the other hypothesis is radically altered; and that it must be admitted, at aU events, that the thermal-elastic theory gives a reasonably ac curate account of the observed facts. On the other hand, Lonsdale,27 who, jointly with Smith, is responsible for the majority of the experi mental work on this subject in England, is a much bolder champion of the "classical" theory than others. She calls attention to some inac curacies in experiment, and to some errors in interpretation which affect the conclusions of Raman and his colleagues, and emphasizes the validity of the classical interpretation of the Indian work. The real importance of the anomalous x-ray spots is not, however, in this argument about their origin but in the use to which they may be put to help in the solution of other problems. As Lonsdale hinted nearly two years ago, the diffuse spots are very closely bound up with the elastic properties of the crystal. The original theory of 66 Waller has been interpreted and applied by Jahn28 in such a way as to relate the appearance and size of the spots to the elastic constants of the crystal. In general, the more inelastic the crystal, the bigger and fuzzier the spots are. The pattern for tungsten, for example, which is highly elastic, is much tidier and less intense than that for sodium or lead.29 The diffuse spots of tungsten, indeed, appear to be only one thirtieth as strong as those of sodium; while those for lead are half as intense as those of sodium. These values are in agreement with estimates made from J ahn's calculations. This is by far the most important result which has emerged from these studies to date, and it promises to be of very great practical importance in the future. It will allow elastic constants to be determined without straining the crystal. VII. ACTIVE NITROGEN There is a fascination about active nitrogen which brings its devotees back after years of absence. First discovered in 1900, the main features of its behavior are well known, but in the detailed explanations which have been offered for these there were many contradictions. The customary mode of excitation is by a high frequency electrodeless discharge. Under its influence the nitrogen glows with a yellowish light, which, under the proper conditions, remains visible for several hours after the discharge ceases. Lord Rayleigh30 has recently published a new series of papers on the subject, which provide some definite facts to be taken into account in any theoretical explanation of the phenomenon. In the first place he has, by means of a simple photometer, measured the luminosity of a known quantity of active nitrogen at intervals of a few seconds during its initial rapid decay; and from the data has arrived at the integrated light emitted per cm3 of nitrogen under the most favorable conditions. This turns out to be 3.18 candle-sec. per cm3 under standard temperature and pressure. Thence, knowing approximately the average wave-length of the emitted light, it is possible to calculate how many quanta are emitted per molecule of nitrogen under the con ditions of the experiment. The answer is 1.3 X 10-3• JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19If active nitrogen be compressed, its brightness increases, and vice versa. Raleigh finds that if the gas be handled in a simple manner as in a pump, the brightness varies inversely as the cube of the volume. This result is in agreement with two other series of experiments. In one of them, additional ordinary nitrogen was admitted to a bulb con taining the glowing gas, increasing the pressure about fivefold. The increase in brightness was very closely of the same order of magnitude, so that the brightness is proportional to the con centration of inert nitrogen, because here the concentration of active nitrogen remained fixed. In the other series, two bulbs containing nitrogen at equal pressures were connected by a stopcock. First, the gas in one was activated. Upon opening the stopcock, the active component diffused throughout the double volume of inert gas, reducing the active concentration to one-half of its original value. The intensity of the light was found to drop to one-fourth of its original intensity. Hence the luminosity is proportional to the square of the concentration of active nitrogen. Therefore, in a simple "pump" experi ment, wherein the concentrations of the active and inert gas must be increased together, the luminosity would be expected to vary as the cube of the concentration, in agreement with the result quoted above. A difficulty was found here, however, for upon allowing partially active nitrogen to expand into a vacuum, so that both concentrations were diminished by the same factor, the cube law was found to be a-somewhat inaccurate prophet of the inte~sity. - Rayleigh noticed that pieces of sheet gold, silver, copper, or platinum could be made red hot or even melted by immersion in active nitrogen, and he used this property to make an estimate of the amount of energy which must be abstracted from the gas in the process. The results are startling. In a "continuous flow" ex periment, in which the active gas passed in a stream from the site of the discharge to another bulb containing the small sheet of metal, it was found that the power radiated by the hot metal was such that it was necessary to assume that every molecule which passed through the dis charge carried some 5 or 10 electron volts to the metal foil. This might conceivably be interpreted as implying that the energy is carried by dis- VOLUME 14, FEBRUARY, 1943 socia ted atoms which associate again to N 2 at the surface of the foil, but only on the assumption that every molecule was dissociated. At first sight this seems so unlikely that it might be ruled out of consideration. It would follow that the luminous energy radiated as the familiar after glow, under favorable conditions, represents only a tiny fraction of the total energy transported by the gas to the metal. The addition of a little oxygen to glowing nitrogen has an important effect on the intensity. In one case it was increased 32-fold. Rayleigh shows that this must be a wall effect. The logical thing to do, then, is to experiment with various treatments of the walls. As