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1.1702707.pdf | Influence of Preferred Orientation on the Hall Effect in Titanium
Louis Roesch and R. H. Willens
Citation: J. Appl. Phys. 34, 2159 (1963); doi: 10.1063/1.1702707
View online: http://dx.doi.org/10.1063/1.1702707
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Downloaded 10 Mar 2013 to 131.170.6.51. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 8 AUGUST 1963
Influence of Preferred Orientation on the Hall Effect in Titanium*
LOUIS ROEscHt AND R. H, WILLENS
W. M. Keck Laboratories, California Institute of Technology, Pasadena, California
(Received 14 January 1963)
The effect of preferred orientation on the Hall effect in titanium was examined in an attempt to resolve
the inconsistent results of previous investigators. Three specimens of iodide titanium were prepared with
different textures and the Hall coefficient of each was measured between 4.20 and 295°K. The Hall coefficient
was found to depend on both temperature and preferred orientation. At room temperature, it was deter
mined to be -1.8X lO--l1m3 jC in two specimens and + 1.2 X lO--l1m3 jC in a third one. From the x-ray analy
sis of the texture and the Hall coefficient data, it was concluded that the positive component of the Hall
coefficient was associated with the hexagonal axis of titanium, being parallel to the magnetic field. An
approximate calculation estimated the principal components of the Hall tensor to be RII = +4.2 X lO--11
and Rl=-7.7XlOm3jC, where RII is the Hall coefficient when the magnetic field is parallel to the c axis
and Rl the Hall coefficient when the magnetic fiela is perpendicular to the c axis. The values of RII and Rl
given above can account for the spectrum of previously published Hall values for titanium.
I. INTRODUCTION
THE results published on Hall effect in titanium
are inconsistent in both their values and signs.
Figure 1 shows the Hall coefficient as a function of
temperature as determined by some previous investi
gators.1-4 At room temperature both positive and nega
tive values have been reported. The broad spectrum of
Hall effect data might be attributed to the influence of
impurities, specimen size effects, and crystallographic
texture. The purpose of this investigation was to study
the effect of crystallographic texture. Since single crys
tals large enough to enable the measurement of the
Hall effect in various crystallographic directions are
very difficult to obtain, this work was carried out with
polycrystalline specimens having a high degree of pre
ferred orientation. The crystallographic character of the
Hall effect can then be deduced from measurements on
the orientated samples and a subsequent determination
of the texture by an x-ray diffraction method.
In an anisotropic media (single crystals, hereafter)
the Hall effect is no longer described by a scalar quantity
RH, the Hall coefficient, but by a set of galvanomagnetic
tensors Rk(B). In the phenomenological theory de
veloped by Okada,5 the components of the electric field
in the presence of a magnetic field are
(1)
where Ei are the components of the electric field to the
crystal, Jj the components of the current density, Pij(B)
the resistivity tensor modified to take magnetoresistance
effects into account, and €iik is the antisymmetric Kro
necker. The last term in the preceeding equation is
* This work was sponsored by the U. S. Air Force Office of
Scientific Research. t Present address: not available.
1 S. Foner, Phys. Rev. 107, 1513 (1957).
2 G. W. Scovil, J. App!. Phys. 27, 1196 (1956); and 24, 226
(1953) .
3 T. G. Berlincourt, Phys. Rev. 114, 969 (1959).
4 J. M. Denney, Ph. D. thesis, California Institute of Technology
(1954) .
D T. Okada, Mem. Fac. Sci. Kyusyu Univ. B 1, 157 (1955). referred to as the Hall field EHi• When the Hall field
is proportional to the magnetic induction, as is the case
for titanium, its components are
(2)
The components of the second-rank tensor Rk•m are
called galvanomagnetic coefficients. The number of
independent components of this tensor depends on the
point group symmetry of the crystal. When a coordinate
axis is chosen parallel to the principal axis of hexagonal
symmetry, this tensor is diagonal and has only two inde
pendent components: Rl.1= R2.2 = -Rl and Ra•3=R!!.
The subscripts II and ..1 correspond to the Hall coef
ficient when the magnetic induction is, respectively,
parallel and perpendicular to the hexagonal axis. For a
general orientation of the magnetic field with respect to
the hexagonal axis, it can be shown that the Hall coef
ficient, measured in the usual way, with EH, B, and J
mutually orthogonal, is
(3)
where () is the angle between magnetic induction and
the hexagonal axis.
5 RH X lOll mo/COULOMB
•
a S. FONER
G G. SCOVIL
• J.DENNEY
'" T. BERLINCOURT. 8 0 [000
TEMPERATURE ("K)
FIG. 1. The Hall coefficient of titanium versus temperature
as determined by some previous investigators.
2159
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II. SAMPLES AND EXPERIMENTAL METHODS
The material used in this investigation was iodide
titanium obtained from the U. S. Army Ordnance Corps,
Watertown Arsenal Laboratories. The latice parameter
measurements (el a= 1.5874) and the ratio between the
resistances at room temperature and liquid helium
(ratio = 30) indicated that the material was of high
purity. Three specimens were prepared by cold-rolling
a piece of titanium, initially 0.4 cm thick, to a thickness
of 0.02 cm. Part of this strip was further reduced by
longitudinal rolling to about 0.008 cm and became
specimen No. 1. The remaining part of the strip was
cold-rolled to a thickness of 0.010 cm in the transverse
direction in order to produce a different texture and
became specimen No.2. The sample was later annealed
at 950°C (above the a -7 (3 transformation) and be
came specimen No.3. Before the measurements were
made, the samples were heat treated to relieve the
residual stresses (one hour at 450°C). To prevent con
tamination, the samples were sealed in a Pyrex or quartz
tube filled with an atmosphere of purified helium. In
addition, a titanium "getter" was enclosed in the tube
to capture the remaining traces of oxygen and nitrogen
prior to heat treating the sample.
The quantitative (10.0) pole figure of each sample was
obtained by the x-ray diffractometer method.6 The
samples were studied only in transmission because the
central part of the pole figure was not essential for an
unambiguous interpretation of the texture. The speci
mens were mounted on an automatic pole figure goniom
eter which performed the necessary rotations auto
maticallyand continuously. The diffracted intensity was
detected by a proportional counter, analyzed by a pulse
height discriminator, and printed on a paper tape. The
numbers thus obtained were corrected for absorption
and background and used to determine the pole figure.
The Hall voltage was measured using a method very
similar to that used by Foner.7 A three-probe geometry
was used to obtain zero voltage between the Hall po
tential probes in the absence of the magnetic field. The
Hall voltage was balanced to the nearest 10-7 V by
means of a Wenner potentiometer. The remaining part
(less than 0.1 }.IV) was read by interpolation on the
galvanometer scale or on a dc voltmeter after amplifica
tion by a dc amplifier. When appropriate electrical and
thermal shielding was provided, the rms noise level was
about 0.005 }.IV. In order not to include parasitic ther
moelectric or transverse galvano- and thermomagnetic
voltages (e. g., the Ettingshausen effect) in the Hall
measurement, titanium wires were used as potential
leads and changed to copper in an isothermal oil bath
outside the magnetic field. The Hall voltage was taken
as an average of the voltage measured with the magnetic
field in one direction and then reversed twice.
6 B. D. Cullity, Elements of X-ray Diffraction (Addison-Wesley
Publishing Company, Reading, Massachusetts, 1956), p. 285.
7 S. Foner, Phys. Rev. 91, 20 (1953). Temperatures between 4.2° and 29SoK were obtained
by putting the specimen inside a metallic liquid helium
Dewar. This double Dewar consisted of two coaxial
cylindrical tanks of stainless steel terminated by a
copper section of a rectangular cross section which
fitted into a 2.7 -in. air gap of a 12-in. electromagnet.
Besides room temperature, isothermal conditions were
obtained at the boiling points of liquid helium (4.2°K),
liquid nitrogen (77.35 OK), and freon 22 (232°K). At
these points the temperature was maintained over a
long period of time and the magnetic field dependence
of the Hall voltage was investigated. Measurements
between these points were obtained as the Dewar slowly
returned to room temperature. Above liquid-nitrogen
temperature, the Hall measurements were inaccurate
because of fluctuations in the measured Hall voltage.
This probably was due to the nonuniform temperature
distribution in the Dewar. The temperature was meas
ured by a copper-Constantan thermocouple calibrated
over the entire temperature range and, below 77°K, by
calibrated carbon resistors. Due to inaccuracies in cali
brations, the temperature was measured with an un
certainty of about 1°C,
III. EXPERIMENTAL RESULTS
The textures of each sample are described in Table 1.
The angle of tilt of the basal planes (e) given in Table I
was the predominant texture. There was a distribution
of poles about this value. In specimen 1 more than 90%
of the (00.1) poles were within an angle of ±15° of 45°.
Specimen 2 had two minor orientations of the basal
planes, (b) and (c), which contained less than 10% of
the poles associated with the main orientation (a). For
this specimen, more than 85% of the (00.1) poles were
within an angle of ±100 of predominant tilt of 30°.
Specimen 3 had two predominant orientations which
can be given approximate relative weighting factors of
0.7 and 0.3 for (d) and (e), respectively. There was a
minor orientation (f) which contained less than 5% of
the poles. For both orientations (d) and (e) 85% of the
poles were within an angle of ±8° of their respective
angles.
The value of the Hall coefficient was found to be inde
pendent of magnetic field strength. The temperature
TABLE I. Predominant textures of the specimens.
Angle between [OO.lJ
Angle between [to.OJ direction and a
Specimen direction and the perpendicular to
longitudinal axis specimen surface
of the specimen (i. e., 0)
0° ±45°
2 a 0° ±30°
b 0° 90°
c 90° 90°
3 d 0° ±30°
e 90° 90°
f 900 90°
Downloaded 10 Mar 2013 to 131.170.6.51. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsPRE FER RED 0 R lEN TAT ION AND HAL L E F FEe TIN TIT A N I U M 2161
variation of the Hall coefficient between 4.2° and 295°K
is shown in Fig. 2. As can be seen in Fig. 2, the tempera
ture variation was very similar in all the specimens. The
Hall coefficient varied strongly between 20° and 700K
and little outside this range. The major difference be
tween specimen 2 and the other specimens was the
positive value of the Hall coefficient above 77°K. This
difference is attributed to crystalline anisotropy.
IV. DISCUSSION
The scatter of the previously published data for the
Hall coefficient of titanium had been associated with
impurities, specimen size effects, and crystallographic
texture. In the present experiment, the three specimens
had comparable purity and the fact that specimens 2
and 3 (which were actually the same piece of material
but with different textures) gave drastically different
Hall constants (even the sign of RH changed above
77°K) seems to rule out the influence of impurities as
predominant cause of differences in RH• It is likely that
variations of the Hall coefficient due to impurities would
only be important at low temperatures where impurity
scattering is the predominant scattering mechanism.
Size effects cannot explain the observed differences since
specimens 1 and 3 had different thicknesses but similar
Hall coefficients. Furthermore, size effects become im
portant only when the electron mean free path becomes
comparable with the thickness of the sheet. This was not
the case with the specimens studied, especially near
room temperature.
The experimental results have a consistent interpreta
tion by assuming a dependence of the Hall coefficient on
crystallographic direction. In each grain the Hall coef
ficient is a function of the angle between the magnetic
field and the hexagonal axis and varies from R[[ to Rl in
a continuous fashion. In the case of a polycrystalline
sample, the measured Hall effect is a weighted average
of the contributions from the different grains depending
on their orientation with respect to the magnetic field.
By replacing each sample by a single crystal having its
ideal orientation, an order-of-magnitude estimate can
be made for RII and R1• Substituting in Eq. (3) the
measured predominant textures, the following equations
relate the Hall coefficients of specimens 1, 2, and 3,
respectively, to RII and R1•
and RIll = HRII+R 1),
RIl2=t(3RII+R1), (4)
(5)
(6)
If the values of RHI and RH2 are used to calculate RII
and R1, at room temperature RII= +4.2 and R1= -7.7
in units of 1O-llm3/C. Substituting these values into
Eq. (6), RIlB= -1.5 instead of the measured value
of -1.84. Similarly, at liquid-nitrogen temperature
RII = + 2.63 and Rl = -7.9 and a calculated value for
RIl3 of -2.4 as compared to the measured value of RH X lOll mYCOULOMB
1.0 --"~ ------
~------------------------
-1.0
I--2.0
Z
!!! -3.0
U it -4.0
~ -5.0
.J .J
~ -6.0
o 50 .. SPECIMEN I
'" SPECIMEN 2
• SPECIMEN 3
100 150 200 250
TEMPERATURE ('K)
FIG. 2. The Hall coefficient versus temperature for
the three titanium specimens. 300
-2.75. At liquid-helium temperature, the corresponding
values are R,,= -2.4 and R1= -11.4. The calculated
value for RIl3= -6.75 as compared to the measured
value of -6.45.
These results are qualitatively consistent. For speci
men 3 one of the components of the texture favors a
positive Hall coefficient and the other a strongly nega
tive value; the over-all effect is a negative RH3. The
values of RII and Rl given above are somewhat question
able because of the crude approximation of a textured
sample to a perfect single crystal. A refined calculation
with a more detailed expression for the distribution of
the (00.1) poles could be made, but this probably would
not produce much more reliable numbers. Also, the
effect of grain boundary scattering and all lattice imper
fections has been neglected. However, it is believed that
the difference in sign of the two coefficients is significant.
It should be noted that the values of R" and Rl given
above can account for the spectrum of Hall coefficient
values previously published.
The shape of the Hall coefficient versus temperature
curves is essentially the same for all three specimens.
The semiclassical theories on which the interpretation
of Hall effect data is usually based, do not seem appro
priate to describe the results obtained for titanium. An
isotropic two-band model could explain the results for
one particular sample by a proper choice of the densities
of states and mobilities in each band and of their tem
perature variation, but the same set of parameters can
not be used to interpret measurements made on other
properties (e. g., magnetoresistance) of the same sam
ple.3 In addition, this model cannot predict different
values for RIl for different specimens of the same
material. The similarity between the Hall coefficient
curves and the lattice specific heat curves suggests the
influence of a changing mechanism of scattering with
temperature. The mechanisms of impurity, small angle
electron-phonon, and classical phonon scattering, each
predominating in their respective temperature regions
Downloaded 10 Mar 2013 to 131.170.6.51. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions2162 L. ROESCH AND R. H. WILLENS
may lead to different values of the relaxation time and
he~ce the Hall coefficient. However, if this is true, a
similar effect should be observed for all metals. This is
not the case (e. g., the results on thorium and niobium,
Ref. 3). A more satisfactory explanation may be found
in very sensitive overlap conditions of the Fermi sur-face and filling of pockets between the Fermi surface
and Brillouin zones. Other measurements, to determine
some of the qualitative features of the Fermi surface in
titanium, seem to substantiate this possibility.S
8 R. H. Willens, California Institute of Technology, Pasadena,
California (to be published).
JOURNAL OF APPLIED PHYSICS VOLUME H. NUMBER 8 AUGUST 1963
Grown-In Dislocations in Calcium Tungstate Crystals Pulled from the Melt
A. R. CHAUDHURI AND L. E. PHANEUF
SPerry Rand Research Center, Sudbury, Massachusetts
(Received 15 January 1963)
Etch pits on (001) surfaces of calcium tungstate crystals, formed on treatment with hot dilute solutions
of hydrochloric acid, were identified to be dislocation etch pits. The technique is able to follow the motion
of dislocations during annealing by the formation of new sharp-bottomed pits adjacent to flat-bottomed pits
at the initial dislocation positions. Inclusions, both solid and gaseous, are an important source of dislocations
in crystals of calcium tungstate grown from the melt.
INTRODUCTION
ETCH-PIT techniques exist for a large number of
nonmetallic crystals.! In this paper a dislocation
etch-pit technique is described for the tetragonal ionic
crystal calcium tungstate. Some observations are also
presented on densities of dislocations in the crystals
pulled from the melt.
CRYSTAL GROWTH
Calcium tungstate powder (99.99%), purchased from
A. D. McKay, was used as the starting material for the
crystals. The general recommendations of Nassau and
Sur'Qce, B
(1'~'."
(001) CleovaOt Plan.
Gr."th.
DlrlCtion k------------9 11O
Norm~1 t. 110
SUrfacII 8
FIG. 1. Stereographic projection showing growth direction
of crystals. Insert shows specimen geometry.
1 W. G. Johnston, Progress in Ceramic Science, edited by J. E.
Burke (Pergamon Press, Inc., New York, 1962), Vol. 2, p. 1. Broyer2 were adopted in gro!ing the crystals in air from
a rhodium crucible by the Czochralski technique. The
crucible (l!-in. diameter X l!-in. heightXO.06-in. wall)
was also the susceptor for coupling with rf power from
a lO-kW generator. The top turn of the water-cooled
copper rf leads was kept about i-in. below the rim of the
crucible to prevent the possiblity of melting the crucible
by a concentration of current at the edge.3 The crucible
was in direct contact only with high-purity Alundum
and an arrangement of mullite bricks around the cruci
ble was employed to minimize radiation losses away
from the crucible.
A thermocouple was not used in the growth of the
present series of crystals. Rather, it was attempted to
maintain the diameter of the crystals by manually
changing the power output from the rf generator; hence,
the constancy in diameter of the calcium tungstate
crystals was by no means comparable to that of silicon
and germanium crystals grown by the Czochralski
technique. Most of the crystals studied in the present
work were grown at the rate of ! in./h, although the
effect of growth rates of up to 4 in./h was also studied.
The crystals were not passed through after heaters
subsequent to growth, aside from the zone of radiation
from the crucible. The rf power was gradually decreased
after growth over a period of about an hour in order to
obtain a slow cooling of the crystal.
A small piece of calcium tungstate crystal was used
as the initial seed; subsequent seeds were cut from the
crystals that were grown. The growth direction of the
crystals is shown in Fig. 1. The crystals were about
3 in. in length and about! in. in diameter.
2 K. Nassau and A. M. Broyer, J. Appl. Phys. 33,3064 (1962). 3J?r. ~enry Albert, Baker Platinum Company (private com.
mUnICatiOn) .
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1.1736103.pdf | Calculation of the Maximum Efficiency of the Thermionic Converter
John H. Ingold
Citation: Journal of Applied Physics 32, 769 (1961); doi: 10.1063/1.1736103
View online: http://dx.doi.org/10.1063/1.1736103
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/5?ver=pdfcov
Published by the AIP Publishing
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] IP: 141.218.1.105 On: Mon, 22 Dec 2014 23:30:35JOURNAL OF APPLIED PHVSICS VOLUME 32. NUMBER 5 MAV. 196\
Calculation of the Maximum Efficiency of the Thermionic Converter
JOHN H. INGOLD
General Electric Company, Vallecitos Atomic Laboratory, Pleasanton, California
(Received June 13, 1960; in final form January 9, 1961)
A theoretical analysis of the efficiency of a thermionic converter is made in terms of the following
parameters: Va, the potential difference between the top of the potential barrier in the interelectrode space
and the Fermi level of the anode; V L, the potential drop across a load impedance in series with the converter;
and V z, the potential drop in the necessary electrical connection to the cathode. The analysis is carried out
by developing an expression for the efficiency of the converter and then maximizing this expression with
respect to V Land Vz. This method yields optimum values of load impedance, cathode lead geometry, and
cathode work function in terms of Va, cathode temperature, cathode emission constant (usually denoted by
A), and effective emissivity of the cathode. A hypothetical example is worked out numerically and the
results show that (1) a low value of Va is required for high efficiency, and (2) relatively low values of cathode
work function are required for maximum efficiency at ordinary cathode temperatures.
INTRODUCTION
THE purpose of this paper is to show how to
choose the optimum values of the appropriate
parameters which will allow a thermionic converter to
be operated at maximum efficiency at a given cathode
temperature. It is assumed that the reader is familiar
with the concept and terminology of thermionic
conversion. Those who are not may refer to papers
such as those by Wilsonl and Houston,2 and to the
book edited by Kaye and Welsh.3
An analysis of the efficiency of a thermionic converter
can be made in terms of the following parameters:
Va, the potential difference between the top of the
potential barrier in the interelectrode space and the
Fermi level of the anode; V L, the potential drop across
a load impedance in series with the converter; and
V I, the potential drop in the necessary electrical
connection to the cathode. The analysis is carried out
by developing an expression for the efficiency of the
converter and then maximizing this expression with
respect to V L and V I. This method yields optimum
values of load impedance, cathode lead geometry, and
cathode work function in terms of Va, cathode temper
ature, cathode emission constant (usually denoted by
A), and effective emissivity of the cathode. A hypo
thetical example is worked out numerically and the
results are summarized in the form of a contour map
(see Fig. 3) which gives directly the value of cathode
work function required for a desired efficiency for
various cathode temperatures and Va'S.
ANALYSIS
Figure 1 shows the potential diagram used in this
analysis. Subscripts c and a denote cathode and anode . .' respectlvely, and ¢ denotes work functlOn. Vc the
potential difference between the top of the pot~ntial
barrier and the Fermi level of the cathode, is seen to
1 V. C. Wilson, ]. App!. Phys. 30,475 (1959).
2 J. M. Houston, J. App!. Phys. 30,488 (1959).
3 J. Kaye and J. A. Welsh, Direct Conversion of Heat to Electricity
(John Wiley & Sons, Inc., New York, 1960). equal Va+ V L + V I. The net current density in the
system is equal to fe-fa, where fe and fa are the
current densities due to the electrons from the cathode
and anode, respectively, which get over the potential
barrier. Ie and Ia are given by the Richardson-Dushman
equation:
Ie=ATe2 exp[ -(eVclkTe)]
Ia=ATa2 exp[ -(eVa/kTa)] (1)
(2)
where e=electronic charge, k=Boltzmann constant,
T= temperature in degrees Kelvin, and the theoretical
value of A is 120 amp/cm2 deg2.4 It should be remarked
that recent experiments on tungsten" and tantalum6
show that when the temperature dependence of the
work function is taken into account, experimentally
determined values of A for these metals compare well
with the theoretical value.
Efficiency is defined as the useful electrical power
output per unit area of cathode divided by the heat
input per unit area of cathode. The useful electrical
power output is given by (Je-Ia)VL. The case of
practical interest, of course, is that for which Ia«Ie,
for otherwise there would be negligible power output
Anode L.--L--ic--Fermi
Level
Cathode Vf Fermi -"-_-L_--' __________ 1: __ _
Level
FIG. 1. Potential diagram for thermionic converter.
4 E. Wigner, Phys. Rev. 49, 696 (1936). This work discusses
possible reasons why experimentally determined A values may
vary from the theoretical value.
D A. R. Hutson, Phys. Rev. 98, 889 (1955).
6 H. Shelton, Phys. Rev. 107, 1553 (1957).
769
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] IP: 141.218.1.105 On: Mon, 22 Dec 2014 23:30:35770 JOHN H. INGOLD
from the device. The following analysis is restricted to
the case for which Ja is very small in comparison with
Je• [Consideration of Eqs. (1) and (2) shows that
Ja«Je when
(8a/8e)2 exp[ (Vc!8e) -(Va/8a) J«l,
where 8{== kT i/ e. Assuming an anode temperature of
SOOoK, the maximum value of the left side of this
inequality for that portion of Fig. 3 to the right of
Te= 10000K and above Va= 1 v is found to be 7.SX 10-4•
For practical purposes, therefore, the neglect of J(> in
comparison with Je in the following1analysis is justified].
In the steady state, the heat input to the cathode
must equal the heat 105s from the cathode. The heat
loss from the cathode consists chiefly of three terms7:
(1) Pc=electron emission cooling in w/cm2 of cathode;
(2) Pr=w/cm2 radiated from hot cathode; and (3)
P1=w/cm2 of cathode conducted away from the cathode
through its electrical connection. In the case of the
gas-filled converter, there is an additional loss P g due
to the conduction of heat in the gas. However, since
this term is probably very small, it has been neglected
in the following analysis.
The electron emission cooling term is calculated in
the following way. Only those electrons emitted from
the cathode with an x component of velocity greater
than The radiation loss term is given by
Pr= ~(J(e/k)4(8eC8a4),
where ~= effective emissivity8 of the cathode and (J is
the Stephan-Boltzmann constant.
For Ja«JC) that cathode lead heat-loss term IS
given by
Pl= (4 ae) (a/I) (e/k)(8e-8o)-V/acp(l/ a),
where
K= thermal conductivity of cathode lead,
p= electrical resistivity of cathode lead,
a= cross-sectional area of cathode lead,
I = length of cathode lead,
ae=cathode surface area,
8o=kTo/e with To=ambient temperature (the load
is assumed to be at ambient temperature).
This result is obtained by solving the heat flow equation
(d/dx)K(dT/dx) = -Jc2(aHa2)p
for constant K and p, in which J2R heating is taken into
account but radiation from the lead is neglected.
Since
can get over the potential barrier (Ve-CPc) to the for metallic conductors, PI can be written
anode, and each such electron takes away from the PI=1r2/6[(8c2-802)/(acRI)J-!Je2aeRI
cathode an energy equal to
where u, v, and ware the x, y, and z components of
velocity, respectively. Then, if n is the total number of
electrons per unit volume just outside the cathode, the
total energy taken away from the cathode per unit
area IS
where Xexp(-~TJ2)dudVdw,
2kTc
a2= 2 (e/m) (V.-CPc)
U2=U2+V2+W2.
Thus, the electron emission cooling term is
There is an additional term in p. to account for the
energy received by the cathode from the electrons
emitted from the anode which get over the potential
barrier, but for Ja«Je, this term is negligible.
7 It is assumed that the cathode requires no physical support
other than its electrical connection. where HTe+To) has been used for T in the Wideman
Franz value for KP, and RI has been written for pel/a).
Then since Ve= Va+ VL+ VI, the efficiency is given by
JcVL
Upon dividing the numerator and denominator of the
right side of this equation by J fie and noting that
VI=JeacRI, one can write for the efficiency
h
~=-----------------------------
if;L +if;a+2+ (Pr/ Jc8c) + (tr2/3if;1) +!if;z' (3)
where if;i= V;j8e, 8U2 has been neglected in comparison
with 8e2, and Je is given by
(4)
with Jo=A (e/k)28c2. According to Eq. (3), the efficiency
can be interpreted as the ratio of power delivered to
the load to the sum of powers delivered to the load
8 For radiation between infinite, plane-parallel electrodes, the
effective emissivity is given by
Eeff= 1/[ (1/ Ee) + (1/ Ea) -lJ,
where Eo and Ea are the emissivities of the cathode and anode,
respectively.
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] IP: 141.218.1.105 On: Mon, 22 Dec 2014 23:30:35M A X I MUM E F FIe lEN C Y 0 F THE THE R M ION ICC 0 N V E R T E R 771
and to the anode. In optimizing I/;L and 1/;1, i.e., V Land
VI, it is convenient to work with the reciprocal of the
efficiency, which from Eq. (3) is
where I/;a, Oe, and Pr are constant parameters. For 1/ to
be a maximum (1/1/ to be a minimum), it is required
that
a (1/1/) Pr ale 7r2 1 0=--=-----+- (5)
al/;l I}Oe al/;l 31/;12 2
and
a (1/1/) 1 Pr ale 0=--=----
aI/;L I/;Llc20eoI/;L
From Eq. (4),
so that Eqs. (5) and (6) become, respectively,
1/;1= 7r(i)!/[1 +2 (Pr/ I elle)]!,
I/;a+ 2+ (Pr/leOe) +I/;D + (PrlleOe)] (6)
(7)
(8)
Unfortunately, Eqs. (7) and (8) are not explicit
solutions for the optimum values of 1/;1 and I/;L because
Ie depends exponentially on these two parameters
[d. Eq. (4)]. Instead, one has two equations which
must be solved simultaneously for the optimum values
of 1/;1 and I/;L. It turns out, however, that this can be
done indirectly by first working with Ie alone.
Substituting Eqs. (7) and (8) into Eq. (4), taking the
logarithm of each side, and then simplifying, gives
Pr l/;a+2+7r(213)![1 +2 (Prl leOe)]!
leOe In(JoOc/Pr)+ln(Pr/leO e)-(l/;a+l) (9)
which is the condition on Ie, and hence on 1/;1 and I/;L,
for which 1/ is a maximum.
Substituting Eqs. (7) and (8) into Eq. (3) and
simplifying the result gives maximum efficiency in
terms of the optimum value of PrlleOe obtained from
Eq. (9):
1
1/mal< (10)
This can be written Thus, the maximum efficiency for particular values
of Va and Te depends only on the ratio of the radiation
loss Pr to the optimum value of 21 cOe, which is the
kinetic energy of the electrons which reach the anode
from the cathode.
The optimum values of cathode lead resistance RI
and load impedance RL can be obtained in terms
of a from Eqs. (7) and (8) by using the relation
Ri= (Oellcae)I/;,:
(12)
(13)
In other words, for the efficiency to be a maximum,
the following interrelated conditions must be fulfilled:
(a) the current in the circuit must satisfy Eq. (9);
(b) the cathode lead resistance and the load impedance
must satisfy Eq. (12) and (13), respectively.
The optimum cathode lead geometry II a can be
obtained directly from Eq. (12) and the relation
RI= p(l/ a); that the result agrees with that obtained
by Rasor9 can be seen immediately on substituting
Eq. (10) into Eq. (12) and simplifying.
NUMERICAL EXAMPLE
For the purpose of illustrating the results of the
preceding section, some arbitrary numerical values
were assigned to the constants so that some theoretical
efficiencies could be computed. The value of A was
taken to be 120 amp/cm2 deg2, while ~ was taken to be
the emissivity of bare tungsten10 radiating to a black
body. With these values, Eq. (9) was solved by an
iterative method for different values of Tc and Va.
The resultant values of (PrllcOe) opt were used in
connection with Eq. (10) to calculate maximum
efficiencies for various values of Tc and Va. The maxi
mum efficiencies obtained in this way are presented
in Fig. 2, which clearly shows that a low Va is necessary
for a high efficiency. The equicurrent lines in Fig. 2
give an idea of the current density required for maxi
mum efficiency. Thus, for Te=1780oK and Va=1 v,
the maximum efficiency is about 38% at a current
density of about 50 amp/cm2•
The approximation made in order to simplify Pl-
namely, neglect of To2 in comparison with Te2-leads
to an underestimation of the maximum efficiency. In
addition, the assumption that the cathode radiates to
a blackbody leads to an underestimation of the maxi
mum efficiency because the effective emissivity of the
cathode is less than its actual emissivity by a factor of
l+~e[(1/~a)-IJ
1/rnax= II (1 +2a), (11) 9 N. S. Rasor, J. App!. Phys. 31, 163 (1960).
where 10 American Institute of Physics Handbook, edited by Dwight E.
Gray (McGraw-Hill Book Company, Inc., New York, 1957),
Sec. 6.
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] IP: 141.218.1.105 On: Mon, 22 Dec 2014 23:30:35772 JOHN H. INGOLD
Vo=1.0
.5,1--__ -l--__ +-_-----'\+-::;"'"""\--+:-~_iV 0'=1.25
.41-----+
Vo=2.50
, Vo=2.7S
." . 31-----I--Il---I-''h':;.-,~_V---,,<0\''''-f--r-_'I V 0=3.0
.1 "-._-LJ._L.JJ.....{'_..L-L:>LC+ __ C+_---l
.0 OL-~l,.---2~0!"0.".0---,,,30:l.:0""0-......,.,40:l.:0"'0--.J
FIG. 2. Efficiency vs cathode temperature for optimum
values of lead geometry, Vc and VL.
if the cathode and anode can be considered as two
infinite, parallel planes. However, the effect of these
conservative approximations is probably balanced by
other factors such as: (a) the neglect of radiation in
cathode lead heat-flow equation, and (b) the neglect
of the heat conduction loss in the gas, if any, used to
neutralize space charge.H Therefore, in the author's
opinion, the maximum efficiencies presented in F~g .. 2
represent the upper limits of efficiency of a thermlOmc
converter with a cathode A value of 120 amp/cm2 deg2,
and an effective cathode emissivity of that of tungsten
radiating to a blackbody.
A convenient way of presenting the information
obtained from this analysis is in the form of a contour
map, which is shown in Fig. 3. This contour map is
used as follows. Suppose an efficiency of 40% were
desired and a Va of 1 v were available. Then Fig. 3
shows that a cathode with a Vc of about 2.62 v (or less)
and Tc of about 20000K are required. The corresponding
current is about 90 amp/ cm2; for this current and a Tc
of 2000oK, Eq. (7) gives Vz=0.22v; therefore, VL=~c
-Va-Vl= 1.4 v. Then the useful power output IS
JcVL=126 w/cm2 of cathode surface.
11 The gaseous heat conduction loss P u may be taken i~to
account simply by replacing Pr by Pr+P g III t~e ~oregolllg
analysis. This would in no way alter the analysIs; It would
merely result in: (a) different values for t~e optimum curr~nt Jc
for given values of Tc and Va; and (b) slightly lower maxImum
efficiencies. 3.0
-0 2.5 >
.~
0 2.0
>
1.5
o
FIG. 3. Guide for choosing the cathode work function which gives
desired efficiency for particular Tc and Va.
SUMMARY AND CONCLUSIONS
The maximum efficiency of a thermionic converter
has been calculated for various values of cathode
temperature and anode work function plus voltage
drop in the interelectrode space by optimizing two
parameters which are at one's disposal. These
parameters are the voltage drop across a load impedan~e
in series with the converter and the voltage drop In
the electrical connection to the cathode. A convenient
guide in the form of a contour map has been prepared
which gives, for the appropriate parameters, the
cathode work function required for maximum efficiency.
Of course, this guide does not apply to every thermionic
converter because an A value of 120 amp/ cm2 deg2 and
the emissivity of bare tungsten were used in the
numerical computations. On the other hand, the
optimization procedure presented in this paper is quite
general in that an analysis of the efficiency of a specific
thermionic converter would involve the same equations
with different numbers.
In conclusion it may be stated that the results of
this analysis show that a low Va is required for high
efficiency and that relatively low cathode work functions
are required for maximum efficiencies at ordinary
cathode temperatures.
ACKNOWLEDGMENTS
The author wishes to acknowledge the assistance of
Dr. T. M. Snyder, who made helpful suggestions
throughout the preparation of this report, and of
Barbara A. Kerr, who programmed Eq. (9) for the
IBM-650 computer (both are of this Laboratory).
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1.1729837.pdf | Effect of Magnetic Field Reversal on the Determination of Certain Thermo
magnetic Coefficients
John A. Stamper
Citation: Journal of Applied Physics 34, 2919 (1963); doi: 10.1063/1.1729837
View online: http://dx.doi.org/10.1063/1.1729837
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/9?ver=pdfcov
Published by the AIP Publishing
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For certain voltage and temperature ranges,2 the tunneling cur
rent density J(T) at temperature T(OK) is
J (T) =J (O)+aT". (1)
It can be shown3 that, as a first-order approximation,
a=[(87rmtk2)/h3]exp{ -(47r/h)J[2m",(x)]!dX} (2)
where 111 is mass of electron, t is charge of electron, It is Planck's
constant, k is Boltzmann's constant, and ",(x) is the potential
barrier measured from the Fermi level. The ratio of the incre
mental current density [defined as t:..J(T)=J(T)-J(O)] to
J(O) is, roughly,
-y = t:..J (T)/ J (0) = 32r11ls2k2T"/ (3h2<p), (3)
where s is insulating film thickness and <p is the average value of
",(x).
The tunnel current leT), of the BeO structure, was measured
as the temperature was varied from approximately 100° to 4OO0K
and the applied voltage was kept at 1 V. Two measurements
(7.6 p.A at 148°K and 9.49 p.A at 345°K), were used in (1) to cal
culate for 1(0). The resultant value of 1(0) was 7.17 p.A, which
was then subtracted from all measured values of leT) to obtain
the incremental currents t:..l(T)=l(T)-l(O) that are plotted on
a log-log scale in Fig. 1. The plot is a straight line of slope 2.04,
thus agreeing with the P dependence.
Measured by the technique of Simmons and Unterkofler,4 the
oxide thickness s was estimated to be 33 A. With the reasonable
assumption that <p= 1 eV, the value of -y calculated from (3) is
about 0.25 at 300°K. The data in Fig. 1 give -y=0.237.
Stratton obtained the TO dependence by expanding (6-y )!csc (6-y)!
for small -y. The present experimental result indicates that the P
dependence is still valid for values of -y somewhat higher than
those implied by Stratton's equation.'
1 J. G. Simmons. G. J. Unterkoller, and W. W. Allen, Ap]>!. Phys. Letters
2, 78 (1963).
, R. Stratton, J. Phys. Chern. Solids 23, 1177 (1962).
3 C. K. Chow. "Temperature Dependence of Tunnel Current Through
Thin Insulating Films," Burroughs Corporation, Burroughs Laboratories.
Internal Technical Report, TR62-57. December 1962 (unpublished). 'J. S. Simmons and G. J. UntNkoller, App!. Phys. Letters (to be
published).
Effect of Magnetic Field Reversal on the
Determination of Certain Thermo
magnetic Coefficients
JOHN A. STAMPER
Texas Instruments, Incorporated. Dallas, Texas
(Received 9 May 1963)
IN the measurement of the Nernst, Righi-Leduc, and magneto
Seebeck coefficients, it is customary to take data with the
magnetic field in each of two opposite directions. For the adiabatic
conditions generally assumed, it is then possible to eliminate cer
tain errors. It should be emphasized that the effect of magnetic
field reversal depends on experimental conditions and that there
are important cases where the coefficients can be evaluated even
under nonadiabatic conditions.
In these cases it is possible to separate the symmetric and anti
symmetric contributions to the temperature gradient and measur
able electric field. Errors due to superposition of effects and mis
alignment voltages can then be avoided. Adiabatic conditions (no
heat loss from the sides of the sample) can be closely approximated
in the laboratory and allow the separation of symmetric and anti
symmetric contributions. The constancy of heat current density
w on reversal of the magnetic field is a more general condition
which permits the separation. This is discussed below.
The components of w normal to the sides of the sample are de
termined by the temperatures of the sample surface and the sample surroundings. Reversal of the magnetic field B causes negligible
change in these temperatures when SB«1 where S is the Righi
Leduc coefficient. Thus, at the sample surface, w is the same for
both directions of the magnetic field. The following analysis shows
when this must be true throughout the volume of the sample.
Consider a sample in the form of a rectangular parallelepiped.
Let the magnetic field be applied in the Z direction and a tempera
ture gradient VT be applied in the X direction. It is assumed that
electric current density is zero. If V2T = 0 for both directions (in
dicated by + and -) of the magnetic field (allowing time for
steady-state conditions to be reestablished) then V2w = 0 for both
directions so that if w+=w-at the sample boundaries then
w+=w-throughout the volume of the sample.
An expression for V2T can be obtained from the equation
V·w=O. This relation comes from the theory of steady-state pro
cesses' and can be written
V2T= [1-K'(B)/ K(B)](l2T /(lz', (1)
where K'(B) and K(B) are thermal conductivities parallel to and
normal to the magnetic field, respectively. Thus V2T is zero if
either factor in the right-hand side of (1) is zero. Other than in
the adiabatic case ((IT /(lz=O), the condition (l2T /az2=0 is not
likely to be met experimentally. A positive heat flux into both
xy-faces or out of both xy-faces implies a2/Taz2~o. However, the
relation w+""'w-is valid in materials for which K'(B) =K(B) even
under nonadiabatic conditions.
Equations suitable for the evaluation of the adiabatic coefficients
can be derived from the relation w+-w-and the equations 2 for
heat current density and measurable electric field.
1 H. B. Callen. Phys. Rev. 73, 1349 (1958).
2 J. B. Jan. Solid Stale Physics (Academic Press Inc., New York. 1953).
Yol. 5, p. 8.
Influence of the Silicon Content on the Crystal
lography of Slip in Iron-Silicon Alloy
Single Crystals
S. LIBOVICKY AND B. SESTAK
I11slilute of Physics. Cuchoslovak Academy of Sciences.
Prague. Cuchoslovakia
(Received 29 April 1963)
IN earlier papers we found that the occurence of slip on single
crystals of Fe-3% Si alloy depended on the deformation rate
along the crystallographic or non crystallographic planes.,-a At
room temperature, during bending at deformation rates in the
surface of up to about 2X10-' sec-', the slip planes approach the
maximum resolved shear stress planes. At velocities above 10 sec'
slip occurs along the {110} planes. The transition from one type
of slip to another was studied at a lower temperature.4 At 78°K,
at velocities of about 10-7 sec-', the slip remains generally 11011-
crystallographic but there is a clear tendency of parts of the slip
planes to become {11O} planes, particularly on the tensile side of
the bent samples.
Up to now it has been universally accepted that, in an alloy of
iron with more than 4% Si, slip occurs only along the {110}
planes .. We have now found that when the silicon contents are
higher the slip planes differ markedly on the compression and
FIG. 1. Orientation
of samples.
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] IP: 130.113.76.6 On: Wed, 03 Dec 2014 17:11:502920 COM 1\1 U ~ 1 CAT ION S
tensile sides of the same sample while on the compression side
the slip remains noncrystallographic up to higher deformation
rates than with an alloy with 3% Si.
The samples with the orientation shown in Fig. 1, cut from
single crystals grown by the Bridgman method, were chosen for
the study. They were deformed by three-point and four-point
hending at different deformation rates in an Instron tensile ma
chine and by the impact of a falling weight as in our earlier
papers.2,3 The samples were deformed at room temperature and
the slip bands were observed on the compression and tensile sides
of the samples. Figures 2 and 3 show photographs of part of the
surface on the compression and tensile sides of the same sample of
an alloy with 7.5% Si after deformation at two different deforma
tion rates. It is clearly seen that slip occurred quite differently on
the compression and tensile sides. On the compression side, slip
occurred along the maximum resolved shear stress planes in a
range of deformation rates of 6X 10-7 sec' to 5X 10-"2 sec'. With
the orientations used the planes with maximum resolved shear
stress are identical with the (H2) planes but it should be em
phasized that this is not crystallographic slip along these planes
since the character of the slip lines does not correspond to that of
crystallographic slip. The slip lines are slightly wavy according to
local inhomogeneities and deviate from the correct direction under
the influence of neighboring bands and as a result of inhomogen
eous stress. This is best seen in samples in which the middle knife
edge during three-point bending pressed the edge of the sample; as
a result of the inhomogeneous external stress field, the direction of
the slip bands changed like a fan [similarly as in Figs. 13(a), (b)
of Ref. 1]. Only at deformation rates of 4X 102 seC' do we ob-
Ca)
Cb)
),FIG. 2. Slip bands on compression (a) and tensile (b) side of same three
point-bent sample of Fe-7.5 % Si alloy with orientation shown in Fig. 1.
Deformation rate 1 XIO-6 sec-I, Direction of tensile and compression stress
are indicated. Oblique illumination. Magnification Xt50. Ca)
Cb)
FIG. 3. Same as Fig. 2. Deformation rate 4 X 102 sec'.
serve on the compression side a transition to slip along the {llO}
planes [Fig. 3 (a)]. In similar samples of the same orientation but
with 3% Si the slip under the same deformation conditions was
only crystallographic on both sides [cf, Fig. 13(c) in Ref. 1]. On
the tensile side of the samples with 7.5% Si slip always occurred
exclusively along the {llO} planes in a range of deformation rates
from 6XlO-7 sec' to 4X102 seC'.
A similar difference in the slip geometry was also observed on
samples containing 5.5% silicon. On the tensile and compression
sides of these samples at a deformation rate of 5 X 10-6 seC' the
slip is non crystallographic along the maximum resolved shear
stress plane [the same character as in Fig. 2 (a)]. When the defor
mation rate is increased to 4X 10-2 seC' sections appear on the
tensile side of the sample apart from non crystallographic slip
where the slip occurs exactly along the {llO} planes (Fig. 4).
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] IP: 130.113.76.6 On: Wed, 03 Dec 2014 17:11:50COMMUNICATIONS 2921
FIG. 4. Slip bands on tensile side of sample of Fe-5.5 % Si alloy with
orientation shown in Fig. 1. Deformation rate 4 X10~2 sec-I. Direction of
tensile stress b indicated. Oblique illumination. Magnification X150.
while on the compression side it remains quite non crystallographic.
When the deformation rate is raised still further to 4X102 sec1
slip occurs only along the {110} planes on both sides.
It follows from the above results that the range of deformation
rates, in which there is a difference between the slip on the tensile
and compression sides of samples, expands with increasing silicon
content.
The difference in the character of slip on the tensile and compres
sion sides of the same sample support the conception1,3 that slip
along the {110} planes is caused Ly the extension of dislocations
on these planes since it is plausible that the energy of suitable
stacking faults in samples of our orientation may decrease on the
tensile side as a result of deformation and increase on the compres
sion side, Several modes of extension have been proposed.6" It is
not yet possible to decide which of them plays a role in the plastic
deformation of the crystals studied hy us. Since it has not yet been
possible to observe extended dislocations in bcc metals after plas
tic deformation, it can be deduced that the extension is small or
exists only in the stress field. Due to the insignificant extension it is
Letter to consider the anisotropy of the dislocation core char
acterized by extension.
In the case of small energy of the stacking fault on the (110)
planes, the dislocations in this plane dissociate into their partials
and can then move only along these planes. It is not yet clear
whether the dislocations move also atomically along the non
crystallographic planes during noncrystallographic slip or whether
they alternately move along small sections of nonparallel {110}
planes. In the first case one would have to assume a pronounced
influence of the deformation rate and the influence of temperature
011 the energy of the stacking fault. In the second case the temper
ature and deformation rate would influence the alternation of the
sections of the {110(planes,
1 B. Se8tiik and S. Libovicky, Proceedings of Symposium on the Relation be·
tween the Structure and the Mechanical Properties of Metals (National
Physical Laboratory. Teddington, England. 1963).
2 B. Sestak and S. LibovickY. Czech. J. Phys. BI2, 131 (1962).
3 B. Sestak and S, Libovicky, Czech, J, Phys, B13, 266 (1963).
'B. Sestak and S. Libovicky, Acta Met. (to be published).
5 C. S. Barrett, G. Ansel, and R. F. Mehl, Trans. Am. Soc. Metals 25, 702
(1937).
r, J. Friedel, see discussion in Ref. 1.
7.1. B. Cohen, R. Hjnton, K. Lay, and S. Sass, Acta Met. 10,894 (1962). Observation of Continuous-Wave
Optical Harmonics
s. L. MCCALL AND 1,. \'1. DAVIS
Wt~stan Development Laboratories, Philco Corpora/ion,
Palo Alto, California
(Received 5 June 1963)
INVESTIGATORS previously have used the intense light from
a pulsed solid-state laser to generate optical harmonics in vari
ous substances, Here we report use of the light beam from a high
intensity gas-discharge laser to observe the production of continu
ous-wave (cw) second-harmonic light in potassium dihydrogen
phosphate (KDP).
For comparing experimental results with certain aspects of the
theory of nonlinear optical phenomena, the cw light from a gas
laser has some important advantages over the pulsed light from a
solid-state laser. For example, (1) harmonic conversion efficiency
could be measured more accurately with gas laser cw excitation
than with pulsed light; (2) Franken and Wardl suggest that ex
tremely monochromatic light, as is provided by the gas laser,
would permit further study of the possibility that phonon inter
actions shift or broaden the frequency of harmonic radiation.
For the experiment we used a helium-neon gas laser ,,·ith
",-,20-mWoutput (recently constructed by Spectra-Physics, Inc.).
The laser cavity was 3.5 mm in diameter and approximately 3 m
in length, Confocal mirrors, with focal lengths of 3 m, were used
as end reflectors. Oscillation was restricted to TEMoo modes,
Laser light of wavelength 6328 A was passed through a red-pass
filter and focused into a KDP crystal oriented at the index match
angle.2,3 The emergent light was passed through a NiS04 solution
filter to remove the 6328 A component, and then was detected.
Polaroid color film photographs gave a blue image having the
striking intensity pattern reported by Maker et aI.' The second
harmonic light was also detected by a photomultiplier. When the
KDP was rotated, the intensity of harmonic light was highly sen
sitive to the angle between the crystal z axis and the incident beam
direction, with the half-power rotation angle being less than S°.
On attenuating the intensity of the incident light with neutral
density filters, oscilloscope traces were obtained as shown in Fig,
1. The data agree well with the expectation that the second-
5msec -! -"
J ~ V"t'\, "-..;.,;',. 0rv'~
-- --.
.L
50mV
..... ./' T
FIG,!' Oscilloscope traces of lP28 photomultiplier output. Vertical scale
is 0.050 V /div; horizontal, 0,005 sec/div. The 120·cps ripple is due to modu
lation of the laser light and also to pickup in the detector circuit. Top to
bottom: Laser light unattenuated, attenuated by 0.3 (10-0 •• transmission)
and 0,5 neutral density filters, and completely attenuated. Maximum second
harmonic generated is approximately 8 X 10-1' W for 20 mW of excitation
light, corresponding to 5 X 10' red photons required to produce one ultra
yiolet photon.
harmonic production efficiency is proportional to the intensity of
the incident light.
The experiment was made possible through use of the ~3-m
laser developed by W. E. Bell and A. L. Bloom of Spectra-Physics,
Inc,
1 p, A. Franken and J. F, Ward, Rev, Mod. Phys, 35, 23 (1963). 'J, A. Giordmaine, Phys. Rev. Letters 8, 19 (1962).
3 P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys.
Rev, Letters 8, 21 (1962).
• See Ref. 3, Fig, 3,
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1.1696283.pdf | Radiolysis of Hexafluoroethane
Larry Kevan and Peter Hamlet
Citation: The Journal of Chemical Physics 42, 2255 (1965); doi: 10.1063/1.1696283
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128.248.155.225 On: Mon, 24 Nov 2014 11:43:43THE JOURNAL
OF
CHEMICAL PHYSICS
VOLUME 42, NUMBER 7 1 APRIL 1965
Radiolysis of Hexafluoroethane
LARRY KEVAN AND PETER HAMLET
Department of Chemistry and Enrico Fermi Institute for Nuclear Studies, University of Chicago, Chicago, Illinois
(Received 5 October 1964)
The radiolysis of C.F6 at 3-atm pressure has been examined. The products and 100 eV yields are CF4
(1.6), cyclo-CaF6 (0.30), CaFs (0.21), C.H10 (0.14), and C.F. (0.03). Good material balance is obtained;
the F IC ratio in the products is 3.0. Experiments using radical scavengers indicate that 50% of the CF4
comes from radical reactions and 50% from nonradical reactions, that CaF sand C4H10 are entirely formed
by radical reactions, and that C.F. and cyclo-CaFe are probably formed by ionic reactions. In Ar-C.Fe
mixtures energy transfer, thought to be charge transfer, is observed and ionic production of CaFs is seen.
C.F6 is about one-fifth as sensitive to radiation decomposition as is C.He. It is concluded that excited mole
cule decompositions, particularly those giving molecular fluorine, are relatively unimportant in the radiolysis
of perfluoroalkanes.
INTRODUCTION
IN the present work we examine how the general
chemical characteristics of different compound types
affect the radiolysis mechanism. Fluorocarbons and
hydrocarbons are ideal for this purpose because fluoro
carbon chemistry is strikingly different from hydrocar
bon chemistry. For example, hydrocarbon radicals dis
proportionate whereas fluorocarbon radicals do not,r
the C-H bond is weaker than the C-F bond, the H-H
bond is stronger than the F-F bond, etc.
The radiolysis of saturated hydrocarbons has received
much study; recent attention has been focused on the
relative contributions of ionic, free radical, and excited
molecule reactions to the over-all radiolysis mechanism.2
However, the radiolysis of perfluoroalkanes has re
ceived no detailed study. A few results have been re
ported for perfluoroheptane3,4 and for perfluorooctane.5
These studies were carried out in the liquid phase at
very high radiation doses and the products were only
incompletely analyzed. Therefore, to see how the chem
istry of fluorocarbons is reflected in the radiolysis mech
anism we have investigated the gas-phase radiolysis
of a simple perfluoroalkane, hexafluoroethane.
1 G. O. Pritchard, G. H. Miller, and J. R. Dacey, Can. J.
Chern. 39,1968 (1961).
• See, for example: R. P. Borkowski and P. J. Ausloos, J. Chern.
Phys. 39, 818 (1963).
a J. H. Simons and E. A. Taylor, J. Phys. Chern. 63,636 (1959).
4 R. E. Florin, L. A. Wall, and D. W. Brown, J. Res. Natl.
Bur. Std. MA, 269 (1960).
6 R. F. Heine, J. Phys. Chern. 66, 2116 (1962). EXPERIMENTAL
Hexafluoroethane was generously provided by D. G.
Hummel of DuPont's Jackson Laboratory. It was
found to be more than 99.9% pure by gas chroma
tography. CF4 and CO2 were obtained from Matheson
Company and CaFs and C4F10 were from Columbia
Organic Chemical Company. C2F4 was prepared
from the thermal decomposition of Teflon tape (Wilkens
Company) at 490°C under vacuum.6 Cyclo-CaF6
was prepared from the mercury-sensitized photolysis
at 2537 A of C2F4,7 purified by gas chromatography,
and identified by its infrared absorptionS at 1275 and
868 em-I.
Irradiations were performed in a 600-Ci 6OCO gamma
source. The dose rate was measured by ethylene do
simetry [G(H2) = 1.2J9 in brass and in Pyrex sample
cells. In brass cells ethylene dosimetry gave a dose
rate 25% higher than in Pyrex cells. This difference
deserves further comment. For gas-phase samples under
our experimental conditions the sample cells approxi
mately meet the requirements of a Bragg-Gray cavitylO
so that the energy absorbed in the gas depends on the
cell wall material. From the Bragg-Gray relationship,
EI1= EwSg/ StD, and the relationships, Ew= En~, and
6 S. L. Madorsky, V. E. Hart, S. Straus, and V. A. Sedlak, J.
Res. Natl. Bur. Std. 51, 327 (1953).
7 B. Atkinson, J. Chern. Soc. 1952, 2684.
8 W. Mahler (private communication).
9 K. Yang and P. L. Gant, J. Phys. Chern. 65, 1861 (1961).
10 L. H. Gray, Proc. Roy. Soc. (London) A156, 578 (1936).
2255
Copyright © 1965 by the American Institute of Physics
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128.248.155.225 On: Mon, 24 Nov 2014 11:43:432256 L. KEVAN AND P. HAMLET
TABLE I. C2FS radiolysis products (100-eV yields) at 3 atm
pressure and 7 Mrad.
C2Fs:02 C.Fs:O.
Product C.Fs 200:1 100: 1
CF4 1.6 0.80 0.84
c-CsFs 0.30 0.57 0.59
C2F• ? 0.03 0.16 0.16
CsFs 0.21 <0.01 <0.01
C4F'0 0.14 <0.01 <0.01
CO. <0.01 1.0 1.1
F/C ratio: 3.02 2.45- 2.46-
• Without Co..
Sw= nS. the ratio of energy absorbed by the gas in
cells of material 1 and 2 is given bylO:
Eol EwlSw2 Se2 -=---=-
Eol is the energy absorbed per gram in the gas phase
in a cell of material 1, Ew is the energy absorbed per
gram in the cell wall, E is the gamma-ray flux, n is the
number of electrons per gram, IT is the Compton ab
sorption coefficient per electron (this is independent
of atomic number), Sw is the stopping power per gram
in material 1. The photoelectric contribution to the
absorption coefficients has been neglected. Values of
S. decrease slowly with Z and are essentially independ
ent of the primary electron energy from 0.17 to 2.2
MeVY To compare the energy absorbed by a gas in
glass (2= 10) and brass (2= 29.3) cells we use the aver
age experimental values of S. from 32p and 35S sources
given by Baily and Brownl1: S. (glass) / S. (brass) =
1.20. This is in good agreement with our value,
Eo (brass) / Eo (glass) = 1.25. In brass cells the photo
electric contribution is estimated to be 4% which
further improves the agreement.
The dose rate was also measured by FeS04 dosimetry
[G(Fe3+= 15.5J12 in Pyrex cells. The result was com
pared with ethylene dosimetry in Pyrex by use of the
relation
in which the subscript s refers to the FeS04 solution
and w to the Pyrex wall. The two methods of dosime
try agreed to within 15%.
The average dose rate in brass cells, measured by
ethylene dosimetry and corrected for the electron den
sity and electron stopping power of C2F6 relative to
C2H4, was 0.105 Mrad/h. This was corrected for decay
as necessary.
Most of the irradiations were carried out in brass
cells of 1O-cc volume and equipped with a Matheson
11 N. A. Baily and G. C. Brown, Radiation Res. 11, 745 (1959).
12 A. J. Swallow, Radiation Chemistry of Organic Compounds
(Pergamon Press, New York, 1960), p. 42. lecture bottle, cylinder valve. A pressure of three at
mostpheres was used so as to have adequate sample
for analysis.
The radiolysis products were analyzed with an Aero
graph A90P gas chromatograph on a 2-m silica gel
column at 60° and 120°C. Identmcation was made by
retention times and in some cases (CF4, cyclo-C3F6)
by mass-spectrometric or infrared analysis. For quan
titative analysis cyclo-C3F6 and the product tentatively
identmed as C2F2 were assumed to have the same
thermal conductivity response as C2F6. Reported G
values were reproducible to less than 10% or 0.01 G
units, whichever is larger (G units are units of mole
cules produced per 100 e V absorbed).
RESULTS
Irradiation in Pyrex cells always gave large yields
of CO2 and SiF4 which indicates that wall reactions
were occurring. All of the reported results refer to
irradiations in brass cells. In the presence of small
amounts of oxygen, CO2 is readily formed; therefore
the CO2 yield was always measured as a check on our
handling procedures. The first two or three irradia
tions in newly made brass cells always contained CO2
and were disregarded. After the cells has become" con
ditioned" subsequent irradiations were usually CO2 free.
The product yields are reported as G values in
Table I. The yields were linear in the dose range
of 7 to 16 Mrad. The identification of C2F2 is tenta
tive and is based on its gas chromatographic retention
time. Table II shows the observed retention times for
several fluorocarbons including all of the products. Since
the silica-gel column separates compounds essentially
in order of their boiling points it seems that the 15-
min product peak could represent only C2F2. C2F2 has
been reported13 as a pyrolysis product of difluoromaleic
anhydride and is apparently a stable compound at
low partial pressures; at higher pressures it dimerizes
TABLE II. Effect of rare gases on C2Fs radiolysis products
(100-eV yields) at 7 Mrad.
C.Fs:Xea C.Fs:Ara C"Fs:Ar:O. a
Product (55:45) (35: 65) 40:60:1
CF4 1.4 1.8 1.3
c-CsFs 0.21 0.34 0.58
C.F. ? 0.03 0.06 0.16
CsFs 0.22 0.46 0.24
C4F'0 0.15 0.15 <0.01
CO. 0.91
F/C ratio: 3.01 2.97 2.66b
a Calculated on basis of energy absorbed in C2F, only. No assumptions about
energy partition were required since the ratio of product to C2F. pressure was
always measured.
b Without CO,.
13 W. J. Middleton, U. S. Patent 2,831,835 (22 April 1958).
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128.248.155.225 On: Mon, 24 Nov 2014 11:43:43RADIOLYSIS OF HEXAFLUOROETHANE 2257
and polymerizes. No C2F4 or other fluorocarbon olefins
are observed as stable products; if present they are
formed with a G value <0.01. Also no evidence for
stable F2 formation is found. The F IC ratio of the
products is 3 within experimental error as it is in the
parent compound. This good material balance indi
cates that all the major products are observed.
The results with added O2 are also tabulated in
Table I. CF4, CsFs, and C4FlO are decreased but C2F2
and cyclo-C3F6 are increased in the presence of O2.
The effects of added xenon and argon are given in
Table III; in general argon increases the product yields
while xenon decreases them.
DISCUSSION
Radiolysis mechanisms are generally incompletely
understood and are often discussed in terms of separate
free radical, ionic, and excited molecule reactions, all
of which may be independently studied. Several stud
ies of perfluoroalkyl radicals have been made1 but at
present there is no independent knowledge of ionic or
excited molecule reactions in perfluoroalkanes. There
fore, what we may infer about such reactions in the
present discussion must be considered as somewhat
speculative.
Gamma radiation produces an initial spectrum of
excited and ionized molecules. The mass spectrum pro
vides an indication of the ionic fragments formed by
dissociation of the parent ion. The most abundant
ions in the mass spectrum of C2Fa are CF3+ (58%),
C2F.+ (24%), CF+ (11%), and CF2+ (5%).14 The
associated neutral fragments are presumably CF3, F,
and perhaps a small amount of CF4.
Excited C2FS molecules, if not deactivated, probably
dissociate into two CF3 radicals or into CF2+CF4.
Dissociation of a C-F bond requires about 35 kcal
more energy and is thought to be less probable. The
absence of C2F4 among the radiolysis products also
TABLE III. Gas chromatographic retention times of fluorocarbons,
2-m silica-gel column, flow rate 40 ml/min.
Compound
Air
CF4
C2F6
CO2
C2F4
C2F2?
C3Fs
C-C,F6
C3F6
C4F1O Retention time (±0.3 min)
60°C
1.9 min
3.0
6.8
10.8
12.5
14.7
28 1.5 min
2.8
6.8
9.0
10.4
14.7
14 F. L. Mohler, V. H. Dibeler, and R. M. Reese, J. Res. Natl.
Bur. Std. (U. S.) 49,343 (1952). implies that the endothermic expulsion of F2 is not a
main process. One could argue, however, that C2F4
is formed and then rapidly consumed by further re
actions.
The good material balance indicates that little if
any stable F2 is formed. Fluorine atoms might be ex
pected to combine to form F2. However, the 80-kcal
exothermic reaction of F2 with perfluoroalkyl radicals
will keep the F2 concentration very low.
A. Formation of CF4, CsFs, and C4HlO
The yields of CF4, CsFs, and C4FlO are partly or
wholly scavengeable by 0.5% to 1.0% oxygen. Fifty per
cent of the CF4 yield is scavengeable. The scavenge able
CF4 yield IS attributed to the free radical combina
tion (1) :
F+CFs~CF4,
CFS+C2F5~CsFs,
2C2F.~C4F10. (1)
(2)
(3)
The nonscavengeable CF4 must arise from molecular
dissociation of an excited molecule, from an exothermic1 5
ion-molecule reaction such as (4) or (5), or from both
sources,
CFS++C2F6~CF4+C2F.+,
C2F5++C2FrCsF7++CF4. (4)
(5)
A mass-spectrometric study is planned to determine
whether Reactions (4) and (5) do occur at an appreci
able rate.
In pure C2F6 the CvFs yield is completely scavenged
by oxygen and accordingly is attributed to Reaction
(2). In mixtures of C2Fa with Ar (see Table III), how
ever, the CsFs yield is doubled; furthermore, the addi
tion of 1% oxygen to the C2Fa-Ar mixtures only scav
enges half of the total CsFs yield. This behavior is to
be compared with C2Fa-Xe mixtures in which the CsFs
yield is unchanged from its value in pure C2Fa and is
completely scavenged by oxygen. Thus it appears that
in Ar-C2Fa mixtures we have a new reaction which
produces CsFs. This reaction does not occur in C2F6
alone or in Xe-C2Fa mixtures. In all cases CsFs is
formed by free radical reactions with G=0.2. Since
the additional CsFs formed in the Ar-C2F6 mixtures is
not scavengeable by oxygen it is apparently formed
by a nonradical process. The most plausible possibili
ties would appear to be CF2 insertion into C2Fa or an
ion-molecule reaction of CF2+ or C2Fa+ with C2Fa.
CF2 does not appear to insert into a C-F bond even
at 120°C la so we favor an ion-molecule reaction.
15 Reaction (4) is 20 kcal exothermic while Reaction (5) is
thermo neutral as calculated from data tabulated or referred to
by C. R. Patrick in Advances in Fluorine Chemistry, edited by
M. Stacey, J. C. Tatlow, and A. G. Sharpe (Butterworths Scien
tific Publications, Inc., Washington, 1961), Vol. 2, p. 1. In
saturated fluorocarbons ionic reactions producing F2 are endo
thermic by ",,120 kcal whereas ionic reactions yielding CF4 are
thermoneutral or endothermic.
16 W. Mahler, Inorg. Chern. 2, 230 (1963).
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128.248.155.225 On: Mon, 24 Nov 2014 11:43:432258 L. KEVAN AND P. HAMLET
TABLE IV. Total decomposition yields in
C2F6 radiolysis.
Initial mixture Atom ratio G(-C2Fs)
C2F6 100:0 1.9
C2F6:02 100:1 2.0·
C2F6:Xe 55:45 1.7
C2Fs:Ar 35:65 2.5
C2F6:Ar:0! 40:60:1 2.5&
a This assumes that all radicals reacting with 0. are detected as Co..
The C4F10 product is completely scavenged by oxy
gen in pure C2Fe and in the rare-gas mixtures. In all
cases it is believed to be formed by the free radical
reaction (3).
B. Formation of C2F2 and cyc1o-CaF6
The yields of C2F2 and cyclo-CaFe are not decreased
with added oxygen; to the contrary they are increased.
This fact indicates that these products are not formed
by thermal free radical combinations. It is suggested
that ion-molecule reactions may be involved in the
formation of these products. With respect to ionic re
actions it is interesting to speculate that the yield
enhancement in the presence of oxygen may be due
to electron capture by oxygen molecules; however, the
details of such an effect are not understood at present.
C. Evidence for Energy Transfer in Ar-C2F6 Mixtures
The total yield of C2F6 decomposition is tabulated
for different mixtures in Table IV. The yield is given
in molecules of C2Fe decomposed per 100 eV absorbed
in C2Fe. No approximations of energy partition were
made since the ratio of product to C2Fe pressure in the
irradiated samples was measured. The enhanced de
composition yield in the Ar-C2Fs mixtures clearly shows
that energy transfer is occurring in these mixtures.
Energy transfer may be occurring by charge transfer
or by excitation transfer. The lack of information on
excitation energy levels in C2F6 prevents further evalu
ation of excitation transfer, but charge transfer may
be considered. The ionization potential of Ar is 15.8 eV
while that of Xe is 12.1 eV. The ionization potential
of C2FS is not known but the lowest appearance poten
tial in its mass spectrum is 14.4 eV 17 for CF3+; its
ionization potential is thought to be not much above
14.4 eV. Thus in mixtures of C2FS with either Ar or
Xe we may expect Reactions (6) or (7), respectively,
to occur. The cross section for charge transfer depends
on the
Ar++C2FrAr+C 2Fe+
Xe+C2F6+-7Xe++C 2Fs (6)
(7)
17 W. H. Dibeler, R. M. Reese, and F. L. Mohler, Phys. Rev.
87, 213 (1952). near matching of energy levels.ls Since the ionization
potential of Xe is below the ground-state energy level
of C2Fs+ and since vibrational and rotational energy
levels are lacking in Xe Reaction (7) may be expected
to be inefficient.
D. Comparison of Fluorocarbon and Hydrocarbon
Radiolysis
Here we compare the principal features of difference
between the radiolysis of saturated fluorocarbons and
hydrocarbons:
(a) F2 is a reactive product in perfluoroalkanes
whereas H2 is unreactive in alkanes. This fact may be
attributed to the great difference in bond strength and
to the exothermic reaction of F2 with perfluoroalkyl
radicals.
(b) Perfluoroalkanes are characterized by large yields
of CF4 while linear alkanes are characterized by large
yields of H2.
(c) Olefins are not formed in perfluoroalkanes but
are abundantly formed in alkanes. This may be at
tributed to the absence of disproportionation between
perfluoroalkyl radicals, to the fact that ejection of F2
from fluorocarbon molecules is highly endothermic, and
to the thermodynamic stability of the isomeric cyclic
compounds. Olefins may also be highly reactive.
(d) Cyclic compounds may be formed in linear fluo
rocarbons but not in linear hydrocarbons.
(e) Based on the material balance C2F6 is about
one-fifth as sensitive to radiation decomposition as is
C2H6• G(-C2FS) = 1.9 while G(-C2H6) =9.9 This appar
ent radiation stability may be attributed to several
causes including the absence of F abstraction in fluoro
carbons, the possibility of back reactions of free radicals
to reform C2F6, and the absence of excited molecule
decompositions which give molecular F2. It has been
shown that molecular ejection of H2 from excited mole
cules makes important contributions to product forma
tion in alkanes.19 We suggest that similar excited mole
cule reactions in perfluoroalkanes are unimportant.
ACKNOWLEDGMENTS
We gratefully thank Dr. D. G. Hummel for providing
a sample of hexafluoroethane, the U.S. Atomic Energy
Commission for support under Contract No. At(11-1)-
1365, and the Louis F. Block Fund for their support.
We also thank the referee for pertinent comments con
cerning the dosimetry.
18 D. Rapp and W. E. Francis, J. Chern. Phys. 37, 2631 (1962); J. L. Magee, J. Phys. Chern. 56, 555 (1952).
19 H. Okabe and J. R. McNesby, J. Chern. Phys. 37, 1340
(1962) ; ibid. 34, 668 (1961).
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1.1713322.pdf | Diffusion and Solubility of Copper in Extrinsic and Intrinsic Germanium,
Silicon, and Gallium Arsenide
R. N. Hall and J. H. Racette
Citation: J. Appl. Phys. 35, 379 (1964); doi: 10.1063/1.1713322
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Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissionsJ 0 URN ,\ L 0 F ,\ P P LIE D PH Y SIC S VOLUME 3.1, :-..rUMBER 2 FEBRUARY 19(,4
Diffusion and Solubility of Copper in Extrinsic and Intrinsic Germanium,
Silicon, and Gallium Arsenide*
l{, N, HALL AND J. H. RACETTE
General HieD/ric Research Laboratory, Sclleneciady, l'few 1'''1'''
(Received 29 August 1963)
The soluhilities of substitutional and interstitial copper (Cu' and CUi) have been measured in intrinsic
and extrinsic n-and p-type Ge, Si, and GaAs, using Cu6'. These measurements show that Cu' is a triple
acceptor in both Ge and Si, and that CUi is a single donor in all three semiconductors. Charge compensa·
tion experiments show that Cu' is a double acceptor in GaAs. In intrinsic semiconductor near 700°C the
ratios of suhstitutional to interstitial solubilities are 6, ,...,10-" and 30, respectively. Electrically active
donors due to CUi have heen ohserved in Si by Hall-effect measurements. Pairing hetween Cu' and donors is
pronounced in n-type Ge hut relatively unimportant above 600°C in Si. There is no evidence for pairing
hetween CUi and acceptors in any of these semiconductors.
The interstitial diffusion coefficient Di is independent of doping in extrinsic p-type material. At 500°C,
Di = 2.8, 0.76, and 1.0X 10-5 em' /sec with activation energies of 0.33, 0.43, and 0.53 (± 10% in each case)
cV, respectively. Di and the single positive charge of CUi in p-type GaAs have been confirmed by drift
experiments.
The decrease in energy gap with acceptor doping is 0.28 eV in Ge and 0.14 eV in Si at 4X 1020 cm-3 ac
ceptors, as deduced from CUi solubility data.
KCN solutions remove Cu effectively from surfaces and from liquid Ga. Ga is much more effective than
molten KCN in extracting Cu from the interior of semiconductors. KC1\ contains Cu and may actually
introduce Cu instead of removing it.
1. INTRODUCTORY REMARKS
COPPER exhibits an unusually large diffusion
coefficient in many nomnetallic crystals. While
early indications of this behavior were given by experi
ments with zinc sulfide crystals,! the first definitive
studies of rapid copper diffusion were conducted using
single crystals of germanium.2 Similar behavior has
since been observed in silicon,:! indium antimonide,4
indium arsenide,5 gallium arsenide,6 aluminum anti
monide} cadmium sulfide,8 lead sulfide,9 bismuth tel
luride,lO and silver sclenide,u It is generally agreed
that the rapid diffusion is clue to singly ionized inter
stitial copper, and drift experiments have confirmed that
copper does incleed migrate as a positively charged ion
in germanium and silicon.!2,!3
Substitutional copper, on the other hand, is relatively
immobile and consequently the effective diffusion rate
for copper is determined by the relative abundance of
the two species. Furthermore, since a vacancy is re-
* This work was supported by the U. S. Air Force Cambridge
Research Laboratories, Office of Aerospace Research, under Con
tract No. AF 19(604)-6623.
11\. Riehl and H. Ortmann, Z. Phys. Chern. A188, 109 (1941).
2 C. S. Fuller, J. D. Struthers, J. A. Ditzcnberger, and K. B.
Wllifstirn, Phys. Rev. 93, 1182 (1954).
:t J. D. Struthers, J. Appl. Phys. 27, 1560 (1956).
, H. J. Stocker, Phys. Rev. 130, 2160 (1963).
c, C. Hilsull1, hoc. Phys. Soc. (London) 83, 685 (1959).
6 C. S. Fuller and J. M. Whelan, J. Phys. Chell1. Solids 6, 173
(1958).
7 R. H. Wieber, H. C. Gorton, and C. S. Peet, J. Appl. Phys.
31, 608 (1960).
8 R. L. Clarke, J. App!. Phys. 30, 957 (1959).
9 J. Bloem and F. A. Kroger, Philips. Res. Rept. 12,281 (1957).
10 H .. O. Carlson, J. Phys. Chern. Solids 13, 65 (1960).
11 K. W. Foster and M. V. Milnes, J. Appl. Phys. 33, 1660
(1962). I' C. S. Fuller and J. C. Severins, Phys. Rev. 96, 21 (1954).
13 C. J. Gallagher, J. Phys. Chern. Solids 3, 82 (1957). quired for the transition of a copper atom from an
interstitial to a substitutional site, the migration of
copper may depend strongly upon the perfection of the
host crystal.
This dependence is most striking when the solubilities
of the interstitial and substitutional species are of com
parable magnitude, as is the case in germanium, Be
cause of the interesting phenomena displayed by this
system it has been SUbjected to considerable study and
the basic mechanisms are understood in considerable
detail.!4-!8 )Jevertheless, these investigations leave
several important problems unsettled: In spite of the in
terest in interstitial diffusion, there have been no
quantitative measurements of the diffusion coefficient
of interstitial copper or of its temperature dependence.
There is disagreement as to the equilibrium ratio of
the solubilities of the two species of copper in intrinsic
germanium.15,!7,18 Also, it had been predicted19,20 that
this solubility ratio should depend strongly upon
the electron or hole concentrations in extrinsic ger
manium and there is interest in verifying such effects
experimen tally.
The understanding of the behavior of copper in other
semiconductors has been much less complete. In silicon,
it was established that an appreciable fraction of the
total amount of copper present was interstitial, but no
upper limit was obtainedY The electrical activity pro-
14 F. C. Frank and D. Turnbull, Phys. Rev. 104, 617 (1956).
'" A. G. Tweet, Phys. Rev. 106, 221 (1957); 111, 57 (1958);
111,67 (1958).
16 A. G. Tweet and W. W. Tyler, J. App\. Phys. 29, 1578 (1958).
17 C. S. Fuller and J. A. Ditzenberger, J. Appl. Phys. 28, 40
(1957).
18 P. Penning, Philips Res. Rept. 13, 17 (1958).
19 R. L. Longini and R. F. Greene, Phys. Rev. 102, 992 (1956).
'0 W. Shockley and J. L. Moll, Phys. Rev. 119, 1480 (1960).
379
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissions380 R.:\T. H.,\LL AI\D J. H. RACETTE
duced by copper in silicon is much less than would be
expected from the amount known to be present in
crystals which had been saturated with copper, pre
sumably as the result of rapid precipitation during cool
ing. Consequently, the impurity levels which had been
assigned to this impurity21 are probably due to the
presence of precipitates rather than to atomically dis
persed copper. Among the III-V compound semicon
ductors, the behavior of copper has been studied in
greatest detail in gallium arsenide and indium anti
monide, but only in the latter has an estimate been
made of the relative abundances of the different species
of copper and of their rates of diffusion. Moreover, some
experiments22 have indicated that substitutional copper
is a single acceptor in gallium arsenide whereas valence
considerations lead one to expect that it should be a
double acceptor as it is in indium antimonide.23
In the work described below we have studied the
behavior of copper in samples of germanium, silicon,
and gallimn arsenide which contained sufficient con
centrations of shallow donors or acceptors to render
them extrinsic at the diffusion temperature. In such
samples the ratio of interstitial to substitutional copper
can be made either large or small compared to unity,
thereby permitting an independent study of each
species. We have measured the diffusion coefficient of
interstitial copper and the solubilities of both species in
each of these semiconductors as functions of tempera
ture. Our experiments confirm that interstitial copper
is a shallow donor carrying a single positive charge,
and show that substitutional copper is a triple acceptor
in both germanium and silicon, and a double acceptor
in gallimn arsenide. In pure germanium and gallium
arsenide the solubility of substitutional copper is
greater than that of interstitital copper, whereas the
opposite is true in the case of silicon. We have observed
rapidly precipitating electrically active donors due to
interstitial copper in silicon. We find evidence for
pairing between substitutional copper and donors in
strongly n-type germanium and perhaps also in silicon,
but no evidence for pairing between interstitial copper
and acceptor impurities in any of the semiconductors
studied. Rather unexpectedly, we were also able to deter
mine the rate at which the energy gaps of gennaniul1l
and silicon decrease with acceptor concentration from
the copper solubility measured in degenerate p-typc
crystals.
2. OBJECTIVES
The initial objective of this work was to determine
the behavior of copper in heavily doped gallium arsenide
in order to determine whether this impurity was re
sponsible for forward injection failure24 in gallium
., C. B. Collins and R. O. Carlson, Phys. Rev. 108, 1409 (1957).
22 J. M. Whelan and C. S. Fuller, J. App!. Phys. 31 1507 (1960).
23 W. Engeier, H. Levinstein, and C. Stannard, 'Jr., J. Phvs.
Chern. Solids 22, 249 (1961). .
24 A. Pikor, G. Elie, and R. Glicksman, J. Electrochem. Soc. 110,
178 (1963); H. J. Henkel, Z. Naturfsch. 17a, 358 (1962). arsenide tunnel diodes. While our results show that
copper exhibits behavior which can account for forward
injection failure, we have not yet been able to demon
strate that copper is responsible for thi~ phenomenon.
It could be caused by some other rapidly diffusing
defect such as a vacancy, for example. During the course
of this investigation we found that experiments in
volving copper dissolved in extrinsic 11-and p-type
gallium arsenide afforded a powerful means of studying
the characteristics of this impurity. The techniques
and equipment which were developed for this system
were then applied to similar studies of copper in ger
manium and silicon.
3. ANALYTICAL REMARKS
The experimental results to be presented are most
conveniently discussed if the model which is used to
interpret them is first described. We find that two
species of copper are of importance in our experiments.
Interstitial copper is a rapidly diffusing impurity which
gives rise to a single donor level dose to the conduction
band. Consequently it carries a single positive charge
under all of the experimental conditions with which we
are concerned. Substitutional copper, on the other hand,
is a multiple acceptor having a much smaller diffusion
coefficient. In germanium and silicon it is a triple
acceptor and in gallium arsenide it is a double acceptor.
The behavior of the copper can be strongly influenced
by the presence of other impurities.19•2o,25 \Ve are con
cerned here with the relatively simple case in which
only one other impurity is present and this is a shallow
(hydrogenic) donor or acceptor having a negligibly
small diffusion coefficient. Two kinds of impurity in
teraction may be distinguished. The simplest is a purely
electronic effect in which the copper solubility is
affected only to the extent that the position of the
Fermi level is changed by the presence of the donor or
acceptor. The other kind of interaction is known as ion
pairing. It involves the physical association of the
copper atoms with the donor or acceptor atoms and the
establishment of chemical bonds between them. These
two mechanisms are discussed in more detail below.
3.1. Dependence of Copper Solubility upon
Donor or Acceptor Concentration
The physical principles involved in the electronic
interaction between an impurity such as copper and the
shallow donors or acceptors have been clearly elucidated
by Shockley and MoI1.20 This interaction is particularly
simple, inasmuch as it depends only upon the ratio of
the free hole (or electron) concentration to the intrinsic
carrier concentration at the temperature in question
and is thus independent of the particular donor or
acceptor impurities used. One of the important findings
2. H. Reiss, C. S. Fuller and F. J. Morin, Bell Systems Tech. J.
35, 535 (1956).
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of this research is that this electronic interaction, rather
than pairing, is the one that is most important in
determining the solubility of both species of copper over
a wide range of temperature and donor 0: acce'p~or
concentrations. We found clear evidence for IOn pamng
only when copper Vias diffused into n-type germaniu:n.
For convenience in later discussions, we summanze
here the equations which describe this electronic con
tribution to the solubility for the specific case of the
interstitial species of copper which we kno~ to ~e. a
shallow donor impurity. When an atom of mterstltIal
copper is dissolved in an intrinsic semicon~uctor the
enthalpy of solution includes the energy gamed when
the electron which accompanies it drops down to the
Fermi level. If the semiconductor is extrinsic p-type the
Fermi level is lower, and the electron gains additional
energy !J.E in reaching its equilibrium energy. Con
sequently the solubility is increased by the Boltzmal!n
factor, exp(!J.E/kT). Since in nondegenerate matenal
the hole concentration is larger than ni by this same
factor, it is evident that the solubility Ci is directly
proportional to the hole concentration p:
Ci=C/P/ 17i. (1)
. . t "." d " " Throughout this paper we use superscnp s ~ an s
to distinguish interstitial and substitutional copper, and
a subscript "i" to indicate the value of a parameter in
intrinsic semiconductor. c,' is the solubility of inter
stitial copper in intrinsic semiconductor and ni is the
intrinsic carrier concentration.
Tn pure germanium and silicon the solubilities of both
species of copper are much less than ni and the total
measured solubilities correspond closely to C'+C'.
However, in gallium arsenide the solubility of copper is
greater than ni and, therefore, the free carrier concentra
tions are appreciably disturbed by the introduction of
the copper. Consequently, it is necessary to distinguish
the solubilities observed in pure gallium arsenide from
the intrinsic solubilities. Thus, in applying Eq. (1) to
gallium arsenide, it is to be noted that C,' is the s?lu
bility of interstitial copper in a sample of gallIum
arse~ide which contains a sufficient concentration of
shallow donor impurities to neutralize the net number
of acceptor levels introduced by the copper, thereby
making the electron and hole concentrations equal to 11,
at the saturation temperature.
If the semiconductor contains a degenerate concen
tration of electrons or holes at the saturation tempera
ture, then the rate at which the Fermi level changes
with carrier concentration becomes more rapid and
Eq. (1) is no longer valid. In degenerate p-type material,
for example, p= (2/v/7r)N"Fl(l;ikT), where Nv is the
densitv of states in the valence band, FI; is the Fenni
integr~l/6 and i; is the amount by which the Fenni level
lies below the valence band edge. Using tl.i= (N eN,,) i
exp (-Eg/2kT), it is readily shown that Eq. (1) takes
26 J. S. Blakemore, Proc. Phys. Soc. (London) 71, 692 (1958), 'T ! \ I j illlj
IOO~
,~"~ ... ,J
IO".~1 ==±==:L.L.Ll-'-ll_-'--L.L 10 50
PIN,.
FIG. 1. Degeneracy correction factor ~ as a f~nction of hole
concentration divided hy valence hanel denslty of states.
the fonn
(2)
where
~= V7r exp(i; jkT)j2F~(1 jkT).
Since l/kT is a known function of p/N., we can write ~
as a fu'nction of this same quantity, i;CpiN v). This
function is shown in Fig. 1.
Equation (2) can also be modified to take into account
shrinkage of the energy gap due to heavy doping. Such a
decrease in energy gap would not affect the concentra
tion of that fraction of the interstitial copper which is
neutral. Using the neutral species as a reference point,
we can calculate the concentration of ionized interstitial
copper using arguments patterned after Shockley and
MoI1.20 We find that this effect can be taken into account
in extrinsic material by multiplying the right side of
Eq. (2) byexp( -!J.Eg/kT), where !J.Egis the decrease in
energy gap due to heavy doping.
Ci=Cii(p~/ni)exp( -!J.EgjkT). (3)
In this equation C;i and fti are the values appropriate
for the undisturbed energy gap.
An equation analogous to (1) describes the solubility
of substitutional copper in extrinsic n-type semicon
ductor. If we make the assumption that the acceptor
levels introduced by substitutional copper are all
located below the middle of the energy gap, then in the
nondegenerate case the solubility varies as20
C'=C/(n/n,)r, (4)
where 1l is the free electron concentration and r is the
mUltiplicity of the acceptor level: 3 for germanium and
silicon and 2 for gallium arsenide.
Ion pairing may further affect the solubility of copper
in the presence of added donors or acceptors.25 Since
this mechanism has been found to play little part in
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most of our experiments, (it was important only in the
case of copper in ll-type germanium) we need only note
a few of its characteristic features. The functional
dependence upon the concentration of added donor or
acceptor impurities is generally different from that given
by the foregoing equations. Since pairing only occurs if
the chemical bonds formed are exothermic, it can only
enhance the solubility of the copper. Finally, it is to be
expected that the dit1usion coefficient of substitutional
or interstitial copper would be decreased by pair for
mation, inasmuch as the donor or acceptor impurities
themselves have such small diffusion coefficients.
3.2. Diffusion of Copper in the Presence of
Donor or Acceptor Impurities
Complex penetration profiles are often observed when
copper is diffused into semiconductors because of the
simultaneous presence of appreciable concentrations of
Loth substitutional and interstitial copper and the re
action bet ween vacancies and interstitials to form the
substitutional species.14-18 In certain cases, however, the
penetration obeys the simple diffusion equation.
One such case is where the copper solubility is small
compared with the free carrier concentration and the
crystal is sufficiently disordered that a plentiful supply
of \'acancies is available so that local equilibrium exists
between the substitutional and interstitial copper.
Under these circumstances the dissociative diffusion
process of Frank and Turnbull14 applies and the copper
exhibits an effective diffusion coeffici('llt III intrinsic
material given by
(5)
D' is the difiusion coefficient of interstitial copper and
a, is the ratio of substitutional to interstitial copper in an
intrinsic sample. 'Ve have assumed the diffusion rate of
substitutional copper to be negligible. In extrinsic
material Ci, must be replaced by
(6)
in accord with Eqs. (1) and (4), and the effective
ditTusion equation becomes,
(7)
In extrinsic /I-type samples such thata»1, the diffusion
coefficient is expected to decrease as the inverse fourth
power of the donor concentration in germanium and
silicon, and as the inverse cube in gallium arsenide.
A simpler case is that of p-type semiconductors which
are sufficien tl)' extrinsic that a« 1. vVe then have
(8)
This relation results when most of the copper is in
terstitial and evidently it remains valid even in highly
perfect crystals. These are the experimental conditions
under which we have measured the diffusion coefficient
of interstitial copper as reported below in Sees. 6 and 7. 1000
1020 ,
~
r I I
10"l:-
~
1018~ t
c
."
~ [ .;:
"1
'016~
f:
~ I
lols-_L.
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
laaO/T CK
F1G. 2. High-temperature intrinsic carrier concentrations.
This equation is, of course, only valid in the absence of
ion pairing.
In strongly n-type material the copper solubility may
be large enough to appreciably reduce the free electron
concentration. This causes a corresponding increase in
the interstitial/substitutional ratio, and hence in the
diffusion coefficient. The penetration profile in such a
case consists of a near! V uniform laver near the surface
where the donors are ~'losely compensated and Deff is
large, followed by an abrupt concentration decrease
where Fick's law diffusion takes place with a much
smaller diffusion coefficient.
It is important to recognize that although copper can
be diffused into the n-type crystal to a considerable
distance because of this nearly compensated layer, the
copper cannot be removed again at the same tempera
ture in a comparable time. As soon as the copper is
depleted near the surface to the point where the material
is no longer closely compensated Doff drops to a small
value there and consequently the out-diffusion process
takes a much longer time.
3.3. Intrinsic Carrier Concentrations
at High Temperatures
The electronic interaction which is the basis for the
foregoing equations depends critically upon the intrinsic
carrier concentration. Curves giving n, as a function of
temperature have been constructed for a number of
semiconductors and are shown in Fig. 2.