found by Herzberg, strong preliminary heating in vacuum diminishes the subsequent afterglow. It is now found that heating in nitrogen at atmospheric pressure does the same. Heating in oxygen, even at 1-mm pressure, restores the container to its original helpful state. It seems clear that these effects cannot be explained easily by the formation or removal of surface layers of gas. The behavior of the glass surface is therefore complex. The behavior of the gas itself away from the wall is of more fundamental interest. By studying the glow at the center of a large flask, it was shown that the best results were obtained with very pure nitrogen, and that the addition of a trace of oxygen had no favorable effect in promoting the active nitrogen phenomena. Nearly all these details of behavior now receive an explanation in a paper by Debeau,31 who learned by repeating an old observation of J. J. Thomson's that nitrogen is practically com pletely dissociated in the electrodeless discharge. Once this is established, it is possible to construct a very definite picture of the process of formation and decay. After the dissociation, according to Debeau, the first stage is the formation of a "collision complex," which may then react with another molecuI-e to form nitrogen in the B3II excited state with 9.84 electron volts of energy, which subsequently degenerates into the A 3~ metastable state with emission of afterglow radiation. Alternatively, the collision complex may descend directly to the l~ state, the ground state of the nitrogen molecule, with liberation of energy as in Rayleigh's experiments on the heating of metal foils. This latter is much the 67 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 132.174.254.155 On: Tue, 23 Dec 2014 01:33:19more probable course of events, so that the heating phenomenon is a parallel and inde pendent rather than a previous or subsequent step in the life of active nitrogen. vm. GENERAL TRENDS The physics journals of this country grow slimmer and slimmer as an increasing number of her scientists turn from academic problems to investigations of military, naval, and aeronautical importance, the results of which cannot be broadcast to the world. Very few reports on entirely new topics are being published, and most recent papers, like those summarized in this article, are concerned with gathering up the loose ends of research programs initiated several years ago. Publications in the field of applied physics, such as appear monthly in this journal, are frequently devoted to general topics viewed from different angles. An experimental observa tion, being a matter of fact, need not be of recent date to be interesting. What makes it interesting is the light in which it is viewed, and here it often happens that modern theories from exceedingly diverse branches of physics can be trained on a single problem. For this reason the various reports of conferences which are pub lished from time to time take on an added importance for industrial work. These trends of the professional journals will undoubtedly be accentuated in the immediate future, and as time goes on, the need will grow greater for that peculiar type of intuition which is able to see the significant cross-relations between what may be superficially unrelated topics. In a year when physicists have little time to spare, the writer is grateful for the cooperation of those men who have lent him photographs to illustrate this article. He is also deeply in the debt of the authors of those papers which he has read recently, and from which he has quoted freely, often without specific acknowledgment. Finally, thanks are due to his colleagues, espe cially C. D. Hause and J. W. McGrath, who have read the manuscript and have given him many helpful suggestions. BIBLIOGRAPHY (1) N. F. Matt and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, 1940). (2) F. Seitz, The Modern Theory of Solids (McGraw-Hill Book Company, 1940). (3) A simple account is given by J. M. Kendall, Nature 150, 136 (1942). (4) R. C. Jones, Phys. Rev. 58, 111 (1940), and other references quoted there. (5) S. P. Frankel, Phys. Rev. 57, 661 (1940). (6) K. E. Grew, Nature 150, 320 (1942). (7) G. L. Ragan, W. R. Kanne, and R. F. Taschek, Phys. Rev. 60, 628 (1941). (8) G. Breit, H. M. Thaxton, and L. Eisenbud, Phys. Rev. 55, 1108 (1939). (9) H. Carroll, Phys. Rev. 60, 702 (1941). (10) H. Tatel, Phys. Rev. 61, 450 (1942). (11) W. Rarita and J. Schwinger, Phys. Rev. 59, 556 (1941). (12) H. F. Dunlap and R. N. Little, Phys. Rev. 60, 693 (1941). (13) J. S. Allen, Phys. Rev. 61, 692 (1942). (14) Kan Chang Wang, Phys. Rev. 61,97 (1942). 68 (15) B. Rossi and K. Greisen, Rev. Mod. Phys. 13, 240 (1941). (16) N. Nereson, Phys. Rev. 61, 111 (1942). (17) S. A. Korff and E. T. Clarke, Phys. Rev. 61, 422 (1942). (18) W. H. Bostick, Phys. Rev. 61, 557 (1942). (19) P. V. Auger, Phys. Rev. 61, 684 (1942). (20) B. Rossi and K. Greisen, Phys. Rev. 61, 121 (1942). (21) K. Greisen, Phys. Rev. 61, 212 (1942). (22) P. Auger and J. Daudin, Phys. Rev. 61, 549 (1942). (23) B. Rossi, K. Greisen, J. c. Stearns, D. K. Froman, and P. G. Koontz, Phys. Rev. 61, 675 (1942). (24) R. A. Millikan, H. V. Neher, and W. H. Pickering, Phys. Rev. 61, 397 (1942). (25) W. H. Zachariasen, Phys. Rev. 57, 597 (1940). (26) G. D. Preston, Nature 149, 373 (1942). (27) K. Lonsdale, Nature 149, 698 (1942). (28) H. A. Jahn, Proc. Roy. Soc. A179, 320 (1942). (29) K. Lonsdale and H. Smith, Nature 149, 21 (1942). (30) Rayleigh, Proc. Roy. Soc. A180, 123 (1942). (31) D. E. Debeau, Phys. Rev. 61, 668 (1942). JOURNAL OF ApPLIED PHYSICS [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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1.1712767.pdf

Stellar Temperatures G. P. Kuiper Citation: Journal of Applied Physics 11, 241 (1940); doi: 10.1063/1.1712767 View online: http://dx.doi.org/10.1063/1.1712767 View Table of Contents: http://aip.scitation.org/toc/jap/11/4 Published by the American Institute of PhysicsStellar Temperatures* By G. P. KUIPER Yerkes Observatory, University of Chicago, Chicago, Illinois 1. Introduction THE temperatures found m astronomical literature may be classed into two main groups, referring, respectively, to (1) the ob servable surfaces of the stars and planets, and (2) the interiors of these bodies. The first group depends directly on observations, although some well-established theoretical results are usually needed in the reductions; the second group depends entirely upon theoretical deductions, in connection with observed values for the boundary in order to make the problem defined. In addition we have temperature determinations of the highly diluted material filling the large spaces between the stars; they depend on observations interpreted with the aid of atomic physics. The total range of temperatures thus covered runs from about 30 absolute to about 50,000,000°. The lower limit is well exceeded by that obtain able in the physical laboratory; but the upper limit remains reserved (or the domain of as tronomy, as are the upper limits of pressure and density found in astronomical bodies. The reason is that all three limits can only be reached or approached as a result of the action of gravity on large compressible masses. Although the high pressures and densities found in the interiors of some of the stars can in no way be duplicated in the laboratory, some aspects of the high temperatures (the high kinetic energy of the particles) can now be covered by the fast particles produced in the study of nuclear processes. The energy generation in the stars, which is a result of the high tempera tures and densities prevailing near the center, is therefore accessible to physical interpretation. The progress made in interpretating this process of energy generation may in turn be considered to provide a general confirmation of the tempera tures computed for the interiors, on which the energy generation appears to depend in a very sensitive manner. * Presented at the American Institute of Physics Tem perature Symposium, New York, New York, November 2-4,1939. VOLUME 11, APRIL, 1940 2. Surface Temperatures of the Stars Obviously it is not possible to measure the surface temperatures of the stars by contact methods; but two other methods are applicable, depending, respectively, (1) on the laws of radiation, and (2) on the laws of ionization, in combination with those of atomic physics. The observed spectral-energy curve, corrected for absorption in our atmosphere, provides the empirical data for both methods. The first method deals largely with the "continuous spectrum," i.e., the intensity distribution of the smoothly-varying background in the spectrum on which absorption and emission lines appear. The second method deals with the total intensities (or profiles, if high dispersion can be used) of these absorption and emission lines. The empirical data are restricted to the wave length interval of about 2900A to 140,000A, the absorption in our atmosphere being practically complete outside these limits. The lower ab sorption limit is caused by the ozone in our atmosphere. From about 3000A to 25,000A the atmosphere is comparatively transparent, al though several heavy absorption bands due to water vapor and carbon dioxide occur in the near infra-red. These absorptions become even heavier between 25,000 and 80,000A (2.5 and 8 microns). Finally from 8 to 14 microns our atmosphere is comparatively transparent, at least if it is dry. The long-wave cut-off at 14 microns is due to carbon dioxide. The trans mission curve of the atmosphere for the Mount Wilson Observatory, published in 1930 by Pettit and Nicholson,! is reproduced in Fig. 1. It was computed for 0.7 em of precipitable water. Since that time many new details of the transmission curve, and their identifications, have been pub lished by AdeJ.2 As is well known, much of the earlier work on the transmission of the atmos phere is due to Langley, Abbot, and Fowle of the Smithsonian Institution; Fig. 1 is largely based on their work. 241 ! O.&1-1~;1HI~U.. I0.4~'4!1.<H~C" ,M'---\,HI-.-I' 3. Stellar Temperatures Derived from the Continuous Spectrum The continuous spectrum has so far been the more important of the two sources of information on stellar temperatures. With the transmission curve of our atmosphere known (Fig. 1) it is possible to reduce the observed energy curves of the Sun and the brightest stars to true energy curves (as they would be observed outside our atmosphere). It now appears that these energy curves roughly, but not exactly, correspond to those due to "blackbodies," i.e., to Planck curves. We must distinguish now between two cases: (1) the diameter of the star is known (like for the Sun); (2) the diameter is not known. In the first case the ordinates of the energy curve may be reduced to absolute units (ergs emitted per cm2 per sec.); but in the second case the ordinates are only relative. In the latter case, therefore, in fitting a Planck curve to the ob served energy curve only the shape can be used; and if the wave-length interval is rather small, only the slope. Nevertheless we shall find only one value of the temperature giving the best fit, because the slope varies in a monotonic way with temperature. This temperature is called color temperature. 3 If the star should radiate like a blackbody this color temperature (Tc) would be the true temperature of the emitting layer. Actually deviations from a Planck curve always appear if the wave-length interval covered is large. This shows that Tc will depend on the wave length used, and has no fundamental significance. I ts significance is a purely practical one: Te is easily determined empirically for a great number 242 FIG. 1. Transmission curves of (a) the atmosphere above Mount Wilson (the shaded curve); (b) water vapor 0.082 cm precipi table water; (c) microscope cover glass, 0.165 mm thick; (d) fluorite 4 mm; and (e) rocksalt 2 mm. of stars. Because after it has once been de termined for a few bright standard stars, further determinations can be made relative to these standards by making observations at the same zenith distance; then the atmospheric absorption is the same at corresponding wave-lengths for the stars compared, and is therefore eliminated in the ratio of the energies. If these spectrophoto metric comparisons are made photographically, with large telescopes, color temperatures may thus be determined for stars 10,000 times fainter than the