In general, when only a single conduction and single
valence band are involved, it is expected that n, has the
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form
1l1=ATI exp( -Eyo/2kT), (9)
where Ego is the value of the energy gap extrapolated to
T= 0, and A involves the densities of states masses and
a factor due to the temperature coefficient of the energy
gap. The procedure adopted for these semiconductors
was to fit the above formula to the highest temperature
data published for ni that appeared reliable in order to
extend these data to higher temperature. Well estab
lished values for th and Eyo are available for Ge and
Si. Data for most of the III-V compounds were taken
from recent literature.27,28 In some cases the calculation
of 11 i required further elaboration:
GaSh. Leifer and Dunlap29 give til data for tempera
hlres up to 640°C, but they do not apply Sagar's two
band correction30 to their Hall measurements. The curve
which we show was calculated by assuming that all of
the electrons are excited to the (111) conduction band
which is like that of Ge except 0.108 eV farther from the
valence band. We thus multiplied the Ge curve by
exp(-0.108/2kT). At MO°C this curve is higher by 1.6
than the data of Leifer and Dunlap with the two-band
correction applied (a factor of 1.85), but below soooe
the agreement is good.
GaAs. A two-band correction similar to that used for
GaSb is required at high temperatures in GaAs. The
curve. shown was calculated from the energy gaps and
effective masses reported for this material using a den
sity of states for the (100) band that is 100 times that
of the (000) band as indicated by the temperature
dependence of the Hall data.28 The results agree ,veIl
with the Hall measurements of Whelan and Wheatley 31
with the two-band correction applied. ' ,
Note added in proof. .I. O. McCaldin [.I, Appl. Ph\'s.
34,1748 (1963)J finds n.i=4X1018 cm-3 at 1000°C. '
AlSb. Since the band structure of this material is
apparently similar to that of Si/8 and there are no
other.nearby band edges, we estimated ni by correcting
the SI curve for the difference in energy gaps as in the
case of GaAs.
4. EXPERIMENTAL PROCEDURE
Radioactive copper CU64 was used as the principal
means of measuring the amount and distribution of
copper in the samples. It is supplied as an acid solution
containing this isotope mixed with a much larger
amount of stable copper. For convenience, we often
refer to the mixture simpley as CUM. It was obtained
fro~l. Oak ~idge National Laboratory with a specific
act!vlty whIch was sometimes as high as 60 Ci/g upon
arrIval.
_ Th~cay of CU64 is accompained by both 'Y and f3
.27 C. Hilsum and ;\. C. Ro&;-[nnes, Semiconducting III-V
Compounds (Pergamon Press Inc. New York 1961)
2" H. Ehrenreich. J. :\ppl. Phys: 32 2155 (1961) ..
29 H.~. Leifer and W. C. Dunlap, J~., Phys. Rev. 95, 51 (1954).
30 A. Sagar, Phys. Rev. 117, 93 (1960).
31 J. M. Whelan and G. H.Wheatle v J Phy's Chern Soll'ds 6 169 (1958). J" • • , emission, the latter being 4500 times more numerous.
The total amount of copper in a sample was determined
by measuring the 'Y activity, using a deep-well KI
scintillation counter, which was shielded from the f3
particles. The distribution of the copper within a sample
was determined by means of a mica-window (1.4
mg/cm2) Geiger counter or by densitometer measure
ments of radioautograms obtained by placing the
sectioned sample in contact with Kodak x-ray film. The
latter two methods are primarily sensitive to the f3
radiation, which has a mean range of 0.08 mm in
GaAs and Ge and 0.2 mm in Si.
The free carrier concentrations in the semiconductor
samples were determined by measuring the Hall
coefficient R in the sample to be diffused or in an im
mediately adjoining region of the same crystal, assum
ing the concentration to be given by l/eR (except as
noted in Sec. 8) in view of the large impurity concentra
tions which were used in most of our experiments. Be
fore adding the radioactive copper, the samples were
lapped with fine carborundum over most of their sur
faces and then soaked for several minutes in a 20%
solution of KCN in order to remove any natural copper
which might have been deposited there. They were next
rinsed several times in triply distilled water and then
in reagent grade methyl alcohol and blotted dry using
tilter paper. CUM was then electroplated on all surfaces
(or along a narrow stripe in the case of the drift experi
ments) and the activity due to the total amount of
radioactive copper present was measured. The samples
were then placed in quartz tubes which had been
similarly cleaned with KeN and rinsed. In most cases
the samples were heated under one atmosphere of
hydrogen in these tubes for the required time interval.
At the end of the diffusion period the ends of the tubes
containing the samples were quenched in water, thereb\
bringing the samples to room temperature within 10 s~r
or so. Diffusion of gallium arsenide samples at tempera
tures above 800°C was usually carried out with added
arsenic in evacuated sealed quartz tubes to prevent
decomposition. Following diffusion the activity was . " agam counted, and then the sample was etched lightlY,
weighed, and rinsed in KCN, followed by distilled wat~r
and alcohol, and again counted. This sequence was
repeated a second and third time or more if necessan'
until it appeared that all of the surface copper had bee~
removed. A radioautogram of the sample was then
taken as a means of detecting the possible presence of
undissolved copper which might still remain. Thicker
samples were sectioned prior to obtaining the radio
autogram in order to measure concentration gradients
within them. Further experimental details related to
the detem1ination of diffu"ion coefficients are de,;crilwd
later.
5. DISCUSSION OF EXPERIMENTAL ERRORS
The activity of the CU64 is assayed by the Oak Ridge
Laboratory to an accuracy which is claimed to be
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TABLE 1. Copper solubility in extrinsic p-type germanium.
N A (1019 em-3) Temp (deg C) Ci (1016 em-a)
3.5 (Ga) 670 1.6
6.3 (Ga) 670 3.8
20.0 (Ga) 670 8.2
40.0 (Ga) 670 26.0
1.6 (AI) 600 0.25
1.6 (AI) 600 0.20
3.7 (Ga) 600 0.54
6.3 (Ga) 600 1.6
7.7 (Ga) 444 0.135
13.4 (Ga) 444 0.26
7.7 (Ga) 348 0.010
13.4 (Ga) 348 0.015
better than 5%. Each lot of CU64 was calibrated in terms
of the sensitivity of our own counters, and the decay was
observed to conform accurately to the 12.8-h half-life
over the duration of each series of experiments, which
sometimes lasted as long as 10 days. It was thus estab
lished that the counter sensitivities remained constant
and that no other isotopes were present in significant
amounts. An error of 0.05 h in the half-life of the CU64
would result in a 5% error in copper concentration over
this period.
Solubility measurements were obtained from samples
diffused to saturation as shown by radioautograms of
the sectioned samples. The sample surfaces were lapped
before being electroplated with CU64 in the belief that a
highly damaged surface would be less likely to exhibit
a barrier to the entrance of the Cu into the interior of
the sample as has been observed for antimony in
germanium.32 \Ve are not aware that such an effect
actually exists in the case of copper and found no
evidence for it in our experiments, but lapped the
samples as a precaution. \Ve also required that the total
amount of CUM measured immediately after diffusion
be at least five times the amount in the sample after re
moval of the surface copper in order to make certain
that sufficient excess copper was present to insure
saturation. This was only a problem in strongly extrinsic
samples exhibiting high copper solubilities. The amount
of copper required for these samples was clearly visible
after electroplating, and the plating was continued until
a reasonably uniform distribution of copper was ob
served over the surface of the sample. The radioauto
grams showed the samples to be free of surface copper
after etching, otherwise the measurement was dis
carded. Densitometer measurements of these radio
autograms were usually made and checked against the
y-counter detel1l1inations as further evidence of the
reliability of the measurements. We estimate the pro
bable error in individual measurements of the absolute
value of the copper concentration to be about 20%. In
many cases the same solubility was determined in
several different experiments and using more than one
32 R. C. Miller and F. 11. Smits, Phys. Rev. 107,':65 (1957). ~ exp(.6.E g/kT) Cii (em-3)
1.52 1.54 2.3X1014
2.10 2.12 3.0XlO14
8.4 7.8 1.9X 1014
60.0 34.0 1.85X 1014
1.23 1.24 4.5XI013
1.23 1.24 3.6X 1013
1.64 1.64 4.3X 1013
2.27 2.25 7.3X1013
3.6 3.3 1.1 X 1012
8.8 6.9 LOX 1012
4.8 3.9 2.2X 1010
14.4 9.2 1.6X 1010
lot of CU64 and the agreement was usually within this
20% uncertainty.
Diffusion data were obtained by several methods, and
the associated errors are discussed later in connection
with the results.
Statistical errors were often appreciable due to the low
solubilitv and short half-life of the copper. In such cases
the sta~dard deviation is indicated by vertical bars
through the data points.
6. EXPERIMENTAL RESULTS ON THE SOLUBILITY
AND DIFFUSION OF COPPER
6.1. Solubility and Diffusion of Copper
in Germanium
p-type Ce. Enhanced interstitial solubility has been
observed in accord with Eqs. (1) to (3) using samples
which are extrinsic p-type at the saturation tempera
ture. For samples which are sufficiently extrinsic, inter
stitial copper is the dominant species and the fraction
that is substitutional can be neglected. Our results are
summarized in Table 1. The first two columns give the
sample doping and saturation temperature. Ci is the
interstitial solubility at the saturation temperature as
given by the radioactivity detemlination. From these
measurements we wish to calculate the interstitial
solubility in intrinsic germanium C,J.
As a first approximation, Eq. (1) can be applied
directly without taking into account the degeneracy or
energy gap corrections, giving a tentative solubility,
C,:i*, which should be nearly correct for the more lightly
doped samples. The values thus obtained are found to
fall along a single solubility curve, essentially inde
pendent of the acceptor concentration. However, it is
clear that the degeneracy correction factor I; is quite
large for some of the samples so that Eg. (1) is invalid.
But if Eq. (2) is used to calculate C,i, assuming a val
ence band density of states mass ratio of 0.36, it is
found that the results obtained from the heavily doped
samples fall far below the curve obtained from the more
lightly doped samples. This discrepancy can be resolved
by assuming that the energy gap decreases with in
creasing acceptor concentration by an amount that is
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700 500 300'C
1O'1,---,-L-y,L! -,,;-'--,--t\--,-'r---,-.,-.!::-r-.....----,
'0"
lOOOIf
FIG. 3. Substitutional and interstitial Cu solubilities in intrinsic
Ge. Circles give C,i calculated from p-type samples. Diamonds
give C'+c" obtained from intrinsic samples. The upper 4500
diamond may be high due to Sb (see text). The curve for Ci' is
from Woodbury and Tyler.".
twice as great for the same acceptor concentration as
it is in silicon (see Fig. 10) where more extensive data
were obtained. The last column of Table I gives C ii
corrected for both degeneracy and energy gap shrinkage,
using Eq. (3). These values are plotted in Fig. 3. They
are in good agreement with the higher temperature
measurement'> of C i by Fuller and DitzenbergerY
It is noted that the two correction factors, ~ and
exp(t:.E.ikT), are found to be almost equal and can
sequentIy C;i"",C/* except in the most strongly degen
erate specimens. This is also the case with silicon. The
decrease in energy gap which we require in order to
explain our experimental results is in good agreement
with that given by Sonuners:J3 (0.03 eV for 4X 1019 cm-3
Ga or As) and by Fowler, Howard, and Brock34 (0.15
eV for 2X1020 Lm-3 Ga+As). It is about half that re
ported for n-type germanium.35•36
Intrinsic Ge. For completeness, we also made a few
determinations of Ci+C; by diffusing CUM into single
crystal samples of pure gelmanium. The samples were
approximately 4 mm thick and were diffused for 14 h,
which is sufficient to achieve saturation. The results are
shown in Fig. 3. The lower of the 450°C points was
obtained using germanium that was nearly intrinsic at
room temperature. The higher one was obtained from a
3.SX 1014 em-3 antimony doped crystal. At the time
33 H. S. Sommers, Jr., Phys. Rev. 124, 1101 (1961).
34 A. B. Fowler, W E. Howard, and G. E. Brock, Phys. Rev.
128, 1664 (1962).
3. J. I. Pankove and P. Aigrain, Phys. Rev. 126, 956 (1962).
36 C. Haas, Phys. Rev. 125, 1965 (1962). these experiments were performed it was felt that this
was of sufficient purity to give the intrinsic copper
solubility, but in view of the n-type germanium results
to be described next it seems likely that pairing may
have taken place, accounting for the anomalously high
value. Aside from this discrepancy, these data are in
satisfactory agreement with the curve for C; obtained
by Woodbury and Tyler37 using electrical measure
ments. Near 700°C the ratio of substitutional to inter
stitial solubility is 6, in agreement with the findings of
Tweet/5 and Fuller and DitzenbergerP
n-type Ge. Shockley and Mo1l20 calculated the solubility
of substitutional copper in germanium as a function of
donor concentration, assuming that ion pairing was
unimportant. This calculation suffered from failure to
take into account the temperature dependence of the
energy gap of germanium and from a serious error in the
value of ni which was used. When performed correctly
the calculation indicates that the solubility enhance
ment should be observable, although much smaller
than their estimates had indicated. In order to complete
our observations of the behavior of copper in germanium
we undertook to test the predictions of these calculations
by measuring the copper solubility in a series of arsenic
doped samples. The results showed quite substantial
departures from the calculated behavior.
Solubilitv enhancement is more difficult to demon
strate in g~rmanium than in silicon or gallium arsenide
because of the much larger value of 11, at comparable
temperatures. The largest effect is observed at the low
est temperature at which the solubility is sufficient to
be measured. On the basis of the results of some pre
liminary experiments, 23 arsenic-doped n-type wafers
about t mm thick and having donor concentrations from
1.8 to 52 X 1018 cm-3 were diffused at 600° and 650°C for
times ranging from 28 to 140 h. Only partial penetration
of the copper through the wafers was obtained in most
cases, and by analysis of the curves of activity vs wafer
thickness it was possible to deduce both the surface
concentration and the effective diffusion coefficient for
many of the samples. In the case of the samples diffused
at 650°C the solubility values may be in error by as
much as a factor of two due to difficulty in interpreting
the penetration profiles with a limited number of data
points. At 600°C the difficulties were even greater, and
data from several of the samples were too uncertain to
plot. Those results which were judged to be reliable are
shown in Fig. 4. They show several unexpected features.
The solubility rises smoothly as the donor concentra
tion increases, but far more rapidly than the nonpairing
theory would predict. At a donor concentration equal
to Iti the increase should be no greater than (n/ni)2= 2.6,
the upper acceptor level being unoccupied, whereas the
observed enhancement at 650°C is more like a factor of
100. Ion pairing is evidently playing a major role in
increasing the solubility in this range. However, the
37 H. H. Woodbury anu W. W. Tyler, PhO's. Rev. 105,84 (1957).
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissions3cS6 R. N. HALL Al\D ]. H. RACETTE
1020,--_-.._== .... --,----,...----,------,10-5
.... \
\ SOLUBILITY OF Cu IN n -TYPE Ge
\
\
\
\
\
\
\ \ 0,650'
\
\
\
\
\
\
\
\
\
\
\ I
I
I
I
I
/!
I
I
I
I I
I
I
I
I
I
I
FIG. 4. Solubility and diffusion of Cu in n-type Ge. The two
high-concentration diffusion points (triangles) are lower limits,
since the samples were uniformly saturated.
data are not sufficiently extensive or accurate to attempt
a quantitative fit to a combined ion-pairing and elec
tronically enhanced solubility theory. The Hall coeffici
ent of one of the 650° samples was remeasured after
copper diffusion, and the free electron concentration was
found to have decreased from 3.37 to 1.18XlOI9 cm 3.
This is just sufficient to account for the 7AX 1018 cm-:l
copper atoms which had been diffused in, assuming that
they still behave as triple acceptors even though paired
or forming higher complexes with the arsenic atoms.
Pairing between copper and donors in germanium has
also been observed by Potemkin and Potapov.38
The behavior of the effective diffusion coefficient in
these n-type samples is also surprising. At first it
decreases rapidly with donor concentration as would be
expected in view of the rapidly increasing substitutional
solubility. However, beyond arsenic concentrations of
about 10IU cm-3 it suddenly increases to much larger
values again. Just how great this increase is has not been
determined, since the samples were uniformly saturated,
and, therefore, the data only provide a lower limit for
the diffusion coefficient. \Ve have observed this rapid
diffusion of copper in several other samples of heavily
doped n-type germanium as well. Enhanced sub
stitutional diffusion due to an increase in the vacancy
38 A. Y. Potemkin and V. I. Potapov, Soviet Phys.-Solid
State 2, 1668 (1960). concentration induced by the high donor concentra
tion39 might account for this behavior. It seems more
probable that precipitation of the arsenic was beginning
and giving rise to vacancies or other defects which
aided the diffusion of the substitutional species.40
Interstitial diiJusion in Ce. Germanium cubes, 1 cm on
an edge and gallium doped to 1.34 X 1020 cm-3 were Cu 64
plated and diffused at 348°, 444°, and 7S0°C for 30,14,
and 3.1 min, respectively, in hydrogen. Radioautograms
were made after sectioning, and the copper distribution
was determined by densitometry. Families of curves
shown in Fig. 5 were calculated for diffusion into a cube,
using the methods given by Olson and Schultz41 and
values for the interstitial copper diffusion coefficient
were obtained by fitting the densitometer data to these
curves. The parameter, Dt/a2, which gave the best flt
to these curves had the values 0.07, 0.063, and 0.070,
respectivel y. The diffusion coefficient is plotted in
Fig. 6.
The uncertainty indicated for the 348°C point is due
to the faintness of the radioautogram because of the low
copper solubility. At 7S0°C the diffusion time required
was short. A reasonably "square" temperature profile
was obtained by heating the sample in an 850°C furnace
and moving the sample to a 7S0°C zone when it reached
750°C. The tube containing the sample was quenched in
water after diffusion.
C
Co 0.1
0.01
'10 0.4
0.3
0.2 ___ ~
FIG. 5. Calculated concentration profile along a cube axis
for difIusion into a cube of edge 2a.
39 M. W. Valenta and C. Ramasastry, Phys. Rev. 106, 73 (1957).
40 W. G. Spitzer, F. A. Trumbore, and R. A. Logan, J. Appl.
Phys. 32, 1822 (1961).
41 F. C. W. Olson and O. T. Schultz, Ind. Eng. Chern. 34 874
(1942). '
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissionsDIFFUSION AND SOLUBILITY OF Cll IN EXTRINSIC Ge 387
While the diffusion measurements were carried out
using strongly p-type germanium, our more extensive
experimental measurements in gallium arsenide show
that the interstitial diffusion coefficient is independent
of the acceptor concentration. Presumably, this is also
the case in germanium. Any pairing which might take
place between interstitial copper and gallium would
reduce the diffusion coefficient by the greatest amount
at lower temperatures, and consequently the true inter
stitial diffusion coefficient would have, if anything, a
smaller activation energy than the one which we report
here.
6.2. Solubility and Diffusion of Copper
in Silicon
Undoped silicon. Samples of high-purity floating-zone
silicon from Merck and du Pont were saturated with
eu 64 in the temperature range 400° to 8S4°e, giving
solubilities shown by the square data points in Fig. 7.
Ten samples were measured at 4000e and eight of
these fell in the range 1.3 to 3.7X 1012 cm-3• The activi
ties of these samples were close to the limit of detecta
bility, and much of this scatter is of statistical origin.
The other two showed somewhat higher activity pre
sumabl.:-" due to concentrations of copper which had
not been removed. Ten similar samples were measured
at SO(}Oe and eight exhibited solubilities in the range
3.4 (0 4X 101:! cm-3, the other two being only slightly
\ ~il..--"r--'I--'IJ,I-~I--'D;-. ~::LI °e~,lp-(~~~1 k TI~ 10C 'C
Do E
(em2/see) (eV, I
Ge 0.0040 0.33
SI 0.0047 0.43
\ \
\ \
\ \
\ \
\ \
\ \
\ '\
\ (' \ (' \' y GoAs 0.030
• •
GaAs
• \~ {~
\ \ .
\ \ ..
\ \ 66 0.53
\ r \ \
10-8::- \ \\
"J,~~ ~ ~, c ~ ~ ,_ ,_ '_\--'\_\J~_\.J.I---!-=--'-_
, a 1.5 2.0
10001T
FIG. 6. Interstitial Cu diffusion measured in extrinsic p-type
Ge(.), SiC e), and GaAs(.). 0 Out-diffusion in GaAs. D. Drift
in GaAs. 0 eu diffusion in intrinsic Si by Struthers.' Diffusion of
Li in Ge and Si is included for comparison (See Ref. 43). I
E 14
u 10 ,
>
~
-"
'" '" ~ 1013
10" \
\
\ 500 300'C
------.----lr '-'---'-I T -. .----,
\
\
\ C '. I
\ C 1 (SEE TEXT)
\
\
\ ",
•• o FROM PURE Si
o FROM P-TYPE Si
\
\
10 10 L.--'---+O--L-'--L..--'----'--,',~L..--'----'-'-...J._7.
2.0
1000 IT
FIG. 7. Copper solubilities in intrinsic Si. Square points are
actually the total CU64 activity measured in pure Si. Circles give C,' calculated from solubilities measured in extrinsic p-type sam
ples. C,' is estimated from extrinsic n-type samples.
higher. Another group of nine samples including six
which were grown from quartz crucibles and had meas
ured oxygen concentrations between 3 and 13X 1017
cm-a were diffused at soooe in a separate experiment.
The results were substantially the same, with no evi
dence of any effect due to the presence of oxygen. The
points at other temperatures in Fig. 7 were obtained
using individual samples.
Abnormally high copper solubilities were invariably
exhibited by samples which had previously been satur
ated with copper at a higher temperature. Presumably
the copper introduced during the first diffusion formed
precipitates upon cooling, and these were not dissolved
upon reheating at the lower temperature but simply
exchanged with the fresh radioactive copper to give a
high apparent solubility. \Ve also found that the "as
deposited" rods of high purity polycrystalline silicon
obtained from the decomposition of silane or trichloro
silane gave copper solubilities that were typically ten
times higher than that exhibited by monocrystalline
silicon in the range 40(}0 to SOO°e.
The results of the solubility mesaurements in n-and
p-type silicon to be described below show that sub
stantially all of the solubility in intrinsic silicon is due to
interstitial copper. We conclude, therefore, that this
impurity is a donor which precipitates so rapidly that
the electrical activity which it produces is not normally
observed. Experiments carried out by Soltys of this
laboratory were able to detect the presence of this
rapidly precipitating interstitial copper. Monocrystal
line samples of p-type silicon containing 0.9 to 1.0X 1013
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissions388 R. :\. HALL A~D ]. H. RACETTE
cm a boron were cut with enlarged electral contact
regions extending from the ends and sides in order to
eliminate contact problems during Hall effect and re
sistivity measurements. They were copper plated and
heated in hydrogen for several hours in the temperature
range 400° to 575°e and then quenched by dropping
directly into chilled ethylene glycol. Their surfaces were
then removed by sandblasting, gallium-aluminum paste
contacts were applied to the contact regions, and Hall
effect measurements were made as a function of time at
room temperature. Figure 8 shows the results. Near
400°C the decrease in hole concentration at the be
ginning of the measurements is about 40% of the inter
stitial copper solubility given by Fig. 7. At 475°e it is
about 30% and at 575°e it: is only about 3%. These
experiments are consistent with the interpretation that
electrical activity due to interstitial copper has been
observed, and that when saturated with copper near
400"C where the degree of supersaturation is small after
1.0
. 8
~
Po
.6
450·C,3,5h ..
.2
0
" " 1 1 10 100 500
t.lINUTES AFTER QUENCH
FIG. 8. Room temperature precipitation of interstitial Cu in
floating-zone monocrystalline Si after saturation at indicated times
and temperatures. po and p are the hole concentrations measured
before and after Cu diffusion. Samples contain boron in the range
0.9 to 1.0X 101:< cm-3•
quench the time constant for precipitation is a few
minutes but becomes much shorter with increasing
supersa tura tion.
p-type Si. Silicon samples doped with boron in the
range 0.95 to 430X 1018 cm-3 were diffused to saturation
with eu64 at 300°, 400°, 600°, and 700°e. The solubili
ties determined from these experiments are shown in
Fig. 9. The solid curves are calculated using Eq. (3),
with C;i chosen to give the best fit. At low boron
concentrations these curves approach the limiting value,
C,i+C;"~C;i, the substitutional solubility being negligi
bly small, as shown below by experiments with n-type
samples. The linear extension of these curves at high
boron concentrations corresponds to Eq. (1). Taking
into account the degeneracy correction using Eq. (2)
with a valence band density of states mass ratio of 0.6
gives calculated solubilities [shown by the dashed
curves (a) at 300' and 7000e] which rise much more
rapidly than do the measured solubilities. I
~
~IOI6
:::;
iii
~
10" 600
500
FIG. 9. Copper solubility in p-type Si. Solid curves are calculated
from Eq. (3) with I:;Eg from Fig. 10. Curves "a" show the cal
culated solubility assuming l:;£g=O .
The decrease in energy gap required to satisfy Eq. (3)
was calculated for each of the samples having N> 1020
cm-3 and is plotted in Fig. 10. The points obtained at
different temperatures all fall along a single curve within
experimental error. This curve for I:l.Eg was used in
calculating the solid curves of Fig. 9.
The values of Cii which were obtained by fitting
Eq. (3) to these groups of data points are plotted as
circles in Fig. 7. The fact that a good fit can be obtained
and that these values of C;i agree very well with the
solubilities obtained by saturating high purity silicon
with copper is evidence that: (a) most of the solubility
in pure silicon is due to interstitial copper, (b) inter-
15
~.lO
.05
'" 300·C
• 400'C
• 600'C
• 100'C
~~------L-~--~2L-------L-------.L,~IO'~O----~
NA-cm-l
FIG. 10. Decrease in energy gap of Si due to B doping, deduced
from Cu solubility data. For Ga-doped Ge, I:;Eo is found to be
twice as large for a given acceptor concentration.
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissionsDIFFUSION Al\D SOLUBILITY OF Cll 1:\ EXTRINSIC Ge 389
stitial copper is a single donor, and (c) ion pairing is not
playing an important part in these experiments.
n-type silicon. Experiments similar to those just
described were also carried out using extrinsic n-type
silicon in order to test the conclusion reached on the
basis of the preceding experiments that C :«Ci and
to obtain evidence for the electrical charge states of
substitutional copper. It was expected that as the donor
concentration was increased the total copper solubility
would first decrease due to the reduced solubility of the
interstitial species, but that if substitutional copper were
a multiple acceptor this decrease would be followed by a
rapidly increasing solubility such as is exhibited by
germanium, Fig. 4. Such behavior has been observed
as shown in Fig. 11.
Single crystals of silicon doped with arsenic or phos
phorus in the range between lOl8 and 1.35 X 1020 cm-3
as determined bv Hall coefficient measurements were
plated with CU64 and ditTused in hydrogen at 500°,
600°, and 700°C. Since diffusion occurs very slowly in
strongly extrinsic n-type silicon because of the reduced
interstitial concentration, as described by Eq. (7), the
more heavily doped samples were diffused for longer
periods of time. Si wafer thicknesses of t to i mm were
used and each wafer was etched and counted repeatedly
until the thickness was reduced to about half the initial
value to make certain that a true volume concentration
was being measured. This proved to be the case in all of
the samples except for two doped with 1.35 X 1020 cm-a
phosphorus. At 700°C the copper penetration at this
doping level was sufficiently deep to permit an estimate
of the surface concentration, but it was too small to
10"r--------,-------,------.------,--~
• P -DOPED, 700 'C
• As-DOPED, roo °c
10,r 0 P -DOPED, 600'e
o As-DOPED, 600'C
A A.-DOPED, 500'C
700'C 'j
I .b -.h
FIG. 11. Solubility of Cu in n-type Si. To the left of the minima
the solubility is primarily due to singly charged interstitial eu; to
the right it is dominated by triply charged substitutional Cu. measure by the methods used in our experiments at
600°C even after diffusing for 74 h. Longer times were
ruled out by the 12.8-h half-life of CUM.
The results, which are shown in Fig. 11, exhibit
several interesting features. The curves clearly exhibit a
minimum at donor concentrations of a few times lOl8
cm-a, showing a very small ratio of substitutional to
interstitial solubility in intrinsic silicon. At higher donor
concentrations the curves rise very sharply, approxi
matelyas the cube of the donor concentration, indicat
ing that substitutional copper is a triple acceptor,
contrary to previous indications,2l Finally, we find that
the solubility in phosphorus-doped silicon appears to be
somewhat greater than in silicon doped with the same
concentration (as indicated by Hall coefficient measure
ments) of arsenic. The latter observation suggests that
some degree of ion pairing may be taking place.
The curves drawn through the phosphorus-doped
data points are calculated from the formula, obtained by
combining Eqs. (1) and (4),
C=C;i(11i/1n)+C/(n/nJ\ (10)
which would be expected in the absence of ion pairing
under the assumption that three acceptor levels are
introduced by substitutional copper and that tbese
levels are below the Fermi level in the doping range
where the substitutional solubility dominates. We con
clude that all three acceptor levels are more than 0.25
eV below the conduction band edge. The fit with these
data is good and gives the estimate of C/ which is
plotted in Fig. 7. An earlier conclusion that C8>C/ was
incorrect.42 The fact that somewhat lower solubilities
are indicated by the arsenic-doped samples would make
it appear that the curve for C: should be taken with
some reservation.
At high donor concentrations the solubility appears to
be increasing at a rate that is even faster than ND3.
Such a rapid rate of increase may perhaps be explained
by the more rapid movement of the Fermi level as the
samples approach degeneracy, as in the case of the
p-type specimens discussed in the preceding section.
However, we do not feel that the present experiments
are sufficiently extensive to justify such a treatment.
The diffusion rate of copper in strongly n-type silicon
was observed to decrease rapidly with increasing donor
concentration in an earlier series of experiments using
shorter diffusion times and thicker samples. This be
havior was at least qualitatively in accord with the
behavior expected on the basis of the solubility depend
ence for the two species given by Eq. (7), but the de
pendence was not tested in detail.
Interstitial diffusion in silicon. Samples of silicon in
the form of slabs approximately 4 mm thick and 1 cm
sq and doped with 5 X lOW cm-a boron were diffused with
CUM at 4000 and 480°C for 30 and 20 min, respectively,
42 R. N. Hall and J. H. Racette, Bull. Am. Phys. Soc. 7, 234
(1962).
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissions390 R. N. HALL A~D J. H. RACETTE
in hydrogen and quenched in ethylene glycol. They were
sectioned and a radioau togram exposure was made from
half of each. Cubes were cut from the other halves and
they were similarly rediffused at 600° and 680°C for
5 ~nd 2 min, respectively, sectioned, and exposed.
Densitometer traces made from these radioautograms
were fitted to the curves of Fig. 5 (or similar curves for
diffusion into a slab) to determine the diffusion coeffici
ent, with results shown in Fig. 6. The diffusion times and
temperatures and sample thicknesses were such that the
copper concentration at the ~urface was 5 to 100 times
larger than the concentration at the center of the various
samples. In this range the accuracy with which D may
be determined is quite high. We estimate the total error
associated with the individual D determinations to be
less than 20%. The relative error among these measure
ments should be less than this.
Heavily doped silicon was used in order to make the
interstitial wlubility of copper high enough to measure
at low temperatures, on the assumption that interstitial
diffusion would be unaffected by the presence of the
acceptors. Experimental justification for this assump
tion is to be found from measurements of interstitial
copper diffusion in p-type gallium arsenide to be re
ported below, and from the fact that a measurement of
copper diffusion in undoped silicon at 900°C by Stru
thers3 gave a value shown in Fig. 6, which falls reason
ably close to the extrapolation of our data. For com
parison we have included curves showing the interstitial
diffusion coefficients of lithium in germanium and
silicon.43
Our conclusions as to the diffusion coefficient and
charge of copper in silicon are consistent with those of
Gallagher, based upon drift experiments.13
6.3. Solubility and Diffusion of Copper
in Gallium Arsenide
Solubility in undoped gallium arsenide. Early copper
solubility measurements were reported by Fuller and
Whelan6 in the temperature range between 700° and
1100°e. These data showed no evidence for the exist
ence of a solubility maximum which is to be expected
near 1100°C. Furthenuore the validity of these experi
ments might be subject to question due to the use of
samples which were not truly intrinsic at the tempera
tUres involved. For these reasons we remeasured the
copper solubility using the higher purity material that
is now available, and extending the measurement range
to both higher and lower temperatures.
Single crystal samples of semi-insulating and of 0.1-
n-cm n-type gallium arsenide (SX 1015 electrons/ cm3)
obtained from Monsanto Chemical Company, were
plated with CU64 and diffused either in 1 atm of hydrogen
or sealed in vacuum in quartz ampules of 4-ml volume
with 4 mg of additional arsenic to prevent decomposi-
43 C. S. Fuller and]. C. Severiens, Phys. Rev. 96, 21 (1954). I
I
1016<--
E -
~
'OJ I
0.6 0.8 ~
\ Cu in GaAs
• ND EXTRA As
o SEALED WITh 4 mg As
• FULLER AND WHELAN
\
\ \.
\
\
\
\
~\ \ B
\
\
I \
1.0 1.2
1000/T ~
I
,
1
~
1
I I
1.4
FIG. 12. Solubility of Cu in undoped (but not intrinsic) GaAs.
Data by Fuller and Whelan6 and a portion of the eli curve from
Fig. 14 are also shown.
tion. Diffusion times ranged from t h near llOO°C to
over two days near 500°C, and the samples were verified
as having been uniformly saturated. There was no dis
tinguishable difference between the solubility in the
semi-insulating and the n-type material.
The results are shown in Fig. 12. Above 700°C the
measurements are consistent with a retrograde solubility
curve having a reasonably shaped solubility maximum
as shown by the solid curve. While the accuracy is not
sufficient to demand such a maximum, it is clear that
one should exist, and it is felt that the curve which is
drawn is close to the true solubility. Below the solu
bility maximum, the results obtained from the samples
sealed with 4 mg of arsenic (curve A) fall below those
of samples diffused in hydrogen without additional
arsenic. The Jatter are in good agreement with the
findings of Fuller and Whelan6 who diffused their sam
ples in vacuum but without additional arsenic. We
suspect that the reduced solubility represented by
curve A may be due to the presence of a condensed
arsenic-rich phase which dissolved most of the copper
at the lower temperatures and thereby reduced the
concentration within the gallium arsenide below satura
tion by a corresponding amount.
The curves of Fig. 12 do not represent the solubility
of copper in intrinsic gallium arsenide since this solu
bility (which is mostly due to substititional copper) is
greater than 1Zi. Consequently the material, when
saturated with copper, is p-type. This must be taken
into account in applying Eqs. (1) to (4).
Downloaded 07 Aug 2013 to 129.187.254.46. 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_permissionsDIFFUSIOK A:--JD SOLUBILITY OF Cu IN EXTRINSIC Ge 391
1000
E
~ 10" 500 100'C
Cu SOlUSI LlTY IN p -TYPE Go A, • 6 X 10" cm-J Zn
° 4X1019 cm-J Zn
o
0
01.8
1000/r o AS INDICATED.IN
UNITS OF 10" cm-J
• 1020 Zn
0° • •
° § 8
10"
10"
FiG. 13. Copper solubility in p-type GaAs vs temperature for
various immobile acceptor concentrations. Data points are meas
ured using CU64• Curves for 1017 to 10'0 Zn are calculated. Undoped
curve is from Fig. 12, curve. B.
Comparison of solubility curve B with the families of
curves expected from thermodynamic arguments in
dicates that the melting-point distribution coefficient of
copper in gallium arsenide is approximately 7X 1O~4.
This value is probably correct within a factor of two.
Below 700°C the solubility levels off at a value near
1.5X 1016 cm~3 instead of continuing to decrease in the
manner indicated by curve B as would be expected for
a pure compound. This abnormally high solubility in
dicates the presence of a corresponding number of de
fects which form complexes with the copper, thereby
enhancing its solubility. These imperfections are as
yet unidentified.
Solubility in p-type gallium arsenide. Copper exhibits
a high solubility in extrinsic p-type gallium arsenide,
due to the enhanced solubility of the interstitial species.
This increase can be very large, often amounting to
factors of 106 or 1010, because of the large energy gap and
correspondingly small value of ni in this semiconductor.
Figure 13 gives experimental data illustrating this
effect. On the left is the solubility curve for undoped
gallium arsenide from Fig. 12. The data points give
the total solubility, measured using radioactive copper,
in a number of samples of extrinsic p-type galliwn
arsenide. vVhen the acceptor concentration is large
enough to make interstitial copper the dominant species
the solubility increases linearly with acceptor concentra
tion in accord with Eq. (1). Using this equation we can
calculate the interstitial solubility in intrinsic gallium
arsenide, giving the curve shown in Fig. 14. Because of
the high solubility and rapid diffusion of copper in p-type galliwn arsenide, we have been able to measure
this curve down to lOO°C.
Near 700°C the ratio of substitutional to interstitial
solubility of copper in intrinsic gallium arsenide is about
30. While this ratio does not appear to be very tempera
ture-dependent, it should be noted that the regions
where the two curves have been measured do not over
lap in temperature.
In samples containing increasing amounts of an
immobile acceptor such as zinc, the solubility of sub
stitutional copper decreases while that of interstitial
copper increases. Thus the total solubility goes through
a minimum in slightly extrinsic p-type material. This
behavior is illustrated by the family of curves shown
in Fig. 13 which were calculated from the curves of
Figs. 12 and U under the assumption that in this
doping range the substitutional copper is only singly
charged. This solubility minimum corresponds to the
minimum exhibited by the curves for n-type silicon
shown in Fig. 11. In extrinsic n-type gallium arsenide
where both acceptor levels of copper would be occupied
the solubility would be expected to increase as the
square of the free electron concentration.
In calculating Ci we did not make any corrections for
degeneracy or for a decrease in energy gap, such as were
required in the case of p-type germanium and silicon,
since the range of available acceptor concentrations was
insufficient for the determination of these corrections in
gallium arsenide. I t is reasonable to suppose that these
500 200 IOO'C
GoA'
cj
la' • 6 • 1019 cm-J Z n
o 4 x 1Q19cm-3 Zn o
o < 1019 cm-3 Zn
IOOO/r
FIG. 14. Interstitial Cu solubility in intrinsic GaAs calculated
from data of Fig. 13. B is the extrapolated total Cu solubility from
Fig. 12.
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two correction factors almost cancel each other as they
do in germanium and silicon for not too strongly cle
generate material.
Solubility in n-type gallium arsellide. It would have
been desirable to measure the solubility enhancement of
copper in extrinsic n-type gallium arsenide, since this
would provide a means of determining the charge state
of substitutional copper which would supplement the
experiments to be reported in Sec. 8. However, this
cannot be done at the (opper solubility limit above
500De since in this temperature range Ci'>nj2 and
the copper would compensate the donors. At low tem
peratures diffusion is too slow. The experiment could be
done using a diluted source such as a reservoir of
gallium doped with a few per cent copper to make the
concentration of substitutional copper small compared
with ni, but we have not tried this experiment.
Interstitial diffusion in gallium arsenide. Measure
ments of copper diffusion in several strongly extrinsic
p-type samples were carried out, using the methods pre
viously described for germanium and silicon. Figure 15
shows densitometer determinations of the copper dis
tribution in two of these p-type samples and one n-type
sample. The data obtained from the p-type samples can
be fitted quite well by the curves of Fig. 5, except for a
tilt which may have been due to a slightly non-uniform
zinc concentration.
On the other hand, diffusion into extrinsic n-type
gallium arsenide is quite different in character. It is
much faster near the surface where the copper has been
able to neutralize the donors making the material
nearly intrinsic or slightly p-type, whereas in the still
extrinsic interior the interstitial solubility is greatly
reduced and therefore diffusion is slow.
We also measured copper diffusion at 100De in two
crystals doped with 4X 1019 cm-3 zinc using a precision
lapping fixture and counting the activity on the lapping
paper using a (3 counter. These samples were diffused
10'8 r.---·--
XI, FIG. 15. Diffusion of
CU64 along axes of cubes
of GaAs. Above, extrin
sic n-type doped with
2.7X 1018 Sn, diffused
12.5 h at 870°C, cube
edge 2a=0.44 cm. Be
low, extrinsic p-type
doped with 5.8X101s
cm-a Zn, diffused at
300°C for 1 and 5 h;
curves are best fit from
Fig. 5. If)
l ~ Umm
T'19/'C o Oi:12xIO-7cm2/sec !
~ __ -L __ ~ ____ ~ __ -L ___ J ____ ~I __ ~I
10 20 30 40 50 60 /0
TIME -h
FIG. 16. Through-diffusion of CUM for wafer thickness 1.3 mm.
C is the counting rate measured at the unplated surface corrected
for radioactive decay. C", is the rate at infinite time. The curve is
calculated and Di chosen to give best fit.
in hydrogen for 10 h and gave diffusion coefficients of
3.3 and 3.5 X 10-19 cm2/sec.
These measurements were checked at the lower tem
peratures using other diffusion configurations, in order to
rule ou t possible unsuspected sources of error. "Through
diffusion" experiments were carried out in which eu 64
was plated onto the central region of one side of several
p-type wafers. These samples, which ranged in thickness
from 0.5 mm to 1.3 mm, were diffused in hydrogen and
the radioactivity was measured at the unplated side
using a {3 counter, which measures the copper concentra
tion in the immediate vicinity of the unplated surface.
The data were fitted by theoretical curves, calculated
assuming that copper which diffused to the unplated
surface does not accumulate in a surface layer at this
surface. The data gave reasonably good fits to these
curves, as shown by Fig. 16. The large uncertainty of
the data taken later than 50 h is due to the ShOI t half
life of CU64 and to its limited solubility at low tempera
tures where these data were taken. It was shown using
radioautograms that the copper went through the
wafers rather than around the edges. Three such experi
ments gave Di= 12 and 4.2X 10-8 cm2/sec at 197°e,
and 5.6X 10-8 cm2/sec at 210°C, The results of these and
the preceding diffusion measurements are summarized
as solid squares in Fig. 6.
"Out-diffusion" experiments were also performed by
first saturating a sample with copper and then removing
the copper again by gettering it at the surface of the
sample and observing the decrease in total radioactivity
remaining. This method must be used with cautio~
since, if the sample is saturated at a temperature that is
higher than the out-diffusion temperature, precipitation
may take place at internal nucleation sites, thereby
slowing down the rate of removal of copper and leading
to an anomalously low diffusion coefficient. A sample
0.12 mm thick and doped with 4 X 1019 cm-3 zinc was
out-diffused in this way at 100De and the total radio-
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activity was observed to decrease by a factor of 4 in 3 h,
giving Di= 1.6X 10 9 cm2/sec. This point is shown as all
open square in Fig. 6.
The higher temperature results for gallium arsenide
shown in Fig. 6 were obtained using samples containing
2jnc concentrations ranging from 2X 1018 cm-3 to 6X 1019
cnr;\ with no indication of any dependence of diffusion
coefficient upon acceptor concentration. Heavily doped
samples were required below 250°C in order to obtain a
sufficiently high copper solubility.
The diffusion data for gallium arsenide show consider
able scatter. These results were obtained during the
initial stages of this investigation when measuring
techniques were still being worked out, and emphasis
was placed upon measuring diffusion in several different
ways to avoid possible misinterpretation rather than in
refining a particular method in order to achieve the
greatest accuracy. Judging from the scatter of the data
we estimate that Do is correct within a factor of 3 and
that the activiation energy for diffusion is 0.S3±O.05 eV.
7. DRIFT OF COPPER IN AN ELECTRIC FIELD
IN p..TYPE GALLIUM ARSENIDE
The linear dependence of copper solubility upon
acceptor concentration in extrinsic p-type gallium
arsenide is to be expected if interstitial copper is a
singly charged donor. To verify that this is indeed the
case, we carried out experiments to observe the drift of
copper in an electric field, as has been done for german
iuml2 and silicon.l3•
Samples of gallium arsenide containing 4X 1019 cm-3
zinc were cut and lapped to dimensions of approixmately
12 J11J11 long, 4 mm wide, and t mm thick. The surfaces
were not etched, since experience had shown that copper
could be more reliably electroplated on the lapped sur
faces. Ohmic electrical contacts were made along the full
width at each end using an indium-zinc alloy. The
gallium arsenide was masked with polystyrene cement
except for a !--mm stripe across the center of one
face of the sample in the direction of the 4-mm
dimension. CU64 was electroplated on this stripe and
the poylstryene was dissolved away. At this point it
was verified that CUM was present only on the plated
area. The location of the copper was determined by
radioaut.ograms and by measuring the activity along the
length of the sample using a i3 counter behind a Pb slit
2 cm thick with a t-mm slit opening and 60° total
acceptance angle. The profile of counting rate vs dist
ance measured on the opposite side of the sample
showed a weak and broad maximum due to the 'Y rays
which are emitted during the decay process, the i3
counter being quite insensitive to 'Y rays.
The samples were next heated to 200°C for several
hours in hydrogen to saturate the gallium arsenide
under the electroplated area with copper. After this
treatment the counting profile on the side opposite the
plated stripe showed a strong and symmetrical peak, T
--- - -------.• ----"SACKGROllND
oL-.---L_--"----_-!:-_--"------!:-_-'---!:-_-'-- __ ' DiSTANCE-'mm
FIG. 17. Drift of Cu in GaAs. Radioactivity corrected for decay
vs distance along sample after diffusion at 200° (curve A), followed
by drift for 17.5 h at 255°C and 1.94 V/cm (curve B).
showing that copper had diffused completely through
the !-mm thickness of the wafer. Such a profile is shown
by curve A of Fig. 17. The samples were then drifted in
hydrogen by applying an electric field which in a typical
case was 2 V/cm for 20 h, with the sample temperature
held near 200°C. Following this treatment the copper
profile was again measured, and it was invariably found
to be shifted towards the negative end of the sp~cimen.
An extreme case in which the copper was drifted the
full distance to the end of the bar is shown by curve B
of Fig. 17, which was drifted at a relatively high temper
ature. The flat-topped shape of this curve is due to the
presence of the electroplated copper which remained as
a continuing source of copper to replenish that which
was drifted away by the field. Radioautograms showing
the distribution of CU64 in this sample were taken after
diffusion and after drift and are shown in Fig. 18 (a)and
(b), respectively.
Figure 18(c) shows the results of a similar drift
experiment using a sample which was cut so as to leave
small extensions under opposite ends of the electroplated
stripe of CUM. These extensions were used to produce
small field-free regions which would prevent the copper
from being drifted away and thus serve as markers. The
copper remaining in these extensions shows up strongly
in the radioautogram whereas in the center of the sam
ple most of it has been swept away.
During the drift experiments the samples were
clamped between metal blocks using thin mica spacers,
in order to remove the Joule heat (typically 12 W) as
effectively as possible. The sample temperature was
monitored by observing its resistance, the temperature
dependence of resistance having been determined in a
separate experiment. Earlier attempts to observe drift
in gallium arsenide were carried out in air but were
unsuccessful due to oxidation of the copper at the
sample surface.
The results of the drift experiments are summarized
in Table II. Diffusion coefficients were calculated from
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TABLE U. Drift of copper in extrinsic p-type gallium arsenide.
-
Drift Electric Diffusion
Sample distance Time Temp field coefficien t
(mm) (h) (Oe) (V /cm) (cm'/sec) --------. -------
1 0.64 16 195 1.45 3.1 X 10-8
2 0.56 16 200 1.08 3.7X10-s
3 >5 17.5 255 1.94 > 1.9XIO-7
-+ 5 17.5 255 2.00 1.8X 10-7
5 >5 20.5 266 1.87 >1.5XIO-7
6 5 20.5 220 1.77 1.6XIO-7
----------
these data using D= kTp./e, and assuming that the
diffusing species carried a single positive charge. The
results, except for samples 3 and 5, are shown as
triangles in Fig. 6. The agreement with the diffusion
data is quite satisfactory. The results from samples 3
and 5 were omitted, since it appeared that the copper
had actually drifted beyond the end of the samples,
and as a re~ult only lo~er limits could be determined
for the diffusion coefficients. These limits are consistent
with the other determinations.
8. DOUBLE ACCEPTOR BEHAVIOR OF COPPER
IN GALLIUM ARSENIDE
Substitutional copper has been shown to be a triple
acceptor in germanium37 and the experiments described
above in Sec. 6.2 indicate that it behaves similarly in
silicon. It is generally believed to be a single acceptor
in the II-VI compounds. In each case, this behavior is
consistent with the valence of copper, and it is reason
able to expect that substitutional copper should behave
as a double acceptor in the I II-V semiconductors. It
has been reported to behave this way in indium anti
monide.z3 Initial experiments by Fuller and \Vhelan,6
in which copper was diffused into gallium arsenide near
600°(" appeared to confirm these expectations. How
ever, from later experiments in which copper was
dit1used into gallium arsenide in the 1000° to 1200°C
temperature range, these authors concluded that copper
only introduced a single acceptor level.z2 The experi
ments described below in which copper was introduced
between 600° and 8000e confirm the double acceptor
behavior of copper in gallium arsenide.
Samples were cut from neighboring regions of a single
crystal of boat-grown gallium arsenide. Two were
rectangular, 3.SXS mm, with thicknesses t and 1 mm,
respectively. Their Hall coefficients were measured at
3(JOoK, giving carrier concentrations of S.O and 8.2X 1016
cm-\ assuming the formula R= 37r/Sen. The third was
of irregular shape with its minimum dimension approxi
mately 3 111m, and it was presumed that it had a similar
electron concentration, 8X lO16 cm-o. They were elec
troplated with CU64 and ditTused in Hz for 46 h at
600°C. Following this diffusion treatment they were
cooled to room temperature in less than 30 sec, and all
were found to have gone high resistivity throughout.
The gamma ray activity gave copper concentrations of (O) (b) (c)
r-i cm--;o.j
FIG. 18. Radioautograms showing eu" distribution in GaAs
before and after drift in an electric field. The outline of the sample
and holder has been revealed by exposure to light. (a) Sample 3,
diffusion only. (b) Sample 3, drifted 17.5 h at 255°C and 1.94
V /cm. (c) Sample 6, drifted 20.5 h at 2200e and 1.77 V /em.
5.0, 4.5, and 4.3X lOI6 cm-3, respectively. The loss in
electrons per copper atom is thus 1.60, 1.82, and 1.9,
respectively. Since impurity scattering contributed
substantially in these samples, it seems reasonable that
the factor in the Hall coefficient formula should have
been somewhat greater than the 37r/S value which
was used, thus bringing the above results closer to 2
electrons per copper atom.
In a second experiment a tin-doped sample was used,
having an electron concentration of 3.19X 1018 cm-3 as
determined by measurement of the Hall coefficient,
again assuming R= 37r/8en. This sample was gettered
in liquid gallium for 67 h at 600°C to remove any nor
mal copper or other rapidly diffusing impurities, and
remeasured. It then had an electron concentration of
2.S4X 1018 cm-3• It was next saturated with CU64 at
SOO°C for 42 h after which it was found to be converted
to high impedance throughout. The copper concentra
tion at this point was l.S6X lO18 cm-3, corresponding to
1.S3 electrons lost per copper atom. The sample was then
gettered in liquid KCN at 670°C for 4.5 h which re
duced the copper content to 1.39X lO18 cm-3, and again
for 16 h more, after which it contained 1.33X1018 cm-3
Cu. At this point the sample had turned slightly n-type,
and a Hall measurement indicated an electron concen
tration of 1.07 X 1017 cm-o• Thus 1.33 X lO18 cm-3 copper
atoms gave rise to a net loss in electron concentration
of 2.73X 1018 cm-:\ or 2.05 per copper atom.
In a third experiment six samples were used, three
containing 5X 1017 cm-3 tin, and three doped with
2.6X 1017 cm-3 tellurium. These samples were out
diffused for 63 h at 670°C in KCN to determine whether
changes in electron concentration would result from
such a treatment, and to eliminate any normal copper
that might be present. Changes which were observed
were of the order of ± lO%, which is about the experi
mental error involved in the Hall measurement. One
of the tin-doped and two of the tellurium-doped samples
were saturated with copper at 650°C, and the others
at 7000e after which they were still n-type but reduced
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in electron concentration by factors of 10 to 50 times.
At this point the ratio of electrons lost to copper atoms
introduced ranged from 0.76 to 1.18. One of the 650°C
Te-doped samples and one of the 700°C Sn-doped sam
ples were out-diffused in KC~ for 20 h at 680°C. After
this the electron concentration was measured and found
to be still low, while the copper content was reduced by
about a factor of two, corresponding to ratios of electron
loss to copper present of 1.75 and 2.44, respectively.
The other four samples were not similarly out-diffused
because their activities had already fallen to too Iowa
value.
Different shipments of copper were used in each of the
three groups of experiments, and the samples were ob
tained from several different crystals and covered a
wide range of electron concentration. While we do not
understand the results obtained during the third experi
ment prior to the final KC~ gettering, we believe that
our results show quite clearly that substitutional copper
introduces two acceptor levels in gallium arsenide.
9. REMOVAL OF COPPER FROM GALLIUM
ARSENIDE AND OTHER MATERIALS
During the course of our investigations it was
necessary to evaluate the effectiveness of various pro
cedures for the removal of copper from semiconductors
and associated equipment. Our findings are reported
below.
9.1. Removal of Copper from Surfaces
In agreement with the work of Larrabee44 we have
found that as much as 1015 atoms/cm2 of copper are
adsorbed on gallium arsenide surfaces from alcohol or
water containing a few ,ug/ml of Cu64• Approximately
1/100 as much may be deposited on glass or polyethy
lene under similar circumstances. This copper can be
removed by soaking the sample for a few minutes in a
10% solution of KC~ to the limit of detection, less
than 1012 atoms/cm2• Other agents which we have tried
such as NH40H solutions, various chelates, and acids,
have proved to be far less effective.
We repeated the above experiments with 6 ,ug/ml of
CU64 (6X 1016 atoms/cm3) added to the KCN washing
solution. This did not affect the ability of this solution
to reduce copper contamination on a gallium arsenide
surface below the 1012 atoms/cm2 detection limit. Thus,
the fact that the best reagent grade KC~ currently
available often contains as much as 100 parts per
million of copper would not appear to reduce the
effectiveness of such KCN solutions for cleaning
purposes.
9.2. Removal of Copper from the Interior
of Gallium Arsenide
High-temperature heat treatments of semiconductors
in contact with a liquid phase (solvent extraction) have
44 G. B. Larrabee, J. Electrochem. Soc. 108, 1130 (1961). been used to reduce the concentration of rapidly diffus
ing impurities.45-47 Gallium and liquid KC~ in particu
lar have been used to remove copper from gallium
arsenide, and we conducted experiments to measure the
effectiveness of these agents.
Two kinds of experiments were performed. In the
out-diffusion experiment, a gallium arsenide sample
was saturated with CU64 and then heated with a known
volume of the solvent. The CU64 content of the sample
was measured at intervals until a steady state appeared
to have been reached. At this point the ratio of copper
in the sample to that in the solvent per unit volume
gives the distribution coefficient for out-diffusion k"ut.
r n the in -diffusion experiment the Cu 64 is added to the
solvent and a similar steady-state distribution coeffici
ent kin is measured. If true equilibrium has been reached,
one should find kout = kin.
Our results are collected in Table III. Thev show that
TABLE III. Removal of copper from GaAs by solvent extraction.
Solvent TempoC GaAs doping cm-3 kout kin ._-------- ------ ------ -
Ga 550 4X1019 Zn ~0.c)026
Ga 600 2X 1019 Zn 0.002
Ga 940 2X10i9 Zn 0.004
Ga 940 5X lOI7 n-typc 0.001
KC~ 650 4XlOI9 Zn 0.06
KCN 650 4XlOI9 Zn 0.034
KC~ 650 5XlOI9 Zn 0.15 KeN 650 2.7XlO1S Zn 0.026 KeN 650 Semi·ins. 0.024 KeN 650 3X 101r. n-type 0.017
the copper concentration can be reduced by a factor of
several hundred using a volume of galli~m equal to
that of the sample. KCN, on the other hand, is less
effective by a factor of 20 or so.
The distribution coefficient. at any given temperature
should vary with donor or acceptor doping in the same
way that the copper solubility shown in Fig. 13 does.
vVhile there may be some tendency for the strongly
p-type samples to yield higher k values, this dependence
is much less than expected, at least among those sam
ples where KCK was used as the solvent. \Vhen these
results were obtained the copper concentration in the
samples ranged from 3X 1014 to 2X 1016 cm--3, which is at
or below the level of anomalous solubilitv shown in
Fig. 12. This suggests that the k values whi(~h we report
may be determined to some extent by defects in the
GaAs rather than by the intrinsic properties of the
semiconductor. An alternative explanation is that the
45 A. Goetzberger and W. Shockley J. Appl Phvs. 31 1821 (1960). • 0,
46 M. Aven and H. H. \Voodbury, J. AppJ. 1>h)'s. Letters 1,
53 (1962).
47 J. Blanc, R. H. Bube, and L. R. Weisberg, Phys. Rev.
Letters 9, 252 (1962).
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KCN or impurities in it (lithium, for example4S) may
be reacting with the gallium arsenide, introducing
imperfections which affect the copper solubility.
KCN must be used several times in succession to
achieve the same degree of copper removal that is
afforded by a similar quantity of gallium. KCN has
the further disadvantage that the presently available
reagent grade sources contain approximately 100 ppm
or lOIS atoms/em3 of copper, so that after gettering in
molten KCN a sample of gallium arsenide can be ex
pected to contain approximately 1017 cm-3 copper atoms
whether it had any to begin with or not! This should
be taken into account in interpreting experiments in
volving extraction with molten KC~Y Extensive
purification of the KC~ would be required in order to be
effective, and this appears to be a difficult procedure.
On the other hand, high-purity gallium is available and
probably other less expensive liquid metals could be
used with similar effectiveness.
9.3. Extraction of Copper from Gallium
Using KCN Solutions
Radioactive copper was added to 19 of high-purity
gallium which was then treated by washing in several
successive 20% KCN solutions. The removal of copper
was measured by observing the activity of the gallium
at various stages. The rate of removal was accelerated
by warming the solution to 70° or 80°C and by rapid
stirring. The copper content of a sample of gallium was
reduced from 1017 to 2X1012 cm-3 in 2 h at 75°C, this
being the limit of detection. We did not try the cor
responding in-diffusion experiment to see how much
copper will leave the KCN solution and enter the
gallium.
9.4. Extraction of Copper from Tin Using
Molten KCN
Tin is generally used in forming the alloyed junction
of gallium arsenide tunnel diodes. We have found that
when tin containing CU64 is saturated with gallium
arsenide at 600°C and cooled to 400°C, the volume
concentration of CU64 in the recrystallized gallium
arsenide is at least 100 times greater than tha t in the tin.
Thus traces of copper which may be present during
construction of tunnel junctions are greatly concen
trated and deposited in the degenerate n-type regrowth
material adjoining the junction. Accordingly we inves
tigated the possibility of removing copper from tin by
heating it in molten KCN.
5X 1016 atoms of CU64 were dissolved in 5 g of tin and
then heated in 10 g of molten KCN for 4.5 h at 650°C.
The activity in the tin was found to have decreased by
17% which may not be significant compared with the
experimental error. We conclude that copper k~not
removed at an appreciable rate by this treatment.
~s C. S. Fuller and K. B. Wolfstirn, AppJ. Phys. Letters 2, 45
(1963). 9.5. Extraction of Copper from Silicon
Planar Junctions
As an illustration of the manner in which copper
accumulates in heavily doped (extrinsic) regions of a
crystal, a simple experiment was performed using oxide
masked planar silicon wafers. Each wafer contained a
pattern of seven round windows through which a
strongly p-type boron diffused layer was produced.
These wafers, which were approximately 0.2 mm in
thickness, were plated on the backs with CU64 and
diffused to saturation at 700°C. The excess copper was
then removed and exposures were made by placing the
oxide-masked side of the wafer against a negative. These
radioautograms have been published elsewhere.49 They
illustrate clearly the accumulation of copper in the
extrinsic p-type regions of the wafer.
The significance of such solubility enhancement
should not be overlooked. It causes trace amounts of
Cu to be concentrated in the heavily doped regions of a
junction structure, often where they can produce the
most serioLlS consequences. Upon cooling, these regions
become supersaturated and under some circumstances
they can produce precipitates in the regions adjoining
the junction. It is probable that a number of other im
purities behave in a similar manner.
Copper was extracted by applying a gettering agent
to the backs of the wafers and removing it after the
gettering was completed in order to measure the residual
activity. This was repeated a second time, at which
point the low activity ruled out further experiments.
Oxide getters were painted on the backs of the wafers
and the extraction was carried out in air. KC~ was
applied similarly, the extraction being done in hydrogen.
The results are shown in Table IV, where the last
TABLE IV. Copper extraction from silicon planar junctions.
1st 2nd
Getter Temp Time decrease decrease
B20a 900°C 1 h 37
B2Oa-PzO. 900 1 22
B2Oa-P 2Os 870 0.6 10 3.6
B2Oa-P,Os 1000 1 12 37
KCN 700 0.85 52 7.3
None 870 0.6 1.1
two columns give the factor by which the activity
decreased after each extraction. The measurement of
this reduction factor was subject to considerable ex
perimental error because of contamination with stray
CU64 during handling and incomplete removal of the
getter, but the results show that the reduction can be
quite substantial and that a second extraction can
49 S. W. lng, Jr., R. E. Morrison, L. L. Alt, and R. W. Aldrich,
J. Electrochem. Soc. 110, 533 (1963). The data in Fig. 8 are in
correctly described. They actually represent the activity measured
after two successive getterings rather than during the course of a
single gettering treatment.
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again reduce the copper concentration significantly.
KeN appears to be at least as effective in removing cop
per as the oxide getters, even at the low temperature
where it was used.
Radioautograms taken after the second extraction
showed very little copper remaining in the boron
diffused regions, most of it being concentrated at
numerous isolated points. Since the copper solubility
is highest in the p-type regions of the silicon, we con
clude that most of this copper was left in the Si02 layer
or at the interface between it and the silicon.
10. SUMMARY
Solubility enhancement of interstitial copper in p
type material and of substitutional copper in n-type
material has been observed in germanium, silicon, and
gallium arsenide. The effects are particularly pro
nounced in silicon and gallium arsenide where n, is
small.
The enhanced solubility which is observed in ex
trinsic p-type samples is in good quantitative agreement
with the theory of electronically enhanced solubility,
assuming that interstitial copper is a singly charged
donor. Measurements of the drift of copper in gallium
arsenide in an electric field also show that interstitial
copper carries a single positive charge.
The diffusion coefficient of interstitial copper has been
measured as a function of temperature in extrinsic p
type crystals. The diffusion coefficient is independent
of acceptor concentration. The temperature dependence
shows an activation energy of 0.33, 0.43, and 0.53 e V
(± 10% in each case) in germanium, silicon, and gallium
arsenide, respectively. These values do not agree very
well with Weiser's calculations.50
In p-type germanium and silicon which is degenerate
at the saturation temperature, the solubility enhance
ment does not agree with that observed in nondegen
erate samples unless it is assumed that the energy gap is
smaller in the heavily doped material. At an acceptor
concentration of 4X 1020 cm-3 the decrease is 0.28 eV
in germanium and 0.14 eV in silicon.
The ratio of substitutional to interstitial solubility in
intrinsic germanium is 6 near 700°C, while in gallium
arsenide it is about 30. In silicon this ratio is of the order
of 10-4, and consequently most of the solubility observed
50 K. Weiser, Phys. Rev. 126, 1427 (1962). in pure silicon is due to interstitial copper. The smallness
of this ratio in silicon accounts for the absence of climb
of dislocations in copper diffused crystals, whereas in
the case of gold, where Ci8»C;i, climb does occur. 51
Electrical activity due to rapidly precipitating inter
stitial copper has been observed in high purity silicon
saturated with copper in the range 400° t.o 575°C.
In extrinsic n-type germanium and silicon the copper
solubility increases approximately as the cube of the
free electron concentration as would be expected for a
triply charged acceptor impurity, and the effective
diffusion rate decreases correspondingly. Pairing be
tween substitutional copper and donor impurities is
pronounced in germanium, but does not greatly affect
the solubility in silicon. These experiments show that
substitutional copper is a triple acceptor in silicon, as it
is in germanium.
Measurements of the compensation of n-type gallium
arsenide with radioactive copper show that substitu
tional copper introduces two acceptor levels between
the middle of the energy gap and the valence band edge.
Copper is effectively removed from surfaces and from
liquid gallium using KCN solutions. Copper can be ex
tracted from the interior of gallium arsenide very effec
tively using gallium as a getter at 600°C or higher
temperatures. Molten KCN is less effective, and may
actually introduce copper rather than remove it, be
cause of the high copper content of current.ly available
KCN. Molten KCN does not remove copper from tin.
Thin layers of B203 and P205 as well as moltenKCN
are effective in extracting copper from planar silicon
junctions. Copper which accumulates in the p-type
regions can be reduced in concentration bv about a
factor of 10 or 20 per extraction. .
ACKNOWLEDG MENTS
T. J. Soltys measured the precipitation of interstitial
copper in silicon which is reported in Fig. 8. He also
measured the Hall effect and resistivity of the many
samples which we used. D. J. Locke made the measure
ments of through-diffusion of copper in gallium arsen
ide as shown in Fig. 16. The oxygen content of the
silicon samples was determined by E. M. Pell. Miss S.
Schwarz was helpful in many ways throughout these
investigations.
51 W. C. Dash, J. App!. Phys. 31, 2275 (1960).
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1.1725152.pdf | Linewidth of the Exciton Absorption in Naphthalene Crystals
H. J. Maria
Citation: The Journal of Chemical Physics 40, 551 (1964); doi: 10.1063/1.1725152
View online: http://dx.doi.org/10.1063/1.1725152
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128.83.63.20 On: Thu, 27 Nov 2014 10:19:08THE JOURNAL OF CHEMICAL PHYSICS VOLUME 40, NUMBER 2 15 JANUARY 1964
Linewidth of the Exciton Absorption in Naphthalene Crystals
H. J. MARIA
Physics Department, American University of Beirut, Beirut, Lebanon
(Received 11 February 1963)
The crystal spectrum of naphthalene near 3200 A is studied in the temperature range 4° to 80°K. The
intensity and half-linewidth of the a band of the first purely electronic transition are found to be strongly
dependent upon temperature and proportional to the density of phonons with energy of 14 cm-I. The b
band shows very little temperature dependence. It is concluded from the line shape and linewidth of the
a band that mobile excitons are possible in naphthalene.
INTRODUCTION
ALTHOUGH the crystal spectrum of naphthalene .ft near 3200 A is one of the best studied of the
aromatic band systems,1-4 no detailed investigation has
been reported of the effect of temperature on the exciton
bands. The importance of such an investigation to an
understanding of the origin of the band intensities and
broading has been recently pointed out in connection
with the crystal spectrum of benzene.5 This paper
reports the results of a study of the temperature de
pendence of the linewidths and intensities of the bands
of the first purely electronic transition in naphthalene.
EXPERIMENTAL
Eastman Organic Chemicals naphthalene (recrystal
lized from alcohol) was used in these experiments with
out further purification. The concentration of im
purities, which are assumed to be predominantly
anthracene, was estimated as 10-6 moles/mole from
the impurity fluorescence yield.6
Crystals suitable for absorption experiments were
prepared by sublimation, by growing from n-hexane
(spectroscopic grade) on water,7 or by pressing a small
quantity of molten naphthalene between two quartz
plates and allowing it to cool slowly. The thinnest speci
mens ( < 1 J.I.) were obtained by the last method, but
this method generally gave polycrystalline films and
therefore was used only for the thinnest crystals in
which the a band did not appear.
The apparatus and experimental procedures have
been described elsewhere.5 The resistance thermometer
has since been checked against a copper-constantan
thermocouple above 200K. The exposure times used
were 2 or 3 min and the slitwidth was 10 J.I. in all the
experiments. The crystal spectrum was photographed
at temperatures from 40 to 800K with intervals of a few
degrees.
I D. P. Craig, L. E. Lyons, and J. R. Walsh, Mol. Phys. 4, 97
(1961).
2 D. P. Craig and S. H. Walmsley, Mol. Phys. 4,113 (1961).
3 D. S. McClure and O. Schnepp, J. Chem. Phys. 23, 1575
(1955) .
4 A. F. Prichotjko, Zh. Eksperim. i Teor. Fiz. 19, 383 (1949).
6 H. Maria and A. Zahlan, J. Chem. Phys. 38, 941 (1963).
6 A. A. Kazzaz and A. B. Zahlan, Phys. Rev. 124, 90-95 (1961).
7 H. C. Wolf, Solid State Phys., 9, 1 (1959). RESULTS
The crystal spectrum of naphthalene in nonpolarized
light has only the two bands of the first purely electronic
transition completely resolved. These two bands were
observed in several grown or sublimed crystals and are
shown as a function of temperature in Fig. 1. The posi
tions of the a-polarized and b-polarized bands are in
good agreement with the results of earlier workers and
therefore are not given here. It should be noted, how
ever, that the positions of the bands do not change ob
servably with temperature up to 400K and that at
higher temperatures they appear to draw together until
they merge and appear as one band at about 100oK.
The observed a-b separation at SDK is 153 cm-I, in very
good agreement with the result of Craig et aU Above
40oK, the a-b separation begins to decrease with tem
perature; but now both bands are quite broad and it is
difficult to determine the separation with sufficient
accuracy. Our main interest will be in the intensities
and linewidths of the two bands. The intensities and
linewidths were obtained from the densitometer re
cording of optical density (in arbitrary units) vs fre
quency.
1. Band Intensities
The integrated band intensity was determined for a
number of crystals from 4.20 to 60oK. The temperature
dependence of the integrated intensity (in arbitrary
units) of the a band is shown in Fig. 2. In all the crystals
studied, the b-band intensity did not change with
temperature although it varied from crystal to crystal
depending upon the thickness. The b: a intensity ratio
is 40:1 at SDK and decreases to 9:1 at SO oK.
2. Half-linewidths
Figure 3 shows the half-linewidth W! of the a band
as a function of temperature while in Fig. 4 lnW~ is
plotted against l/T. Above 200K W~ increases linearly
with temperature. Below 20oK, it shows some tempera-
ture dependence but not as strongly as above.
For the b band the half-linewidth is found to be
dependent upon the thickness of the crystal in agree
ment with the observation of McClure.3 In a crystal
that is less than 1 J.I. thick (the estimate of the thickness
551
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128.83.63.20 On: Thu, 27 Nov 2014 10:19:08552 H. j. MARtA
2Scm-1
>----<
>- T. 6.4' K I-
iii z
W D
...J T. 23' K <{
~
I-a.
0
w T. 39' K I-<{
...J a.
T. 53' K
T. 73' K
FREOUENCY-
FIG. 1. The first 0-0 transition in naphthalene.
of the crystal is here based upon McClure's observa
tion that the a band appears in a crystal 2 p. thick but is
too weak to appear in a crystal whose thickness is 0.5 p.),
Wi of the b band was found to be 78 cm-I and constant
to within ±2 em-lover the temperature range 4° to
SOCK. For a thicker crystal (2-4 p.) Wi increased to
180 cm-I but again was constant to within ±8 cm-I
over the same temperature range.
DISCUSSION
The observed features of the temperature dependence
of the first purely electronic transition in naphthalene,
with the exception of the intensity and bandwidth of
the b band, are qualitatively similar to those of the
corresponding transition in benzene.5 The 2600-1
system in benzene is electronically forbidden whereas
the 3200-1 system in naphthalene is allowed although
very weak. Both systems are in the class of very weak
transitions (f~0.002) and are more or less strongly
effected by lattice perturbations. The b band in naph
thalene derives its intensity from the mixing of the Bau
with the higher B2u. molecular state by the crystal field.8
The transition to the B2u is strong and therefore the
intensity of the b band is not expected to show signifi
cant temperature dependence.
Craig and Walmsley2 have calculated the crystal
spectrum of naphthalene in the framework of the simple
rigid lattice exciton theory and concluded that the
splitting of the bands of the purely electronic transition
of the 3200-1 system is explained if the intermolecular
coupling is taken to be through transition octupole
moments. The Davydov splitting of the purely elec
tronic transition in the 2600-1 system of benzene was
calculated theoretically by Fox and Schnepp9 who also
find that an octupole-octopole interaction (which is
the first nonvanishing term in a multipole expansion of
the interaction energy) gives splittings of the right
8 D. S. McClure, Solid State Phys. 8,1 (1959).
v D. Fox and O. Schnepp, J. Chern. Phys. 23, 767 (1955). order of magnitude. Their predicted order of the bands
is a<c<b. The experimentally observed order was con
sidered to be c<a<b 8 until recently Broudelo has re
assigned the bands to the order a<c<b in agreement
with the calculation of Fox and Schnepp. This suggests
that the assignment of these bands is not to be con
sidered as resolved although the observations on the
temperature dependence of the positions of the a and c
bands5 may be taken to favor the assignment c<a<b .
Experimentally, the a-c splitting in benzene is found to
be dependent upon temperature above 200K, whereas
the present study shows that the a-b splitting in
naphthalene crystals is independent of temperature up
to 400K.
McClure8 states that the b: a intensity ratio in the
0--0 band is of the order of 100: 1, whereas the ratio
expected from the molecular orientation for a long axis
transition AIq-+B3u is 1: 4.2. Our observed ratio of
40: 1 at SOK is in satisfactory agreement. The ratio of
12: 1 calculated by Craig and Walmsley2 gives too small
an intensity to the b band. The lack of temperature de
pendence for the intensity of the b band suggests that
this band receives almost all of its intensity from
mixing of the molecular states by the crystal field.
The intensity of the a band (Fig. 2) may be fitted to
an equation of the form
j=A[{ (exp (Iiw/k T) -ll-I+!J=A (N+t), (1)
with liw/k=20oK, where N is the thermal equilibrium
number of phonons of frequency wand the additional
term t represents the number of phonons at OaK. It is
this term which leads to "zero-point" energy.u The
intensity of the a band is therefore proportional to the
density of phonons with frequency 14 em-I.
3.0
2.0
• •
c • •
...J
• 1.0
•
oL-______ ...JL ______ ~ ________ L-______ ~
o 10 15 20
100/T
FIG. 2. The integrated intensity I (arbitrary units) of the a
band. Points are experimental and the full curve is obtained
from Eq. (1) with liw/k=20oK.
10 V. L. Broude, Usp. Fiz. Nauk 74, 577-608 (1961) [English
trans!' Soviet Phys.-Uspekhi 4, 584 (1962)].
11 R. E. Peierls, Quantum Theory of Solids (Oxford University
Press, London, 1956), pp. 25 and 27.
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128.83.63.20 On: Thu, 27 Nov 2014 10:19:08EXCITON ABSORPTION IN NAPHTHALENE CRYSTALS 553
The first electronic transition in naphthalene is
formally allowed by the symmetry of the molecule and
crystal, but is very weak and is therefore subject to
perturbation by the surroundings. It is not known what
perturbation is operative, and it may be that second
order perturbations as well as first-order ones are im
portant. However, the present study suggests that the
perturbation which gives the a band its intensity is a
14-cm-1 phonon.
Such low-frequency phonons have been predicted by
Hexter and DOWSl2 who showed that the deviations of
observed dichroic ratios in the infrared spectra of
molecular crystals from the ratios predicted by the
"oriented gas model" may be accounted for by the
librational motion of the molecules. Their results were
later obtained more rigorously by Hexter.13 They calcu
lated a libration of 16 cm-l for crystalline benzene and
used the bands at 687 and 705 cm-l in the infrared
spectrum to support their calculation. A librational
motion of about this frequency was also inferred from
the temperature dependence of the intensity of the first
electronic transition.5 Thus, although Person and
Olsenl4 later showed that the 687-and 705-cm-1 bands
are components of a correlation-field doublet, it is still
valid to seek evidence for librations in the infrared
spectrum of crystalline naphthalene.
The infrared spectrum of crystalline naphthalenel5
contains bands at 1129 and 1142 cm-l with a separation
of 13 cm-I• The indistinct contour of the gas-phase
band at 1125 cm-I and the absence of a similar pair of
<,0
30
• 10
•
o 20 • •
<,0
TO K •
• • •
•
60
FIG. 3. The temperature dependence of the half-Iinewidth, Wi
of the a band.
12 R. M. Hexter and D. A. Dows, J. Chern. Phys. 25,504 (1956).
13 R. M. Hexter, J. Mol. Spectry. 3, 67 (1959).
14 W. B. Person and D. A. Olsen, J. Chern. Phys. 32, 1268
(1960) .
Ii G. C. Pimentel, A. L. McClellan, W. B. Person, and O.
Schnepp, J. Chern. Phys. 23, 234 (1955). 4.0
3.0 • • " •
~
c
--'
2.0 • • •
1.0 L-__ -L ____ i' ___ --'---__ -----' ___ ---"
o 10 15 20 2S
100fT
FIG. 4. Ln Wi of the a band plotted against liT. The points
are experimental and the full curve is obtained from an equation
of the same form as Eq. (1) with nwlk=20oK.
bands with about the same splitting in the spectrum of
solid deuterated naphthalene (contrast the case of
benzenel4) lead Pimentel et al.15 to conclude that these
two bands are independent features. It is here suggested
that the 1142-cm-1 band is a combination of the 1129-
cm-l band and a 13-cm-1 librational mode. Unfortu
nately, no calculation similar to the one for benzenel2
exists for naphthalene.
The naphthalene crystal, with two molecules per
unit cell, has six rotational lattice modes all of which
are Raman active. The Raman spectrum has been ob
served by several investigators. Kastler and Roussetl6
report six lines at 46, 54, 74, 76, 109, and 127 cm-I at
room temperature. Their observations enabled them to
decide that the lines at 46, 74, and 109 cm-l correspond
to the two molecules in the unit cell vibrating in phase
while for the lines at 54, 76, and 127 cm-l the two mole
cules vibrate out of phase. Ichishimal7 observed five
lines at 47, 52, 74, 106, and 126 cm-l at 20°C. Cruick
shankl8 suggests that Ichishima's line at 74 cm-l may
be an unresolved doublet corresponding to the 74-and
76-cm-1lines of Kastler and Rousset and that therefore
the agreement between these two investigations is good .
However, Mitra,I9 relying on the results of Bhagavan
tam and Venkatarayudu,20 reports four broad bands at
45, 73, 109, and 124 cm-l• He assigns the bands at 45
and 73 cm-I as fundamentals and suggests that the
remaining bands result from a Fermi resonance be
tween the combination tone (45+73) cm-I and a third
fundamental which are accidentally degenerate. This
can not be reconciled with the findings of Kastler and
Rousset that the 109 and 127 cm-I lines correspond to
the two molecules in the unit cell vibrating in phase and
out of phase, respectively. Mitra also argues that the
three remaining fundamentals have energies which do
16 A. Kastler and A. Rousset, J. Phys. Radium 2, 49 (1941).
171. Ichishima, J. Chern. Soc. Japan (Pure Chern. Sec.) 71,
607 (1950).
18 D. W. J. Cruickshank, Rev. Mod. Phys. 30, 163 (1958).
19 S. S. Mitra, Solid State Phys. 13, 1 (1962).
20 S. Bhagavantam and T. Venkatarayudu, Theory of Groups
and its Application to Physical Problems (Andhra University,
Waltair, India, 1951), p. 146.
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128.83.63.20 On: Thu, 27 Nov 2014 10:19:08554 H. J. MARIA
not differ much from the first three and that this is
consistent with the fact that the observed bands are
broad. It is clear then that the assignment of the lattice
rotational fundamentals is not definite and a possible
librational mode of 14 cm-I is not ruled out.
It must be pointed out that for some of the crystals
studied the temperature dependence of the intensity
and linewidth of the a band correspond to a dominant
phonon of 28 em-I. In these crystals Wt does not show
any variation with temperature below 20oK, unlike the
case shown in Fig. 3. It is not clear how to account for a
phonon of this energy, or for the observation that the
crystals studied seem to fall into two classes leading to
two different phonon energies. Undoubtedly the imper
fections in the crystal play an important role.
WolF quotes a value of 10-20 em-I for the half-line
width at 20°K. Our observed value of 11-14 em-I at
the same temperature is in good agreement. The half
linewidth of the a band may also be fitted to an equa
tion of the same form as Eq. (1) with hw/k =20°K. That
is, W~ is also proportional to the density of phonons with
frequency of 14 em-I. This result is reasonable since the
halfwidth is expected to be affected by the ground state
perturbation and the value of 14 em-I may be in error
by as much as 15%. According to Toyozawa,21 Wt de
pends on a complex sum of the phonon density for the
entire phonon spectrum of the crystal.
Toyozawa21 has developed the theory of the exciton
phonon interaction in the weak coupling limit. In the
case where k = 0 is neither the top nor the bottom of the
exciton band, or when there is a finite density of levels
in the other exciton bands at the energy value of the
k=O exciton being considered (k is the wavenumber
of the exciton), he obtains for the half-width of the
exciton absorption the relationship:
Wl= (2/rr)gkT (2m*u2<kT<kTo) (2a)
= (4/1r)gm*u2 (kT:52m*u2), (2b)
where g is the exciton-phonon coupling constant, m* is
the effective mass of the exciton, u is the velocity of
sound and To is defined by To=2.6m*u2/kg. The con
stant g may be obtained from a plot of Wt vs T. Then,
using Eq. (2b) and putting 105 em/sec for the velicoty
of sound, one gets a value of about 200 for m* in units
21 Y. Toyozawa, Progr. Theoret. Phys. (Kyoto) 20, 53 (1958);
27,89 (1962). of the electron mass. For benzene, using the data of
Ref. 5, m* is about 70. Both these values are quite
large and it would be interesting to compare them with
the predictions of the mobile exciton model of Frenkel.22
Furthermore, if k=O is at the bottom or the top of the
exciton energy band and there are no states with the
same energy in other exciton bands, then Toyozawa21
finds that W! is proportional to P:
(3)
Our Wt data do not distinguish between the two
cases represented in Eqs. (2) and (3), and it is believed
that it would be very difficult to obtain data that
would. However, if one considers the line shapes the
situation is more hopeful. Toyozawa finds a line shape
that is Lorentzian for the first case if the temperature is
not too high, whereas in the second case the line shape
is strongly asymmetrical. The observed line shape is
very nearly Lorentzian. It is also interesting to note
that the line shape obtained by Toyozawa for the
second case (k=O at the bottom of the exciton band)
is very similar to the line shape obtained by Davydov
and Lubchenk023 for the case of localized excitations
which may be treated in the strong coupling limit.21
This last observation serves to emphasize that the
understanding of linewidths and line shapes is by no
means satisfactory.
Toyozawa21 also gives expressions for the mean free
path of the exciton. Using his results one calculates
mean free paths of the order of 10 A, which suggests that
the exciton would be scattered frequently. Gold and
Knox24 calculate mean free paths of the same order in
solid argon where they conclude, however, that mobile
excitons are likely.
ACKNOWLEDGMENTS
The author thanks Dr. A. Zahlan for helpful corre
spondence. He also acknowledges with thanks a grant
from the Arts and Sciences Rockefeller Research Fund
of the American University of Beirut, and support from
the Research Corporation by a grant to the Physics
Department.
22 J. Frenkel, Physik. Z. Sowjetunion 9, 158 (1936) and earlier
papers.
23 o. S. Davydov and A. F. Lubchenko, Ukr. Fiz. Zh. 1, 111
(1956) .
24 A. Gold and R. S. Knox, J. Chern. Phys. 36, 2805 (1962).
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1.1728525.pdf | Elastic Wave Propagation in Piezoelectric Semiconductors
A. R. Hutson and Donald L. White
Citation: J. Appl. Phys. 33, 40 (1962); doi: 10.1063/1.1728525
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Downloaded 09 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 1 JANUARY, 1962
Elastic Wave Propagation in Piezoelectric Semiconductors
A. R. HUTSON AND DONALD L. WHITE
Bell Telephone Laboratories, Inc., Murray Hill, New Jersey; Whippany, New Jersey
(Received June 16, 1961)
A plane elastic wave propagating in a piezoelectric crystal may be accompanied by longitudinal electric
fields which provide an additional elastic stiffness. When the crystal is also semiconducting, these fields
produce currents and space charge resulting in acoustic dispersion and loss. A linear theory of this effect is
developed, taking into account drift, diffusion, and trapping of carriers for both extrinsic and intrinsic
s~mic?nductors. C0n.d~ctivity modulati?n sets an upper limit on strain amplitude for a linear theory. The
chrectIOnal. charac~enstics and t~e magmtud~ of the effects are illustrated for CdS and GaAs. The Appendix
treats the mteractIOn of an arbItrary acoustIc plane wave with the electromagnetic fields in a piezoelectric
crystal (based on a treatment .by Kyame U. J. Kyame, J. Acoust. Soc. Am. 21, 159 (1949); 26, 990
(1954).J) and further shows exphcItly that only the effects of longitudinal electric fields need be considered.
I. INTRODUCTION
A STRONG piezoelectric effect has recently been
discovered in some semiconducting materials.l·2
The interaction of these properties can effect the
velocity and attenuation of acoustic waves. The light
sensitive ultrasonic attenuation in photoconductive
CdS observed by Nine3 has been ascribed to this inter
action by Hutson.1 The contribution of internal electric
fields to the elastic stiffness of a piezoelectric medium
was pointed out by Voigt.4 This effect has been treated
in more detail for plane waves by Kyame,5 Koga et at,6
and Pailloux.7 Kyame required that plane wave solu
tions for the material displacements and internal elec
tric fields satisfy both the mechanical-piezoelectric
equations of state and Maxwell's equations. For an
arbitrary direction of propagation he arrived at a
secular determinant coupling two transverse electro
magnetic waves to three acoustic waves. Depending
upon the direction of propagation and the piezoelectric
tensor, the acoustic waves could be accompanied by
longitudinal electric fields which effectively increase
the elastic constants.
The solutions of the five-by-five determinant corre
spond to two transverse electromagnetic waves travel
ing at the speed of. light and three acoustic waves
traveling at the speed of sound. Since the material is
piezoelectric, the electric field of the electromagnetic
wave may create stresses in the material and be
accompanied by an acoustic wave. Since this acoustic
wave is a forced vibration traveling at the speed of light
its amplitude is very small and its effect on the propaga~
tion of the electromagnetic wave is also very small.
Depending on the piezoelectric tensor and the direction
~ A. R. Hut~on, Phys. Rev. Letters 4, 505 (1960).
Jaffe, Berlmcourt, Krueger, and Shiozawa Proceedings of the
14th Annual Symposium on Frequency Controt' (Fort Monmouth
New Jersey, May 31, 1960). '
8 H. D. Nine, Phys. Rev. Letters 4, 359 (1960).
4 W. Voigt, Lehrbuch der Kristallphvsik (Teubner I el'pzig 1910). ., ~ ,
, J. J. Kyame, J. Acoust. Soc. Am. 21, 159 (1949).
6 I. Koga, M. Aruga, and Y. Yoshinaka, Phys. Rev 109 1467 (1958). . ,
7 H. Pailloux, J. phys. radium 19, 523 (1958).
40 and mode of propagation, each acoustic wave may
create an electric field with transverse and longitudinal
components. The transverse electric field (and the
consequent magnetic field) corresponds to an electro
magnetic wave forced to travel at the velocity of sound;
hence it is small and has very little effect on the acoustic
wave. The longitudinal electric field is electrostatic in
nature and has an effect in most piezoelectric materials
large enough to be observable, and it is this type of
interaction with which we will be concerned.
In a second paper, Kyame8 specifically introduced
electrical conductivity and showed that it relaxed the
piezoelectric stiffening. In the Appendix of this paper,
we briefly sketch Kyame's derivation of the five-by-five
secular determinant including a constant conductivity
(eliminating some minor errors). Then we show that
the effect of piezoelectric coupling to the electro
magnetic waves on the square of the acoustic wave
phase velocity is smaller than the effect of longitudinal
piezoelectric fields by the square of the ratio of the
velocity of sound to the velocity of light. Thus, for the
acoustic waves only the usual three-by-three secular
determinant need be considered, provided the elastic
constants are properly modified to take account of the
piezoelectricity and conductivity.
The effects of the interaction between piezoelectricity
and semiconduction on acoustic wave propagation can
be effectively discussed in terms of a one-dimensional
model. In Sec. II we consider the effects of drift
diffusion, and trapping of carriers in an extrinsic semi~
conductor or photoconductor. Intrinsic semiconductors
are discussed in Sec. III.
The directional properties of these effects and their
magnitudes are illustrated in Sec. IV for two examples:
CdS (hexagonal) and GaAs (cubic).
II. EXTRINSIC SEMICONDUCTORS
A. Basic Equations in One Dimension
Consider acoustic wave propagation in the x direction
of a piezoelectric semiconducting medium, and define
8 J. J. Kyame, J. Acoust. Soc. Am. 26, 990 (1954).
Downloaded 09 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsWAVE PROPAGATION IN SEMICONDUCTORS 41
a strain 5, a stress T, and a displacement u such that
au aT a2u 5=-and -=p--, (ILl)
ax ax at2
where p is the mass density. Further, assume that the
medium is characterized by a piezoelectric constant e
such that 5 produces an electric field in the x direction.
Then the equations of state corresponding to this one
dimensional problem are
T-=c5-eE,
D=e5+fE, (II.2)
(II.3)
where c is the elastic constant at constant electric field,
f is the dielectric permittivity at constant strain, and
the mks rationalized units are used.
Clearly, if E can be expressed in terms of 5 in (II.2),
a wave equation is obtained for u upon differentiating
(II.2) with respect to x. The simplest case would be to
require D to be a constant; then
(11.4)
and the medium possesses an additional electric stiffness
due to the longitudinal electric field. This condition of
constant D may be related through Poisson's equation
aD/ax=Q (II.S)
to a condition of zero space charge in the medium. From
the continuity equation
aJ /ax= -aQ/at, (II.6)
one can see that in this case the varying current density
due to the piezoelectric fields is zero and hence this
situation obtains for very low conductivity in the
medium.
The opposite situation of very high conductivity
implies that the E field accompanying the wave must
be zero. Thus, the electric stiffness is absent, and the
wave is accompanied by D fields, currents, and varying
space charge. For intermediate values of the conduc
tivity, one makes use of (II.S), (11.6), and an expression
for J to obtain D in terms of E, which together with
(II.3) can be used to eliminate E from the wave
equation.
The expression for the current density in an extrinsic
semiconductor (which we shall assume to be n type)
may be written
(p.) an. J=q(n+fn.)p.E+ -j-,
{3 ax (II.7)
where the first term is due to drift, and the second
due to diffusion. Here q is the electronic charge,
{3= (kT)-l, n is the mean density, and (n+ fn.) is
the instantaneous local density of electrons in the conduction band. The fraction f accounts for a division
of the space charge between the conduction band and
bound states in the energy gap. (The evaluation of f
is considered m Sec. D.) Poisson's equation may be
written
aDjax= -qn., (II.S)
and the equation of continuity may be written
aJjax=qansjat. (11.9)
Combining Eqs. (II.7-II.9), the desired relation
between D and E is
a2D aE aD aE a2D fp. a3D -= -qnp.-+ fp.--+ fp.E-+--. (II. 10)
axat ax ax ax ar q{3 axB
Assuming the plane wave time and space dependences
D= DoeiCkx-wt); E= EoeiCkx-wt),
Eq. (ILlO) becomes
-j(nqp./w)E
D= (ILll)
[1 +2 (k/w)fp.E+ jW(k/W)2(p.fjq(3)]
For a linear (small-signal) theory, the terms in the
product of D and E must be negligible. This will be the
case if the conductivity modulation is small, (fn.«n).
The drift term may then be neglected in the denomi
nator of (II.ll) so that,
(IIol2)
where the average conductivity is u=nqp..
The condition of small conductivity modulation
(fn.«n) is satisfied when the effective drift velocity of
the carriers in the piezoelectric field f p.E is much less
than the velocity of sound. This restricts the strain
amplitude to be
(For longitudinal wave propagation along the c axis of
CdS at room temperature, the strain amplitude must
be small compared with 3XlO-s.)
B. Acoustic Propagation
To obtain the propagation properties of acoustic
waves, the electric field is obtained in terms of the
strain using Eqs. (ILl2) and (11.3), and then substi
tuted into (II.2) to obtain an effective elastic constant.
It is convenient at this time to define a "conductivity
frequency" We=UjE and a "diffusion frequency"
The reciprocal of We is often called the dielectric relaxa
tion time. WD is that frequency above which the wave-
Downloaded 09 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions42 A. R. HUTSON AND D. L. WHITE
length is sufficiently short for diffusion to smooth out
carrier density fluctuations having the periodicity of the
acoustic wave. We shall treat WD as a constant, ignoring
in it the small changes in (w/k) which characterize the
acoustic dispersion which we are investigating. Then,
(II.13)
First consider the case where diffusion of carriers out of
their space-charge bunches can be neglected, WD»We,W.
Then,
eS 1-j(we/w) E=-- ,
E 1 + (wc/w)2 (II. 14)
and substituting in (11.2),
[ e2(1-j(we/w»)] T=c 1+- S
CE 1+(wc/w)2 (11.15)
or
T=c'S.
The propagation of the wave is characterized by
dispersion and loss of the usual relaxation type with
relaxation frequency We. In terms of the complex elastic
constant c', the velocity and absorption constants are
Real V c' Real V c'
v =vo yp yc
W a=-ct Im(c'-l),
Vo (II. 16)
where Vo= (c/p)t. Since (e2/cE) is a small quantity,9 one
obtains
v = vo[ 1 +_e2_/_(2_CE)_]
1+(wc/w)2 ' (11.17)
At very low frequencies, v=vo and a=O; while at very
high frequencies,
and V=Voo=VO[l+~],
2CE
We e2
a=aoo=--·
Vo 2CE
A simple relaxation-type dispersion occurs around
W=We, at which point v= (vo+voo)/2 and the quantitylo
9 The usual quasi-static definition of the electromechanical
coupling coefficient K is (CD-CE)/C D= K2, thus the quantity
e2/(cf) =K2/(1-K2)~K2.
10 This quantity is very nearly the absorption in nepers/radian
since v differs from Vo by only a small amount. 0.5
N I~ Q) c';l 0.4
IL
0
'" 0.3 ...
Z
::J 0.2 1.0 ~
~ I
~~ ~ 0.1 05 ..... .g
I >
log 01
FIG. 1. Acoustic velocity and loss, in nepers/radian,
vs log frequency neglecting carrier diffusion.
(avo/W) has a maximum value of (e2/4CE). This dispersion
and loss are illustrated in Fig. 1. The interesting point
here is that the dispersion frequency is controlled by
the conductivity of the material.
C. Effects of Carrier Diffusion
The complete expression for the complex elastic
constant with diffusion effects included is
{ e2[ 1 + (We/WD)+ (W/WD)2- j(wc/w)]} c'=c 1+-
CE 1+2(we/wD)+(w/wD)2+(wc/w)2 '
(11.18)
so that the velocity and absorption constant are
and
a=-- .
Vo 2CE 1+2 (WeIWD) + (W/WD)2+ (We/W)2
If WD»We, the velocity behavior is adequately described
by (11.17) and Fig. 1, at all frequencies. The quantity
(avo/w) behaves in the same manner in the dispersion
region; however, it should be noted that a approaches
the constant value (we/vo) (e2/2cE) in the frequency
range between We and WD and then drops to zero as W
becomes larger than WD.
If the diffusion frequency is comparable to or less
than We, a somewhat different behavior is to be expected.
This may be illustrated by considering the case where
We»WD. The velocity and absorption constant may then
be written as
(II. 20)
The velocity goes from Vo to v"" as before, but the half
way point now occurs at w= (WDWe)t. The absorption, in
nepers/radian, has its maximum at w= (wDwe/3)t, thus
Downloaded 09 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsWAVE PROPAGATION IN SEMICONDUCTORS 43
N I~ 0.2 Q)N 1.0
"- ~ 0 1
(/) ,J !:: 0.1 0,5 "- z
:::> ~
'!': 1 >
~I~ ° ° 0.001"'0 0,01"'0 1°"'0
log '"
FIG. 2. Acoustic velocity and loss when wc= 10wD.
the loss occurs at frequencies below that of the maxi
mum velocity change. Furthermore, the maximum
value of the absorption per radian is
so that the losses associated with the dispersion may be
considerably reduced. This is illustrated in Fig. 2 for
We= 10wD and Fig. 3 for We= 100wD. In Fig. 4, the dis
persion and loss characteristics are shown for the case
We=WD, computed from (II.19). This striking difference
in the behavior of the losses associated with the dis
persion in the two cases We»WD and We«WD may most
easily be understood by considering the phase relation
ship between current and electric field. From Eqs. (II.S)
and (II.6) and the relation between D and E (II.12),
uE J=----
1+j(W/WD) (J1.21)
When We«WD, current and voltage are in phase
throughout the dispersion region centered at We. For
frequencies below We, the current flows for a long enough
time to "short" the electric field by a space-charge
buildup. For frequencies above We, the time of current
flow is too short to allow a space-charge buildup, hence
the electric field is unaltered by the current. The
acoustic loss is proportional to J. E, hence is low for
W <We, and rises to a constant value as W becomes greater
than We. The loss finally goes to zero as W becomes
greater than WD because current and field are no longer
in phase.
When WD <We, current and field start to go out of
phase for W':::'.,WD and the space-charge variation is no
longer phased for optimum "short-out" of the electric
field. Thus, the dispersion occurs at frequencies less
than We. The losses associated with this dispersion are
smaller than in the previous case because current and
field are not in phase.
D. Fraction of Space Charge Which Is Mobile
The effects of diffusion and drift have been shown to
be proportional to the fraction f of the acoustically
produced space charge which is mobile. Therefore, it is
useful to obtain an expression for f in terms of the
appropriate parameters of the semiconductor. N I~ 0.2 Q)N ~--r---.---=...-----,1.0
"-0
(/)
!:: z
:::>
'!':
~I~ ~
1
0,1 f----t---/---+----j 05 ~
° 0,001"'0
log 01 ~
I >
FIG. 3. Acoustic velocity and loss when wc= 100wD.
Consider an extrinsic semiconductor in thermal
equilibrium, and suppose it to be n-type for the purpose
of discussion. There will be one-electron bound states
of various types at various energy levels in the forbidden
band. Let the densities ofithese states be N 1, N 2, • • • j
their energy levels E1, E2, • • • j and their statistical
degeneracy factors g1, g2, .... The total space-charge
density Q may then be written as
Q Ni -=-Neexp(3(Ef- Ee)-L:------
q i 1+gi-1 exp(3(Ei-Ef)
+terms independent of Ef• (11.22)
The first term is the concentration of electrons in the
conduction band which will be nondegenerate for all
cases of practical interest. The condition of electrical
neutrality, Q=O, yields the equilibrium Fermi level Efo.
The acoustically produced space charge is then a
periodic variation in Q about zero describable by a
periodic variation of Ef about Efo in the conduction
band term and those bound state terms for levels which
equilibrate with the conduction band in times short
compared with the frequency of the sound wave. Levels
which equilibrate with the conduction band in times
long compared with the sound frequency will have a
constant electron occupancy in equilibrium with the
time average conduction band electron concentration.
The fraction f of the space charge which is mobile is
then
(r1.23)
where ne=N e exp(3(Ef-Ee), and (Q'/q) contains only
0,3
NI~ "'u N
"-0,2 1.0 0
(/)
~ t-
Z I
:::>
05 ~ '!': 0,1
~I~ ~
I
>
° ° O.OIOle 0,1"'0 I.Owe 10"'e
log '"
FIG. 4. Acoustic velocity and loss when WC=WD.
Downloaded 09 Jun 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions44 A. R. HUTSON AND D. L. WHITE
the states with fast equilibration times. After some
manipulation, we may write
1 n/Ni-nj)
-=1+ L , 1 noNj (IJ.24)
where the sum is over all states j which have fast
equilibration times; nj are the equilibrium numbers of
electrons in these states, and no is the equilibrium num
ber of conduction band electrons.
As a practical case, consider a sample containing
donors only, then
(11.25)
and the smallest value of f is one-half in the case that
there is very little ionization. As a second case, consider
a concentration N D of shallow donors highly com
pensated by a concentration N A of deep acceptors, then
1 nD(N D-nD) nA (N A -nA) -=1+ +-------1 noND nONA (II. 26)
Here, nA"::: . .N A and (N A -nA)«nO; hence, the last term
is small and the acceptors do not take part in the
periodic variation of space charge. The donor term
becomes
(N D-NA-no)(N A+nO)
noND (II.27)
Here the limiting cases are: (1) ionization of donors to
the conduction band nearly complete, in which case
no'" (N D-N A) and f = 1 ; and (2) very slight ionization
so that n6«(N D-N A) in which case 1«1. In this latter
case, there is no periodic modulation of conductivity by
the acoustic wave, and therefore no diffusion or drift
effects.
The rate of equilibration of the conduction band
electrons with a given type of center will depend upon
the capture cross section of the center, the density of
the centers, and the mean thermal velocity of the elec
trons. Thus a given type of center may contribute to
1/ f for only a restricted range of concentration and
temperature at a particular frequency.
For photoconductivity in which only one type of
carrier plays an appreciable role, the fraction 1 may be
determined from (II.22) and (II.23) by replacing the
Fermi level by the quasi-Fermi level for the mobile
carriers.
III. INTRINSIC SEMICONDUCTORS
For intrinsic material, ionization of states within the
energy gap will be complete, so that no space charge is
stored in them. As in the extrinsic case, a relation
between D and E is obtained from the continuity
equation and Poisson's equation. However, an addi
tional complication results from having to take account
of both hole and electron currents and recombination. We shall briefly consider the two limiting cases of
recombination time long and short compared with the
period of the acoustic wave.
When recombination is fast compared with the
acoustic frequency, the electron and hole concentrations
may be written as
and
The electrons and holes then satisfy a single continuity
equation
dJ / dx = 2qdn./ dt.
The expression relating D and E analogous to (11.1)
is then
D
From a comparison of (IILl) and (II.ll) one can see
that the propagation properties of acoustic waves will
be the same as for the extrinsic case with an effective
diffusion frequency
WD= 2q{3v2/ (,un+,up),
and that nonlinear (drift) effects will be negligible if
S«Ev/ei,un-,upi·
For long recombination times, the hole and electron
currents will separately satisfy continuity equations.
The expression relating D and E is then
(III. 2)
If the theory is to be linear, the limit on strain amplitude
IS
5« Ev/,ue ,
where,u is the larger of ,un,,up. If diffusion frequencies for
electrons and holes are written
the expression for the complex elastic constant is
e2/e c' = c+ . (III.3)
wc[,upj (,un+,up) ,un! (,un+,up)] l+j- +-------
W l+j(w/wD p) l+j(W/WDn)
If diffusion can be neglected (WDp,WDn»W,W c), a simple
relaxation occurs at W=Wc. Rather complicated diffusion
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effects are possible if diffusion frequencies are lower
than We.
IV. EXAMPLES
In order to apply the one-dimensional considerations
of Sees. II and III to particular cases, one must obtain
the appropriate elastic constant for the chosen direction
and type of wave motion and also the piezoelectric
constant which determines the electric field in the
direction of wave propagation. In the Appendix it is
shown that the propagation of plane waves in the 1
direction of an orthogonal coordinate system is deter
mined by the solution of a three-by-three secular
determinant
Cllll' _ W:)
P k-
C1211 ,
p
Cl3ll ,
P
where C1112 ,
CUl3 ,
p p
(CI212' _ w) Cl213 ,
p k? P
Cl312 ,
(C1313' _ W)
P P k2
e11ie11k
Clilk' = Clilk+----
(fll+ jUll/W) =0, (IV.1)
in the case of constant conductivity. Note that these
elastic and piezoelectric constants must be obtained
from the usual ones by a coordinate rotation in the case
that the wave propagation is not along the crystal
lographic 1 direction. If CjZmn * and eZmn * are the constants
referred to the usual crystallographic axes, then
Clilk= aljaiZalmaknCjZmn*,
el1i= aUalmaineZmn *,
where the a's are the appropriate direction cosines.
To avoid complication we shall limit our practical
considerations to cases where the wave motion is either
pure longitudinal or pure shear, that is, where (IV.1)
is diagonal.
Wurtzite Crystals-CdS
These semiconducting compounds of crystal class
6 mm possess three independent piezoelectric constants
e31l*=e322*, e33a*, and e113*=e223*,
where the 3 axis is the axis of hexagonal symmetry.
From inspection of (IV.1) it can be seen that for propa
gation along the hexagonal axis, longitudinal waves
(c= C33*) are piezoelectrically stiffened by e333*, and
that no shear waves (C=C44*=C66*) will be affected.
When propagation is perpendicular to the hexagonal
axis, shear waves for which' the displacement is along
the hexagonal axis (c= C44*= C56*) will be stiffened, TABLE 1. Parameters for n-type CdS.·
W,(CT in 0-1 cm-1)
WD at 3000K
"JD at 1000K
e2/2c.
'Va Type of wave
Longitudinal C44 shear
II to axis .L to axis
1.2X 1012 CT
2.9XlO10
1.1 X 1010
0.015-{).025
4.3XI0· 1.25X1012 u
4.8XlO"
1.8X 10"
0.018
1.75 X 100
• Values of e obtained from values reported for d by Hutson,l and by
Jaffe, Berlincourt, Krueger, and Shiozawa.' Elastic data from D. I. Bolef,
N. T. Melamed. and M. Menes, Bull. Am. Phys. Soc. 5, 169 (1960), and
McSkimin et a/.ll Mobility from Miyazawa, Maeda. and Tomishina,
J. Phys. Soc. Japan 14. 41 (1959).
whereas longitudinal waves (C=C11*=C22*) and shear
waves for which displacement is perpendicular to the
hex axis (c = C66 *) will be unaffected. These directional
properties agree with the photoconductivity-modulated
acoustic attenuation measurements by Nine3 and with
additional measurements by McSkimin, Bateman, and
Hutson,!l on CdS.
Particular note should be taken of the shear wave
propagation characterized by C4/~C66*. For high con
ductivity crystals, where the piezoelectric fields are
completely "shorted" W«We, the velocities of propaga
tion along and perpendicular to the hex axis will be
equal, and characterized by C44*E. For low-conductivity
crystals, w»wc, the velocity of propagation along the
hex axis is unchanged, but the velocity perpendicular to
the hex axis is increased by e24*2/2fl1' This apparent
violation of classical elastic theory arises simply from
the electrical condition, imposed by Maxwell's equa
tions, that only longitudinal electric fields may accom
pany plane acoustic waves.12
A compilation of the parameters which playa role in
the relaxation-dispersion for longitudinal and shear
waves in CdS is presented in Table I.
Zinc-Blende Crystals-GaAs
For AIIIBv semiconducting compounds (zinc-blende
structure, crystal class 43m), there is only one inde
pendent piezoelectric constant referred to the cubic
crystallographic axis: eijk, * i~j~k. Therefore, there is
no longitudinal electric field for either longitudinal or
shear waves propagating along [100J directions.
For propagation in a [110J direction, one can obtain
the appropriate elastic and piezoelectric constants for
the three-by-three secular determinant in which the 1
direction is the direction of propagation, by a 450
coordinate rotation about the 3 axis. This determinant
is diagonal, and only the shear wave with displacement
in the 3 direction is piezoelectrically stiffened. The
appropriate elastic constant is just C44* and the appro-
II H. J. McSkimin, T. B. Bateman, and A. R. Hutson, J. Acoust.
Soc. Am. 33, 856(A) (1961).
12 The Lyddane-Sachs-Teller relation describes an analogous
electrical stiffening of the longitudinal optical-mode branch of the
elastic spectrum which is absent for the transverse branch.
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TABLE II. Parameters for GaAs stiffened [110J shear wave.B
W,(O' in 0-1 em-I)
WD at 3000K
WD at nOK
e2/2cE
vo(cm/sec) 1012 0'
7XI08
lXlOS
1.2X1O- a
3.35XW
• eH~O.12 coul/m'. determined by D. L. White. Elastic data from T. B.
Bateman. H. J. McSkimin, and J. M. Whelan, J. Appl. Phys. 30. 544
(1959). Mobilities and dielectric constant from J. M. Whelan (private
communication) .
priate piezoelectric constant is e14*' Parameters charac
terizing the [110J shear wave in GaAs are presented
in Table II.
ACKNOWLEDGMENTS
The authors are indebted to E. O. Kane, W. P.
Mason, and J. H. McFee for helpful comments on the
manuscript and to D. E. Collins for preparing the
figures.
APPENDIX
For completeness we shall sketch the derivation of
the fifth-order secular equation for the phase velocities
of acousto-electromagnetic waves propagating in an
arbitrary direction in a conductive piezoelectric crystal.
(This is basically the derivation given by Kyame5•8
except that the index errors of the piezoelectric tensor
have been corrected, space charge has been properly
taken into account, and the derivation has been
simplified by the practical assumption that the magnetic
permeability is isotropic and equal to that of free space.)
Let Xl, X2, and Xa be orthogonal axes arbitrarily
oriented with respect to the crystal axes and consider
the propagation of plane waves in the Xl direction.
Derivatives with respect to X2 and Xa are then zero, and
the force on a volume element in the i direction in terms
of the stress tensor is (iJTdiJXl). The usual definition
of the strain tensor in terms of the displacements is
but for the plane wave problem it IS convenient to
define a strain
S lk' = S lk+ SkI = aUk/ aXI'
Under adiabatic conditions, the applicable mechanical
and electrical equations of state can then be written
(with repeated indices as summation indices)
T H= Clil~l/ -ekl;Ek,
Dk=eklvS1i'+EikEi. (Al)
(A2)
Here Cijkl is the elastic tensor for constant electric field,
eijk is the piezoelectric tensor, and Eik the dielectric
permittivity tensor, all referred to the axes Xi and in mks
units. The constitutive equations for the medium are
(A3) The electromagnetic field quantities must simul
taneously satisfy Maxwell's equations
vXH=aD/at+J; VXE= -aBjiJt (A4)
v·B=O; V·D=Q.
For the plane wave condition, Bl and II 1 are constant
in space and time and (vxHh=O which yields the
continuity equation for II and Q upon differentiating
with respect to Xl. From the curl equations, one can
obtain equations for E2 and Ea from which D and J can
be eliminated using (A2) and (A3); thus,
(AS)
'b,], k=l, 2, or 3, but p, q=2 or 3. Differentiation oJ
(Al) yields three equations
--=p---=Clilk---ekl,--. (A6)
iJXl at2 aX12 aXl
The Eqs. (AS) and (A6) are five coupled wave equations
in the six variables Ui and Ei. Using (vXHh =0 and
(A2) and (A3), El can be expressed as
(A7)
By assuming plane wave solutions of the form
and eliminating El from (AS) and (A6) using (A7), one
obtains
and
(Al0)
where
elliellk
Clilk'=Clilk+ ,
(EU+ juu/w) (A11)
(A12)
(A13)
The secular determinant of the five Eqs. (A9) and (A10)
may be manipulated by row and column multiplication
until all of its elements have the dimension velocity
squared. To accomplish this, we make use of the average
dielectric permittivity of the medium E and define
velocities (which may be complex) corresponding to
the primed piezoelectric tensor elements as
(A14)
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The secular determinant is then
(Cllll' _ w) CU12 ,
CIU3 ,
V211'W V3ll'W
P k~ p p k k
, e1
:U
'_ ::) ,
V212'W ,
C12ll C1213 V312W
P P k k
C1311 ,
C1312 ,
(C1313' _ W2) V21a' W V31a'w
p p k2 k k =0. P (A1S)
V2U'W V21Z'W V213'W
k k k
Vall'W V312'W Val3'W
k k k
If the piezoelectric tensor is zero, (A1S) splits into the
usual third-order acoustic wave determinant and
second-order electromagnetic wave determinant.
The piezoelectricity may be taken into account in the
propagation of the acoustic waves to a very good
approximation by solving only the third-order acoustic
determinant with the piezoelectric conductivity modi
fied elastic constants Cljlk', neglecting the coupling to
the electromagnetic waves. The modified elastic con
stants introduce a term in the solutions for (W/k)2 of
the order of some average Vilk'2. The additional correc
tion to (W/k)2 which would result from considering the
coupling to the electromagnetic waves is smaller by the
/)2 0 0
0 (Vl-VI2+02) 0 (W2 E2Z' 1) W2E32'
k2 E E}J.o k2E
W2E3Z' (W2 E33' __ 1 )
k2e k2 E EjJ.O
square of the ratio of the velocity of sound to the
electromagnetic wave velocity.
To demonstrate this, suppose that the three-by-three
acoustic determinant has been solved (W/k=Vl,V2,V3)
and diagonalized in terms of new displacement vectors.
To find the magnitude of the correction IF to VI2 resulting
from the coupling to the electromagnetic wave, set
(w/k)=ZII in the fifth-order determinant for the new
displacements and EI and E2• Letting c= (jJ.OE)-! be the
average velocity of electromagnetic waves, one may
write the determinant to within the correct order of
magnitude of the various terms as
,
V VI V'VI
V'VI V'VI
0 0 (Z'32_VI2+02) V'VI V'VI =0. (A16)
V'VI V'VI V'VI (V12-C2) VI2
V'VI V'VI V'VI VI2 (VI2-C 2)
Expanding (A16),
Thus,
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1.1714394.pdf | Heat Flow as a Limiting Factor in ThinFilm Devices
David Abraham and T. O. Poehler
Citation: Journal of Applied Physics 36, 2013 (1965); doi: 10.1063/1.1714394
View online: http://dx.doi.org/10.1063/1.1714394
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/6?ver=pdfcov
Published by the AIP Publishing
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IP: 155.33.16.124 On: Sun, 30 Nov 2014 07:11:28REF R ACT I V E I N D E X 0 F 5 i 0 2 F I L M 5 G ROW NON 5 T L I CON 2013
tive indices and thicknesses of a wide range of films, we
have eliminated the calculations involved in Eq. (6)
by tabulating the thicknesses in various sets. The
thickness is given by
d=NAJ/2p.j COSf. (7)
Each set is identified by the refractive index at 5459 A
and the corresponding indices used in tabulating the
thicknesses associated with the other filters within the
same set are based in one system of sets on the disper
sion of fused quartz and in another system on the
dispersion of lead silicate glasses. In compiling the
tables, corrections were made for the stereo angle of
the microscope and for phase change differences at the
silicon-film interface. The refractive index and thick
ness are then determined by interpolation between sets
to obtain consistent thicknesses for different readings.
This simplified method has been applied to thin films
JOURNAL OF APPLIED PHYSICS where different filters are necessary to obtain at least
two readings and to thicker films where two or more
minima can be obtained with the same filter. By these
techniques, we have measured films with refractive
indices from 1.30 to 1.70. When the refractive index of
the film is known and the order known (as can be easily
determined by one of the previously described tech
niquesl) it is only necessary to obtain one reading for
a thickness determination from the tables based on
Eq. (7).
From consideration of the accuracies of minima
determinations and filter values, it is concluded that a
single calculated refractive index for very thin films
has an accuracy of 0.001 to 0.002. However, when the
refractive index is known or a precise refractive index
assumed, different experienced operators can obtain
the same measured thickness within 2 A on a sooo-A
film of Si02 on silicon.
VOLUME 36. NUMBER 6 JUNE 1965
Heat Flow as a Limiting Factor in Thin-Film. Devices*
DAVID ABRAHAM AND T. O. POEHLER
The Johns Hopkins University, Applied Physics Laboratory, Silver Spring, Maryland
(Received 12 November 1964; in final form 4 March 1965)
The performance of thin-film devices can be limited by thermal failure. A model for thermal power dis
sipation of a thin-film device on a glass substrate has provided the basis for the solution of the heat flow
equation. Solutions for the maximum temperature as a function of power input and rectangular device
and substrate dimensions have been obtained by digital computations. These results have been applied to
the TFT and to the Mead device, and it is concluded that the Mead device will suffer thermal destruction
before it can achieve a figure of merit greater than 10 Me/sec whereas the TFT is not limited by thermal
considerations until a figure of merit of several hundreds of Me/sec has been achieved.
THE gain-bandwidth product M is a figure of merit
often applied to active devices. It has been de
fined as follows:
(1)
where for a voltage controlled device, gm= a10/OV I is
the transconductance, 10 is the output current, V I is
the signal input voltage, and C[ is the input capacitance.
It is our purpose to determine an upper bound on the
value of gm, and through it M, for thin-film active de
vices by imposing the limitation that the active ma
terial cannot perform above a temperature Terit, which
in turn is a function of the material and the physical
mechanism of the current modulation. T crit will be
exceeded if the power to be dissipated due to the product
of the current and voltage drop across the device exceeds
a value which we intend to deduce for some typical
thin-film active device configurations. We do not intend
to imply that power dissipation is the exclusive factor
in evaluating the upper limit on gm. However, some
* Work supported by the U. S. Bureau of Naval Weapons,
Department of the Navy under Contract NOw 62-0604-c. devices have been proposed where the application of
this criterion would have provided an upper limit on
gm and M so small as to dampen if not extinguish the
intense interest in their development.
Let us examine the relationship between gm and power
dissipation. Figure 1 characterizes four relationships
between output current and input voltage which can
result, depending on the modulation mechanism. The
transconductance in each case is the slope of the curve.
In Fig. 1 (a) the gm is independent of the operating
point; and in Fig. l(b), gm is maximum when 10=0.
For these cases power dissipation is not a limiting factor
and we need not consider them. In Fig. l(c), gm in
creases with increasing 10 and in Fig. l(d) an optimum
value ofIo exists for which gmis maximum. Figure 1 (c) is
characteristic of the hot-electron device proposed by
Meadl and Fig. 1 (d) is characteristic of theTF T2 in which
10 saturates with V I. In the former case the thermal dis
sipation will always be a major determining factor in the
limit on gm. TFT's in which 10 does not saturate with VI
1 C. A. Mead, Proc. IRE 48, 359 (1960).
2 P. K. Weimer, Proc. IRE 50, 1462 (1962).
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(0)
<Cl (d)
FIG. 1. Current-voltage relationships possible in active
devices with different modulation mechanisms.
also fall in this category. In the latter it would depend
on whether thermal breakdown occurs before or after
the otherwise realizable maximum gm is achieved. Should
it occur after gm maximum is reached we need not con
cern ourself with the thermal problem, or else we might
try to alter the characteristic to push gm maximum to a
coincident with that thermally allowed. In a real TFT
this might be achieved by increasing the applied source
drain voltage to a saturation value.
In order to determine gm maximum as limited by
the power dissipation we shall employ the following
procedure:
1. We shall determine from the geometry and thermal
conductivity, the relationship of temperature rise to
power dissipation.
2. From the physical properties of the device and its
modulation mechanism we shall estimate the maximum
permissible temperature rise and thus the maximum
power which can be dissipated.
3. Because we do not have an a priori knowledge of
the outcome of our calculation and are not likely to have
experimental data to the ultimate limits of device
operation we must make an educated guess as to the
operating point corresponding to the maximum allow
able power dissipation.
4. We then compute gm at that operating point. If
it is less than the gm maximum derived from other con
siderations, the device is thermal-dissipation-limited.
The following calculations will demonstrate that
severe thermal limits on the hot-electron device pro
posed by Mead exist, whereas some TFT's are not
subject to loss of performance due to thermal limits on
gm' This is not to imply that the TFT is immune to
thermal degradation or destruction. In fact, the calcu-lations relating temperature to power dissipation can be
used to establish the upper limit on power dissipation
for any thin-film element, active or passive, whose
geometry is compatible with one of the models used.
HEAT FLOW PROBLEM
In order to relate the maximum operating temperature
and the power dissipation for any given thin-film ele
ment, we must solve the heat-conduction equation for
this type of configuration. The problem which must be
solved is dependent on the geometry, boundary con
ditions, and any other assumptions used to express the
physical situation in mathematical terms.
The geometry with which we are concerned is that
of very thin rectangular thin-film element which lies
on substrate in the form of a rectangular parallelepiped.
This is illustrated in Fig. 2. It will be assumed that the
thickness of the thin-film element is quite small with
respect to all other quantities involved.
The actual calculation that is made is that of the
temperature at the interface between a thin-film element
and the substrate. The thin-film element is considered
to be a source of uniform heating at this surface. All
heat loss is assumed to be by conduction through the
substrate. The other face of the substrate is considered
to be a uniform temperature T= O.
There are a number of assumptions about the physical
nature of the problem implicit in these conditions.
By concerning ourselves with the temperature at
the interface and assuming all heat losses to occur by
conduction to a cooler back surface, we are neglecting
possible cooling by convection and radiation into the
medium at the upper surface. However, at the tempera
tures which most thin-film elements can tolerate the
latter assumption is not unreasonable.
The thin-film element is assumed to be a uniform,
constant source of heat at the film-substrate interface.
By use of this condition we can solve for the temperature
at the interface using Laplace's equation in the region
enclosed by the boundaries of the substrate. Since the
assumption has been made that the thin-film element
is very thin with no cooling at the upper surface, the
resulting solution will be representative of the tempera
ture of the element for any reasonable power input. In
FIG. 2. Geometry of thin-1ilm element.
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some thin-film elements, specifically the TFT, the power
dissipation may not be entirely uniform over the area
of the device, but this possibility will be neglected in
this analysis.
The bottom surface of the substrate is considered to
be held at a unifonn temperature. In the interest of
computational simplicity this temperature is taken to
be 0° j solutions for any other higher value can be ob
tained by adding a constant temperature to the results
for this case. Such a condition represents a uniform
cooling or heat sink to which the substrate under-surface
is attached. Except for the possibility of some small
additional cooling from the upper surface this presents
the best method of extracting heat from the device.
The final boundary condition which must be satis
fied is that at the periphery of the element of the sub
strate which contains each device. That is, in the nomen
clature of Fig. 2 we must specify the conditions at
x= ±R/2 and z= ±L/2. Two basic choices exist for
these conditions depending on whether we are interested
in the case of one device per substrate or a number
placed on the substrate in a regular array. Assuming the
latter to be a more practical situation, then for devices
of equal power dissipation there will be no heat flow at
the points x= ± (R/2) and Z= ± (L/2).
Finally, we will only be interested in the solution of
the heat equation under steady-state conditions. The
problem which will be solved is that of heat flow after
thermal equilibrium has been achieved in the system.
Under transient conditions devices might be able to
dissipate for very short times more power than derived
here, but for most applications the condition of thennal
equilibrium is the best available description.
Having specified the problem as outlined in the pre
vious paragraphs, we are now concerned with the
solution of Laplace's equation
V2T(x,y,z)=0, (2)
where T(x,y,z) is the temperature of the substrate as a
function of position including the area of greatest
This expression gives the equilibrium temperature as a
function of position throughout the substrate and at the
interface between the device and the substrate. As
suming a device dissipating power unifonnly throughout
its volume, the point of maximum temperature will then
be at x=O, y=O, z=O. We shall choose a number of
values for the dimensions of the device, substrate
thickness and size of the element of the substrate on
which the device is placed; for these values the tempera
ture T(O,O,O) has been computed.
The solutions of the conduction equation for certain
values of the parameters h, R, L, S, W a,re illustra,ted in interest-the interface between the substrate and the
device. We can solve the problem by the standard
technique of separation of variables.3 Using this pro
cedure we obtain a series solution of the form
T= L ai cosl;x sinhpi(h-y) coskiz, (3)
1-
where pl=kl+ll. From the boundary conditions
u=-K(aT/ax)=O at x=±R/2
and
u=-K(aT/az)=o at z=±L/2,
where u is the heat flow across the boundary and K is
the thennal conductivity, we find
T= L L amn cos[2(m-1)1I'/RJx
m=ln=l
Xsinhpmn(h-y) cos[2(m-l)1I'/LJz;
pmn = 211'{[ (m-l)2jR2J+[(n-l)2j DJ} t.
From the condition that the heat flow into the substrate
from the active device located in the plane at y= ° must
be equal to the flow out through the face at y= h
we have
fWI2 f8/2 fLI2 fRI2 dz Udx= dz -K(aT/ay)dx,
-W/2 -8/2 -L/2 -R/2
where U is the heat flux flowing from the active device
per cm2 per sec. Performing the necessary substitutions
and integrations we are able to evaluate the coefficient
amn yielding
where
lm=[2(m-l)lI'J/R and kn=[2(n-1)lI'J/L.
Finally we obtain
(4)
Figs. 3-7. Variations in device temperature as these
parameters are allowed to vary are plotted using the
normalized temperature parameter T(K/U) when K is
the substrate conductivity and U is the heat flow per
unit area. Figure 3 shows the variation in temperature
versus the width of the film W for selected values of
the width of the substrate element L. The temperature
is seen to increase slowly with increasing film widths
until this dimension approaches the width of the sub
strate element. When the width of the active element
3 H. S. Carslaw and J. C. Jaeger, CondllCtion of Heat in Solids
(Oxford University Press, Inc., New York, 1959), p. 163.
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O.OG
0.05
C.O'l
0.0'3
o.oz / I
/
/ I
I
/ ,
./'" ,.." ...... '
",,/' ~./.
",.."",. . .",'
0.01 ._..:::;.:;::-:.':/ I I
I
I
I
I
I
I
./ ./ ./ / ./ ./ I
./ .'
I f
i
f
L (em) Ul ____ _
Us--·-·-2.5--
O·a:;,'-::.OO::-,----'O''''O''l------;;O~,-----n]..r--
FIG. 3. Normalized temperature of thin-film elements of dif
ferent widths Wand substrate widths L when length of device
approaches length of substrate. (R=2.5 ern, S=2.0 ern.)
becomes an appreciable fraction of the substrate element
on which it is contained, the temperature of the active
element is seen to rise sharply to a high value. The
behavior illustrated in Fig. 3 is for the case where the
length of the active element S is almost equal to that of
the substrate element R. In the case where R is several
times longer than S, the increase in temperature as W
approaches L is much less pronounced, as can be seen
in Fig. 4. Figure 5 shows the temperature variation of
an element with change in length for several values of
width. This is an almost linear variation for all values
of practical interest in these parameters.
Figure 6 illustrates the variation in temperature of
an active device for different values of substrate thick
ness, h. For values of h large with respect to the device
width the increase in temperature with h is a linear
function. However, as h is reduced to the same order
of magnitude as W, the temperature drops quite sharply.
Some caution must be exercised in extrapolating such a
result too far since the final result of this process would
be to apply the heat generated almost directly to the
cooling surface over a very small region. Such a physical
situation would not be in accord with the boundary
condition of a substrate with a uniform-temperature
0.0'\
0.03
o.oz
0.01
Wlem)
FIG. 4. Temperature vs width curves for thin-film elements of
Jen~ths mUcl~ less than llubstrate. \R=O.5 ern, S=O.l ern.) back surface since no cooling medium could remove heat
at a sufficient rate to satisfy this condition. However,
for the ratios of W to h shown in Fig. 6, the results are
not subject to this objection.
The effect of changing the power density, keeping the
total input constant, rather than a fixed density as in
the previous cases is shown in .Fig. 7. As the width of
the active element decreases (increasing power density)
the temperature increases first rather slowly and then
rapidly at low values of W.
HOT·ELECTRON TRIODE
The results described in the previous section can be
used to estimate the limits on thin-film active element
performance determined by thermal considerations. In
this discussion we are interested in applying the results
to the thin-film hot-electron device originally proposed
by Mead.
The hot-electron amplifier described by Mead is
0.6
05
0 .•
O,I~
6----~0~5-----~I.~O------T.1.5,--____to
5 (e .. )
FIC. 5. Normalized temperature vs thin-film-element length
for several values of width. (a) W=1.0 cm (b) W=O.lO ern
(c) W=O.01 cm.
based on a tunneling of electrons through a dielectric
film separating two conductors. Those electrons which
penetrate the film due to an applied field suffer no loss
of energy in transmission through the barrier and, hence,
have significantly greater energy than electrons at
thermal equilibrium with the lattice. When these "hot"
electrons collide with the lattice they give up this excess
energy which in turn raises the over-all temperature of
the device.
In such a physical situation the current increases
exponentially with an increase in the applied field. Since
the figure of merit for such a device depends on the
current density as in Eq. (1), then for modest input
voltages useful figures of merit might be expected.
If the second metal electrode is sufficiently thin,
electrons can penetrate this electrode and pass into a
third electrode. If the second electrode (gate) is sepa
rated from the third (collector) by an insulating film,
then only "hot" electrons will reach the collector.
1deally, most of the el~ctrop.s leavin~ the emitt~r as !l,
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result of a voltage applied to the gate will pass through
the gate and arrive at the collector. On arriving at the
collector, the hot electrons will give up the energy they
have acquired from the field with a heating power equal
to V Je. Those electrons which which penetrate the
structure only as far as the gate will give up power equal
to V of o' Because of the nature of the structure this
power will also be effective in heating the collector. The
heating power per unit area will then be related to the
current density of the device in a manner similar to the
figure of merit.
For a uniform current density, the total dissipated
power per unit area will be given by
P=J.Vo(1-a)+J.V ca, (5)
where P= power density, J.= emitter current density,
V g= gate potential, V c= collector potential, and a
= emitter-to-collector transfer ratio. If 0~a:$1 and
V 9 ~ V c, then the lower bound on P will be J. V g= P min.
0'
FIG. 6. Variation in normalized thin-film-element temperature
with substrate thickness h. tW =0.001 em, L=0.25 em.)
It is possible to relate the power dissipation and the
figure of merit M [Eq. (1)J if we know the current
voltage relationship for the device. Conduction through
thin insulating films which are of interest here is princi
pally attributed to two mechanisms: tunnelint and
thermionic emission. 5 Assuming that the tunneling
current obeys the Fowler-Nordheim formula and emis
sion current is given by Schottky's theory, we have the
current-voltage relationships shown in Fig. 8. On the
basis of these curves we may then plot the lower bounds
of P vs M as shown in Fig. 9. These curves are shown
for relatively optimistic values of a since transfer ratios
of 10-2 are the highest reported values. On the basis of
these curves and the calculations of the previous section
it is possible to determine the maximum figure of merit
obtainable before thermal runaway will occur.
Thermal runaway and device failure will occur when
4 C. A. Mead, J. App!. Phys. 32, 646 (1961).
5 P. R. Emtage and W. Tantraporn, Phys. Rev. Letters 8, 267
(1962). <1.
/.0
0.9
0.1ii
o.
0.00/ 00.1
FIG. 7. Effect of varying power density on thin-film-element
temperature by varying element width and maintaining total
input power constant at iUS UW watts. (R=2.S em, S=2 em.)
heating in the device is sufficient to cause an ever
increasing amount of current generation due to thermal
emission. The initial current which causes this heating
may be due either to tunneling or the Schottky effect.
To estimate where such a condition may begin, consider
a typical device with dimensions 1 mmX 1 mmX 50 A.
From the calculations in the previous section the temper
atureof such a structure is seen to be equal to 0.02 U /K,
so that for a substrate with K = 0.002 the temperature is
10 V or 2.5 V if the input energy is expressed in terms of
W -sec. From Fig. 9 we see that for even a modest figure
of merit the device must be operated at 100 W /cm2•
This yields an operating temperature of 250°C above
the temperature of the back of the substrate. If the
back side is maintained at O°C then the device will
be operating at 250°C which is sufficient to cause a sig
nificant amount of thermal emission. The Richardson
Dushman equation with the Schottky effect included is
given by
J=A'P exp[ -e/kT(¢o- (eE/41re)i)J, (6)
where k= Boltzmann's constant, e= dielectric constant,
16 r ~ l~\ ..
Q..
J.'+
h
g>
-J
Z
0
-t-o Z 4 6 8 10
Vl)lt~/cm l( IO~
FIG. 8. Current-voltage relationships for hot-electron triodes.
(a) Fowler-Nordheim tunneling (metal--oxide barrier of 2.75 eV),
(b) Schottky emission (metal-insulator barrier of 0.62 eV).
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IP: 155.33.16.124 On: Sun, 30 Nov 2014 07:11:282018 D. ABRAHAM AND T. O. POEHLER
"
....... <4
"'E
~ 1: '-'
ll-l
O"!
0
0
"'CIOl
Zo Z 4-..rectos eu 10 , , ,
I I I
I I ,
, , I
,1 " ,
~ " ~"
, , ,
I , ,
s
log M (CpS) 10
FIG. 9. Power vs figure of merit for hot-electron triodes. (Tunneling
triodes-dashed lines; Schottky emission-solid curves.)
A = Richardson constant, T= absolute temperature,
tPo=metal-dielectric work function, and E=applied
field. If we know the work function tP we may calculate
the amount of Schottky emission at a device tempera
ture of 250°C. From the work of Emtage and Tantra
porn, we know the work function for anAI-AhOa barrier
to be 0.74 eV.5,6 Using this value and the device temper
ature of 250° corresponding to an initial current of 100
A/cm2, we find the thermal current density to be
140 A/cm2• This thermally generated current added
to the initial current will, of course, further increase
the device temperature leading to even higher currents
and eventual thermal runaway. Such a mechanism
would seem to be of paramount importance as a failure
mechanism in thin-film hot-electron triodes.
As is evident from the calculations of the earlier
sections, one can improve this situation by reducing the
dimensions of the devices to obtain better cooling. A
structure 10 p. wide would be about the minimum width
obtainable using presently available technology. For a
device of 0.1XO.001 cm the temperature is given by
0.002 U /K, so that for a conductivity of 0.002 at runa
way condition T= 250°C will not be obtained until the
power input is 1000 W /cm2• This would then seem to
be close to a maximum figure for the device. Assuming
that the device is operated with 0:= 1 and that the cur
rent is tunneling current, the maximum figure of merit
obtainable would be about 10 Mc/sec (Fig. 9).
These calculations would seem to indicate that even
should unity transfer ratios be obtained, it would be
impossible to obtain hot-electron thin-film triodes with
high figures of merit. Moreover, thermal limitations
rather than dielectric failure would seem to be respon
sible for device failure in both present and future at
tempts to obtain high-performance devices.
THIN-FILM TRANSISTOR-TFT
In this section an estimate of TFT performance will
be made applying first a field excitation model and then
6 M. Hacskaylo, J. App!. Phys. 35, 2943 (1964). an injection model as proposed by Weimer. The charac
teristics of a field excited TFT are shown in Fig. 1 (d) .
The transconductance is at its theoretical maximum
value when the drain current (Id) vs voltage (Vd)
characteristic is saturated in source voltage and the
slope of the drain current vs gate voltage characteristic
is greatest. The minimum power dissipation associated
with this gm maximum is estimated by requiring that
V d be just larger than the maximum V g in the operating
range to insure saturated operation and then multiplying
that value by Id• Now this procedure is simple when the
complete saturated characteristics have been measured,
but in that case it is obvious that the device is not dis
sipation limited. What we really want to know is the
ultimate limitation on the TFT's in the domain of our
concern. To do this we invoke a model relating the
curves of Fig. 1 (d) to the physical parameters of the
device, and then extend our choice of values for those
parameters to those which both maximize g", and are
compatible with the model. We have found that the
curve corresponding to saturated operation in Fig. led)
can be fitted quite well with the relation between field
excitation probability and electric field if a single valve
for trap depth, AE, is judiciously chosen from the actual
distribution of shallow traps. 7
The transconductance
gm= aId =Nl,oTep.(WT)(Vd) (7)
aVg L Vg
X[8?1' (2m*)! (D.E)!(T+d)]p(Vu),
3 he Vg
where the excitation probability
p(vu)=exp[-8?1' (2m*)! (AE)t(T+d)] ,
3 he Vg
and
NT=Shallow trap density (l/cm3),
OT= Occupancy of the shallow traps,
e= Electronic charge (coulombs),
,u=Mobility (V-sec/cm2),
W=Breadth of the semiconductor film (em),
t= Thickness of the semiconductor film (em),
L=Length of the semiconductor film (cm) from
source to drain,
m*= Electronic effective mass (g),
n= Planck's constant,
d= Thickness of the dielectric film between gate and
semiconductor (em).
It has been assumed that the device is not yet sutu
rated and the applied gate field is distributed uni
formly, normally across both the dielectric and semi
conductor films.
7 T. O. Poehler and D. Abraham, J. App!. Phys. 35, 2452 (1964) .
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] IP: 155.33.16.124 On: Sun, 30 Nov 2014 07:11:28HE A T FLO WAS A LIM I TIN G FA C TOR IN T HI N - F I L M DE V ICE S 2019
or The transconductance has its maximum where
a21 D/avg2=0,
v g= 47r/3(2m* /he) (.1E)!(T+d).
Then
gm maximum= 2NTOTe,u(Wt/L) (Vd/Vg) exp( -2).
The maximum value of Vd for which operation is not
quite saturated is V d= Vo' Then, by choosing these
compatible values for the constants,
NT=SXlO20, OT=1O-2, ,u=1O, W=lXlO-I,
t= 2X 10-5, L= 10-3, and .1E= 0.04 eV,
we estimate that gm maximum per mm of breadth is of
the order of SX 103 /oImhos. Values of gm maximum
for devices 200 mil wide (a typical dimension for devices
reported by Weimer) would then be 2.SX 1()4 ,umhos.
For the dimensions chosen this would correspond to a
gain-bandwidth product of approximately 100 Me/sec.
Some improvement might be realizable but it almost
certainly would be less than an order of magnitude.
The coefficient of P(V g) in the relation for gm given
above varies directly as the number of occupied states
represented as being at a depth .1E below the conduc
tion band. Let us now determine the upper limit on
TFT operation imposed by thermal dissipation con
siderations. The Fermi level in the device described
lies at about 0.2 eV below the conduction band. In this
way, both the density of states and the temperature
combine to give an equilibrium value for gm, while the
occupancy can be enhanced by such phenomena as
photo excitation from the valence band into the states.
As a direct consequence, an operating temperature
above 200°C will generate sufficient current to further
raise the device temperature to the point at which the
modulation mechanism will cease to operate.
At this maximum operating temperature with device
dimensions of 1 mmX 10 /01 the thermal computations
show T= 0.002 U /K, or for K = 0.002 cal/cm2-sec/Co we
have T= U. Expressing U in W /cm2-sec, T= U /4.17 so
that for a maximum temperature of 200°C the maximum
device power density would be approximately 1000
W /cm2• The total power for the specified dimensions
would be 100 mW.
Going back to the model of device operation, the drain
current-voltage product corresponding to the operating
point for maximum gm and the values chosen for the
other parameters would be approximately 70 mW.
Thus, we find that the heat generated within the device
at its optimum operating point is less than that which
would lead to unstable operation.
The analysis performed above was based on a model
derived from the physical properties of the many CdSe
TFT's constructed in our laboratory. A similar analysis
can be applied to those TFT's which may operate ac
cording to the theory proposed by Weimer; the charac-teristic is represented by Fig. l(c). In that case the
drain current will not saturate with gate voltage and
the transconductance will rise until failure occurs with
increasing V g' According to this model the maximum
figure of merit M will be
M=,uVg/27rL2= 1.6X107Vg,
for L= 10-3 em and /01= 100 cm2/V-sec. This maximum
figure of merit is obtained when the gate voltage V g is
equal to the source-drain voltage V D. The power to be
dissipated will be
P= V DID= VgI D,
where ID=,uCV g2/2L2. The capacitance C is the gate
capacitance of the device which for W = 10-1 em,
L= 10-3 em, d= 10-5 em, and E= 4X 10-11 F /m will be
4X 10-12 F. Then the power will be given by
P= ,uCV N2L2= 2X 10-4 V g3 W.
The maximum values of V g and P will be determined
by the maximum allowed temperature. This tempera
ture will be limited either by the bandgap or the semi
conductor, or by that temperature at which any of the
materials comprising the device fail to perform their
functions. For intrinsic CdSe at a temperature of ap
proximately 700°C, sufficient thermal excitation of elec
trons from the valence band to conduction band will
take place to cause thermal runaway. However, experi
ence with thin-film dielectrics suggests that even 500°C
might be regarded as an optimistic limiting value for the
gate dielectric. The temperature will be determined by
the power dissipation P as
T m= U/4.17=P/4.17A=P/4.17LW.
Using the expression for power dissipation developed
in the previous paragraph we find
Tm=2XlO-4VN4.17LW.
If we insert the dimensions used earlier and a limiting
temperature of 500°C we find the maximum allowed
power dissipation to be 0.330 Wand maximum V g to
be 11.2 V. This leads to a maximum figure of merit M
of 1.8X 108 cps.
The resulting calculated value of M maX for the Weimer
model TFT is somewhat higher than it should be if one
considered that the resistivity of the semiconductor is
decreased much less near the drain than near the source.
This would in turn lead to thermal dissipation in a
narrower region than the gap dimension.
A field excitation TFT, on the other hand, can be
constructed with the transverse field applied between
the two field plates in a circuit isolated from the source
and drain,S and the semiconductor resistivity is modu
lated uniformly over its entire length. No correction in
M max need be applied in this case.
8D. Abraham and T. O. Poehler (to be published).
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IP: 155.33.16.124 On: Sun, 30 Nov 2014 07:11:282020 D. ABRAHAM AND T. O. POEHLER
CONCLUSIONS
The preceding discussion has once again clearly
demonstrated the importance of considering the power
dissipation in relation to those physical mechanisms
upon which device operation is postulated. The authors
wish to emphasize that a similar analysis can be applied
to any thin-film device, passive or active, for which a
limiting temperature of operation can be defined.
The results presented have been based on a uniform,
constant input of power. When a dev:ice is to be used
for small signal operation, the average power dissipation
can be used for purposes of calculation. However, when
the percent modulation is large the instantaneous
temperature can deviate from the average by a sub
stantial fraction because we are dealing with very small
thennal masses with resulting small thermal time con
stants. Unless such time constants are much larger than
the signal risetime, it would be prudent to compute
the thennal limits on operation on the basis of peak,
rather than average power.
JOURNAL OF APPLIED PHYSICS Two calculations based on independent models of
TFT operation have yielded upper limits on device
performance that are remarkably close to each other
and point to M max = 108 cps for the geometry chosen
when CdSe is the semiconductor and the substrate is
glass. Changes of substrate material to those with
higher thennal conductivity can increase that limit.
Hot electron triodes would benefit in the same way from
an increase in the thermal conductivity of the substrate.
In support of the validity of the particular device
calculations performed, we point to the experimental
results appearing in the literature; hot-electron devices
have suffered thennal degradation and failure9 with
small figures of merit reported, whereas the TFT has
been operated with values of M as high as 75 Mc/sec
and failure usually resulting from breakdown of the
gate insulator rather than excessive heating due to drain
current.
9 H. Kanter and W. A. Feibelman, J. Appl. Phys. 33, 3580
(1962). '
VOLUME 36, NUMBER 6 JUNE 1965
Velocity Dependence during the Stick-Slip Process in the Surface
Friction of Fibrous Polymers
W. JAMES LYONS AND STANLEY C. SCHEIER*
Textile Research Institute, Princeton, New Jersey
(Received 16 November 1964)
. Examination of chart tracings representing frictional force during the slip phase of the stick-slip process,
m. fibers of two polymer types, revealed that these tracings do not have the sinusoidal character associated
With a cons~ant .coefficient of kinetic friction. Considerations of the dynamics of the system of fiber surface
and ~easurmg ~nstrument l;d to the suggestion that a velocity-dependent term be introduced into the
equ~tlOn of motion o~ the shder..The new force repre~ented by this term was assumed to be directly pro
portIOnal to the velOCity of the slider. Curves for the displacement of the slider as a function of time based
on solutions. of the modified equatio~ of m?tion, for both the underdamped and overdamped case~, were
found to be m excellent agreement With tYPical examples of the experimentally observed curves for the two
fiber types. An expression is obtained for the coefficient of friction, which predicts the well-known experi
mental fact that, for most materials, the kinetic coefficient is less than the static.
INTRODUCTION
IN the measure~ent of the s~face fricti?n of textile
fibers under situable condItIOns, the mtermittent
~otion characteristic of the "stick-slip" phenomenon
IS observed,! as it is with many other materials. A record
is usually obtained for the displacement S of a slider
as a function of time, S being taken as a measure of
frictional force, through a calibration. An idealized
version of a portion of a typical chart tracing obtained
for S is shown in Fig. 1 (top). In the bottom illustration
is a model of the mechanical measuring system (which
* P~esent address: The Kendall Company, Charlotte, North
Carohna.
I H. G. Howell, K. W. Mieszkis, and D. Tabor Friction in
Textiles (Interscience Publishers, Inc. New York 'and Butter
worths Scientific Publications Ltd., Lo~don, 1959).' may actually be of many different designs), showing the
positions So, SB, and S8 of the slider corresponding to
points on the tracing. The segment SoSB of the tracing
represents the "stick" phase of the stick-slip cycle.
At SB slipping starts, at time to, and continues until
the displacement of the slider is reduced to S8 at time
t1• The time interval t1-to is exaggerated in Fig. 1; as
observed w~th chart speeds normally used, the portion
of the tracmg for the slipping would appear in this
figure as a nearly vertical straight line.
The occurrence of the stick-slip process reflects the
definite difference between static and kinetic friction
of a material. As Bowden and Tabor2 point out, static
2 F. P. Bowden and D. Tabor, The Friction and Lubrication of
Solids (Oxford University Press, London, 1950), pp. 106ff.
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] IP: 155.33.16.124 On: Sun, 30 Nov 2014 07:11:28 |
1.1695779.pdf | Unrestricted Hartree—Fock Calculations. II. Spin Properties of PiElectron Radicals
Lawrence C. Snyder and Terry Amos
Citation: The Journal of Chemical Physics 42, 3670 (1965); doi: 10.1063/1.1695779
View online: http://dx.doi.org/10.1063/1.1695779
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:06THE JOURNAL OF CHEMICAL PHYSICS VOLUME 42, NUMBER 10 15 MAY 1965
Unrestricted Hartree-Fock Calculations. II. Spin Properties of Pi-Electron Radicals
LAWRENCE c. SNYDER
Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey
AND
TERRY AMos
Department of Mathematics, University of Nottingham, Nottingham, England
(Received 9 November 1964)
Unrestricted Hartree-Fock wavefunctions have been computed for a large number of conjugated-hydro
carbon pi-electron radicals. Pi-electron spin-density and charge-density functions have been computed
with and without annihilation of the major contaminating spin multiplet in the wavefunction.
Three different empirical relations have been used to relate our results to proton isotropic hyperfine
splittings measured in ESR experiments. These are the simple McConnell expression; the Colpa and Bolton
relation which includes a term depending on the excess charge density at the carbon atom; and that due to
Giacometti, N ordia, and Pavan which introduces a contribution from the spin densities in the bonds between
the carbon and its nearest neighbors. When the Pariser-Parr method with semiempirical electron-repulsion
integrals is employed in the unrestricted Hartree-Fock calculation, then the experimental splittings are
almost always bounded by those computed with spin densities before and after the annihilation of the major
contaminating spin state, but spin densities after annihilation are closer to the experimental values derived
with the empirical relations. This was found to be true for all the three relations used, but the simplest one
gave slightly less satisfactory results.
The general problem of the interpretation of spin densities obtained from an unrestricted Hartree-Fock
wavefunction which is not an eigenstate of spin is discussed. It is found, however, that unrestricted compu
tations with semiempirical integrals give small splittings of the underlying "closed"shell orbitals, so that a
single annihilation is a good approximation to projection and the spin densities after annihilation should be
satisfactory from a theoretical point of view.
INTRODUCTION
AN improved method of computing spin properties
1"i.. from unrestricted Hartree-Fock (uhf) wavefunc
tions was described in the first paper of this series.1
There we derived convenient formulas for computing
spin and charge density functions and the expectation
value of 82, after proper annihilation of the major
contaminating higher spin component from a uhf single
determinant. We give here an account of extensive
applications to compute spin densities and proton hy
perfine splittings in conjugated-hydrocarbon pi-electron
radicals, a limited report of which was given in a recent
communication.2
The radicals for which we have constructed un
restricted Hartree-Fock wavefunctions and report here
are all doublet pi-electron radicals. They are illustrated
in Fig. 1, which includes odd-alternant (a), even-alter
nant (b), and nonalternant (c) conjugated hydrocar
bons. With the exception of triphenylmethyl, they may
reasonably be assumed to be planar, as we have assumed
for all.
The objective of this study is to show the quali
tative properties of unrestricted Hartree-Fock wave
functions for such radicals. In particular, we investi
gated whether the semiempirical integral scheme of
Pariser and Parr, which has been so successful in cor
relating and predicting the optical spectra of the parent
aromatic hydrocarbons, predicts qualitatively correct
1 A. T. Amos and L. C. Snyder, J. Chern. Phys. 41,1773 (1964).
2 L. C. Snyder and A. T. Amos, J. Am. Chern. Soc. 86, 1647
(1964). spin densities when incorporated into an unrestricted
Hartree-Fock calculation.
Secondly, we continue the discussions begun in the
first paper of the series on the effects introduced by
the presence of the unwanted spin components in the
unrestricted wavefunction. We therefore seek informa
tion on the amount and type of contaminating higher
multiplets, and thus on the degree to which an annihi
lation approximates a full projection. We wish to know
the nature and amount of splitting of underlying
"closed-shell" molecular orbitals, and the implications
of the amount of this splitting for qualitative discus
sions of properties computed from these functions. We
also seek to determine the reliability of such calcula
tions for the prediction of proton isotropic hyperfine
splittings in conjugated pi-electron radicals.
There are several basic ways to compute spin and
charge-density functions for radicals. These include the
restricted Hartree-Fock method,3-5 which keeps elec
trons paired in underlying orbitals; a restricted method
augmented with configuration interaction6•7; the va
lence bond procedure6•s; and the unrestricted Hartree
Fock method without or with annihilation or projec-
a C. C. J. Roothaan, Rev. Mod. Phys. 32, 179 (1960).
4 O. W. Adams and P. G. Lykos, J. Chern. Phys. 34, 1444
(1961).
6 J. R. Hoyland and L. Goodman, J. Chern. Phys. 34, 1446
(1961).
6 H. M. McConnell, J. Chern. Phys. 28, 1188 (1958); H. M.
McConnell and D. B. Chestnut, ibid., p. 107.
7 G. J. Hoijtink, Mol. Phys. 1, 157 (1958); G. J. Hoijtink, J.
Townsend, and S. I. Weissman, J. Chern. Phys. 34, 507 (1961).
8 P. Brovetto and S. Ferroni, Nuovo Cimento 5,142 (1957).
3670
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:0617
III UNRESTRICTED HARTREE-FOCK CALCULATIONS. II 3671
ALLYL
1""-../3
2
BENZYL
4~
!!> II
TRIPHENYLMETHYL
4
I!!> 11 10
(a) PENTADIENYL
3
I~!!>
2 4
PERINAPHTHENYL
2 6)3
II 10 13 11 4
8 12 !!>
7 8
FIG. 1. Conjugated hydrocarbons studied. TRANS BUTADIENE
1~4
NAPHTHALENE 7C08
I 2
II 0 3
5 4
NAPHTHACENE
10 11 12 I CIS BUTADIENE
ANTHRACENE
891 7~2
6~3
!) 10 4
PHENANTHRENE
9 10
1I~2
1I~3 7 2
7 8 !!> 4
PYRENE
7 2
PERYLENE
3 4
!!>
II
12
11
10 II
AZULENE
<4 !!> 200· 187
ACENAPHTHALENE
4Q9!!> 9 II 7
3 0 IS
12 11
2 1 (b) BIPHENYL
II 8 2 3 loOD4
11 12 6 !)
BIPHENYLENE
8 1 7~2
1I~3
!!> <4
DIBENZOBIPHENYLENE
10 11 12 1 9~2 8~3
7 II !!> <4
FLUORANTHENE
3 4
2
(c)
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:063672 L. C. SNYDER AND T. AMOS
tion.l.2.9.10 Each method has advantages in particular
situations. We think, however, that the unrestricted
Hartree-Fock calculation is probably most convenient
for large conjugated-hydrocarbon pi-electron radicals,
and that annihilation is necessary and more convenient
than projection in these cases.
BASIC THEORY
The unrestricted Hartree-Fock method is based on
a single-determinant wavefunction in which the or
bitals occupied by electrons with a spin may be differ
ent from those occupied by electrons with (3 spin:
'¥uhf= [(p+q) !]-l det{lfl(l)a(l)'"
Xlfp(p)a(p)epl(p+l)(3(p+1)" .epq(p+q)(3(p+q) I·
(1)
The p orbitals {If;} occupied by electrons with a spin
can be taken as orthonormal among themselves as can
the q orbitals {ep,} occupied by electrons and with (3
spin. Without loss of generality we assume p> q and
for the radicals considered here p = q+ 1.
Usually the orbitals in (1) are written as a linear
combination of a basis set of atomic orbitals {wr}. For
pi-electron radicals they will be of 2p. type localized
about the carbon atoms. In our derivations we take
the set {wr} to be orthonormal
(2)
with
Later it is convenient to use the unrestricted charge
and bond-order matrices for electrons of spin a and
(3, P and Q, respectively. These are defined by11:
Prs= tar/a.i
;-1 (3)
The first-order density matrix corresponding to the
wavefunction '¥"hf can be written in terms of the ma
trices P and Q and the basis set of orbitals {wr} and
from this, following McWeeny and Mizuno,t2 we can
obtain the spinless density matrix
E(p+Q) .. Wr *(l')w.(l)
and the spin-density matrix
E(p-Q),.w r *(1')w.(1). (4)
(5)
In the case of pi-electron systems, with the approxima
tion of zero differential overlap for the {wr}, we can
DH. M. McConnell, J. Chern. Phys. 29, 244 (1958).
10 A. T. Amos, Mol. Phys. 5, 91 (1962).
11 A. T. Amos and G. G. Hall, Proc. Roy. Soc. (London) A263,
483 (1961).
12 R. McWeeny and Y. Mizuno, Proc. Roy. Soc. (London)
A295, 554 (1960). interpret the diagonal elements of the matrices (P+Q)
and (P-Q) as the pi-electron charge and spin densi
ties at the appropriate carbon atoms, i.e.,
q.dr= Prr+Qrr (6a)
is the charge density at the carbon atom rand
(6b)
is the pi-electron spin density at the same atom. Simi
larly the off-diagonal elements of these matrices will
be the charge and spin densities in the CC bonds.
Unfortunately as was elaborated in Paper I, for a
doublet with p=q+l, '¥uhf is not a pure spin state
but the sum of spin multiplets
q
'¥uhf= ECHm'¥Hm (7)
m=O
of which the lowest '¥! is the largest and the one of
interest to us. Lowdin's projection operator technique13
can be used to select '¥~ and '¥ uhf but this is rather
difficult to do. As a compromise, the major unwanted
component '¥t can be removed from '¥uhf using the
annihilator
(8)
since the remaining contaminating components '¥h
'¥7/2, "', have little effect.
In earlier work by Amos and Hall,11 when such a
procedure was first suggested, alternative formulas
were derived for the first-order density matrix from
which we can deduce
(9a)
(9b)
where the matrices Rand S are also defined in terms
of P and Q.11 As we pointed out in Paper I the results
given by (9b) are only approximately correct, since
the derivation assumed af to be idempotent.
In Paper I we derived improved expressions for the
charge and spin densities after annihilation; these are
qaar= Jrr+Krr
Paar= Jrr-Krr, (lOa)
(lOb)
where formulas for the matrices J and K were given
in Paper I in terms of P and Q. We also give in Paper I
a discussion of the generalization to a nonorthogonal
basis set. We use all three formulas (6b), (9b), (lOb)
in order to obtain spin densities to correlate with pro
ton isotropic hyperfine splittings.
METHOD OF CALCULATION AND INTEGRAL
VALUES
To calculate the wavefunction '¥ ... hf we seek those
orbitals {If;} and {ep;} in the form of Eqs. (2) which
minimize the total energy, E, where the total energy
13 P.-O. L6wdin, Phys. Rev. 97,1474 (1955).
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:06UNRESTRICTED HARTREE-FOCK CALCULATIONS. II 3673
of a radical is taken in this study to be the sum of a
pi-electron energy (E7I') plus a term representing repul
sion of core charges. We define the core charge of an
atom equal to the number of pi-electrons donated by
the atom and it is taken to be at the corresponding
nucleus;
E=E7I'+ LZrZ.(K/Rrs). (11)
r<.
Here we take energies in electron volts and lengths in
angstroms; thus k= 14.395 eV/ A. The expression for
the pi-electron energy is
where the matrices H, Fa, FP may be written in terms
of integrals over the basis set of atomic orbitals (see
Amos and Hall,ll Berthier,14 andiPople and Nesbet14).
The coefficients {ard and {brd of the orbitals which
minimize (12) satisfy equations of eigenvalue form:
LF"aa'i= Eiaari
(13)
and this provides a convenient method for calculating
the orbitals using a procedure similar to the Roothaan
technique for the ordinary SCF equations.
First of all, however, we have used the assumptions
of the Pariser and Parr1• approximation to simplify
the formulas for H, Fa, FP. Thus the definition of Hrr:
( 14)
has been simplified to
(15)
where the 'Yrs are electron repulsion integrals. Similarly
we assume tration or kinetic energy integrals between carbon
atoms bonded to differing numbers of hydrogen atoms.
We have taken for neighboring atoms rand s, H,,=
f3 .. = -2.39 eVj for nonneighbors we have assumed
H.,=O. The semiempirical one-center Coulomb repul
sion integral 'Ycc has been given the value 11.0 eV,
while for nearest neighbors C and C' we have 'Ycc,=7.1
eV. Otherwise we have adopted a classical electrostatic
model, representing the charge distribution of an elec
tron in a pi orbital by ! unit charge 0.82 A above and
below the C nucleus, to find the remaining electron
repulsion integrals.
For all the molecules we have studied, we have
taken all CC bond lengths to be 1.40 A and all bond
angles about carbon to be 120°, except in azulene for
which symmetrical five-and seven-member rings are
assumed. Although we have made these assumptions
and those made in regard to the values of the integrals
we think that the major features we are most interested
in, namely spin and charge distributions, should be
given correctly. We comment on cases where a relaxa
tion of these assumptions may lead to somewhat dif
ferent results as they arise.
Our numerical procedure for solving Eq. (13) has
been to construct an initial P and Q matrix from
HUckel molecular orbitals for the aromatic pi-system
and to use these in Eq. (17) to obtain the correspond
ing Fa and FP. These are diagonalized and the new
orbitals corresponding to the lowest p eigenvalues Eia
of Fa used to construct a new P matrix, and with
those corresponding to the lowest q eigenvalues of FP
a new Q matrix. The process is then repeated and in
the calculations reported here 15 iterations were made
for each system. This was sufficient for the diagonal
elements of P-Q (atom spin densities) to have con
verged to two significant figures and similarly for (S2).
The energies converged to six significant figures after
15 iterations.
RELATION BETWEEN ISOTROPIC PROTON HY
PERFINE SPLITTINGS AND PI-ELECTRON SPIN
r~s, (16) AND CHARGE DENSITIES
while the F matrices become
Fr.a= Hr.-Prs'Yrs+Or.L (Ptt+Qtt)'Yr" ,
Fr.P=H,,-Qr8'Yr.+Or.L(Ptt+Qllhr,. (17)
t
The values used for the integrals were semiempirical
ones like those suggested by Pariser and Parr,ts and
which would give a qualitatively correct prediction of
the optical spectra of aromatic molecules. We have
taken Ucc=O and thus make no distinction by pene-
14 G. Berthier, J. Chim. Phys. 51, 363 (1954), 52, 141 (1955);
J. Pople and R. Nesbet, J. Chern. Phys. 22, 571 (1954).
15 R. Pariser and R. G. Parr, J. Chern. Phys. 21, 466; 767
(1953) . The origin of the isotropic hyperfine splittings by
protons in the ESR spectra of organic radicals is the
Fermi contact interaction as suggested early by Weiss
man.Is This splitting by a proton is proportional to the
electron spin density at the proton.
It was early noticed by McConnell, Weissman, and
Bersohn17 that the isotropic hyperfine splittings by pro
tons in doublet pi-electron hydrocarbon radicals should
be approximately proportional to the pi-electron spin
density on the carbon atom to which the proton is
bonded:
ar= _Qpr. (18)
16 S. 1. Weissman, J. Chern. Phys. 22, 1378 (1954).
17 H. M. McConnell, J. Chern. Phys. 24, 764 (1956); S. 1.
Weissman, ibid. 25, 890 (1956); R. Bersohn, ibid. 24,1066 (1956).
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:063674 L. C. SNYDER AND T. AMOS
Here Q is a constant, about 25 G per unit atomic spin
density. Using this relation, the observed splittings
were shown to be correlated fairly well with simple
Huckel spin densities in a large number of radicals. ls
McConnell and Chestnut6 have employed perturba
tion theory to give a detailed discussion of the relation
of the spin density at a proton to elements of the
pi-electron spin density matrix of the organic fragment
to which it is bonded. McConnell6 concluded that the
isotropic hyperfine splitting by a proton is proportional
to the diagonal element of the pi-electron spin-density
matrix at the carbon to which it is bonded, where a
nonorthogonal pi basis set was assumed. This is the
basic theoretical justification for empirical relations
which take the proton hyperfine splitting proportional
to this diagonal matrix element, even with an orthogo
nal basis as is assumed in the calculations reported
here and which use the zero differential overlap
approximation. McConnell also estimated that off
diagonal elements of the spin-density matrix would
make a much smaller contribution to the splitting.
As is well known, the simple Huckel method pre
dicts the spin densities in the positive and negative
ions of an altern ant hydrocarbon to be identical. Mc
Lachlanl9 has shown that a similar result holds to
the Pariser-Parr-Pople level of approximation for un
restricted Hartree-Fock wavefunctions. However, it
was pointed out by Carrington and co-workers20 that
the observed proton splittings in aromatic positive
ions, e.g., anthracene, are generally greater than in
the corresponding negative ions. Clearly the simple
relation (18) cannot account for this fact and recently
more complicated relations have been introduced to
do so (for a review see Ref. 21).
For instance Colpa and Bolton22 showed that a semi
empirical relation which assumes that Q depends line
arly upon the pi-electron atomic charge density gives
a much improved correlation of proton splittings with
Huckel spin densities. They found the expression
(19)
where Er= (qr-1), accounts nicely for the difference
between positive and negative ions with the param
eters chosen to be A = -31.2 and C = 17 to give the
best correlation with splittings. The value of C has
been rationalized by Bolton in terms of a dependence
of the Slater 2p. orbital exponent upon E. He reports
that the original derivation of C by Colpa and Bolton
was incorrect.22
The dependence of ar on pr and Er has also been
18 E. DeBoer and S. I. Weissman, J. Am. Chern. Soc. 80, 4549
(1958).
19 A. D. McLachlan, Mol. Phys. 2, 271 (1959).
20 A. Carrington, F. Dravnieks, and M. C. R. Symons, J.
Chern. Soc. 1959,947.
21 G. C. Hall and A. T. Amos, Advan. Atomic Mol. Phys.
(to be published).
22 J. P. Colpa and J. R. Bolton, Mol. Phys. 6, 273 (1963); and
private communication by J. R. Bolton. studied theoretically by Higuchi23 who followed the
usual derivation of Eq. (18) but allowed for the effects
of an excess charge on the carbon atom upon the
polarity of the C-H bond. He found, contrary to Colpa
and Bolton, that the splittings should be larger in the
negative ion so that if Eq. (19) is used C should be
negative. Nevertheless, at the present time it is our
opinion that an empirical relationship of the form of
Eq. (19) is very satisfactory for the comparison of
our computed atomic spin and charge densities with
observed splittings provided C is taken to be positive
which, in spite of the rather persuasive argument of
Higuchi, we take it to be.
An alternative expression which can explain the dif
ferences between hyperfine splittings of the positive
and negative ions of alternants equally as well as Eq.
(19) is due to Giacometti, Nordio, and Pavan.24 Their
relationship is
(20)
where pnnr is the sum of the spin densities in the bonds
linking the carbon r to its nearest-neighbor carbons.
For Huckel spin densities, good agreement with experi
ment is found with B= -31.5 and D= -7.0; which
have the orders of magnitude of their theoretical esti
mates. The expression (20) corresponds to the off
diagonal elements of the pi-electron spin density ma
trix between the pi orbital on the carbon to which the
proton is bonded and the pi orbitals on nearest-neighbor
carbons being more important contributors to the split
ting than estimated by McConnell.
In our study we take, for the unrestricted wave
function after annihilation,
(21)
Here s is summed over nearest-neighbor carbon atoms
only. We note that the consistent application of zero
differential overlap would make the off-diagonal ele
ment of the pi-electron spin density matrix effectively
zero.
The most general treatment of this problem by
McConnellS suggests that there should also be a term
in the formula for a of the form Q' p' where p' is the
sum of the spin densities at the carbon atoms which
are neighbors to the one attached to the proton. Such
a term has to be included in the case of l3C splittings
but it has always been neglected for proton splittings
since Q' is presumably rather small. On the other hand
in many cases p' is an order of magnitude greater than
pr so that it may not always be possible to neglect the
term Q' p'. Until the situation is clarified by further
investigation of U-'1r interactions in pi-electron radicals
we think it best to present our results using the three
expressions (18), (19), and (20) since these seem well
established and rather easy to work with.
23 J. Higuchi, J. Chern. Phys. 39, 34455 (1963).
24 G. Giacometti, C. L. Nordio, and M. V. Pavan, Theoret.
Chern. Acta 1, 404 (1963).
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:06UNRESTRICTED HARTREE-FOCK CALCULATIONS. II 3675
TABLE 1. Comparison of observed and theoretical proton splittings for rnonocyclic radicals."
Radical aesptl
Cyclopentadienyl -5.60b
Benzene- -3.750
Cycloheptatrienyl -3.91d
Cyclo-octatetraene- -3.21"
a The quantities subscripted sym are computed on the assumption that all
carbon atoms have equal charge and spin density.
b S. Ohnishi and I. Nitta, J. Chern. Phys. 39, 2848 (1963); P. S. Zandstra
reported a=5.98, ibid. 40, 612 (1964).
Considering only the three relations (18), (19), and
(20) therefore, we note that in the case of odd alternant
radicals pnnr=er=O for all r, so that to be consistent
we must have -Q=A=B. Bolton and Fraenke}25 re
cently studied both the 13C and proton splittings in
the ESR spectrum of anthracene positive and nega
tive ions. By making use of the theory of Karplus and
Fraenkel26 for 13C splittings they were able to show
that Q = 27 was most consistent with their observa
tions. However from recent studies by Fessenden and
Schuler,27 it appears that Q depends on the number of
carbon atoms to which the central carbon is bonded.
Thus, although Q= 27 for conjugated hydrocarbons
having two carbons bonded to the central Sp2 carbon,
we adopt their conclusion that Q = 24.4 if only one
carbon is bonded to the central Sp2 carbon as is the
case at C1 of allyl, C7 of benzyl, C1 and C4 of butadiene
and C1 of pentadienyl radicals. Although methyl radi
cal is not part of this study we would take Q = 23.04
for it since, in methyl radical, no carbons are bonded
to the central sp carbon.
In their work on the ions of anthracene, Bolton and
Fraenke}25 also concluded that the spin density is very
probably the same in the positive and negative ions
and they computed the spin density at the 9 position
to be 0.22. We have used qaa9 computed for this posi
tion to calibrate C of Eq. (19). Since eaa9=0.212 and
a9=6.65 in the positive ion and a9=5.41 in the nega
tive ion, we choose C = -12.8. In the same way since
Paann9= -0.0986 we choose D of Eq. (20) to be D=
-6.3.
Therefore the three relations we use are
aCBT= -27pT-12.8eTpT, (22a)
(22b)
(22c)
For several cyclic radicals the average values of pT and
er are determined by symmetry. For these radicals the
25 J. R. Bolton and G. K. Fraenkel, J. Chern. Phys. 40, 3307
(1964).
26 M. Karplus and G. K. Fraenkel, J. Chern. Phys. 35, 1312
(1961).
27 R. W. Fessenden and R. H. Schuler, J. Chern. Phys. 39,
2147 (1963). aSymM aSym CB Psym Esym
-5.40 -5.40 0.200 0
-4.50 -4.15 0.166 0.166
-3.86 -3.86 0.143 0
-3.38 -3.18 0.125 0.125
CT. R. Tuttle, Jr., and S. I. Weissman, J. Am. Chern. Soc. 80,5342 (1958).
d S. Arai, S. Shida, K. Vomazaki, and Z. Kuri, J. Chern. Phys. 37, 1885 (1962);
A. Carrington and I. C. P. Smith, Mol. Phys. 7,99 (1963).
e T. J. Katz and H. L. Stevens, J. Chern. Phys. 32, 1873 (1960).
experimental aT and those computed using (22a) and
(22b) are given in Table I. Except for benzene nega
tive ion, on which we comment later, the agreement
is very good. For benzene- and cyclo-octatetraene
where aMT and aCBT are different, the Colpa and Bolton
relation gives better results.
SUMMARY AND COMPARISON OF COMPUTED
AND EXPERIMENTAL PROPERTIES
We have computed unrestricted Hartree-Fock wave
functions for doublet radicals of the conjugated sys
tems shown in Fig. 1. Where proton isotropic hyper
fine splittings are known for both the positive and
negative ions we have compared calculated and ex
perimental splittings for both ions. The comparison
with experiment takes two basic forms. We use the
empirical relations (22a), (22b) , (22c) with theoreti
cal spin and charge densities after annihilation to com
pute "theoretical" proton splittings. The sign of the
experimental splittings is generally unknown but we
take it to be that computed. In addition, we have
used the semiempirical Eq. (22) with the experimental
values for aT and where necessary the charge density
qaaT and the nearest-neighbor bond spin density Paannr
to compute an "experimental" atom spin density. In
Tables II, III, and IV, we compare these experimental
splittings and spin densities with the theoretical val
ues calculated for the single determinant and the two
methods of annihilation. For the comparisons, the con
jugated hydrocarbons studied are divided into the types
important in grouping their spectra: even-alternant,
odd-alternant, and nonalternant.
In Table II theoretical and experimental splittings
and spin densities are compared for odd-alternant radi
cals. As we have already explained, in these radicals
pnnr=er=0, and the three relations (22) are equiva
lent. Generally the observed splittings fall between
those computed with PaaT and PasaT and as a rule closer
to those computed with PaaT• The unannihilated single
determinant appears to give too large a separation of
positive and negative spin density.
Note that for the benzyl radical the computed para
(4) proton splitting is less than the ortho (2) splitting
in contrast to the opposite experimental result. A simi-
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:063676 L. C. SNYDER AND T. AMOS
TABLE II. Comparison of experimental splittings and spin densities with those computed using semiempirical repulsion
integrals for odd-alternate radicals.
Radical Atom ae:l:ptl aaaM aa8aM a.dM pM P •• Pa8a P.d Ph P"A
Allyl
1 -14.38- -13.35 -14.74 -15.88 0.589 0.547 0.604 0.651 0.500 0.594
2 +4.06 +2.52 +5.61 +8.16 -0.150 -0.093 -0.208 -0.302 0 -0.187
Pentadienyl
1 -8.99b -9.34 -11.49 -13.30 0.333 0.383 0.471 0.545 0.333
2 +2.65 +2.55 +5.69 +8.30 -0.098 -0.094 -0.211 -0.307 0
3 -13.40 -11.39 -12.95 -14.15 0.496 0.422 0.480 0.524 0.333
Benzyl
1 -0.060 -0.127 -0.189 0
2 -5.100 -4.23 -5.54 -6.87 0.189 0.157 0.205 0.254 0.143 0.161
3 +1.60 +1.35 +2.84 +4.26 -0.059 -0.050 -0.105 -0.158 0 -0.063
4 -6.30 -3.44 -4.73 -6.06 0.233 0.128 0.175 0.225 0.143 0.137
7 -16.40 -17.51 -18.35 -18.81 0.672 0.718 0.752 0.771 0.571 0.770
Perinaphthanyl
1 -7.30d -5.91 -7.48 -9.03 0.270 0.219 0.277 0.334 0.167 0.226
2 +2.20 +1.74 +3.61 +5.50 -0.081 -0.064 -0.133 -0.204 0 -0.070
10 -0.054 -0.113 -0.172 0 -0.052
13 0.044 0.079 0.122 0 +0.006
Triphenylmethyl
1 -0.043 -0.089 -0.136 0 -0.045
2 -2.53- -2.81 -3.79 -4.83 0.093 0.104 0.140 0.178 0.077 0.114
3 +1.11 +0.96 +1.96 +2.99 -0.041 -0.036 -0.073 -0.111 0 -0.044
4 -2.77 -2.32 -3.28 -4.31 0.103 0.086 0.121 0.160 0.077 0.101
19
a See Ref. 27.
b These are splittings of cyclohexadienyl radical given in Ref. 27.
oW. T. Dixon and R. O. C. Norman, J. Chem. Soc. 1964, 4857.
lar incorrect prediction is made for the triphenylmethyl
radical. Configuration interaction methods also give
incorrect results for benzyl,28 A previous unrestricted
Hartree-Fock calculation29 gives results rather differ
ent from ours and in fact predicts the spin densities at
Positions 2 and 4 to be almost equal. This may be due
to differences in integral values used in the calculations,
or their transformation to orthogonalized atomic orbit
als. The valence bond method30 gives the correct rela
tive splittings of ortho and para protons although the
over-all spin distribution given by this method is in
very poor agreement with experiment. Also in Table II
are collected for odd-altern ant radicals, experimental
and SCF spin densities, and spin densities computed
by the simple Hiickel (Ph), and perturbed Hiickel (pph)
method reported by McLachlan.31 In general, uhf spin
densities after annihilation (Paa), are in much better
agreement with experiment than the simple HUckel,
which give only positive spin densities. The spin densi
ties computed by McLachlan are in equally good agree
ment with experiment. This is probably a result of
compensating errors, the use of a single SCF iteration
and the failure to project or annihilate.32 Similar com-
28 r. C. Smith (to be published).
29 J. Baudet and G. Berthier, J. Chern. Phys. 60, 1161 (1963).
so H. H. Dearman and H. M. McConnell, J. Chern. Phys. 33,
1877 (1960).
31 A. D. McLachlan, Mol. Phys. 3, 233 (1960).
:!2 Alternatively, it may be preferable to regard McLachlan's
method as a most convenient way to do approximate configuration
interaction calculations rather than approximate unrestricted
Hartree-Fock (cf. Ref. 21). 0.462 0.497 0.520 0.308 0.413
d P. B. Sogo, M. Nakazaki, and M. Calvin, J. Chern. Phys. 26, 1343 (1957).
• D. B. Chestnut and G. J. Sloan, J. Chern. Phys. 33, 637 (1960).
ments apply to the results summarized in Tables III
and IV.
In Table III we compare theoretical and experi
mental splittings and spin densities for even-alternant
radicals. We also list there the pnn required for the
empirical relation (22c). Since we have found that
there is generally good agreement between experimen
tal splittings and those calculated using Pal and that
in any event splittings calculated from p.,l are in poor
agreement we only give the values of aaaM, aaaOB, aaaGNP
computed from spin and charge density after annihila
tion. This gives some opportunity to compare the three
different expressions relating spin densities and hyper
fine splittings. Since the more complicated expressions
give a better description of the differences between the
positive and negative ions of anthracene, napthacene,
perylene, and biphenylene they must be adjUdged su
perior to the simpler McConnell relation. It is, how
ever, impossible to say that either of (22b) or (22c)
is preferable to the other. In this context it is a great
pity that the 2 position of all polyacenes is predicted
so poorly. Because the magnitude of the experimental
splitting at this position is larger in the negative than
the positive ions of anthracene and naphthacene in
contrast to what is found at the other positions. A
satisfactory theoretical explanation of this would be
an excellent test both of the calculated spin and charge
densities and the expression relating these to the ex
perimental splittings. It certainly seems that the cal
culated spin density at the 2 position is much too low,
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155.247.166.234 On: Sun, 23 Nov 2014 10:52:06TABLE III. Comparison of experimental splittings and spin densities with those computed using semiempirical repulsion integrals for even-alternant radicals.
Radical Atom aesptl 4GaM a CB aa aaaGNP pM pCB pGNP Paa Pa.a Pld qaa p"" Ph pph
Trans-butadiene-
1 -7.62" -9.49 -7.58 -7.96 0.312 0.391 0.368 0.389 0.424 0.457 -0.385 -0.243 0.362h
2 -2.79 -3.00 -2.84 -2.57 0.103 0.109 0.119 0.111 0.076 0.043 -0.115 -0.069 0.138
~
Cis-butadiene- Z 1 -7.62" -9.91 -8.07 -8.44 0.312 0.383 0.366 0.406 0.442 0.475 -0.354 -0.233 0.362 ::0 2 -2.79 -2.54 -2.36 -2.06 0.103 0.111 0.121 0.094 0.058 0.025 -0.146 -0.076 0.138 i:'1
Naphthalene-Ul
1 -4.90b -5.81 -5.29 -5.11 0.181 0.199 0.207 0.215 0.239 0.262 -0.184 -0.111 0.181 0.222i 8
2 -1.83 -1.30 -1.23 -1.02 0.068 0.071 0.078 0.048 0.037 0.026 -0.097 -0.044 0.069 0.047 ~
9 -0.024 -0.051 -0.076 +0.062 0 -0.037 ......