faintest stars visible with the naked eye. We shall later mention some cases where this technique is of special interest. More progress can be made if the diameter of the star is known. \Ve saw that then the ordinates of the energy curve may be expressed in ergs/cm2 sec. We can still derive the color temperature, as before, but in addition the total amount of radiant energy leaving the star per cm2 per sec. may be found, by integration over the energy curve. This total flux is obviously a fundamental quantity; we may express it in terms of tempera ture by means of Stefan's law. In astronomy this temperature is called the effective temperature,4 abbreviated as Te. It is a precisely defined quantity, of great importance for the equilibrium of the stellar atmosphere, and related to the total luminosity, L, by the relation: (1) (R is the stellar radius, (J the radiation constant). The aim of stellar temperature determinations is the evaluation of Te for as many stars as possible. The color temperatures, Te, are only JOURNAL OF APPLIED PHYSICS derived as an intermediary step, simply because they are more easily determined for a great number of stars, and are already a first approxi mation to effective temperatures. They may be reduced to effective temperatures once the rela tion between Tc and Tc is established by means of those stars where both can be determined empirically. There is well-founded hope, however, that in the near future this relation may be reliably determined from theoretical consider ations, after the theory of the continuous ab sorption coefficient for the stellar atmosphere has been completed satisfactorily. From Eq. (1) it follows that Te may be found for those stars for which both Land R are known (or for which the emission per unit surface area, L/47rR2 may be derived). Only for the Sun are these quantities accurately known; Te(Sun) = 5713° absolute. For three. eclipsing binaries the data are still useful. Finally there are half a dozen stars for which the angular diameter could be measured interferometrically at Mount Wilson Observatory. They can be used also, since the apparent brightness gives L/D2, if D is the unknown distance to the Sun, and the angular diameter gives R/ D; hence L/ R2 is found independently of D. These ten objects exhaust the list at present, and they would be entirely inadequate if the color temperatures were not already good ap proximations to the effective temperatures, so that only some adjustments remain necessary. On the basis of the foregoing discussion it would seem natural to tabulate the Te just derived against Te. But there is an observational quantity even more easily determined with accu racy than Tc and closely related to it. It is the so-called spectral type. Originally the stars were ordered by their spectra according to the symbols A, B, ... etc.; but later several symbols were dropped as superfluous, and the order of the remaining ones, if arranged according to de creasing surface temperature, is now 0, B, A, F, G, K, M, if we omit a few rare types from con sideration. These types are further subdivided into 10 smaller steps: BO, B1, B2, .. ·B9, AO, A1, ... A 9, FO, etc. Most of the stars in a volume of space belong to the sequence 0, B, ... M, in which the radii of the stars progressively de crease, and also the luminosities. But the latter VOLUME 11, APRIL, 1940 TABLE I. The stellar temperature scale.' d = dwarf; g = giant. sP. AO A5 FO F5 dGO dG2 dG5 dKO dK2 dK5 dM2 Temperatures in centigrade absolute. T, 10,700° 8500 7500 6470 6000 5710 5360 4910 4650 3900 3200 sP. gGO gG5 gKO gK5 gMO gM2 gM4 gM6 gM8 5200° 4620 4230 3580 3400 3200 2930 2750 2590 1 For the details of the derivation, see a paper by the author. Astrophys. J. 88, 429 (1938). Table I of this paper corresponds to Table 13 of the paper quoted. decrease very much faster than the former, because most of the decrease of the luminosities is due to the decrease in Te (d. Eq. (1)). These stars are said to belong to the main sequence (or the dwarf stars) ; the Sun is one of them. For this sequence there is practically a one to one corre spondence between spectral type and Te, and also between Tc and Te. Hence the most practical procedure is to give Te immediately as function of the spectral type. This is done in Table I. In addition to the main sequence (or dwarfs) there are giant stars which are roughly lOO times larger in diameter than the dwarfs, and have somewhat larger masses. Since the value of the surface gravity is proportional to M / R2 it follows that the surface gravities, and consequently the pressures and densities in the atmospheres, are roughly 10LlO4 times lower in giants than in dwarfs. In general the spectral type is based on the strength of the spectral lines, and is therefore a measure of the degree of ionization and excitation in the stellar atmosphere. With the greatly diminished densities in the giants lower temperatures than existing in the dwarfs are sufficient to produce the same degree of ionization (or the same spectral type). This difference in temperature between giants and dwarfs is con siderable for types G and K, as is shown in Table I. Certain complications arise at the ends of the spectral sequence, connected with the sig nificance of the criteria used in the classification. We have indicated how L/D2 is obtained from integration over the energy curve (L = lumi nosity, D=distance of star to Sun). In practice it appears to be sufficient to do this for a number 243 of standard stars of different spectral type, and then to determine which corrections should be applied to the apparent visual brightnesses in order to obtain the so-called apparent bolometric brightness, L/D2. Such corrections are called bolometric corrections. The word bolometric refers in both cases to the nonselective recording of energy by the bolometer. For the Sun the energy curve has been deter mined in considerable detail,5 but for stars it is not possible to use high dispersion in the infra red, where the sensitive photographic methods cannot be used. For cool stars (T. < 4000°), which have a considerable infra -red in tensi ty, the energy curve is most readily studied by means of a sensitive thermocouple placed in the focus of a large telescope; the different intervals of wave length are then separated by means of filters.6 If the known atmospheric absorption is allowed for the true energy curve is at once obtained. For stars of about the solar temperature (with the maximum of the energy curve near 5000A) either the photographic or the photoelectric method of recording energies in the spectrum are useful; but both methods need a calibration by means of a standard having a known energy distribution, before the true energy curve is found. For still higher temperatures the same two methods may be used, but for temperatures much higher than 10,000° the results become uncertain, since the maximum of the energy curve then falls below 3000A. Fortunately for these high temperatures additional information may be obtained by the method discu~sed in Section 5. 