()
Anthracene'Fd 8
1 -2.76· -2.84 -2.72 -2.65 0.102 0.107 0.109 i:'1
-3.12· -2.84 -2.97 -3.03 0.116 0.111 0.109 0.105 0.122 0.138 =F0.089 =F0.029 0.096 0.118 t:t
2 -1.53 -0.76 -0.73 -0.65 0.057 0.059 0.061
~ -1.40 -0.76 -0.80 -0.87 0.052 0.050 0.048 0.028 0.021 0.014 =F0.088 =F0.018 0.048 0.032
9 -5.41 -7.02 -6.32 -6.40 0.200 0.223 0.223 ;..-
-6.65 -7.02 -7.74 -7.64 0.246 0.224 0.223 0.260 0.290 0.319 =FO.212 =F0.099 0.193 0.256 ~
11 -0.014 -0.038 -0.061 ±0.035 0.004 -0.028 8
NaphthaceneT ::0
t"1 1 -1.55· -1.49 -1.46 -1.33 0.057 0.059 0.058 t"1 -1. 72· -1.49 -1.53 -1.65 0.064 0.062 0.063 0.055 0.066 0.078 =F0.046 =F0.006 0.056 I 2 -1.15 -0.43 -0.42 -0.35 0.043 0.044 0.045 >:rj -1.06 -0.43 -0.45 -0.51 0.039 0.038 0.037 0.016 0.011 0.006 =F0.076 =F0.007 0.033 0 5 -4.25 -5.32 -4.91 -4.81 0.157 0.171 0.169 () -5.17 -5.32 -5.73 -5.83 0.191 0.178 0.179 0.197 0.225 0.252 =F0.163 =F0.053 0.147 ~ 13 -0.005 -0.023 -0.041 =F0.017 0.012
17 -0.028 -0.059 -0.088 ±0.056 ()
Phenanthrene-;..-
1 -3.60- -3.92 -3.70 -3.46 0.133 0.141 0.150 0.145 0.182 0.219 -0.113 -0.073 0.115 0.159 " 2 +0.72 +0.92 +0.89 +0.90 -0.027 -0.027 -0.028 -0.034 -0.070 -0.106 -0.307 +0.003 0.002 -0.042 ()
3 -2.88 -3.05 -2.85 -2.69 0.107 0.114 0.121 0.113 0.141 0.170 -0.133 -0.058 0.099 0.125 ~
4 -0.32 -0.70 -0.68 -0.75 0.012 0.012 0.010 0.026 0.008 -0.009 -0.058 +0.008 0.055 0.038 " 9 -4.32 -5.40 -4.96 -4.70 0.160 0.174 0.186 0.200 0.209 0.215 -0.178 -0.112 0.172 0.196 ;..-
11 0.012 -0.010 -0.033 +0.025 0.027 0.003 8
12 0.037 0.041 0.045 -0.007 0.030 0.021 ......
0
Pyrene- Z
1 -4.751 -4.70 -4.38 -4.67 0.176 0.189 0.177 0.174 0.213 0.252 -0.146 -0.005 0.136 0.187 Ul
2 +1.09 +1.13 +1.13 +1.11 -0.040 -0.041 -0.041 -0.042 -0.088 -0.132 -0.018 +0.003 0 -0.052
4 -2.08 -2.30 -2.19 -2.12 0.077 0.081 0.084 0.085 0.089 0.092 -0.099 -0.029 0.087 0.092 ......
11 0.001 0.001 0.001 +0.028 0 -0.012 ......
13 0.011 -0.009 -0.029 -0.010 0.027 0.002
Biphenyl-
1 0.134 0.122 0.108 -0.074 0.123 0.128
2 -2.75b -2.94 -2.85 -2.46 0.102 0.106 0.120 0.109 0.132 0.155 -0.079 -0.076 0.089 0.105
3 +0.45 +0.49 +0.48 +0.54 -0.017 -0.107 -0.015 -0.018 -0.054 -0.089 -0.046 -0.007 0.019 -0.023
4 -5.50 -4.97 -4.55 -4.31 0.204 0.222 0.228 0.184 0.223 0.261 -0.178 -0.105 0.159 0.208 w 0-
~
~
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Radical Atom aexpt) aaaM aaaCB aaaGNP pM pCB pGNP Paa Paaa P.d qaa
Perylene'f
-2.67 -2.57 1 -3.08· -2.46 0.114 0.119 0.122
-3.10· -2.67 -2.80 -2.88 0.115 0.110 0.107 0.099 0.121 0.144 =t=0.090
2 +0.46 +0.54 +0.53 +0.55 -0.017 -0.017 -0.017
+0.46 +0.54 +0.55 +0.53 -0.017 -0.017 -0.017 -0.020 -0.054 -0.088 =t=0.036
3 -3.53 -3.94 -3.72 -3.89 0.131 0.139 0.133
-4.10 -3.94 -4.18 -3.99 0.152 0.143 0.149 0.146 0.185 0.225 =t=0.123
13 -0.031 -0.064 -0.097 ±0.029
15 0.037 0.022 0.008 =t=0.030
16 0.007 0.013 0.019 ±0.029
Biphenylene'f
1 +0.21" +0.14 +0.13 +0.24 -0.008 -0.008 -0.004
+0.211 +0.14 +0.14 +0.04 -0.008 -0.008 -0.012 -0.005 -0.026 -0.046 =t=0.048
2 -2.86 -2.08 -1.95 -1.84 0.106 0.112 0.115
-3.69 -2.08 -2.19 -2.32 0.137 0.129 0.128 0.077 0.085 0.094 =t=0.122
9 0.178 0.190 0.202 =t=0.079
Dibenzobiphenylene-
1 -1.621 -2.03 -1.95 -1.87 0.060 0.062 0.066 0.075 0.091 0.108 -0.073
2 -0.93 -0.54 -0.53 -0.46 0.034 0.036 0.037 0.020 0.019 0.018 -0.065
5 -4.31 -4.83 -4.46 -4.32 0.160 0.173 0.179 0.179 0.217 0.254 -0.162
13 -0.012 -0.026 -0.041 +0.034
17 -0.012 -0.051 -0.088 +0.011
• D. H. Levy and R. J. Myers, J. Chern. Phys. 41, 1062 (1964). I See Ref. 7.
b See Ref. 20. g A. Carrington and J. Dos Santos-Veiga, Mol. Phys. 5, 285 (1962).
C See Ref. 22. h Simple Hiickel spin densities.
d Where pairs of numbers are given, the upper value is for the negative ion and the lower for the positive ion. i Perturbed Hiickel spin densities by McLachlan from Ref. 19.
e S. H. Glarum and L. C. Snyder, J. Chern. Phys. 36, 2989 (1962). p"n Ph
=t=0.034 0.083
=t=0.002 0.013
=t=0.008 0.108
0
0.046
0
=t=0.016 0.027
=t=0.038 0.087
0.136
-0.026 0.079
-0.013 0.034
-0.081 0.117
0.002
0.018 Pph
0.115
-0.031
0.145
0.003
0.034
-0.028 Vl 0-~ 00
t-I
(")
en
Z
><:
t:1
M
:;d
>
Z
t:1
>-3
>
~
0
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: 155.247.166.234 On: Sun, 23 Nov 2014 10:52:06TABLE IV. Comparison of experimental splittings and spin densities with those computed using semiempirical repulsion integrals for nonaltemant radicals.
Radical Atom aexptl aM aCB aGNP pM pCB pGNP P •• Pa'lJ P.d q •• pnn Ph Pph
Azulene-
1 +0.27- +0.03 +0.04 -0.01 -0.010 -0.011 -0.011 -0.001 -0.006 -0.011 -0.167 +0.005 0.0040 -0.027d
2 -3.95 -2.16 -2.00 -1.98 0.146 0.158 0.153 0.080 0.099 0.118 -0.151 -0.028 0.100 0.120
4 -6.22 -6.37 -6.29 -5.88 0.230 0.234 0.248 0.236 0.276 0.313 -0.030 -0.077 0.221 0.292
5 +1.34 +1.24 +1.21 +1.29 -0.050 -0.051 -0.048 -0.046 -0.115 -0.178 -0.060 -0.008 0.010 -0.081
6 -8.82 -9.61 -8.93 -8.72 0.326 0.352 0.359 0.356 0.397 0.434 -0.153 -0.141 0.261 0.368
9 0.093 0.097 0.099 -0.091 0.084 0.071
Acenaphthalene-
1 -2.70b -2.62 -2.42 -2.54 0.100 0.108 0.103 0.097 0.104 0.111 -0.156 -0.013 0.104 0.101
3 -5.40 -4.27 -4.11 -3.95 0.200 0.207 0.212 0.158 0.187 0.216 -0.073 -0.051 0.151 0.196
4 (0) +0.78 +0.77 +0.81 (0) (0) (0) -0.029 -0.077 -0.123 -0.075 -0.004 0.014 -0.041
5 -5.40 -6.37 -6.00 -5.99 0.200 0.212 0.214 0.236 0.277 0.317 -0.123 -0.060 0.178 0.245
9 -0.044 -0.092 -0.138 +0.032 0 -0.045
10 0.017 0.033 0.050 -0.016 0 -0.012
11 0.052 0.038 0.023 -0.079 0.053 0.027
Fluoranthene-
1 -3.90b -3.78 -3.65 -3.46 0.144 0.149 0.156 0.140 0.164 0.189 -0.070 -0.050 0.122 0.157
2 (0) +0.54 +0.52 +0.59 (0) (0) (0) -0.020 -0.061 -0.102 -0.069 -0.009 0.022 -0.023
3 -5.20 -6.16 -5.80 -5.75 0.193 0.205 0.208 0.228 0.266 0.304 -0.124 -0.066 0.163 0.227
7 (0) +0.08 +0.09 +0.13 (0) (0) (0) -0.003 -0.014 -0.024 -0.025 -0.008 0.015 0
8 -1.30 -0.78 -0.70 -0.70 0.048 0.050 0.051 0.029 0.035 0.041 -0.092 -0.013 0.040 0.037
11 -0.041 -0.087 -0.130 +0.036 0 -0.039
12 0.012 0.024 0.026 +0.011 0 -0.013
13 0.070 0.061 0.051 -0.105 0.077 0.063
14 0.070 0.080 0.089 -0.038 0.060 0.064
-I. Bernal, P. H. Rieger, and G. K. Fraenkel, J. Chem. Phys. 37, 1489 (1962). d The spin densities Pph are computed by perturbed Huckel method of McLachlan and are taken from Ref.
b See Ref. 18. 19.
C The spin densities Ph are computed by the simple Huckel method. c::::
Z
:;0
t":I
Ul
>-:l
:;0
....
(j
>-:l
t":I
t1
iI1
;.-
:;0
>-:l
:;0
t":I
t":I
I
"%j
0
(j
~
(j
;.-
t"'
(j
c::::
t"'
;.-
>-:l ....
0
Z
Ul
.... ....
W 0\
~ \0
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TABLE V. Dependence of expectation values of S' and energy upon annihilation in calculations with
semiernpirical repulsion integrals.
Radical (S'} ••
Allyl 0.75000
Pentadienyl 0.76762
Benzyl 0.75584
Perinaphthenyl 0.80997
Triphenylrnethyl 0.77157
trans-butadiene- 0.75000
cis-butadiene- 0.75000
Naphthalene- 0.75147
Anthracene- 0.75284
Naphthacene- 0.75450
Phenanthrene- 0.75416
Pyrene- 0.75844
Biphenyl- 0.75432
Perylene- 0.76563
Biphenylene- 0.75070
Dibenzobiphenelene- 0.76202
Azulene- 0.75418
Acenaphthalene- 0.75600
Fluoranthene- 0.75505
• t;.E=E ••• -E,d.
a fact which seems to be true for all calculations based
on the molecular orbital method as was pointed out
by Schug, Brown, and Karplus.33 They suggest that
to get a more satisfactory result it may be necessary
to use different values for Urr [d. Eq. (15) ] at carbon
atoms bonded to three other carbons as compared with
those bonded to two carbons and one hydrogen to
allow for the difference in penetration and kinetic en
ergy integrals. In the same way, a variation in the (3
integrals and the 'Y integrals to allow for different bond
lengths may be useful. It is notable that this 2 position
is the para position of a benzyl fragment, when the
aromatic molecule is subdivided as suggested by Dewar34
in his surprisingly good perturbation theory of the spec
tra of aromatic molecules. Thus this low theoretical
spin density for the 2 position may be related to the
low computed ratio of para to ortho spin densities for
benzyl and triphenylmethyl radicals. As in the case of
benzyl the valence bond method gives a larger spin
density at the Position 2 for naphthalene than does
the molecular orbital method in any of its forms.33
Apart from this position the agreement between the
ory and experiment is probably as good as can be
expected. However, the positions of largest spin densi
ties do tend to have Paa too large. The agreement with
experiment for butadiene- is surprising since it is not
a benzenoid system as are all the other even alternants.
The results for this ion provide rather weak evidence
that the trans conformation is more stable than the cis.
In Table IV, the results are given for nonalternant
radicals. There is generally fair agreement with split
tings computed from Paa for all three relations except
at the 2 proton of azulene-.
33 J. C. Schug, T. H. Brown, and M. Karplus, J. Chern. Phys.
35, 1873 (1961).
34 M. J. S. Dewar, J. Chern. Soc. 1952, 3532. (S'} ••• (S'}.d t;.E"
0.75000 0.84133 -0.10740
0.73938 0.95625 -0.16491
0.74652 0.84425 -0.09015
0.71623 1.00010 -0.22561
0.73777 0.89351 -0.13118
0.75000 0.80296 -0.01392
0.75000 0.80234 -0.01626
0.74912 0.79618 -0.10306
0.74834 0.80974 -0.13940
0.74739 0.82300 -0.17333
0.74758 0.82193 -0.06992
0.74508 0.85290 -0.05714
0.74746 0.82334 -0.05503
0.74098 0.88663 -0.02638
0.74958 0.77021 -0.06053
0.74302 0.88116 -0.18893
0.74751 0.83609 -0.06851
0.74646 0.83777 -0.09147
0.74705 0.82920 -0.08847
To provide a basis for a more quantitative analysis
of the contributions of higher multiplets to the uhf
single determinant, we summarize in Table V the ex
pectation values (82) computed for the radicals in
cluded in this study. They indicate that the contribu
tions of multiplets higher than quartet are very small
for these radicals. The energy difference between E.d
and Easa has been computed by the method of Amos
and Hall.ll
As was shown in Table I, the observed splittings in
benzene- are lower than one would estimate on the
basis of the empirical relation (22b) and symmetrical
charge and spin densities of i on each carbon. We
present in Table VI a summary of our calculations
for an SCF treatment of the even and odd members
of the degenerate pair of states for the benzene nega
tive ion. Here we refer to even or oddness with respect
to reflection of the electron coordinates through a plane
perpendicular to the benzene plane and passing through
carbons on opposite sides of the molecule, which is
assumed to have a sixfold axis of symmetry. The SCF
computations were not entirely satisfactory from a
symmetry point of view. As is shown in Table VI,
the energy and expectation values of 82 were not quite
identical for the even and odd states, a situation which
we understand has also been obtained by Colpa. How
ever, if we assume that interactions with its environ
ment send it into the even and odd states, and if we
compute average proton splittings for these states,
using the empirical relations (22b) and (22c) , then
we find the difference between the experimental and
theoretical splittings to be greatly reduced. Hobey has
made a Pariser-Parr configuration interaction descrip
tion of the even and odd states.35 His computed spin
densities fall between our Paa and pasa.
33 W. D. Robey, Mol. Phys. 7, 325 (1964).
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TABLE VI. Detailed study of benzene negative ion.
Atom au:ptl aaaM aaa OB aaa GNP
Proton hyperfine splittings
Even state
1 -10.422 -8.871 -8.362
2 -1.539 -1.472 -1.016
Av -3.750 -4.500 -3.938 -3.464
Odd state
1 +1.080 +1.073 +1.086
2 -7.290 -6.441 -5.685
Av -3.750 -4.500 -3.936 -3.428
Computed spin and charge densities
Atom Ph Paa Pasa P,d qaa
Even state
1 0.333 0.386 0.411 0.435 -0.314
2 0.167 0.057 0.044 0.033 -0.093
Odd state
1 0 -0.040 -0.084 -0.125 -0.009
2 0.250 0.270 0.292 0.313 -0.246
Expectation values of S2 and E
(S2 }aa (S2}a,a
Even state 0.75000 0.75000
Odd state 0.75059 0.74964
INTERPRETATIONS
It appears from these comparisons that a spin-density
distribution intermediate between Paa and Pasa (and
generally a distribution closer to Paa) gives best agree
ment with experiment. Clearly the spin densities of
the raw single determinant are less satisfactory. We
would like to find a theoretical meaning for these
observations.
Marsha1l36 has suggested that the spin distribution
of the single determinant should be compared with
experiment. He has shown that under the assumption
that the underlying orbitals split in a symmetrical way
in a uhf calculation, that the spin density resulting
from that splitting is about t as great after projection
as before. This is in agreement with our result. He
also has shown that if exchange splitting of the added
single excitation doublet and quartet components is
small relative to the promotion energy to these con
figurations, then the unprojected spin density should
be closer to experiment than the projected.
On the other hand, Lowdin13 has recommended the
use of an extended Hartree-Fock method in which
the energy of the single determinant after projection
is minimized. Besis, Lefebvre-Brion, and Moser3"7 have
concluded that spin densities computed from uhf func
tion are close to those computed from configuration
interaction of doublet single excitation of a HF function.
They suggest that this is equivalent in first order to
Lowdin's extended-HF scheme.
36 W. Marshall, Proc. Phys. Soc. (London) A78, 113 (1961).
37 N. Bessis, H. Lefebvre-Brion, and C. M. Moser, Phys. Rev.
124,1124 (1961). (S2}.d E.d Ea,"
0.77429 -91.456406 -91.492457
0.78619 -91.462660 -91.511600
We now attempt to analyze the meaning of our
computed spin densities in terms of the qualitative
features of our uhf wavefunction. As we stated before,
a uhf wavefunction may be expressed as a linear com
bination of spin components, as in Eq. (7). These
have spin ranging from s=!(p-q) to s=!(p+q). In
the first paper of this series we showed that the un
restricted MO's if; and cp of 'l' uhf may be transformed
to corresponding MO's X and 7], which are closely
related to the natural orbitals A and p and p. of the
unres tricted single determinant:
where
and Xi= Ai( 1-Ai2) !+PiAi i=l, "., q;
i=l, "', q; (23)
(24)
i=q+l, "', p, (25)
(26)
f xi7]jdT= T/jij. (27)
As we have shown in the previous paper of this series,
for pi-electron radicals the Ti are quite close to unity,
usually greater than 0.98 and almost always greater
than 0.90, when semiempirical repulsion integrals are
used.
If p=q+l, then the lowest spin component is a
doublet and we may write 'It uhf in the form
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where (rf) stands for restricted function, (se) for single
excitation, and (de) for double excitation. We now
write the terms containing up to the first power of ~;
in more detail38
with
q
Clr= [II (1-~l) ] (30)
i=1
q q
Iflse=A-l.L~j(1-~;)1[ II (1-~i2)J6-t
1=1 i=I,i»'j
x {+ 1 A jOi.----+1I)"OI 1 -1 A j{3----+1I j{3 1
+21 Aj{3----+1IjOi.; p.pa:----+p.p{3l}' (31)
with Here H is the Hamiltonian operator. We note that
(lflrfHy;:)= 1/v3" (lfrfHlf1se) and that Ec=!Else+~EIB •.
We thus may write
(1/v3") (If{fHlfIBe) Cc"'-=--:--,,:-:,:-'----::-,:-'E{f- tElse-~Else (43)
We now define a wavefunction 'IIext which we con
sider to be a perturbation-theory analog of the ex
tended Hartree-Fock function
(44)
By analogy, we take a perturbation-theory estimate
of C·,
Ce= (If{fHlf1se)/ (E{L Else), (45)
(32) and thus we write
q q
Ifise= A-l.L~j(1-~;)![ II (1-~?) J(1/v3")
1=1 i=-I,i»'j
x {+ 1 AjOi.----+1IjOi. 1 -1 Aj{3----+1Ij{31
-1 Aj{3----+1IjOi.; p.pa:----+p.p{3I}' (33)
with
(34)
where
Ai= :t~j(1-~;)1[ IT (1-~?)J. (35)
j=1 i=I,i¢j
The arrows in Eqs. (31) and (33) signify substitu
tions in the determinant of Eq. (29).
If we assume that terms of (28) involving double
and higher excitations are unimportant, we obtain by
equating terms of (7) with terms of (28);
C!'III= C{flf{f+C1B"lflBe,
C!'II1 = CtB"lftBe, (36)
(37) (E,rf-.l E1se-~ E.se) Ce=v3"Cc • 3. 3 ,
(EirLElse) (46)
where
K= (E{L!E!se-~E~se)/(E{LElse). (47)
In summary, we find by perturbation theory that
and (48)
(49)
To make more explicit the meaning of these func-
tions we write the corresponding spin densities
puh/=pr(rf! 1 rf!) + 3V2cjsepr (rf! 1 set), (SO)
Pa8ar=pr(rf! 1 rf!) + 2V2CjBepr (rf! 1 se!) , (51)
PaaT= pr(rf! 1 rf!) +V2C1sepr(rf! 1 se!) , (52)
Pext'=pr(rf! 1 rf!)+3KV2C 1sepr(rf! 1 se!) , (53)
and since Ctse=2!c1se we may to this approximation where we have used the fact that
write
(38)
We now consider 'IIuhf in terms of a perturbation
theory mixing of the higher functions with 1f'f. To do
this we define a normalized composite function Ifc:
Thus we may write
'II uhf= C{flf{f+ C"lfc,
where
As a perturbation theory estimate of Cc we take
38 In Ref. 1, p. is misprinted for" in Eqs. (51) and (52). (40)
( 41) pr(rf! 1 se!) =V2pr(rf! 1 set). (54)
Here pr(a 1 b) is the contribution to the rth diagonal
element of the spin density matrix by a matrix element
of the spin density operator between functions a and
b. These expressions suggest the following use of un
restricted spin densities with and without projection or
equivalent annihilation. If K = 1 as when the single
excitation doublet component is degenerate with the
single-excitation quartet component, so that E1se=
EtBe, then the unprojected spin densities should be
used, as suggested by Marshall. On the other hand
if the single-excitation quartet component is degener
ate with the restricted doublet part, so that Erf=Etse
and K =!, then the spin densities after annihilation
should be used. Most radicals will fall between these
extremes. The conjugated hydrocarbon radicals appear
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to approximate the second case. It may be possible to
estimate or compute Erf, E!s., and E!s. and thus to
fix for general classes of radicals the proper linear
combination of projected and unprojected spin densi
ties for comparison with experiment.
These possibilities may seem rather surprising to
anyone used to configurations based on Huckel orbit
als for with these one would expect Ejse,.....,E!·· with
K,.....,1. However, it is important to realize that the
orbitals we use are the natural orbitals of the un
restricted wavefunction so the situation is rather dif
ferent. Moreover, even if we assume that the orbitals
are Huckel orbitals then Eqs. (50) -( 53) correspond
only to first order perturbation theory. When the
Huckel polarizabilities of Coulson and Longuet-Higgins
are used to calculate pr(rf! I se!) as in McLachlan's
method, it is in fact found that the spin densities given
by (50) are very close to those obtained from con
figuration interaction wavefunctions corresponding to
(53). They are both very different from these calcu
lated exactly from 'Ir uhf.
The reason for this is that to calculate 'Ir uhf a self
consistent procedure is used so that any first-order
terms should be calculated in a self-consistent way
also. Thus for the spin densities calculated by McLach
lan we can write
PMcLr=pr(rf! I rfi)+P{""'Pext', (55)
where p{=3Y2C··pr(rf! I set) is the first-order "Huckel"
term, while corresponding to (50)-(52) we should have
pr=pr(rf! I rf! D+MpsCFr, (56)
where PSCFr is the first-order self-consistent correction
and M = 1 for Puh/, i for Pa8ar, and j for Paar. Comparing
a McLachlan type calculation for naphthalene using
first Huckel and then self-consistent polarizabilities we
find p{"""'!PSCFr although this will, of course, vary from
atom to atom and molecule to molecule. Nevertheless,
using this as a rough estimate we find
PMcLr"""'Pex{"""'!CPa8ar +Paar) , (57)
which seems to agree reasonably well with the results
we have in the previous section.
Finally in this section we should like to return to a
point we discussed in some detail in Paper 1. This
concerns the error in the spin densities after annihila
tion due to the unwanted components which still re
main in the wavefunction. We showed that this error
would be of third order in the .1; which is usually
rather small. Harriman39 has now given what amounts
to a series expansion for the spin densities after a full
39 J. E. Harriman, J. Chern. Phys. 40, 2827 (1964). projection up to second order in the .1;. When his
method is applied to our allyl uhf function the spin
densities are pl=0.545 and p2= -0.091. When applied
to our pentadienyl function the spin densities are
p1=0.380, p2= -0.091, and p3=0.422. The difference
between these and the results in Table V represent
third-order effects and, as expected, these are very
small. For the naphthalene ions where the .1; are
smaller and more representative than those found for
allyl and pentadienyl the results using Harriman's
formula and our own are identical. This helps to con
firm that a single annihilation is a good approximation
to the complete projection.
CONCLUSIONS
The relation between spin and charge densities and
proton isotropic hyperfine splitting in aromatic pi
electron radicals is not entirely clear. The McConnell
relation (22a) is qualitatively good but poorer than
either that of Colpa and Bolton, or of Giacometti,
Nordio, and Pavan. It is not yet possible to conclude
which one of these latter two relations is better. Clearly,
further investigation of the semiempirical relation be
tween hyperfine splittings and pi-electron spin densities
is called for.
It appears that for aromatic pi-electron radicals
uhf wavefunctions computed with Pariser-Parr-Pople
semiempirical integrals give spin densities which, after
annihilation of contaminating higher multiplets, are in
fair agreement with experiment and which are clearly
superior to those computed in the Huckel approxima
tion. There are systematic errors which may be char
acteristic of all relatively simple molecular orbital de
scriptions of these radicals and which may be absent
from a valence bond description. These are the low
spin densities of para protons relative to ortho protons
in benzyl radical and derivations, and the excessively
low spin density computed for the 2 position of poly
acenes.
The uhf wavefunctions for aromatic pi-electron radi
cals have nearly closed underlying shells, when com
puted with semiempirical integrals. The paired molecu
lar orbitals are split only slightly so that the uhf single
determinant doublet component is effectively obtained
by a single annihilation.
It is suggested that the spin properties of an extended
HF calculation may be approximated with quantities
derived from a uhf calculation with and without an
nihilation.
ACKNOWLEDGMENTS
We wish to thank Dr. J. E. Harriman and Mr. 1. C.
Smith for useful discussions.
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1.1722617.pdf | Estimate of the Time Constant of Secondary Emission
A. Van Der Ziel
Citation: Journal of Applied Physics 28, 1216 (1957); doi: 10.1063/1.1722617
View online: http://dx.doi.org/10.1063/1.1722617
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IP: 129.174.21.5 On: Tue, 23 Dec 2014 08:34:341216 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: 129.174.21.5 On: Tue, 23 Dec 2014 08:34:34LETTERS TO THE EDITOR 1217
where q= (eEo/kT,)t. For semiconductors q=O; hence, C=I, and
(7a)
We finally write
N=nod, (8)
where d is an equivalent depth.t Substituting into (3) yields
T=C-'(211"m/kT,)18d[(E,+E h)/El'o] exp(ex/kT,). (9)
We shall apply this to two cases. As a first example consider a
heavy metal having 8=1.5 at El'o=500 v. Assuming d=100A,
(kT,/e)=2 v, Eo=EJ=5 v, x=10 v, and Eh=5 vi; we have
C-'=0.17, E,=7.5 v, and T=3.3XlO-H sec. As a second example
consider an insulator having 8=10 at Epo=500 v. Assuming
d=100A, (kT,/e)=2 v, (ex/kT,)«1 and Eh=15 v:j:; we have
E,=3 v and T=2.5XlO-14 sec.
The estimated values of T are not very accurate; they indicate,
however, that it will be difficult to account for a time constant of
secondary emission that is larger than 10-13 sec, unless other forms
of energy storage (trapped electrons, exciton generation) are
important. The best experimental evidence indicates that T is
indeed very small.
* Work supported by U. S. Signal Corps Contract.
t If Xo is the range of the primaries, then nex) should increase with
increasing x for x <xo, because the rate of production of secondaries increas('s
toward the end of the range. whereas n (x) gradually decreases to zero for
x >xo. If Xl is the escape depth of the secondaries, then d~(XO+Xl). t In an insulator Eh should be larger than the gap width between the
bottom of the conduction band and the top of the filled band. In a metal
Eh should be smaller than the gap width, since the holes generated in the
conduction band have a negative energy. Assuming a gap width of 10 v
in either case, the two estimated \-alues of Eh seem quite reasonable.
Phenomena Associated with Detonation in
Large Single Crystals*
T. E. HOLLAND,t A. \Y. CA"PBELL, AKD M. E. MALI"+
University of Calijornt"a, ["os Alamos Scientt"jic T,aboratory,
Los Alamos, New Mexico
(Received June 19. 1957)
VERY little information is recorded in the literature concern
ing the detonation behavior of large single crystals of
explosive compounds. The opinion has been expressed that it may
not be possible to produce stable detonation in such media, since
the compressional heating at the shock front (in the absence of
air-filled voids or lattice defects) may be too low to provide a
reaction rate sufficient for detonation. Experimental support for
this view is found in the well-known facts that pressed explosive
is made harder to initiate by pressing to higher density; and that
TNT castings are harder to initiate and show larger failure
diameters as the crystal size is increased. On the other hand, it is
known that the primary explosive, lead azide, when prepared
in the form of large crystals detonates very easily. In this note we
report our observations on single crystals of PETN.
A measure of the sensitivity of large crystals of PETN relative
to powdered PETN was obtained by the use of a rifle hullet test.
Crystal specimens were mounted on plywood with the minimum
dimension of the crystal parallel to the path of the bullet;
powdered specimens were prepared by spreading a uniform layer
upon a cardboard support and covering the layer with a thin
cellophane sheet. When subjected to the impact of a soft-nosed
hullet traveling at approximately 4000 It/sec, crystals with a
minimum dimension of I} in. failed to detonate, but detonated
reliably when this dimension was increased to 1 ~ in. whereas the
powdered material detonated reproducibly in layers as thin as
0.092 in.
Evidence that single crystals can be detonated at full velocity
was obtained from charges arranged as diagramed in Fig. 1. A
plane detonation wave was generated in a 2-in. thick piece of
Composition B. This pressure wave was attenuated by passage
through a I-in. steel plate and used to initiate a crystal of PETN.
The latter was essentially a 45°-90°-45° right-angled prism made
by passing a plane through a cube of PETN three-quarters of an RW GENERI\TOR
COfPCSITION B RETN
CRYSTAL
FIG. 1. Smear camera record showing three distinct velocity regimes in
an uncterinitiaterl. PETN crystal.
inch on a side. In order to brighten the firing trace, the slant face
01 the prism ,vas covered with a Lucite plate so as to form a small
air-gap.
At the right in Fig. 1 is shown the firing trace with the PETN
crystal sketched in to give a corresponding space scale. Time zero
lies slightly to the left of the left edge of the print of the firing
record. In region I low-order detonation is seen. The rate of
detonation is estimated to be 5560 m/sec. The detonation rate
changes abruptly to an estimated value 01 10450 m/sec in region
II, accompanied by observable radiation in the interior of the
crystal. There is a final, apparently steady, detonation rate
established in region III with a value of 8280 m/sec. Finally, in
region IV, the detonation wave emerges from the top of the crystal.
Efforts were made to mea-sure the single-crystal failure diameter
using rods of PETN ground from single crystals. These efforts are
as yet incomplete, but show that the failure diameter is greater
than 0.33 in.
Failure of the detonation process takes place through the action
of "dark waves'" originating at the periphery of the detonation
wave. In a typical experiment the charge was a rod of PETN
0.252 in. in diam by 0.438 in. long. Beginning at the boostered end,
the rod was encased with brass foil for a distance of 0.287 in. The
foil served to prevent the occurrence of dark waves in the first
part of the stick. When the detonation wave passed the foil, it was
choked-off by dark waves. The latter waves are believed to be
hydrodynamic rarefactions characteristic of detonation in homoge
neous explosives.
* \Vork done under the auspices of the U. S. Atomic Energy Commission. t The George \Vashington University Research Laboratory. Camp
Detrick. Frederick. Maryland.
::: Advanced Development Di\'iRion, .\vco Manufacturing Corporation.
Stratford, Connecticut.
1 Campbell, Holland, Malin, and Cotter. Nature 178,38 (1956),
Growth of Tellurium Single Crystals by the
Czochralski Method
T . .T. DAVIES
Il(l1J('Y'l1'ell Research Center, Hopkins, Alinnesota
(Received June 3, 1957)
SEVERAL Te single crystals have been grown reproducibly by
the Czochralski technique. Although insufficient experimental
data are available to establish optimum growing conditions, any
future improvements would probably be of minor significance.
The important consideration at this time is that single Te crystals
have been obtained by seed dipping. To the author's knowledge
this has been reported only once before, by J. Weidel' in Germany.
Molten Te when allowed to cool slowly tends to freeze into
single crystals along the c axis of the hexagonal structure. Due to
the presence of bubbles and the polycrystalline nature of a free
frozen ingot, these crystals are quite limited in size and quality,
but do provide an initial source of seeds. Cleavage is easily
accomplished because the valence binding energy between atoms
along the spiral chains in the c direction is much stronger than the
binding energy between chains.' One indication of crystal quality
is the degree of perfection of the resultant cleaved planes.
In the vertical pulling process Te purified by vacuum distil-
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 129.174.21.5 On: Tue, 23 Dec 2014 08:34:34 |
1.1722616.pdf | Erratum: Electrical Conductivity of Fused Quartz
Julius Cohen
Citation: Journal of Applied Physics 28, 1216 (1957); doi: 10.1063/1.1722616
View online: http://dx.doi.org/10.1063/1.1722616
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
Electrical and optical measurements on fused quartz under shock compression
AIP Conf. Proc. 78, 299 (1982); 10.1063/1.33328
Electrical measurements on fused quartz under shock compression
J. Appl. Phys. 52, 5084 (1981); 10.1063/1.329459
The Precise Determination of Thermal Conductivity of Pure Fused Quartz
J. Appl. Phys. 39, 5994 (1968); 10.1063/1.1656103
Strength of Bulk Fused Quartz
J. Appl. Phys. 32, 741 (1961); 10.1063/1.1736084
Electrical Conductivity of Fused Quartz
J. Appl. Phys. 28, 795 (1957); 10.1063/1.1722858
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 155.33.120.209 On: Mon, 08 Dec 2014 02:08:451216 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: 155.33.120.209 On: Mon, 08 Dec 2014 02:08:45LETTERS TO THE EDITOR 1217
where q= (eEo/kT,)t. For semiconductors q=O; hence, C=I, and
(7a)
We finally write
N=nod, (8)
where d is an equivalent depth.t Substituting into (3) yields
T=C-'(211"m/kT,)18d[(E,+E h)/El'o] exp(ex/kT,). (9)
We shall apply this to two cases. As a first example consider a
heavy metal having 8=1.5 at El'o=500 v. Assuming d=100A,
(kT,/e)=2 v, Eo=EJ=5 v, x=10 v, and Eh=5 vi; we have
C-'=0.17, E,=7.5 v, and T=3.3XlO-H sec. As a second example
consider an insulator having 8=10 at Epo=500 v. Assuming
d=100A, (kT,/e)=2 v, (ex/kT,)«1 and Eh=15 v:j:; we have
E,=3 v and T=2.5XlO-14 sec.
The estimated values of T are not very accurate; they indicate,
however, that it will be difficult to account for a time constant of
secondary emission that is larger than 10-13 sec, unless other forms
of energy storage (trapped electrons, exciton generation) are
important. The best experimental evidence indicates that T is
indeed very small.
* Work supported by U. S. Signal Corps Contract.
t If Xo is the range of the primaries, then nex) should increase with
increasing x for x <xo, because the rate of production of secondaries increas('s
toward the end of the range. whereas n (x) gradually decreases to zero for
x >xo. If Xl is the escape depth of the secondaries, then d~(XO+Xl). t In an insulator Eh should be larger than the gap width between the
bottom of the conduction band and the top of the filled band. In a metal
Eh should be smaller than the gap width, since the holes generated in the
conduction band have a negative energy. Assuming a gap width of 10 v
in either case, the two estimated \-alues of Eh seem quite reasonable.
Phenomena Associated with Detonation in
Large Single Crystals*
T. E. HOLLAND,t A. \Y. CA"PBELL, AKD M. E. MALI"+
University of Calijornt"a, ["os Alamos Scientt"jic T,aboratory,
Los Alamos, New Mexico
(Received June 19. 1957)
VERY little information is recorded in the literature concern
ing the detonation behavior of large single crystals of
explosive compounds. The opinion has been expressed that it may
not be possible to produce stable detonation in such media, since
the compressional heating at the shock front (in the absence of
air-filled voids or lattice defects) may be too low to provide a
reaction rate sufficient for detonation. Experimental support for
this view is found in the well-known facts that pressed explosive
is made harder to initiate by pressing to higher density; and that
TNT castings are harder to initiate and show larger failure
diameters as the crystal size is increased. On the other hand, it is
known that the primary explosive, lead azide, when prepared
in the form of large crystals detonates very easily. In this note we
report our observations on single crystals of PETN.
A measure of the sensitivity of large crystals of PETN relative
to powdered PETN was obtained by the use of a rifle hullet test.
Crystal specimens were mounted on plywood with the minimum
dimension of the crystal parallel to the path of the bullet;
powdered specimens were prepared by spreading a uniform layer
upon a cardboard support and covering the layer with a thin
cellophane sheet. When subjected to the impact of a soft-nosed
hullet traveling at approximately 4000 It/sec, crystals with a
minimum dimension of I} in. failed to detonate, but detonated
reliably when this dimension was increased to 1 ~ in. whereas the
powdered material detonated reproducibly in layers as thin as
0.092 in.
Evidence that single crystals can be detonated at full velocity
was obtained from charges arranged as diagramed in Fig. 1. A
plane detonation wave was generated in a 2-in. thick piece of
Composition B. This pressure wave was attenuated by passage
through a I-in. steel plate and used to initiate a crystal of PETN.
The latter was essentially a 45°-90°-45° right-angled prism made
by passing a plane through a cube of PETN three-quarters of an RW GENERI\TOR
COfPCSITION B RETN
CRYSTAL
FIG. 1. Smear camera record showing three distinct velocity regimes in
an uncterinitiaterl. PETN crystal.
inch on a side. In order to brighten the firing trace, the slant face
01 the prism ,vas covered with a Lucite plate so as to form a small
air-gap.
At the right in Fig. 1 is shown the firing trace with the PETN
crystal sketched in to give a corresponding space scale. Time zero
lies slightly to the left of the left edge of the print of the firing
record. In region I low-order detonation is seen. The rate of
detonation is estimated to be 5560 m/sec. The detonation rate
changes abruptly to an estimated value 01 10450 m/sec in region
II, accompanied by observable radiation in the interior of the
crystal. There is a final, apparently steady, detonation rate
established in region III with a value of 8280 m/sec. Finally, in
region IV, the detonation wave emerges from the top of the crystal.
Efforts were made to mea-sure the single-crystal failure diameter
using rods of PETN ground from single crystals. These efforts are
as yet incomplete, but show that the failure diameter is greater
than 0.33 in.
Failure of the detonation process takes place through the action
of "dark waves'" originating at the periphery of the detonation
wave. In a typical experiment the charge was a rod of PETN
0.252 in. in diam by 0.438 in. long. Beginning at the boostered end,
the rod was encased with brass foil for a distance of 0.287 in. The
foil served to prevent the occurrence of dark waves in the first
part of the stick. When the detonation wave passed the foil, it was
choked-off by dark waves. The latter waves are believed to be
hydrodynamic rarefactions characteristic of detonation in homoge
neous explosives.
* \Vork done under the auspices of the U. S. Atomic Energy Commission. t The George \Vashington University Research Laboratory. Camp
Detrick. Frederick. Maryland.
::: Advanced Development Di\'iRion, .\vco Manufacturing Corporation.
Stratford, Connecticut.
1 Campbell, Holland, Malin, and Cotter. Nature 178,38 (1956),
Growth of Tellurium Single Crystals by the
Czochralski Method
T . .T. DAVIES
Il(l1J('Y'l1'ell Research Center, Hopkins, Alinnesota
(Received June 3, 1957)
SEVERAL Te single crystals have been grown reproducibly by
the Czochralski technique. Although insufficient experimental
data are available to establish optimum growing conditions, any
future improvements would probably be of minor significance.
The important consideration at this time is that single Te crystals
have been obtained by seed dipping. To the author's knowledge
this has been reported only once before, by J. Weidel' in Germany.
Molten Te when allowed to cool slowly tends to freeze into
single crystals along the c axis of the hexagonal structure. Due to
the presence of bubbles and the polycrystalline nature of a free
frozen ingot, these crystals are quite limited in size and quality,
but do provide an initial source of seeds. Cleavage is easily
accomplished because the valence binding energy between atoms
along the spiral chains in the c direction is much stronger than the
binding energy between chains.' One indication of crystal quality
is the degree of perfection of the resultant cleaved planes.
In the vertical pulling process Te purified by vacuum distil-
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 155.33.120.209 On: Mon, 08 Dec 2014 02:08:45 |
1.1735302.pdf | XRay and Expansion Effects Produced by Imperfections in Solids: Deuteron
Irradiated Germanium
R. O. Simmons and R. W. Balluffi
Citation: Journal of Applied Physics 30, 1249 (1959); doi: 10.1063/1.1735302
View online: http://dx.doi.org/10.1063/1.1735302
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
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Diffusion of Deuterium in DeuteronIrradiated Copper
J. Appl. Phys. 31, 1474 (1960); 10.1063/1.1735866
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:59JOURNAL OF APPLIED PHYSICS VOLUME 30. NUMBER 8 AUGUST. 1959
X-Ray and Expansion Effects Produced by Imperfections in Solids:
Deuteron-Irradiated Germanium*
R. O. SIMMONS AND R. W. BALLUFFI
University of Illinois, Urbana, Illinois
Important information concerning atomic models of the defects in damaged crystals can be obtained from
measurements of bulk length changes, aL/ L, and x-ray lattice parameter changes, aa/ a, during irradiation
and subsequent recovery. These quantities are not necessarily equal, and their magnitudes and signs may
serve to discriminate between possible models. A detailed description of the aL/ Land aa/ a effects to be
expected from models consisting of point imperfections (either uniformly distributed or clustered), displace
ment spike regions, and dislocation loops is given. Additional discussion is given of the effects expected in
measurements of low-angle x-ray scattering, Laue-Bragg reflection broadening, Laue-Bragg reflection in
tensities, and x-ray diffuse scattering. The currently available experimental results on irradiated germanium
are discussed in an attempt to discriminate between the various models. Comparatively simple models con
sisting of clusters of vacancies and clusters of interstitials are probably consistent with present experiment.
No information is available regarding the shapes of these regions. More complicated models are not excluded,
of course. The aforementioned techniques are compared with electrical measurements in semiconductors,
and some of the strengths and weaknesses of the various methods are assessed. The need for further meas
urements of all types is emphasized.
I. INTRODUCTION
DESCRIPTIONS of the state of an irradiated solid
fall roughly into two groups. In one group are
phenomenological models which involve only measur
able macroscopic properties; in the second group are
physical models which attempt to interpret the meas
ured properties in terms of a detailed atomic picture of
the damage. The present paper is mainly concerned
with the establishment of models of the second type.
The majority of the electrical measurements on
damaged semiconductors has led most directly to
models of the first group in terms of such parameters
as carrier concentrations and mobilities, defect energy
levels, and carrier trapping cross sections of defects.
These measurements have been of great importance in
establishing the basic electrical properties of damaged
semiconductors. Under certain conditions when the
damaged structure is stable over a considerable tem
perature range, it is possible to make combined elec
trical measurements which lead to approximate atomic
models of the damage sites.! In general, such measure
ments can readily be carried out only in stabilized
structures after considerable thermal recovery has oc
curred. The damage state investigated is then one
resulting from the combined effects of the primary
collision processes during irradiation and of the sub
sequent redistribution of the defects by motion during
thermal recovery. At low temperatures thermal in
stability of the defect structure combined with possible
long-lived trapping processes 2 makes interpretation less
straightforward. In addition possible chemical impurity
* This work was supported by the U. S. Atomic Energy
Commission.
1 See, for example, G. K. Wertheim, Phys. Rev. 105, 1730
(1957); 110, 1272 (1958); 111, 1500 (1958).
2 H. Y. Fan and K. Lark-Horovitz, in Report of the Bristol Con
ference on Defects in Crystalline Solids, JuJy, 1954 (The Physical
Society, London, 1955), p. 232. effects are difficult to investigate independently by
electrical means. Thus while many of the electrical
effects are fairly well established,2-4 our knowledge of
the atomic structure in damaged semiconductors is
still rudimentary.
The primary purpose of the present paper is to de
scribe the use and present status of other methods,
viz, bulk expansion and x-ray lattice parameter meas
urements, in establishing the atomic model of damaged
semiconductors. Additional discussions of the effects
expected in measurements of low-angle x-ray scattering,
Laue-Bragg reflection broadening, Laue-Bragg reflec
tion intensities, and x-ray diffuse scattering are also
included.
II. BULK AND X-RAY EXPANSIONS
(a) Principles of the Measurements
Simultaneous measurement of changes in length and
x-ray lattice parameter yields information which may
be used to discriminate between several models of the
radiation damaged structure. In the following discus
sion we consider the length and lattice parameter
changes which should occur upon bombardment and
subsequent thermal recovery in several models which
have been proposed in the literature. The expected
effects in the following models will be discussed: I, iso
lated vacancies and interstitials distributed homogen
eously throughout the volume; II, clustered vacancies
and interstitials; III, displacement spike regions where
the atoms are in a different structure than the matrix
with a different coordination number (for example, an
amorphous or a liquid-like structure); IV, dislocation
3 H. Brooks, in Annual Review of Nuclear Science (Annual
Reviews, Inc., Stanford, 1956), Vol. 6, p. 215.
4 F. Seitz and J. S. Koehler, in Solid State Physics, edited by
F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1956),
Vol. 2, p. 305.
1249
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:591250 R. O. S I M M 0 N SAN DR. \V. B ALL U F F I
loops; and V, combinations of the foregoing. We do not
consider explicitly here possible additional effects arising
from the presence of foreign atoms of different valency
or size. It is possible, of course, that such chemical im
purities can form complexes of different types in several
of the models.
The length and lattice parameter changes may be
analyzed using a generalized sphere-in-hole model for
the defects. A defect region may be constructed in the
following way: (1) cut out the region to be damaged;
(2) produce the damage in the region; (3) force the
damaged region back into the original cavity. The
volume of the cutout region will, in general, be changed
by the damage and it will act as a center of dilatation
in the solid causing length and lattice parameter
changes. The dilatation may be analyzed using elas
ticity methods which have been described in detail by
Eshelby."
Model I
We consider a general case in which random uniform
distributions of vacancies and interstitials are present
in concentrations not necessarily equal. Such a situation
could develop if Frenkel pairs were created during the
knock-ons followed by loss of some of the imperfec
tions. Imperfections could be destroyed by a variety of
mechanisms: for example, diffusion to sinks or collapse
into dislocation loops. When considering the relative
changes in length, !J.Lj L, and lattice parameter, !J.aj a,
account must be taken both of the dilatation of the
lattice caused by the imperfections and of the fact that
an atomic site is destroyed for each vacancy destroyed
and created for each interstitial destroyed. The length
measurement measures both the average lattice dilata
tion and the change in the number of lattice sites,
whereas the x-ray lattice parameter measurement is
capable only of measuring the average lattice dilata
tion. Let us consider the lattice dilatation first. If the
equilibrium vacancy position in the diamond lattice is
centered at the tetrahedral position, the vacancy will
act as a center of dilatation with an effect which is
elastically equivalent to the application of four point
forces along the tetrahedral directions. An interstitial
in the tetrahedral position should act similarly, and the
dilatation should be isotropic in each case because of
the symmetry.t Eshelby" has shown that the dilatations
due to a random uniform distribution of such point
centers of dilatation will produce a dilatational strain
which is uniform and isotropic throughout the entire
specimen. For example, in an elastically isotropic ma-
6 J. D. Eshelby, in Solid State Physics, edited by F. Seitz and
D. Turnbull (Academic Press, Inc., New York, 1956), Vol. 3,
p. 79; and J. Appl. Phys. 25,255 (1954); 24, 1249 (1953). t See, however, the discussion in Sec. IV, concerning possible
small deviations from tetrahedral symmetry. If the individual
defects exhibit nonsymmetric dilatations, we still expect the over
all average dilatation to be isotropic for a random distribution of
many defects. terial containing N centers per unit volume the average
strain is
(E)=4?r(1-u)Ncj (1+u)
where u is Poisson's ratio and C is the strength of the
center of dilatation. For a prototype defect consisting
of a sphere of radius ro of the same material where the
relaxed misfit strain is Ed
and we see that c is proportional to the misfit strain and
to the volume of the defect region. If several types of
defects are present then effects are simply additive.
When the lattice is dilated by the presence of defects,
the fractional change in lattice parameter due to the
average dilatation, (!J.aja) (.), should equal the frac
tional change in length due to the dilatation, (!J.Lj L)(.),
since the reciprocal lattice undergoes a uniform strain
which is equal and opposite to the strain of the specimen
lattice." If Q is the atomic volume and jvQ is the volume
change due to the dilatational field of one vacancy, and
jiQ is the volume change due to one interstitial we then
have
where Cv and c i are the atom fractions of vacancies and
interstitials.
We must now consider the effects due to the creation
or destruction of atomic sites. These events will gener
ally produce volume changes which are equivalent to
the addition (interstitial destruction) or subtraction
(vacancy destruction) of atomic sites at the specimen
surface. This result follows from the fact that any
geometrical change in the internal sink during its op
eration will generally produce a lattice dilatation which
is negligible compared to the other processes being
considered. The net result, therefore, is the addition or
subtraction of atomic volumes to the crystal without
any significant lattice dilatation. The total changes in
length and lattice parameter are, therefore, given by
3!J.Lj L=3(!J.Lj L)(.)+Cv-Ci
= cv(jv+ 1)+C;(ji-1), (1a)
3!J.aj a= 3 (!J.aj a)(.)= cvjv+Cdi' (lb)
The foregoing relations assume that the volume changes
in the sink operation are randomly directed. This as
sumption should be quite good for cubic crystals in the
absence of external constraints.
Equations (la) and (lb) have been found to be con
sistent with at least one experiment involving uniformly
distributed point defects. Precise measurements of
!J.Lj Land !J.aj a were made on aluminum as it was
heated to near its melting point. 6 At high temperatures
an appreciable concentration of vacant lattice sites
should be present in thermal equilibrium. When only
6 R. O. Simmons and R. W. Balluffi (to be published).
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:59LATTICE AND LENGTH CHA:"JGES I:"J DEjUTERON DAMAGED Ge 1251
19
18
17
FIG. 1. Identification of 16
the predominant thermally
generated lattice defect in 15
aluminum as the lattice va
cancy, from measurements 14
by Simmons and Balluffi.6
When the formation or an
nihilation of lattice defects 13
requires the creation or de
struction of atomic sites,
the observed length and 12
lattice parameter changes
will be different. For va-I I
candes in a cubic sub
stance the atom fraction is
3 (!!.L/L-!!.a/a). The ob-10
served length expansion due
to defects is small compared 9
to the thermal expansion of
the lattice. For the figure
!!.L/L=!!.a/a=O at 20°e. e
7
6 .. o
~I..J III CURVE
l COOLING
HEATING
COOLING
Ilaa CURVE
T E M PER AT U R E (oG) {.o COOLING RUN
HEATING RUN
vacancies are present we may subtract Eq. (lb) from where
(1a) to obtain gijk= (1-2(1) (oijlk+Oiklj-Ojkli)+3l;lk
cv= 3 (IJ.L/ L-IJ.a/a). (2)
IJ.L/ L should then become greater than IJ.a/ a as vacan
cies are added to the crystal because of the length
changes associated with the creation of new atomic
sites. The experimental results are shown in Fig. 1,
and an increase in IJ.L/ Lover IJ.a/ a is apparent at the
elevated temperatures. Vacancy concentrations were
obtained from Eq. (2) which agreed within a factor of
two with those expected on the basis of quenching
experiments.7
Model II
In this case the point defects are closely associated
in configurations of arbitrary shape. The dilatation in
this case may be analyzed by considering an entire
cluster as a center of dilatation which has suffered a
misfit strain due to the presence of the point defects
within it. EshelbyB has solved for the elastic displace
ments at large distances, rl, from such a defect region
of arbitrary shape to obtain
(3)
7 W. DeSorbo and D. Turnbull, Acta Met. 7, 83 (1959).
8 J. D. Eshelby, Proc. Roy. Soc. (London) A241, 376 (1957). fjk is the misfit strain of the defect region and V is the
volume of the region. If the point defects are distributed
in the cluster, fjk will be a uniform dilatation and the
displacement u is radial and is independent of the
cluster shape. At large distances, therefore, a cluster of
arbitrary shape behaves like a spherically symmetric
one of the same strength. This result could, perhaps,
have also been seen qualitatively by applying Saint
Venant's Principle. The strength of a cluster containing
n distributed defects is just n times the strength of an
individual defect and we conclude that the average
lattice dilatation is independent of the fine scale dis
tribution of the point defects in the elastic approxima
tion when no overlap of the nonelastically strained
cores of the point defects occurs.
In cases where such overlap may occur, the strength
of the centers of dilatation is not determined by simple
addition; no calculations on realistic models have yet
appeared in this case. Two types of effects may enter
here; they can be illustrated by considering a divacancy
for example. The first arises because one nearest neigh
bor of each component vacancy is missing, contributing
an appreciable modification of the, strength of the
monovacancy elastic field. The second arises because
the defect is no longer even approximately cubically
symmetric. The axis of symmetry would be one of the
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20
~
...J
<J 15
;;
o 010
<J
o o LATTICE PARAMETER
'" LENGTH
'" 0
o
0"'----'---2---.0-3 --4L----.'5c----'cS-'IO""IS
NUMBER OF DEUTERONS leM"
FIG. 2. Comparison of lattice parameter and length changes in
deuteron-irradiated copper. The relative changes in each property
are equal within experimental error. X-ray lattice expansion meas
urements near lOOK by Simmons and Balluffi10; length measure
ments near 15°K by Vook and Wert,9 corrected to effective
deuteron energy 7 Mev.
crystallographic axes, however, and the symmetry axes
of many such defects would tend to be randomly di
rected, again leading to isotropic dilatation.
When the clusters are uniformly distributed through
out the specimen the over-all dilatation should be uni
form and isotropic as in model 1. Experimental results
which are consistent with Eqs. (la) and (lb) for models
I or II have been obtained in deuteron irradiation
studies of copper at low temperatures followed by an
nealing. Precise measurements of !1L/ D and !1a/ a10 in
dicated that !1L/ L= !1a/ a during bombardment and
during low temperature thermal recovery as shown in
Figs. 2 and 3. The !1L/ L measurements between 15 and
400K were obtained during a warmup of time com
parable with the annealing half-life and hence appear
displaced somewhat to larger values. There is consider
able accumulated evidence for copper that suggests that
Frenkel pairs are created during irradiation and that
the low temperature thermal recovery occurs predomi
nantly by mutual annihilation of vacancies and inter
stitialsY The vacancy and interstitial concentrations
should always be equal, therefore, during the low tem
perature recovery, and the observed equality of !1L/ L
and t:..d/d is consistent with Eqs. (la) and (lb). The ex
perimental result that !1L/ Land !1a/ a are positive is
also consistent with theoretical calculations for copper
which indicate that 1v"'-0.5 and 1i"'1.7.12 We con
clude, therefore, that Eqs. (la) and (lb) should hold for
models I and II.
Model III
Defect regions of arbitrary shape may be present
containing a highly disarranged atomic structure which
9 R. Vook and C. Wert, Phys. Rev. 109, 1529 (1958).
10 R. O. Simmons and R. W. Balluffi, Phys. Rev. 109, 1142
(1958).
11 F. Seitz, Institute of Metals lecture, 1959 (to be published in
Trans. of the Am. Inst. Mining Met., Petrol. Engrs.).
12 L. Tewordt, Phys. Rev. 109, 61 (1958); E. Mann and A.
Seeger (to be published). will generally produce a misfit strain. For example,
germanium melts with a volume contraction of about
5%. These regions will act as centers of dilatation, and
if the damaged regions tend to dilate uniformly the
situation is similar to that described for model II where
the lattice dilatation due to misfit regions of arbitrary
shape was described. However, in the present model
no new atomic sites are created or destroyed at such
sources or sinks as dislocations, and !1L/ Land !1a/ a
are then completely due to lattice dilatation. Therefore,
!1L/ L = !1a/ a. The results on deuteron-irradiated copper
shown in Figs. 2 and 3, therefore, are seen to be con
sistent with this model as well as with model I.
If the misfit dilatation is nonuniform due to possible
directional effects during bombardment the over-all
expansion of the specimen may no longer be isotropic.
The displacements around each defect region will then
vary with direction according to Eq. (3) and the defects
100 0
~ 80 z 0 o LATTICE PARAMETER
I LENGTH
0
iii z
f x
L&J
oJ
c( :) c
iii
L&J
II: 60
40 o 00
20
0~1~0~~20~~3~0--4~0~~50~~6~0~~70~~8~0-~90
RECOVERY TEMPERATURE OK
FIG. 3. Comparison of thermal recovery of lattice parameter
and length changes in deuteron-irradiated copper, from measure
ments of Simmons and Balluffilo and Vook and Wert.9 Congruence
of the residual expansion in the two cases indicates that recovery
is due either to interstitial-vacancy pair annihilation or to localized
recovery in separated centers of dilatation, as discussed in Sec. II
of the text.
will have a preferred orientation relative to the direc
tion of irradiation. However, a uniform distribution of
defect regions throughout the specimen should still
produce a homogeneous dilatation, and the length and
lattice parameter changes in any particular direction
should then be equal.
Model IV
Small dislocation loops may be produced in the stress
fields set up by displacement spikes4 or by the formation
of platelets of imperfections. The loops will produce a
lattice dilatation, since it is estimated that nonlinear
core effects produce a volume expansion of about 1 to
2 atomic volumes per atomic plane cutting the disloca
tion line.13 It is easily seen, however, that the dilatation
13 A. Seeger and P. Haasen, Phil. Mag. 3, 470 (1958); W. M.
Lomer, Phil. Mag. 2, 1053 (1957); H. Stehle and A. Seeger, Z.
Physik 146, 217 (1956).
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:59LATTICE AND LENGTH CHANGES IN DEUTERON DAMAGED Ge 1253
due to any reasonable number of dislocations is ex
tremely small and is almost certainly too small for
measurement with x-rays. For example, if the disloca
tion density has the unreasonably high value of lOlO
per cm2, the average dilatational strain is only "-'3 X 10-6•
It seems therefore that dislocation effects can be safely
ignored in measurements of ALI Land Aal a.
Model V
Various combinations of the above models are, of
course, possible. The various contributions to ALI L
and Aal a may then simply be added.
(b) Available Experimental Results
Extensive measurements of the length changes14 and
limited measurements of lattice parameter changes15
have been made on deuteron-irradiated and annealed
germanium. High purity germanium single crystals
were irradiated at 25°K with lO.2-Mev deuterons and
'V RUN I 25°K I).
12 I). RUN II 85°K
I).
-' 8 'V ~'V
.... .f'V -' <I vlSl'V V
4
176
0 fP
0 2 4 6 8 10 k 10"
,..UMBER OF DEUTERONSICM2
FIG. 4. Linear expansion of deuteron-irradiated germanium at
different bombardment temperatures, according to Vook and
Balluffi.14 Effective deuteron energy is 10 Mev. The rate of ex
pansion is only one-half that observed in copper near 15°K.
warmed slowly to 308°K, then again irradiated near
85°K and annealed to above room temperature. Length
change measurements were made during all of these
treatments,14 and the data for the expansion during the
bombardment and subsequent annealing above 85°K
are shown in Figs. 4 and 5. Lattice expansion measure
ments15 were made only above room temperature near
the end of the annealing experiments and are shown in
Fig. 6. They were carried to higher temperatures than
the length measurements.
The length expansions upon bombardment are pro
portional to deuteron flux and appear to be almost in
dependent of the bombardment temperature. The
length expansion upon bombardment is consistent with
models I or II if equal numbers of vacancies and inter
stitials are present due to Frenkel pair production, and
if the expected expansion around an interstitial is
14 F. L. Vook and R. W. Balluffi, Phys. Rev. 113, 62 (1959).
16 R. O. Simmons, Phys. Rev. 113, 70 (1959). 16xIO·'rr~~-'--'--,,~,,~,,"T,,~~-'--'~-'--"~~-'--'~-'--'-~~~
~ 14 A A
.J 12
<I 10
...J 8
::l 6 o 4
[3 2
0:: 0 80 After annealing
to t43·K
120 160 200 240 280
RECOVERY TEMPERATURE OK 320 360
FIG. 5. Thermal recovery of expansion in deuteron-irradiated
germanium, according to Vook and Balluffi.14 Other measurements
at lower temperatures set an upper limit of 20% on the amount of
thermal recovery between 25 and "'200oK. On the other hand,
large recovery of electrical resistivity occurred, centered in the
range indicated by t;T, following bombardment at 85°K.
greater than the possible contraction around a vacancy. §
The expansion of the close-packed metal copper upon
bombardment (Fig. 2) can be explained on these models,
since detailed calculations indicate that a Frenkel pair
produces a net positive dilatation in that metal:
ji+ jv"-'1.2,12 There is very little corresponding infor
mation concerning the dilatations around point defects
in valence crystals. In a rigid sphere model the large
unoccupied tetrahedral holes can accommodate inter
stitials without any distortion, but the disruption of
the covalent bonding will undoubtedly produce an ap
preciable lattice distortion.
Calculations based upon the simple theory of dis
placements 4 indicate dilatation strengths of the point
defects which are physically reasonable. We should
expect about l.4X 1020 Frenkel pairs for 1017 deuterons/
cm2 with lO.2 Mev energy if the displacement energy
is 30 ev, and if there are on the average 6 progeny per
primary. Putting this value into Eq. (la) and using the
6.10.5 ~
~ 5 -' <I 4
:; 3
~ 2
C
<I
...J 0
<I :;, ·1 a
in ·2 III a::
-3 o LATTICE PARAMETER 0
A LENGTH I).
I).
I).
I). o
A
0
0
0
300 320 340 360 380 400 420 440
RECOVERY TEMPERATURE OK
FIG. 6. Thermal recovery of lattice parameter changes in
deuteron-irradiated germanium, according to Simmons. 16 The re
covery in the range 320 to 3800K closely parallels that of the
macroscopic length, but the residual effect appears to be smaller
in magnitude, becoming in fact a slight lattice contraction near
385°K. Such a contraction is consistent with the presence of
excess centers of dilatation of negative strength.
§ Note added at Conference.-It has been suggested by G. J.
Dienes, on the basis of necessarily crude calculations, that the
dilatation due to a vacancy may actually be positive in german
ium. In this case, of course, there is no problem in explaining the
observed expansion.
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:591254 R. O. SIMMONS AND R. W. BALLUFFI
observed length expansion, we find that !i+ !v=0.14.
The experiments with copper9,tO indicate that the simple
theory overestimates the number of defects produced
by a factor of about 6 in that metal. !i+!v may then
be as large as about 0.8 in germanium. It is possible, of
course, that Ci~Cv directly after bombardment. This
possibility seems rather remote, however, since it re
quires mechanisms for the preferential annealing out of
one type of defect at the low temperature (25°K).
The observed bombardment expansion is also clearly
consistent with model III. All that we require is a
volume expansion in the defect regions due to a local
change in atomic structure. The structure of the defect
region, however, must be rather different from that of
a frozen-in liquid, since melting causes a 5% volume
contraction in germanium. Further experimental evi
dence is cited in Sec. III(a) which indicates that model
III probably does not hold for germanium. We note
here that, by itself, the expansion of copper during
irradiation is consistent with model III; moreover,
copper expands about 4% upon melting. However,
there is a body of evidence which indicates that model
III does not represent a predominant contribution to
the primary damage in deuteron-irradiated copper.
First, the observed rate of increase of electrical resis
tivity during bombardment16 combined with theoretical
estimates of the resistivity contribution due to Frenkel
pairs of about 3.6,uQ cm/atom percent17 give a very
consistent result; the density of Frenkel pairs is ob
tained from combining expansion measurements9,tO with
theoretical work.I2 Second, the detailed correspondence
of thermal recovery of electrical resistivity at low tem
peratures observed between deuteron18 and 1.4-Mev
electron19-irradiated copper indicates a similar fine
structure in the annealing spectrum. Simple displace
ment theory shows that for l.4-Mev electron-irradiated
copper insufficient energy is available to produce model
III.
Thermal recovery of the length expansion (Fig. 5)
occurred gradually over a broad temperature range and
was almost complete at 360°C. No detectable anneal
ing, within the experimental error of ±20%, was ob
served below about 200°C. The residual length and
lattice parameter values during final annealing above
room temperature are shown in Fig. 6. The lattice
parameter showed a slight expansion (3 X 10-5) after
warming to 3200K which annealed out by further
heating to about 430oK. A slight lattice contraction
(-3X 10-5) developed near 385°K.
The precision of the 11 a/ a measurements is somewhat
greater than the I1L/ L measurements, and it is difficult
16 Cooper, Koehler, and Marx, Phys. Rev. 97, 599 (1955).
17 For a review see F. J. Blatt, in Solid State Physics, edited by
F. Seitz and D. Turnbull (Academic Press, Inc., New York, 1957),
Vol. 4, p. 199. Also R. J. Potter and D. L. Dexter, Phys. Rev. 108,
677 (1957); H. Stehle and A. Seeger (to be published).
18 Magnuson, Palmer, and Koehler, Phys. Rev. 109, 1990 (1958).
19 Corbett, Smith, and Walker, Phys. Rev. (to be published). to conclude whether the apparent difference between
these quantities is real, since the disagreement is near
the sum of the respective estimated experimental errors.
It should be noted that the samples studied for lattice
parameter changes had been subjected to a rather
complex sequence of damage and annealing operations.
The main conclusion to be reached is that both the
length and lattice parameter expansion are small and
nearly equal after annealing to room temperature.
This result is consistent with models I and II if both
the vacancies and interstitials are largely annealed out.
Such a state could be reached by mutual annihilation
or else by separate anriealing at sinks. The slightly
negative 11 a/ a values near 385 OK are of particular
interest, and can be explained in models I or II if a
small excess vacancy concentration develops during the
final stages of annealing and if !v is negative. It is pos
sible that interstitials are more mobile than vacancies
and that they have a greater chance of annealing at
sinks. The slight contraction is difficult to explain purely
in terms of a simple form of model III, since a compli
cated volume change reversal would be required.
It would be valuable at this point to have a complete
set of 11 a/ a measurements during bombardment and
annealing to complement the available I1L/ L measure
ments. Such measurements are currently being carried
out by the writers at the University of Illinois. The
very broad range of the thermal recovery of length
changes in germanium is similar to that observed in
other predominantly covalently-bonded materials such
as diamond, silicon carbide, and fused silica.20 For the
latter substances, the measured activation energy spec
tra are correspondingly broad, however having one
well-defined peak. Stored energy measurements on
germanium and silicon would furnish additional valu
able information.
It is of interest in connection with the present dis
cussion to mention briefly some results of combined
measurements of I1L/ Land l1a/ a in radiation damage
studies in other materials. Binder and Sturm21 measured
I1L/ Land 11 a/ a on pile-irradiated LiF and found ex
pansions which agreed within about 6%. They assumed
on this basis that models I or II held and that the
vacancy and interstitial concentrations were closely the
same. Adam and Martin22 have examined molybdenum
pile-irradiated at 30°C and found that I1L/ L increased
about 2.S times as rapidly as l1a/ a during irradiation.
They concluded that model I or II held and that a
higher concentration of vacancies than interstitials de
veloped during irradiation. This situation could occur
if the interstitials were more mobile than the vacancies
20 Primak, Fuchs, and Day, Phys. Rev. 103, 1184 (1956); W.
Primak and H. Szymanski, Phys. Rev. 101, 1268 (1956); J. H.
Crawford, Jr., and M. C. Wittels, in Proceedings of the Inter
national Conference on Peaceful Uses of Atomic Energy (United
Nations, New York, 1956), Vol. 7, p. 654.
21 D. Binder and W. J. Sturm, Phys. Rev. 96, 1519 (1954);
107, 106 (1957).
22 J. Aclam and D. G. Martin, Phil. Mag. 3, 1329 (1958).
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:59LA TTl C E AND LEN G THe HAN G E S I ~ 0 E UTE RON 0 A MAG E 0 G e 1255
at the pile temperature. Further examples of t:.L/ L
= t:.a/ a in various solid solutions are given by Berry.23
III. OTHER X-RAY MEASUREMENTS
(a) Low Angle X-Ray Scattering
Low angle x-ray scattering appears to be a promising
tool for the study of radiation damage. Appreciable
scattering of x-rays at small diffraction angles will
generally occur when small regions are present which
have a different electronic density from the matrix.
The diffracted intensity does not depend upon the de
tailed atomic configuration in the defect region but
depends only upon the size, shape, and average absolute
difference in electronic density. Appreciable scattering
may be expected up to angles given to an order of
magnitude by
cp=A/2l,
where A is the x-ray wavelength and l is the average
size of the scattering regions.24 For example, with
Cu Ka radiation, cp""-'2° when l= 20 A. This technique,
therefore, should be useful for investigating the possible
existence of models II and III. We also note that double
Bragg scattering, which has often given an unwanted
contribution in scattering experiments in cold worked
metals,25 may be readily avoided in radiation damage
studies. This may be accomplished by using single
crystals with orientations well away from any Bragg
reflecting orientation, since appreciable misorientation
tilting should not occur during irradiation. Experi
mental difficulties still remain of course. Among them
is scattering by irregularities in the crystal surface.26
Fujita and Gonser7 have recently reported fairly
strong low-angle x-ray scattering from deuteron-irra
diated germanium at about 900K after irradiation near
90oK. Their scattering curve of intensity versus the
square of the scattering angle cp was approximately
Gaussian indicating that defect regions with a radius
of about 30 A were present. Upon annealing to room
temperature the scattered intensity decreased by a
factor of about S. This experiment indicates that model
I does not hold for germanium under these conditions,
and we must conclude that the point defects are either
present in clusters (model II) or that model III holds.
Model IV is excluded here because the scattered in
tensity from dislocations is weak.28 Vook and Balluffi29
23 C. B. Berry, J. Appl. Phys. 24, 658 (1953).
24 A. Guinier and G. Fournet, Small Angle Scattering of X-Rays
(John Wiley & Sons, Inc., New York, 1955), p. 3.
25 See, for example, M. B. Webb and W. W. Beeman, Acta Met.
7, 203 (1959).
26 W. H. Robinson and R. Smoluchowski, J. Appl. Phys. 27,
657 (1956); S. N. Zhurkov and A. I. Slutsker, Zhur. Tekh. Fiz.
27, 1392 (1957).
27 F. E. Fujita and U. Gonser, J. Phys. Soc. Japan 13, 1068
(1958).
28 See, for example, H. H. Atkinson and P. B. Hirsch, Phil. Mag.
3,313 (1958); Phil. Mag. 3, 476 (1958). .
29 F. L. Vook ann R. W. Balluffi, Phys. Rev. 113, 72 (1959). have attempted to reconcile the small length expansion
of deuteron-irradiated germanium with the fairly strong
x-ray scattering on the basis of models II and III. In
general it might be expected that the electronic density
differences which are required for strong low angle scat
tering would require large mass density differences and
hence correspondingly large length changes. Detailed
examination29 of this problem appears to favor model II.
If separate clusters of vacancies and interstitials are
present their dilatations may largely cancel, since they
act as negative and positive centers. However, regions
of both excess and deficit electron density will make
positive contributions to the x-ray scattering. The
small length change and strong low angle scattering
are, therefore, readily explained with model II. Con
siderable difficulty is encountered with model III in
this respect, since unrealistically large strains (td""-'0.2)
are required in the defect regions.29
Further low-angle scattering measurements at lower
temperatures would be desirable at this point. An im
portant question raised by the available data near
900K concerns the origin of the defect clusters in model
II. Presumably they could arise either by diffusive
motion of the defects or by displacement processes
directly upon bombardment. The displacement me
chanism appears questionable, since large displacement
distances of about 60 to 90 A would be required.
(b) Line Broadening, Diffuse Scattering and
the Artificial Temperature Factor
In heavily irradiated materials additional x-ray
effects may be observed due to the presence of the
localized static defects. Theory30.31 and experiment32.33
indicate that the following effects may be observed
under certain conditions; (1) a decrease in the intensity
of the Laue-Bragg maxima, (2) no broadening of the
Laue-Bragg maxima, and (3) diffuse scattering. These
effects bear a close relation to the influence of thermal
agitation of the lattice on x-ray scattering, since the
effects of the atomic displacements around a random
distribution of static dilatation centers approach those
of frozen heat motion. We will not consider cases where
the irradiation is so extensive that the Bragg maxima
are practically wiped out.34
It is of interest to discuss the use of these phenomena
in determining radiation damage models. None of
the x-ray experiments on homogeneously damaged
materials have shown well-established line broaden
ing.10,15.32,33,35,36 The present theory has been mainly
worked out for uniformly distributed dilatation centers
30 K. Huang, Proc. Roy. Soc. (London) A190, 102 (1947).
31 W. A. Cochran and G. Kartha, Acta Cryst. 9, 944 (1956).
32 C. W. Tucker, Jr., and P. Senio, Acta Cryst. 8, 371 (1955).
33 C. W. Tucker, Jr., and P. Senio, Phys. Rev. 99,1777 (1955).
34 See for example G. E. Bacon and B. E. Warren, Acta Cryst.
9, 1029 (1956).
35 M. C. Wittels, J. Appl. Phys. 28, 921 (1957).
36 E. A. Wood and B. W. Batterman (private communication),
reference in G. K. Wertheim, Phys. Rev. Ill, 1500 (1958).
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of approximately atomic size (model I). The line
breadth situation will not be appreciably changed in
models II or III where considerably larger defect
regions may be present. In general, the line width will
be affected only when strain fields are present with a
range of the order of the dimensions of the diffracting
region of the specimen. The localized strain fields in
any reasonable model should, therefore, be of too short
range to affect line widths. Both experiment and
theory, therefore, indicate that little positive informa
tion can be gained from line breadth measurements.
Strong decreases in the intensity of the Bragg
maxima, and increases in diffuse scattering have been
observed in a number of materials subjected to com
paratively heavy irradiation.32,33 These materials have
generally been rather tightly bound compounds which
exhibited very little annealing during irradiation and
which were irradiated to lattice expansions of the order
of 0.1 to 1%. Refined measurements of these effects
presumably could lead to a model of the damage, since
they can be directly related to the atomic displace
ments. However, these effects have not yet been ob
served in metalslO or semiconductors!· during or after
low temperature bombardment. The failure to observe
such effects has undoubtedly been due to the com
paratively low defect concentrations present, since the
lattice expansions were smaller by one to two orders of
magnitude. It appears that it would be quite imprac
tical to damage germanium and silicon (and metals)
sufficiently at low temperatures to give strong diffrac
tion effects of this type. For example, calculations31
indicate barely detectable effects from the presence of
one percent of interstitials in copper. Concentrations of
this magnitude would require a bombardment of the
order of 103 hr near helium temperature with deuterons
in a typical cyclotron experiment. In addition, there
are formidable problems in reconstructing the atomic
positions from the diffuse scattering data.
Recent work37 proves that additional effects may be
produced by the introduction of point centers of dilata
tion if the crystal is initially highly perfect. With
highly perfect crystals the integrated intensity of a
Laue-Bragg reflection may be increased by a reduction
of primary extinction caused by the strain. This effect
was not considered in the previously cited work. We
conclude that attempts to use Laue-Bragg reflection
and diffuse scattering intensities for investigating spe
cific damage models in germanium and silicon (and
metals) at low temperature will be difficult both ex
perimentally and theoretically and may be less re
warding than other available methods.
IV. COMPARISON WITH ELECTRICAL
MEASUREMENTS
We make no attempt here to review the various
electrical effects present in irradiated semiconductors,
31 B. W. Batterman, J. Appl. Phys. 30, 508 (1959). but content ourselves with a number of observations
on relationships to be expected between them and the
measurements of the type considered in this paper.
First of all, we mention several difficulties which are
encountered in electrical measurements which are either
absent or can be avoided in length and x-ray measure
ments. Difficulties arise in applying most electrical
techniques at low temperature where the state of the
solid is presumably more nearly that due to the primary
damage alone. Observed changes in electrical properties
may be due not only to changes in the number, type, or
arrangement of structural defects but also to secondary
factors. One is the presence of the relatively high con
centrations of chemical impurities which have been
used in many experiments to manipUlate the position
of the Fermi level. While the impurities themselves
may neither participate directly in the primary damage
processes nor be electrically active parts of the damage
sites, their role in stabilizing the imperfections whose
properties are investigated is unknown, and, presently
at least, no detailed and independent study of chemical
impurity effects is available. Additional complications
may be caused by varying thermal ionization of the
chemical impurities. Expansion measurements, of course,
are best made in materials of the highest possible
purity. Another difficulty with electrical measurements
is that trapping effects with long time constants may
make it difficult to obtain thermal equilibrium condi
tions for carriers in times convenient for measurement.2
It is of interest to consider the possibility that
changes in the occupation of particular electronic states
in semiconductors will generally cause slight volume
dilatations of the material. In this case there is a direct
and close relationship between electrical properties and
expansions. In general, however, changes in the electron
distribution will exert a much larger influence on the
electrical properties than on the volume. Some slight
readjustment of the equilibrium nuclear positions
around a point defect may be expected depending on
the particular electronic states occupied. This can be
seen from crude consideration of electrostatic forces on
neighboring nuclei using the hydrogenic model in a
dielectric medium, a case in which the electronic wave
functions are relatively diffuse.3s Alternatively, a mole
cular orbital type construction of possible electronic
states of a neutral vacancy in diamond39 indicates that
the ground state may be degenerate and hence not
stable unless J ahn-Teller40 distortions occur which re
sult in less than tetrahedral symmetry for the defect.
The magnitude of the readjustment to be expected has
not been calculated, but it is probably small compared
to the mean distortion of the lattice characteristic of the
38 For a general discussion of this approach see J. C. Slater,
Phys. Rev. 76, 1592 (1949).
39 c. A. Coulson and M. J. Kearsley, Proc. Roy. Soc. (London)
A241,433 (1957).
4<l H. A. Jahn and E. Teller, Proc. Roy. Soc. (London) A161,
220 (1937).
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:59LATTICE AND LENGTH CHANGES IN DEUTERON DAMAGED Ge 1257
point defect itself. It is apparent that careful theo
retical estimates of lattice distortions would be of con
siderable interest, as well as more detailed models giving
electrical properties.41 The small volume dilatations just
considered do not affect the validity of the results in
Sec. II(a). We simply have a case where the strength of
the centers of dilatation depends slightly upon the
electron occupation. The experimental result that little
length change was observed while annealing irradiated
germanium over a broad temperature range where large
changes in electrical properties occurred14 (Fig. 5) in
dicates that electronic distortional effects must be rela
tively small.
Any detailed comparison of thermal recovery of
electrical effects with volume changes and x-ray meas
urements requires consideration of another point. Vari
ous investigators have ascribed intermediate stages in
thermal recovery as due to association of the point
defects produced by irradiation.42 Association is de
fined as presence of the defects so close together that
the effects produced are characteristic of the complex
and are not simply effects which can be treated as
small perturbations from the sum of the contributions
of widely separated individual defects. It seems quite
certain that association occurs at considerably larger
defect separation distances for electrical effects than for
dilatational effects. The range of electrical interactions
in semiconductors can be relatively large, because of the
large dielectric constant. On the other hand, the effec
tive size of the nonelastic core of a point defect is much
smaller j dilatational effects are probably simply addi
tive for defect spacings as small as distances of the order
of the interatomic spacing. These properties have been
used by Vook and Balluffi29 as a possible means of
reconciling the observed large changes in resistivity of
deuteron-irradiated germanium between 85 and 2000K
with the observed absence of appreciable volume
changes. Vook and Balluffi29 conclude that compara
tively simple forms of model II, consisting of clusters
of vacancies and clusters of interstitials, are probably
consistent with the experiments which have been made
with deuteron-irradiated germanium to date. Unfor
tunately, we cannot deduce the shapes of these regions
from the presently available data. Other more compli
cated models are not excluded, of course.
In a comparison of electrical measurements with
volume change measurements it is of interest to con
sider the available measurements below liquid nitrogen
temperature. The measurements of Vook and Balluffi14
on germanium irradiated with 10.2-Mev deuterons at
"-'25°K showed no indication of length change recovery
41 Crude models for isolated vacancies and interstitials in ger
manium based on the two types of assumptions mentioned in the
text have been put forward by H. James and K. Lark-Horovitz,
Z. physik. chern. (Leipzig) 198, 107 (1951) and E. I. Blount, Phys.
Rev. 113,995 (1959).
4l! See, for example, J. W. Cleland and J. H. Crawford, Jr.,
Phys. Rev. 98, 1942 (1955), and reference 29. below "-'200oK within an experimental accuracy of
about ±20%. Also, the electrical measurements of
Cleland and Crawford43 on n-type germanium irradiated
with pile neutrons at "-' lOOK showed no recovery below
95°K. However, Gobeli44 irradiated n-and p-type
germanium with 3.7-Mev alpha particles at 4.2°K and
found "-' 25% recovery of electrical effects during warm
ing to 78°K. Also, MacKay, Klontz and Gobeli45 irra
diated n-type germanium with 1.1O-Mev electrons at
"-' WOK and found "-' 50% recovery of electrical effects
upon warming to 80oK. The latter authors44.45 inter
preted their results in terms of the annihilation of
carrier trapping centers. It is possible, of course, that
this process produced only a small length change. It
was further suggested,44.46 however, that these trapping
centers disappeared by the mutual annihilation of close
vacancy-interstitial pairs. It is also possible that this
mechanism is not inconsistent with the present experi
mental results. It has been found in copper, for example,
that the relative amount of recovery below liquid nitro
gen temperature depends upon the type of irradiation
and increases in the order: neutrons,46 deuterons,18 and
electrons.47 Presumably, the low-temperature recovery
is due to close-pair annihilation in copper,1l·18 and the
relative number of close pairs increases with decreasing
average energy of the primary displaced atoms. It is
also possible that the lower temperatures achieved in
the alpha particle and electron bombardments increased
the number of thermally unstable defects in the low
temperature recovery range. The experimental error in
the length change recovery measurements was large
enough to mask recovery effects of the order of 20%
which are of the same magnitude as those found after
the alpha particle irradiation.44 It should be noted that
the accuracy of volume change and dilatation measure
ments will always be lower than the highly sensitive
electrical measurements. Complications in the electrical
measlirements due to different impurity concentrations
may also have been significant. Greatly different fluxes
were also used in several of the experiments. The recent
result that p-type germanium, lightly irradiated with
1.10 Mev electrons near WOK, exhibits no annealing
below 130oK48 is of particular interest. The material of
Vook and Balluffi, became p type after heavier irradia
tion14 and presumably became converted to p-type rela
tively early during bombardment. 2 The sensitivity of
the IlL/ L measurement was, of course, inadequate to
indicate clearly a possible corresponding change. in the
43 J. W. Cleland and J. H. Crawford, Jr., J. App\. Phys. 29,
149 (1958).
44 G. W. Gobeli, Phys. Rev. 112, 732 (1958).
45 MacKay, Klontz, and Gobeli, Phys. Rev. Letters 2, 146
(1959).
46 Blewitt, Coltman, Holmes, and Noggle, Creep and Recovery
(American Society for Metals, Cleveland, Ohio, 1957), p. 84.
.7 Corbett, Denney, Fiske, and Walker, Phys. Rev. 108, 954
(1957).
48 J. W. MacKay and E. E. Klontz, J. App!. Phys. 30, 1269
(1959,) this issue.
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IP: 132.236.27.111 On: Mon, 15 Dec 2014 14:40:591258 R. O. S I M M 0 N SAN DR. \V. B ALL U F F I
rate of expansion. The absence of low temperature re
covery in the two cases may therefore have a common
explanation. On the other hand, Gobeli44 reported low
temperature recovery in p-type as well as n-type
material.
The previous discussion emphasizes the need for
further measurements of both the electrical and struc
tural types. The structural measurements of the types
described in the present paper have certain advantages and disadvantages. The measurements can be made at
arbitrary temperatures on pure material, and they are
not strongly dependent upon the secondary details of
the electronic structure of the damage sites. However,
the structural measurements require a mean density of
defects of about 1018/cm3 to allow detailed analysis,
and even then, the measuring accuracy is lower than
that of electrical measurements. Their usefulness as a
complement to other methods is clearly apparent.
JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 8 AUGUST, 1959
Annealing of Radiation Defects in Semiconductors*
W. L. BROWN, W. M. AUGUSTYNIAK, AND T. R. WAITEt
Bell Telephone Laboratories, Inc., Murray Hill, New Jersey
Radiation induced defects studied through changes in conductivity and Hall coefficient have been ob
served to anneal in a number of different temperature ranges. Only those processes occurring above SocK
and involving defects created by electron irradiation have been considered in this paper. It has been found
that the first annealing process in n-type germanium occurs at about 50°C and is structure sensitive, ap
parently to the original chemical donor impurity. Higher temperature annealing processes, observed at
about 200°C and previously interpreted as due to direct annihilation of vacancies and interstitials must also
be sensitive to other crystal defects. In p-type germanium there is a process of rearrangement of a defect
center at about 200oK, exhibiting first order kinetics, but with a time constant which is strongly dependent
upon the charge state of the defect. At about 120°C the defects in p type apparently anneal out completely,
in striking contrast to the n-type case. Less extensive silicon measurements, showing lifetime recovery be
tween 200 and 400°C again indicate through their kinetics the importance of other impurities or defects in
the annealing process.
INTRODUCTION
THE study of annealing of irradiation produced
defects is an aspect of the over-all problem which
has tremendous possibilities. It offers the opportunity
for examining the motion of vacancies and interstitials,
the simplest of point defects, under conditions in which
their density is subject to precise control. An under
standing of the defects themselves must necessarily in
clude an understanding of the kinetics of their motion
in a crystaL
In the semiconductors the potentialities are par
ticularly intriguing. There is tremendous variety and
sensitivity in the techniques available for detecting the
defects. The understanding of the structure and the
electronic properties of the crystals in which they are
studied is at a very high leveL With this background it
should be possible to examine in detail the interaction
of these defects with each other as they annihilate and
cluster. It should also be possible to examine their inter
action with other types of imperfections, the chemical
impurities, dislocations, and surfaces that compete with
annealing as the defects move in the crystal. There is
little doubt that an understanding of these processes
* This work was supported in part by the Wright Air Develop
ment Center of the U. S. Air Force. t Present address: California Institute of Technology, Pasadena
California. ' will evolve. This paper will describe a few systematic
attempts that have been made at achieving an under
standing of annealing. It will allude to some of the clues
which have been obtained in experiments aimed less
directly at this aspect of the radiation defect problem.
It will not succeed in giving a complete and consistent
picture of the phenomena involved.
Almost all of the remarks to follow will be limited
to the specific cases of germanium and silicon because
they have been most widely studied and offer, or seem
to offer, the best chance of being understood. We will
primarily consider the case of defects introduced by
electrons, with energies the order of a Mev or by
gamma rays of similar energy whose ultimate inter
action with the nuclei in producing defects occurs
through photo-or Compton electrons. This restriction
amounts to assurance that the vacancy interstitial
pairs are produced at random through the crystal and
not in small regions containing a large number of other
pairs. In principle at least this should represent the
simplest possible situation. In actuality there are many
similarities between the results for electron and for
heavy particle irradiations, for example in some of the
prominent energy levels they introduce.1-s Comparison
I!I. Y. Fan and K. Lark-Horovitz, Proceedings of the Inter
natlOnal Conference on Semiconductors, Garmisch-Partenkirchen
(1956).
• Cleland, Crawford, and Holmes, Phys. Rev. 102, 722 (1956)_
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1.1744052.pdf | Theory of Isotropic Hyperfine Interactions in πElectron Radicals
Harden M. McConnell and Donald B. Chesnut
Citation: The Journal of Chemical Physics 28, 107 (1958); doi: 10.1063/1.1744052
View online: http://dx.doi.org/10.1063/1.1744052
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Published by the AIP Publishing
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130.113.111.210 On: Fri, 19 Dec 2014 08:42:14THE JOURNAL OF CHEMICAL PHYSICS VOLUME 28. NUMBER 1 JANUARY. 1958
Theory of Isotropic Hyperfine Interactions in ?t-Electron Radicals*
HARDEN M. MCCONNELL AND DONALD B. CHESNUTt
Gates and Crellin Laboratories of Chemistry,t. California Institute of Technology, Pasadena, California
(Received August 1, 1957)
Indirect proton hyperfine interactions in 1l'-electron radicals are
first discussed in terms of a hypothetical CH fragment which holds
one unpaired 1" electron and two O'-CH bonding electrons. Molecu
lar orbital theory and valence bond theory yield almost identical
results for the unpaired electron density at the proton due to ex
change coupling between the 1" electron and the 0' electrons. The
unrestricted Hartree-Fock approximation leads to qualitatively
similar results. The unpaired electron spin density at the proton
tends to be an tiparallel to the average spin of the 1" electron, and
this leads to a negative proton hyperfine coupling constant.
The theory of indirect proton hyperfine interaction in the CH
fragment is generalized to the case of polyatomic lr-electron radical
systems; e.g., large planar aromatic radicals. In making this
generalization there is introduced an unpaired 1l'-electron spin
density operator, ~N, where N refers to carbon atom N. Expecta
tion values of the spin density operator ~N are called "spin densi
ties," PN, and can be positive or negative. In the simple one
electron molecular orbital approximation a l1'-electron radical
always has a positive or zero spin density at carbon atom N,
0:( PN:( 1. In certain lr-electron radical systems; e.g., odd-alternate
hydrocarbon radicals, the spin densities at certain (unstarred)
carbon atoms are negative when the effects of 1r-1l' configuration
interaction are included in the lr-electron wave function.
I. INTRODUCTION
THIS paper presents further studies on the relation
between the molecular electronic structure of
1T-electron radicals and the isotropic proton hyperfine
splittings which are observed in the electron magnetic
resonance spectra of these radicals in liquid solutions.l
By 1T-electron radicals we refer especially to the positive,
neutral, and negative ion radicals of planar aromatic
hydrocarbons where, to a first crude approximation, the
unpaired electron(s) moves in a 1T-molecular orbital
antisymmetric with respect to the molecular plane. The
hyperfine splittings of interest are those arising from in
plane aromatic protons «(J' protons) which have now
been observed in numerous aromatic molecular radicals.1
As has been pointed out by McConnell/-4 Bersohn,6
Weissman,6 and Fraenkel,7 the observed proton hyper
fine splittings result from (J'-1T electron exchange inter
action. In fact, it has been suggestedl-4 that the nature
of this (J'-1T interaction is such that the proton hyperfine
splittings can be used to measure unpaired electron
distributions on the carbon atoms. In particular, it was
proposed that if aN is the hyperfine splitting due to
* Sponsored by the Office of Ordnance Research, U. S. Army.
t Shell Oil Company Predoctoral Fellow. t Contribution No. 2264.
1 H. M. McConnell, Ann. Rev. Phys. Chern. 8, 105 (1957).
2 H. M. McConnell, J. Chern. Phys. 24, 764 (1956).
3 H. M. McConnell, J. Chern. Phys. 24, 632 (1956).
• H. M. McConnell, Proc. Nat!. Acad. Sci. U. S. 43, 721 (1957).
6 R. Bersohn, J. Chern. Phys. 24, 1066 (1956).
6 S. I. Weissman, J. Chern. Phys. 25, 890 (1956).
7 B. Venkataraman and G. K. Fraenkel, J. Chern. Phys. 24, 737
(1956). The previously proposed linear relation between the hyperfine
splitting due to proton N, aN, and the unpaired spin density on
carbon atom N, PN,
is derived under very general conditions. Two basic approxima
tions are necessary in the derivation of this linear relation. First,
it is necessary that 0'-1l' exchange interaction can be treated as a
first-order perturbation in lr-electron systems. Second, it is neces
sary that the energy of the triplet antibonding state of the C-H
bond be much larger than the excitation energies of certain doublet
and quartet states of the 1l' electrons. This derivation of the above
linear relation makes no restrictive assumptions regarding the
degree of 1l'-1l' or 0'-0' configuration interaction. The validity of
the above approximations is discussed and illustrated by highly
simplified calculations of the proton hyperfine splittings in the
allyl radical, assuming the 1l'-1r configuration interaction-and
hence the negative spin density on the central carbon atom-to be
small.
Isotropic hyperfine interactions in molecules in liquid solution
can also arise from spin-orbital interaction effects, and it is shown
that these effects are negligible for proton hyperfine interactions in
aromatic radicals.
aromatic proton N, then aN is related to the "unpaired
electron density" at carbon atom N by the simple
equation,
(1)
Here Q is a semiempirical constant, Q= -22.S gauss, or
-63 Mc, and Q is assumed to be approximately the same
for all aromatic CH bonds.8(a) A recent review of experi
mental and theoretical work bearing on (1) is given
elsewhere.l Equation (1) has recently been derived using
the Dirac vector approximation for the (J' and 1T elec
trons.4 The purpose of the present paper is to give a
much more general derivation of this equation. A pre
liminary account of this aspect of the present work has
been published.8(b)
Brovetto and Ferroni9 have recently assumed a linear
relation similar to (1) in order to interpret the proton
hyperfine splittings in the triphenylmethyl radical in
terms of a spin distribution calculated using the Pauling
and Wheland10 valence bond functions for this molecule.
These authors9 were the first to propose on theoretical
grounds that the proton hyperfine splittings are both
positive and negative in this (odd-alternate) radical.
Because Brovetto and Ferroni did not recognize the
indirect character of the (1T-electron spin-proton spin)
hyperfine interaction, their calculated coupling con
stants are here predicted to be in error by a factor of
8 (a) H. M. McConnell and H. H. Dearman, J. Chern. Phys. 28,
51 (1958); (b) H. M. McConnell and D. B. Chesnut, J. Chern.
Phys. 27, 984 (1957).
9 P. Brovetto and S. Ferroni, Nuovo cimento 5, 142 (1957).
lOL. Pauling and G. Wheland, J. Chern. Phys. 1,362 (1933).
107
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minus one, aside from other possible intrinsic errors in
the valence bond approximation itself, or in the linear
relation between aN and PN, or in the assumed planarity
of triphenylmethyl.
All of the conclusions reached to date on the origin of
the proton hyperfine splitting in radicals which are
observed in liquid solutions have been based on
Weissman's conclusions that these splittings arise from
the Fermi contact interactionY Weissman's work was in
turn based on neglect of spin-orbit interaction effects; in
the present paper a simple model is used to show that
spin-orbit interaction effects can lead to isotropic
hyperfine splitting in solutions, but that these effects are
'" 100 times too small to account for the observed
splittings in aromatic radicals. On the other hand, spin
orbit effects may be significant in giving pseudo-contact
hyperfine splittings in other paramagnetic molecules in
solution.
II. CONTACT HYPERFINE HAMILTONIAN
For large applied magnetic fields (Paschen-Back re
gion), we need only consider the hyperfine interaction
between the z components of the electron and nuclear
spins; in this approximation the Fermi contact Hamil
tonian which gives the nonvanishing isotropic hyperfine
splitting for molecules in solution is
JCN= (81rgl.6I/3)(J.tN/I)Lk o(rkN)SkzI Nz, (2)
where 1.6 I is the absolute magnitude of the Bohr
magneton, o(rkN) is the Dirac delta function of the
distance rkN between electron k and nucleus N, J.tN is the
magnetic moment of proton N, and INz is the z com
ponent of the spin of proton N, in units of h. Equation
(2) can also be expressed in terms of the "coupling
constant" for proton N, aN,
(3)
where S z is the total z component of the electron spin
angular momentum. Thus, if WI is the ground-state
electronic wave function,
aN= (81rgl.6I/3h)(J.tN/I)oN, (4)
ON=(W11 L k o(rkN)S kz I W1)/ SZ
=(w1IoNlw1)/Sz. (5)
We seek to study the relation between aN and the
unpaired electron distribution.
Absolute values of aN, I aN I, are easily deduced from
hyperfine splittings in high field electron magnetic
resonance spectra. The signs of the aN'S are considerably
more difficult to determine experimentally but are of
considerable theoretical importance, as will be shown
later in the present work. The signs of the aN'S can in
principle be determined if the nuclear resonances of
protons N can be observed, and if shifts in these
resonances are sufficiently large and are dominated by
11 S. I. Weissman, ]. Chern. Phys. 22, 1378 (1954). the contact hyperfine interaction. This method is de
scribed below.
In a system characterized by a paramagnetic relaxa
tion time Tl, or an exchange time Te such that Tel or
Te-l»aN, then proton N will see a single average
hyperfine magnetic field corresponding to the effective
spin Hamiltonian for proton N
JC= -J.tNz(Ho-21raN(Ih/J.tN)(Sz»), (6)
where (Sz) is the time average value of the z component
of the electron spin. For systems obeying the Curie law,
(Sz)
so that -gl.6IS(S+l)H o
3kT (7)
X= -J.tNz(1+21raN{ Ih}gl.6IS(S+l))Ho. (8)
J.tN 3kT
Since J.tN is positive for the proton it is seen from (8)
that when the proton resonance of N is observed at a
fixed frequency, the contact shift will be to lower applied
fields when aN is positive, and to higher applied fields
when aN is negative. In substances with unpaired
electrons in open shell s orbitals, the electron spin
nuclear spin interaction is direct and the quantity
corresponding to ON in (5) is positive and aN has the sign
of J.tN. In such cases aN/J.tN is always positive and the
second term in the parentheses in (8) is positive. Thus,
"normal" nuclear paramagnetic shifts are to lower
fields. As will be seen, a characteristic feature of the
calculated aN'S in aromatic radicals is that they are
most often, but not always, negative; the predicted
proton shifts are to higher applied fields, at constant
(applied rf) frequency.
It is interesting to note that the above conditions on
the validity of (6) and (8) for the nuclear resonance
shift with regard to the relative value of aN and T 1-1 or
T .-1 imply the absence of observable hyperfine splittings
in the same system for which (6) and (8) are valid.
That is, if Te-1»aN, the hyperfine splittings are ob
scured by exchange narrowing, and if T1-1»aN the
paramagnetic line widths are greater than the splittings.
This is clearly a convenient experimental criterion for
the observability of contact nuclear resonance shifts.
III. ELECTRON INTERACTION IN A
CH FRAGMENT
To begin our treatment of hyperfine splittings in
planar aromatic radicals, it is convenient to consider, as
before, a hypothetical CH fragment which is abstracted
from an aromatic radical. We consider the U-1r electron
interaction between the two u-bonding electrons, and
the single unpaired 1r electron which is considered to be
in a pure 2pz atomic orbital. The occurrence of unpaired
spin density at the proton can be understood qualita
tively in terms of three approximate treatments of
molecular electronic structure: (a) The valence bond
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model with a single configuration, (b) a molecular orbital
model with U-7r configuration interaction, and (c) the
"unrestricted Hartree-Fock method." The basic in
gredients of these three methods in their application to
the three electron problem of the CH fragment are
summarized below.
(a) Valence Bond Approximation
McConnell2 has applied simplified valence bond theory
to the CH fragment. Jarrett12 has proposed quantitative
improvements on these calculations. Bersohn6 has carried
out a rather elaborate calculation of (j-7r electron
interaction in planar C2H4+, using integrals obtained by
Altman13 in his work on ethylene. We summarize here
the basic ingredients of the valence bond approximation
in its application to this problem, and particular refer
ence will be made to the theoretically deduced sign of
the hyperfine coupling.
Let h denote an Sp2 hybrid orbital centered on the
carbon atom and directed towards the proton; p is a 2p.
atomic orbital centered on the carbon atom and is used
in building up the 7r-molecular orbitals for the complete
7r-electron system. The hydrogen atom ls orbital, s, is
centered on the proton in the CH fragment. A single
covalent CH bond is represented by the normalized
doublet state eigenfunction,
(9)
In (9), ~ is the projection operator for antisymmetriza
tion and renormalization and So is the h-s overlap
integral:
So=(hls). (10)
An excited doublet state of the same electron configura
tion which corresponds to antibonding between the
carbon and hydrogen atoms is
In the three electron functions (9) and (11), and in the
polyelectron functions considered later, the labels for
the electron coordinates always appear in serial order.
For example,
phsaa{3= p(1)h(2)s(3)a (1)a(2){3(3).
H denotes the three-electron Hamiltonian for the elec
tronic kinetic energy and electrostatic potential energy,
including nuclear attraction terms. As shown previously,
the "first-order" mixing of if/1o with if/20, gives for the
ground-state wave function,
(12)
12 H. S. Jarrett, J. Chern. Phys. 25, 1289 (1956),
13 S. L. Altman, Proc. Roy. Soc. (London) A210, 327, 343 (1951). where it is assumed 1X,12«1, and where
X,= -H21/AE21, (13)
tJ.E21=H22-Hll, (14)
Hij=(if/lIH! if/l), (15)
H21=[ -v.'f/2(1-So4)t]eJ ph-Jps), (16)
Jph=(phle2/r!hp), (17)
Jps= (ps! e2/rlsp). (18)
The exchange integral in (17) is (p(1)h(2) I e2/t'I2l h(1)p(2»
where r=r12 is the interelectronic distance. From (5)
and (12) we obtain for ON for the CH fragment
Let (aCH)~b denote the theoretically calculated valence
bond hyperfine coupling constant for the CH fragment,
and let aH denote the hyperfine coupling constant in the
hydrogen atom.
-1 (Jph-Jp.)
(acH)vb=-- aH, (20)
1-S04 AE21
aH= (87rgl{3I/3h)(~N/I) I s(o) 12
=1420 Mc. (21)
A safe order-of-magnitude estimate of the energy differ
ence between the two states <PI0 and if/20, tJ.E21, is 5-15 ev.
Altman gives J ph= 1.81 ev, J p.=0.745 ev. We previ
ously estimated J ph= 1.17 ev from the work of Voge.2 In
any event (J ph -J ps) / tJ.E21 is positive and of the order
of 0.1-0.01. Assuming Jarrett's value for the overlap,
So"",0.8, we obtain an estimate of aCH= -20 to -200
Me, which is in good order-of-magnitude agreement
with the "best" semiempirical value for Q, Q= -63 Mc.
(b) Molecular Orbital Theory
Weissman6 has used molecular orbital theory for the
(j electrons in a discussion of aromatic proton hyperfine
splittings. In this section we further develop this theory
with particular reference to the CH fragment, and to the
sign of the hyperfine coupling constant.
As a first approximation to the configuration inter
action problem for the CH fragment we take for the
lowest energy configuration:
(22)
where the u-bonding orbital is a linear combination of
the atomic and hybrid orbitals, sand h:
1
(j (s+h).
(2 (1+So))1 (23)
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In (23) we have, for simplicity, assumed equal pro
portions of sand h. That is, ionic character or charge
transfer (other than the overlap effect) has been neg
lected. Configuration (22) gives no proton hyperfine
splitting, but admixture of the doublet state excited
configuration 1/12° does.
Here 0"* is the normalized antibonding orbital orthogonal
to 0":
1
0"*= (s-h).
(2(1-So»$ (25)
A second excited doublet state with the same configura
tion is
(26)
The doublet state functions 1/11°, 1/12°, and 1/13° are taken
here as the basis of a variational calculation for the
mixing of 1/12° and 1/13° with 1/11°. The variational parame
ters 1/2 and 1/3 describing this mixing are assumed to be
small; 11/212«1, 11/312«1
1/11 = 1/11°+1/21/12°+1/31/13°,
1/2= -H21jt:.E21i 1/3= -H~l/t:.E31. (27)
(28)
Here, as before, t:.E21=H22-Hu. The mixing of 1/13°
with 1/11° does not affect the hyperfine splitting as long as
1/3 is small. Corresponding to (15) we have here
HZ1 = -(3h/6)J*, (29)
J*=(0"*ple2/rlpO") (30)
1
=~ (Jps-J ph). (31)
(I-S02)!
From (5) and (27)
4
OCR =----7]20"(0)0"*(0), y6
2 1
=----7]2 Is(0)12 y6 (I-S02)t
= 2(J* / t:.E21) 0" (0)0"* (0) (32)
(assuming I k(O) 12"",0) or From a comparison of (20) and (34) it is clear that the
valence bond and molecular orbital methods, in the
approximations we have used, lead to almost identical
results.
It may be noted that under certain conditions the set
of functions 1/IP, j = 1, 2, 3, 4 (1/14°= 'liO"*O"*pa{3a) consti
tute a "complete" set of functions as far as the present
calculation is concerned. Thus, if 1/IP(j> 1) contributes
to the hyperfine interaction in the above calculations, it
is necessary that (1/I1°IJCI1/Il)r"'0 in order that 1/1/ mix
with 1/11°; it is also necessary that (1/I,.o! ON I 1/11°) be different
from zero in order to obtain a finite contribution to the
hyperfine interaction. Thus, we have a complete set 1/1/
if we include in the calculation all those 1/1 l for which
(1/I1oIJCI1/Il)(1/Ill ON 11/11°) is different from zero. We may
test the completeness of our functions 1/1/ by explicit
evaluation of the left-and right-hand members of the
matrix equation
(1/11°1 JCON 11/11°)= Li(1/I10j JC j 1/1/>(1/1/1 ON I 1/11°). (35)
For the set 1/1/, j= 1,2,3,4, only (1/12°1 ON 11/11°) is different
from zero, so that for our particular set of functions,
(if10 I JCON 11/11°)= (1/11° I JC I 1/12°)(1/12° ION 11/11°). (36)
Evaluation of the left-and right-hand members of (36)
yields the equation,
where Jo-(O) = -J*O"*(O),
J = (O"p I e2jrl pu). (37)
(38)
Equation (37) is obviously not true in general; this can
be seen by noting that (a) J and J* are simple numbers,
and (b) 0"(0) and 0"*(0) refer to a point in space for
which rkN=O. However, the derivation of (37) is equally
valid for any point in space X which lies in the nodal
plane of the p orbital. Therefore (37) is valid only for
points X, and in particular for the point corresponding
to the position of the proton N, when J= -J*, 0"(0)
=0"*(0), or, J=J*, 0"(0)= -0"*(0).
In terms of atomic orbitals, these conditions are met
when k(O) ""'0, (psi e2/rl sp) "'" 0, and {sl h)=O. Since these
conditions on the completeness of the set 1/Il for
j= 1, 2, 3, 4 are only rather crude approximations for
the CH problem, it is clear that in any serious attempt
to make a quantitative calculation of Q one must be
extremely careful to include all interacting excited state
functions. An important feature of the general de
velopment in Sec. V of the present paper is that this
explicit enumeration of the interacting excited states is
avoided.
(c) Unrestricted Hartree-Fock Method 1 (J P.-J ph) OCH=t-- s(O) 12
I-S02 t:.E21 (33) Nesbet14 has pointed out that the unrestricted Hartree-
or,
(34) Fock methodHi can be used to approximate indirect
U R. K. Nesbet (private communication).
16 R. K. Nesbet, Proc. Roy. Soc. (London) A230, 312 (1955).
See P.-O. LOwdin, Phys. Rev. 97, 1509 (1955) and references
therein.
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130.113.111.210 On: Fri, 19 Dec 2014 08:42:14HYPERFINE INTERACTIONS IN 1r-ELECTRON RADICALS 111
hyperfine interactions. A brief sketch of the application
of this approximation to the problem of the CH frag
ment is given below; this qualitative development to a
considerable extent parallels the application of this
method to the lithium atom which is given by PratUS
Using a Hartree-Fock self-consistent field method we
minimize the energy of the CH fragment using 'P as a
variational antisymmetrized single product function,
(39)
In (39), u and u' are two u CH bonding orbitals which
are varied independently of one another in the Hartree
Fock procedure of minimizing the ground-state energy.
It is readily shown that the minimum energy consistent
with the form of (39) corresponds to two distinct
functions, u and u'. Thus, from our previous discussion
of the valence bond and molecular orbital approxima
tions, we expect that 'P will describe the a CH u bonding
electron as being in an orbital preferentially concen
trated near the carbon atom, and the fJu electron as
being more strongly localized near the proton.
The function 'P is not acceptable for calculating either
an electronic energy (kinetic+electrostatic), or a hyper
fine magnetic energy because (39) is not an eigen
function of S2. Two eigenfunctions of S2 can be generated
from 'P as follows:
if;=RlJ1uu'p(afJa-fJaa), (40)
R= 1/ (2(1 + r2»)!, (41)
if;'=R'lJ1uu'p(2aafJ-a/3a-/3aa), (42)
R'=I/(6(1-r2))t, (43)
where the "overlap" integral r is
r={ulu'). (44)
Note first that when u=u', r= 1, if; becomes identical
with 'P, and if;' does not exist. This is the situation when
the U-1r electron interaction is negligible relative to the
u-u electron interaction. When the U-1r electron
interaction is a significant perturbation, u:;t.u', and if;'
mixes in with the ground state:
'It=if;+"Aif;'. (45)
When 1"A12«1,
oN=4"ARR'(lu'(0) 12-lu(O) 12), (46)
ON= (2"A)/(3(1-r4»!(lu'(0) 12-lu(0) 12). (47)
The exchange energy argument again points to a nega
tive value for "A("A<O) and thus negative ON and aN,
since 1 u' (0) 12> 1 u(O) 12. Note that as a first approxima
tion, u and u' could be regarded as different linear
combinations of sand h, so that I u' (0) 12_1 u(O) 12 would
again be proportional to I s(O) 12, We conclude, there
fore, that the unrestricted Hartree-Fock method in its
16 G. W. Pratt, Jr., Phys. Rev. 102, 1303 (1956). application to the problem of indirect hyperfine splittings
involves essentially the same qualitative ideas as those
developed for the valence bond and molecular orbital
methods. In all three cases, the "first-order" mixing of
an "excited state" is required to give indirect hyperfine
splitting, and it is this first-order mixing that we use in
the following to relate to the unpaired spin distribution.
We shall use molecular orbital theory with the formalism
of the restricted Hartree-Fock method although either
of the other two methods could be used to obtain
essentially the same semiquantitative results.
IV. UNPAIRED ELECTRON DISTRIBUTION AND
HYPERFINE SPLITTINGS
The observed isotropic hyperfine splittings in 1r
electron radicals give a direct measure of the unpaired
spin densities at the u protons. In the present work we
seek a quantitative relation between these indirect
hyperfine splittings and the unpaired 1r-electron distri
bution in the 1r-electron radical. In our previous work
we used the concept of the "unpaired electron density at
carbon atom N, PN." This was essentially the proba
bility of finding an unpaired electron in a p.-1( atomic
orbital centered on carbon atom N, and O~PN~ 1 for,
say, an aromatic radical having one unpaired electron.
This concept of the unpaired electron density is only
good in the one electron approximation. In our present
work we must introduce a more general expression of
unpaired electron distribution. This is given by the spin
density operator for carbon atom N, (IN, which is defined
by the operator equation
f}N Lk Skz= Lk AN (k)Skz. (48)
In (48), AN(k) is a previously introduced17 "atomic
orbital delta function" which is essentially a three
dimensional step function such that AN(k) = 1 when
electron k is in a pz atomic orbital on carbon atom N,
and AN(k)=O elsewhere. In the present approximation,
neglect of 1r-1r overlap implies that
(PNz(k) I AN' (k) 1 PN" (k»)= ONN'ON'N'" (49)
As shown later, expectation values of PN can sometimes
be negative, meaning that the unpaired spin density at
carbon atom N has a polarization which is opposite to
the total spin polarization of the molecule. To within the
limits of the neglect of 1(-orbital overlap,
LN (IN= 2S, (50)
where S is the total electron spin of the molecule;
S=!, 1, i, ....
The present task is to show the conditions under
which there exists a simple proportionality between PN
and the hyperfine splitting due to proton N, aN. We
shall use molecular orbital theory to describe the 1(
electron wave functions and for uniformity we also use
molecular orbital theory for the u CH bonds.
17 H. M. McConnell, J. Molecular Spectroscopy 1, 11 (1957).
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130.113.111.210 On: Fri, 19 Dec 2014 08:42:14112 H. M. McCONNELL AND D. B. CHESNUT
V. GENERAL THEORY WITH CONFIGURATION
INTERACTION
In this development we first solve the one-electron
Hamiltonian problem for the 'Ir-electron motion in an
average effective field due to the other 'Ir and rr electrons,
and the nuclei. As before, we let 'lri denote a self
consistent field one-electron orbital for a 'Ir electron,
with equivalence restriction. We let Ui denote a nor
malized configurational wave function for the 'Ir elec
trons; Ui is a simple product of a space function and a
spin function. The antisymmetrized functions
(51)
are then used as a basis for a variational treatment of
the 'Ir-'Ir configuration interaction problem, still neg
lecting rr-'Ir exchange interaction. The eigenfunctions of
this Hamiltonian, which includes 'Ir-'Ir electron inter
action explicitly but uses an effective field for the rr
electrons, are the <Pj:
(52)
Uj= Lk CkiUk. (53)
We let <1h, UI correspond to the lowest energy doublet
state of the 'Ir-electron system.
Precisely the same type of calculation can then be
carried out for the rr electrons. In the present calcula
tions we consider only two rr electrons, which are
assumed to be localized in the region of one CH bond,
denoted as before by N. The generalization of the
calculation to more than one noninteracting CH bond is
trivial. Also, the inclusion of other rr electrons introduces
no important neW feature in the final result. Corre
sponding to the above Ui and Uj for the 'Ir electrons, we
now have for the two rr electrons,
(54)
It is most convenient to choose each of the single
configuration functions Ui, Vj to have permutation
orthogonality. That is,
(udPlui)=O,
(viIPlvi)=O. (55)
(56)
In (55) and (56), P is any electron permutation opera
tor, except the identity. Under these conditions we may
use the same antisymmetrization operator in all the
calculations:
(57)
Here K is the total number of electrons in the function
to be antisymmetrized. We consider a very general CH
rr-bond function, which includes rr-rr configuration
interaction:
VI = rr(JI.)rr(v )a (JI.)(3 (v),
V2= rrCJi.)rr*(v) (a (JI.)(3(v)-(3CJi.)a(v»/V'l. (58)
(59) A third configurational rr-electron state,
Va = rr* (JI. )rr* (v)O' (JI.)(3 (v),
could also be included in the subsequent calculations
without any difficulty as all hyperfine cross terms be
tween Vt and Va vanish. The final expression for Q [Eq.
(77) ] can easily be generalized to include contributions
from Va since they have the same form as do the contri
butions from VI.
Let tfi denote the complete set of antisymmetric
functions for the entire molecule; the only deviations
between the tfi and the exact eigenfunctions are due to
the neglect of rr -11" exchange coupling in deriving the tf i.
In general, the tfi are obtained from antisymmetrized
linear combinations of the products U m V n:
(60)
(61)
We only consider those functions Wi corresponding to a
doublet electronic state. These doublet state functions
Wi may be divided conveniently into two classes, I and
II. The class I functions contain only V n functions
belonging to a singlet electronic state, and U m functions
belonging to a doublet state of the 'Ir electrons. The
functions in class II are derived from linear combina
tions of the V n belonging to a triplet state, and U m
functions belonging to doublet and quartet states of the
'Ir electrons. For brevity, we refer to the Wi in class II as
representing "rr-'Ir excited states." We assume that
tfI=2{W t=2{U 1VI is a good approximation to the
ground electronic state wave function, W G. The first
order contribution of excited states tfi to 'IF G is then
(62)
We now use (62) to calculate the expectation value of ON
to first order.
ON= (w G ION I W G)
= -2 Li(tfllJC I tfi)(tfi I 'ON Itfl)/ ~Eil' (63)
Only rr-1I" excited states contribute to the summation in
(63). We assume here that the rr-electron triplet excita
tion energy is so large that all of the effective excitation
energies ~Eit are approximately equal to ~E. Equation
(63) can then be simplified to give
-2
ON=-Li{tfl1 JCltfi){tfil 'ON Itft).
~E
Further simplification is possible;
1 (64)
(1fil 'ONltfI)=-(WiILk o(rkN)Skzl LP( -l)'YPW t). (65)
Sz
Since the 0 (r kN) factor gives zero unless electron k is in rr
orbitals in both < I and I) terms, the above equation can
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be written
(fi P)N Ifl)
1
=-(Wil (0 (r.N)S .. +O (r"N)S".) (l-P",) I WI). (66)
Sz
The matrix elements of the complete Hamiltonian JC can
be simplified by recalling that the f /s are by assumption
eigenfunctions of JC except for the u-i exchange terms:
(fd JC Ifi)
=-< WI/L"(~+~) (P""+p,,.) I Wi). (67)
r1r1J r7rp
In (67), 7r is a running index which labels the 7r electrons
in the functions Wi. The following expression for ON, is now formally equivalent to a quantum mechanical
problem involving distinguishable particles in a repre
sentation Wi. The rules of matrix multiplication give,
ON=_2_( wlj (~+~) (P""+P,,.)
IlES. r"" r".
X (0 (r.N)S.z+o(r"N)S"z) (1-P".) I w). (69)
In the above operator expression, cross terms of o(r.N)
with p". and e2/r". can be dropped.
ON =_2_ L;"< WII (~+~) (P""+P,,.) /w,) +( wt'P'~("N)S,.(l-p")lw,) 1 (70)
IlES. r "" r 1rP
X(Wil (0 (r.N)S.z+O(r"N)S"z) (l-P".) I WI) (68) From (70), since WI = UI VI,
ON =_2_ Lm",,[CnICml< unVlj~p""o(r'N)s .. (l-P".) I[UmVI)+ ... J.
IlESz r .. " (71)
The operator P "" can be written in the form,
(72)
where P .. / is the operator which acts on the space variables of electrons 7r and p,. Explicit calculation shows that
P "" can be replaced by 2P "/S".S,,.; since (1-P".) VI is an eigenfunction of S".S .. with eigenvalue-i, we obtain
for ON:
(73)
The functions Un and Um are products of the space (R) and spin (T) variables of the 7r electrons only: un=R"Tn
ON=~ Lmn .. [cm1Cn1{< RnVII~p .. ,,80(r'N)(1-P".)IRmVl)(TnIS?rZ1 Tm)
IlES. r .. "
+< Rn Vllr:2. P".'o(r"N) (1-PI") jRm VI )(T n I S",I T m) } J (74)
If we make the eminently plausible assumption that the exchange interaction between 7r and p, (or II) is negligible
except when the 7r electron is on carbon atom N, and further assume that the 7ri orbitals are linear combinations of
atomic orbitals, then (74) can be simplified to give
ON=~[<PN(7r) VI (p"II) /~p,,/o(r~N)(l- P".) /PN(7r) VI (P"II»
IlES. r""
. Lmnr Cm1Cnl(R,,/ IlN(7r) I Rm)(Tn I S .. z I T m)+··1 (75)
ON= :~[ <PN(7r)VI(P"II)/r:: P .. /O(r.N) (l-P".) IPN(7r) VI (p"II) )PN+ ... J (76)
Equation (76) expresses the linear relation between spin density, PN, and hyperfine splitting, aN, which we have
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130.113.111.210 On: Fri, 19 Dec 2014 08:42:14114 H. M. McCONNELL AND D. B. CHESNUT
sought. Equation (76) is thus equivalent to Eq. (1), and the theoretical expression for Q is, in terms of the funda
mental matrix elements,
It may be noted that in the absence of u-u configura
tion interaction, d2=0, and Eqs. (77) and (37) yield the
same proportionality constant (Q, or OCH) as that given
in (32).
VI. DISCUSSION AND ILLUSTRATIVE
CALCULATIONS
There are two essential approximations in the pre
ceding derivation of Eq. (1). First, we have assumed in
Eq. (62) that the U-7r exchange interaction can be
treated as a first-order perturbation. Second, in making
the essential jump from Eqs. (63) to (64) it was assumed
that all the effective U-7r excited states have approxi
mately the same excitation energy. Both of these ap
proximations are satisfied if the triplet excitation energy
of the u CH electrons is very large compared to the
U-7r exchange coupling, and also large compared to the
excitation energies of the doublet and quartet electronic
states of the 7r electrons which give rise to the U-7r
excited states. These conditions are here referred to as
the "tight u-shell approximation" since as a rule the
more strongly two electrons are engaged in exchange
binding the higher the corresponding triplet excitation
energy will be. This triplet excitation energy is un
fortunately not known for the CH bond, nor for the CH
diatomic molecule nor, in fact, is it known for the great
majority of diatomic molecules. In molecular hydrogen
this triplet state is 3~u+ and is about 12 ev above the
1~g+ ground state. We suggest that this number is also a
reasonable value for the corresponding triplet state of
the CH u-bonding electrons. The tight u-shell approxi·
mation is therefore probably adequate as far as U-7r
exchange coupling is concerned since the U-7r exchange
integrals are of the order of magnitude of 1 ev. The
adequacy of the second part of the tight u-shell ap
proximation is a much more formidable problem. The
essential aspects of this problem are summarized below.
The excitation energy /:,.Eil in (63) can be divided into
a contribution from the triplet u-excitation energy
/:"E(u)("-'10 ev) and a 1I"-excitation energy, /:"Ei(7r).
/:,.Ei1=/:,.E(u)+/:,.E i(7r). (78)
The tight u-shell approximation then requires that
/:,.Ei(7r)«/:,.E(u) for all those states 1/Ii which make an
appreciable contribution to the summation in (64). It
is certainly true that a 7r-electron system may have
several high-energy excited states which, in the one
electron approximation, correspond to the excitation of
more than one electron to higher one-electron energy
levels. However, these excited states yield (1/Ii[ X [1/11)=0
in the one electron approximation. It is therefore clear that many high-energy 7r-electron states will not make
significant contribution to (64) because of the smallness
of the (1/Ii [X [1/11) term, and therefore we need not con
sider these energies relative to /:"E(u) in (78). On the
other hand, one can imagine some highly energetic U-7r
excited state 1/Ii which involves (in one-electron termi
nology) the excitation of a very low-energy bonding 7r
electron to a high-energy antibonding 7r orbital, with
"one-electron" excitation energy as large as 10 ev. In
this case, a detailed examination of the (1/Ii [ON I 1/11) term
in (64) reveals that this matrix element will be small to
the extent that the corresponding highly energetic 7r
configuration makes only a small contribution to the
ground state function 1/11, and such contributions are
indeed expected to be small. This argument suggests
then that all highly energetic U-7r excited states with
large /:,.Ei(7r), comparable to /:,.E(u), should make a
relatively small contribution to the sum in (64). On the
basis of these arguments we believe that there is a good
chance that the important /:,.Ei(7r) in (78) do indeed
correspond to relatively low lying 7r-electronic states
with excitation energies appreciably less than /:"E(u).
Because the above discussion is admittedly quali
tative and at the same time rather involved, and be
cause the development in Sec. V is highly generalized,
we give below some illustrative calculations on the
hyperfine splittings in the allyl radical. Several rather
arbitrary approximations are freely introduced into the
calculation solely for the purposes of brevity of presen
tation, while at the same time the essential aspects of
the above discussion regarding 11"-7r configuration inter
action and /:,.Ei(7r) excitation energies are retained in the
development. The allyl radical was selected because this
is the simplest odd alternate hydrocarbon which illus
trates the problem of negative spin densities.
We number the carbon atoms in the allyl radical as
follows:
(I)
A twofold symmetry axis passes through carbon atom
C2 and it is assumed here that the ground electronic
state of (I) is antisymmetric with respect to space
rotation about C2. (Spectroscopic state 2B.) This system
has three 7r-molecular orbitals (J.I.= 1, 2, 3) :
3
11",,= L aN"pN. (79)
N~l
The two orbitals with lowest and highest one-electron
energies, 7rl and 7r3, are symmetric to rotation about C2,
and 7r2 is antisymmetric. That is, a12= -a32, a22=0. The
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lowest energy doublet state configuration Ul for the 11"
electrons is
Other doublet state configurations are
U2= 11"111"211"31/ y'6 (2a{3a-aa{3- (3aa) ,
U.= 1I"111"211"31/v'2 (aa{3-(3aa) ,
U4= 1I"211"311"3a{3a,
U5= 1I"11l"11I"3a{3a, (80)
(81)
(82)
(83)
(84)
(85)
Since Us and U6 are symmetric in the space variables
with respect to rotation about carbon atom C2, and we
are only concerned with antisymmetric states, U5 and U6
need not be considered further. According to our general
program in Sec. V, the calculation of the allyl hyperfine
constants requires that we take into account all the
configurations UI-U4, in the ground state Ul. This is a
difficult task for even this simple problem, and so we
shall arbitrarily simplify our illustrative calculations by
assuming for simplicity that the 11"-11" configuration is
very weak, so that
(86)
where I e I, I f:' 1 «1. Equation (63) is now
+e(~uua{3u21 JC I if;i)(if;i ION 1 ~uua{3ul)
+e(~uua{3ud JC 1 if;i)(if;il ON I ~uua{3u2)]. (87)
In (87), terms in e2 have been assumed to be negligibly
small, and terms in e' are dropped because the detailed
calculations (not given here) show that all terms in e'
eventually cancel one another and give zero contribu
tion to ON. In general, the 11"-11" configuration interaction
in the U-1I" excited states is different than in the pure 11",
or U excited states. Therefore, we assume to further
simplify the calculation that this configuration inter
action (both 11"-11", and u-u) is exactly zero in the U-1I"
excited states.
Let us first consider the hyperfine splitting due to a
proton attached to one of the outer carbon atoms, say
carbon atom Cl. In this case, the matrix elements with
coefficient eO dominate the summation in (87), and terms
in e can be dropped. For this particular situation, the
only U-1I" excited state which contributes to 01 is
The only other doublet U-1I" excited state belongs to
class I and makes no contribution to 01: The necessary matrix elements are then
(~uua{3ull JC 1 if;2)
= -3/y'6(u*1I"2 1 e2/rI1l"2u), (90)
= -3/y'6 LN'N" aN'2aN"2(u*pN' 1 e2/rl PN"U), (91)
"-'-3/y'6aI22J*. (92)
In passing from (91) to (92) we take advantage of the
fact that the product of the three functions, u*u(e2/r), is
small everywhere except in the region of proton N ( = 1)
and carbon atom N so that all terms in the summation
in (91) are dropped except those for which N' = Nil = 1.
This approximation is essentially the same as that made
in going from (74) to (75). The second matrix element
to be evaluated is
(if;21 ON 1 ~uua{3uI>= 2/y'6u(O)u*(O). (93)
Thus, from (87), to terms of order eO,
(94)
Equation (94) can also be written in a form similar
to (1),
(95)
since, for carbon atom 1, the spin density is Pl=aI22=!
to terms of order eO. Note that in (94) the energy
denominator is simply t!..E(u) since no 11" excitation is
involved in if;2.
Consider now the considerably more complex problem
of the hyperfine splitting by proton 2 attached to carbon
atom 2. In this case one must consider terms of order e
because all the terms of order eO give zero since a22=O.
Consider the first term in € in (87). One can show that
because of the ON operator, the U-1I" excited state 1/Ii
(assumed to have a single 11" configuration) must have
the same 11" configuration as does Ul in order that
(1/1;1 oNI ~uua{3ul) be different from zero. There are only
two U-1I" excited states satisfying this condition (if;2 and
1/13) and of these only if;2 belongs to class II. This
hyperfine component of the first term in e in (87)Zis
therefore given by (93). The associated electronic
matrix element is
(~uua{3u21 JC 11/12) = -(U*1I"11 e2/ r 11I"3U > (96)
= -a21a23J*. (97)
Therefore the first term in € in (87) gives a contribution
to 02 of
(98)
Note again that the energy denominator involves only
t!..E(u) since if;2 involves no 1I"-configurational excitation.
Next we consider the second term in € in (87). Here
now U2 involves a completely open shell configuration, so
the interacting 1/1; must also have completely open shell
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130.113.111.210 On: Fri, 19 Dec 2014 08:42:14116 H. M. McCONNELL AND D. B. CHESNUT
configurations, which are of the form,
I/; j= ~(T(T*1l'11l'21l'a'Y j, (99)
where j = 4, 5, .. " 8 since for the completely open shell
configuration five doublet state spin functions 'Yj are
possible. For completeness we list a set of these doublet
state 'Yj spin functions below.
'Y4=V2jVJ (jI-jn),
'Y6=V2jVJ(hv- jv),
'Y6=V2jvl5(h+ jn), (100)
1'7= -4(1O)tj15(jI+ jn)+ (10)!j3(jIV+ jv),
1'8= -2V2j3(jI+ jn)+V2j3(jIV+ jv)+V2hv,
where
h = ![ aa/3a/3 -a/3f3aa -aaa/3/3+a/3a/3a],
hI = ![ aa/3a/3 -aa/3/3a -/3aaa/3+ /3aai3a] ,
jIll = ! [aa/3a/3 -a/3/3aa-/3aaa/3 + /3/3aaa], (10 1)
hv = ! [aa/3a/3 -aa/3/3a -a/3aa/3+a/3a/3a],
jv = ![ aa/3ai3 -aaa/3/3 -/3a/3aa+ f3aa/3a].
The states I/;i for j=4-8 will therefore have 1l'-excita
tion energies corresponding to the one electron jump
'll'1"'-~1l'a, together with the various possible 1l'-electron
exchange integrals. If we now make the tight O'-shell
approximation and assume all of these 1l'-electron ener
gies to be small relative to AE(O'), then explicit evalua
tion of all of the terms arising from the second term in E
in (87) can be carried out and it is found that the result
is exactly the same as that given in (98). The equality
of the two terms in E in (87) (for carbon atom C2) can be
seen immediately by first making the tight O'-shell ap
proximation, in which case (87) is
-2E
02 = --L i[ (~O'O'a,8ull X I I/; i) (I/; i I ON I ~O'O'a/3u2)
AE(O')
+(~O"O'a/3u21 X I l/;i)(l/;il ON I ~O'O'ai3Ul)], (102)
-2E [(~O'O'a/3ull JCON I ~O'O'a/3u2) ] =-- (103)
AE(O') +(~O'O'a/3u21 XON I ~O'O'a/3uI) .
Therefore, we obtain for 02, to terms of order E,
Also, to terms of order of E, the unpaired spin density at
carbon atom C2 is
4E
Comparison of (104) and (105) yields
2
02 =--0' (0)0'* (0)J*P2,
AE(O') (105)
(106) which is equivalent to (1), (32), and (94) or (95). This
calculation shows how the linearity in (1) can hold even
though the interacting O'-1l' excited states are rather
involved, provided the tight O'-shell approximation is
valid. The present calculation also illustrates the pro
duction of negative spin densities in odd alternate
hydrocarbons due to 1l'-1l' configuration interaction.
This is discussed briefly below.
The term in E in (86) modifies the spin densities on all
the carbon atoms, and to order E all these configurational
densities sum to zero. This can be seen by noting that if
PN' is the spin density at carbon atom N due to the 1l'-1l'
configuration interaction, then to order E,
4E
PN'=-aNIaNa y6 (107)
and LN PN'=O because of the orthogonality of 1l'1 and
1l'a, i.e., LN aNIaNa= O. The sign of the spin density on
carbon atom C2 can be determined by estimating E with
first-order perturbation theory for the 1l' electrons. It is
found that P2' is negative, and PI' and pa' are positive.
The above highly simplified treatment of spin densities
and hyperfine splittings in the allyl radical clearly indi
cates the nature of the tight O'-shell approximation in the
limit of weak 1l'-1l' configuration interaction. In this
particular calculation this approximation holds es
sentially perfectly for the hyperfine splittings on carbon
atoms CI and Ca. For carbon atom C2 we obtain the
correct spin density from Eq. (1) and the observed a2
only if the tight O'-shell approximation is valid for this
particular atom, and in this case the approximation re
quires that all the 1l'-electron states 1/;4-1/;8 [Eqs. (79)
through (101)] have 1l'-excitation energies that are
small relative to the triplet excitation energy, AE(O'). In
the limit of zero 1l'-1l' configuration interaction, the
tight O'-shell approximation is automatically fulfilled,
provided of course the O'-1l' exchange coupling can be
treated as a first-order perturbation. It is rather difficult
to say without detailed quantitative calculations just
how well the O'-shell approximation holds in the limit of
strong 1l'-1l' configuration interaction. The present dis
cussion of the allyl radical may well grossly underesti
mate the strength of the 1l'-1l' configuration interaction.
This can be seen by simply noting that the negative spin
density on C2 in the present calculation is of order E,
whereas a simple valence bond treatment of the 1l'
electrons in the allyl radical which includes only the two
covalent structures,
CI=C2-Ca·
'Cl-C2=Ca
gives spin densities PI = Pa= j, P2= -i.
In conclusion we may make the following remarks on
Eq. (1). The validity of this equation is important in
theoretical chemistry to the extent that it can be used to
obtain unpaired spin distributions in 1l'-electron systems
from observed proton hyperfine splittings, and these
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130.113.111.210 On: Fri, 19 Dec 2014 08:42:14HYPERFINE INTERACTIONS IN 'II"-ELECTRON RADICALS 117
unpaired spin distributions can be compared with those
obtained with various approximations to 1I"-electronic
structures. Unfortunately, the validity of (1) depends,
in general, on the "tight a--shell approximation" which
we suspect is a good approximation in general, but
which really needs detailed quantitative calculations for
verification. On the other hand, it is interesting that (1)
can be used to test the adequacy of simple HUckel
1I"-electron spin distributions because in this case of an
assumed zero 11"-11" configuration interaction, the tight
a--shell approximation is exact as far as the 1I"-excitation
energies are concerned.
VII. PSEUDO-CONTACT HYPERFINE INTERACTION
In addition to the contact hyperfine interaction, the
combined magnetic interactions between electron spin,
electron orbit and nuclear spin can give rise to an
isotropic component of the effective electron spin
nuclear spin hyperfine coupling. For brevity this contri
bution to the isotropic coupling is here referred to as the
"pseudo-contact" coupling. Pseudo-contact interaction
is of course well known from the work of Abragam and
Pryce18 on the theory of hyperfine interactions in
transition metal ions. Bloembergen and Dickinson19
have suggested that pseudo-contact coupling is responsi
ble for nuclear resonance shifts in certain paramagnetic
solutions. In the present section we use a simplified
model similar to that used by Bloembergen and
Dickinson in order to make an order-of-magnitude
estimate of the pseudo-contact coupling in 1I"-electron
radicals.
The simplest and most realistic model to use here is
18 A. Abragarn and M. H. L. Pryce, Proc. Roy. Soc. (London)
A20S, 135 (1951).
19 N. Bloernbergen and W. C. Dickinson, Phys. Rev. 79, 179
(1950). the eHa radical, assumed planar. Except for the
hyperfine terms, the effective spin Hamiltonian for
planar eHa is of the form,
JC= gil 1.81 H ,sr+g! 1.81 (H uSu+HvSv) , (108)
where r is the principal symmetry axis of eHa and u and
v are two axes perpendicular to each other and to r. In
(108), gil and g! are the spectroscopic splitting factors
when r is parallel and perpendicular to H. As is well
known, (108) can be interpreted as though the electron
spin had a z component of magnetic moment, J.l.z, which
was a function of the angle if between rand H.
(109)
The dipolar interaction between J.l.z and the nuclear spin
does not equal zero when averaged over the spacial
orientations of eHa, but gives a pseudo-contact coupling
equal to -2
aNP=-(R-a)AV(gc gil) 1.8 I 'YN. (110)
1511"
For the eHa planar radical, and for 1I"-electron radicals
in general, one expects20 I g! -gill < 10-2. If we assume
the unpaired electron to be localized primarily on the
carbon atom, and set the average inverse cube electron
proton distance, (R.-a)AV, equal to 10-24 cm-3, we obtain a
pseudo contact coupling of 0.1 Mc. This interaction is
then completely negligible relative to the indirect
coupling mechanism to which we have assigned the
coupling Q= -63 Mc. The pseudo-contact coupling by
proton N in an extended 1I"-electron system should be
roughly proportional to the spin density at carbon atom
N, and therefore the pseudo-contact coupling is negli
gible for all spin densities.
20 H. M. McConnell and R. E. Robertson, J. Phys. Chern. 61,
1018 (1957).
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1.1740002.pdf | Exchange Potential in the Statistical Model of Atoms
C. J. Nisteruk and H. J. Juretschke
Citation: The Journal of Chemical Physics 22, 2087 (1954); doi: 10.1063/1.1740002
View online: http://dx.doi.org/10.1063/1.1740002
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/22/12?ver=pdfcov
Published by the AIP Publishing
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130.88.90.110 On: Sun, 21 Dec 2014 00:34:46LETTERS TO THE EDITOR 2087
Paramagnetic Resonance Absorption of
Violanthrone and Violanthrene
Y. YOKOZAWA A:-l"D I. TATSUZAKI
The Research Institute 0/ Applied Electricity,
IIokkaido University. Japan
(Received September 20, 1954)
THE diamagnetic susceptibilities and anisotropies of con
densed polynuclear aromatic hydrocarbons have been
measured by Akamatu and ),Iatsunaga,l their results for the
violanthrone and the violanthrene being shown in Table I in
which the small diamagnetic anisotropy of the violanthrone
compared with that of the violanthrene is noteworthy.
Present investigation was undertaken to examine a view that
this small diamagnetic anisotropy was due to the cancellation by
the hidden paramagnetism involved in the violanthrone. This
hidden paramagnetism was detected by the method of microwave
paramagnetic resonance absorption using a 3.2-cm wave at
room temperature. The magnetic absorption was measured using
a rectangular reflecting cavity operating in TEol2 mode. To
eliminate the crystal detector noise at audio-frequencies, the
reflected power was balanced using a magic tee to a level at which
superheterodyne receiver and a local oscillator, followed by a
intermediate-frequency amplifier at 30 :\ic/sec, could be used.
FIG. t. Oscilloscope trace: absorption spectrum of violanthrone.
The absorption of the violanthrene is also observed (see Fig. 1).
The g values, half-widths 6.Hj, and magnetic susceptibilities are
shmm in Table II.
The paramagnetic part of the susceptibility involved in the
violanthrone ,,,as obtained from a comparison of the integrated
intensity of the absorption curve with that of a single crystal
of euso,· 5H20. The absorption intensity of the violanthrene
was very small, and the paramagnetic contribution to the total
magnetic susceptibility could be neglected. By adding these
contributions, the results are obtained: diamagnetic part of the
susceptibility -n[=264X10~G, diamagnetic anisotropy -6.K=
330X 1O~6 and averaged ... orbital radius (t~2)! = 1.5, 1A in the
violanthrone. These magnitudes are the same orders with those
of the violanthrene. Assuming this paramagnetism is originated
TARLE I. Diamagnetic susceptibilities and anisotropies
of violanthrone and yiolanthrene.
Violanthrone
Violanthrene Mole sllscept.
-x.v·IO'
204.8
273.5 Anisotropy
per mole
-"'K·IO'
141
320 Average orbital
radius Cy2)t
(A)
1.05
1.49 TABLE II. Paramagnetic resonance data.
g "'H; (Oer) Paramag.
suscep. per mole
x·IO'
Viol anthrone
Violanthrene 2.00
2.00 15
13 63
in unpaired 7r electrons, the fractional magnetic population of
these ... electrons x is obtained from the relation,
Nxg2{32S(s+n
x= 3kT ~-
where N is Avogadro's number. From this eCJuation, it is found
x~1/100.
The authors are greatly indebted to Professor Akamatu for
providing them with these organic compounds.
1 H. Akamatu and Y. Matsunaga. Bull. Chern. Soc. Japan 26, 3M (1953).
Exchange Potential in the Statistical Model
of Atoms*
C. J. NrSTERUK AND H. J. JURETSCIIKE
Polytechnic Institute of Brooklyn, Brooklyn, ~Yew York
(Received October 7, 1954)
THE statistical model of the atom which includes the free
electron exchange potential of Dirac! has the shortcoming
that it leads to electron distributions decreasing to zero discon
tinuously at a finite radius Ro from the nucleus." We want to
report some results obtained with a model based on a modifIcation
of Dirac's potential in the atomic surface region.'
The exchange potential can be interprete(1 as the potential
arising from a distribution of unit positive charge, the exchange
hole. Slater' has given a simplified expression for such a distribu
tion representing an averaged exchange hole common to electrons
of all energies. For electrons described by plane waves this
exchange hole has spherical symmetry and is always centered at
the position of the electron in CJuestion. \Vave functions proper
to atomic boundary conditions yield an exchange hole of consider
ably more complex shape. In the interior of the atom the hole is
concentrated around the electron position. As the electron
distance from the nucleus increases the hole tends to remain in
the outermost shell, at first concentrated around the nucleus
electron axis but later distributed more uniformly throughout
the shell.
This behavior of the hole suggests that in the atomic interior a
density dependent free-electron exchange potential is adequate,
while in the far outer region of the atom the exchange potential is
more nearly that due to a concentrated unit charge located at
first near the outermost shell. but approaching the nucleus as
the electron moves far away.
We have extended the variational approach of statistical theory
to include a simplified exchange potential with the above general
properties. If r is the distance of the electron from the nucleus,
then for r ~Ri the exchange potential is given by the usual
electron density-dependent expression of Dirac +e(3n/ ... )I. For
r ?,Ri it is represented by a density-independent function con
tinuous with the inner expression at Ri and approaching e/y at
large r. Ri is an additional parameter in the variational problem.
Its equilibrium value indicates the extent to which the exchange
hole follows its electron in an atom.
The differential equation for the density obtained in the
variation is identical with that set up by J ensen5 on physical
grounds. The outer boundary condition, derived formally, differs
from that assumed by Jensen. It requires that the density vanish
continuously at RD. Thus, the substitution of a density indcpend-
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130.88.90.110 On: Sun, 21 Dec 2014 00:34:462088 LETTERS TO THE EDITOR
ent exchange potential near the surface removes in a natural
manner the anomaly of the usual Thomas-Fermi-Dirac boundary
condition.
The resulting value of Ri depends on the particular function
chosen for the density-independent potential. Actually, there is
little choice, since, as Jensen has pointed out, the potential is
practically completely determined by its boundary values. We
have found that, for all reasonable potentials, Ri lies very close
to the nucleus. Thus, for Kr, Ri <0.6ao. This results because a
potential asymptotic to l/r is stronger than Dirac's potential
over most of the atom. In the interior of the atom the same
relationship is maintained because the density there is independent
of the exact form of the exchange potential.
The small value of Ri indicates that the exchange hole remains
stationary near the nucleus for all positions of its electron. There
fore the electron distributions to be expected in this model are
not very different from those obtained in the Fermi-Amaldi6
modification of Thomas-Fermi's theory. Instead of using a
reduced effective number of electrons we substitute an increased
effective nuclear charge.
The statistical approach does not lead to an asymptotic ex
change hole stationary somewhere in the surface region of the
atom. This property of the exchange hole in the more exact theory
is intimately connected with atomic shell structure, and one may
expect that a statistical model will describe atomic surface prop
erties accurately only when it also exhibits shell structure.
* This work has been supported in part by the Office of Naval Research.
I P. A. M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930).
2 P. Gombas, Die statistische Theorie des Atoms (Springer Verlag, Vienna,
1949) p. 80.
3 C. J. Nisteruk, M. S. thesis, Polytechnic Institute of Brooklyn (1954).
• J. C. Slater, Phys. Rev. 81,385 (1951). 'H. Jensen, Z. Physik 101, 141 (1936).
'E. Fermi and E. Amaldi, Mem. Accad. Italia 6, 117 (1934).
Different Ice Forms under Ordinary Conditions*
R. M. VANDERBERGt AND J. W. ELLIS
Department of Physics, University of California, Los Angeles, California
(Received October 1, 1954)
WHILE determining infrared birefringences of single
crystals of ice by means of channeled interference spectra,
it was observed that a group of interference maxima and minima
near 2.0).1 replaced a similar group recorded a half year earlier at
approximately 0.1).1 shorter wavelengths. The earlier results were
obtained with several crystals grown at that time, the later with
a crystal prepared approximately two months before recording
the results. A comparison of the absorption spectrum of the new
crystal with the spectrum which had been recorded of the earlier
material also showed a pronounced difference. Fortunately, some
of the earlier crystals had been preserved in a refrigerator. When
the absorption and birefringence spectra of the older material
were now reinvestigated it was found that both had changed so
that they conformed with the spectra of the later crystal. Thus
it seems that we have had at least two crystal types. We shall
designate these earlier and later grown types by A and B, respec
tively. Although it is certain that the earlier ice changed from
type A to type B during, or at some time during, the half year it
was in the refrigerator, unfortunately there is no way of knowing
whether the later ice had been grown as type B, because no study
had been made of it during the first two months of its existence.
The original type A material which had changed to type B
showed no further spectral changes during the ensuing four
months.
All of the crystals used were grown in the following manner.
A small seed crystal was cut from a large ice block and was
placed on the lower end of an aluminum rod whose upper end
projected into a refrigerating unit kept at approximately -10°C.
The seed was dipped into a container of distilled water kept at O°C by an ice jacket. Crystals grown by this method assume a
roughly hemispherical shape, exhibiting no plane faces associated
with the usual hexagonal nature of the crystal. The orientation
of the optic axis is always the same as that of the seed crystal.
Working plates of ice were cut from these larger crystals as
desired.
Near the end of our experimental program, after the existence
of types A and B had been clearly revealed, another crystal was
grown and immediately studied. Its absorption spectrum, although
more nearly like type A than type B, shvws distinct differences.
Hence we designate it type C. Its absorption spectrum was
occasionally recorded over a four months' period but no observable
change occurred. It is possible that this type C crystal was grown
more slowly than the others. These results indicate that a more
detailed investigation of crystal forms of ice could profitably be
made, with careful attention to conditions of growth.
The absorption differences among ice types A, B, and C consist
of changes in the structure of absorption bands near 2.0jl, pre
sumably associated with hydrogen bridging between water
molecules. In general the shift from A to B involves a displacement
of certain absorption maxima to longer wavelengths. Whether the
change is from greater order to disorder or vice versa in the
crystal structure seems impossible to say.
The changes involved are not associated with strain in the
crystals. We have subjected ice plates to stress and have shown
that the uniaxial form changes to biaxial without any appreciable
change in the absorption spectrum or in the dichroism which,
contrary to the findings of Plyler,l is small or lacking for all
wavelengths in the very near infrared, and with only a slight
general shift in the channeled spectrum.
Independently of the several well-known forms of ice produced
by Bridgman under extreme conditions, references to two forms
of ice found under ordinary conditions occur. Thus in the Hand
book of Chemistry and Physics' a and f3 forms are tabulated, with
hexagonal and rhombohedral symmetry, respectively. Seljakov3
believed he had shown the existence of two forms by means of
x-ray diffraction. However, Berger and Saffer< think they have
demonstrated an error in Seljakov's technique and hence seriously
question his interpretations.
* The material of this letter was taken from the Ph.D. thesis of R. M
Vanderberg. t Now at Sacramento State College, Sacramento, California.
IE. K. Plyler, J. Opt. Soc. Am. 9, 545 (1924). . . .
'Handbook of Chemistry and Physics (ChemIcal Rubber Pubhshlng
Company, New York, 1950-51), 32nd edition, p. 2225.
3 N. Seljakov, Compt. rend. acado sci. U.S.R.R. 10,293 (1936); 11, 92
(1936); 14,181 (1937). 'C. Berger and C. M. Saffer, Science 118,521 (1953).
Formation of Negative Ions in Hydrocarbon Gases*
T. L. BAILEY, J. M. MCGUIRIJ:, AND E. E. MUSCHLITZ, JR.
College of Engineering, University of Florida, Gainesville, Florida
(Received August 23, 1954)
IN connection with studies of collisions of gaseous negative
ions with neutral molecules,! negative ions produced by
electron bombardment of methane, ethane, and acetylene gases
have been investigated in a mass spectrometer. The ions observed
and their relative intensities under similar source conditions are
shown in Table I.
TABLE I. Relative negative ion intensities.
Mass Mass Electron
Gas H- 25 12-15 energy
CH, 120 12 1.5 35 ev
C,H. 73 27 0.5 70 ev
C,H, 8.5 55 0.0 70 ev
(100~1O-12 amp)
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1.1722417.pdf | pn Junction Theory by the Method of δ Functions
Howard Reiss
Citation: Journal of Applied Physics 27, 530 (1956); doi: 10.1063/1.1722417
View online: http://dx.doi.org/10.1063/1.1722417
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/5?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 137.149.200.5 On: Sun, 23 Nov 2014 07:02:42JOURNAL OF APPLIED PHYSICS VOLUME 27, NUMBER 5 MAY, 1956
p-n Junction Theory by the Method of 0 Functions
HOWARD REISS
Bell Telephone Laboratories, Murray Hill, New Jersey
(Received December 2, 1955)
This article describes a concise new method for calculating current-voltage phenomena in structures in
volving p-n junctions. In fact, the problem of the current-voltage characteristics of anyone-dimensional
p-n junction structure with any number of junctions and contacts is solved in a form general enough so that
one only needs to insert the physical parameters of the structure into the formulas to write down
its characteristics.
I. INTRODUCTION
THIS article deals with a concise presentation of
the theory of current flow in p-n junction
structures. The original thoughts of Shockleyl on this
matter were so thorough that little can be done in
improving the formulas which he derived. In fact we
shall (except for minor alterations) arrive at the same
expressions. The motivating force for this paper is then
not so much the presentation of new formulas as it is
the introduction of an efficient means of arriving at
older ones. Besides being concise, the new formulation
has the advantage of focusing attention on the central
parameters of the theory, i.e., the space derivatives of
the hole and electron currents near junctions. These
derivatives behave as reduced currents associated only
with their respective junctions, irrespective of the
proximity of other junctions. In fact if the reduced
currents are used each junction (even though junctions
in series may be separated by less than a diffusion
length) behaves like an isolated ideal rectifier.
By judicious use of the reduced currents the problem
of calculating the current-voltage characteristics of a
wide variety of semiconductor structures is made to
depend upon the solutions of sets of linear algebraic
equations. In this way it is possible to give general
formulas for the characteristics of any unidimensional
p-n junction structure involving any number of contacts.
Some of the formulas derived in this way are not easily
derived by manipulation of the equations which appear
first in the natural course of the Shockley method.
An example of this kind is the relation between the
collector current and the total voltage across a hook
multiplier2 when the reverse junction of the multiplier
is not saturated. In this case one has two transcendental
equations at his disposal, each involving the collector
current and the differences in quasi-Fermi levelsl at each
of the hook junctions. These equations can actually
be replaced by a single transcendental relation between
the collector current and the total voltage in the hook,
but it is not easy to see this by manipulation of the
two equations available originally.
Another example concerns the current voltage
characteristics of some complicated sequence of p-n
I W. Shockley, Bell System Tech. ]. 28, 435 (1949).
2 Shockley, Sparks, and Teal, Phys. Rev. 83, 151 (1951). junctions. Such a sequence might be of interest in
studying the effect which random fluctuations in the
distribution of impurities has on the resistance of a
supposedly homogeneous semiconductor. In this case
the usual method leaves one with many transcendental
relations involving the current and the differences in
quasi-Fermi levels at the various junctions. Again, these
can be replaced by a single equation involving the
current and the total voltage.
II. THE METHOD OF 0 FUNCTIONS
We shall assume that the rate law for hole-electron
recombination is "bimolecular," and use the notation
employed by Shockley.l Thus, the net rate of re
combination is specified by
r=g[pn -1],
ni2 (2.1)
where g is the rate per unit volume at which hole
electron pairs are generated thermally, ni is the density
of carriers of one kind in an intrinsic sample, and p
and n are, respectively, the local densities of holes and
electrons.
Following Shockleyl we specify the quasi-Fermi
levels, rpp and rpn, for holes and electrons in terms of the
local densities, p and n. Thus,
p=ni expq(rpp-1/;)kT,
n=ni expq(1/;- rpn)kT, (2.2)
(2.3)
where q is the negative of the electronic charge,1/;, the
local electrostatic potential, k, the Boltzmann constant,
and T, the absolute temperature. It is an easy matter
to show that the hole and electron currents Ip and
In are determined by
Ip= -MP'ilrpp,
In= -JJ.bqn'il rpn, (2.4)
(2.5)
where JJ. is the hole mobility, and b is the ratio of electron
to hole mobility.
In the steady state the equations of continuity
3 W. Shockley, Electrons and Holes in Semiconductors (D. Van
Nostrand Company, Inc., New York, 1950).
530
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demand
V· 11'= -V· (p.qpV IPp) = -qr
=_qg[pn -1]=qg[ex pq(lPp-lPn)/kT-1] (2.6)
n?
(2.7)
These relations are in accord with
(2.8)
where I, the total current, is divergenceless. Retaining
the second and fifth terms of both (2.6) and (2.7)
yields two equations in IPp and IPn. By differentiation,
rearrangement, and suitable use of (2.4), (2.5) and the
fact that p and n are already related to each other by
(2.2) and (2.3) it is possible to replace (2.6) and (2.7)
by the following separated current equations
p bDqnp [ 1 ] Ip=--I---v In 1--V·l p
p+bn p+bn qg (2.9)
bn bDqnp [ 1 ] In=--I+--v In l+-v·l n ,
p+bn p+bn qg (2.10)
where the Einstein relation,3
(2.11)
has been used to replace the mobility by the diffusion
coefficient, D. Equations (2.9) and (2.10) can be cast
in an alternative form by carrying through the indicated
gradient operations and making use of the relations
furnished by the first and fourth terms of (2.6) and
(2.7). Thus,
p bDn;2 Ip=--I+ V(V·lp) (2.12)
p+bn g(p+bn)
bn bDn;2 1,,=--1+ V(v·I,,). (2.13)
p+bn g(p+bn)
A second conversion can be achieved by employing
the relation furnished by the first and fourth terms of
(2.6) and (2.7) to replace v·lp and v·l" in (2.12) and
(2.13). Thus,
p qbD Ip=--I---(pvn+nVp) , (2.14)
p+bn p+bn
bn qbD In=--I+--(pVn+nVp). (2.15)
p+bn p+bn
In the n-region, p«n, so that (2.14) becomes
11'= -qD"Vp, (2.16) TRANSITIQN
FIG. 1. Elementary p-n junction structure.
and in the p-region, n«p, so that (2.15) becomes
In=qbDVn, (2.17)
i.e., the minority currents are merely diffusion currents;
a result used to great advantage by Shockley.!
Equations (2.9) and (2.10) are the basic equations of
the method we wish to develop here. A number of
conclusions arrived at previously by Shockley! are
evident immediately from them. For example when g
tends to infinity the last terms on the right vanish and
the current divides in the ratio
(2.18)
If we specialize to the one-dimensional case
(2.19)
in which 1 is now constant. If (2.19) is integrated over a
p-n junction between x=a and x=b we have
(2.20)
If b and a are sufficiently far from the junction, on
either side, then
(2.21)
where V is the voltage across the junction. Substi
tution of (2.21) into (2.20) makes it evident that in the
limit of infinite g the resistance of the junction structure
is simply the integral on the right, i.e., the integrated
local resistance.
Our development will be called the "method of 0
functions," a name which has its origin in the functions
and p/(p+bn)
n/(p+bn), (2.22)
(2.23)
which appear in Eqs. (2.9) and (2.10). If we imagine a
p-n junction structure (Fig. 1) with a p-region extending
from x= a to x= -11" with n-region between x= 1" and
x= b, and with the transition regions between x= -11'
and x= 1", then (2.22) and (2.23) manifest the following
behaviors as we move from left to right. Consider first
(2.22). In the p-region bn in the denominator can be
neglected and (2.22) has the value unity. It retains
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p/(p+bn)
FIG. 2. Illustration of the
step functions. 1 6 n/(p+bn) r-------------
I , _________ _J..
x"a x "-1..p
this value even into the transition region until bn
becomes comparable with p. At this point it rapidly
passes to zero as p becomes negligible in comparison
to bn, and retains the value zero as we pass into the
n-region. (2.22) is therefore a step function with a
step down in the positive x-direction. The step actually
begins and ends deep in the transition region (see
Fig. 2) so that it is very sharp indeed. By the same
argument (2.23) is a step function with a step up of
height, 1/b, in the positive x-direction.
The derivatives of (2.22) and (2.23) must be sharply
peaked in the vicinity of the step. They therefore have
the characters of {j functions, and it is this property
of the fundamental equations which will be used.
Before proceeding to the task it is appropriate to
examine how sharp the {j functions are in a particular
numerical example.
For this purpose we have chosen to examine the
derivative of (2.22) for the equilibrium situation in an
abrupt junction such that the chemical density of
acceptors is 1016 cm-3 on one side while the chemical
density of donors is the same on the other side. The
calculation has been made for T= 3OQoK under the
assumptions that ni= 1013 cm-3 and that K, the di
electric constant is 16. With these conditions, the
thickness, 2l, of the transition region (computed
according to the Schottky exhaustion layer theory)
turns out to be 36X 10-6 cm. The derivative of (2.22)
for this case has been plotted in Fig. 3. The width of
the figure is just 21, the thickness of the transition
region. We see that the function is actually very
sharply peaked, its effective width being about 4X 10-6
cm or about ~ the distance across the transition region.
When large forward currents are driven through the
junction the exhaustion layer narrows and the {j
functions broaden. When they have broadened so
much that they are no longer {j functions relative to the
task at hand the formulas derivable by the present
technique (and by identity, those derivable by the
Shockley technique) will no longer be valid. x"b
III. THE ISOLATED JUNCTION: C~NT
DERIVATIVES AS REDUCED CURRENTS
In the remainder of this paper we shall only have use
for the one-dimensional versions of (2.9), (2.10),
(2.12), and (2.13). Confining attention to I P' we obtain,
in place of (2.9) and (2.12),
p bDqnp d [ 1 dIp]
Ip= P+bnI-p+bn dx In 1-qg dx (3.1)
and
p bDnl d2Ip Ip=--I+ -, (3.2)
p+bn g(p+bn) dx2
where I is now constant. Replacing the left member of
(3.1) by (2.4) and rearranging yields
dcpp I kT bn d [ 1 dIp] In 1 (3.3) --;;:; qp.(p+bn) -;; p+bn dx -qg a; .
Now integrate (3.3) across the junction between the
limits X= a and x= b both of which are sufficiently
removed from the transition region so that
cpp(b)-cpp(a) = V, (3.4)
where V is the voltage across the junction, and
(dI p) (dIp) --=--=0.
(dx)_a (dX)_b (3.5)
The result is
-lcpplab=V=Ifb
dx
a qp.(p+bn)
_ kTfb(~) ~ln[l-~ dIpJdX. (3.6)
q a p+bn dx qg dx
The first term on the right is the potential drop due
to the integrated local resistance. The integral in this
term is the quantity which Shockleyl calls R1, which
he asserts is not less than half the;integrated local
resistance. We see in (3.6) that it is .. exactly the inte-
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grated local resistance. The second term on the right
can be evaluated through an integration by parts. Thus,
kTfb bn d [ 1 dIp]
--;; a (p+bn) dx In 1-qg ax dx
kT/ (bn) [ 1 dIPJ/b
= --;; (p+bn) In 1-qg ax a
kTfb [ 1 dIp] d (bn) +-;; a In 1-qg dx dx (P+bn/x, (3.7)
The first term on the right of (3.7) vanishes in view of
(3.5). The second contains, in the integrand, a slowly
varying factor multiplied by a ~ function whose peak
lies in the transition zone. The slowly varying factor is
slowly varying not only because it is a logarithm, but
because it can be demonstrated (as we show below)
that I p must have a point of inflection in the transition
region. Consequently, if the latter is narrow enough
(dIp) - = S"'" constant.
dx trans (3.8)
Henceforth the derivatives of the hole current shall be
denoted by S when they refer to transition regions.
It is now possible to express the integral on the right
of (3.7) in the form
kT [ 1]f b d ( bn ) -In 1--S ---dx,
q qg a dx p+bn (3.9)
where the slowly varying function has been extracted
and given the value it assumes at the ~-function peak.
Since
---1
Ibn Ib
p+bn a-, (3.10)
Eq. (3.9) is simply
kT [ 1] -In 1--S ,
q qg
and, since this is the value of the integral on the right of
(3.6), that equation takes the form
fb dx kT [ 1] V=I +-In 1--S .
a qp,(p+bn) q qg (3.11)
As Shockley has indicated1 the contribution to V
of the integrated local resistance can be ignored for the
case of narrow junctions. The evaluation of the current
voltage characteristic depends then upon a knowledge
of the relation between S and I. This relation will be
developed shortly, but for the moment it suffices to
say that it is of the form
S=-KI, (3.12) .
'2 I(:t&
4,0
3.
3,0
2.'
'.0
V \ 0.'
o
I -------- -8 16 14 12 10 B 6 4 2 0 2 4 6 a 10 12 14 fa 18
I X IN eM x 106 .1
wrOTH OF TRANSiTiON ZONE ---_ • ..J.
FIG. 3. Typical 0 function for the case ,,= 16, T=300oK, n,= 1013
em-s, and N=1018 em-a.
where K is a constant. We thus arrive at the rectifying
characteristic
qg
1= K[eQV/kT -1J, (3.13)
which when K is evaluated will prove to be identical
with Shockley's.
Regarding S as a reduced current we obtain a V-S
characteristic
V= kT In[1-~S].
q qg (3.14)
For the simple case under consideration these equations
exhibit the key position occupied by S. In order to
obtain the current-voltage characteristic the dependence
of S on I must be known. In more complicated one
dimensional structures involving k-junctions in sequence
it will be proved that (3.14) is replaced by
kT i-k [1] V==F-' L (-1)'ln 1--Si ,
q i=1 qg (3.15)
where the minus and plus signs are to be used according
to whether the first junction is p-n or n-p. Now (3.15)
is a sum of terms like (3.14) and if each term is assigned
the symbol Vi we have
i-k
V= LVi. (3.16)
i=1
If one wishes to do so, Vi can be defined to be the
voltage spanning the ith junction and from this point
of view the entire structure behaves as a sequence of
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independent ideal rectifiers, no matter how closely the
various junctions are spaced. It must be emphasized,
however, that Vi will not be the voltage measured over
the ith junction unless the latter is separated from
junctions i-1 and i+ 1 by fairly large distances.*
On the other hand in any application we generally
need to know only the sum, V, in its dependence on the
currents flowing in the various contacts, so our in
ability to observe Vi is of no consequence.
When the junctions are far apart, they may be
regarded as isolated junctions in series. In this circum
stance S will be shown to depend linearly [as in (3.12)J
only on the current through the junction to which it
applies. The structure consists then of a sequence of
ideal rectifiers. As the junctions are moved closer
together, so that they interfere, S proves to be a linear
combination of the currents through all the junctions
(these currents may be quite different depending on the
number and location of external contacts) but the
relation (3.15) applies. Thus the structure continues to
behave as a sequence of ideal rectifiers provided that
the S's are regarded as the reduced currents appropriate
to each junction. We shall be able to write general
formulas for the linear combinations composing the
various S's and so present the general solution for the
current voltage characteristics of an arbitrary structure
containing p-n junctions.
We return now to the problem of evaluating K in
(3.12). As an aid in accomplishing this we shall fulfill
an earlier promise by proving that I p has a point of
inflection in the transition region. An inflection point
occurs when
dHp/dx2=0. (3.17)
By (3.2) this demands that
(3.18)
Since I p can never be greater than I, nor less than zero,
its behavior must resemble that shown in Fig. 4.
Furthermore, the right member of (3.18) is a step
(p:bn)l --------.... ....
...... ........
................ Ip ---- -x--lp
FIG. 4. Diagram for use in proving the existence
of a point of inflection.
* However it can be shown that Vi measures the separation of
the quasi-Fermi levels at the ith junction-no matter what the
spacing is. function having a step down of magnitude, I, in the
transition zone. It, too, is plotted in Fig. 4. It is obvious
that the two curves must intersect at the step which
lies within the transition zone; hence the point of
inflection occurs there.
Equation (3.3) can be used to indicate the manner
in which ({Jp changes with x. If the local resistance term
(the first on the right) is ignored,
(3.19)
in the p-region where the step function bn/(p+bn) is
zero. d({Jp/ dx achieves a finite value only in the n
region where the step function is unity. Similar con
siderations show that ({In varies only in the p-region.
Like Shockley we then come to the conclusion that
(3.20)
If the transition zone is sufficiently narrow, it follows
that I p in it, which shall be denoted by I pT, can be
approximated by the form
IpT=AT+Sx. (3.21)
In the p-region, between x= a and x= -lp, I P' denoted
by I pp, is governed by a specialization of (3.2). Thus,
ni2 ni2
--~-=np, p+bn pp (3.22)
where pp is the equilibrium density of holes in the
p-region and np the equilibrium density of electrons.
Of course,
p/(p+bn) = 1. (3.23)
When these relations are substituted into (3.2), the
result is
(3.24)
where the relation,
(3.25)
employed by Shockleyl has been used. Ln is the diffusion
length for electrons in the p-region.
Similar considerations specialize (3.2) In the n-
region to
where
is the diffusion length for holes in the n-region . (3.26)
(3.27)
The solutions of (3.24) and (3.26) are, respectively,
and (3.28)
(3.29)
Since I p cannot be infinite both Bp and A" must vanish.
This leaves four constants Ap, AT, S, and Bn to be
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determined by application of the continuity conditions, The last form is obtained by application (3.25) and
1 pp( -lp)=1 pTe -Ip), (3.30)
dlpp dlpT
(x= -lp), --=--, (3.31)
dx dx
dlpT dIpn
(x=ln), --=--, (3.32)
dx dx
1 pT(l,,) = I pn(ln). (3.33)
The results are
LJelpJLn
Ap=- ,
L,,+Lp+l,,+lp (3.34)
AT= (Lp+1,,)1
(3.35) ,
L,,+Lp+ln+1 p
I S=- ,
Ln+Lp+l,,+lp (3.36)
B,,= LplelnlLp
(3.37)
L .. + Lp+ln+1p
If
L»l, (3.38)
as will always be so in the cases of practical importance,
S, given by (3.36), is closely approximated by
S=-I/(Ln+Lp). (3.39)
In fact, the continuity conditions might have been
applied directly between 1 pp and 1 pn at x=O. Ignoring
the transition region in this manner yields the formulas
obtainable from (3.34) through (3.37) by setting
l,,=lp=O. We shall follow this practice from now on.
Substitution of (3.39) into (3.14) yields
I=qg(L .. +Lp)[eQVlkT-l]
[P" bnp] =qD -+- [eQVlkT-l].
Lp L ..
FIG. 5. General
ized one-dimensional
p-n junction struc
ture. (3.40)
2 (3.27) to the first. Equation (3.40) is identical with
Shockley's result contained in Eq. (4.13) of reference 1.
IV. JUNCTIONS IN SEQUENCE, CURRENT VOLTAGE
CHARACTERISTICS FOR AN ARBITRARY
ONE-DIMENSIONAL STRUCTURE
Structures of the general type illustrated in Fig. S
will now be discussed. They consist of sequences of
homogeneous regions of alternating type, all of which
have unit cross sections. If in an actual case the cross
section is uniformly A, then this can be accounted for
by multiplying all of the currents derived below by A.
The homogeneous regions are numbered from left to
right, the first on the left being assigned the index 1,
while the last on the right is assigned the number k.
Junctions are distinguished by the indices of the
homogeneous regions to their various lefts. A non
rectifying contact, passing only the majority carrier,
will have access to each homogeneous region. The
current flowing in this contact (which may have ,<ero
magnitude, e.g., when there is actually no contact)
is denoted by Ji where the subscript i is the index of
the homogeneous region. Ji is positive when flowing
toward the semiconductor.
The width of the ith homogeneous region is symbol
ized by Wi. In all cases we assume
(4.1)
The diffusion length for the minority carrier in the ith
region is denoted by Li•
V is the potential spanning the sequence; positive
when it drives positive current from left to right. F; is
the current crossing the ith junction, from left to right.
It follows, immediately, that
J1=F1 (4.2)
J,,= -Fk-l (4.3)
J.=Fi-F i_1 (4.4)
l' r VIH
FL-, Fi" F K-l
L
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;-i
F,= :E I;-i-I (4.5)
By integrating (3.3) over the sequence of junctions
we obtain for the total voltage, V,
V=TkT
:E (-1)!:ln[1-~Si]'
q i qg (4.6)
The ith 0 function selects the ith S, and provides a
series of terms with alternating signs because the steps
are alternately steps up and steps down. The minus sign
is to be used if the first junction is p-n. If it is n-p the
plus sign should be employed. Equation (4.6) remains
valid even when the various junctions are close enough
to interfere with each other (although not so close that
their space charges interact). In terms of Si, therefore,
the sequence behaves like an assembly of ideal rectifiers,
each having the characteristic
kT [ 1 ] Vi=±-ln l--Si ,
q qg (4.7)
where the upper and lower signs refer, respectively, to
p-n and n-p junctions.
The problem of the detailed current-voltage relation
ships, possible for the structure of Fig. 5, is solved by
relating the various Si and Ji. Si proves to be a linear
combination of the J/s, and it is possible to give the
general formula for Si. To do this we make use of
(2.12) in n-regions and (2.13) in p-regions. At a junction
the following conditions [based on (2.8), and ideas of
continuity] are satisfied
Ip+In=F,
df pldx=S= -dI,./dx. (4.8)
(4.9)
Consider (2.12) or (2.13) specialized to an n-or p-region,
respectively. Call it the ith region. Then, in analogy to
(3.26) we obtain
(4.10)
where we understand that f i is I p if i refers to an
n-region, and In, if a p-region is involved. Similarly,
L, is a diffusion length for holes if an n-region is involved,
and for electrons in a p-region. The derivatives of Ii
111 'Yl -Fl
a2 112 'Y2 +F2
aa 113 'Y3 -F3
ai +F; at the junctions bounding the ith region are
Si-l at Ai-I (4.11) Si at Ai
if it is an n-region, and
-Si-l at Ai-l (4.12) -Si at Ai
if it is a p-region.
The solution of (4.10) subject to these boundary
conditions is
LiS i cosh (x-Ai-II Li) -LiS i-1 cosh (x -Ail L,)
fi=----------------------------------
±sinh(Ai- Ai_lILi)
(4.13)
where the positive sign refers to an n-region and the
negative to a p. This formula applies even to the first
region if we set
AO= -00
So=O
and to the last region if (4.14)
(4.15)
If (4.8) is applied at the ith junction (at X=Ai) then
the use (4.13) to represent fp and In, explicitly, yields
aiSi-l+l1;Si+'Y;S>+I= TFi, (4.16)
where the upper sign is used when the ith junction is
p-n and the lower when it is n-p. Furthermore
ai=Li CSCh(Ai~:i-I) =Li CSCh(:i), (4.17)
l1i= -[ Li coth(::)+Li+1 coth(:i::)]. (4.18)
'Yi=L>+l cSCh(W>+I).
Li+1 (4.19)
There are as many Eqs. (4.16) as there are
junctions, and, therefore, as many as there are S;'s. As
a result, a unique function for each Si is determined
which proves to be a linear combination of the Fi and
hence, through (4.5), of the Ji. We can, in fact, write
the formula for Si. It is
'Y.
-Fk-3 ak_3 11k-3 'Yk-3
+Fk-2 ak-2 I1k-2 'Yk-2 -Fk_1 ak-l 11k-I Si=± (4.20)
111 'Yl
a2 112 'Y2
as I1z 'Y3
ak-a 11k-3 'Yk-3
ak-2 11k-2 'Yk-2
ak-l 11k-I
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and in which the sequence of F's in the upper
determinant has alternating signs. With regard to the
indicated choice of signs the upper sign is to be used
when the first junction is p-n and the lower when it is
n-p.
We can remove g from (4.7) by using the relation
ship, easily proved,
D(bC.C i+1)1
g=
L.Litl (4.21)
where Ci and Ci+1 are the equilibrium densities of
minority carriers in the ith and i+ lth regions,
respectively.
When
Eq. (4.20) shows that
Si==F·--
Li+Litl (4.22)
(4.23)
the upper sign referring the p-n junctions and the lower
to n-p. However, as Wi and Wi+1 become finite, i.e.,
(4.24)
Si becomes a linear combination of all the Fi.
Equation (4.20) represents the complete solution of
the problem, for by use of (4.5) all the F's can be
expressed in terms of the observed currents. Further
more only observed voltages need enter the problem.
For example if Ii is zero then the ith homogeneous
region floats and V i-1 as well as Vi are not individually
observed. Only their sum is measured. But by (4.7)
[1-(l/Qg)Si-l] Vi-1•i=Vi-l+V i=±kTln . (4.24) 1-(1/qg)Si
Thus the combination represented by (4.5), (4.7),
and (4.20) is not only general but has the additional
advantage of not requiring the inclusion of unobserved
voltages and currents, although the latter can be
included if desired by reference to the F i.
V. CONCLUSION
The preceding text contains a concise formulation of
the theory of current flow in p-n junction structures.
The order of approximation in most cases is identical
with that found in the original calculations of Shockley,!
and when comparable problems are treated the present formulation reproduces Shockley's results. The chief
merit of the present method lies in the fact that it
emphasizes the central mathematical features of the
phenomenon so that not only do all of Shockley's
theorems fall out of the basic equations immediately,
but hitherto unrecognized relationships are disclosed.
In particular the importance of the space derivatives
of the hole currents in the neighborhoods of junctions is
brought to light. If the space derivative at the ith
junction is denoted by Si it can be shown that the
voltage spanning a one-dimensional sequence of arbi
trarily spaced junctions with an arbitrary number of
ohmic contacts is
kT [¢ ] V==F-L (-1)iln 1--Si .
q qg
The minus or plus signs are to be used according as
the first junction is p-n or n-p. If Si is regarded as a
reduced current associated with the ith junction the
preceding equation has the form expected for a sequence
of independent ideal rectifiers connected in series.
As a matter of fact when the junctions are far apart
Si is simply proportional to the current flowing through
the ith junction. As they are moved closer together S.
becomes a linear combination of all the currents
flowing in external contacts. It is possible to write a
general formula for Si which specifies each linear
combination in terms of structural parameters and
diffusion lengths.
In this manner, general formulas for the current
voltage characteristics of an arbitrary p-n junction
structure are obtained. Judicious use of the reduced
rectifier formula, given above, permits one to eliminate
unobserved floating potentials from these charac
teristics.
The only disadvantage of the new method is that it
does not focus as much attention on physical processes,
e.g., injection processes, etc., as Shockley's method.
On the other hand the )nteraction of such processes
may become very complicated in complex structures
and it is doubtful whether the visualization of the
individual effects is of much value there.
ACKNOWLEDGMENTS
The author is indebted to R. G. Shulman and D.
Kleinman of Bell Laboratories for detailed helpful
discussions which have done much to improve the value
of this paper.