4. Applications The previous discussion indicates how the energy curve, L/D2, Tc, and T. are determined. We have also mentioned the importance Te has for the interpretation of the steIIar spectrum, and the mechanism of the atmosphere. But the most urgent reason for making precise Te determi nations is a different one. The three most fundamental empirical data about a star are its mass, M, its luminosity, L, and its radius, R. It is the object of theoretical astrophysics to use these data for as many stars as possible in a study of stellar structure, stellar energy generation, and stellar evolution, which are perhaps the most fundamental astronomical 244 problems. The determination of masses is re stricted to certain suitable groups of binaries. For one of these groups (visual binaries with well-known D), L may be found, but R has to be computed from Eq. (1), after T. has been determined; whereas for the other group (eclipsing binaries) R is found in absolute units, but L can only be found by (1) from Rand Te, since D for these objects is too large for a determination from the parallax. 7 Another application of T. determinations is made in the study of white-dwarf stars. They are stars of low luminosity (dwarfish), rather high surface temperature (white), and hence, by Eq. (1), of very small radius. Since the masses of these stars are roughly equal to that of the Sun, the average densities (being proportional to M/R3) are excessively high, 10LI08 times water. The matter of these objects, except for an outer fringe which is gaseous, is in the degenerate state, obeying Fermi-Dirac Statistics. Here again we have conditions far outside the range obtainable in physical laboratories. Because the essential conclusions depend directly on the results of T. determinations the writer's recent results on eight of these objects are given in Table II. They are the first measures of temperature made on white dwarfs (most of which are very faint), and are only approximate and somewhat provisional except for the second star, which is well de termined. There is no doubt, however, that the orderof magnitude is correct for all stars included. It is necessary to point out that for white dwarfs the relation between the observed Tc and the desired T. cannot be established empiricaIIy. Fortunately the theory of the continuous ab sorption coefficient is sufficiently well advanced to be useful here; it appears that with exception of a certain range of temperatures, the relation of Tc to T. is the same as for the main sequence, where this relation is known empirically. In Table II we have expressed Land R in terms of the Sun; they are fairly well established. The mean densities, depending on "~1/R3, are less well known, first, because errors in R are in creased more than threefold in these values, and second, because M is known empirically only for the first three stars. For the remaining stars M has been computed from R by means of Chandrasekhar's theoretical relation between JOURNAL OF ApPLIED PHYSICS these two parameters. This relation involves, however, the mean molecular weigh t per electron, which may vary between 1 and 2 depending on the amount of hydrogen present. We have given the minimum values of the mean density, based on the extreme case in which no hydrogen is present. The other extreme, valid for pure hydrogen stars, would give values about 4 times higher; the actual value is probably closer to the lower limit than to the upper limit. The foregoing discussions will suffice as illus trations of the importance of the temperature problem in astronomy. In the next section we mention briefly how surface temperatures ranging from 10,000° to 100,000° are determined. 5. Stellar Temperatures Derivedfrom Spec tral Lines The strength of an absorption line in a stellar spectrum depends on a great number of factors, of which the temperature is only one: it depends on the ratio of the absorption coefficient inside the line (taken for each wave-length separately) to that just outside the line, in the continuous spectrum. This means that a theoretical predic tion of the strength of a line requires the knowl edge of the mechanism responsible for the con tinuous spectrum as well as of the concentration of atoms in the level from which the absorption takes place, the atomic transition probability, and the various effects which broaden the line (radiation damping; Doppler effect of the atoms, due to temperature motion as well as to con vection currents of the gases; and pressure effects). The concentration of atoms in the level under consideration depends on the abundance of the element, the degree of ionization' (which depends on temperature and surface gravity), the excitation potential of the level, and again the temperature. Since, however, in a stellar atmos phere no strict thermodynamic equilibrium ob tains (since otherwise no lines would be visible!) the populations of atoms in different levels is not strictly governed by the Boltzmann law. The precise allowance for this deviation from thermo dynamic equilibrium is very difficult. Hence it is in general not practicable to obtain temperatures from absolute intensities of absorption lines, even apart from the difficulty caused by the uncer tainty of the abundance of the element. VOLUME 11, APRIL, 1940 TABLE II. Surface temperatures of white dwarfs, and results derived therefrom. NAME Te L R MEAN DENSITY Sirius B 95000 0.0030 0.020 170,000 40 Eridani B 13,500 .0080 .016 160,000 van Maanen 2 8200 .00016 .0062 12,000,000 AC 70°8247 35,000± .021 .0039 >30,000,000 Ross 627 15,000± .0010 .0046 > 19,000,000 Wolf 489 5070 .00007 .0105 > 1,000,000 Wolf 457 10,000± .00011 .0034 >50,000,000 Wolf 219 15,000± .00052 .0033 >50,000,000 Several of these difficulties are avoided if the behavior of a line or a series of lines originating from the same level (as the Balmer series of hydrogen) is studied along the main sequence (0, B, A, etc.). Assume for a moment that the continuous absorption coefficient were constant along this sequence. Then the variation in the strength of the line (or the series) would indicate the variation of the number of atoms in the appropriate level. Now consider as an example the Balmer series, which does not originate from the ground level of hydrogen, but from the second level. For low temperatures nearly all of the atoms are in the ground state, and very few in the second state. For very high temperatures nearly all of the atoms are ionized, and again very few atoms will be in the second state. In either case the Balmer lines will be weak. But there will be an intermediate temperature at which the ionization is not yet too far advanced, but which is yet high enough to raise many of the neutral atoms to the second quantum level. Obviously this interplay of two opposing tend encies will lead to a maximum of intensity of the Balmer series at medium-high temperatures. Actually the position of the maximum will be somewhat affected if we now take into account the known variation with temperature of the continuous absorption. But the assumption of the constant abundance along the main sequence remains unavoidable. Fortunately we have evi dence that most of the stars of the main sequence have roughly the same composition; furthermore, the position of the maximum is not based on one star, but on many stars, so that individual variations of abundance are mostly eliminated in the mean result. Finally we have not one, but several elements which are suitable for this analysis: H, He, N, Si, C, and 0, some of them in 245 TABLE III. The stellar temperature scale. Temperatures in centigrade absolute. SPEC. LOG Te T, SPEC. LOG Te T, ---- OS 4.9 80,000° B2 4.31 20,000° 06 4.8 63,000 B3 4.27 18,500 07 4.7 50,000 B4 4.23 17,000 08 4.60 40,000 B5 4.19 15,500 09 4.50 32,000 B6 4.16 14,500 BO 4.40 25,000 B8 4.09 12,300 B1 4.36 23,000 AO 4.03 10,700 various stages of ionization. These different ele ments give independent results, in which the small variations in abundance remaining in the determination of the intensity maxima enter as accidental errors. From the remarkable accord between the temperature determinations as a function of spectral type, obtained from the different elements, we conclude that the effect of variations in abundance on the temperature scale may be considered to be negligible.8 The resulting temperature scale is found in Table III; it has been taken from a paper by the writer,9 and is largely based on computations by Pannekoek (cf. reference 8). 6. Planetary Temperatures For stars we have no a priori knowledge of the surface temperatures. But for the planets and the Moon we are able to predict the surface tempera tures with fair accuracy if we assume that the planetary heat is due entirely to absorbed solar radiation, and not partly to internal sources. Suppose the planet absorbed and emitted as a blackbody. Then, if L is the known luminosity of the Sun, the energy received by the planet per unit area at a point where the Sun is in the zenith, is L/47rD2, if D is the known distance of the planet from the Sun. Since by hypothesis all this energy is absorbed and re-radiated, the surface temperature T is given by ~=fJT\ or T= (~)}.~= 390:K, (2) 471'D2 471'fJ D' D, if the numerical values of Land fJ are substituted, and if D is expressed in astronomical units (distance Sun -Earth = 1). Actually the planets and the Moon are not black; but their reflectivity can be determined from the measured brightness and the known distances of the planet from the Sun and the 246 Earth, together with the known brightness of the Sun. These reflectivities vary from about 0.07 for the Moon and Mercury, to about 0.59 for Venus. The light that is not reflected is absorbed, and its amount is therefore also known. We may call the absorbing power A. The emissivity for planetary radiation, E, can only be estimated. Eq. (2) now becomes AL ---= fJET4 or 471'D2 ' (3) This relation holds for a nonrotating planet, for the so-called subsolar point (the point that has the Sun in the zenith). If we take a point where the zenith distance of the Sun is Z the amount of heat received and emitted per unit area will obviously contain the additional factor cos Z, and the temperature (cos Z)l. If the planet rotates fast (so that the daily temperature variation is small) an additional factor 1/7r is introduced in the amount of heat received, and 71'-1 in the temperature. If the planet rotates slowly the temperature fluctuations around this "average" will be greater, depending on the properties of the planetary atmosphere, and the speed of rotation. The principle of determining planetary temper atures empirically is the same as that described in Section 3 for stars. Although at first sight the case might seem more complicated because the radiation received from the planet does not only consist of the heat radiated by the planetary surface but also of reflected sun light, yet this complication disappears in practice since the two kinds of radiation are practically entirely sepa rated in wave-length, so that a filter may be used to distinguish between the two. This separation is a result of the difference in temperature be tween the two sources, about 6000° and 3000K, so that the wave-lengths of the maxima fall at about 5000A and 100,000A (0.5 and 10 microns), respectively. Fortunately our atmosphere is fairly transparent between 8 and 14 microns if it is sufficiently dry (Fig. 1), so that the surface temperatures of most of the planets, and the Moon, may be measured. Only for the planets beyond Saturn the distance to the Sun becomes so large (D> 10) that the temperatures