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1.1731359.pdf | Inter and IntraAtomic Correlation Energies and Theory of CorePolarization
Oktay Sinanoğlu
Citation: The Journal of Chemical Physics 33, 1212 (1960); doi: 10.1063/1.1731359
View online: http://dx.doi.org/10.1063/1.1731359
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:33THE JOURNAL OF CHEMICAL PHYSICS VOLUME 33, NUMBER 4 OCTOBER,1960
Inter-and Intra-Atomic Correlation Energies and Theory of Core-Polarization
OKTAY SINANOGLU*
Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley 4, California
(Received March 18, 1960)
Atomic and molecular energies depend strongly on the correla
tion in the motions of electrons. Their complexity necessitates the
treatment of a chemical system in terms of small groups of elec
trons and their interactions, but this must be done in a way con
sistent with the exclusion principle. To this end, a nondegenerate
many-electron system is treated here by a generalized second-order
perturbation method based on the classification of all the Slater
determinants formed from a complete one-electron basis set. The
correlation energy of the system is broken down into the energies
of pairs of electrons including exchange. Also some nonpairwise
additive terms arise which represent the effect of the other elec
trons on the energy of a correlating pair because of the Pauli
exclusion principle. All energy components are written in ap
proximate but closed forms involving only the initially occupied
H.F. orbitals. Then each term acquires a simple physical interpre
tation and becomes adoptable for semiempirical usage. The treat-
I. INTRODUCTION
THE total energy of a many-electron system is well
approximated by independent particle theories, in
particular by the Hartree-Fock (H.F.) method.I,2 But
the remaining error, which results from the tendency
of the electrons to avoid one another, is usually of the
order of chemically interesting quantities. This error
i.e., the difference between the completely self-con
sistent field (SCF) H.F. solution to the energy and
the exact solution of the many-electron nonrelativistic
Hamiltonian, will be taken here as the precise defini
tion2 of "correlation energy" unless otherwise indicated.
Considerable effort has been devoted to obtaining the
correlation energies of atoms and molecules by varia
tional techniques, especially by the method of con
figuration-interaction. 2 In spite of slow convergence,
computers now allow some progress by these techniques
for simpler systems, e.g., molecules from the first row of
the periodic table. But, generally, it is necessary to
have schemes that lend themselves to simple physical
interpretation and generalization. It is also important
in view of the complexity of most molecules and atoms
that such schemes should be usable in a semiempirical
but still well-defined way.
There exists a special class of problems where it has
been possible to treat the correlation effects very simply.
These involve the correlation energies between non
overlapping charge distributions; or more precisely, in
these cases different electron groups are assumed to be
localized in totally isolated regions of space, each group
* University of California Lawrence Radiation Laboratory
Postdoctoral Fellow 1959-60. Present address: Sterling Chemistry
Laboratory, Yale University, New Haven, Connecticut.
1 For reviews see D. R. Hartree, Calculation of Atomic Struc
tures (John Wiley and Sons, Inc., New York, 1957).
2 P. O. L6wdin, Advances in Chern. Phys. 2, 207 (1959). ment is applied in detail to two particular problems: (a) The
correlation energy between an outer electron in any excited state
and the core electrons, e.g., in the Li atom, is represented by a
potential acting on the outer one. This potential can be regarded
as the mean square fluctuation of the Hartree-Fock potential of
the core, and applies even when the outer electron penetrates into
the core. The magnitudes of some of the correlation effects are
calculated for Li. (b) Starting from a complete one-electron basis
set of SCF MO's, the energy of a molecule is separated into those
of groups of electrons and of intra-molecular dispersion forces
acting between the groups. The assumptions that are usually made
in discussing dispersion forces at such short distances are then re
moved and generally applicable formulas are given. Some three
or more electron-correlation effects and limitations in the use of
"many electron group functions" for overlapping systems are
also discussed.
with its own complete set of eigenfunctions. The best
example is the familiar London dispersion attraction8,4
between two atoms in their ground states.s This energy,
which has the very useful property of being pairwise
additive for a system of atoms, also has a very simple
physical interpretation. It arises from the mean
square fluctuations of the instantaneous atomic dipoles.
A different case concerning correlation energy
between separable groups arises in the theory of atomic
spectra. In alkalilike configurations optical transitions
occur by the excitation of a "series," i.e., valence,
electron, outside an inner "core" giving a hydrogenlike
spectrum. The H.F. SCF method for such an atom,
takes the average effect of the outer orbital on the core
into account, but the correlation between the in
stantaneous position of the series electron and the core
electrons is left out. This correlation effect has come to
be known under the name of "core polarization."l The
simplification in this case arises when the series electron
is in a highly excited "nonpenetrating" orbit, i.e.,
when the core and the valence electron can be sepa
rated. Then it is possible to represent the correlation
energy between the core electrons and the series electron
by an additional effective potential which is deter
mined only by the core and acts on the outer electron
in anyone of its higher orbits. A classical argument
indicates that at large separations this potential is
given by -!ar4 where a is the polarizability of the ion
core and r is the distance of the series electron from the
nucleus. Several quantum mechanical derivations have
3 F. London, Z. Physik. Chern. Bll, 222 (1930).
4 H. Margenau, Revs. Modern Phys. 11, 1 (1939).
6 In this case, the definition given for correlation energy is
somewhat altered; it still applies, however, if the electrons of one
system are considered to move in the self-consistent field of the
other.
1212
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:33CORRELATION ENERGIES AND CORE-POLARIZATION 1213
been given for this asymptotic potential. The "adia
batic approximation"6 method distinguishes a tightly
bound core electron from an outer one in any orbital
with high quantum numbers, e.g., in the excited states
of He, and replaces the exact two-electron wave func
tions by
( 1)
where rl and r2 refer to the coordinates of the inner and
outer electrons, respectively; u.k( r2) 's are the various
states of the outer electron; uc(rl; r2) describes the core
electron and depends only parametrically on r2. This
approximation is very similar to the Born-Oppenheimer
separation of nuclear and electronic motions in mole
cules. Since the outer electron is much less tightly
bound than the inner one, it can be considered, in view
of the virial theorem, to move slowly compared to the
latter. Then the energy of the core can be determined
for various fixed values of r2, thus depending on r2
parametrically; and it, in turn, acts as a potential energy
for the motion of the outer electron. Bethe7 has treated
the highly excited states of He by this approach, con
sidering the stationary charge of r2 as a perturbation on
the free "core" function, ucO(rl)' This discussion should
emphasize that the name "core polarization" refers to
the polarization of the core by the outer electron at its
instantaneous position. This is the entire 'correlation
effect and should not be confused with the polarization
of the core only by the smeared-out over-all charge
distribution of the outer orbital. The latter which we
shall call the "orbital average polarization" is a much
smaller effect and in fact vanishes if completely SCF
H.F. orbitals are used (see Sec. II).
Mayer and MayerS and Van Vleck and Whitelaw9
have treated the core-polarization energy by conven
tional second-order perturbation theory with hydrogen
like orbitals for the outer electron and by assuming
the valence electron and the core to be essentially inde
pendent systems. This treatment shows that the analogy
between core polarization and the interatomic London
forces mentioned previously, is perfect. We may even
be tempted to refer to the attraction arising between
the outer electron and the core in the former case, as a
Van der Waals force.
The particular advantage of both London forces and
the use of a correlation potential, is that they both
relate the interactions in the composite system to the
properties of the component parts. This advantage has
induced attempts at using both theories under more
general conditions.
6 I am indebted to Professor W. T. Simpson for a stimulating
discussion on this approach.
7 (a) H. A. Bethe, Handbuch der Physik (Edwards Brothers,
Inc., Ann Arbor, Michigan, 1943), Vol. 24/1, p. 339; see also
Bethe and E. E. Salpeter, Encyclopedia of Physics (Springer
Verlag, Berlin, Germany, 1957), Vol. 35, p. 223 ff; (b) C. Ludwig,
Helv. Physica Acta 7, 273 (1934).
8 J. E. and M. G. Mayer, Phys. Rev. 43, 605 (1933).
9 J. H. Van Vleck and N. G. Whitelaw, Phys. Rev. 44, 551
(1933). PitzerlO has reviewed the use of London forces not
only at the usual large separations, but even within the
same molecule, e.g., to estimate the correlation energies
between the lonepair electrons in the halogen series,
F2 to 12, and to account for the isomerization energies
of the hydrocarbons. Similarly, Douglassll has assumed
the feasibility of using a core-polarization potential
even inside the core and has determined such a poten
tial semiempirically by comparing the observed and
H.F. energy values for the first few series levels in
alkaline ions. Various other empirical attempts at
obtaining such a potential for cases where the outer
electron penetrates well into the core are summarized
by Hartree.1.12 If it were possible to write a corre
lation potential which could act over the whole range
of an outer orbital coordinate another interesting
application would be justified: In molecules where
the inner cores may be assumed to be quite unchanged
by the binding, the correlation energy between the
cores and the valence electrons could be obtained simply
by taking the expectation value of the core polariza
tion potentials (obtained perhaps partly from atomic
spectra) over the outer molecular orbitals. Callaway13
did this for Li, Na, and K atoms14 in their ground
states and for their metals, but he used Bethe's7
method and neglected the penetration effects. This
type of application of the core polarization idea is of
course also implied in the "IT-electron" approximation15
used for organic molecules.
Both the use of London forces within a single molecule
and the representation of inner-outer electron, or
sigma-pi correlation effects by effective potentials when
there is penetration, are open to question. Some of the
difficulties involved in the case of intra-molecular
London forces have already been mentioned by Coul
son.16 On the other hand, the need for the clarification
of the form or the feasibility of a correlation potential
for penetrating orbitals has been emphasized by
Hartree.1.12 In both cases, multipole expansions are often
used at short distances and, sometimes, only the
dipole terms are retained. But this is a defect whose
elimination is relatively straightforward (see Secs.
II and VI). All the other difficulties have to do with the
exclusion principle and the validity of separating elec
trons into distinguishable (either by strong localization
around different centers or by large differences in
energy) groups.
The preceding, discussion indicates that important
simplifications have been possible in dealing with
correlation effects whenever a system of electrons could
10 K. S. Pitzer, Advances in Chern. Phys. 2, 59 (1959).
11 A. S. Douglas, Proc. Cambridge Phil. Soc. 52, 687 (1956).
12 D. R. Hartree, Repts. Progr. Phys. Kyoto 11, 113 (1946).
13 J. Callaway, Phys. Rev. 106, 868 (1957).
14 The adiabatic method has also been applied to alkali by G.
Veselov and I. B. Bersuker, Vestnik Leningrad Univ. Ser. Fiz. i
Khim. No. 16,55 (1957).
16 R. Pariser and R. G. Parr, J. Chern. Phys. 21, 466 (1953).
16 C. A. Coulson, Symposium on Hydrogen Bonding, Ljubljana,
1957 (Pergamon Press, New York, 1959), p. 349.
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:331214 OKTAY SINANOGLU
be divided up into groups in a clear-cut way, but that
as soon as this condition is relaxed difficulties are
encountered. Now if similar simplified treatments of
correlation effects are to be possible in the general case
of a system of many electrons all spread out in nearly
the same region of space, the following question, which
is fundamental to all quantum chemistry, must be
answered:
Given an atom or molecule with some N electrons, can
we consider this system in a nonarbitrary way as made up
of certain groups of electrons in spite of the fact that
all these electrons are indistinguishable from one another?
The presence of a periodic table of the elements and
chemistry suggest that the answer should be yes, not
only in single particle theories, but also after the in
clusion of correlation energies. Clearly if this is to be
done in a nonarbitrary way: (1) All effects of the Pauli
exclusion principle must be included, (2) If we start
with a method based on the expansion of a many
electron function in terms of single particle functions,
then all electrons must use the same complete basis set
of one-electron orbitals. Previous separations of an
electronic system into simpler systems do not satisfy the
second condition and therefore neither the first. In these
treatments, either a one-electron basis set is divided into
two mutually exclusive subsets as in connection with the
sigma-pi problem,17 or else, antisymmetrized products
of many-electron group functions satisfying generalized
orthogonality conditions are used.18 These conditions,
which we shall refer to again in Sec. VI, are too re
strictive and they imply the subdivision of the basis
set, too.
The general problem of formulating a scheme such
that the correlation energy of a system with a large
number of electrons can be obtained in terms of its
simpler components, can now be broken down into two
phases: (a) The separation of the total correlation
energy into that of small groups of electrons. (b) The
evaluation of the interactions within or between these
small groups. The extensions of London-force or core
polarization type treatments to systems with no
strongly localized groups now become special cases
of (b).
With the objectives (a) and (b) in view, we shall now
present a generalized second-order perturbation theory
of a many-electron system whose zero-order wave
function is a single Slater determinant of Hartree-Fock
orbitals. The perturbation method will be based
on the use of all "ordered configurations,"2 i.e., all
unique Slater determinants that can be formed from a
complete one electron basis set. By a systematic
classification of all the virtual transitions represented
by the ordered configurations, the correlation energy
of the system will be broken down into the energies
of pairs of electrons including exchange. Some non-
17 P. G. Lykos and R. G. Parr, J. Chern. Phys. 24, 1166 (1956).
18 R. McWeeny, Proc. Roy. Soc. (London) A253, 242 (1959). pairwise additive terms will also arise. These represent
the tendency of a correlating pair of electrons to avoid
the other electrons of the system, because of the
Pauli exclusion principle. These extra terms will be
defined here as the "exclusion" terms and in future
work they may turn out to be the most important three
or more particle effects. The use of Hartree-Fock
energy as a starting point is convenient, but is not a
requirement. The approach presented here also pro
vides a link with the recent "many-body" methods
developed in different fields of physics.19 Definition of
"mean excitation energies" for each pair of electrons
will allow the estimation of both the pair energies and
the exclusion effects, using only the initially occupied
spin-orbitals. This procedure, although semiquanti
tative, is useful for giving many simple physical inter
pretations, e.g., as in London forces. For instance, in
Sec. II, a core-polarization potential which is applicable
even near the nucleus will be derived and shown to be
the mean-square fluctuation of the Hartree-Fock po
tential acting on an electron. A large portion of this
article will be devoted to the core polarization problem,
not only because it is of interest in itself, but also be
cause a system with a "series" electron is more general
than a system of closed shells, and the excited states,
as well as the ground state of the former, can also be
treated. Detailed derivations will be given, for con
venience, with specific reference to the lithium atom,
although generalization to any other appropriate
atom or molecule is straightforward. Section III
discusses the "exclusion" effect of an outer orbital, e.g.,
in Li, on the correlation energy of the core (Li+) itself.
In Sec. IV, numerical magnitudes of some of these
effects will be calculated for Li and the penetration
parts of the core-polarization potential will be examined.
In Sec. V, molecules are discussed, in general, and
both the use of intramolecular London-type energies
and of "core polarization" justified eliminating the
usual approximations mentioned previously by start
ing from a complete one-electron basis set of molecu
lar orbitals, SCF MO's. In the last section, some
higher-order correlation effects are mentioned; a
"dispersion" energy formula including such effects, but
for use only with asymptotic intermolecular forces, is
derived, and the use of many-electron group functionsI8
is discussed.
II. SEPARATION OF VARIOUS CORRELATION EFFECTS
AND THEORY OF CORE POLARIZATION
The basic theorem2 of the method of "superposition
of configurations" is that if {Uk(X) I form a complete
orthonormal basis set for the space of a single electron
(x including both spatial and spin coordinates), than
any antisymmetric N electron function can be ex-
19 For a brief introduction, see D. ter Haar, Introduction to the
Physics of Many-Body Systems (Interscience Publishers, Inc.,
New York, 1958).
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panded as
Y;(XI' X2," 'XN) = f:,CK.1/IK(XI, X2," 'XN) , (2)
K
where Y;K represents the normalized Slater determinants
y;xCXI, X2, ••. XN) = (N !)-1 det :Ukll Uk., •• 'UkN}, (3)
and K runs over every unique selection of the one
electron indices kl <k2<· .. <kN, i.e., all "ordered
configurations."2 The Y;K now form a complete ortho
normal basis set for the space of N electrons. In this
representation, the energy eigenvalues are the solu
tions of the secular equation
I H-EI I =0, (4)
with H=: (Y;K I H I h))=: (K, HL)} and I the unit
matrix. If anyone of the nondegenerate Y;K in Eq. (2),
say, Y;a, is chosen as the first approximation to y;, then
E can be written2 rigorously as the solution of
E=Haa+Hab(Elbb-Hbb)-IHba, (5)
where the matrix H has been partitioned into four
submatrices. A great variety of perturbation methods
can be derived2,20 from Eq. (5) by making various
approximations in the exact remainder after taking Haa
as the "unperturbed" energy. Thus, if the off-diagonal
elements of Hbb are neglected and E replaced by
Haa, a Schrodinger-type generalized perturbation equa
tion is obtained. This equation, first given by Epstein,21
is
E~HMM-f: [HMKHKM/(HKK-HMM)J
K;>"M
=HMM+EM(2). (6)
Here, among all the diagonal elements of H, HMM is
assumed to be the closest to the exact eigenvalue Ep..
In the summation, K covers the entire orthonormal
many-electron basis set, as in Eq. (2), except Y;M.
Because of the orthonormality of : Uk} the only HMK
that contribute to Eq. (6) are those in which K differs
from M [Eq. (3) J by only one or two spin-orbitals.
These will be referred to as single and double virtual
excitations from M. Further, if Iud are chosen as
Hartree-Fock functions, then the contribution of single
virtual excitations vanishes. This use of Eq. (6) was
proposed by Brillouin22a and MjiSller and Plesset22b
for nondegenerate Y;M.22c Extension to degenerate
states has been discussed by Nesbet.23 In this article
we shall deal only with systems whose zero-order wave
function can be taken as a single Slater determinant
and use a more general basis set, {ud.
20 See also; W. B. Riesenfeld and K. M. Watson, Phys. Rev.
104,492 (1956).
21 P. S. Epstein, Phys. Rev. 28, 695 (1926); see also; L. C.
Pauling and E. B. Wilson, Introduction to Quantum Mechanics,
(McGraw-Hill Book Company, Inc., New York, 1935), p. 191.
22 (a) L. Brillouin, Actualites sci. et ind., No. 159 (1934);
No. 71 (1933); (b) C. M¢ller and M. S. Plesset, Phys. Rev. 46,
618 (1934); (c) see, also, footnote reference 2, p. 283.
23 R. K. Nesbet, Proc. Roy. Soc. (London) A230,312 (1955). Consider first an N electron system of closed shells.
For these, the first N spin-orbitals will be the occupied
H.F. SCF orbitals of the system. The rest of the basis
set {Uk} (k>N) may be assumed to be completed, e.g.,
as described by Lowdin,22o by taking independent
functions and using the Schmidt orthogonalization
process. As we shall see, in the following, the specifica
tion of {Uk} for k>N will be unnecessary. Now, con
sider the more general case of a system of closed shells
plus an outer electron, e.g., an alkali atom. The treat
ment of this case will include the treatment of closed
shells only, and, in addition, will allow a theory of core
polarization for the outer electron in its various states.
We shall return to a discussion of all closed shells in the
section on molecules (Sec. V). For the more general
case, if a polarization potential independent of the
state of the outer electron is to be obtained, the initially
occupied core orbitals must be chosen so as to be inde
pendent of the outer orbitals. The following modified
basis set {ud is the most suitable one for the core
polarization problem: Let the system contain N core
electrons and the (N+l)st outer electron, and let us
suppose that we are interested in, say, n actual states
of the outer electron including the lowest one corre
sponding to the ground state of the atom. Then the
first N spin-orbitals UI, U2, "', UN are to be obtained
from a complete H.F. SCF solution of only the closed
shell part of the system after stripping the (N + 1) st
outer electron. The next n orbitals U(N+l) , U(N+2),
"', UN+n will be taken as the solutions of the H.F.
equation for an electron in the field of the already
determined fixed core orbitals. The rest of the basis set
may be completed again, e.g., by a Schmidt ortho
gonalization process. The complete basis set of spin
orbitals thus obtained will now be denoted by {Uk(X) l =
{k} = 1, 2, 3, "', all odd integers designating the spin
orbitals with spin a, and all even integers those with
spin {3. For concreteness and convenience, we now
continue the treatment on the simplest case, the Li
atom, although extension to larger systems is straight
forward. For any state [1/(3 !)!J det (lsals,3Uka) or
del' (12k) [det' denoting the normalized determinantJ
of Li, the first few spin orbitals by our choice satisfy
the equations
Mls(rl) +(1 11s(r2) 12rI2-1dT2)ls(rl) =EI81s(rl), (7)
where
(8)
and
hcoree ffUk = hlOUk ( rl) +2(/11S( r2) 12r12-ldT2)Uk( rl)
-(I ls*( r2) Uk( r2) r12-1dT2 )lS( rl) = EkUk ( rl)
for uka=k2::3. The second-order correlation energy is
given by Eq. (6) in conjunction with Eq. (2) and the
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one-electron basis set just defined. To separate this
energy into parts, all the ordered configurations (with
any orthonormal {Uk}) that would give matrix ele
ments in Eg. (6) can be classified according to which
one or two spin-orbitals of the initial state det' (12k)
have been virtually excited to give the particular
configuration. Consider first the ground state det' (123),
i.e., (1/v' 6) det (ls"lsI12s,,). From this state, keeping in
mind that the indices kl' k2, k3 which make up K in
Eg. (2) must satisfy the condition k3>k2>k1(k1=1
to 00), the following types of virtual excitations are
possible:
Single excitations:
1, 2, 3~1, 2, I 1>3
~1,k,3 k>3
--+m,2,3 m>3 (9)
Double excitations:
1, 2, 3~1, k, I l>k>3
~m,2,1 l>m>3
~m,k,3 k>m>3. (10)
All higher excitations l>k>m>3 give HOK=O. We
have labeled the indices differently depending upon
which initial orbitals remain unchanged. The first
thing to notice is that with our choice of the orbitals all
H OK for the single virtual excitations of the outer
electron24 vanish by the generalized 22e Brillouin
theorem,25 i.e.,
H OK = (det' (123), H det' (121) ) = (I, hcoreeff3 )
=f3(1,3)=O; (l7"'3). (11)
On the other hand, single excitations from the core and
double excitations lead to
(det'(123), H det'(lk3) )= (det'(23) , g12 det'(k3) )
(det'(123), H det'(m23) )= (det'(13), g12 det'(m3) )
(12)
where Eg. (7) has been used; and,
(det' (123), H det' (lkl) ) = (det' (23), g12 det' (kl) )
(13)
and, similarly, for the rest of Egs. (10). Here g12 =r12-1
and det' denotes 0/v'6) det on the left and (1/v'2) det
on the right of Egs. (12) and (13). Notice that Egs.
(12) do not yield zero, because the core orbitals 1 and
2 have been chosen as the H.F. SCF solutions of Li+
24 Because of complete antisymmetry, we cannot refer to a
definite electron. What we mean by "outer" electron is anyone
of the electrons occupying the particular orbital 3.
25 Notice that with this generalization,22 only the initially occu
pied orbital 3 must be an eigenfunction of h.orceff to make Eq.
(11) vanish. There is no such condition for I. rather than of Li. This point will be discussed in detail
after stating Eg. (17c).
Eguations (9) through (13) allow the separation of
the second-order .energy in Eg. (6) into correlation
energies of pairs of electrons. Of course it is a general
result that, whenever, orthonormal functions are
used and overlap, and exchange effects are neglected,
second-order energies come out "pairwise additive"4 as
in intermolecular "dispersion" forces. Three-body and
higher correlations appear in higher orders of per
turbation. However, we shall see shortly that, here, the
Pauli exclusion principle has already introduced some
many-body correlations into the second-order. The
first-order energy in Eg. (6) can be written as the
energy of the ion core in the field of the bare nucleus
and the energy of the outer electron in the SCF field of
the ion core. For the ground state
3 3
Hoo= (det'(123), (~:::hiO+ Lgij) det'(123) )
i~1 '>i
where
feoreO( 12) = 2 (Is, h10ls)+ (Isis, gds1s).
With the systematic classification of the virtual transi
tions given by Egs. (9) and (10), EO(2) in Eg. (6) can be
broken down into the following terms, depending upon
which initial orbitals are involved:
Again numbers in parentheses label the orbitals, and
not the electrons; and,
-E
O(2) (12) = L (det'(12), g12 det'(mk) )2 (16a)
k>m>3 ~(1, 2, 3~m, k, 3)
_ E
O(2) (23) = L (det' (23), g12 det' (k3) )2
k>3 ~(1, 2, 3~1, k, 3)
+ '" (det' (23), g12 det' (kl) )2
L...J (17a)
l>k>3 ~(1, 2, 3~1, k, I)
_ &(2)(13) = L (det'(13) , g12 det'(m3) )2
m>3 ~(1, 2, 3~m, 2, 3)
'" (det' (13) , g12 det' (ml) )2
+L...J • l>m>3 ~(1, 2, 3~m, 2, I) (18a)
The ~'s in the denominators represent the energies of
the respective virtual transitions and are given from
Eq. (6) by
~(1, 2, 3-4m, k, I) == (det'(mkl), H det'(mkl) )
-(det'(123),Hdet'(123». (19)
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In particular, when m is odd and k is even,
~(1, 2, ~m, k, 3) =~hO(1~m)+~hO(2~k)
+(J mk-J12) +(JkS- J2S)+(J -K)ms-(J -K)13,
(16b)
~(1, 2, 3)~1, k, 3) =MO(2---,>k)+(Jk1-J 21)
+ (hs-J23) , (17b)
~(l, 2, ~1, k, l) = ~hO(2~k) +~hO(3~l) + (Jk1-J21)
+(Jkl-J 2S)+(J-K)n-(J-K)31, (17c)
and, similarly, for Eq. (18a). J's and K's are the usual
coulomb and exchange integrals,
and ~hO(r---'>s) = (s, hOs)-(r, hOr). The numerical
evaluation of E(2) for anyone case is a lengthy pro
cedure because large contributions to such sums are to
be expected2 from the continuum (or from what would
take the place of the continuum) part of the basis set
{k I. Our objective here, instead, is to investigate,
semiquantitatively, various physical aspects of Eqs.
(16a)-(18a) and their variation with respect to the
different actual excited states of the "series" electron.
(See, however, Sec. IV.)
The first part, Eq. (16a), corresponds to the correla
tion energy of the ion-core (ls"ls~) as it exists in the
neutral atom. The effect of the "outer" electron by
means of the exclusion principle, on this core energy
will be taken up later in Sec. III. Equations (17a)
and (18a) give the correlation energy between an
electron occupying 3, i.e., 2s" and those in 1 (i.e., ls.,)
and 2 (i.e., ls~), respectively. The first terms on the
right hand sides of Eqs. (17a) and (18a) represent the
inner electrons in orbitals 1 and 2 making virtual transi
tions in the average field of the orbital 3. These are
the "orbital average polarization" terms as we re
ferred to them in the Introduction. Notice that they
arose only because 1 and 2 were chosen as the SCF or
bitals of Li+ without introducing the H.F. field of 2s into
Eq. (7). [See also Eq. (12).] Actually "orbital average
polarization" is a very small effect and, therefore, in
Hartree-Fock calculations on alkalilike configurations
the same core orbitals are often usedl,26 for the free ion
and the atom as we have done for Li+ and Li. On the
other hand, the real correlation energy that remains
after a completely SCF calculation has been made, is
given by the second terms of Eqs. (17 a) and (18a).
These terms are due to one core electron at a time
making virtual transitions simultaneously with the
outer electron. Such double excitation effects are
sometimes referred to as "dispersion" energy in analogy
to London forces.18 Here we have referred to the to
tality of Eqs. (17a) and (18a) , including "disper-
26 See for example: D. R. Hartree and W. Hartree, Proc. Cam
bridge Phi!. Soc. 34, 550 (1938). sion," by "core-polarization" energy (see Introduc
tion). Unfortunately, the possibility of confusion
exists, because sometimes the "orbital average polari
zation" is simply referred to as "polarization. "18 ,23
Such "orbital average polarization" effects, which also
arise in going from various "restricted" types of H.F.
calculation to "unrestricted" types, have been dis
cussed2s for Li. As it has been shown,27 however, these
are much smaller effects than the correlation energy
(in fact, about lO-S of it), i.e., than the "core-polariza
tion" energy given by Eqs. (17a) and (18a). Although
a small effect, here, we have included the "orbital
average polarization" energy in the total "core-polariza
tion" for completeness and convenience.
The sums in Eqs. (16a) to (18a) can be put into ap
proximate but closed forms which are physically
interesting by taking out the energy denominators
as "mean excitation energies" in each case. This type of
approximation has been made in many different con
texts since U nsOld28 and is used, for instance, in London
forces.s,lo In the first parts of Eqs. (17a) and (18a),
~'s consist of excitations of a single core electron at a
time, and we write
<~(l, 2,3---'>1, k, 3) )Avk~ <~(1, 2, 3---'>m, 2, 3) )Avm=5c,
(20)
where < )AV denote averages. The second, i.e.,
"dispersion" parts, of Eqs. (17a) and (18a) involve a
similar single core excitation, but, in addition, virtual
transitions of the outer electron in the field of the
already excited core. [See Eq. (17c).] Thus to define
a "mean excitation energy" for the outer electron, an
averaging over both k and l in Eq. ( 17 c) is necessary,
i.e.,
<~(1, 2,3---'>1, k, l) )Avk,l~ <MO(2~k) + (Jk1-J21) )AV"
+ <MO(3---'>l) +(J -K) 11-(J -K)al+(hl- J2S) )A,k,l
and
<~(1, 2,3---'>1, k, l) )Avk,l~ <~(1, 2, 3---'>m, 2, l) )A,m,l
:=5c+5v*(3) (21)
where 5v*(3) refers to the "mean excitation energy" of
the valence, i.e., the "series" electron initially occupy
ing orbital 3. For our purposes only very rough ideas of
the magnitudes of 5's are necessary. The best use of 5's
would be as semiempirical parameters. If the part of
the basis set {Uk} corresponding to the continuum 2 had
not made a large contribution to the second order
energy sums, 5's would have been of the order of ioniza
tion potentials, I. But actually more than half the
second order energy, e.g., in He, comes from the
continuum. Thus in general 5's will be several times the
ionization potentials. The use of ionization potentials is
27 R. E. Watson and R. K. Nesbet, Mass. Inst. Techno!. Quart.
Progr. Rept., October 15, 1959, p. 35.
26 A. Unsiild, Z. Physik. 43, 563 (1927).
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permissible, however, only where lower limits to energy
terms are desired. Thus we write Be> f(Li+) and Bv*;(;
f(LI) and defer further discussion of B's until Sec. IV.
In Eq. (21), Bv* represents a small fraction of Be;
moreover, the single excitation terms in Eqs. (17a)
and (17b) are small as was mentioned earlier. Thus
neglecting the absence of Bv* in Eq. (19) [compare
Eq. (20) ] all the energy denominators may be equated
and Eqs. (17a) and (18a) combined into one "series"
electron-core correlation energy. For the ground state
ofLi
-Eev(2) (3) =-EO(2)(23) -Eo(2)(13)
""[Be+Bv(3)]-1[ 1: (det'(23),g 12det'(kl»)2
l>k;;';3
+ 1: (det'(13),g 12det'(ml) )2]. (22)
l>m~3
For convenience, so far we have considered only the
ground state [det (123) ] of the Li atom and in Eqs.
(9) and (10) analyzed the various types of virtual
transitions that are possible from this state. Since,
however, one of our objectives is to examine the
validity of deriving an effective potential for the outer
electron, before giving closed expressions for the sums
in Eq. (22) it is necessary to consider the excited states
of Li, [det' (12n)] where now the "series" electron
occupies an orbital (n) different from 2sa•
With a different state of Li, the second-order sum in
Eq. (6) again involves the same complete set of
"ordered configurations," except that whereas before,
det' (123) had been excluded and singled out as the zero
order wave function, now the sum includes this deter
minant but instead excludes det'(12n), the new zero
order wave function. E(2) in Eq. (6), or in operator
form
-EN(2)= (N, {H 1: M)(HNN-HMM)-l(M,Hl1'vT
)
M,cN
(23)
where N denotes det' (12n), can be put also in the form
of an expectation value over the "series" electron
orbital n. Denoting
N=det'(12n) =a{nu c(12) I
where
uc(12) = (1jv2!) det(12),
and a is the operator antisymmetrizing n with the two
core electrons, we obtain
-EN(2)= (n{uc(12),aH 1:M)(H NN-HMM)-1
M,cN
X (M, Hauc(12)}n >
(24)
The curly brackets in the second term have been
placed to indicate that integrations over the coordinates
in n are to be performed last. As it stands the formal core "series" electron interaction operator ten (2) is far
from being a potential energy for n. First, the summa
tion over M includes all double virtual excitations of the
core orbitals, (12); thus the part corresponding to Eq.
(16a), the correlation energy of the core itself, has not
yet been separated. As we have mentioned earlier, this
part actually depends on n, due to the "exclusion"
principle and it will be taken up later. In addition if an
arbitrary one-electron basis set were to be used, ten (2)
would depend on which N had been excluded from the
sum in Eqs. (23) or (24). On the other hand, with an
SCF H.F. basis or the basis {k I we have chosen above,
this type of dependence is eliminated, because the
"series" electron orbitals satisfy Eq. (11). Thus in
EO(2) single virtual excitations from det' (123), i.e.,
the first of the Eqs. (9), would include
1,2,3---->1,2, n,
whereas in EN (2) , the same type of excitations from
det'(12n) would exclude 1, 2, n but include
1,2, n---->1, 2, 3.
But, both of these virtual transitions have zero matrix
elements by Eq. (11) and, hence, are without effect on
tcn(2). The rest of Eqs. (9) and (10) can be generalized
to any state of Li, det'(12n), keeping in mind the
ordering of configurations in Eq. (2) by k>l>m
(m = 1 to 0Cl). Some transitions that were single core
transitions for one state become "core"-n double
transitions for another state and vice versa, as, for
example, in
1,2,3---->1,5,3
1,2,n---->1,3,5 (n~3, 5);
otherwise the totality of the virtual transitions involving
either a single core orbital or a single core orbital and
the outer one, n, are unchanged. Thus, after making the
same approximations in the energy denominators, the
core-valence electron correlation energy in Eq. (22) can
be written for any "series" state of Li as
-Eev(2)(n) = -EN (2) (2n) -EN(2) (1n)
""[t3c+t3v(n)]-I[ 1: (det'(2n), g12 det'(kl»2
l>k~3
+ L: (det'(1n),g 12det'(ml»)2], (25)
l>m~3
where n> 2. The independence of the indices in the
above sums, from n, is the biggest advantage of the way
the basis set {k I was chosen previously. [See Eqs. (12)
and (7)]. If the orbitals 1, 2, 3 had been taken as the
completely SCF solutions of Li, the first terms in Eqs.
(17a) and (18a) would have vanished and both the
core orbitals 1, 2 and the summing indices in Eq. (25)
would have been made to depend on the particular
outer orbital. We can now anticipate putting the sums
appearing in Eq. (24) into closed forms and hope to
obtain the desired potential (independent of n) were it
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not for the appearance of Bv(n) in the denominator.
Bv(n) as in Eqs. (21) and (22) represents the "mean
virtual excitation" energy of the outer orbital n. For
n>3, in some terms such as
1, 2, n--l, 3, 5 (n>3, 5),
of the equations corresponding to Eqs. (17a) and
(18a), n--5 would actually be a "deexcitation" and
make a negative contribution to Bv(n). However, there
are only a few such terms for low lying n, and the weight
of anyone term to the sum, e.g., in Eq. (17a), is small.
As mentioned earlier, Bv(n) represents a small fraction
of Be, so that it can either be neglected in comparison
with Be or replaced by a reasonable average,1 as, for the
few states det' (12n) of interest. Since Bv( n) is small,
the errors made by replacing it by as will be even
smaller. Assuming for the moment that the sums could
be put in the suitable form, the requirement
(8v(n)/8c) <1 is necessary if any "core-polarization"
potential derived from Eq. (25) is to be independent
of n. Notice that this requirement is similar to the
necessity of having the outer electron move slowly
compared to the core electrons in the adiabatic ap
proximation, except that here it does not enter in a
fundamental way.
The summations in Eq. (25) can now be carried out.
Since the core (lsals~) is a closed shell, it is necessary
to discuss only those n that have either all ex or all f3
spin. Take all n to be odd, i.e., with spin ex, and in
Eq. (25) consider first the pair involving opposite
spins, i.e., det'(2n). Then
L (det'(2n), g12 det'(kl) )2= (det'(2n), g122 det'(2n) )
l>k<:3
-L (det'(2n), g12 det'(2l)2
1>2,k=>2
-L (det'(2n),gI2 det'(11»)2 (26)
1>I,k=1
This follows from a matrix multiplication relation of the
type
I:(M, AL)(L,BN)= (M, (AB)N), (27)
L=1
where A and B are operators acting on the space for
which the set of orthonormal vectors {L l = 1, 2, 3, .•• 00
form a complete basis. In Eq. (26) the set of all nor
malized two-electron determinants corresponding to the
ordered configurations l>k~l, as in Eq. (2), form a
complete orthonormal basis for the space of two-electron
coordinates (including spin) on which g12 acts. The
spins of 2 and n being opposed, the determinantal
matrix elements on the right-hand side of Eq. (26)
reduce to the direct integrals only; e.g., for the ground
state of Li,
(det'(23), g122 det'(23) )= (1s2s, glns2s). (28) Similarly,
L(det'(2n),g12det(21»2= 2: (2n,g1221)2, (29)
z>2 1>2,(I=odd)
where to conserve spin, l must have spin ex, i.e., be odd.
The last sum can also be evaluated by carrying out the
integrations in each matrix element over one set of the
electron coordinates and, thus, obtaining a function
(W2,2) of the coordinates of the other electron; i.e.,
where
Then, making use of Eq. (27) we get
2: (2n, g122l)2 = 2: (n, W2,2l)2
1>2,(I=odd) 1>2, (l=odd) (30)
since 1= 1, 3, 5, ... odd··· 00 form a complete one
electron basis set and W2,2 of Eq. (30) is a function of
the coordinates of a single electron. In the same way,
in Eq. (26)
L (det' (2n) , g12 det' (1l) )2 = 2: (2n, g12l1 )2
1>1 l>l,(l=even)
2: (2, Wn,11)2= (2, (Wn,I)22), (32)
1>1,(I=even)
since now to conserve spin, l must be even (spin f3) and
1 = 2, 4, 6, ... even .•. 00 form a complete basis set for
functions of the spatial coordinates of one electron.
Similarly for the parallel spin pair (1n) in Eq. (25),
the use of Eq. (27) leads to
2: (det'(1n), g12 det'(ml) )2
l>m>3
= (det'(1n), g122 det'(1n) )
-L (det'(1n), g12 det'(2l) )2
1>2
-L (det'(ln), g12 det'Cll) )2. (33)
1>1
Now both of the spin orbitals 1 and n have spin ex, so
that
(det' (In), g122 det' (In) ) = (In, g1221n)- (In, gllnl ).
(34)
Also, since 2 has spin {3,
L (det'(ln), g12 det'(2l) )2=0. (35)
l>2
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The last term of Eq. (33)
L (det'(1n) , g12 det'(1l) ?
1>1
L [ (In, gdl)-(In, gdl)]2
1>I.Cl=odd)
L (n, W1,1I )2+ L (1, Wn,11)2
1>I,Codd) 1>I,Codd)
1>I,Codd)
defining the one electron functions as in Eq. (30).
Then, using Eq. (27) in each term of the last expres
sion,
L( det' (In), g12 det' (11) )2
1>1
= (n, (W1,1)2n)+ (1, (W n,l)2l )
-2(n, (W1,I)(Wn,I)1), (37)
and, from Eq. (33),
L (det'(ln),gI2 det'(ml»2
l>m"'3
=( In, g1221n)- (In, glz2nl)
-(n, (Wl,l)2n)-(1, (Wn,I)21)
+2(n,(W 1,l)(Wn,I)1). (38)
Before going into the correlation energy of the core
itself and deriving expressions for it similar to those
given above, let us examine the meaning of the various
terms in the results for Ecv(2)(n), Eq. (25). The major
part of the core-valence electron correlation energy is
due to the pair with opposing spins, i.e., EN(2)(2n),
since the other aa pair electrons are already kept
apart by the Pauli exclusion principle. By Eqs. (26),
and (29) through (32), the a!3-pair energy for some
state det'(12n) of Li is given by
-EN(Z) (2n) """ (bc+os)-l[ (2n, glz22n)-(n, (WZ,2)2n)
+ (n, (Wd 1)2-(2, (Wn,I)22)]. (39)
Or, for the ground state of Li with n=3, replacing
1 = (1S)a, 2 = (1s)/l and 3 = (2s)a,
-EO(2J[ (ls) /l(2s) aJ""" (8C+OS)-1
X [( (2s), F(lB) (2s) )-RClsp)(28,,)J, (40)
where we have defined
«2s), F(lB) (2s) )=( (2s) (ls), gI22(2s) (ls»
-«2s), (WCls)(18»2(2s» (4la)
and
R(18a)(lsp)= « Is), (WC2S) ,(18»2(1S) )-(1s2s, gdsls )2.
(41b)
Clearly F(lB) represents a true potential acting on the
outer electron, and depends solely on the core orbital (ls) ; it is given by
F(18) (rz) ={J !ls(rl) !Z[1/(! r2-r1!2)JdTl}
-{J !ls(rl) !Z[l/(! r2-r1!)]dTlf, (42)
where r2 and r1 are the position coordinates in (2s) and
(ls), respectively, and the potential acting at r2 is
obtained by integration over all r1' Fc1s) (r2) in Eq. (42)
can be identically written as
F(lB) (r2)
= «(1s) , [(1/rI22) -«(1s) , (1/rI2) (1s) )12J(ls»1
= «ls), [(l/rd -«ls), (l/rd (1s) )IJ2(ls»1
= «gI2-(gI2)Av,ls)Z)Av,ls, (43)
where ( )1 means that all integrations in Eq. (43)
are over the same coordinates rl as in Eq. (42). In the
last term we have denoted the quantum mechanical
averages of the "source point" rl over (1s) with the
symbol, ( )AV' It will be noticed that (glZ-(glZ )Av,IB
simply represents the "instantaneous" (in the virtual
sense) deviation of the electrostatic potential, produced
at the point rz by the electronic charge at rl, from the
orbital average [i.e., the expectation value over ls(rl)]
potential of the electron (rl) produced at r2. Thus as
we have done in the theory of Van der Waals' inter
actions between molecules and solid surfaces,29
F(Is)(r2)/bc may be called the fluctuation potential
since F(ls)(rz) is simply the mean square fluctuation
of the coulombic potential of the orbital (ls) at the
point rz.
In Eq. (40), there still exists a remainder term
R(18)p(Z8)a which cannot be put in the form of an ex
pectation value of a potential. A close examination of
the Eqs. (26) through (32) leading to Eq. (40) shows
that R(l8)p(Z8a) arises because the closed inner shell
(lsals/l) of Li prevents the outer electron occupying
2sa from making virtual transitions to the already
occupied inner levels. This is one of the exclusion
effects that were mentioned in the introduction and
represents the nonpairwise additive effect of (ls",) on
the pair (ls/l2s .. ). Rc18p) (n) has been neglected in the
previous treatments of core-polarization for non
penetrating orbitals.7-9,13,14 However, especially for the
ground state, its magnitude requires examination
(See Sec. IV) .
Likewise, the correlation energy of the aa pair
(ls .. 2sJ of the ground state of Li in Eq. (25) is derived
from Eqs. (38), (40), and (41) by
-Eo(Z)(1sa2s .. )~(bc+os)-I[ «2s), F(18) (2s) )
-«2s), F(18)ex(2s) )-RIBa28,:x] (44)
29 O. Sinanogiu, Ph.D. thesis, Part I, University of California,
August, 1959; O. Sinanogiu and K. S. Pitzer, J. Chern. Phys. 32,
1279 (1960).
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:33CORRELATION ENERGIES AND CORE-POLARIZATION 1221
where
«2s), F(ls)ex(2s) )== (2s1s, g1221s2s)
-(2s, (W1s,ls) (Wls,2s) is) (4Sa)
and
RIsa2s,:x== (is, (W28,ls) 21s)-(2s, (W1s,ls) (Wls,2S) is).
(4Sb)
Comparison of the Eqs. (38) and (44) with (39) and
( 40) shows that .£0(2) (lsa2sa), in addition to the
fluctuation potential of EO(2) (ls,s2sa) also contains
F(ls)ex/(~c+os) or what may be called the "exchange
fluctuation potential." Note that the latter is a "po
tential" only in exactly the same sense2 as the exchange
part of the Hartree-Fock field [see Eq. (8) J is. The
"exclusion" term in Eq. (4Sb) is similar to that in Eq.
(41b). Combining Eqs. (6), (7), (14), (is), (40),
and (44), the total energy of Li in any state,
N ==det' (12n), is given as
EN(12n)'"""HNN+EN(2)'"""~coreO(12) + (n, hcoreeffn)
+ EN(2) (12) + (n, U tn )+Qn
= EN (core) + (n, (hcore"ff+U,)n)+Qn, (46)
where
and
Qn = (R1spn+ Risanex) / (~c+os).
EN (core) contains the H.F. energy [Eq. (14) J of the
free ion Li+, and, in addition, the correlation energy
EN(2)(12) [given for the ground state by Eq. (16a)J
which refers not to free Li+ but to the core as it exists
in state LV of Li. The dependence of EN(2)(12) on the
outer orbital n through the "exclusion" principle is
discussed in Sec. III. Qn constitutes the rest of the
"exclusion" effect. It depends upon the exchange
charge density of n with (is), so that it will be small
for excited states of Li, i.e., "nonpenetrating" n. Uj
is the desired correlation (fluctuation) potential. With
a larger atom, similar results can easily be written
down. In general, there will be a contribution from each
electron of the core to Uj and Qn. Mter estimating the
"exclusion" effects, Uj can be determined semi
empirically (see e.g., Douglasll) for instance, by leaving
(~c+os) as a parameter. Aside from the "exclusion"
effects, Eq. (46) has the variational form for the outer
electron with an effective "core-Hamiltonian." It is
important to realize however that in this form n
cannot be varied to improve the energy even when
Qn"'O and EN (core) "'constant, because the result was
derived for a specific choice of orbitals for n, namely,
those satisfying the H.F. condition, Eq. (11). With
any other choice, the single virtual transitions of n as in
Eq. (9) would lead to nonvanishing matrix elements
and make a new contribution to Eq. (46). This point
brings out the connection between the present treat-ment and the recent nuclear manybody theory.19.30 In
fact, Eq. (46) corresponds to a starting approximation
of that theory with the neglect of higher-order cor
relations (Sec. VI) as can be seen, e.g., in the work of
Bethe31 and Rodberg.32 Improved choices for n can be
made and perhaps restrictions, as in Eq. (11), removed
by going to higher orders of perturbation, but such
generalizations will be deferred to a future date.
III. "EXCLUSION" EFFECT OF AN OUTER ELECTRON
ON THE CORE ENERGY
The dependence of the core correlation energy
EN(2) (12) of Eq. (46) on the outer orbital n in Li
can be examined by a careful classification of all the
"ordered configurations" entering Eq. (6) and gen
eralization of Eq. (16a) to any n. In Eq. (10) some of
the configurations that correspond to double virtual
transitions from the core (ls,xis,8) when n was 3 (i.e.,
2sa) become triple transitions from another initial state
det'(12n) with n>3, and vice versa. Including all
such configurations in E(2), one obtains the expected
result that
-EN(2)(12) = L
k>m;;'3,(k,"""n) (det' (12), g12 dee (mk) )2
,:l(1, 2, n---+m, k, n)
(47)
Le., all the double core transitions 1, 2--'>m, k are missing
when m or k is the already occupied orbital n. This may
be compared with the second-order energy of the free
ion core Li+ using the same one-electron basis set I k}
that was defined previously for Li:
-ELi+(2) (12) = L (det'(12), g12 det'(mk) )2. (48)
k>m>2 ,:l(1, 2--+m, k)
The energy denominator in Eq. (47) differs from that
in Eq. (48) by the presence of n [See Eq. (19)].
Nevertheless a semiquantitative estimate of the varia
tion of EN(2)(12) with n and its difference from the
energy of Li+ can be obtained by replacing both ,:l's by
one average, Llcore[3core> (I Li++ hiH)]. Then com
paring Eqs. (47) and (48),
(E<2) Li core(n) == EN(2) (12) '" ELi+(2) (12)
+3core-1t(det'(12),g12 det'(nk) )2, (49)
k;;'3
orusingEq. (27) as in Eq. (32),
EN(2) (12) '" ELi+(2)+Llcore -l[ «ls), (W(ls),n' )2(1S) )
-«is) (n'), gI2(ls)(ls) )2J, (SOa)
30 K. A. Brueckner, C. A. Levinson, and H. M. Mahmoud,
Phys. Rev. 95, 217 (1954); for later references see H. Yoshizumi,
Advances in Chern. Phys. 2,323 (1959).
31 H. A. Bethe, Phys. Rev. 103, 1353 (1956).
32 L. S. Rodberg, Ann. Phys. 2,199 (1957).
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:331222 OKTAY SINANOGLU
with n'a=n, and from Eq. (46),
EN( core) '" ELi++Aoore -1[ «(ls) , (W1s,n') 2(lS) )
-«(ls)n', gI2(ls) (ls) )2J
== E(L i+) + (Rcore-n/ ~core) . (SOb)
Thus, the total energy of the core in the Li atom is
given by the total energy of the free Li+ ion including
its correlation energy plus the last term which is the
desired "exclusion" effect of n' on the core. With any
larger system, the evaluation of all such "exclusion"
effects is similarly possible from a classification of
ordered configurations and the use of Eq. (27) for
summations.
IV. MAGNITUDES FOR THE GROUND STATE OF
LITHIUM
In Sec. II, Eq. (46) we have obtained an expression
for the total energy of the Li atom in anyone of its
"series" states. That derivation shows, that aside from
the "exclusion" effects, Uf is the desired "core-polariza
tion" potential including exchange and it may be
regarded as the mean-square fluctuation of the Hartree
Fock potential of the core per unit of "mean excitation
energy." Notice that Uf is a complete potential and is
not dependent on a multipole expansion of g12. In the
previous treat~ents7-9,I3,I4 of "core polarization" (a)
the "exclusion" effects, (b) the exchange part of Uj, i.e.,
FI:x[5c+os)' has been neglected (see, however,
Ludwig7b) and (c) after making a multipole expansion
of gI2, as
X (3 cosL 1) +... (51)
mainly the dipole term (with estimates of quadrupole
terms) has been considered and the first part r>-1
dropped. In Eq. (51) r> denotes the greater of the two
distances rl and r2, and w is the angle between the radius
vectors of the two electrons. For highly excited states,
i.e., with larger n, assumptions a to c approach validity.
For instance, as the portion of the outer orbital that is
inside the core becomes negligible, we get r> =r2 only,
so that the ,>-1 part of g12 no longer contributes to U f
[See Eqs. (41) and (45) J. For "penetrating" orbitals,
however, such is not the case. To get an idea of the
magnitudes of the previously neglected penetration and
"exclusion" terms, we shall consider here the ground
state of Li for which the effects should be largest. We
take for 2s the orthogonalized Slater orbital,
2s0= 1.0148 (oN311')t, exp( -02') -0.1742(oNn-)t
Xexp( -Olr) (5Za)
with 01 = Z.65 and 02 =0.65, which sufficiently approxi
mates the Hartree-Fock (Zs) orbital of Fock and Petrashen,33 and for ls,
ls= (IlNlI') I exp( -oIr) (5Zb)
with oI=Z.65. The "exclusion" terms in Eqs. (41b),
( 45b) , and (SOb) involve g12 in the W integrals. These
are like the usual atomic integrals34; upon substitution
of Eq. (51) for g12, only the r> -1 term contributes due
to the spherical symmetry of ls and Zs. The first parts
of F(1s) and F(Is)ex in Uf on the other hand, contain
gI22. Uf can be obtained completely, without any
expansions, from Eqs. (43) and (46). However, in this
article we shall consider only the penetration terms.
To compare the penetration effects with the pre
viously considered dipole terms on an equivalent basis,
we must also expand gI22 and not consider higher
multipoles. g122 can be conveniently expanded in terms
of the Gegenbauer35 polynomials, Cn (1) (cosw) .
g122 = 1/rI22 = (1/r>2) fer <Ir» nCn (1) (cosw). (53)
n~
These polynominals are analogous to the Legendre
polynominals and have similar addition theorems. The
first term of Eq. (53) is r> -2, neglected previously. It is
responsible for most of the penetration effects of Zs, as
can be seen e.g., from the fact that only r> -1 contributes
to the "exclusion" terms and will be calculated here.
With the orbitals of Eq. (5Z), the desired integra
tions can be performed analytically and yield
(ZSO, (W1S,ls)2ZS0)=<Zso, (f lls(rI) l2r>-IdrJ2s0)
=0.16144(a.u.)2
(ls, (W2so,Is)21s )=0.018537(a.u.)2
(ls2s0, gdsls )2= (ls2sO, ,>-11s1s )2=0.013855(a.u.)2
(2s0, (Wls,Is) (W280,Is) ls ) =0.OZ9806( a.u.) 2
(2s01s, r> -22so1s ) =0.17385 (a.u.) 2
(2s01s, r> -21sZs0) =0.053256(a.u.)2
(54a)
where l(a.u.) =27.202 ev and, e.g.,
+ to lls (rI) l2dri. (55) r,
As it was mentioned in Sec. II, ionization potentials
provide lower limits to the "mean excitation energies,"
33 v, Fock and M. J. Petrashen, Physik. Z. Sowjetunion, 8,
547 (1935).
34 E. U. Condon and G. H. Shortley, The Theory of Atomic
Spectra (Cambridge University Press, New York, 1957), p. 174.
36 For a quite detailed account of these polynominals, see I.
Prigogine et al., Molecular Theory of Solutions (North-Holland
Publishing Company, Amsterdam, 1957), p. 265.
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:33CORRELATION ENERGIES AND CORE-POLARIZATION 1223
and therefore, we write
5e+5s= Xes[I(Li)+ I(Li+)]"'3Xes(a.u.) (54b)
and
Keore= Xeore[I(Li+)+ I(Li'+)] = 7.24Xcore(a.u.),
where Xcore> Xes> 1. Estimates show that in the
(Li+) ion, where two electrons are in the same orbital,
Xcore is approximately four due to the large contribu
tion of the continuum in this case. On the other hand,
when two electrons are in different orbitals X may be
about two. We will take Xes"'2. For results more
quantitative than we are aiming at here, it is also
possible to obtain X values for each of the pair sums in
Eqs. (16a) to (18a), e.g., by a procedure devised by
Kessler.36
We may also remark that, once the total correlation
energy of a large system has been separated into those
of pairs of electrons, the E(2) of each pair can be ob
tained in a number of ways. Here we have emphasized
the semiempirical approach. One may also formulate
"variation-perturbation" methods for each pair as has
been possible for He7&.
Substituting the foregoing results, Eqs. (54a, b), into
Eqs. (41b), (45b), and (SOb), we obtain
and (5c+5s)-IRI8~28a=0.04245Xcs-1 ev,
(8e+5s)-IRIs,,2sa = -0.1022X cs-1 ev, (56a)
(56b)
(Kcore)-IRcore-2sa=0.0176JCcore-1 ev. (56c)
U :oing only the first terms of gl2 and gl22 from Eqs. (51)
and (53) in Eqs. (40) and (44) and denoting the corre
sponding parts of F(ls) and F(ls)ex by
(2s0, Fr>2s0)= (2s01s,'> -22s01s)-(2s0, (Wla,ls)22s0)
(2s0, Fr>ex2s0)= (2s01s, ,>-21s2s0)
-(2s0, (W1s,ls) (W1s,2s) Is) (57)
we get
(8e+5s)-1 (2s0, Fr>2s0) =0.1125X cs-1 ev. (58a)
(8c+5s)-1 (2s0, Fr>ex2sO) =0.2126X cs-1 ev. (58b)
Essentially the same values are obtained by the use of
51 = 2.70 instead of 2.65 in 2s0 and Is so that the results
are not very sensitive to our specific choice of the
orbital parameters in Eqs. (52).
Most of the "penetration" effects of 2s are included
in Eqs. (56) to (58). Callaway13 has obtained a "core
polarization" potential in Li using only the dipole part
of g12, equivalent to taking the second term of Eq. (53)
in F(ls) and neglecting F(1.)ex. He finds a contribution
of 0.1 ev to (2s, Uf2s). Actually the results of Ludwig7b
suggest that the exchange term in the dipole part may
be negligible. By comparison, several interesting con-
36 P. Kessler, Compt. rend., 242, 350 (1955). clusions follow from the Eqs. (56)-(58). First, the
"exclusion" effect of the outer orbital on the core
correlation energy is only 0.0176 JCcore-1 ev with
JCcore -1< 1, hence negligible even for 2s. In Eq. (50) we
can then take EN (core) =E(Li+). Secondly, the total
contribution to the correlation energy of the lsfJ2sa
pair from the r> -1 terms is (8c+5s)-I( (2sO, Fr>2s0)
RIs~2.J or (0.1125-0.0423 =0.0702) Xcs-1 ev. With
Xes-I",!, this is still appreciable compared to 0.1 ev.
the dipole contribution from both of the core electrons. IS
Also the "orbital average polarization" effect of the
orbital 2s appears only in the ,>-1 part of gl2 due to the
spherical symmetry of Is and 2s. This effect, however.
as was mentioned earlier, is not strictly a correlation
effect since it results in converting the Li+ H.F. SCF
orbitals to the completely H.F. SCF orbitals in Li.
The term, Fr>, which we have calculated previously, on
the other hand, corresponds to the fluctuation of ,>-1
i.e., (,>-2_ (,>-1 )12 )1, and, hence, to the inclusion of
the "dispersion" effect. It is much larger than the
"orbital average polarization" energy (see Sec. II).
Thirdly, combining all the '> or "penetration" terms
for the ls)sa pair we find that
(8c+5s)-I( (2s0, Fr>2s0)-(2s0, Fr>ex2s0)-Rlsa2saex)
=0.OO208Xcs-1 ev,
an entirely negligible value. Thus the "Fermi hole" is
very effective in keeping the electrons of the aa pair
apart and not necessitating a "Coulomb hole." Hence
to obtain the over-all "core-polarization" potential \ve
need to add the ,>-1 terms only for the lsp2s" pair.
Then neglecting exchange in dipole and higher-order
terms, Uf in Li may be taken as
(59)
V. MOLECULES
The treatment that was given in previous sections
and demonstrated in detail for the case of the Li atom
can be applied to any N-electron system whose zero
order wave function is a single Slater determinant of
H.F. orbitals. Thus, the second-order energy of most
molecules can be separated into "pair correlations"
and nonpairwise additive "exclusion" effects by taking
the H.F. SCF molecular orbitals (MO) as the one
electron basis set {k}. These orbitals are obtainable by
Roothaan's procedure.37 Each energy component can be
obtained in closed form by taking out the denominators
as "mean excitation energies" for each electron pair.
Although rather crude, this procedure has the ad
vantage that the various energy components can then
be estimated using only the same H.F. orbitals as in the
initial single determinant. Contrary to the use of an
average energy denominator for the over-all second
order energy36 here each "mean excitation energy" has
37 C. C. J. Roothaan, Revs. Modern Phys. 23, 69 (1951).
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a more physical basis and can be left as a semiempirical
parameter especially for those electron groups that are
relatively unchanged in going from one atom or
molecule to another. [See also the discussion following
Eq. (54b)].
More directly, two specific applications are sug
gested by this approach as was mentioned in the
Introduction. Both of them may be demonstrated
with reference to the Li2 molecule, for convenience.
In this molecule, the first four MO's of the one electron
basis {kl are (O'g1S)2(0',,1s)2. When the atomic orbitals
(AO) that make up such inner shells do not overlap
appreciably as in Li2, we can perform a unitary trans
formation on the (O'g1S)2(O'u1s)2 part of the basis only
and convert the MO determinant det'[(O'g1s)2(O'u1s)2]
into the ion core description det'[(1sa)2(1sb)2], i.e.,
KaKb, where a and b refer to the two nuclei, assuming
that admixture of other AO's is negligible. Then taking
(1sa) 2( 1Sb) 2 equivalently, as the first four spin-orbitals
of {k I with the rest of the MO's unchanged, a classifica
tion of all "ordered configurations" as in Eqs. (9) and
(10) into various types of virtual transitions leads to a
separation of the correlation energy as in Eqs. (16)
(18). We get the total energy of the molecule separated
as in Eq. (46) into the energy of two free Li+'s (includ
ing their individual E(2)'S), the energy of the two H.F.
MO valence electrons (Ug2S)2, each in the field of both
cores including the core "fluctuation potentials" [the
effective "core-hamiltonian" from one Li+ is (hcore"ff+
U I) a] the energy of the two valence electrons (as in
H2), (E(2» (~.28)2, and finally the correlation energy
between the two cores Ka and Kb. There are also
the "exclusion" terms associated with each of these
components.
The first application concerns the "core-polarization"
energy between a valence electron and the ion-cores.
\\Then these cores can be assumed quite unchanged, the
expectation value of the potential, UI determined from
Eq. (59) in conjunction with the "series" levels of the
atom, may be calculated over the gnmnd or an excited
state valence MO in the molecule. Thus the calculation
of the contribution of "core polarization" to the al
ready small binding energy of a diatomic alkali molecule
is possible. Callaway13 has made such calculations on
alkali metals and found appreciable values even with
just the dipole part (see Sec. IV) of Ufo In this type of
application the change in the "exclusion" energy
(8c+Os)Rlo~na [Eq. (56a)] may also need to be esti
mated.
The second application, although a small effect in the
case of Li2, concerns the correlation energy between
the two cores, Ka and Kb themselves. This energy
which can be written in the "fluctuation" form similar
to that in Eq. (41) and including the exchange part,
is just the "dispersion" (plus "orbital average polari
zation") energy which on making a multipole ex
pansion [or more conveniently, Gegenbauer expan
sion35 as in Eq. (53) but now for two centers] for g122 and taking the second term would simply lead to
London's formula.3 Aside from not requiring such an
expansion, the approach presented here now includes
the exchange terms as well as the "exclusion" effects
similar to those in Eq. (50), but with the appropriate
orbitals. It is particularly important to recognize that
the discussion given here does not require the two cores
to occupy completely isolated spaces, each with its
own distinct basis set as has been considered necessary
in previous discussions of London forces.3s We have
started from a complete set of MO's and have made
assumptions only about the first four MO's, (O'g1s)2
(0' uls) 2. The assumption that only the AO's, (1sa),
(1Sb), should not overlap appreciably is necessary so
that we get into the KaKb description. In general, such
an assumption is much more plausible (and may even
be improved on by considering some overlap) than the
requirement of essentially complete localization of the
core electrons around different centers, each group
with its own distinct set of eigenfunctions.
Finally we observe that the second-order method
which has been presented in detail in Sec. II, not only
provides an approximate but very convenient way of
estimating the energy of a many electron system, but
also allows one to discuss many of the correlation
effects in simple physical terms.
VI. HIGHER-ORDER CORRELATION EFFECTS
For simplicity the treatment presented in this article
has been so far confined to the second order. Here,
correlations among more than two particles at a time are
introduced only by the "exclusion" terms. For three
and more body "Coulomb" correlations it is necessary
to go to higher orders of perturbation.39 In the system
of a single electron outside closed shells, some third
order correlation effects influencing the "core-polariza
tion" energy (e.g., in Na) can be introduced into Uf
as an additional potential by methods similar to those
in Sec. II, or by letting the mean-excitation energy,
Be, absorb the higher order effects semiempirically.
(A similar situation usually occurs in various Van der
Waals forces10•29.)
An interesting case where higher-order correlations
deserve further examination is a nondegenerate system
of two electrons outside large closed shells as in Ca.
Here, aside from the "exclusion" effects, there would
be a U I from the core acting on each of the 4s electrons,
and a "fluctuation potential" (r12-2-(r12-1 i2AV,4,) /~
[as in Eq. (41)] acting between the two (4s) electrons
[similar to the correlation of (1S)2 in He]. However,
the presence of a large polarizable core inside introduces
additional effective interactions between the two outer
electrons in higher orders. This can be seen by a crude
38 See, e.g., H. C. Longuet-Higgins, Proc. Roy. Soc. (London)
A235,537 (1956).
39 Actually, at least for the light atoms, the empirical work of
Arai and Onishi suggests that the pairwise additivity of correla
tion effects may turn out to be a quite good description. See
T. Ami and T. Onishi, J. Chern. Phys. 26,70 (1957).
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132.236.27.111 On: Mon, 15 Dec 2014 14:18:33CORRELATION ENERGIES AND CORE-POLARIZATION 1225
but suggestive classical argument: Consider the core
as a charge sphere with polarizability a and assume the
two outer electrons to be momentarily at rest [see
Eq. (1) ] at distances r1 and r2 from the nucleus with
an angle 812 between them. Then, each electron induces
a dipole moment of air i2 at the core with which it
interacts to yield an energy -a/2r;4. This is the limit
ing form of U, in Eq. (46) as ri approaches infinity.
But, in addition, the dipole induced by the electron at
rl acts on the electron at r2 giving
[tI2(3)", -a Cos812/rI2r22].
The introduction of such effective interactions between
two electrons to account for the influence of a "medium"
(the core in this case) is possible quite generally by
essentially an extension of the methods given for only
the second order in Sec. II. It is also interesting to note
the similarity of the additional interaction, tI2(3), to the
new third-order Van der Waals interaction that is
introduced by a solid surface between two inert mole
cules adsorbed on it29 (the new force is in addition
to the London force between the two molecules alone).
In general, the correlation energy between two groups
of electrons will be influenced by the internal correlation
of each group, as in the example just mentioned, where
the (4s2) shell and the core in Ca could be considered
as the two groups. The resulting higher order correla
tion energy can also be treated very simply for the
special class of problems pointed out in the introduc
tion. These were the cases where the two groups could
be assumed to be totally separated and localized
individually, each group with its own set of eigenfunc
tions. To see how "many electron group functions" can
be used in such cases, consider, for instance, two many
electron atoms A and B far apart, and their mutual
Van der Waals attraction. Here we can use ordinary
second-order Schrodinger perturbation theory. Let 1/IAk
or 1/IRI be the complete set of exact many electron
eigenfunctions for each of the unperturbed "independ
ent" systems A and B, respectively. Then the un
perturbed Hamiltonian Ho equals HA+HB, the com
posite basis set 1/Ikl=1/IA~Bl and the perturbation is the
total interaction between A and B given by
V.4B=GAB+UAB, (60)
where
i,i i.i
and
GAB=L[ZrZJ/(\ RrA-RJB \)]. r,J
GAB refers to the interaction between the nuclei in A
and the nuclei in B. U AB is the electrostatic potential
between pairs of electrons, one in A (at r ~) and the
other in B (at rl) .1/IAk and 1/IBl, being the set of exact
eigenfunctions of the isolated systems A and B, include
the coordinates of all the electrons and even the nuclei localized at A or B, respectively.29 Then the intergroup
"dispersion" energy is given by
Edisp(2) (AB) =_ L <1/IA~BO, VAB1/IA~Bl)2
k;o<O,l,.<O OAOk+OBOI
where OAOk=EAk_EAo and HA1/IAk=EA k1/lAk. Replacing
the denominators by mean excitation energies and
using Eq. (27) we get
Edisp(2)(AB) = (8A+8B)-1( (00, VAB2OO)-(0, WA20)
-(0, W B20 )+ (00, V ABOO )2) (62)
where
and
with RA denoting all r~ and RrA. If A and Bare
neutral systems, the last three terms of Eq. (62) may
be negligible since they depend on the static charge
distributions of A and B. Then Eq. (62) takes on the
form of the "fluctuation potential" of one system
acting on the other; i.e.,
Edisp(2)(AB)=- (00, VAB200)/(5A+8B). (64)
This description of the Van der Waals forces, that
they are the result of the mean-square fluctuation of the
electrostatic potential between two systems, is the
generalization of the usual fluctuating dipoles picture
(Sec. I). Note also that the form of Eq. (64) is the
same as the potential in the core polarization problem.
Similar considerations, of course, apply to the scattering
of electrons40 by atoms as well.
Now, consider only the UAB part of VAB [Eq. (601]
and in Eq. (64) write U AB2 in detailed form as
UAB2= "--=" -+ " (65) ( 1)2 (1)2 (1) ~ . .AB ~ . .AB L.... .AB AB' '1,,1 r t] '1..1 r tJ i,i¢.r,lt r iJ r ra
Here i and r designate any two electrons in A and j
and s any two electrons in B. The (r ilB) 2 terms in
Eq. (65) involve the coordinates of only one electron
at a time from each group. Their contribution to Eq.
(64) can be written in terms of the first-order density
matrices of A and B. On the other hand, two electrons
i and r from A enter along with one or two electrons
(j and s) from B into the (riIBrr8AB) terms and their
contribution is in terms of the second-order density
matrices of A and B.29.41 Now 1/IAO and 1/IBo were exact
many-electron eigenfunctions and included the internal
correlations of each group, respectively. Thus the pre
ceeding examination of Eq. (65) along with Eqs. (60)
40 See A. Temkin, Phys. Rev. 107, 1004 (1957).
41 J. Bardeen, Phys. Rev. 58, 727 (1940).
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and (64) brings out the desired result, i.e., the effect of
the internal correlations of A and B on the "inter
group" correlation energy has been taken into account
in Edisp (2) (AB) .
If, instead of the special case considered, we now have
any two groups that cannot be assumed to be sepa
rately localized, the treatment of higher-order correla
tions by the use of many-electron group functions is no
longer straight-forward. The difficulty is again mainly
due to the exclusion principle (see Introduction). To
circumvent this difficulty, previous treatments have
been restricted to the use of very special many electron
group functions, 17,18 i.e., those satisfying "generalized
orthogonality conditions." However, the subdivision
of a one-electron basis set into mutually exclusive
subsets, apparently implied by these conditions, is too
restrictive. The degree of restriction becomes par
ticularly apparent if we consider the Be atom as a two-
THE JOURNAL OF CHEMICAL PHYSICS shell system (1S2 and 2s2) with correlated group func
tions for each shell, instead of the "sigma-pi"
problem. In the former case where there is no nominal
symmetry difference between the two groups, it is more
evident that both groups would have to use the same
spin-orbital set. On the other hand, it seems that a
treatment based on a single complete basis set can be
given not only for two, but also for many-body correla
tion effects, by going to higher orders of perturbation
and classifying all possible virtual excitations as in
Sec. II. Some of these excitations will now involve
more than two electrons at a time.
ACKNOWLEDGMENTS
The author wishes to thank Professor W. T. Simpson
and Professor K. S. Pitzer for various helpful discus
sions. This research was carried out under the auspices
of the U.S. Atomic Energy Commission.
VOLUME 33, NUMBER 4 OCTOBER, 1960
Charge Transfer between Atomic Hydrogen and N+ and O+t
R. F. STEBBINGS, WADE L. FITE, AND DAVID G. HUMMER*
John Jay Hopkins Laboratory for Pure and Applied Science, General Atomic Division of General Dynamics Corporation,
San Diego, California
(Received June 13, 1960)
The cross sections for charge transfer in collisions between atomic hydrogen and singly charged atomic
ions of nitrogen and oxygen have been measured within the energy range from 400 to 10000 ev, using
modulated-beam techniques. The results are compared with recent calculations.
I. INTRODUCTION
THE cross sections for resonant charge transfer in
collisions between atomic hydrogen and positive
and negative atomic ions of hydrogen within the energy
range from a few hundred ev to about 40 kev were
presented in previous papers.l In the present work,
measurements of a similar nature were made for the
singly charged ions of atomic nitrogen and oxygen. These
two charge-transfer processes are of particular interest,
as they characterize nonresonant and almost exactly
energy-resonant collisions. Since the colliding particles
are all atomic in nature, no complication is introduced
through dissociation. On the basis of the near-adiabatic
theory2-4 the transfer cross section between 0+ and H
t This research was supported by the Advanced Research
Projects Agency through the United States Office of Naval
Research.
* Present address: Department of Physics, University College,
London, England.
1 W. L. Fite, R. F. Stebbings, D. G. Hummer, and R. T.
Brackmann, Phys. Rev. 119, 663 (1960); and D. G. Hummer,
R. F. Stebbings, W. L. Fite, and L. M. Branscomb, ibid. 119,
668 (1960).
2 D. R. Bates and H. S. W. Massey, Pbil. Mag. 45, 111 (1954). (for which the energy defect dE=0.019 ev) should
show a maximum value at a near-threshold energy,
while for collisions between N+ and H(dE=0.94 ev)
the cross section should be small at low energies and
should rise to a maximum value in the region of a few
thousand ev.
II. APPARATUS
The apparatus used in these experiments was bas
ically the same as that used in the earlier ion experi
ments and is shown schematically in Fig. 1. An arbi
trarily highly dissociated beam of hydrogen issued from
a tungsten furnace in the first of three differentially
pumped vacuum chambers and was modulated at 100
cps by a mechanically driven, toothed chopper wheel
located in the second chamber. On entering the third
chamber, the beam passed between two deflector plates,
between which an electrostatic field swept out any
charged particles accompanying the neutral beam, and
3 J. B. H. Stedeford and J. B. Hasted, Proc. Roy. Soc. (Lon
don) A227,466 (1955).
4 H. B. Gilbody and J. B. Hasted, Proc. Roy. Soc. (London)
A238,334 (1956).
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1.1703683.pdf | Quantization of Nonlinear Systems
I. E. Segal
Citation: J. Math. Phys. 1, 468 (1960); doi: 10.1063/1.1703683
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Downloaded 08 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissionsJOURNAL OF MATHEMATICAL PHYSICS VOLUME 1, NUMBER 6 NOVEMBER-DECEMBER, 1960
Quantization of Nonlinear Systems
I. E. SEGAL*
Department of Mathematics, University of Chicago, Chicago, Illinoist
(Received April 25, 1960)
. '!'-dfrect method of quantiz.ation, ap~licable to a !1iiven nonlinear hyperbolic partial differential equation,
'~ mdicated. From such classIcal equations alone, wlthout a given Lagrangian or Hamiltonian, or a priori
hnear r~ference system ~uch as ~ bare or incoming field, a quantized field is constructed, satisfying the
conve~tlOnal commutat~on relatlOns. While mathematically quite heuristic in part, local products of
quantiZ/!d fields do not mtervene, and there are grounds for the belief that the formulation is free from
nontrivial divergences.
1. INTRODUCTION
THERE has been interest recently in the develop-
ment of purely nonlinear quantum field theories,
i.e., theories which are formulated without the use of
such physically somewhat dubious and mathematically
linear notions as those of a "free" field or of an
"elementary" particle.l Despite the promise of this
work, the difficulties are such that there has sometimes
been lacking a satisfying demonstration of the internal
consistency of the formal structure on which the theory
is based, or, on occasion, a reasonably clear-cut physical
interpretation of the formalism. The purpose of the
present work is to indicate a method of quantization
that seems on the whole somewhat less subject to these
defects. The main result is a new framework for co
variant quantum field theory, which appears to be
convergent, although mathematically quite heuristic.
From any of a fairly wide class of given manifolds of
"classical" wave functions, there is constructed an
associated quantum field, as well as a possible means
of determining theoretically vacuum expectation values
of functions of field operators, and aspects of a formal
elementary particle interpretation. In particular, the
work provides some basis for a renewal of the traditional
intuitive belief-which has been strongly tempered by
the persistence of divergences during the past 30 years
that for any simple covariant coupling of the conven
tional elementary particles of relativistic quantum
field theory, there should be a corresponding quantum
field theory of their interaction; but at the same time
casts further doubt on the rigorous relevance to such
theories of the notion of elementary and/or physical
(dressed) particle, as well as the possibility of expressing
such a theory in terms of an a priori type of incoming
field.
* Research supported in part by the Air Force OSR and
conducted in part at the University of Copenhagen while a-d NSF
fellow. t Present address: Massachusetts Institute of Technology
Cambridge, Massachusetts. '
1 See notably W. ~ei.senberg, Revs. Modern Phys. 29, 269
(1957), and S. Deser, ~bid. 29, 417 (1950) (and other articles in
addition !o t~e last-named, reporting t~e Chapel Hill Confere~ce
on GraVltation); and especlally articles by Heisenberg and
Yukawa, Proc. Internat!. Conf. High-Energy Nuclear Phys
Geneva, 1958. ., Besides the perturbation-theoretic divergences of
quantum field theory, and its use of an a priori linear
reference space, there is another feature that is rather
unsatisfactory from a foundational viewpoint. This is
the dependence of the theory on a notion, the product
of local fields [e.g., q,(x)if;(x)if;(x)* in conventional
notation] which seems inevitably remote from any
physical measurement. As is clear from a line of work
originating with the well-known classical paper of
Bohr and Rosenfeld, a suitably smoothed average
fq,(x)f(x)d 4x (f=a "test" function, corresponding to
a probe into the field) is the most that one can hope
to measure even in principle. However, no way has
been found to express such a product as q,(x)if;(x)if;(x)*,
or averages of it, in terms of such smoothed averages
of individual fields; and quite apart from the di
vergences which such products directly lead to, it is
odd that a notion so lacking in direct physical meaning
(as well as in rigorous mathematical significance, so
that it rests purely on traditional formalism and
metaphysics) should play the essential role in the
construction of the field dynamics. The attempt to
bypass this kind of difficulty by a purely axiomatic
approach as in the work of Haag, Kallen and Wightman,
Lehmann et al., and some others, has clarified the
logical situation, but on the whole the results are still
rather inconclusive, and certain of the axioms are
rather strong from a physical standpoint. A more
constructive (in the technical sense) line of attack is
given by Segal,2 the essential presently relevant idea
being the use not of the q,(x)if;(x)if;(x)* themselves, but
only of integrals of the type H = .It-t'q, (x)if; (x)if; (x)*d3x,
involving only commuting (and so more amenable)
fields; and the use of these not as operators, but as
generators of motions of the dynamical variables of
the field. That is, roughly speaking, only the [H,X]
need be finite for any field observable X, and not H
itself, which leads to a mathematically quite well
defined category of objects H materially broader than
the class of self-adjoint operators in a Hilbert space.
Although some definite results in quantum electro
dynamics, of a rigorous character, can be obtained in
2 1. E. Segal, Kg!. Danske Videnskab. Selskab Mat.-fys. Medd.
31, No. 12, 1-39 (1959).
468
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this way (d. footnote reference 3), this work has the
important limitation that the physical vacuum is not
constructed, and there are very substantial, if relatively
well-understood difficulties in showing that H[ =H(t)]
has the required properties for the rigorous existence
of the time-ordered (product) integral exp[ifH(t)dt]
that defines the transformation taking the in-into the
out-field observables. It is hard to believe that a
definitive foundational treatment of a system whose
dynamics are conceptually as simple as those of
quantum electrodynamics, say, must depend on the
resolution of the intricate and special problems that
arise here.
At any rate, it seems reassuring to have a general
scheme for setting up quantum field interactions in
which the singular products of the type cf>(x)if;(x)if;(x)*
play no role whatever. The formalism involves only
quantities which are in principle capable of being
related to direct physical measurement. It rests
mathematically on the combination of certain simple if
abstract ideas from the theory of differentiable mani
folds with the intrinsic (representation-independent)
theory of operator systems applied to field theory in,2
and the development of mathematical analysis in
function space. On the other hand, many of the relevant
mathematical developments are presently available
only in highly rudimentary form (e.g., it is not yet
proved that the relevant classical partial differential
equations have any nontrivial solutions-even of a
generalized character-in the large), and we merely
assume their eventual existence on the basis of plausi
bility considerations. Also, the particle interpretation
of the present scheme refers in the first instance
essentially to the "primary" particles, and it seems
doubtful whether a precise "empirical" particle inter
pretation will exist with any generality, in view of the
applicability of the scheme to both renormalizable and
nonrenormalizable field equations, and to fields involv
ing bound states and unstable particles. In particular,
when a theory of the present sort is specialized, say,
to quantum electrodynamics, it gives no a priori
labeling of the states of the incoming field in terms of
finite aggregates of "free physical" electrons and
photons. Whether or not such labels can be rigorously
established-as is a well-defined mathematical question
according to the present framework, along with the
question of the existence and character of bound states
and unstable particles-it is difficult to make specific
computations of real empirical effects without them or
some approximate equivalent. Without these various
developments there is no assurance either that the
theory can be made mathematically irreproachable
or can be accurately correlated with the crucial experi
mental results pertinent to field theory. It is only from
a theoretical physical point of view and relative to the
present state of quantum field theory that the present
3 I. E. Segal, Ann. Math. (to be published). work appears to represent a contribution of possible
significance. There have after all been extremely few
truly unambiguous theoretical developments in the
subject since it was set up by Dirac, Heisenberg, and
Pauli, despite the large number of fragmentary con
tributions that have been made. It seems that for
foundational purposes only a quite comprehensive
attack employing conservative but global methods has
much hope of ultimate success. As this has never really
precisely been undertaken, there is no reason for undue
pessimism, but the scope of such a development is
necessarily such that it is unrealistic to begin highly
explicit analytical computations until the fundamental
design is well established. It is to the settlement of
this design question-of what is actually a quantum
field theory-that this article is intended to contribute.
The present theory is related to linear quantum field
theory (or the theory of noninteracting fields) in
roughly the same way that the theory of differentiable
manifolds is related to the theory of linear vector
spaces-the interaction has its source in the nonlinear
structure of the manifold representing the classical
states of the system being quantized. The conventional
type of theory of interacting fields (which may be
called quasi-linear) is related to the present theory in
somewhat the way that the theory of a Riemannian
manifold as described by normal coordinates at a
distinguished point is related to the intrinsic theory
of the manifold. The vectors in the tangent space to
the manifold at that point represent the bare particles
of the theory, which would make the extrinsic theory
convenient for giving a particle interpretation, if the
apparent need for infinite mass and charge renormal
ization did not make it impossible then to give ab in#io
in the theory the precise relation between the manifold
and the tangent space. The extrinsic theory is also
disadvantageous from a theoretical point of view in
its use of ad hoc assumptions as to the structure of the
incoming field, which make the role of bound states
and unstable particles in the theory highly elusive.
For example, in the case of quantum electrodynamics
it is conventionally assumed implicitly that the in
coming field is describable by the Fock representation,
with a renormalized tangent space as single-particle
space; in general, such an assumption overdetermines
the theoretical structure of a quantum field, and may
well lead to internal inconsistencies.
In its simplest form the nonintrinsic character of
conventional theory is exemplified by the ad hoc
separation of the total Hamiltonian into "free-field"
and "interaction" parts, a separation that is required
for the usual analytical treatment of scattering. The
kinematics of the interacting field is derived from the
free-field part and is linear, while the dynamics is
superimposed on the kinematics through the statement
of the interaction Hamiltonian or Lagrangian (or more
operationally, of the S operator). In the present work,
no Hamiltonian or Lagrangian (or S operator) needs
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to be specified; the theory is built up entirely from the
classical equations of motion. The distinction between
the free-field and total Hamiltonians is seen to be
essentially that between the linear motion in the
tangent plane at a fixed point in a manifold under a
group of transformations that is induced in the natural
manner by the group action, and the nonlinear motion
that is obtained by transferring, through the use, e.g.,
of normal coordinates, the given group to the tangent
plane. In general relativity there is no distinguished
invariant point of the manifold of solutions of the
equations that is physically analogous to the point
defined by vanishing fields in the case of elementary
particle theory, and hence no covariant separation of
the motion into two parts, but the theory may still be
quantized by the intrinsic approach.
From the viewpoint of general analytical dynamics,
a theory of the present type is determined primarily
by the specification of a differentiable manifold B
representing the classical phase space of the system
under consideration, together with a second-order
Hermitian differential form D on B, and a correspond
ing notion of multiplication by complex scalars in the
tangent spaces of B. In classical mechanics the funda
mental bilinear covariant is quite analogous to D, but
complex scalars in the tangent spaces of phase space
have apparently not been used. In the case of a field,
where B is infinite-dimensional, D is better known in
the form of the singular functions D(x,x') that arise
as field commutators in the quantization of a linear
equation. The tangent space at any point cp of B is
parametrized by functions f,g,··· on space-time
satisfying the first-order variations of the coupled
field equations in the infinitesimal vicinity of cp (taking
the case of a scalar field for simplicity), while D is
determined by its imaginary part Di, which is by
definition a rule that assigns to a point cp a bilinear
form in the tangent vectors at cp, and is given by the
equation
D;(j,g; cp) = j jf(x)g(X')Dq,(x,X')d 4xd4X'.
In conventional theory only the singular functions of
covariant free fields seem to have been used, and in
this case D(x,x') depends only on x-x', but here the
singular functions defined by similar Cauchy data for
all first-order variations of the coupled field equations
are relevant, and D(x,x') will have the usual type of
dependence only in the special case cp=O (or other
constant solution, if any, of the equation defining the
manifold). This canonical construction for Di in the
case of a manifold in function space defined by a non
linear hyperbolic partial differential equation has been
explored in certain cases and in a rather different form
by Peierls.4 The fundamental symmetry group G of
the theory may be any group of transformations on B
4 R. E. Peierls, Proc. Roy. Soc. (London) A214, 143 (1952). that leaves D invariant, as does, e.g., the Lorentz
group in the case of a manifold defined by a Lorentz
invariant equation of the foregoing type. A complete
set of primary quantum numbers of the usual type--of
group-theoretic origin-will exist if, and only if, the
induced action of G in the tangent space at cp=O is
such that the linear operators in the tangent space
that are left invariant commute with one another
(or equivalently, the irreducible constituents of this
representation are all distinct). When the generators
of G are suitably labeled as "energy," "angular
momentum," etc. (the Lorentz group is by no means
the only one for which this is possible; d., e.g., Segal")
the resulting physical theory, in particular the formal
S operator, is in essence completely determined.
The idea of constructing a purely nonlinear quantum
field theory has been developed in recent years most
extensively by Heisenberg (see footnote reference 1
where further references are given), with whose stand
point the theory described in the foregoing is in general
harmony. While it thereby lends some support to
Heisenberg'S idea that a purely nonlinear theory should
be convergent, its specific form deviates in some
important respects from that suggested by Heisenberg'S
program, notably in the significant role played in it
by singular functions associated with linear partial
differential equations. It has been a cardinal principle
of Heisenberg to avoid the use of such functions, with
the aim of eliminating the divergences of conventional
theory, which arise from the a priori meaningless
character of their products. The latter are involved
in computations based on perturbation theory, as well
as, in essence, in the formulation of conventional
dynamics. As indicated in the foregoing, in the present
work no such products arise, so that the use of these
singular functions introduces no divergences. But this
does not in itself indicate that a fully satisfactory
theory may be based on the Lorentz group and con
ventional space-time, for divergences may well be
introduced by the use of ad hoc labels for the states of
the incoming field. Such desiderata as the observation
of stable single-particle states of sharply-defined mass
may well ultimately require the introduction of a
fundamental length into the structure of B and/or lead
to the replacement of the Lorentz group by another
to which it is a partial approximation in the sense
considered in footnote reference S. We may also note
a rather obvious rough analogy between the role of
the infinite-dimensional tangent spaces to nonlinear
function manifolds in the present work and those of
the finite-dimensional tangent spaces of the space-time
manifold in general relativity. Partial parallels with
important ideas of Feynman concerning the use of
functional integration, of Dirac dealing with covariance
questions, and of Wiener concerning nonlinear analysis,
will also be evident to the knowledgeable reader.
s 1. E. Segal, Report of Lille Conference on Quantum Fields
(C.N.R.S., Paris, 1959), pp. 57-103.
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Briefly, we show how a generalized canonical
variable R(X) can be associated with an infinitesimal
generator X of the group of classical contact trans
formations on the given classical system. The con
struction of these variables depends inessentially on
the choice of a first-order differential form whose
covariant differential gives Oi, and which is analogous
to the form Lk pkdqk-Hdt employed in classical
mechanics. In the case of a linear manifold, the R(X)
associated with infinitesimal translations X in phase
space give the conventional (Heisenberg) commuta
tion relations, while the present generalized ones
satisfy the rule
[R(X),R(Y)]= -iR([X,Y])+O(X,Y),
where [X,Y] denotes the usual bracket of two vector
fields. When 0 vanishes (and only then), R gives a
representation of the infinitesimal contact group, which
is in fact in the finite-dimensional case the well-known
one introduced by Koopman and studied by him and
von Neumann, in connection with classical mechanics.
It may also be noted that in not being concerned
with an actual representation of the contact group, as
well as in a number of other respects, the specialization
of the present approach to the case of a finite
dimensional linear manifold differs from the note
worthy investigations of van Hove6 directed toward
a basis for a rational correspondence between classical
and quantum mechanical Hamiltonians.