drop below 100oK; for these planets the radiation of JOURNAL OF APPLIED PHYSICS FIG. 2. March of absolute temper ature, T, of energy received from the sun by the moon, ER, and of energy radiated, E, during the total lunar eclipse of June 14,1927. o , / " I ! ! : :' ,I I , , I , , I I ,- / ~\ 0 / / \ 0 /1 0. larK wave-length shorter than 14 microns is too weak to be measured. For the planets Mercury, Venus, Mars, Jupiter, and Saturn, and for the Moon, temperatures have been determined empirically, chiefly at the Mount \N'ilson and Lowell observatories. For Mercury (which has no atmosphere) Pettit and Nicholson1o found about 330°C for the subsolar point when the planet has its mean distance from the Sun. This value is only about 10°-15° lower than computed theoretically for a nonrotating planet (as indicated above). Part of the difference may be due to conduction. For Venus Coblentz and Lampland,ll and Menzepz found about 50°C at noon. Pettit and Nicholson13 found lower values, but at both observatories the day and night temperatures on the planet were found to be rather similar, indi cating that the planet is rotating slowly. (A fast rotation is excluded on the basis of measured Doppler shifts.) The temperature found for Venus may well refer to some average layer in the atmosphere, since this atmosphere is quite opaque to visual, and even infra-red, light. On Mars the temperature at the subsolar point is near O°C, probably somewhat higher in certain dark areas. The polar caps are about -70°C. The night at Mars is probably very cold, perhaps -60°C. The dew point at Mars is probably about as low as this latter value, in view of the ex tremely small water-vapor content of the atmos phere. The existence of liquid water on the planet (for which there is no direct observational evi- VOLUME 11, APRIL, 1940 \~ -- -___ 0 __ ..j../ ~~ -"'-.. ... --_ ... -... .., ....... -..... -........... -I"'. """ -.. -... -~ l II 11. I 2- PAc-me STA/lPJ\Rl> TIl'!!. I !lAM. dence either) can therefore probably be excluded. Jupiter appears to have a temperature of about -135°C, Saturn -150°C, and Uranus less than -185°C. These temperatures refer to an atmos- pheric layer in each case. For none of the planets is there definite evidence that the measured tern pera tures are higher than those to be expected theoretically; hence there is no evidence of internal heat raising these temperatures. For the Moon, Pettit and Nicholson14 found for the sub solar point + 134°C when viewed from above, and only +85°C if viewed horizontally. The difference is explained by the rough surface of the Moon which allows more heat to escape vertically than horizontally. Taking into account the solar radia tion lost by reflection the theoretical temperature at the subsolar point comes out to be 101°C. The dark side of the Moon was found to be about -150°C, a value difficult to determine with precision because of its extreme lowness. A series of very interesting observations was made by Pettit and Nicholson during a lunar eclipse. A point near the edge of the Moon was kept under observation for about 5 hours, during which it was first illuminated by the Sun, then passed into the shadow of the Earth, and finally emerged again. The diagram showing the temper ature variation is reproduced in Fig. 2. It shows the extremely rapid cooling of the surface (from about 70°C to -115°e) which must be the result of a small conductivity, comparable to that of pumice or volcanic ash which, on the basis of measures of the polarization of the reflected light, 247 had already been assumed to cover the surface of the Moon.15 7. Interstellar Temperatures Although interstellar space is a far better "vacuum" than the best obtainable in the phys ical laboratory, it is by no means devoid of matter. The dark obscuring clouds in the Milky Way, as well as the "interstellar" absorption and emission lines prove this conclusively. These two lines of evidence point at once at the two kinds of matter occupying space. The obscuring clouds are composed of dust particles of which a good fraction have diameters between 0.1 and 1 micron; this follows from the scattering properties of these particles, and estimates that may be made of the mass of absorbing clouds. The interstellar absorption and emission lines prove the existence of gases in the atomic state. Hydrogen is by far the most abundant interstellar element, as was recently shown by Struve. Calcium, sodium, titanium, and potassium have been found from their absorption lines; oxygen and nitrogen are present in emission. Some interstellar lines are still unidentified. The temperatures to be ascribed to the inter stellar particles need some explanation. There is no more difficulty in assigning a definite tempera- . ture to a dust particle than to a planet, and it may be computed by an equation similar to (2) or (3). A particle far away from anyone star would be exposed, on the average, as we are on a clear night by the starry skies except, of course, that no atmosphere would absorb the ultraviolet light, and that the stars would cover a sphere, not a hemisphere. Eddington has shown16 that such a particle would assume a temperature of about 3°K. But the radiation in interstellar space is very far from being blackbody radiation corresponding to 3°K. It contains much ultraviolet light due to the 0 and B stars in the sky. Hence ionization and excitation of atoms will take place in spite of the low energy density. Since the density of matter is also very low, recombinations will occur at a very slow pace. Without computations it is seen, therefore, that a fair proportion of the atoms may well be ionized. The ionizations will in turn lead to considerable velocities of the ions and electrons. Since collisions between particles 248 are still sufficiently frequent the velocities of the particles forming the interstellar gas will approxi mate a Maxwellian distribution corresponding to a high temperature, roughly 10,000°, and in the proximity of hot stars even higher. Obviously there is no universal temperature of the inter stellar gas; it depends on the position with respect to the stars, as the planetary tempera tures depend on the distance from the SunY 8. Internal Temperatures of Stars