The avoidance of convergence difficulties depends in
part on the elimination of any ad hoc Hilbert space in
the foundation of the theory for the representation of
the states of the incoming field, such a space being,
however, convenient for correlation with experiment
and also used, implicitly or explicitly, in most of the
recent literature on quantum fields in a rigorous
direction (d. the authors already cited). Rather, the
incoming field becomes one of the objects whose
structure the theory is to determine. The method is
roughly to work with the system of all bounded func
tions of finite sets of the canonical variables, together
with their limits in the sense of uniform convergence,
as in footnote reference 2; this gives a covariant class
of observables that is representation-independent, in
contrast to the set of bounded observables obtained
by using functions of infinite sets of canonical variables
and/or limits in the sense of so-called strong or weak
convergence. States are defined through their expecta
tion value functionals on the foregoing system, which
is both more physical, and mathematically more
effective than their a priori representation by vectors
in a Hilbert space. Yet ultimately a Hilbert space can
be constructed which represents the states of the
incoming field, by the use of the physical vacuum
expectation values, which are in turn connected with
6 L. Van Hove, Acad. roy. Belg. Classe sci. Mem. Collection
in 8° 29, No.6, 1-102 (1951). a process resembling integration over the classical
manifold B (such integration is made fully rigorous
in the case of infinite-dimensional linear manifolds
by Segal,7a and a formal adaptation of this work to the
relevant nonlinear manifolds will be indicated later).
In more analytical terms, the main ideas of the
present work may be indicated in their simplest form
as follows. The manifold B of all real solutions of a
given Lorentz-invariant hyperbolic nonlinear pa~tial
differential equation in four-dimensionsl space-tIme
carries a distinguished Hermitian structure. Quantiza
tion involves in essence analysis over this manifold
(in contrast to classical mechanics, which is concerned
with the construction of the manifold and the action
of various groups on it), i.e., the study of certain
operators (in particular the values of the "qua~tum
field") in spaces of functionals over the mamfold.
Canonical variables may be attached to the vector
fields on the manifold through the use of a differential
form of first order related to the given Hermitian
structure. The field itself arises from the projection of
the variational derivative a/aj in function space onto
the (sub-) manifold B; taking j as a delta function at
a point yields formally the field at the point. The
quantum-theoretical physical vacuum is represen~ed
by the unit function on B, the vacuum state ?emg
characterized by invariance under the group of Isom
etries of B leaving invariant the vanishing field cp=O.
The primary elementary particle species of the theory
are given by the irreducibly invariant subspaces of
the tangent space to B at the point cp=O (or other
Lorentz-invariant point of B, if any), and formally the
theory may be expressed entirely in terms of this
tangent space, which corresponds essential~y to the
most conventional procedure. The conventIOnal free
fields (those given by the quantizations of the Klein
Gordon, Maxwell, etc. equations in empty space)
correspond precisely to the special case in which the
manifold B is a complex Hilbert space and the Lorentz
group action is unitary, the Hermitian structure being
that given by the fundamental inner product, and the
physical vacuum as characterized before being unique
and the familiar one associated essentially with an
isotropic normal distribution in Hilbert space.
In considerable part, the foregoing description is
valid only for Bose-Einstein fields. While it appears
that the Fermi-Dirac fields can probably be treated in
a rather analogous way, it will presumably be necessary
to replace vector by spinor fields (over function
manifolds), and the notion of integration by that
treated in the linear case in footnote reference 7b, etc.
In view of the substantial character of such modifica
tions, the present paper is confined to the Bose-Einstein
case.
In Sec. 2, the nonrelativistic quantum mechanics of a
finite number of degrees of freedom is extended to the
7 (a) 1. E. Segal, Trans. Am. Math. Soc. 81, 106 (1956); (b) 1. E.
Segal, Ann. Math. 63, 160 (1956).
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case when space is not necessarily fiat; this involves
in particular the reformulation of the conventional
quantum conditions so as to be covariant under general
point transformations in physical space. Section 3
completes the preliminaries by showing how canonical
variables and commutation rules may be set up in
terms of phase, rather than physical, space, in a form
covariant under contact transformations. The hydrogen
atom and harmonic oscillator problems in the presence
of nonvanishing curvature are briefly discussed in
these sections. In Sec. 4, the earlier developments are
combined with methods previously developed in con
nection with certain aspects of infinite systems to
obtain a quantization scheme for a class of infinite
nonlinear systems, represented by the case of a classical
system defined by a nonlinear hyperbolic partial
differential equation. The concluding Sec. 5 discusses
the present results in relation to some of the existing
literature and possible further developments.
2. FINITE SYSTEMS AND POINT
TRANSFORMATIONS
Consider a quantum-mechanical system whose posi
tion is described by a point of an n-dimensional
manifold M (and so is a system of 2n degrees of
freedom). If as in conventional theory M is a linear
manifold, one proceeds by introducing Hermitian
operators PI,P2,'" ,pn and ql,q2,'" ,qn satisfying the
Heisenberg commutation relations. If however, M is
nonlinear, it is unclear a priori to what extent it is
possible to proceed in a suitably parallel way. In the
case of a sphere or torus, results can be obtained by
making use of the simple natural parametrizations
available for these manifolds. However, physically the
availability of a suitable special parametrization
appears as a rather technical restriction on M; intui
tively it would appear possible to quantize a classical
system whose position is represented by a point of a
relatively arbitrary manifold.
To develop an appropriate quantization method,
we note that the canonical p's are naturally associated
with vector fields on M, and the canonical Q's with
position coordinates; the SchrOdinger representation
for the linear case associates Pj with a/iax;, and Qj
with the coordinate Xj. To handle the nonlinear case
we merely allow the P's to be associated with arbitrary
vector fields-i.e., linear forms in the iJ/iiJxj, with
variable rather than constant coefficients (which are
undefined in the absence of distinguished coordinates
or related special features of M)-and the Q's with
. arbitrary functions on M, and not merely linear func
tions (which are likewise undefined on a general
manifold). The commutation relations are virtually
automatically generalized thereby; any commutator
of canonical variables is required simply to be that
associated with the commutator of the corresponding
transformations on the functions over M. To make this approach mathematically effective, it
is necessary to formulate the P's and Q's as well-defined
operators in a Hilbert space. To set up an appropriate
Hilbert space, take a measure m on the given manifold
M that has a continuous nonvanishing density at
every pointS; in general there will be no distinguished
measure analogous to the Euclidean volume element
used in conventional theory, but we proceed, tenta
tively, with an arbitrary measure of the foregoing
type; it will develope that actually the theory is
independent of the choice of measure.
The Hilbert space X is then defined as consisting of
all square-integrable functions f on M (the values of f
being complex numbers) with the inner product
(j,g) = ff(x)g(x)*dm(x).
Now if T is a general vector field on M, the associated
canonical momentum peT) might be provisionally
defined as the operator in X taking f into (li/i)Tf;
this is appropriate from a formal algebraic viewpoint,
but it gives rise to difficulties originating in the non
Hermitian character of T as an operator in X. With
the modified definition
P(T)= (1i/2i) (T-T+),
the fundamental commutation relations are unchanged,
and peT) is now manifestly Hermitian. A simple
computation shows that the foregoing definition works
out concretely as
P(T)= (h/i) (T+K T),
where K'J' denotes the operation of multiplication by
the function kT, which is defined by the equation
2kT(x) = Tw+l(T),
where m has the element wIIdxj (locally), and leT) is
defined as Li (iJaj/iJxj) for T of the form Li aj(a/iJxJ).
A straightforward computation that is here omitted
yields the commutation relation
[P(S),P(T)]= (h/i)P([S,T]). (1)
The symbol [S,T] denotes the commutator of the two
vector fields Sand T in the usual sense of the theory
of manifolds. In the case of a linear manifold this
vanishes for two infinitesimal translations, and (1)
specializes merely to the commutativity of the con
ventional linear momenta. For an infinitesimal trans-
3 For convenience, it is assumed, as seems no essential loss of
generality from a physical standpoint, that the manifold M is
infinitely differentiable, i.e., that it is possible near each point to
choose local coordinates in such a manner that whenever a point
is assigned two sets of coordinates, then near the point the one
set may be expressed as infinitely differentiable functions of the
other set. It is known (virtually as a matter of definition) that
the existence of a measure with a nowhere vanishing continuous
density function is mathematically equivalent to the orientability
of M, which will be assumed in the present section.
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lation and an infinitesimal rotation, however, (1) gives
the conventional commutation relations between linear
and angular momenta.
The canonical Q's are defined more simply: if j is
a general function on M, Q(f) is defined as the operator
in H taking h into jh (i.e., the operation of multiplication
by j). For real j, Q(j) is Hermitian, and there is no
difficulty in verifying the additional commutation
relations
[P(T),Q(J)]= (h/i)Q(Tj),
[Q(J),Q(g)]=O. (2)
(3)
The first of these relations includes the conventional
commutation relations between an angular momentum
and a coordinate, as well as the basic relations between
a linear momentum and a coordinate given explicitly
in the Heisenberg form. The second merely asserts the
commutativity of all the Q's.
The foregoing construction is essentially merely an
adaptation of the Schrodinger representation to an
arbitrary manifold, together with a reformulation that
makes manifest the invariance of the scheme under
arbitrary coordinate transformations. When M is
three-dimensional Euclidean space, as in the conven
tional theory of a single particle, the basic canonical
variables are taken as the peT) with T restricted to
be a first-order linear differential operator with constant
coefficients, and as the correspondingly restricted Q(j)
(i.e., j, a linear function on the space). Since such
peT) and Q(f) however already suffice to give an
irreducible set of operators on L2(M), the additional
peT) and Q(J) defined earlier are already observables
in the conventional scheme, so that the phenomeno
logical structure of the theory-the observables, states,
and notions defined in terms of these-is unaltered by
the present reformulation. The broadened definitions
of the peT) and Q(j) merely amount to a labelling of
certain of the observables, which facilitates a general
treatment of kinematics, in which transformations that
do not have constant coefficients are treated on the
same footing as those that do.
Thus, as far as phenomenology and kinematics are
concerned, the present formalism is quite equivalent
to the conventional one of nonrelativistic quantum
mechanics for the case of a system of finitely many
particles i~ three-dimensional Euclidean space. Si~ce
our ultimate aim is to treat systems whose dynamICs
is implicit in their kinematics, that is all that is primarily
relevant. Nevertheless, it is of interest to consider
how the application of the correspondence principle
to the determination of the quantum dynamics is
affected. This may also serve to clarify and make more
concrete the development just described.
Conventionally, the quantum-theoretic Hamiltonian
is derived from the classical one by a familiar, although
generally somewhat ambiguous, process of substituting
variables satisfying the Heisenberg relations for the commuting classical canonical variables. From the
present standpoint, this means that a special frame
(or class of frames) of reference is used in the manifold
that will have no analog on a general manifold. The
substitution method thus appears as less applicable
in the case of a nonlinear manifold, but there is another
effective method of implementing the correspondence
principle, notably that of matching the invariance and
other formal features of the classical Hamiltonian.
Consider for example the problem of the hydrogen
atom in an arbitrary Riemannian manifold. The
relevant classical Hamiltonian is (or, strictly speaking,
is defined as) the sum of the kinetic energy with the
Coulomb potential (the latter being defined in general
as proportional to the elementary. solution for the
Laplace equation for the manifold). There is no need
to describe the use of normal coordinates, etc., in
obtaining a precise analog for the conventional classical
kinetic energy, for the Laplacian gives immediately
an operator that satisfies the key desiderata of general
izing the kinetic energy in conventional nonrelativistic
quantum mechanics and of being intrinsically defined
in terms of the Riemannian geometry. It is clear that
any finite number of particles with Coulomb inter
actions may be similarly treated.
This example may be not without some realistic
relevance. The validity of three-dimensional Euclidean
space as a model for macroscopic space at the non
relativistic level is open to direct verification, but that
the same model is valid in dealing with microscopic
space (i.e., that in which it is theoretically appropriate
to consider an electron as imbedded, if indeed such
exists) is quite a different postulate, which can only
be verified experimentally by indirect means, such as
through its implications for atomic spectra (d., e.g.,
Schrodinger9). In particular, in the event that with
increasing precision of measurement discrepancies from
present theory are found in the spectrum of hydrogen,
it might well be of interest to compare them with the
first-order perturbations in the spectrum arising from
a nonvanishing constant curvature, a problem which
seems technically quite accessible.
The correspondence principle as just applied does
not have rigorous mathematical character, but is
based partly on the exercise of judgement as to what is
physically appropriate and mathematically natural.
In involving possible ambiguity, the present form of
the correspondence principle does not, however, differ
from the conventional process, in which the assignment
of the order of factors in a product of canonical operators
is generally quite essentially nonunique. There have
been many efforts toward the solution of this unique
ness problem (see notably footnote reference 6, which
is definitive in certain respects), but no completely
satisfactory mathematical process has yet been pre
sented. Thus the application of the correspondence
9 E. Schrodinger, Naturwissenschaften 22, 518 (1934).
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principle within the present formalism appears to be
fundamentally not more difficult than its application
by means of the conventional formalism, in the case of
a linear manifold. Actually, in the following a unique
method is given for passing from a covariant classical
motion to a quantum-mechanical one in line with the
present approach, but nontrivial applications are
limited to systems of infinitely many degrees of
freedom.
It remains to consider the dependence of the fore
going quantization scheme on the choice of a measure
m on M. In case another measure m' is used, operators
P' (T) and Q' (f) in another Hilbert space X' = L2(M,m')
are obtained. But the transformation
V:f ---t f(dm/dm')!
is unitary from X onto X', and it is straightforward
to verify that
VP(T)V-l=P(T), VQ(f)V-l=Q'(f).
Thus the two systems of canonical variables are
unitarily equivalent, and in fact the equivalence is
implemented by the relatively trivial transformation V.
The following paragraphs of this section concern
questions of rigor, and some readers may prefer to
omit them.
In the foregoing work a certain loophole for irrelevant
mathematical pathology has been left open through
the use of the unbounded P's and Q's, which operate
not on all of X, but on certain dense domains in X
(this domain varying from operator to operator), and
what is more serious, cannot be unambiguously
multiplied and added together freely. The well-known
device of Weyl for eliminating pathological canonical
systems and making in a natural fashion the P's and
Q's mathematically more clear-cut in the case of a
linear manifold can however be adapted to general
manifolds. It consists in the replacement of the con
sideration of the P's and Q's in the foundations of the
theory by the consideration of the one-parameter
unitary groups they generate. Actually it is convenient
to modify this device and consider in place of the one
parameter groups generated by the Q's the smooth
bounded functions of them, for this merely amounts
to using only those Q(j) for which f is such a function.
In this way one is led to make the following definition
reminiscent of that for a representation of a group of
transformations given by G. W. Mackey.
Definition 1. A generalized Heisenberg canonical system
over a finite-dimensional infinitely differentiable mani
fold M is a pair of maps [U,Q], which are respectively
from the group G of all nonsingular infinitely differenti
able transformations in M and the class (Jt of all real
bounded infinitely differentiable functions on M that
vanish at infinity, to the bounded operators in a
Hilbert space 3(, such that: (1) U is a unitary representation of G: U(gg')
=U(g)U(g'), U(e)=I (e=unit of G, I=identity
operator on x), U(g)-l= U(g)*; and is continuous on
finite-dimensional subgroups of G.
(2) Q is an isomorphism: Q(f+f')=Q(f)+Q(f'),
Q(lf)=IQ(f), Q(ff')=Q(f)Q(f'), and Q(f)~O if f~O.
(3) U(g)Q(f) U (g)-l = Q(fg) , where fg(x)=J(g-l(X»
(this essentially gives in finite form the commutation
relations between a P and a Q).
(4) The Q(j) generate a maximal commuting sub
system of the total system of operators generated by
the U(g) and Q(f).
In the case when M is a finite-dimensional Euclidean
space, the only such system, within physical equivalence
(or observables and states) is that in which X=L 2(M),
U(g)h(x)=h[g-l(X)], and Q(f)h=fh. But if M is not
a simply connected manifold, there will be unitarily
inequivalent Heisenberg systems.1° Nevertheless there
is always a fully covariant way to specify the repre
sentation that is relevant here, i.e., to make Definition 2.
Definition 2. A generalized Schrodinger canonical
system over a finite-dimensional infinitely differentiable
orientable manifold M is the pair of maps [U,Q] from
G and (Jt described earlier, to operators on L2(M,m),
where m is an arbitrary measure on M with infinitely
differentiable nonvanishing density function, given by
the equations
U (a)h(x) = h(a-1(x» (dma/ dm)!,
Q(j)h=fh.
Here a is an arbitrary element in G, and ma denotes the
transform of m under the transformation of measures
induced by the transformation a on M.
As noted earlier, all the Schrodinger systems are
unitarily equivalent, and no essential ambiguity will
10 The number of inequivalent such is an invariant of M closely
related to its one-dimensional cohomology in the following way:
if w is any closed first-order differential form on M, then the
equations P'(X)=P(X)+w(X), Q'(j)=Q(f), define a Heisenberg
system [P',Q'] (in infinitesimal terms) which will be equivalent
to the system [P,Q] if w is exact, but not generally otherwise.
Specifically, there is equivalence if, and only if, w is logarithmically
exact, in the sense that w=dF/F for some function F on M. It
follows from a study of the logarithmically exact forms (d. a
forthcoming paper by R. S. Palais; similar but less complete and
unpublished results are due to E. Dyer and R. Swan) that on a
manifold with first Betti number r, there is an r-parameter
family of inequivalent Heisenberg systems.
Mathematically it is interesting to weaken statement (4) by
requiring only (4'), ergodicity: no nontrivial function of the P's
and Q's commutes with all the P's and Q's. The analog of the
Schrodinger representation with square-integrable functions
replaced by square-integrable tensor fields is an example of a
system satisfying (4') but not (4). The foregoing connection with
closed differential forms and cohomology can be extended, but
some of the quantum-mechanical invariants of M obtained in
the indicated fashion may be new, depending in part on the
extent to which the tensor field examples exhaust the possibilities,
within unitary equivalence and the intervention of a closed form.
This is a point having a certain differential-geometric interest,
and conceivably there is a physical role for the tensor, etc.
representations in other physical connections, but in the present
paper only the "scalar" Heisenberg representations given by
Definition 1 are used.
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result if anyone of these systems is referred to as the
Schrodinger system. Thus we may summarize the
foregoing section as:
Principle I. There exists a unique and mathematically
precise scheme for setting up quantization conditions on
an arbitrary orientable finite-dimensional manifold M;
this extends the conventional scheme for the case of three
dimensional Euclidean space, and is covariant under
arbitrary transformations on the manifold. In essence,
the generalized canonical variables are represented by
Hermitian first-order linear differential operators on M,
relative to an arbitrary measure on M.
3. FINITE SYSTEMS AND CONTACT
TRANSFORMATIONS
Let us now consider the method of the preceding
section in relation to the problem of the quantization
of an infinite nonlinear system. At a nonrelativistic
level the problem is that of developing a parallel to
Dirac's extension to a radiation field of Heisenberg's
original quantization procedure. A field whose state at
a particular time is represented classically by a solution
of a certain nonlinear partial differential equation,
rather than by a linear equation as in the case treated
by Dirac, has its nonlinear canonical Q's associated
with functionals on the manifold M of all classical such
solutions of the equation, while the canonical nonlinear
P's are associated with vector fields on M, as indicated
in Sec. 2. If the field equation is first order and hyper
bolic in the weak sense that the values of the solutions
at a particular time t= to determine the solutions
throughout space-time, and if the initial values form
a linear vector space (assumptions which in essence
are frequently made), this set of initial values may be
taken as the manifold M, and the nonlinearity enters
primarily in the nonlinear action of displacement in
time on M. The adaptation of Sec. 2 to this case
requires a notion of integral in M, and the development
of its transformation properties under nonlinear trans
formations of M, of the type presented by Gross,ll as well
as, for a rigorous treatment of certain divergent cases,
an as yet unavailable combination of the transformation
theory of footnote reference 11 with the representation
free approach of footnote reference 2. Basically how
ever-in particular as regards the formulation of the
quantized field itself-the development of this non
relativistic theory is closely related to that of the co
variant theory that is our central concern, and to which
we shall therefore restrict our further consideration.
The quantization of a nonlinear covariant system
involves new formal elements roughly analogous to
those involved in the Heisenberg-Pauli extension of
the Dirac theory to the relativistic case. The circum
stance that there is no separation between the P's and
11 L. Gross, Trans. Am. Math. Soc. 94, 404 (1960). Q's that is invariant under the entire Lorentz group
in the case of a conventional field shows that there is
no fully Lorentz-invariant manifold of classical wave
functions in the covariant case that plays the same role
as the manifold M in Sec. 2. Rather, the manifold of
classical wave functions that is usually given in the
field-theoretic case by a partial differential equation
is analogous to the phase space in the case of a classical
system of finitely many degrees of freedom. An
element of such a manifold (e.g., a particular solution
of Maxwell's equations, as an element of the manifold
of all solutions) completely describes the "classical"
state of the system. A point of the manifold M in Sec.
2, however, merely determined the location in physical
space of the classical system; to specify its state
completely requires in addition the momentum vector
at the point. The collection of all such complete
specifications forms a manifold B of twice the dimension
ofM.
Thus in the relativistic field-theoretic case, one is
given an analog to the classical phase space B, but is
not given any analog for the space M describing the
spatial location of the system, nor is there any explicitly
relativistic way to define such an analog. Therefore,
in passing from the treatment of Sec. 2 to the case of
an infinite covariant physical system it is natural to
attempt to interpolate a treatment of a finite system
directly in terms of its phase space, in such a manner
that the P's and Q's are dealt with on an equal footing.
The point of this interpolation is primarily theoretical;
there are in fact no nontrivial and realistic Lorentz
invariant systems of finitely many degrees of freedom.
But it is useful to be able to develop the formalism
free from the analytical complications that are present
in the case of infinite systems, and in fact the results
for the finite case will be needed in dealing with the
infinite case.
A classical phase space such as B is not at all an
arbitrary space, but has a special structure. In the
case of a conventional classical system of n degrees of
freedom, a point of B is often specified by a vector
(ql,' .. ,qn, Pi,' .. ,Pn) whose first n components give
the spatial location of the system, and whose last n
give its momenta. When the spatial location is de
scribed by a point of a nonlinear manifold M, such a
coordination is generally only locally valid. In intrinsic
terms, a point of the phase space B is a pair consisting
of a point of M together with a vector in M at the
point, the components of the latter being the various
momenta. (Cf., e.g., Veblen and Whitehead.12) The
conventional (ql,'" ,qn) give a nonintrinsic way of
specifying the point of M, while the (h'" ,pn) give a
similar specification for the vector. The key property
of B from the standpoint of dynamical theory is its
covariant association with a distinguished differential
form of second degree, say n, which is defined by the
12 O. Veblen and J. H. C. Whitehead, Foundations of Di./Jerential
Geometry (Cambridge University Press, New York, 1932).
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equation
n= Li=ln dpidqi,
in the vicinity of a point (p,q) of B, where (ql,'" ,qn)
are local coordinates in M near q, and (pl," . ,pn) are
the corresponding coordinates for the vectors at q.
This form is nondegenerate and determines a nowhere
vanishing positive element of measure dm=nn
= ITi dpidqi. There is no difficulty in verifying that n,
and hence also m, are independent of the local co
ordinates, and are globally defined on B. A dynamical
or "contact" transformation is then defined as a point
transformation on B that leaves invariant the form n.
To quantize the system, starting from the phase
space B, observe first that for any vector field X on M
there is, as is well known (d. Whittaker13) a cor
responding contact transformation p(X) on B. It
suffices to defined p(X) locally, in terms of the local
description of X in a particular coordinate system.
Writing X = Li a(djdqi), then
p(X)=X -l:i.i p,[(da.jdqj)(djdPj)]
(this is the contact transformation corresponding to
the Hamiltonian H = l:i Piai). Next, for any function
] on M, there is a corresponding infinitesimal contact
transformation q(J) onB: q(J)= -L;j [(d]jdqj) (djdPj)]
(this corresponds to the Hamiltonian]).
Next observe that the p(X) and q(J) satisfy almost
the same algebraic relations as the P(X) and Q(J).
Specifically, it is straightforward to compute
[p(X),p(X')]= p([X,X']),
[p(X),q(J)] = q (X]) ,
[q(J),q(J')]=O.
There is, however, a certain difference, which is quite
fundamental, namely, that q(J) = 0 if ] is constant; in
particular p(X) and q(J) commute in the linear case
when X is an infinitesimal translation and] a linear
function. Thus the p(X) and q(J) do not directly give
quite a canonical system; but there is an invariant
construction employing them that gives such a system.
If T is an infinitesimal contact transformation, it
defines a Hermitian operator in L2(B,m), where the
measure m is determined by the fundamental form
dm= ITi dpidqi, by its direct action: h ~ -iTh, for any
function h that is square-integrable qver B. Now the
form n is an exact differential: n= dw, where w is the
differential form of first order L;, pidqi which is in
variant on M. Associated invariantly with T and w is
the function on B, weT), and the definition
R(T)= -iT+w(T)
then gives a Hermitian operator in L2(B,m). It follows
from the formula in the theory of differentiable mani
folds for the derivative of a one-form (or alternatively
13 E. T. Whittaker, Analytical Dynamics (Cambridge University
Press, New York, 1959). by direct computation) that the R(T) satisfy the
commutation relations
[R(T),R(T')]= -iR([T,T'])+n(T,T').
Now when T and T' are taken as the p(X) and q(J),
one has, by direct computation
n[p (X),q(J)]= X],
n[p(X),p(X')]=w([X,X']),
n[q(J),q(J")]=O.
In particular, substituting in the foregoing commutation
relations and defining P(X)=R[P(X)] and Q(J)
=R[q(J)], there results
[P(X),P(X')]=P([X,X']),
[P(X),Q(J)]=Q(X]),
[Q(J),Q(J')J=O.
Here P(X) and Q(J) vanish if X vanishes or] is con
stant, respectively, but they have the proper
commutation relations in the case of an infinitesimal
coordinate and a linear function. The last set of
equations are in fact identical with the commutation
relations given at the beginning of the preceding
section.
Now the foregoing commutation relations not only
extend the conventional ones of the nonrelativistic
quantum mechanics of finite systems, but are closely
analogous to those used in footnote reference 2 for the
quantization of general Bose-Einstein fields. In view
of this, and since we seek a formulation in which the
p's and q's are treated symmetrically, we make
defini tion 3.
Definition 3. A SchrOdinger canonical system over a
phase space B with exact fundamental differential
form n is a mapping X ~ R(X) from the infinitesimal
contact transformations on B to the self-adjoint
operators in the space L2(B,nn) of square-integrable
functions over B with respect to the canonical measure
on B, of the form R(X)=X+w(X), where w is a first
order differential form such that dw=n.
In case B is simply connected, any two w's differ by
the differential of a function, multiplication by the
complex exponential of which gives a unitary trans
formation taking the one Schrodinger system into the
other. Assuming now, that B is simply connected, a
rather slight restriction as far as our purposes go, we
may speak of the Schrodinger system on B with no
essential ambiguity, as in Sec. 2.
Any contact transformation on B, say T, gives rise
to a unique transformation of the canonical variables
defined by the property of taking R(X) into R(XT),
for an arbitrary vector field X, where XT denotes the
vector field into which X is transformed by T. In this
way it is possible to pass uniquely from a given classical
kinematics (or even dynamics) to corresponding
quantum-mechanical ones. The foregoing would appear
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to be the simplest quantization scheme that is invariant
under all classical contact transformations, although,
as discussed below, it is open to serious question
whether all the R(X) are truly obse~able, or equiva
lently whether some additional selection principle does
not operate, as well as to what extent the dynamics
just defined agrees with the conventional substitution
rule.
To see the connection with conventional theory,
consider the case when B is the phase space for three
dimensional Euclidean space M. At first glance it
would appear that the Hilbert space L2(B) is far too
large, and that the present theory must be materially
different from the conventional one. The point is
however that the elements of L2(B) serve only to set
up our observable algebra, and have primarily analyti
cal rather than physical significance; our states are
linear forms on our observable algebra, and only
coincidentally expressible in terms of vectors in
specific Hilbert spaces. The R(X) with X restricted
to be the extension to B of a Euclidean motion in M,
or the infinitesimal contact transformation whose
Hamiltonian is a linear function on M, or a sum of
two such vector fields, satisfy the very same commuta
tion relations as the conventional linear and angular
momenta, and position observables. It follows there
fore from the Stone-von Neumann theorem on the
uniqueness of the Schrodinger operators,14 or actually
by a fairly simple direct reduction in this case, that
these R(X) are identical with the conventional
Schrodinger operators, not within unitary equivalence,
but what is physically just as effective, within unitary
equivalence and multiplicity. There is no difficulty in
verifying that the kinematics defined above for the
R(X) is in corresponding identity with the conventional
kinematics. The dynamics is also in agreement in' the
two formulations, for the case when the Hamiltonian
is at most quadratic in the canonical variables; but
for a general Hamiltonian the two formulations are
incomparable a priori because the class of R(X)
singled out in connection with conventional theory is
not invariant under a general contact transformation.
Thus for a free particle or harmonic oscillator the two
theories are in precise agreement (d. SegaP5); but for,
say, the hydrogen atom problem, the relationship is
obscure. This is not of special concern to us because
our primary interest is in the covariant case, and we
could hardly expect to solve in an incidental way the
much considered problem of formulating a unique way
of passing from a classical nonrelativistic Hamiltonian
to a quantum-mechanical one, which is invariant
under contact transformations, etc. It would never
theless be of significant independent interest to
determine in the hydrogen atom case the precise
connection between the theories, which may possibly
14 J. von Neumann, Math. Ann. 104, 570 (1931).
15 L E. Segal, Can. J. Math. (to be published). be in agreement within terms of order h2• [An eigen
state of the present motion in L2(B) gives rise to a
linear form on the subsystem generated by the special
class of R(X) designated before, which in turn gives
a linear form on the conventional system of operators
on L2(M); this should be a pure state within O(h2)
which has a wave function agreeing with a conventional
hydrogen atom wave function within O(h2).]
A natural and general way to pick out the relevant
special class of R(X) seems to be to make use of a
Riemannian structure in B, which it will inherit from
that of M, in case B originates from an M. When Mis
Riemannian and ql,' .. ,qn are normal coordinates at a
point, while PI,'" ,Pn are corresponding vector co
ordinates, the symmetric quadratic form (!) Lk (dpk2
+dqk2) defines a Riemannian structure in B. An in
finitesimal complex structure can be introduced in B
by defining multiplication by i to act in each tangent
space of B by taking the dp's into the corresponding
dq's and the dq's into the corresponding -dp's; this
structure is evidently intrinsic, and in combination
with the form n, gives a positive definite Hermitian
structure to each tangent space of B. When B arises
from an M the infinitesimal complex structure will
be integrable only when M has vanishing curvature,
according to a result obtained by K. Kodaira (written
communication via N. Steenrod) and also by A.
Frolich and A. Nijenhuis (oral communication). The
case of a given Hermitian manifold B not necessarily
originating from an M, is, however, more relevant to
relativistic field-theoretic situations. In any event a
transformation on a Hermitian manifold B may be
called isometric in case it preserves the Hermitian
inner product in each tangent space to the manifold;
and the observables R(X) may, in the case when B is
endowed with a Hermitian structure having n as the
imaginary part of the inner product, be restricted to
those for which X is infinitesimally isometric. This is
natural from a mathematical viewpoint, and it will be
seen later that it gives the conventional theory in the
case of covariant free fields, as well as, as noted earlier,
in the case of elementary quantum mechanics. It
should perhaps be emphasized that, in any case, the
presence of the additional R(X) does not in any way
alter the physical conclusions concerning the sub
system generated by some restricted class of R(X)-the
stationary states and expectation values, transformation
properties, etc., of the subsystem are unaffected by
treating it as a subsystem rather than as a full system
in itself.
A theoretically less severe limitation on the R(X) to
be used in forming the subsystem of interest, although
for many manifolds apparently an equivalent limitation,
is the use only of those for which X is holomorphic,
i.e., commutes with the operation defining multiplica
tion by i in each tangent space. In a formal way one
may in fact describe the relevant states explicitly, as
represented by the holomorphic functions on B.
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Example. Let B be a complex n-dimensional space,
with coordinates ZJ, Z2, ••• , Zn, and fundamental Hermi
tian form Lk dzkdzk*. The isometry group is generated
by the translations together with the homogeneous
unitary transformations. Writing Zk= Pk+iqk, where Pk
and qk are real, and setting Pk=R(ajapk) and Qk
=R(ajaqk), gives the conventional commutation rela
tions. Choosing w= (!) Lk (zkdzk*-Zk*dz k) gives speci
fically Pk=[(lji)(aj8pk)]+qk and Qk=[(lji)(8j8qk)]
-Pk .. These look rather different from the conventional
quantum-mechanical variables, but by the uniqueness
result cited, must be the same, apart from a unitary
transformation, and the introduction of a multiplicity
(visible in the circumstance that B has twice the
dimensionality of the real manifold on which the
Schrodinger representation is based). It is actually not
difficult in this case to exhibit specifically, without the
use of the uniqueness result, the decomposition of the
present operators in the form
Pk=PkoXI; Qk=QkoXI,
where (PkO,QkO) are the conventional Schrodinger
operators and I is the identity operator in a certain
Hilbert space.
For any unitary transformation U on B, there will
be a corresponding unitary transformation r(U) on
L2(B), transforming the canonical P's and Q's into
corresponding linear combinations of themselves, by
virtue of the fact that transformation of a translation
by a unitary is another translation. More generally
this is true of any linear contact transformation
(i.e., so-called "symplectic" transformation). Of par
ticular interest is the case when U ( = U t) is mul tiplica
tion by eit, Zk ----> eitzk ; r (U t) is then a one-parameter
group of operators whose generator is the conventional
harmonic oscillator (isotropic) Hamiltonian. Its ground
state, as an expectation value linear functional on the
algebra generated by the R(X) with X an infinitesimal
isometry is invariant under the r(U) with U unitary.
Now let B be any simple connected Hermitian
manifold with associated closed form Q, for short
phase manifold. There is then a corresponding theory.
Because there is in general no distinction analogous to
that between the translations and homogeneous
transformations, all the various momenta, and even
the position coordinates, are treated on the same
footing. Transformation properties of these canonical
variables under the subgroup Go of isometries leaving
fixed a point CPo of B are quite analogous to those in the
linear case, where CPo is the origin and Go the unitary
group.
To formulate the relevant intrinsic nonlinear analog
to the ground state of the harmonic oscillator, fix a
point CPo, and consider the connection between the
tangent plane T <1>0 at CPo and the manifold B. The
"exponential" map introduced in differential geometry
by Whitehead, taking a vector I of T <1>0 into a cor-responding point exp(l) in B, lying an appropriate
distance from cpo on the geodesic from CPo in the direction
of I, gives a local linear parametrization of B, which will
have, in general, certain singularities in the large.
These singularities will, however, form only sets of
measure zero in T <1>0 and in B, in the case of many
manifolds, particularly those whose deviation from
linearity arises from the non triviality of the funda
mental Hermitian form, rather than from the non
triviality of the connectivity properties of the manifold
B, as is formally the case of basic interest here. (The
manifold of solutions of a nonlinear hyperbolic equation
is from the quite heuristic standpoint usually employed
in theoretical physics topologically flat, as it is generally
implicitly assumed that the admissible Cauchy data
at a particular time do not need to satisfy any special
nonlinear conditions, and determine the solution
throughout space-time.) At any rate, for a fairly
extensive and interesting class of manifolds M, the
map 1----> expl will give rise to a well-defined mapping
of sets into sets, if sets of measure zero are neglected,
and thereby to a linear and multiplicative correspond
ence between the measurable functions on T <1>0 and
those on B.
Any unitary transformation U on T <1>0 will give a
corresponding transformation ro(U) on L2(T <1>0), and
by virtue of the foregoing correspondence, a trans
formation r(U) of L2(B,Qn). Choosing U to be multi
plication by eit (t real) gives then a one-parameter
group on L2(B,Qn), whose generator may be designated
as the Hamiltonian for the generalized harmonic
oscillator on B at CPo. This will not necessarily be self
adjoint relative to the given inner product, but it will
have real, and in fact integral eigenvalues. It may
also reasonably be conjectured that in the cases of
interest, and in particular when B is obtainable by
continuous deformation of a linear manifold, the
spectrum will be bounded from below, and the ground
state will be unique, as an expectation value functional
on the functions of the R(X) for isometric X.
The point of this construction is that it picks out in
a natural and well-defined way a particular state that
is invariant under the group Go of isometries leaving
invariant the point cpo. This will be useful in getting at
the physical vacuum in the case of fields, where such
invariance presumably characterizes the physical
vacuum, although in the finite-dimensional case there
will generally be other invariant states under Go.
Now when B is a complex unitary space, the cor
responding physical situation is considered to be free
of interaction, and in a certain sense this is evidently
true of the situation for a general Hermitian manifold
B. But from the standpoint of an observer who utilizes
as a reference system the tangent plane to B at a
particular point cpo-i.e., the reference system appro
priate for the examination of small displacements from
a particular classical state-interaction is present. In
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particular the ground state of B and the ground state
of the linear system associated with the tangent plane
are not simple transforms of one another, nor does
the isometry group leaving l/>o invariant transform B
in the same way as does the transform via the ex
ponential map of the linear action of the isometry
group on the tangent plane at l/>o.
The preceding line of development may be summar
ized as follows.
Principle II. Let there be given in a simply connected
manifold B of states of a physical system a distinguished
locally exact Hermitian differential form. There is then
a unique and mathematically precise scheme for setting
up quantization conditions, which extends elementary
quantum mechanics as well as the conventional quantiza
tion theory for relativistic free fields. The relevant co
variance group is that of all transformations on the
manifold leaving invariant the fundamental form
(isometries, that is) as well as a distinguished point of B.
The canonical variables R(X) are associated with in
finitesimal isometries, and satisfy the commutation
relations (4); they are Hermitian first-order linear
differential operators on B, relative to the canonical
measure determined by Q. There exists with significant
generality. a ground state on B analogous to the lowest
eigenstate of a harmonic oscillator in elementary quantum
mechanics.
The situation as regards field quantization needs to
be elaborated somewhat; this will be done in the next
section, where the foregoing principle will be applied
to the quantization of a nonlinear hyperbolic partial
differential equation.
4. INFINITE SYSTEMS
This section extends the preceding one to the case
of infinite systems, and indicates how this extension
may be used to quantize a given nonlinear hyperbolic
partial differential equation.
A. Formal Differential Geometry of Nonlinear
Hyperbolic Equations
For simplicity and concreteness we treat here
primarily the classical (unquantized) system M defined
by the equation
Dl/>=m2l/>+p(l/», p a polynomial vanishing at 0; (5)
the extension to rather more general cases appears to
involve no great difficulties. M may be regarded as an
infinite-dimensional manifold that is imbedded in the
manifold (say S) of all scalar functions on space-time.
Consequently, at any point l/> of M there will be a
tangent plane T", 4efined by the equation
(6)
For arbitary given l/> in S, this equation defines a linear
manifold in S, and in fact M may be considered as an integral manifold of this distribution of linear manifolds,
that passing through the point l/>= O. (It is worth noting
that the satisfaction of the relevant integrability
conditions by this distribution of linear manifolds is
purely a matter of linear analysis, and so on a much
more accessible level than the questions of classical
nonlinear analysis involved in the structure of M as
first defined.) At this generally substantially unique
Lorentz-invariant point of M, the tangent plane is
defined by the so-called "free-field" equation
(7)
Since Eq. (6) is linear and hyperbolic, there is for
any fixed function l/> a unique function D",(x,x') of
ordered pairs of points of M, which satisfies (6) as a
function of the first point x, and also the following
initial conditions [employing the notation X= (x,xo)]:
D",(x,x')=0 }
when Xo= xo'.
(iJjiJxo)D",(x,x') = o(x-x') (8)
Now this function also satisfies the differential equation
as a function of x', or more exactly:
Heuristic Proposition 1. D",(x,x') = -D",(x',x) for
arbitrary x and x'.
Argument: It suffices to show that -D(x',x) (sup
pressing the dependence on l/>, which is here irrelevant)
satisfies the defining conditions for D(x,x'). The first
condition of Eq. (8) is obvious, and for the second
condition, it may be noted that
= -lim.-+oc1D(x',xo',x, Xo'+E) iJ[ -D(X"X)JI
iJxo xo=xo'
=lim.-+oc1[D(x', XO'+E, X, Xo'+E)
-D(x',xo',x, XO'+E)]
=~D(x"xo"X,xo) I
iJxo' xo'=xo
=o(x-x').
It remains only to show that M (x,x') vanishes
identically, where M(x,x') = [Dx-V(x)]D(x',x), writ
ing V=m2+p'(l/». To this end it suffices to show that
M (x,x') is the solution to a Cauchy problem with
vanishing initial data. We shall regard it as a function
of x' with initial values given on the hyperplane
xo=xo'. Since [Dx'-V(x')]D(x',x)=O by the definition
of D(x,x'), and since Dx'-Vex') as an operator
commutes with Dx-Vex), we have
[Dx'-V(x')]M(x,x')=O.
Now let us evaluate M(x,x') for xo=xo'. The only
contribution whose vanishing is not apparent is
(iJ2jiJxo)[2D(x',x)] I xo=xo'.
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This may be written as
lim ..... o(2e)-2[D(xo, xo+2e) -2D(xo, xo+e)+ D(xo,xo)],
where for the moment we suppress the dependence on
x and x'. Now
which by Taylor's expansion and the definition of
D(x,x') is
-2eO(X-X')+2e2[~2D(Xo" xo+2e)] +0(e3).
axo xo' =xo+2.
From the fact that D(x',x) satisfies the differential
equation as a function of x', it follows that (a2/axo)
X [2D(x,x')], evaluated for xo=xo', is the same as
[~z-v (x)]D(x,x'), likewise evaluated for xo=xo',
where ~ denotes the Laplacian; and hence the middle
terms in the preceding expression vanishes. A similar
evaluation applies to D(xo, xo+e), from which it results
that ca2/axo) [2D (x',x)] I xo=xo'= lim ..... o(2 e)-2Q( e3)=0.
It remains only to show the vanishing of (a/axo')
X[M(x,x')] for xo=xo'. Writing Lz=~x- Vex), we
need to examine
Lx ---D(x',x). [ aD (x',x) I a2 a I
axo' xo=xo' aX02axo' xo=xo'
Since Lx involves no differentiation with respect to
time, the first term is the same as
[aD (x',x) I L" , =L,p(x-x').
axo xo=xo'
In evaluating the second term, we write xo=t, xo'=t',
and note that
a2a a aa2 aa2 --=-+-----.
at2 at' at at' atat' at' atat'
The term [(a/at') (a2 / at2)] develops as follows:
--D(x',x) =-Lx,D'x',x) a2
a I a I
at'2 at I'~I at 1'-1
but Lz' and a/at commute, and Lx' involves no differen
tiation with respect to time, so that the expression
reduces to
Lz'[~D(X"X)] =-Lz'o(x-x')
at I~I'
by an earlier result. This precisely cancels the first
term, so to conclude the argument it suffices to show
that
[(a a) a2
] -+--D(x' x) =0.
at at' alat' ' I~C,' Now
(~+~)F(t,t') I
at at' I-I'
vanishes identically if F(t,t) is a constant, so it suffices
to show that {[(a2/atat')]D(x',X)}t-I' is a constant,
as a function of t. Actually it vanishes, for it may be
written as
lim ..... oe-lD(t+e, t+e)-D(t, t+e)-D(t+e, t)+D(t,t)]
=lime-2[( e2/2)Lz,D(x,x') -(e2/2)Lx,D(x,x')
+0(e3)]t-t'=0,
since Lz,D(x,x') vanishes for t= t'.
The function D~(x,x') thus determines a skew
symmetric bilinear form B.p(l,l') in the solutions of (6):
Thus for each pair of tangent vectors at cp there is a
skew-symmetric bilinear functional of them; this is by
definition a second-order differential form on M. This
form will be denoted as fl, and called the fundamental
form on M. To see the connection between this form
and the similarly designated form in classical me
chanics, it is useful to observe that
in the case of the Klein-Gordon equation
(p=O identically),
where PI, P2, ... , ql, q2, . .. are "natural" coordinates
onM.
Specifically, the Pk and qk are obtained by choosing
any complete orthonormal set of Klein-Gordon wave
functions invariant under time reversal, say ft,h, ... ,
and writing a general real Klein-Gordon wave function
f as
where the tilde denotes the action of forming the
Hilbert transform with respect to time. The con
vergence of the infinite sum presents no essential
difficulty, as is clear from the following argument,
which also serves to make clear how such sums are to
be interpreted.
In a linear space, any differential form may be
expanded into a product of differentials of linear co
ordinates, showing that two differential forms are the
same if they agree on all generators of infinitesimal
translations. Hence it suffices to show that if j and g
are arbitrary normalizable real Klein-Gordon wave
functions, then
(aa) co (aa) fl -,-= 2: (dpkdqk) -,-;
aj ag ~l aj ag
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here a/af stands for the vector field on M generating
the transformations cp ~ cp+sf( -00 <s< 00). The left
side reduces to
where D denotes the familiar scalar particle commuta
tion function. The right side is
It is readily seen that (apk/af)= (fJk) , etc., so that the
identity of the two forms on infinitesimal translations
reduces to the equation
f ff(x)g(x')D(x-x')d 4xd4x'
=! Lk [(f,fk)(g,M- (f,fk) (g,fk)],
which can be verified without difficulty; here (u,v)
denotes the unique real Lorentz-invariant inner pro
duct between the wave functions u and v (suitably
normalized).
To develop further the differential geometry of the
function manifold M, we require:
Heuristic Proposition 2. The transformation K",:
lex) ~ fD",(x,x')l(x')dx', acting in the tangent plane
T '" to M at cp, has the property that K i = -1"" where 1",
denotes the identity operation in T ",.
Argument: This is easily seen for the case cp=O by
the use of Fourier transforms, or by reduction to a
similar property for the Hilbert transform in one
dimension. Now suppose that cp is "small," in the
sense that
D",(x+su, x' +su) ~ Do(x,x') as s ~ ± 00
for any timelike vector U; this means that for large
times, Do(x,x') behaves like the "free" commutator
function. In the equation
it is reasonable to suppose that if, say, V is smooth
and vanishes outside of a bounded set, then for large
times the situation is asymptotic to that for the
equation
Dl=m2l.
Rigorous results of this sort are not y~t available in
the mathematical literature, but substantial results
in this direction in nonrelativistic cases have been
established by Kato, Cook, and others (d. e.g.,
Kuroda16 and the literature cited therein), and in any
event such results have a high degree of plausibility
from the standpoint of theoretical physics, constituting
16 S. T. Kuroda, J. Math. Soc. Japan 11, 247 (1959). a weakened and classical version of the adiabatic
hypothesis of quantum field theory (d. Yangl7).
Now let 1 be any tangent vector at cpo Then l(x+su)
is aymptotic, for large s and fixed timelike u, to a
solution lo(x) of the free-field equation. Conversely, 1
may be characterized as that solution of (6) that is
asymptotic to the particular free-field wave function
lo for early times, i.e., it may be regarded as the solution
of a Cauchy problem with data given at time -00.
It is evident that
l'(x) = f D(x,x')l(x')dx'
is likewise a solution of (6); and
I' (x+su)= f D(x+su, x')l(x')dx'
= f D(x+su, x'+su)l(x'+su)dx'.
Now as
s ~ -00, l'(x'+su) ~ lo'(x')
and D(x+su, x'+su) ~ Do(x,x'). Assuming now that
the passage to the limit may be made under the integral
sign, it follows that
I' (x+su) ~ f Do (x,x')10 (x')dx'.
Thus
lim._oo(K",l) (x+su) = Ko lim._o,,l(x+su),
where K 0 is the transformation on the free-field wave
functions with kernel Do(x,x'). If we denote by T the
transformation from the solutions of the free-field
equation to those of (6) asymptotic to the given free
field wave function at early times, the foregoing result
means that
T-IK",T=K o•
Hence T-IK",2T=K02, and since K02= -10, it follows
that K",2= -1",.
Now the property K",2= -1 '" is a variety of functional
equation having no explicit reference to the size of cp;
if it is valid for sufficiently small cp, then it should be
valid as a general rule. For example, if cp is a constant,
then the result is evidently valid, although the argument
for small cp certainly is not.
It is now easy to derive:
Heuristic Proposition 3. M becomes endowed with a
positive definite Hermitian metric if the following defini
tions are made:
(1) For any two tangent vectors land l' at cp, the
inner product is given by the equation
(l,l')",= L", (l,l')+iO",(l,l'),
17 C. N. Yang and D. Feldman, Phys. Rev. 79, 972 (1950),
G. Kiillen, Arkiv Fysik 2, 33 (1950).
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where
(2) Complex scalars act on tangent vectors 1 in
accordance with the unique extension of the manner
in which real scalars act together with the rule
il=K",I.
Argument: The only point that is not immediate
from a quite general argument is the definiteness of
the inner product, i.e., that (1,1)",>0 if 1 is not the zero
tangent vector. Since n", is skew-symmetric,
(l,l)",= L", (l,l) =n",(K",l,l)
= f f D(x,y)l(y)D(y,z)l(z)dxdydz
= f (f D(x,y)l(y)dy YdX.
This shows that (1,1)","2.0, and that the equality holds
here only if fD(x,y)l(y)dy vanishes. But this is K",l,
which by Proposition 2 can vanish only if 1=0.
The remainder of the argument is of a simple and
familiar algebraic character-d. Ehresmann18-and
may be omitted.
It may be illuminating to consider the content of
proposition 3 in the case 4>=0, which is easily de~lt
with explicitly. It says that the space of real normahz
able solutions of the Klein-Gordon equation may be
given the structure of a complex Hilbert space. The
action of i is given by the Hilbert transform with
respect to the time variable; the imaginary part ~f
the inner product is given by the form whose kernel IS
the commutator function; and the real part is obtained
by replacing one of the entries in this skew-symmetric
form by its Hilbert transform with respect to tim~,
obtaining thereby a positive definite real symmetrIc
form. This complex Hilbert space is easily seen to be
in one-to-one correspondence, in an essentially unique
Lorentz-invariant fashion, with the conventional space
of normalizable positive-frequency complex-valued
Klein-Gordon wave function (d. SegaP9).
To complete the analogy with a classical mechanical
system, and make available fonnally the apparatus
developed in Sec. 3, we require further:
Heuristic Proposition 4. The form n is closed,-its
covariant differential dn vanishes.
Argument: The evaluation of dn(X,Y,Z) involves
considerable computation which we shall not carry out
here. There is another way of arguing, which while
quite heuristic, throws light on the origin of the form n.
As noted earlier, our manifold M may be considered
as a submanifold of the manifold S of all scalar functions
on space-time. Now any form on S gives rise, by
18 C. Ehresmann, Proc. Int. Congr. Math. 1950 (Providence,
1952).
191. E. Segal, Phys. Rev. 109, 2191 (1958). restriction of the tangent vectors to tangency to M, to
a fonn on M; and this restricted form will be closed
if the original form was such. In particular this is true
of the form Q on S defined by the equation
(8 8) f dtdt' Q -,--= f(x,t)g(x,t')dax--"
8f 8g t-t
or alternatively, as
Q = Lk dpkdqk,
where PI, P2, ... ql, q2· .. are coordinates on S similar
to those defined earlier.
We may formulate S as an infinite-dimensional
Riemannian manifold by assigning to each tangent
space S",-the general element of which has the form
8/81/; for some formally unrestricted scalar function 1/;
the usual inner product, i.e., [(8/81/;),(8/81/;')J
=f1/;(x)1/;'(x)d 4x. Thus any such tangent space is
isomorphic to the real Hilbert space H of all real
square-integrable functions over space-time. This sr~ce
can be decomposed into eigenspaces of the self-adJomt
operator 0, as a so-called "direct integral" of
(infinitesimal) eigenspaces H.( -00 <s< 00), so there
is a corresponding decomposition of S", into eigenspaces
S",(s). Now at the point 4>, Q gives a skew-symmetric
bilinear form Q", in the vectors of S"" which may be
restricted to any eigenmanifold packet, say that
corresponding to the eigenvalues in the interval
(s-~ s+~) yielding a bilinear skew-symmetric form
Q",(s-':"~, s+'~), in the vectors of this eigenII!anifold. Now
as e -> ° the difference quotient (2e)-ln",(s- e, s+e)
has a lim'it, which is a fonn Q",(s) in the vectors of the
eigenspace corresponding to the eigenvalue s. It can
be explicitly verified, by recourse to Fourier transforms,
that if s=m2, this form is the same as that introduced
above with kernel D",(x,x'), for the case p(4))=O, the
eigenspace S(m2) being identical with the T", defined
above.
Now S", may also be decomposed into eigenspaces of
the self-adjoint operator O-p'(4)), and a similar for:n
density Q",(s; p), which is a bilinear skew-symmetrIc
form in the eigenspace of this operator with eigenvalue
s, obtained. This eigenspace is identical with the T "',
the tangent space to M at ¢ discussed before, and if
we permit ourselves to use the plausible conj~ctu~e
that the two intrinsically defined skew-symmetrIc bI
linear forms on this space, Q",(m2; p) and n, agree in
general, as they do in the case p=O, then it follows
(formally) that n is closed, being a limit of closed forms.
B. SUbsumption of the Conventional
Field-Theoretic Formalism
We now assume that we have the manifold M of all
solutions of Eq. (6) set up as a Hermitian manifold
with fundamental form n, and that linearly associated
with each vector field X on M we have an operator
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R(X), the following commutation relations being
satisfied:
[R(X),R(Y)]=R([X,Y])+Q(X,Y).
(The closure of Q enters primarily as a means of assuring
the consistency of these relations.) We wish now to
define the "quantum field" cP(x) so that the conven
tional commutation relations and transformation
properties are derivable.
For any scalar function f on space-time ("weighting
function"), which is smooth and vanishes at infinity,
we consider the vector field X" whose value at c/> is
the tangent vector fD(x,x')f(x')dx'. Evidently, Xf
depends linearly onf, and hence R(Xf) does so also, so
that we may write formally
R(Xf) = f ¢(x)f(x)dx,
for some operator-valued function cP on space-time.
We may also write
cP(x) = R(Xf) for f=a delta-function at x (formally).
We wish to show that cP(x) satisfies the conventional
commutation relations. To this end, note to begin with
that the solution of the Eq. (6), with Cauchy data
<p(x)=f(x) and (a/at)<p(x)=g(x) at t=t1, is
r [_aD_(X_'~_') f(x') + D(x,X')g(X')]d3X', JXO'=t1 axo
for it is evident that this is a solution of (6); that for
t= t1, it attains the value f(x), by the Cauchy data
defining D(x,x'); and using the fact proved earlier that
[ a2 ] --D(xx') =0 " , axoaxo xo=xo'
it follows similarly that its time derivative for t= t1 is
g(x). Now consider the one-parameter group of motions
on M which takes a given wave function c/> into one
having the same values for t= t1, but with (ac/>/axo)
displaced by sg(x) (-00 <s< 00). The generator of
this group of motions will be a vector field on M whose
value at c/> will be the solution of Eq. (6) having cor
responding Cauchy data on the line t= t1; it is,
accordingly,
f D(x,x')g(x')d 3x, (xo' = t1),
or X" wheref(x)=g(x)o(t-t 1). Thus
R(Xf) = fcP(x,li)g(x)dax.
If we take another function g'(x) and consider the
one-parameter group of transformations of M which it
determines in the same manner as g(x), then it is clear that this group commutes with the group determined
by g, since they both act additively on the Cauchy
data at t= t1• The corresponding vector fields therefore
commute, and substitution in the fundamental com
mutation relation, after choosing g and g' as delta
functions, gives the equation
[¢ (x,t),¢ (x' ,t)] = Q(X"X f')'
wheref(y)=o(y-x)o(yo-t) andf'(y) is the same with
x replaced by x'. Substitution now in the equation
defining Q now gives for the right-hand side of the
foregoing equation the value D(x,t,x',t), which vanishes
by the definition of D. Thereby so-called "local
commutativity" (or "microcausality") is established.
To evaluate [cP(x,t), (a/at)¢(x',t)], consider [cP(x),
¢ (x')], where xo' = xo+ E, E being small. Directly from
the fundamental commutation relations we have
[¢(x ),¢ (x')]= R([Xox,Xo x' ]+Q(Xox,X ox').
By an observation made earlier, Xoy has at c/> the value
10'=YO D(x,x')o(x'-y)d 3x',
or D(x,y), as a function of x. Thus
Q(Xox,Xo x')= f f D(u,x)D(v,x')D(u,v)dudv.
Now proposition 2 may be restated as
f D</> (x,x')D", (x',x")l (x")dx'dx" = -lex)
if I is in T</>. In particular, putting l(x)=D</>(x,y) with
y fixed, it results that
f D(x,x')D(x',y')D(y',y)dx'dy'= -D(x,y).
It follows that Q(Xox,Xo z') = -D(x,x').
To evaluate [Xox,Xox']' recall that Xox' is the
generator of the one-parameter transformation group
on M, with the parameter s, which takes a general
element c/> of M into that element c/>' such that
c/>'(x) = c/> (x) } at xo=xo'.
(ac/>'jat) = (ac/>!at)+so(x-x')
From this characterization we shall show that it
commutes with Xox' within terms of order E2. It is
perhaps clearer to deal more generally with a manifold
M defined by an equation of the form
(au/at) = L(t)u,
where L(t) is a nonlinear operator (i.e., L(t) depends
on t, but does not involve differentiations with respect
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to t). From this equation it follows that
U(t') =u(t)+ (t'-t)L(t)u(t)+O[ (t'-t)2J.
If we now consider a one-parameter group dependent
ona parameter s, which displaces M so that u(t) -t u(t)
+sv(t), t and v being held fixed, then the corresponding
displacement of U(t') may be computed as follows:
U(t') -t [u(t)+sv(t)J+ (t'-t)L(t)[u(t)+sv(t)J
+O[ (t' -t)2J,
= U(t')+ (t'-t)[oL(t)Ju(t)v(t)+sv(t)+O(S2)
+O[ (t' -t)2J,
=u(t')+sWv(t)+O(S2)+Q[ (t'-t)2J,
where W is a certain linear operator (dependent on t
and t', but not on s). That is, the infinitesimal displace
ment of u(t) by an amount sv, displaces u(t+ E) by an
amount sWv+O(e2); and the displacement of U(t+f)
acts similarly on u(t). Since vector translations com
mute, it results that X.x and X.x' commute within
terms of order e2•
Such terms contribute nothing to the commutator
[4>(X,t), ~(XI,t)]=~[¢(X)' ¢(x'n~t'.
at at'
The sole contribution is then
a -[ -D(x,x'n~t'= -o(x-x').
at
Now consider the transformation properties of 4>(x).
Designating as a contact transformation one that pre
serves the fundamental form Q on M, it is clear from
the covariance of the construction of .p(x) that for any
such transformation T, the field {;(x)=fi,(Tx) satisfies
the same commutation relations. In a formal way the
existence of an operator U (T) with the property that
{;(x) = U(T)fi, (x)U (T)-I is clear, for U(T) may be
taken as the operator taking a formally square
integrable functional f(x) over M into the functional
f(T-Ix). It is evident that in this quite formal sense
the map T -t U (T) is a unitary representation of the
group of contact transformations, but this is not
strictly the case even for free fields unless T is suitably
res~ricted, e.g., to be an isometry (d. footnote
reference 2).
In conventional field theory it is assumed that the
quantized field "satisfies" the original field equation.
This is an equation involving local products of fields,
and so has no definite mathematical meaning. It has
also . no empirical physical meaning. The present
formalism eliminates these fundamentally objection
able features of the conventional theory, but this
advantageous feature in itself limits the possibility of
showing complete formal equivalence to conventional
theory. It can be stated that the quantized field fi,(x) is here derived in a covariant and unique manner from
the classical system; but the equation that states that
it "satisfies" the original differential equation has no
clear-cut mathematical or physical meaning, and
cannot be stated in the present formalism.
C. Convergence Considerations
Although local products of fields do not occur in the
formulation of the dynamics of the present quantum
fields, so that what have been regarded as the crucial
divergences do not occur at least in the very formulation
of the theory, some substantial emendations to Sec. 3
are required to provide a rigorous framework for the
case of a system of infinitely many degrees of freedom.
Probably the most obvious difficulty is that the space
L2(M) of square-integrable functions over M is not
really well defined in the infinite-dimensional case, so
that the dynamical variables R(X) are not operators
on any well-defined state vectors. There are two
approaches possible here: (i) the extension of the
integration theory in function space presented in a
rigorous fashion for the linear case in footnote reference
7; (ii) the adaptation of the representation-independent
formalism of footnote reference 2, in which the dynami
cal variables are essentially elements in a well-defined
algebra of observables, which however are not operators
in any ad hoc Hilbert space (states being treated
through their expectation value functionals, i.e., as
suitable linear forms on the observable algebra).
The latter approach is simpler from a theoretical
point of view, but it does not so readily lead to an
explicit construction for the vacuum state, as does the
former approach. In addition, much of what is involved
in developing approach (ii) is parallel to part of the
development of approach (i). It should therefore
suffice here to describe (i).
The main idea is to use the approximation of the
infinite system in a physically meaningful sense by
finite systems. For example, when M is a Hilbert space,
it is approximated in a way by subspaces of large
finite dimension; the relevant functionals on the
Hilbert space are those which are essentially carried
by a finite-dimensional submanifold (depend only on
a finite number of coordinates), or can be approximated
by such in an invariant fashion (d. footnote reference
2); and the relevant vector fields are principally those
generating translations, and so are carried by finite
subsystems. In the case of a general Hermitian mani
fold M we may assume, virtually as a definition of a
nonpathological manifold, that it may be approximated
by finite-dimensional Hermitian manifolds, in the
following sense: There exist phase manifolds N of
finite dimension, and maps F of M onto such an N
preserving the Hermitian structure (Le., the induced
map dF from the tangent space of M onto that of N
is isometric in the finite-dimensional orthocomplement
of the subspace of the tangent space on which dF
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vanishes), forming a "directed set"; for two of the
approximations (N,F) and (N',F'), there is another
approximation (N",F') which may be interposed
between M and each of the two, and being ample in
the sense that for no tangent vector to M do all the dF
vanish. A tame functional on M may then be defined as
one of the form f[F(x)], for some function f on N in
the conventional sense. The sum and product of tame
functionals is again such, and an integral on M may
be defined in the manner of footnote reference 7 once
we have a well-defined linear functional on the collec
tion of all tame functionals that is appropriate
specifically, is nonnegative on nonnegative-valued
functionals, and normalized to be unity on the unit
function, identically one on M.
The requisite functional may be obtained from the
ground state of the generalized harmonic oscillator
treated in Sec. 3, assuming the approximating finite
dimensional manifolds satisfy the conditions given
there. This is an intrinsic definition, and in the rela
tivistic free-field case is known to yield the conventional
theory. The R(X) may then be formulated as operators
in the Hilbert space L2(M) if the X are now restricted
to be "tame," in the sense of being carried by a finite
dimensional manifold: for some (N,F), X corresponds
to a vector field on N. This gives a covariant class of
canonical variables of which the conventional ones are
formal functions. In this way all relevant questions
concerning analysis on M may be brought back to
corresponding questions concerning approximating
finite-dimensional manifolds, which do not involve any
nontrivial divergences.
Probably the next most important difficulty is a
purely classical and mathematical one. There is
available at this time virtually no rigorous theory
concerning the global solutions of a nonlinear hyperbolic
equation, so that the manifold M used above of all
classical solutions of Eq. (5) is a rather vague mathe
matical object. As noted earlier, such a manifold may
be defined as an integral manifold of a certain distribu
tion of elements of contact, which are defined by linear
equations, and so accessible by existing methods. To
a noteworthy degree, the manifold itself is not required,
but only such tangent planes to it. On the other hand,
for a complete theory, the problem of the rigorous
formulation of M cannot be evaded. It is not important
to formulate M as a point set, actually, but only as a
certain variety of (inverse) limit of finite-dimensional
manifolds.
The considerable mathematical difficulties here are
in part of an altogether different character from those
which seem relevant to the basic difficulties of quantum
field theory. The relevant solutions of linear equations
such as (6) must be expected to be not ordinary
functions, nor even distributions in the sense of
Schwartz, but quite highly generalized functions whose
character cannot be described in an a priori explicit
manner. These difficulties are connected with the determination of the precise character of the eigen
functions associated with the continuous spectrum of
a given linear partial differential operator, a problem
which is fairly well understood and to a considerable
extent resolved in the case of an elliptic operator,
although not as yet in the case of a hyperbolic operator.
Such rather technical problems may be avoided by the
simple and physical expedient of smearing over the
mass, in nonlinear analogy with the conventional treat
ment of the continuous spectrum through the use of
packets of eigenfunctions.
The operator O-p'(cf» will be a self-adjoint one
when properly formulated in Hilbert space, and will
have a certain spectral decomposition into eigenspaces,
one of which is defined by (6). If we replace this
eigenspace by an eigenmanifold (= eigenspace packet)
corresponding to the masses in the range (m-e, m+e),
we obtain a tangent space whose elements are bona fide
square-integrable functions. There seems no reason to
doubt that in a quite rigorous and rather straight
forward sense, the corresponding distribution of
elements of contact will admit an integral manifold,
which will be locally a Hilbert-space of functions.
Formally this manifold is obtained by joining together
all of the manifolds defined by (6) with m in the range
(m-e, m+e); the manifold M of solutions of (6) is in
a rough sense a limit of the more accessible and well
defined manifolds M. just described.20 It may be noted
incidentally that the global construction of this manifold
should give, in combination with the developments of
the first part of Sec. 4, concrete and nontrivial examples
of quantum fields satisfying axioms similar to those
axioms of Kallen and Wightman21 which do not pertain
to vacuum expectation values, and in addition the
canonical commutation relations for equal times.
The results of the preceding section may be
summarized as
Principle III. The quantization of a given nonlinear
hyperbolic partial differential equation may be accom
plished by utilizing the intrinsic Hermitian structure, as
a diiferentiqble manifold, of the manifold M of all
classical wave functions for the equation, in formal
accordance with principle I I. The infinite-dimensionality
of M is dealt with by suitable approximation of M by
finite-dimensional image manifolds, to which principle II
is directly applicable. The field operators are among the
canonical variables introduced in Sec. 3. The vacuum
state is characterized as that invariant under the group
of isometries of M leaving fixed the vanishing classical
field, and in suitable cases may be more explicitly
described as a limit of ground states of the approximating
finite-dimensional systems.
It ought to be noted that the foregoing isometry group
will include effectively the Lorentz group, in the case of a
20 Cf. the suggestive work of Dirac in a linear case in Proc. Roy.
Soc. (London) A183, 284 (1945).
21 G. KlilU;n and A. Wightman, Kg!. Danske Videnskab.
Se1skab. Mat.-fys. Skrifter 1, No.6, 58 pp, (1958).
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Lorentz-invariant equation involving only real masses.
Any Lorentz transformation then transforms the
elements of M in a fashion leaving invariant the
fundamental form and the infinitesimal complex struc
ture, as well as the vanishing classical field, and so
determines an element of the group in question. The
necessity of the real mass condition is clear from the
impossibility of making a covariant separation of a
free field into positive and negative frequency com
ponents, in the imaginary mass case. Because of the
close connection of this separation with the infinitesimal
complex structure defined above, the latter will not be
Lorentz-invariant in the imaginary mass case. On the
other hand, the assumption that only real masses are
involved is physically plausible and should be mathe
matically demonstrable, when suitably formulated in
rigorous terms, for the relevant equations.
5. PARTICLES, INTERACTION, AND MODELS
A. Quanta of Fields
The correlation of any quantum field theory with
empirical results depends in a practically essential way
on the possibility of giving a particle interpretation
for the theory. However, if we start with such an
equation as
D<I>=m2(/+l¢3(l~0),
and quantize the system and obtain its physical vacuum
in accordance with the preceding section, the Hilbert
space of states of the incoming field is entirely deter
mined (d. footnote reference 2) mathematically; and
it is open to considerable question whether it contains
any vectors transforming like the solutions of a Klein
Gordon equation of mass m, or some other mass, let
alone is equivalent to a free Bose-Einstein field of
Klein-Gordon particles. The justification of an assump
tion of this type must at this time be essentially
empirical; its success in renormalization theory vali
dates it as a physically motivated maneuver in applied
mathematics, but neither bears directly on the mathe
matical question involved, nor does it seeni to involve
a heuristic principle likely to lead into an effective
mathematical development.
At this stage in the present theory we can only give
a formal analysis of the states of the quantum field in
terms of the particles whose wave functions are the
tangent vectors at some fixed classical field <1>0; there
is no mathematical reason to expect this analysis to
be convergent or rigorizable, in fact there are indications
for the opposite; in a sense the present theory does not
so much remove the field-theoretic divergences as
isolate them in the practice of giving an ad hoc
elementary particle analysis of the states of the field.
A fixed classical field <1>0 may be thought of as the
background field of a particular observer, who will be
able to observe directly only small deviations from <1>0,
in the first instance. That is, classically he does not observe the manifold M of all states, but rather the
tangent plane T 4>0 to M at <1>0, the vectors of which
represent fields deviating only slightly from the back
ground field, these being the only fields his apparatus
will be able to prepare, without interfering significantly
with the object of his observations, i.e., without pro
ducing quantum effects. For him a quantum of the
field is naturally represented by a vector in T <1>0, and
the field variables most accessible to him are notably
the occupation numbers for such quanta. To set up
such occupation numbers in a formal theoretical way,
let us suppose that the exponential map of the tangent
plane T <1>0 into the manifold M is globally without
singularities and applicable to the infinite-dimensional
case. Uncertain as this assumption is, it is not the
most questionable assumption needed, which is that,
at least locally, the measure on M obtained by trans
forming by the exponential map the canonical measure
on T 4>0 is comparable with ("absolutely continuous
with respect to") the physical vacuum measure on M.
That this is a harmless assumption when M is finite
dimensional arises from the fact that any two measures
compatible with the manifold structure of M are com
parable (mathematically, any two measures whose null
sets are .invariant under translation are comparable);
in the infinite-dimensional case this is very far from
being true, even very "small" transformations (e.g.,
x~lx, for any l~±l) taking the free-field vacuum
measure into incomparable ones (d. SegaP2 for a
rigorous treatment of this question). But if the two
measures are comparable, then a development similar
to that given in Sec. 3 is possible, and for every unitary
transformation U in T <1>0, there will be a corresponding
transformation r(U) on the state vector space of the
field, the map U ----+ r (U) being intrinsically defined,
and a representation [r(UU')=r(U)r(U')]. Occupa
tion numbers may then be defined as in footnote
reference 2, pp. 27-31, as the infinitesimal generators of
groups r(U,) for appropriate phase transformations
Uti they will then have integral proper values, anni
hilate the vacuum, etc.
The isometry group Go acts naturally as a group of
linear transformations in T <1>0, as in any Hermitian
manifold; in the case of the manifold defined, e.g., by
the equation D<I>=m2<1>+<I>3, with <1>0=0, this includes
the usual action of the Lorentz group on the real
solutions of the equation O<l>=m2<1>. If the action of
Go is irreducible, as in this case, or more generally if
disjoint invariant subspaces are orthogonal, then a
complete set of group-theoretic quantum numbers of
the usual variety may be set up. In this case the
preceding paragraph gives in a formal way a complete
analysis of the states of the field in terms of elementary
particle occupation numbers, the particles being
described by such quantum numbers (d. footnote
reference 2, pp. 27-31).
22 I. E. Segal, Trans. Am. Math. Soc. 88, 12 (1958).
Downloaded 08 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissionsQUANTIZATION OF NONLINEAR SYSTEMS 487
The rigorous validity of an analysis of this type is
both mathematically dubious and physically somewhat
counter to current lines of thought skeptical of any
absolute meaning to the notion of "elementarity" of
an empirical particle. In any event, the foregoing
analysis appears to exhaust the formally simple
particle interpretations applicable to general hyperbolic
equations, although for suitable special equations an
essentially rigorous notion of elementary empirical
particle may conceivably exist (no solid evidence in
either direction being presently known). That is to say,
there may be in some cases a Lorentz-invariant trans
formation of the observables of a certain linear field
into functions of the observables of the (interacting)
field associated with M, although in even the most
favorable of the cases of conventional theory, quantum
electrodynamics, this seems unlikely, except as an
approximation.
B. Covariant Definition of the Interaction
A puzzling feature of conventional theory has been
its dependence upon an apparently artificial and un
physical separation of the total Hamiltonian (or
Lagrangian) into "free-field" and "interaction" con
stituents (d. e.g., van Hove23). Such a separation
appears in the present theory as the concomitant of
the classical observer's limitation to the examination
of relatively small displacements from his background
field CPo. If CPo is time-independent, then time will act
naturally in a linear, "noninteracting," essentially
kinematical fashion on T <1>0; the actual dynamics,
however, refers to the action of time on M, which
will be nonlinear, when M is formally coordinatized by
T <1>0, by the use, e.g., of the exponential map described
earlier. These are classical motions; the corresponding
quantum mechanical motions may be represented
linearly in the function spaces over T <1>0 and M, re
spectively. The latter motion is formally equivalent
to a motion in the function space over T <1>0, by virtue
of the correspondence between T <1>0 and M, making the
assumption of comparability of the measures involved,
as in the foregoing. Thus are obtained two one
parameter groups of operators in the Hilbert space of
square-integrable holomorphic functions over T <1>0. One
of these is mathematically rather well defined, arising
from the linear action of Go on T <1>0, and has as its
generator the so-called "free-field" energy (relative
to CPo). The other is only formally defined, and in fact
the available evidence rather strongly indicates that
it lacks rigorous existence, arising from the nonlinear
action of Go on M, and.having as generator the "total"
energy. In a formal way the S operator is thereby well
defined by the conventional limit. This analysis applies
both to renormalizable and nonrenormalizable theories;
whether any useful numerical results can be obtained
by a maneuver based on the use of partially empirical
2J L. van Hove, Physica 21, 901 (1955). considerations depends of course, as in renormalization
theory, on the equation and the ingenuity of the
maneuver.
In the case CPo=O, where the background field
vanishes, Go includes the Lorentz group when the
defining partial differential equation is Lorentz
invariant and involves only real masses, and the fore
going paragraph applies then not only to translations
in time but to the entire Lorentz group.
In simple conventional terms the foregoing indicates
the following prescription for the separation of a total
Lagrangian into "free-field" and "interaction" parts.
The free-field constituent is the Lagrangian for the
hyperbolic partial differential equation defining the
first-order variation, in the vicinity of the vanishing
field, to the manifold of all classical wave functions for
the total Lagrangian.
C. Models
The short-lived character of the many attempts to
classify in a systematic and economical way elementary
particles on the basis of the Lorentz and conventional
space-time, with or without an independent internal
symmetry group, indicates that a broader attack, on a
physically more conservative and theoretically more
radical basis, would be desirable. One logical approach
is that contemplating the use of alternative symmetry
groups and/or space-time manifolds. However, if this
is to have a reasonably clear-cut physical interpretation,
it must be based on an adequately general field theory.
The present theory, while extensively heuristic, is
quite independent of the assumption that fields must
be described by nonlinear partial differential equations
in space-time, or, in fact, of the physical existence of
quantum fields at all. Any infinite-dimensional phase
manifold may be used as a basis, and other types of
examples of such manifolds having symmetry groups
of the proper orders of magnitudes are easily given.
An example is the set of all smooth maps from a
measure space into a finite-dimensional Hilbert mani
fold, a natural generalization of the much used linear
function spaces of smooth square-integrable functions.
The fundamental symmetry group Go will be that
leaving a designated point CPo of the basic manifold
invariant. The primary elementary particles of the
theory are then represented by the vectors in the
irreducibly invariant subspaces of T <1>0 under the
naturally induced action of G. Group-theoretical
quantum numbers will then be definable in the fashion
indicated earlier. For example, the elementary particle
models described in footnote reference 5 set up certain
representations and quantum numbers for symmetry
groups G having the group constituted by the Lorentz
group together with space-time position coordinates
asa degenerate limiting case. From the present stand
point this means that G is a subgroup of the isotropy
group leaving fixed the vanishing field; and the stated
Downloaded 08 Jul 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions488 1. E. SEGAL
action is the natural linear action of G on the infinitesi
mal classical fields. The description of the manifold M
is necessarily a good deal more complicated than this,
and is not required in the first instance for particle
classification purposes.
It ought to be noted that the manifold M can in
principle be built up from the knowledge of the tangent
space in the vicinity of each background field. It thus
has a certain conceptual quasi-empirical existence.
Insofar as a relatively arbitrary classical background
field can be experimentally maintained, and the re
sponse of the system to relatively arbitrary small
disturbances ascertained, these tangent spaces are
experimentally approximable to an arbitrary degree
of accuracy, quite without any ad hoc assumptions as
to the particle interpretation of the field,the necessity
of a basic partial differential equation, etc. The struc
ture of the tangent space at the vanishing physical
field is of great interest in itself, being the basis for the
classification of "free" particles; conversely, any
empirical linear description of the free particles can
be regarded as an approximate description of this
tangent plane. A given type of conventional quantum
field will in general have no direct empirical classical
analog, but this may be ascribed to a lack of closure
on the part of the corresponding theory. Ultimately
all measurements are reducible to classical ones, and the classical analog to the field of aU elementary
particles may be considered to be the set of all classical
fields, in speculative theory constructible as a manifold
through the examination of the response to all possible
small classical disturbances of an arbitrary background
classical field. There appears to be no practical possi
bility of setting up a useful empirical manifold M in
this fashion, but the foregoing conceptual experimenta
tion serves at least to indicate that the manifold M
has a certain fairly direct intuitional connection with
physical experience that is lacking in the Lagrangian.
The quantum, as contrasted with the classical, field,
plays primarily a formal part in our analysis, and
serves mainly only to connect the present formulation
with the conventional one. Quite without its use the
total energy of the field and the vacuum state, e.g., are
well-defined (through the use of principle III). In
view of the apparently inevitably dubiously physical
character of the quantum field, the possibility that it
may well be theoretically expendable is not very
surprising.
ACKNOWLEDGMENTS
We are much indebted to the following mathemati
cians and physicists for informative and stimulating
conversations: S. Helgason, D. Shale, W. F. Stinespring;
K. Gottfried, W. Heisenberg, G. Kallen, L. Rosenfeld.
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1.1722615.pdf | Addendum: Evaporation of Impurities from Semiconductors
Kurt Lehovec, Kurt Schoeni, and Rainer Zuleeg
Citation: Journal of Applied Physics 28, 1216 (1957); doi: 10.1063/1.1722615
View online: http://dx.doi.org/10.1063/1.1722615
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Diffusion of Impurities into Evaporating Silicon
J. Appl. Phys. 30, 259 (1959); 10.1063/1.1735142
Evaporation of Impurities from Semiconductors
J. Appl. Phys. 28, 420 (1957); 10.1063/1.1722765
Addendum
J. Acoust. Soc. Am. 20, 549 (1948); 10.1121/1.1906409
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IP: 130.18.123.11 On: Thu, 18 Dec 2014 16:26:571216 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)
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IP: 130.18.123.11 On: Thu, 18 Dec 2014 16:26:57LETTERS TO THE EDITOR 1217
where q= (eEo/kT,)t. For semiconductors q=O; hence, C=I, and
(7a)
We finally write
N=nod, (8)
where d is an equivalent depth.t Substituting into (3) yields
T=C-'(211"m/kT,)18d[(E,+E h)/El'o] exp(ex/kT,). (9)
We shall apply this to two cases. As a first example consider a
heavy metal having 8=1.5 at El'o=500 v. Assuming d=100A,
(kT,/e)=2 v, Eo=EJ=5 v, x=10 v, and Eh=5 vi; we have
C-'=0.17, E,=7.5 v, and T=3.3XlO-H sec. As a second example
consider an insulator having 8=10 at Epo=500 v. Assuming
d=100A, (kT,/e)=2 v, (ex/kT,)«1 and Eh=15 v:j:; we have
E,=3 v and T=2.5XlO-14 sec.
The estimated values of T are not very accurate; they indicate,
however, that it will be difficult to account for a time constant of
secondary emission that is larger than 10-13 sec, unless other forms
of energy storage (trapped electrons, exciton generation) are
important. The best experimental evidence indicates that T is
indeed very small.
* Work supported by U. S. Signal Corps Contract.
t If Xo is the range of the primaries, then nex) should increase with
increasing x for x <xo, because the rate of production of secondaries increas('s
toward the end of the range. whereas n (x) gradually decreases to zero for
x >xo. If Xl is the escape depth of the secondaries, then d~(XO+Xl). t In an insulator Eh should be larger than the gap width between the
bottom of the conduction band and the top of the filled band. In a metal
Eh should be smaller than the gap width, since the holes generated in the
conduction band have a negative energy. Assuming a gap width of 10 v
in either case, the two estimated \-alues of Eh seem quite reasonable.
Phenomena Associated with Detonation in
Large Single Crystals*
T. E. HOLLAND,t A. \Y. CA"PBELL, AKD M. E. MALI"+
University of Calijornt"a, ["os Alamos Scientt"jic T,aboratory,
Los Alamos, New Mexico
(Received June 19. 1957)
VERY little information is recorded in the literature concern
ing the detonation behavior of large single crystals of
explosive compounds. The opinion has been expressed that it may
not be possible to produce stable detonation in such media, since
the compressional heating at the shock front (in the absence of
air-filled voids or lattice defects) may be too low to provide a
reaction rate sufficient for detonation. Experimental support for
this view is found in the well-known facts that pressed explosive
is made harder to initiate by pressing to higher density; and that
TNT castings are harder to initiate and show larger failure
diameters as the crystal size is increased. On the other hand, it is
known that the primary explosive, lead azide, when prepared
in the form of large crystals detonates very easily. In this note we
report our observations on single crystals of PETN.
A measure of the sensitivity of large crystals of PETN relative
to powdered PETN was obtained by the use of a rifle hullet test.
Crystal specimens were mounted on plywood with the minimum
dimension of the crystal parallel to the path of the bullet;
powdered specimens were prepared by spreading a uniform layer
upon a cardboard support and covering the layer with a thin
cellophane sheet. When subjected to the impact of a soft-nosed
hullet traveling at approximately 4000 It/sec, crystals with a
minimum dimension of I} in. failed to detonate, but detonated
reliably when this dimension was increased to 1 ~ in. whereas the
powdered material detonated reproducibly in layers as thin as
0.092 in.
Evidence that single crystals can be detonated at full velocity
was obtained from charges arranged as diagramed in Fig. 1. A
plane detonation wave was generated in a 2-in. thick piece of
Composition B. This pressure wave was attenuated by passage
through a I-in. steel plate and used to initiate a crystal of PETN.
The latter was essentially a 45°-90°-45° right-angled prism made
by passing a plane through a cube of PETN three-quarters of an RW GENERI\TOR
COfPCSITION B RETN
CRYSTAL
FIG. 1. Smear camera record showing three distinct velocity regimes in
an uncterinitiaterl. PETN crystal.
inch on a side. In order to brighten the firing trace, the slant face
01 the prism ,vas covered with a Lucite plate so as to form a small
air-gap.
At the right in Fig. 1 is shown the firing trace with the PETN
crystal sketched in to give a corresponding space scale. Time zero
lies slightly to the left of the left edge of the print of the firing
record. In region I low-order detonation is seen. The rate of
detonation is estimated to be 5560 m/sec. The detonation rate
changes abruptly to an estimated value 01 10450 m/sec in region
II, accompanied by observable radiation in the interior of the
crystal. There is a final, apparently steady, detonation rate
established in region III with a value of 8280 m/sec. Finally, in
region IV, the detonation wave emerges from the top of the crystal.
Efforts were made to mea-sure the single-crystal failure diameter
using rods of PETN ground from single crystals. These efforts are
as yet incomplete, but show that the failure diameter is greater
than 0.33 in.
Failure of the detonation process takes place through the action
of "dark waves'" originating at the periphery of the detonation
wave. In a typical experiment the charge was a rod of PETN
0.252 in. in diam by 0.438 in. long. Beginning at the boostered end,
the rod was encased with brass foil for a distance of 0.287 in. The
foil served to prevent the occurrence of dark waves in the first
part of the stick. When the detonation wave passed the foil, it was
choked-off by dark waves. The latter waves are believed to be
hydrodynamic rarefactions characteristic of detonation in homoge
neous explosives.
* \Vork done under the auspices of the U. S. Atomic Energy Commission. t The George \Vashington University Research Laboratory. Camp
Detrick. Frederick. Maryland.
::: Advanced Development Di\'iRion, .\vco Manufacturing Corporation.
Stratford, Connecticut.
1 Campbell, Holland, Malin, and Cotter. Nature 178,38 (1956),
Growth of Tellurium Single Crystals by the
Czochralski Method
T . .T. DAVIES
Il(l1J('Y'l1'ell Research Center, Hopkins, Alinnesota
(Received June 3, 1957)
SEVERAL Te single crystals have been grown reproducibly by
the Czochralski technique. Although insufficient experimental
data are available to establish optimum growing conditions, any
future improvements would probably be of minor significance.
The important consideration at this time is that single Te crystals
have been obtained by seed dipping. To the author's knowledge
this has been reported only once before, by J. Weidel' in Germany.
Molten Te when allowed to cool slowly tends to freeze into
single crystals along the c axis of the hexagonal structure. Due to
the presence of bubbles and the polycrystalline nature of a free
frozen ingot, these crystals are quite limited in size and quality,
but do provide an initial source of seeds. Cleavage is easily
accomplished because the valence binding energy between atoms
along the spiral chains in the c direction is much stronger than the
binding energy between chains.' One indication of crystal quality
is the degree of perfection of the resultant cleaved planes.
In the vertical pulling process Te purified by vacuum distil-
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1.1743298.pdf | Thermoelectric Behavior of Solid Particulate Systems. Nickel Oxide
G. Parravano and C. A. Domenicali
Citation: The Journal of Chemical Physics 26, 359 (1957); doi: 10.1063/1.1743298
View online: http://dx.doi.org/10.1063/1.1743298
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130.113.111.210 On: Sun, 21 Dec 2014 08:32:04THE JOURNAL OF CHEMICAL PHYSICS VOLUME 26. NUMBER 2 FEBRUARY. 1957
Thermoelectric Behavior of Solid Particulate Systems. Nickel Oxide
G. PARRAVANO* AND C. A. DOMENICALIt
The Franklin Institute Laboratories for Research and Development, Philadelphia 3, Pennsylvania
(Received April 19, 1956)
Thermoelectric power measurements have been performed on powdered nickel oxide in the temperature
range 60-220°C, under different gas atmospheres. These include: oxygen, hydrogen, carbon monoxide,
carbon dioxide, nitrous oxide, helium, and water vapor, at different partial pressures. The extent and
direction of the observed changes in thermoelectric power of the oxide following gas chemisorption have
been related to the extent and nature of the electron transfer process taking place between the different
gaseous molecules and the conducting surface. A theoretical analysis of the system is presented. The analysis
shows how the ratio between the thickness of the space charge layer at the surface and the "thickness"
of the thermal gradient affects the thermoelectric power change resulting from chemisorption. Since this
ratio depends on the size and size distribution of the solid particles, this effect provides a further parameter
which can be used to control and modify the electronic characteristics of semiconducting particulate systems.
IT has long been known that unusual physicochemical
properties are often associated with solid particulate
systems. The large surface/volume ratio of these solids
has been held responsible for their high energy content.
A thermodynamic treatment of the energy effects in
these systems has been developed along lines similar
to those followed in the case of liquid droplets.l En
hanced chemical reactivity of finely divided solids
possessing a defective structure can be related to large
stoichiometric deviations of the surface phase. In this
case, concentration gradient may be set up throughout
the solid particle. This concentration gradient gives
rise to an electrical double layer, whose thickness is
dependent upon the particle size. As the size decreases
and becomes comparable to that of the electrical double
layer, the electrochemistry of defective structures in
particulate form may differ appreciably from that of
the bulk phase. It seems therefore interesting to in
vestigate the effect of stoichiometric deviations on the
electrical properties of finely divided solids.