It can be easily shown1s that if a star as a whole is in hydrostatic equilibrium (which is probably true for normal, constant, stars), and if the perfect-gas law holds throughout (this assump tion can be verified afterwards, and found to be justified), it is possible without any further assumptions about the internal structure of a star to obtain a minimum value for the average internal temperature, T: T>3,840,000(,umM/R, (4) in which lvi is the mass of the star, R its radius, both expressed in terms of the Sun, ,u its mean molecular weight, and (3 the ratio of the gas pressure to the total pressure (gas+radiation). For stars with M < 10 we have (3""'1. Since the matter in a star is mostly ionized, and since the atomic weight is roughly double the atomic number except for hydrogen, we have ,u""'2 if no hydrogen is present, and ,u= ~ if the matter is all hydrogen. If the additional assumption is made that the stars are built on the same pattern (except for factors of scale) then the relation T= C· (,u(3)M/R (5) holds for corresponding points in these stars, C depending on the point selected, and on the pattern common to the stars considered. We derive some interesting conclusions from (5). For stars on the main sequence in the neighborhood of the Sun (types A, F, G, and K), having the same hydrogen content (I-' = constant), the internal temperatures will vary only slowly, because (3"'1, as mentioned before, and M and R change in the same direction. But since M changes faster than R, the internal temperatures will increase if M increases. It is this slow in crease in internal temperature which is sufficient, in connection with the high sensitivity of the energy generation on T(,...., P8) to cause a very JOURNAL OF APPLillD PHYSICS • STAR Sirius Mizar Procyon Sun 70 Oph. A Capella A TABLE IV. Central temperatures. TYPE dAI dA2 dF3 dG2 dKI gG4 T (CENTRAL) 24,500,000° 22,000,000 18,000,000 19,500,000 17,500,000 5,100,000 considerable increase of L with M (empirically L = M4i for the range of masses considered). The numerical values of the internal tempera tures depend on numerical integrations which cannot be discussed here.I9 We quote the values for a few stars in Table IV. It follows from (5) that the giants (with large radii) will have low internal temperatures. This gives rise to difficulties with the explanation of the energy generation which have not yet been solved completely. On the other extreme are the massive 0 stars having comparatively small radius. If they were built on the same model as the less massive stars their in'ternal temperatures would exceed 100,OOO,OOO°C, and the energy generation would be too large by a considerable factor. Chandrasekhar has recently removed this difficulty by proposing a shell-source model for these stars, which leads to a more homogeneous density distribution than that of ordinary stars, and requires much lower temperatures for the hydrostatic equilibrium, temperatures not much higher than the minimum values required by (4), about 40,000,000°. Perhaps the highest internal temperatures occur in sub-dwarfs, stars roughly t to t of the diameter of the Sun, having a small hydrogen content (M'-"2). The internal temperatures of some of these stars may well exceed 50,000,000°; their small hydrogen content would not lead to difficulties with the energy generation. White dwarfs have still smaller radii, but Eq. (5) is not applicable to them, because the matter is degenerate. The internal temperatures are still somewhat uncertain, but are probably at most about 20,000,000°. Bibliography 1. E. Pettit and S. B. Nicholson, Astrophys. ]. 71, 104 (1930) 2. A. Adel, Several papers in the Astrophysical Journal, 1938-1939. 3. The astronomical usage agrees here with that recom mended in the glossary of the Temperature Sym posium. 4. It corresponds to the radiation temperature of the glossary. The astronomical usage dates back several decades. 5. Cf. A. Unsold, Physik der Sternatmosphiiren (1938), pp. 27-40. 6. Reference is made to the extensive series of measures by E. Pettit and S. B. Nicholson with the 100-inch telescope at Mount Wilson, Astrophys. ]. 68, 279 (1928), and 78,320 (1933). The instruments used in astronomical measurements of radiation, and their complete theory, are described by B. Stromgren, Handbuch der Experimentalphysik 26, 795 (1937). 7. The reader will find the details of the derivation of M, L, and R for many stars in a paper by the author, Astrophys, ]. 88, 472 (1938). A theoreti cal discussion of the empirical data is found in Chandrasekhar's recent monograph, Stellar Struc ture (University of Chicago Press, 1939). 8. Space does not permit us to indicate more than some of the principles involved. For a more "technical account, d. A. Pannekoek, Astrophys. ]. 84, 481 (1936). (Pannekoek's results for temperatures lower than 8000° have to be modified in view of more VOLUME 11, APRIL, 1940 recent work; but those for higher temperatures, in which we are chiefly interested, still stand.) For a general account on the interpretation of absorption lines, d. O. Struve, Popular Astronomy 46,431-451, 497-509 (1938). Also: D. H. Menzel, ibid. 47, 6-22,66-79, 124-140 (1939). 9. G. P. Kuiper, Astrophys. ]. 88, 429 (1938). 10. E Pettit and S. B. Nicholson, Astrophys. ]. 83, 84 (1936). 11. W. W. Coblentz and C. O. Lampland, Popular Astronomy 30,551 (1922). 12. D. H. Menzel, Astrophys. ]. 58, 65 (1923). 13. E. Pettit and S. B. Nicholson, Popular Astronomy 32, 614 (1924). 14. E. Pettit and S. B. Nicholson, Astrophys. J. 71, 102 (1930). 15. For further information about plaJ1etary temperatures the reader is referred to the papers already quoted, and to: Menzel, Coblentz and Lampland, Astro phys. J. 63, 177 (1926). H. N. Russell, The Solar System and Its Origin (Macmillan, 1935). T. Dun ham, Pub!. Astr. Soc. of the Pacific 51, 253 (1939). 16. A. S. Eddington, The Internal Constitution of the Stars (Cambridge University Press, 1926), Chapter 13. 17. For an analysis of the physical conditions of the interste!lar gas, particularly of hydrogen, cf. B. Stromgren, Astrophys. J. 89, 526 (1939). 18. Cf. B. Stromgren, Ergeb. d. exakt. Naturwiss. 16, 467-470 (1937). Chandrasekhar, reference 7. 19. Chandrasekhar, reference 7. 249