Because of the well-known difficulties associated with
measurements of transport quantities of particulate
systems, we have chosen to investigate the thermoelec
tric power Q of finely divided metal oxides as a function
of their chemical composition under conditions such
that Q can be considered a thermostatic quantity. As a
method for producing stoichiometric deviations in
these compounds, we have used gas chemisorption,
because of the large amount of chemical data already
available on the adsorption of gases on powdered solids.
In particular, we have chosen to study nickel oxide,
because previous work2•3 has shown that the thermo
electric power of single crystal or sintered or powdered
nickel oxide can be simply and directly related to its
electronic chemical potential and, consequently, to its
* Present address: Department of Chemical Engineering, Uni
versity of Notre Dame, Notre Dame, Indiana. t Present address: Honeywell Research Center, Hopkins,
Minnesota.
1 R. Fricke, Angew Chern. 51, 863 (1938).
2 F. J. Morin, Phys. Rev. 93, 1199 (1954).
3 G. Parravano, J. Chern. Phys. 23, 5 (1955). chemical composition. There already exist data on the
effect of oxygen adsorption on the thermoelectric
power of nickel oxide.4 However, these studies were
performed on sintered specimens and were confined to
relatively high temperatures (>500°C), where bulk
diffusion readily occurs.
A simple analysis of our system shows that the change
in apparent Q for a given change in carrier concentration
depends on particle size. This provides an added param
eter which can be used in controlling the electrical
characteristics of these systems. Thus, the results ob
tained lead to interesting implications for different
phenomena in the general area of the physical chemistry
of solid surfaces.
EXPERIMENTAL
Materials
Nickel oxide was obtained by dissolving nickel metal
(Johnson, Matthey & Company) in reagent nitric acid
(Merck). The resulting nitrate solution was evaporated
to dryness, slowly decomposed at 35G-500°C in a
silica crucible, and finally fired in a nickel crucible at
1100°C for three hours in air. The nickel crucible was
made of pure nickel sheet (A. H. Thomas). The high
temperature treatment was performed in a vertical
furnace and the nickel crucible was placed at the
closed bottom of a tubing of Vycor glass. This arrange
ment enabled rapid quenching of the sample in water
after heat treatment. The oxide formed was pale green
in color. Pellets of this material were made at room
temperature in a specially constructed split mold.
After a few trials, a method was devised to prepare
mechanically strong pellets without the need for
additional heating.
Helium was obtained from a commercial tank and
was purified by allowing it to diffuse through heated
copper oxide, calcium chloride, hot copper turnings,
ascarite, magnesium perchlorate, and, finally, through a
charcoal trap immersed in liquid nitrogen.
4 R. W. Wright and J. P. Andrews, Proc. Phys. Soc. (London)
62A,446 (1949); c. A. Hogarth, ibid. 64B, 691 (1951).
359
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;, ........ ; .. ", ••
FIG. 1. Electron photomicrograph of NiO (29000 X).
Nitrogen, from a commercial tank, was purified in
the same way as helium, except that the copper oxide
and charcoal trap were removed.
Oxygen for static runs was prepared as needed by
thermal decomposition of cp potassium permanganate,
and was dried over magnesium perchlorate. For flow
experiments, oxygen was obtained from an electrolytic
cell and purified by passing it over a Pd-A1203 catalyst
and magnesium perchlorate.
Hydrogen was obtained from an electrolytic cell
and purified in a similar way as oxygen. Two electrolytic
cells (20% NaOH) were used to obtain O2+ Hz mixtures
of different compositions. Nitrous oxide and carbon
dioxide, from commercial tanks, were distilled in
< •.. ~
FIG. 2. Electron photomicrograph of NiO (4400 X). vacuum, the middle fraction being taken and dried
over magnesium perchlorate. In flow experiments,
nitrous oxide was purified by passing it over hot copper,
calcium chloride, and magnesium perchlorate. Carbon
monoxide, from a commercial tank, was purified by
passing it over hot copper, ascarite, and magnesium
perchlorate.
Method
Surface area measurements on nickel oxide were
performed using nitrogen adsorption at liquid nitrogen
temperatures. The results were plotted according to
the BET theory. The surface area found was 1.41 m2/g.
Typical electron photomicrographs of the oxide used
in this investigation are presented in Fig. 1 and Fig. 2.5
From micrographs at 4400 X magnification, the particle
size distribution of the oxide was obtained. The particle
size distribution gives the normal probability distribu
tion curve when plotted on a logarithmic scale (Fig. 3),
>u z w
::;)
o w a::
"-30
20
10
o I I
-f--1.41 m2/g
/-\ SURFACE AREA
/ \ / i\
~ ~ v. - y -
2 4 6 8 10 20 40
Average Particle Diameter (microns Xl 0-1)
FIG. 3. Size distribution curve for NiO particles.
but a rather skewed curve when plotted on a linear
scale. This size distribution effect, which was also
found in previous work,6 is probably a result of sintering
of the oxide powder. Assuming a spherical shape of the
oxide particles, and a density for the particles equal
to the bulk density of nickel oxide, the average particle
diameter d is given by
d=6/Ap=5.7X103 AO
where A is the surface area and p the density of the
solid. This value is in excellent agreement with the
value determined electromicroscopicaUy (Fig. 3).
Oxygen chemisorption experiments were made by
means of standard manometric techniques. Owing to
the relatively low pressures and small volumes of gas
employed, these two quantities were measured in a
McLeod gauge, whose calibrated capillary tip served
Ii We wish to thank Dr. R. S. Smith of these laboratories for
taking the electron photomicrographs of nickel oxide.
6 R. L. Farrar and H. A. Smith, J. Phys. Chern. 59, 763 (1955).
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130.113.111.210 On: Sun, 21 Dec 2014 08:32:04THERMOELECTRIC PROPERTIES OF NICKEL OXIDE 361
as a gas burette. The adsorption vessel containing
nickel oxide was fitted into a hole of an aluminum block
which was heated with a Nichrome coil. The tempera
ture of the block was controlled with a Celectray unit.
Oxygen chemisorption was studied under experimental
conditions most similar to those of thermoelectric
experiments. The amount of oxygen adsorbed per cc
of bulk nickel oxide was computed by assuming that
each oxygen atom was adsorbed on a surface Ni+2. The
surface density of Ni+2 was computed as i power of the
volume density, or (5.65X 1022)l= 10.73 X 1014 ions/cm2•
Thermoelectric power measurements were made
using the arrangement shown in Fig. 4. The nickel
oxide pellet was clamped between two platinum elec
trodes in the form of disks (20 mils thickness) by means
TO VACUUM AND
GAS SYSTEM
GLASS CAP
Pt-Pt Rh
THERMOCOUPLES
NiO SAMPLE
CERAMIC SUPPORT--+-Hil
STAINLESS STEEL
SPRINGS Pt FOIL
NICHROME COIL
GLASS RODS
NONEX RING
WAX ----If=t::t±±y-- GROUND GLASS
JOINT
FIG. 4. Thermoelectric celL
of stainless steel springs and glass rods. A Pt-PtRh
thermocouple was welded on each electrode on the side
opposite the nickel oxide. The electrode assembly was
supported by a porcelain rod (i in. in diameter),
fastened to the inner part of a glass joint by means of wax
(Spectro-Vac High vacuum). A heating coil was wound
around the top end of the porcelain rod. The electrode
assembly was covered with a Pyrex container, which
had a standard glass joint at the bottom and a tubula
tion for high vacuum connection at the top. In flow
experiments, an additional tubulation (not shown in
Fig. 1) was provided at the bottom of the container to
enable the gas to flow past the nickel oxide pellet and
escape from the top. Electrodes and thermocouples were
annealed before use, to relieve cold-work stresses. The
system was tested before use, by putting around the TIME (min)
0 100 200 300 400
430 I I I
0 hAC
'" o~ 0 ~ ., 420 C-/ '0 '- 0 > I "l Nio -H2
~ 410 -
PH2 = 3.5 X 10-1 mmHg
o t=14 0c
400~-L-----------~~
FIG. 5. Effect of Hz chemisorption on the
thermoelectric power of NiO.
electrodes and the nickel oxide pellet a jacket with
condensing steam. The thermocouples showed a
difference of less than 1 J.l.V, and no thermal emf~could
be detected across the nickel oxide pellet.
Gas pressures were measured with a calibrated
McLeod gauge, and corrections due to the thermo
molecular effect applied, when necessary.
Emf's were measured with a Leeds and Northrup:type
K2 potentiometer (NiO-Pt couples) and a Leeds and
Northrup Wenner potentiometer (Pt-PtRh couples).
A value of t:.T=3 to 5°C was used. No variations in
thermoelectric power were observed using different
values of t:.T. The observed thermoelectric power Q
is related to the Fermi level J.I. of nickel oxide by means
of the relationship: QT= J.I.+ A, where A is a constant.
Furthermore, J.I. is a function of the hole concentration,
nh, through the equation
nh=N exp(-J.l./kT)
where N is the total level density and k the Boltzmann
constant. As has been shown previously,2 these relation
ships imply a conduction mechanism involving d elec
tron migration in localized Ni+2 levels of the oxide.
RESULTS
Static Experiments
The investigation was first directed to explore the
effect of the adsorption of the different gases on the sign
and extent of the change in thermoelectric power of
the initial sample. Before adsorption, the nickel~oxide
FIG. 6. Effect of CO
chemisorption on the
thermoelectric power of
NiO. Nio-co
Pco = 4.0 X 10-1 mmHg
t = 11°C
500,-----.----,----,
'" ...,,0 .. ./0
~490 0 o~ 3 ~ o/"
o
100 200 300
TIME (min)
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130.113.111.210 On: Sun, 21 Dec 2014 08:32:04362 G. PARRAVANO AND C. A. DOMENICALI
'" ., 580
~
>
~ 5S0
0
540
0 NiO-02
P02 = 4.05 X 10-1 mmHg
t :: "5°C
o --4.._- 0
50 100 150 200
TIME (min)
FIG. 7. Effect of 0, chemisorption on the
thermoelectric power of NiO.
specimen"was evacuated under a defined set of experi
mental conditions, the purified gas admitted, and the
pressure read on the McLeod manometer. The change
In thermoelectric power of the sample was then followed
as a function of time. As expected, hydrogen and carbon
m~noxide acted as electron donors upon chemisorption
(FIgS. 5 and 6) thus increasing the thermoelectric
power of p-type nickel oxide (see Discussion below).
An opposite change in thermoelectric power ensued
upon chemisorption of oxygen, carbon dioxide, and
nItrous oxide (Figs. 7, 8, and 9) which, therefore, acted
as electron acceptors. The data show that different
pr.e~reatments of the nickel oxide sample yield different
InItIal values for the thermoelectric power. The relation
ship between Q and the nature of the gas adsorbed was
further established by surrounding nickel oxide with
helium. No change in Q was apparent in this case even
after long periods of time (Fig. 10). The data also' show
that opposite effects set in upon desorption (Fig. 5).
Flow Experiments
The effect of the adsorption of oxygen and hydrogen
on the thermoelectric power of nickel oxide was further
investigated in a series of flow experiments using
helium as carrier gas. The rate of change of Q upon
hydrogen chemisorption was first measured under
conditions duplicating those of a static run. Results
similar to those obtained under static experiments were
obtained (Fig. 11). We further studied the effect on Q
of the combined adsorption of oxygen and hydrogen
515 .----,r-------~
'" ., 510
~
> 3 o 505 NiO-COz
PCOz = 4.05 X 10-1 mmHg
t = 7SoC
o
~-----o
500+---~-_+--4_-~
o 100 200 300 400
TIME (min)
FIG. 8. Effect of CO2 chemisorption on the
thermoelectric power of NiO. 500
1 I
~ 495-0-
"t:>
~ I~ ,
-~490 -~o
I ___ +-'~I~----_..\:~o:r==~==~ 485-+ o 200 400 SOO 800
TIME (min) 1000 1200
FIG. 9. Effect of N20 chemisorption on the
thermoelectric power of NiO.
at"various temperatures. Results are presented in Fig.
12 where the "equilibrium" values of the thermoelectric
power of nickel oxide are recorded as a function of the
composition of the gas phase. By "equilibrium" value
of thermoelectric power is meant the value of Q com
puted after leaving a mixture of H2+02 of constant
composition flowing, at constant rate, for three or
more days past the nickel oxide sample. This "equilib-
. " I II h num va ue was genera y reac ed, at the temperatures
of investigations, in a matter of a few hours. There
w~s one notable exception to this behavior. A gas
mIxture composed of H2+ He (pH2= 1.3 mm Hg) pro
du.ced, at 115°e, a small rise in the value of Q of nickel
oXlde even after three days of operation. Obviously,
there was a continuous reduction of the sample. But,
apart from this case, the value of Q could be easily
cycled back and forth using different PH2/P02 ratios.
In Fig. 13, the effect of PH2, at p02=O.8 mm Hg and
1.6 mm Hg, 115°e, on the thermopower is presented.
In Figs. 14 and 15, the effect of temperature and PH2
at constant P02 is shown. In an effort to determine
whether water vapor had any effect on the thermo
electric power of nickel oxide, some runs were performed
in which a definite PH20 was added to the carrier gas,
but the results so far have been inconclusive. Thus,
at 600e and PH20=8 mm Hg, Q was found to increase
(531---7545 IN/degree), but at 202°e, PH20=8 mm Hg,
Q showed a slight decrease (56.7---748.9 IN/degree).
DISCUSSION
We shall attempt to analyze the results of the re
ported experiments on the basis of the following simple
Nio -He
PHI = 197 mmHg
t = 750C
{480 t'----,;o:-----o _.;...0 ___ I
_470-+ ___ -+ __ -+ __ -+ __ -+ __ -+ __ 0~1
o 0 200 400 SOO 800 1000
TIME (min) 1200
FIG. 10. Thermoelectric power of NiO in a He atmosphere.
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130.113.111.210 On: Sun, 21 Dec 2014 08:32:04THERMOELECTRIC PROPERTIES OF NICKEL OXIDE 363
700 •
-
~ /.--
l _.
t = 115°C
~ PH2 = 30 mm Hg o 600
550 I I I I
o 100 200 300 400
TIME (mins)
FIG. 11. Effect of H, on the thermopower of NiO.
Flow experiment with He as a carrier gas.
assumptions. The process of chemisorption of an atom
on the surface of nickel oxide leads to an electron
exchange between the atom and the surface states;
electrical neutrality requires that an equivalent change
takes place in the hole concentration in the space
charge layer. An atom which removes (one or more)
electrons from the surface states thereby increases the
space-charge layer concentration of holes, and this
brings about a decrease in the absolute thermoelectric
power of the nickel oxide space-charge region. Such
an atom, of which oxygen, carbon dioxide, and nitrous
oxide are examples, is called an electron acceptor.
Conversely, an atom which contributes electrons to the
surface states and thereby reduces the hole concentra
tion and increases the thermoelectric power is called an
electron donor. Hydrogen and carbon monoxide behave
as donors on a nickel oxide surface. For the purpose
of the present discussion, there is no need to postulate
in more detail a scheme for the chemisorption process.
Discussions on this subject can be found elsewhere.7-9
TIME (days)
FIG. 12. Thermoelectric power cycle on NiO. Pressures in mm Hg.
7 W. H. Brattain and J. Bardeen, Bell System Tech. J. 32, 1
(1953).
8 J. Bardeen and S. R. Morrison, Physica 20, 873 (1954).
9 P. B. Weisz, J. Chern. Phys. 21, 1531 (1953); K. Hauffe and
H. J. Engell, Z. Elektrochem. 56, 366 (1952). '" '" " >
.3-
0 640
620
600 '--
580
560
540
520 -a If L
500 I
o .----0
/0/ VO-
-. •
I 0-po. = 0.8 mmHg
.-po. = 1.6mmHg
t = 115°C
I I 467
PH2 (mmHg) -
I
9 10
FIG. 13. Thermoelectric power isotherms of H, on NiO
in the presence of 0,. Effect of po,.
Because the Tammann temperature of nickel oxide is
approximately 2000oK, we assume that, at the tem
peratures of our investigations, there is no significant
diffusion of the ambient gas into the bulk of the nickel
oxide.
For a p-type semiconductor, the absolute thermo
electric power Q at temperature T is given10 by the
a pproxima tion
TQ=f.L+A (1)
where f.L is the chemical potential of a hole in the semi
conductor, or the energy of the Fermi level referred
to the energy level of the current-carrying holes, which
latter level we take to be that of the occupied d-states
of Ni+2. The constant A is in our case approximately
an order of magnitude smaller than TQ and is elimi
nated, by subtraction, from the analysis. We let QO
be the absolute thermoelectric power of nickel oxide
before chemisorption and Qc be the same quantity after
'" Q)
" ........ > 3
0 650
O-t=600c
i-.-t = 115°C ---P02 = 0.8 mm Hg v·---
600
-/
/
550 .1·
I ° 0 fO 0
°
500 I I I I
o 3 4 5 6 7
PH. (mmHg)
FIG. 14. Thermoelectric power isotherms of H2 on NiO
in the presence of 0,. Effect of temperature.
10 C. Herring, Phys. Rev. 96, 1163 (1954). 8
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130.113.111.210 On: Sun, 21 Dec 2014 08:32:04364 G. PARRAVANO AND C. A. DOMENICALI
100
90
80
0>
'" -0
~ 70
3-
0 60
50
40 I-P02 = 0.8 mm Hg
-t = 202°C V /
/' ./
V
V·
o 2 3 4
PH (mmHg)
2 5 •
6
FIG. 15. Thermoelectric power isotherm of H2 on
NiO in the presence of O2• 7
chemisorption. These quantities in general depend on
position within each "spherical" nickel oxide particle,
and the basis of our analysis is that the chemical
potential J.I. (and therefore the thermoelectric power)
changes in the space-charge layer upon chemisorption.
The space charge thickness may be small compared
with the particle radius or it may be of the same order
of magnitude as the radius; in the latter case we may
think of the particle as consisting almost entirely of
surface. Letting J.l.o be the space dependent chemical
potential within a particle before chemisorption and
J.l.c the same quantity after chemisorption, we can show
that
fX~L J.l.c dT
[Qceff-Qoeff].:lT= --dx
x=O T dx
fX=L J.l.o dT
---dx
X~O T dx (2)
where L is the length of the nickel oxide rod specimen
whose ends are at temperatures T and T+.:lT re
spectively, and Qceff-QDeff is the measured dijJer~ce in
absolute (or relative) thermoelectric power after and
before chemisorption, at temperature T+t~T. The
integration is in general necessary because the quanti
ties J.l.c and J.l.o vary with position.
If we assume that the temperature distribution within
the nickel oxide particles is not significantly changed
upon chemisorption, the derivatives dT / dx would be
the same function of x in both integrals in Eq. (2) and
we can then write
fX~L (J.l.c-I-tO)dT Qceff-Qoeff= 1/.:lT ---dx.
x=o T dx (3)
The hole densities (nh)O before chemisorption and
(nh)c after chemisorption are related to the respective
chemical potentials by the Boltzmann expressions in which N is the number of NiH ions/cc of the crystal.
Substitution of Eq. (4) into Eq. (3) gives
(5)
Since the material studied is in powder form, it is
clear that the temperature gradient within the specimen
is not uniform. If we imagine the particles to be roughly
spherical in shape the temperature distribution in a
collinear row of spheres will appear as in Fig. 16.
The solid curves inside the spheres of Fig. 16(a) repre
sent isothermal surfaces and the dashed curves represent
lines of heat flow, these two families of surfaces being
mutually perpendicular at each point within a spherical
particle. Figure 16(b) indicates qualitatively the tem
perature variation along a line connecting the centers
of a collinear "string" of particles. M is the number of
particles along a line from the end at temperature T to
the end at temperature T+.:lT. Figure 16(c) shows
qualitatively the variation of temperature gradient.
Of course, the nickel oxide particles are not as neatly
packed as to form a regular pattern, but the results
should not be very sensitive to the exact packing
arrangement. Furthermore, even at low temperatures
there will probably be some sintering of the nickel oxide
particles; this effect will not greatly alter our results
except in those situations where the space-charge layer
is very much less than the particle radius. Finally,
although there will be a distribution of particle sizes
(Fig. 3), we shall further simplify the problem by
trea ting the particles as if they were all identical spheres.
We can consider two possibilities as sketched in Fig.
17. The first two sketches, Fig. 17(a) and (b) show
qualitatively the spacial dependence of dT / dx and of
v 11+1
(a)
(b)
(e) !!I t d, V+2 V+3···M
I
I
I
I
I
I
I
I
I
I
I
I
I lUl1
-X
I I
I I
I I
I I
I i X
~-------M PARTICLES -----------.l
(4) FIG. 16. Temperature distribution in idealized spherical particles.
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130.113.111.210 On: Sun, 21 Dec 2014 08:32:04THERMOELECTRIC PROPERTIES OF NICKEL OXIDE 365
the logarithmic factor in the integral of Eq. (5). The
solid curve in Fig. 17(b) shows an example of a very
thin space charge layer while the dotted curve shows
an example of a thick space charge layer. In order to
derive an approximate and simple relationship between
the hole densities (nh)O and (nh)c, we imagine the factors
in the integrand of Eq. (5) to have the "idealized"
or simplified forms shown in Figs. 17(c) and (d). Our
two possibilities can now be expressed as 0 <w [Fig.
17(c)] and o~w [Fig. 17(d)] where 0 is the width of
the space charge region and w is the width of the tem
perature-gradient region. For the shapes of dT/dx and
In(nh)o/(nh)c shown in Figs. 17(c) and (d), the ex
pression (5) takes the forms
(nh)O t.T
QCeff-Qoeff=1/6Tkln--·--+M for o<w
(nh)c w·M
Solving for (nh)c we find the two relations (6)
(7a)
(7b)
If we can measure (nh)c manometrically for a large
range of oxygen partial pressures, for example, and can
measure the corresponding thermoelectric power (at
gas-surface equilibrium), it should be possible to
evaluate the ratio w/o= (thermal gradient thickness)/
X (space layer thickness). This was done by using
oxygen chemisorption data obtained at 115°C, and
fitting the data in a plot of 6Q=Qceff-Qoeff vs log. Oads.
A qualitative agreement was obtained, and enabled to
set a lower limit of about 600 A for the value of 0.
A brief report of similar work on the thermoelectric
behavior of aggregates of 60X 104 A germanium particles
exposed to oxygen and water vapor has recently been
given.l1 In this investigation, it was found that when
the ambient gas was changed from oxygen to water
vapor, the thermoelectric power of n-type germanium
powder "swings from positive to negative over a range
as large as two to three hundred microvolts per degree."
Germanium particles used were roughly one hundred
times larger than the nickel oxide particles used in the
present work. In reference 11, it is mentioned that the
effective thickness of the "constriction regions" (our
thickness w) is comparable to the thickness of the
space-charge layer 0. Germanium particles were packed
under small compressive loads and presumably have
not been heated, so as to prevent sintering. Our Eq. 7(b)
should apply to these results.
11 E. A. Kmetko, Phys. Rev. 99, 1642(A) (1955). FIG. 17. Simplified form of the spacial dependence of
dT/dx and In[(nh)o/(nh)c.]
Since we have the possibility of controlling, at least
qualitatively, both 0 and w, conditions can be set up
such as to use Eq. 7(b) which does not require the exact
knowledge of 0 and w. It is then possible to relate varia
tions of thermoelectric power of finely divided nickel
oxide directly with changes in hole population brought
about by deviations of surface stoichiometry. Thus,
under proper experimental conditions, the present
method can be useful in supplying information on
electron transfer processes, which follow chemical inter
actions between solid surfaces and surrounding phases.
In this instance, special importance attaches to gas
chemisorption processes. These processes have been
the subject of a large number of investigations by
means of measurements of changes in the electrical
conductivity of thin oxide films brought about by
chemisorption of different gases. This technique, how
ever, cannot be easily applied to powdered or sintered
polycrystalline oxide specimens, without making as
sumptions which cannot be justified on physical
grounds.
The data reported have already interesting implica
tions on the activity of nickel oxide powders to catalyze
the reaction:
(8)
It is known that the heat of adsorption of oxygen on
nickel oxide is larger than that of hydrogen, depending
upon surface pretreatment. One should therefore expect
that during reaction (8), at low temperatures, the oxide
surface will be mainly covered with oxygen. This situa
tion is brought out clearly by thermoelectric measure
ments.
Thus, from the data on the combined effects of PH2
and P02 on Q (Figs. 13, 14, 15, and 16), it can be seen
that at low temperatures « 115°C) oxygen tends to
displace hydrogen from the surface. Furthermore, at
the lowest temperature investigated (60°C) and in the
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range of PH2 and P02 used, the data indicate that oxygen
completely covers the oxide surface. These results
point out that the steady-state coverage of the nickel
oxide surface while catalyzing reaction (8) consists
mainly of chemisorbed oxygen species. The amount
of coverage with oxygen varies with temperature, de
creasing as the temperature increases. These deductions
make it possible to suggest a mechanism for reaction
(8) on nickel oxide. It can be readily seen that a
steady-state surface, covered mostly with oxygen,
obtains if it is assumed that the slow step in the kinetic
sequence of reaction (8) consists in the reduction of
oxygenated surface by means of hydrogen molecules
from the gas phase:
H2+ Ni -O~H20+ Ni
where Ni-O represents a surface site covered with
oxygen. Upon assuming some kind of oxygen adsorption
isotherm on nickel oxide, a rate equation for reaction
(8) can be deduced. The choice of a Langmuir type
isotherm produces a rate expression very similar to the
expression derived directly from kinetic data.12
CONCLUSIONS
By means of a theoretical analysis of thermoelectric
effects in solid particulate systems, it has been possible
12 R. P. DonelIy, J. Chern. Soc. 132, 2438 (1929). to show how measurements of thermoelectric power
of powdered nickel oxide can be related to stoichio
metric deviations occurring at the oxide surface.
This analysis provides a method for describing the
nature and extent of surface coverage, during elec
tron transfer processes between solid surfaces and
surrounding phases. For example, the main features
of the catalytic synthesis of water on nickel oxide have
been determined thermoelectrically and found con
sistent with independent kinetic investigations. Further
more, the analysis of the thermoelectric behavior of
solid particulate systems shows that the extent of
thermoelectric power changes of semiconductors upon
electron transfer processes occurring on their surfaces,
are a function of the particle size. This effect suggests
interesting implications for different areas of the
physical chemistry of solid surfaces.
ACKNOWLEDGMENTS
This investigation was made possible by a grant
from the Gulf Research and Development Company
and the Esso Research and Engineering Company to
The Franklin Institute Laboratories for fundamental
research in the field of heterogeneous catalysis. This
support is gratefully acknowledged.
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1.1743012.pdf | Soft XRay Absorption Edges of Metal Ions in Complexes. II. Cu K Edge in Some
Cupric Complexes
F. Albert Cotton and Harold P. Hanson
Citation: The Journal of Chemical Physics 25, 619 (1956); doi: 10.1063/1.1743012
View online: http://dx.doi.org/10.1063/1.1743012
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130.113.86.233 On: Wed, 10 Dec 2014 04:44:41SOFT 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).
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130.113.86.233 On: Wed, 10 Dec 2014 04:44:41620 F. A. COTTON AND H. P. HANSON
The existence of electronic band structure in the
solid depends upon a repeating organization extending
over ranges which are very large compared with atomic
dimensions. One finds, however, in many cases that
the edge structure of a compound in solution is practic
ally identical with the edge structure of the same com
pound in the solid, crystalline state. In solutions,
beyond the first layers surrounding the ions, there is
little significant organization. Electronic bands, as
they exist in solids, are nonexistent under these condi
tions, and the introduction of the exciton concept does
not seem particularly appropriate.
The fact that the K-absorption edge structures of the
manganese ion in solution and in many solid manganese
salts do not differ in any essential fashion was pointed
out by Hanson and Beeman.s The edge structures of
the cations in solution have been determined for Ni++,
Cu++, Zn+,6 Fe++,1 and Mn++.5 The edges all follow a
pattern which is matched quite well in the solid salts
out to at least 20 v. As may be expected, this is partic
ularly true in highly hydrated salts. The unpublished
study by Shurman, referred to by Mitchell and Bee
man,8 showed that even the unique edge structure of
manganese in the permanganate was essentially the
same for both the solid and dissolved forms.
From such considerations we conclude that the edge
structure of an element in a chemical compound may
best be interpreted on a "molecular" basis in some cases.
Thus in species such as the permanganate and chromate
ions it would seem that the metal atom is so well in
sulated by the surrounding tetrahedron of oxygen atoms
that the effect of any more distant atoms on the struc
ture of the edge is negligible. However, in these and
other cases where the nearest neighbors are covalently
bound it would seem that only by considering the elec
tronic structure of the ion, radical, or molecule as a
whole on some sort of molecular orbital basis, can an
adequate interpretation of the edge structure be given.
But this of course poses a difficult problem since the
required knowledge of electronic structure and energy
levels in such complex species is not available. There is,
however, a second instance in which analysis on a
molecular basis seems not only justified but feasible.
In cases where the nearest neighbors of the metal
atom are bound to it largely by ion-ion or ion-dipole
forces, we have a tractable problem since a fairly de
tailed picture of the organization of the optical levels
of the metal atom can be obtained by the application
of the crystal field theory, in which the Stark perturba
tion of the metal orbitals is calculated to the first order.
Examples of such systems are metal ions in aqueous
solutions, crystalline hydrates, ammines, and similar
5 H. P. Hanson and W. W. Beeman, Phys. Rev. 76, 118 (1<;'49).
6 W. W. Beeman and J. A. Bearden, Phys. Rev. 61,455 (1942).
7 Unpublished work.
8 G. Mitchell and W. W. Beeman, J. Chern. Phys. 20, 1298
(1952). derivatives of tnetal ions, and perhaps also some simple
metal salts which are highly ionic in nature.
The aquo and ammine derivatives of Cu (II) were
chosen for this study for several reasons. In the first
place, it has already been shown9 that the optical spectra
of such complexes can be satisfactorily analyzed on the
basis of crystal field theory. Secondly, there was reason
to believe that the effects of varying the nature, partic
ularly the symmetry, of the coordination sphere, would
be quite noticeable. Beeman and Bearden6 had already
measured the edges of Cu(II) in pure water and in
aqueous ammonia and observed a profound difference
in appearance. We have confirmed their observation,
while paying careful attention to the composition of the
solution, so as to be quite certain that the species meas
ured was unequivocably Cu(NH3MHzO)z++. In the
third place, Cu(II) lends itself to this type of study
since x-ray diffraction studies have provided detailed
knowledge of the structures of aquo and ammine com
plexes with a variety of crystal field symmetries.
EXPERIMENTAL
Preparation of Samples
The aqueous cupric ion, Cu(H20)6++ was examined
as a 1M solution of cupric sulfate. The CuS04·5H 20
used was of analytical reagent grade.
An aqueous solution of the Cu(NH3)4(HzO)2++ ion
was prepared as follows: 17.1 g of Cu (N03)z· 3H20
and 39.0 g of NH4N03 were dissolved in 200 ml of
distilled water. To this solution was added 17.6 ml of
concentrated ammonium hydroxide (27% aqueous
solution, analytical reagent), and the resulting deep
blue solution diluted to a final volume of 250 ml.
According to the studies of J. BjerrumlO such a solution
at 300 contains essentially 100% of the copper (II)
in the tetraammine diaquo form.
The samples of Cu(NH3)4S04·H20 and Cu(NH3)4-
(N03)2 were obtained from Professor George Watt of
the University of Texas. The former was prepared by
the standard method, namely by adding an excess of
ammonia to an aqueous solution of CUS04 followed by
slow precipitation with alcohol. The nitrate was crystal
lized from a solution of CU(N03)2 in liquid ammonia.
The composition of both salts was checked by analysis
for copper and nitrogen. Cu en2(N03)z (en=ethylene
diamine) was prepared by adding the stoichiometric
quantity of 72.5% ethylene diamine to a saturated
aqueous solution of Cu(N03)z, followed by chilling to
50 overnight or longer. Beautiful purple crystals,
several millimeters in length, were obtained. These were
filtered off and washed several times with ice cold water.
The anhydrous compound has not previously been
reported.
• C. J. Ballhausen, Kg!. Danske Videnskab. Selskab. Mat.·fys.
Medd. 29, (1954), No. 14; Bjerrum, Ballhausen, and Jorgensen,
Acta Chern. Scand. 8,1275 (1954).
10 J. Bjerrum, Metal Ammine Formation in Aqueous Solution
(P. Haase and Son, Copenhagen, Denmark, 1941), p. 126.
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130.113.86.233 On: Wed, 10 Dec 2014 04:44:41SOFT X-RAY ABSORPTION EDGES. II. eu EDGE 621
Analysis (See Table I)
The bis-(DL-prolinate) dihydrate, (C4HaNCOzh
CU(H ZO)2, was prepared by addition of a solution
of DL-proline (nutritional Biochemicals Corporation,
Cleveland, Ohio) to an excess of freshly prepared
CuCOa• The resulting solution was filtered and slowly
evaporated whereby blue platelets were deposited.
Analyses for carbon, hydrogen, nitrogen, and copper
confirmed the formula given.
The CuClz·2H 20 used was of analytical reagent
grade. It was converted into the anhydrous chloride,
CuCh, by heating for several hours at 120°, and main
tained in the anhydrous form during measurement by a
coating of paraffin.
Measurements
A double crystal x-ray spectrometer was employed.
The crystals used were large, optically clear calcite
crystals. After etching, the (1, -1) width at the CuKx
line was slightly less than a volt. This width is greater
than that of really excellent crystals, but the resolution
is more than adequate for the problem at hand. The
data were taken with a commercial end-window Geiger
tube and recorded by a decade scaling unit.
The x-ray tube was a water cooled unit with a tung
sten target. The system was supplied by a 10 kva
stabilized ac source which was regulated to 0.1%. The
high voltage obtained from this source was adequately
stabilized. The x-ray tube current was stabilized by a
feedback unit employing thyratrons.
Measurements were made on salts and on solutions.
Samples of the solid compounds were made by spread
ing a thin uniform layer between two pieces of adhesive
cellulose tape. The solutions were studied in glass cells
formed by grinding holes in microscope slides. Thin mica
sheets served as windows. The solutions were admitted
through a slit cut in the glass.
Over the range of measurements, the tungsten target
gives an essentially constant intensity. During the
course of a run it was only necessary to check the 10
occasionally to ascertain its constancy. The curves
shown are in every case composites of several individual
runs. The statistical accuracy per point in the individual
runs was about 2%. This in itself is not particularly
meaningful since it must be considered in relation to the
contrast in absorption between the low-and high-energy
sides of the edge. In the curves presented, the statistical
Cu
C' Ha TABLE 1.
Theory for eu en,(NQ,J, Found
20.6
15.7
5.23 20.5
16.0
5.29
• These and other microanalyses for C. H. and N were carried out by
Schwartzkopf Microanalytical Laboratories, Long Island, or by Dr. S.
Nagy. Massachusetts Institute of Technology. FIG. 1. K-ab
sorption edges of
some cupric com
pounds. c .9
C.
~ .c « --- CuClz'ZHzO
CuCIZ (Anhyd.)
CUS04 '5HzO
error is small enough that the error in drawing repre
sentative curves for these samples is probably no larger
than the error involved in regarding these samples as
being truly uniform. Since curves obtained on different
samples reproduce one another in a satisfactory fashion,
one may be confident that these are truly representa
tive curves.
The corrections for background, second-order, and
counter dead time were not made since their effects on
the curves were negligible.
DISCUSSION
The copper(II) K edges of the compounds studied
are shown in Figs. 1 and 2. The energy scale is based on
the first inflection point of the metallic copper edge as
zero in the usual manner. In Part I the effect of crystal
fields in splitting the p band of a metal ion has been
described. We shall now determine whether or not
the structure of the 1st large absorption maximum
corresponding to the 1s-4p transition can be correlated
on the basis of this theory with the symmetry of the
crystal field which is known directly from crystal struc
ture studies or can be reliably assumed for each of these
compounds. The small maxima occurring at 7-9 v on
some of the edges will be discussed later.
The lower curve in Fig. 1 shows the edge for the
copper(II) ion in aqueous solution. The form and
position are just as previously reported.6 The main peak
at ,,-,16.5 v is smooth and nearly symmetric. There is
no evidence of splitting and the width is not unusual
for work in this energy region. Evidence based on the
optical spectrum of the aquo cupric ion has been pre
sented9 which indicates that the ion is surrounded by
six water molecules, four in the xy plane at equal dis
tances, and two on the z axis at slightly greater dis-
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130.113.86.233 On: Wed, 10 Dec 2014 04:44:41622 F. A. COTTON AND H. P. HANSON
t
FIG. 2. K-ab
sorption edges of
some cupric com
plexes of D4h or
lower symmetry.
tances. That is to say, the 1st coordination shell is a
very slightly distorted octahedron of six identical
ligands. It was shown in Part I that a field of octahedral
symmetry will not split the p level, so that this result
is in accord with theory. In Fig. 1 the edge for copper
(II) in solid CUS04' 5HzO is also shown. A complete
structural analysis of this compound by Beevers and
Lipsonll has shown that copper ions occupy two dif
ferent types of lattice site, in each of which the copper
is surrounded by four coplanar water molecules and,
on the z axis, by two oxygen atoms from the sulfate
ions. The six nearest neighbors are thus all oxygen
atoms in a somewhat distorted octahedron. Though
there appears to be no data bearing on just this point,
it is not unreasonable (vide infra) that the two oxygen
atoms from the sulfate tetrahedra would be approxi
mately equivalent electrostatically to the oxygen atoms
in the water dipoles. Thus Cu(II) ions in CUS04' 5HzO
are in an approximately octahedral crystal field so that
the observed peak, broadened but not actually split,
is in reasonable agreement with theory.
In Fig. 2 are the copper(II) edges for complexes of
lower than octahedral symmetry. In the Cu(NH 3)4++
ion, the octahedral coordination shell is completed by
water molecules along the z axis.9 For Cu(NHa)4(NOah
and Cu enz(NOah, there are no crystallographic data.
However, it is very reasonable to assume that the
copper (II) ions are coordinated in square planar
arrangement by four amine nitrogens, with the usual
sixfold coordination being completed by other species
(probably NOa-oxygens) along the z axis. An x-ray
11 C. A. Beevers and H. Lipson, Proc. Roy. Soc. (London)
A146, 570 (1934). study of Cu (NHg)4S04' H20 recently reported12 shows
square planar coordination by the ammonia molecules
at 2.05 A with water molecules along the x axis, one at
2.59 A and the other at 3.37 A.
It is thus clear that in each of these four cases the
symmetry of the crystal field is such that on the basis
of the theory developed in Part I, splitting of the 4p
level and hence of the ls-4p absorption band into two
is to be expected. As Fig. 2 shows this is just what occurs
in each case. Moreover, the observed splittings run
5-7. v, av~r~ging "'5.~ v. In Part I the expected p
orbltal sphttmg was estlmated to be of just this magni
tude.
The structure of copper(II) DL-prolinate dihydrate
has been determined by x-ray studiesY The Cu(II) ion
is octahedrally surrounded by two nitrogen atoms on
the x axis, two carboxyl oxygen atoms on the y axis,
~nd two wat.er molecules on the z axis. Thus rigorously,
III the notatlOn of Part I, Q:x;~Qy~Qz, and splitting of
the p level into three components should in principle
occur. However, experience with splittings in optical
spectra, particularly as summarized in the so-called
spectrochemical series of Fajans and Tsuchida,14
shows that splitting large enough to be clearly dis
cernible. with the relatively poor resolution of x-ray
work WIll only occur when the atoms on different axis
are different. Atoms of the same element differently
bound (e.g., 0 in H20 and in RCOO-) occur close to
gether in the spectrochemical series, and will not result
in such large splittings as those caused by the presence
of nitrogen coordinating ligands on one axis with oxygen
coordinating ligands on another. Thus the theory satis
~actorily expla.ins the observation that the ls-4p peak
In copper prolmate is apparently split into two rather
broad maxima.
T~e small maxima which occur on the steeply rising
portlOn of some of the edges may next be discussed.
It will be noted that this low-energy absorption occurs
prominently in all of the complexes so far discussed of
tetragonal or lower symmetry, not at all for the almost
perfectly octahedral Cu(H 20)6++ ion and slightly for
CuS04·5H 20 where the octahedral field is more
appreciably distorted. The intensity of this absorption
can be roughly estimated as about 5% of the total
ls-4p intensity, which indicates that the transition in
volved is nominally forbidden. It seems reasonable to
suppose that it is one of the even-even transitions
1s-3d or ls-4s. Since the spin quantum number s is
probably not a good quantum number here1f) we must
seek other criteria for deciding between these alterna
tives. The total angular momentum, J, would seem to
12 F. Mazzi; Acta Cryst. 8,137 (1955).
13 A. McL. Mathieson and A. K. Walsh Acta. Cryst 5 599 (1952). ,. ,
~: See, for example, L. E. Orgei, i Chern. Phys. 23,1004 (1955) .
. E. U. Condon and G. H. Shortley, The Theory of Atomic
Spectra (Cambridge University Press New York New York 1951), p. 318. ,,'
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130.113.86.233 On: Wed, 10 Dec 2014 04:44:41SOFT X RAY ABSORPTION EDGES. II. Cu EDGE 623
be a good quantum number especially since Shenstonel6
has observed an LS coupling constant of 828 cm-1 for
Cu*. This indicates that there is considerable J J
coupling and corresponds to a separation of 2070 cm-1
between the ground state 2D6/2 and the 2D3/2level. Now,
it can be seen that there are two J-allowed transitions
from the 2D,/2 ground state to upper states of con
figuration is, 2S2 ••• 3d9, 4s(4D7/2. 6/2. 3/2. 1/2) but none from
2D6/2 to the configuration ls, 2s2 .• ·3dlO(2SI/2). Further,
a consideration of the term values of Zn*17 which
provides a close approximation to the excited Cu* ion
leads to the same conclusion. By means of an energy
cycle utilizing spectroscopic energy data, Beeman and
Bearden6 calculated the energy of the ls-4p absorp
tion in Cu* (on the present energy scale) in good agree
ment with experiment. We may then say that since for
Zn*, the configurations 3d94s and 3d94p are about 10
ev apart, the ls-4s absorption should occur ,...,10 ev
below the ls-4p peak. It will be seen that all of the ob
served small maxima occur ,...,10 ev below the ls-4p
peak. Any correlation of the appearance or non
appearance of this peak with crystal field symmetry
would require knowledge of the interaction of ligand
vibrations with the electronic states of the metal atom.
We do not feel it profitable at present to undertake such
an analysis.
In Fig. 1 we also give the absorption edges for CuCh
and CuCb·2H 20. These are given to illustrate the
limItations of the crystal field theory approach, for
they are examples of compounds in which the assump
tion of nearly pure ion-ion interaction is almost cer
tainly not valid. It is already well known from studies
of optical spectra that halide ion complexes generally
16 C. E. M.oore, Atomic Energy Levels (National Bureau of
Standards, Circular 467, 1951), p. 121, Vol. II.
11 Reference 16, p. 128. show strong charge transfer bands, resulting from elec
tron exchange between anion and cation. This is in a
sense equivalent to saying that the bonding is appreci
ably covalent. The aqueous CuCl4~ ion possesses very
strong charge transfer absorption in the ultraviolet
passing into the visible which accounts for its red color.
Anhydrous CuCh is yellow-brown, doubtless for the
same reason, since it has been foundl8 that the solid
contains infinite chains of copper ions connected by
chloride bridges giving square coordination about each
copper (Cu-Cl, 2.29 A) with chloride ions from other
chains filling out the sixfold coordination shell at 2.98 A.
CuCh·2H 20 also has square planar coordination by
two chloride ions and two H20 molecules in trans
positions.l9 It can be seen that the crystal field splittings
to be expected for these configurations are not in fact
observed, but instead only a broadening (extreme for
CuCh' 2H20) of the structure and also a small shift to
lower energy. If it is valid to say that covalent character
in the bonds may be treated qualitatively as a lowering
of the effective ionization of the metal atom, then the
shift of the first strong absorption, regarded still as
ls~(smeared out) 4p, is reasonable, for it is known that
in Cu+ salts the ls-4p absorption is about 10 v lower
than that for CU*.20
ACKNOWLEDGMENTS
We should like to thank Professor G. Wilkinson for
his interest and encouragement, and Dr. C. J. Ball
hausen and Dr. A. D. Liehr for valuable advice.
Financial support from the U. S. Atomic Energy Com
mission and The Robert A. Welch Foundation is
gratefully acknowledged.
18 A. F. Wells, J. Chern. Soc. 1947, 1670.
19 O. Harker, Z. Krist. 93,136 (1936).
20 Beeman, Forss, and Humphrey, Phys. Rev. 67, 217 (1945).
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1.1743937.pdf | Theory of the Effects of Exchange on the Nuclear Fine Structure in the
Paramagnetic Resonance Spectra of Liquids
Daniel Kivelson
Citation: The Journal of Chemical Physics 27, 1087 (1957); doi: 10.1063/1.1743937
View online: http://dx.doi.org/10.1063/1.1743937
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134.153.52.226 On: Thu, 11 Dec 2014 00:04:29R ESON AN CE STU DY OF LI QU I D C R YST AL TRAN SI TION S 1087
approximate result. When a small amount of PAA is
allowed to cool from the top, from an initial temperature
slightly above the transition, a thin film of anisotropic
liquid forms at the surface, and collects into drops. The
drops grow until they break away from the surface and
fall, at a diameter of one to two millimeters. If it is
assumed that the drops are hemispherical at the be
ginning of the break, and that in that condition their
apparent weight is just balanced by forces due to
THE JOURNAL OF CHEMICAL PHYSICS interfacial tension, a value of 0.02 to 0.08 erg/cm2 is
found for IT.
The maximum increase realized in the thermal ex
pansion coefficient, according to Fig. 3, is to about 2.5
times the value found well above the transition, if we
exclude apparent values where the phase change is
actually under way. This is in agreement with the less
detailed thermal expansion measurements given pre
viously.6
VOLUME 27. NUMBER 5 NOVEMBER. 1957
Theory of the Effects of Exchange on the Nuclear Fine Structure in the
Paramagnetic Resonance Spectra of Liquids*
DANIEL KlVELSON
Department of Chemistry, University of California at Los Angeles, Los Angeles 24, California
(Received April 25, 1957)
A theory of the influence of exchange forces in liquids upon the nuclear hyperfine structure in paramag
netic "spin resonance" spectra is presented. Formulas are derived for two limiting cases: for the exchange J
much larger than the hyperfine separation A, and for A «J. It is found that as J increases the hyperfine lines
broaden (exchange broadening) and start to shift towards the "unsplit" Zeeman frequency. For J"",A, the
hyperfine lines have coalesced into a single broad line and for J>A the line narrows (exchange narrowing)
about the "unsplit" Zeeman frequency. These calculations are discussed in considerable detail. The effect
upon the spectrum of electronic-electronic and electronic-nuclear dipolar interactions and the effect of the
rapid molecular motions in liquids have also been considered, as well as their interactions with the
exchange forces.
INTRODUCTION
THE hyperfine structure in the paramagnetic reso
nance spectra of many substances arises from the
interactions between the nuclear and electronic mag
netic moments. If the sample under consideration is a
liquid or a gas, the rapid reorientations of the magnetic
moments cancel out the major contribution of the
effect of direct dipolar interactions in much the same
manner as the dipolar perturbations are averaged out
in nuclear magnetic resonance spectra.! However, the
Fermi2 interactions are unaffected by the molecular
motions in a liquid and these effects can give rise to
the hyperfine structure mentioned above.3
Another interaction that must be considered in the
study of bulk samples of paramagnetic materials is the
exchange interaction.4 These exchange effects can affect
the paramagnetic resonance spectrum very markedly.
Gorter and Van Vleck6 and later Van Vleck6 demon
strated that the dipolar interactions could be averaged
• A preliminary report was presented before the American
Physical Society in New York on January 31, 1957.
1 Bloembergen, Purcell, and Pound, Phys. Rev. 73, 679 (1948).
2 E. Fermi, Z. Physik 60, 320 (1930).
3 S. 1. Weissman, J. Chern. Phys. 22, 1378 (1954).
• J. H. Van Vleck, The Theory of Electric and Magnetic Suscep
tibilities (Oxford University Press, New York, 1932), first edition,
Chap. XII.
6 C. J. Gorter and J. H. Van Vleck, Phys. Rev. 72, 1128 (1947).
6 J. H. Van Vleck, Phys. Rev. 74, 1168 (1948). out by exchange effects as well as by motional effects
and that this cancellation of dipolar interactions re
sulted in the narrowing of the spectral lines (exchange
narrowing). Anderson and Weiss7 and Anderson8 con
sidered the problem of exchange narrowing in more
detail. They assumed that the exchange terms gave rise
to a random, and in particular a Gaussian, frequency
modulation of the dipolar amplitude and that the
dipolar broadening itself arises from a Gaussian distribu
tion of interactions. Kubo and Tomita9 have proposed
a most elegant and complete theory of motional and
exchange effects; this theory, henceforth referred to as
KT, will provide the theoretical basis for the present
work.
The nuclear hyperfine structure of the spectrum is
affected by exchange interactions. It is this phenomenon
that will be studied here. The problem is somewhat
analogous to that of the magnetic resonance spectra of
paramagnetic crystals with anisotropic g factors, so
ably treated by Yokota and KoidelO using the KT
theory. One would expect the hyperfine lines to broadenll
and to shift toward the unperturbed "Zeeman line" as
the exchange forces are increased. As the exchange gets
7 P. W. Anderson and P. R. Weiss, Revs. Modem Phys. 25, 269
(1953).
8 P. W. Anderson, J. Phys. Soc. Japan 9, 316 (1954).
9 R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954).
10 M. Yokota and S. Koide, J. Phys. Soc. Japan 9, 953 (1954).
11 Ishiguro, Kambe, and Usui, Physica 11, 310 (1951).
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134.153.52.226 On: Thu, 11 Dec 2014 00:04:291088 DANIEL KIVELSON
large compared to the multiplet splitting, the hyperfine
lines coalesce and become narrower.
In this article an analysis similar to that of Yokota
and Koide will be carried out. First the general theory
of KT is rederived in a rather simpler, though restricted
form. The exchange terms are considered next for an
idealized case and finally the effects of the dipolar
interactions are also included. These calculations should
prove useful in the analysis of liquid solutions of
paramagnetic materials in which the hyperfine struc
ture arises from the anomalous Fermi interactions and
in which the dipolar interactions are small. Many free
radicals display just such a spectrum.l2
GENERAL THEORY
In this section the work of Kubo and Tomita will be
outlined with appropriate simplifications and modifica
tions for the present problem.
The goal of the theory is to describe the response of
a paramagnetic system to an applied disturbance. The
disturbance is a magnetic field H ",(t) applied along the
x axis. The system consists of an ensemble of interacting
nuclear and electronic dipoles in a strong constant
magnetic field H applied along the z axis. The first step
in the derivation is to obtain a relaxation function
cp(t-to) which describes the response of the system
after time to if a constant disturbance is applied at
t= -00 and entirely removed at to.
Let 3C be the Hamiltonian of the system after the
removal of the disturbance and let H", be the constant
magnetic field along the x axis. The disturbance thus
enters the Hamiltonian as -PH x where P is the
x component of the magnetic moment operator, that is,
the sum of all individual magnetic moment operators.
The expectation value of P at some time after to can be
expressed asla
(P(t-to» =Tr{g(t-to)P}, (1)
where get-to) is the density matrix at a time t>to. At
t= to, g(O) = g', the density matrix before the removal
of the disturbance.
g'=exp-{l(3C-PH",)/Tr exp-{l(3C-PH",) (2)
where {l= l/kT.
get-to) = [exp-i(t-to)3C/h ]g'[expi(t- to)3C/h] (3)
By noting that a trace is invariant under cyclic
permutations of the operators composing it, one can
rewrite Eq. (1) as
(P(t-to» = Tr{g'P(t-t o)}, (4)
where
pet-to) = [expi(t-to)3C/h]P[exp-i(t-to)3C/h]. (5)
(P(t-to»
12 See the review article by J. E. Wertz, Chern. Revs. 55, 829
(1955). .
13 R. C. Tolman, The Principles of Statistical Mechanzcs (Oxford
University Press, New York, 1950), Chap. IX. describes the response of the system to a step function
disturbance, but the relaxation function cp(t-to), which
is set equal to
is a more convenient function.
The energies involved in magnetic resonance experi
ments are small compared to ;r\ a fact which enables
one to expand g' in powers of {l provided H", is also
small, i.e., for the case of no saturation. At t= -00,
H ",= 0 and the system is not subject to any disturbance;
the average x component of the moment operator at
this time is zero. Thus
TrgoP(t) = TrgoP= 0 (6)
where go, the density matrix at t= -00, is equal to g'
with H ",= o. To first order in {l the relaxation function
is thus given by the relation
cp(t-to) ={l(P(t-to)P)
The pointed brackets
(> and () (7)
indicate that the trace of the operators enclosed, pre
multiplied by go and g, respectively, are to be taken.
We have already made the approximation that the
characteristic energies occurring in paramagnetic reso
nance are much less than thermal energies and, there
fore Q-l can be substituted for go where Q is the unit
ope;ator summed over all quantum states, i.e., the
trace of 1.
The response cp(t-to) to a step disturbance can be
related to the response to an arbitrary disturbance.
If a field H",(t) is turned on adiabatically at t= -00,
i.e., H "'( -00) = 0 and g( -00) = go, we can write -It {d¢(t-to)} (P (t» = H x (to)dto
-00 dto (8)
and after a partial integration, -It dHx(to) (P(t» =cp(O)H x(t) -cp(t-to) dto.
-00 dto (9)
The specific case of interest is the one of sinusoidal
variations since this is what occurs in plane electro
magnetic waves. If the magnetic field is taken to be
H",coswt,
(P(t»= Re(x' -ix")H ",eiwtV, (10)
where V is the volume and x' -ix" the volume sus
ceptibility of the sample, while x' and x" are the
dispersion and absorption, respectively. It is readily
seen that
x' =CPo/V -(w/V) foo cp(T) sinwTdT (11)
o
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134.153.52.226 On: Thu, 11 Dec 2014 00:04:29NUCLEA R FIN E STR UCTURE IN PA RA MAGN ET I C RESO NAN CE 1089
x" = (w/V) foe cJ>(T) c05{,)TdT
o
= (w/2V) f'" cJ>(T)ei"'TdT. (12)
_00
The last equality holds because cJ> (T) is an even function
of T, a fact that is easily ascertained from Eq. (7) and
the invariance of a trace to cyclic permutations of
operators. The absorption coefficient, the fraction of
energy absorbed per cycle, is
a(w) = (W/21r)X" (13)
in rationalized units.
Kubo and Tomita introduce an autocorrelation func
tion G(t), defined by the relation
G(t) = (P(t)P)=fJlcJ>(t). (14)
If pet) is a stochastic process such that its amplitude
is random at any instant and varies randomly with
time, it can be shownl4 that the spectral density I(",,)
is the Fourier transform of the autocorrelation function
I(w) = (1/21r) foo G(t)e-i",tdt, (15)
-<Xl
where G(t) is an average over-all random processes.
ICw) and a(w) are closely related to each other.
where
P(O) (t) = [expi3Clt/li]P[exp-i3C 1t/li] (21)
3C' (t) = [expi(3CI+3C2)t/li ]3C'[exp-i(3C 1+3C2)t/li].
(22)
Kubo and Tomita have demonstrated that the salient
features of exchange and motional effects are described
by the first three terms in the perturbation series.
The exponentials in Eq. (22) can be written as the
products of exponentials since 3CI and 3C2 commute with
each other. 3C' can then be split into the sum of two
terms; the adiabatic, or secular, part which commutes
with 3CI; and the nonadiabatic, or nonsecular, part
which does not commute with 3CI. A further distinction In a discussion of motional and exchange effects the
Hamiltonian 3C can be broken up into several com
ponents:
(16)
where 3CI+3C2 is the zeroth-order Hamiltonian, the
operator that determines the basis functions; and 3C' is
a perturbation. The following commutation rules are
assumed.
[3CI,3C2] = 0
[3C2,P]=0. (17)
(18)
3CI generally consists of the Zeeman effect; 3C' includes
dipolar interactions; and 3C2 is often the motional or
exchange effects. Since 3C2 commutes with both 3CI and
P it cannot affect the spectrum directly, however, it can
interact with 3C' and so have an indirect effect upon the
spectrum. Anderson and Weiss7 have shown that 3C2
effectively modulates the amplitude of the dipolar
interaction 3C'.
In order to evaluate G(t), Kubo and Tomita expand
it in a perturbation series,
(19)
n=O
where A, the coefficient of 3C/, is an arbitrary perturba
tion ordering parameter which will eventually be set
equal to unity. G(t) can then be expressed in terms of
perturbation series
(20)
might be made between the stationary and non
stationary parts of 3C' which commute and do not
commute, respectively, with 3C2.
Kubo and Tomita rewrite Eqs. (20)-(22) in a some
what altered form. 3C/ is a matrix element of 3C', in
the basis that diagonalizes 3CI, between two states
whose energy separation is liw'Y' and
3C/ei"''Yt= ([expi3C lt/li]JC'[exp-i3C lt/li]}'Y. (23)
The effect of 3C2 can be introduced by the following
relation:
3C-/ (t) = [expiJC2t/li ]JC/[exp-i3C 2t/li]. (24)
The relation for G(t) can now be written as
G(t) = £, (Ajih)n it dtl• •• iln
-1
dt" L ... L L L ([ ... [P a(O), 3C'Yr'(tI)} . ·3C'Yn'(tn)]Ptj(O»
n~ 0 0 'Yl 'Ynatj
where Pa(O) and Ptj(O) are matrix elements of P(O) and
the sums are taken over all matrix elements. Since it is
a trace Gn{t} remains invariant under a canonical
14 M. C. Wang and G. E. Uhlenbeck, Revs. Modem Phys. 17,
323 (1945). Xexpi(wat+W'Yltl+· . ·w'Yntn), (25)
transformation; thus
Gn(t) = [expit'3CI/li]Gn (t) [exp-it':JCI/li]
=L LL ... LG(t)
a tj 'Yl 'Yn
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134.153.52.226 On: Thu, 11 Dec 2014 00:04:291090 DANIEL KIVELSON
[See Eq. (23).J In order for Gn(t) to be independent of
t', the following condition must hold:
(27)
In order to make some estimates of line shapes, one
assumes that G(t) can be written as
G(t,A)=G(t,O) eXfJ'l!(t,A) (28)
where G(t,O) is the value of G(t) when A vanishes.
1/;(t,A) is assumed to be regular in A and t which implies
that 1/;(t,A) can be expanded:
1/;(t,A)=L ak(i)tkAijk!
k.i (29)
Since1/;(t,A) must vanish as A~O, ak(O)= O. Furthermore,
in all the cases below, ak (i) vanishes for odd j and hence
the effect of these terms will be neglected. If G(t,A)
approaches zero rapidly for large t, and the perturba
tion A is small, only terms through order A 2 need be
retained in l/;(t,A) and G(t,A). In this approximation the
second-order contribution to G(t), G2(t), is related to
the lowest order contribution to 1/;(t,A) by the relation
[G2(t,A)J[G(t,0)J-l=1/;(t,A) = L ak(2)tklk!. (30)
k
The parameters ak (i) are known as the kth semi
invariants and correspond to the coefficients of tklk! in
the expansion of 1/;(t,A). The coefficients of tn / n! in the
expansion of G(t,A) corresponds to the nth moment of
the distribution. 8
If G(t,A) is the autocorrelation function for a random
process, it approaches zero rapidly as t~oo. This is
the approximation assumed above. It can also be shown
that for a stochastic process an expansion in semi
invariants converges much more rapidly than an ex
pansion in moments. The moments and semi-invariants
since 3'C2 commutes with P a(O) and 3'C!.I. Setting
J.I.= (tl+t2)/2 and integrating it from TI2 to t-T/2,
and letting r=t1-t2 and integrating it from ° to t,
enables one to rewrite G2(t)
G2(t) = -A2 L it dr(t-r)[expiwatJ
a,'Yl 0
where the "perturbation amplitude" O'an is
O'an2 =h-2( 1 [P a (0), 3'Cn' (0) J 12)( 1 P .. (0) 12)-1, (34)
and the correlation function janeT) is
ja·'/l (T) = ([P a (0), 3'Cn' (T) J[3'C -1'1' (0), P -a (O)J)
X ([P a(O), 3'C" t' (0) ] 12)-1. (35)
16 3'C _",' is the complex conjugate of 3'Cn'. discussed above refer to the entire spectral distribution,
but these quantities related to each individual spectral
line Wa are of particular interest. Provided the lines are
suitably spaced, the moments and semi-invariants of
each can be obtained by factoring out [expiwatJ before
expanding G(t,A). Thus the coefficient of (tn/n!) expiwat
in the expansion of G(t,A) is the nth moment of the Wa
spectral line. 8.9
If 3'C2=0 and only the secular contributions to G(t,A)
are considered, i.e., Wn = Wn = 0, it can be seen from
Eq. (25) that
(31)
is the resultant expression for G(t,A) provided all the
approximations discussed above are valid. This is the
assumption of a Gaussian distribution which may be
nearly valid for a randomly distributed unmodulated
ensemble. Note that all but the second semi-invariant
vanishes but that an infinite number of moments are
finite, thus bearing out the convergence statement
made above. If 3'C2~0, the perturbation may be con
sidered to be modulated, and 1/;(t,A) takes on the more
complicated form of Eq. (30).
If the approximations discussed above are valid,
G(t,A) can be expanded through second order by per
turbation theory, the second-order term correlated with
an expansion in semi-invariants, and G(t,A) expressed
as in Eq. (28). Of course, if fourth-order terms in A are
retained, the simple correlation between G2(t,A) and
1/;(t,A), given in Eq. (30), does not hold and the problem
becomes more complicated.
If the amplitudes of 3'C' are randomly distributed or
if the distribution has a high symmetry, Kubo and
Tomita have shown that G2(t) can be simplified. In
this case they assume that not only does Eq. (27) apply
but that Wa= -W{3 and wn= -Wn, which implies that
janeT) is generally assumed to be a Gaussian (Gaus
sian-Gaussian distribution) or a decaying exponential
(Gaussian-Markoff distribution8). If 3'C2= 0, jan(r) = 1.
WEAK EXCHANGE
A set of isolated paramagnetic molecules with iso
tropic g factors, with no nuclear-electronic coupling,
and with no effective orbital angular momentum,
interacts with a magnetic field H applied along the
z axis; this interaction can be represented by the
Hamiltonian Ho,
Ho= -h"/H L SZi,
i (36)
where SZi is the z component of the electronic spin
angular momentum Si for the ith molecule and "/ is the
gyromagnetic ratio of the electron. Equation (36) repre
sents the usual Zeeman phenomenon. There are, how-
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134.153.52.226 On: Thu, 11 Dec 2014 00:04:29N U C LEA R FIN EST Rue T U REI N PAR A MAG NET I eRE SON A NeE 1091
ever, interactions which must be considered between
the nuclear and electronic magnetic moments. Assuming
that all nuclei that couple with the electron moment are
equivalent and that dipolar effects can be ignored, the
Hamiltonian for the Fermi interaction is:
Hp= -itA 1: ScI.,
i (37)
where Ii is the total nuclear spin for the ith molecule
and A is a coupling constant which is independent of
molecular orientation.16 The extension to nonequivalent
nuclei is trivial provided interactions between nuclei
are small. Under the usual "strong field conditions,"
Hp is small compared to Ho, and Hp may be replaced by
the approximate relation
Hp= -itA 1: Sw Iz ..
; (37a)
where Iz; is the z component of Ii. The following discus
sion will make use of Eq. (37a) rather than (37) al
though the omitted terms, the nonsecular terms, as well
as the neglected dipolar terms will be discussed in a
later section.
If the disturbing magnetic field H x(t) is a component
of plane polarized radiation incident on the sample in
such a way that H x(t) is oriented along the x axis,
(38)
If exchange effects are to be considered, the exchange
term He,
(39)
i i¢i
must be included in the Hamiltonian. iij are exchange
integrals that are independent of both the nuclear and
electronic spins but dependent upon the overlap of the
"unpaired" electronic wave functions and hence upon
the distance between molecules. iij=1i. and the sums
in Eq. (39) are taken over all N molecules. It can be
seen that
[Ho,He] = 0
[H.,P] = 0
[Ho,Hp]=O. (40)
(40a)
(40b)
The immediate problem is to solve for the effects of
the exchange terms upon the nuclear hyperfine struc
ture in the "strong field" case, neglecting motional and
dipolar terms in the Hamiltonian. First, the situation
in which the exchange frequency is small compared to
the hyperfine splitting will be treated. This corresponds
to the case of near degeneracy in the KT theory and so
must be approached, in a manner analogous to that
used by Yokota and Koide.1o
16 A is equal to /t-2'Y')'1(2/3)8(r) evaluated in the appropriate
electronic state, i.e., the ground state. 'Y I is the nuclear gyromag
netic ratio and r is the vector distance between the electron and
nuclear momentll. See reference 3, The following identifications can be made with the
symbols in the preceeding section:
:JC2=O.
These relations imply that
[P,:JC']=O. (41)
(41a)
(41b)
(41c)
These identifications suggest that Ho+ Hp be diagonal
ized and:JC e be treated as a perturbation. A suitable basis
is thus one in which S;, Ii, Mi, mi are quantum num
bers which correspond to the spin operators S;, t,
Sz;, Iz;, respectively. All terms in the Hamiltonian are
diagonal in Ii and mi, and the nonvanishing matrix
elements are
(42)
(M.i'niMJi'nj' .. , Hp I M.i'n,M,i'nr .. )
= -itA 1:M kmk (42a)
(MiMi" 'IHeIMiMi"')
= (1/2)1t 1: 1: Jk1MkM I (42b)
k l""k
(MiMi" ·IH.IM;±l, MiF1···)
= (1t/2)JiX{S(S+1)-M i(Mi±1)}
X {S(S+1)-Mj(MiFl)}]i (42c)
(MiMi" ·IPIMi±l, Mj"')
('Y/2){S(S+1)-M,(Mi±1)}1. (42d)
These elements are diagonal in all quantum numbers
not explicitly denoted.
In order to evaluate Go(t) , Eq. (20), one must evalu
ate P(O) (t), Eq. (21), as well as Q or the Trl.
Q= (2S+1)N(21+1)N
P(O)(t)='Y 1: [SXj cos{"yB+AIzj)t (43)
j
-SYjsin('YB+AIZj)t] (44)
where S=Sj and 1=1;. It is then seen that
GO(t)='Y2Q-11: Tr{Sx; cos({JB+AIzj)t}
i
which, when summed, yields (45)
'YWS(S+l)
Go(t) 1: cos('YB+Am)t, (46)
3(21+1) m
where m=I, 1-1, ···-1. The spectral density to
zeroth order, In(w), is [see Eq. (18)]:
NS(S+1}y2
10(w) 1: Il(w-'YB-Am). (47)
3(21+1) m
In zeroth order the spectral density is thus independent
of the exchange effects, and the absorption spectrum
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consists of sharp lines at 'YB+Am. This represents
the unperturbed nuclear hyperfine spectrum.
In order to evaluate GI(t) and G2(t), an expression
must be obtained for :Je/(t) [see Eq. (22)]. Since :Je2=0,
this is a relatively simple task.
:Je' (t) = (1/2)/t L: L: Jjk[S'jS'k+ (SXjSXk+ SlIjSlIk)
i kr'i
Xcos{A (I.j-I.k)t} + (SXjSYk- SXkSlIj)
Xsin{A(I.j-I.k)t}]. (48)
Equation (20) gives the form for GI(t) and G2(t) j
GI(t) =0 and G(t) through second order is given by the
relationl7 :
N'Y2S(S+1)
G(t)
3(21+1)
where { 2S(S+1)1J2
XL:m e'('YH+Am)t 1 (t2/2!)
3(21+1)
S(S+1)K(I,m) } ----(J2/A)it+··· ,
3(21+1)
k(l,m) = L: 1/(m-m')
m'~m
NJ2=L: L:Jjk2.
i kr'i (49)
(49a)
(49b)
Considerable mathematical manipulation is required in
order to obtain Eq. (49). J2 is an average exchange
constant per molecule. If it is assumed that only nearest
neighbors contribute and. that there are z nearest
neighbors, J2=z/,j2 where (,)2 is the average of the square
of the exchange constant between a pair of nearest
neighbors. Equation (49) is obtained by averaging of
Jii's and so the stochastic process approach of KT can
be applied. It can then be seen by referring back to the
preceding section that the coefficient of -t2 /2 ! is the
second moment of the spectral density and the coeffi
cient of (it) is the shift in each of the spectral lines.
The result of these computations indicates that in
the absence of exchange effects the spectral lines are
"sharp lines" at w='YH+Am, and that in the presence
of exchange interactions the lines converge toward the
central frequency, w='YH, and at the same time they
broaden out. When J=A, the spectrum consists of one
very broad line with breadth of order A, centered about
w='YH.
Equation (49) is not adequate for a determination of
line shapes. An estimate of the half width may be made,
17 The 21/ (21 + 1) factor in the coefficient of (l arises because the
sum over all mk'=m; has been performed. This restriction on the
sum over mk arises because the nonsecular terms in G2(t) become
secular terms if mk=m; and, furthermore, these terms cancel the
mk=m; contribution from the remaining secular terms. Thus only
terms for which mk.=m; remain in both the secular a.nd nonsecular
contributions to G;(t), however, if the fourth moment of each line is computed.
The fourth moment, (Aw4)m of the mth line is equal to
the coefficient of (P/4!) expi('YH+Am)t divided by the
intensity factor (21+ 1)-1( / P(O) /2).18 This coefficient can
be obtained by computing the secular part of G4(t)19:
(Aw4)m = [3J a 4(TrS.2)2
where (21)2
-2J b4 TrS.4J-----
(2S+ 1)2(21+ 1)2
NJa4=L: L: L: Jjk2Jjk'2
i kr'i k~;
Nh4=L: L: Jjk4.
i kr"i (50)
(50a)
(SOb)
If only nearest neighbors are considered and the rather
restrictive assumptions20 are made that Ja4=Z2(,)4 and
Jb4=ZJj4, then
[ 21z(,)2 ]2 SZ(S+1)2
(Aw4)m=
(21+1) 3
[ 2S(6S4+15S3+10SZ-1)j
X 1 .
5z(2S+ 1)252(S+ 1)2 (51)
For large values of z, the ratio of the square of the
second moment to the fourth moment is equal to t,
which is the value for a Gaussian distribution. For
small z this is not the case. The greatest deviation from
Gaussian is obtained for Z= 1 and S=! or 1, in which
case the ratio equals one-half. In actual cases the line
shape probably differs somewhat from the Gaussian
shape, but for liquids the deviations may not be too
great.
At first thought it would seem that the hyperfine
lines should tend to move apart in the presence of the
perturbation He since perturbations tend to repel energy
levels. An analysis of the special case S = 1= t, N = 2 is
useful in elucidating this question. From ordinary per
turbation theory one finds each of the two hyperfine
lines that appear in the absence of exchange effects are
split in the presence of exchange effects into three com
ponents of differing intensity. Although the frequency
shifts away from 'YH are larger than those toward 'YH,
in accord with simple perturbation concepts, the in
tensity factors are such that the spectral center for
each of the two groups of lines is shifted towards 'YH,
in agreement with the results obtained above. The prob
lem of large N and arbitrary S and I complicates matters
18 One divides by the intensity factor in order to refer the results
to the normalized spectral density.
19 The factor [21(21+1)-1] enters because the nonsecular con
tributions become secular contributions for m;=mk and m;=mk',
and these contributions cancel the remaining secular terms with
m;=mk and mj=mk" See reference 17.
20 This approximation is very poor for rigid or viscous media but
is probably not bad for liquids in which the correlation time is very
short. See the section on dipolar and motional effects, in particular,
the discussion preceding Eq. (77).
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but does not alter the essential reasoning behind these
arguments.
STRONG EXCHANGE
In the preceding section the case in which J«A was
discussed. As J approaches A in magnitude, the line
breadths become as large as the line separations with
the result that the hyperfine spectrum coalesces into a
single broad line centered about w=-yH. This case of
"intermediate coupling" is difficult to treat, however,
the strong exchange problem, that is the problem for
which J»A, can be readily solved.
More stringent approximations are necessary in the
strong exchange case than the weak one since the inter
actions between molecules become so strong that the
entire sample of material is essentially a supra-molecule.
Fortunately, the invariance of a trace under unitary
transformations which enables one to evaluate G(t) in
an arbitrary basis, in particular, in the basis introduced
in the preceding section. The identification of the vari
ous terms in the Hamiltonian with the terms in the
KT theory is
xI=Ho
x'=H p
X2=He. (52)
(52a)
(52b)
The problem is thus one of diagonalizing Ho+H. and
treating Hp as a perturbation. According to the KT
procedure, He modulates the perturbation Hp.
In order to evaluate G(t) by the perturbation tech
nique of Eq. (20), P(Ol(t) must be evaluated [see
Eq. (21)J:
P(ol (t) =-y L: [SXj cos-yHt- SYj sin-yHt]. (53)
i
Go(t) becomes
-y2N
Go(t)=-S(S+1) cos-yHt. (54)
3
Thus, to zeroth order the spectral density consists of a
single "sharp line" at w=-yH.
It is simple to show that GI(t) vanishes. In order to
calculate G2(t) one should note in the present case that
Xl commutes with 3C' which implies that the time de
pendence of X' (t) arises through a dependence upon X2
but not upon Xl. This fact enables one to perform the
time rotation described in Eq. (33) with the result that
G2(t) may be expressed as
G2(t) = -h-2 il
(t-T) < [[P(Ol (t), X'(T)JP(Ol]. (55)
o
3C'(T) can be expanded as
3C' ( 1') = [expiJC2T /h J3C'[ exp-iJC2T /h J
=X'+ (iT/h) [3C2,X'J
+ (iT/h)2(1/2 !)[X2, [X2,X'JJ+· . '. (56) This series is not simple to sum but it will be necessary
to retain terms only through order 1'2. G(t), through
second order, is given by the relation
-y2NS(S+1) G(t)=---
3
[ { A2I(I+l)} Xcos-yHt 1-(1/3) N t2/2!
_{ (1/3)A2I(I+l) (N;!)}
X~I (t-T)1'(T)dT+"'] (57)
where l' (1') is
1'(1')= 1-(r/2!){ (2/3)S(S+1)N/N-1}J2+··· (58)
J2 is defined in Eq. (49b). A separation has been made
in Eq. (57) between the terms arising from the sta
tionary xs' and motional Xm' terms.
(59)
and the remainder of AL:LjSz j is the motional part of
X'. The approximation described in Eq. (37a) neglects
the nonadiabatic contributions to the problem. For
large N, l' (1') reduces to the correlation function f( 1'),
f(r) = 1-(r/2!){ (2/3)S(S+ 1)J2}+· . " (58a)
and the stationary part of G(t), the term in t2, vanishes.
G(t) is then given by the relation:
-y2NS(S+l) [ G(t) 3 cos-yHt 1-{(lj3)A2I(I+1)}
X il
(t-T)f(T)dT+" -l (60)
The simplest form that can be assumed for f(T), con
sistent with Eq. (58a) and with the requirement that
it vanish as T-Ht:> , is an exponential:
f(T)=exp[ -(1/3)S(S+1)r2J2]. (61)
This expression is, of course, an approximation but as
demonstrated by Anderson and Weiss,1 a rather good
approximation.
The remaining problems are the evaluation of G(t),
Eq. (60), and then the spectral density, Eq. (16).
This cannot be done analytically. In the case of very
strong exchange, the single spectral line is very narrow
which indicates that the important contributions come
from large t in G(t). Under these conditions the integral
JoI(t-T)f(T) can be evaluated. f(r) decreases rapidly
as l' decreases so that JolT f( T)dT is negligible compared
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with tfotf(T)dT provided t is large. Thus
it (t-T)dTfH= It I i'" f(T)dT
o 0
(3'II'Y -I t I I tiT'. (62) -2J[S(S+1)J!
The upper limit may be changed to infinity since f( T)
decreases very rapidly. The absolute value sign is
included because the integral is clearly an even func
tion of t. If
(63)
and it is assumed that the expansion of G(t) in semi
invariants is valid,
-yWS(S+1)
G(t) cos-yHtexp[-u2T'ltIJ, (64)
3
and lew) is
-yWS(S+1) (h'
lew) = (1/'11") .---. (65)
3 (T4T'2+ (W--yH)2 of exchange effects, i.e., 3C2=0 in Eq. (23), 3C/(t) is
3C'(t)X2=O= -1: [(l"jSxj+IyjSyj)hA cos-yHt
-(IxjSYj-IyjSxj)hA sin-yHt]. (68)
Equations analogous to Eqs. (56), (58a), and (60) can be
derived. The resulting expression for the nonadiabatic
contribution to G2(t) is
-yWS(S+1)
G2(t),,= [(1/3)A2l(l+1)J
3
X it (t-T)[cos{-yHt(t-T)}Jf(T)dT, (69)
o
where the correlation function is the same as that de
fined in Eqs. (58a) and (61). Although the integration
cannot be carried out exactly, a procedure completely
analogous to the one employed in Eq. (62) can be ap
plied with the result:
This equation represents a Lorentzian curve with a
half-width given by the relation where (T2 is defined in Eq. (63) and
l(I+1)'II"t A2
~W!=(T2T'= -.
, 2[3S(S+1)J1 J (66)
These results indicate that for J»A, the nuclear
hyperfine lines coalesce into a single line, and that this
line becomes narrower as J increases.
The results obtained above follow directly from the
treatment of exchange narrowing presented by KT,
but in the KT article an approximation is also given for
the half width which is more accurate than Eq. (66) in
the region where J is not much larger than A. [See KT,
Eq. (8.12)]' The approximation used above is that of a
Gaussian-Gaussian distribution.s This is probably
valid for exchange effects in solutions where there
are many randomly oriented molecules.
N onadiabatic Contributions
In the calculations performed in the preceding sec
tions, Hp was approximated by the relation in Eq.
(37a). Under these conditions Hp commutes with the
Zeeman terms Ho. However, the discarded nonadiabatic
terms Hp",
Hp"= -hA 1: [SxjIxj+SYjIYjJ (67)
i
do not commute with Ho.
For "strong exchange" the contribution from these
nonadiabatic terms can readily be computed by a
procedure similar to the one used above. In the absence Tn'=[cos-yHtJ-t f'" f(T)dT cos-yHt(t-T)
o
exp[ -3rW/4S(S+1)J2J = (3'11")1 . (71)
2[S(S+1)JiJ
If once again it is assumed that the expansion in semi
invariants is valid, G(t) through second order, including
both the adiabatic and nonadiabatic contributions is
-y2N S (S + 1)
G(t) cos-yHt[exp-(T'+Tn')(T2ItIJ, (72)
3
and lew) is
1 -yWS(S+1) (T2 (T'+ T .. ')
l(w)=- . (73)
'II" 3 (T4( T' +T .. ')2+ (w--yH)2
The half-width is given by the relation
/(1+1)'11"1 A2
~Wt=(T2(T'+Tn') -
2[3S(S+1)J1 J
X {1+exp[ -3-y2H2/4S(S+1)J2J}. (74)
The expression given above for the half-width at
half-power demonstrates that the result is the sum of
the half-widths from the adiabatic and nonadiabatic
effects. For strong fields, or for relatively weak ex
change, -yH»J»A, the exponential term in Eq. (74)
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is negligible j hence the nonadiabatic effect can be
neglected. If H is not so strong or if J is very large,
that is, if the exchange and Zeeman energies are com
parable, the exponential term must be kept j the non
adiabatic terms are then significant. The ratio of the
half-width for weak fields (or strong exchange) to the
half-width for strong fields (or weak exchange) is 2.
It should be emphasized that these results apply only
if J»A, and 'YH»A, so that the terms weak exchange
and weak field are qualified in this discussion.
If J <A, truly weak exchange, and the field H is
strong compared to A/I', then the nonadiabatic contri
butions are negligible and the weak exchange calcula
tions performed above are valid. The problem for which
'YH=A is, of course, an entirely different one.
J is an exchange integral and is a function of the dis
tances between paramagnetic molecules. It is not a
simple matter to determine the functional form of J j
however, one might expect the magnitude of J to fall
off rapidly with increasing intermolecular distances.
This would imply that J, and hence the effects calcu
lated above, would have a strong concentration de
pendence. At dilute concentrations of paramagnetic
molecules, dipolar broadening of the spectral lines
would predominate. As the concentration is increased, a
point should be reached at which the exchange broaden
ing and coalescing should become a major factor. Such a
phenomenon has apparently been observed2I•22 but no
quantitative study has as yet been performed. At higher
concentrations, J becomes very large, and the single
resonance line has its width determined by a number of
factors which are discussed below.
Dipolar and Motional Effects: Strong Exchange
There exist, of course, other phenomena besides ex
change coupling which may alter the line shapes. In
order to see under what conditions the various effects
are dominant and how they interact with each other
when they contribute equivalently to the spectral
distribution, a brief study of these effects is discussed
in this and the succeeding sections. The perturbations
which may be important are the nuclear-electronic and
electronic-electronic dipolar interactions, the aniso
tropic part of the g factor which owes its origin to spin
orbital coupling, and chemical reactions. And most
important to consider is the superimposed rapid modu
lation attributed to the random motion of molecules
in a liquid.
Many of these other factors can be considered to
gether. The electronic-electronic dipolar interaction
H •• can be written as
H8.='Y2h-2 L 'jk-3
,>k
21 S. Weissman (private communication).
22 H. S. Jarrett (private communication). where l' is the electronic gyromagnetic ratio and, jk is
the distance between unpaired electrons. The nuclear
nuclear dipolar interactions are negligible, but the
nuclear-electronic dipolar interactions are
Hr8='Y'Yrh-2 L Rjk-3
,>k
where 'Yr is the nuclear gyromagnetic ratio and Rjk is
the nuclear-electronic distance. In a liquid the molecules
are randomly and rapidly tumbling and moving about;
this motional effect is represented by the Hamiltonian
HT• The exchange integral J is a function of the
intermolecular distances and of the relative orientations
of the paramagnetic molecules. Thus J depends upon
spatial coordinates and is not isotropic j it does not
commute with HT• It is convenient to divide J into
two components j one that commutes with HT, PO), and
one that does not, PI). J(O) depends upon the average
distance between paramagnetic molecules, and it is
isotropic. For infinitely rapid Brownian motion J = PO).
J(1) is anisotropic and if a random distribution is as
sumed the average value of J(I), (J(I», IS zero but
{{J(l))2) is finite. For a rigid lattice,
(77)
Naturally, He can be divided into two corresponding
components
(78)
A number of commutation rules can now be introduced:
[HT,H.<O)]= [HT,Ho]= [HT,Hp]= [HT,P]= 0 (79)
[Ho,He]=[Hp,He]=O (80)
[He (0) ,He (1)] = o. (81)
The commutation rules introduced earlier are still
valid. For strong exchange the following identifications
with the KT theory can be made:
3CI=Ho (82)
3C2=HT+He(0) (83)
3C'=H.(1)+H r.+H •• +Hp. (84)
Go(t) is still given by Eq. (54) while GI(t) vanishes.
G2(t) involves an average over the square of 3C' but
since there is no correlation between He(l), Hr., H •• ,
and Hp, G2(t) does not depend upon any cross terms
coupling the four components in Eq. (84). Thus the
contributions to G2(t) arising from each of these terms
can be computed independently.
G2(t)p, the contribution to G2(t) arising from Hp,
has already been given in Eq. (70). Since HT commutes
with Hp, the motional effects do not enter into these
calculations. H.(l) commutes with Ho and P as well as
with H.<O) and hence it does not contribute at all to
G2(t)p. It should be mentioned, however, that if H.(I) is
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large, so large that its contribution is appreciable even
in the presence of motional effects, a second-order cal
culation may be inadequate. In fact, under these circum
stances He(l) should probably be included in 3C2. This
will be discussed in a subsequent article.
Weissman 3 has discussed the effect of molecular
motion upon the nuclear-electronic dipolar interactions
but no specific relationship for this effect has been
derived. Such an expression is readily obtained by
means of the KT theory. Following the KT procedure,
Eq. (76) can be rewritten as derivation of f( r) previously considered [Eqs. (61-63),
(77)J; that is, an expansion of the correlation function
is carried out to second order and an exponential form
assumed. If this is done Eq. (35) can be rewritten as
fa'Y' (r)I.= exp-(1/2)we-y2r2 (90)
where the exchange frequency We is
([H.<O), [P _ (0), 3C_'Y/JJ[[3C'Y/' P + (O)JHe(O)J)
We'Y,2= .
1z2([P _ (0), 3C_'Y/J[3C'Y/' P + (0) J)
(90a)
HI.= L L cI>ik-('Y){jk}'Y
'Y i,k (76a) [See KT, Eqs. (8.6) and (8.9).J It is convenient to
introduce the subscript c which is defined by:
where j is summed over all nuclear moments and k
over all electronic moments and
and {jk}±2= I]'±Sk±
{jk}±I=I]'±Szk
{jk}±ll= I.jSk±
{jk}o= LiS'A:
{jk}oo= -(1/4)[Ii+Sk-+IrSk+J,
cI>ik±(2) = -(3/4h''Yllz-2Rik-6(Xik±iYik)2
cI>ik±(l) = cI>ik± (11) (85)
(85a)
(85b)
(85c)
(85d)
(86)
= -(3/2h''YIIz-2Rik-6Zik(Xik±iYik) (86a)
cI>ik(Ob cI>ik(OO) = -'Y'YIIz-2Rik-6(3Zik2_Rik2). (86b)
The operators O± are as usual equal to O,,±Oy. The
"perturbation amplitude" or second moment [uai}.
can be obtained by means of Eq. (34)23:
[uao2}.=I(I+ 1)ud/6=4[u aoo2}. (87)
[u a±12}.=I(1 + 1)ud/18
[u a22}.= 41(1+ 1)u1,2/9
where U1.2 is
ud= L lcI>ik(O) 12/N
i,k (87a)
(87b)
(88)
and the assumption is made that the dipolar interactions
are isotropic so that
L lcI>ik(l) 12=Nud/6
ik
L lcI>ik(2) 12=2Nud/3.
ik (89)
(89a)
fa'Y(I')(r), Eq. (35), has two contributions, one from
HT and one from H.o. One first neglects HT and com
putes the fa'Y(I')(r) arising from the presence of H.(O).
This calculation proceeds in the same fashion as the
23 A more convenient form of Eq. (34) is
ua-y.2=1i~([P + (0), 3C'Y.'][3C-'Y'" P _ (O)])(P + (O)P _ (0»-•• c=O for 1'1=00,2
c=l for 1'1=0, ±1.
If W'YI is the frequency corresponding to the matrix
element 3C'Yl" then it can be shown that
We=O for c=O
We='YH for c= 1.
The combined second moments, ua2(c), resulting from
each of these transitions are:
[u a2(0)}.= [uao2}.+[U a12}.+[U a_12}.
= (5/18)1(1+ 1)u1.2 (91)
[u a2(1)}.= [uaoo2}.+[u a22}.
= (35/72)1(1+ l)ud. (91a)
The correlation function, Eq. (90), for 1" set equal to
zero or one, is identical with the one given in Eq. (61),
provided the following assumptions are made:
L L Jin21cI>ik('Yl) 12=N2J2 L lcI>jk('Yl) 12 (92)
"""i i,k i,k
L L L Jin2cI>k/'Yl)cI>k,,(-'Yl) =0. (92a)
; """i k
The dependence upon HT of the nuclear-electronic
dipolar terms in G2(t) cannot be neglected; in fact, the
motional effect is undoubtedly greater than the ex
change effect for liquids. This implies that there will
also be a motional relaxation effect represented by a
motional correlation function fr(r). Following the
usual procedure1 a single correlation time r c is assumed
and fr(r) is arbitrarily chosen to be
fr(r)= exp-/rI/r e• (93)
Combining all these results, G2(t)r., the contribution to
G2(t) from HI. is24
24 The prime on G2' (t) 1. indicates that it is the result for the
transition of frequency -,,(H. The complex conjugate with fre
quency +,,(H should be added to G2'(t)I. in order to evaluate
G2(th •.
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Xexp[ -(r)/(rc)- (1/2)wh2]
1(1+1)
. O"d[20+35 exp(i'YHr)]. (94)
36
Since the correlation function is real the approximation
used in Eqs. (62) and (70) may be applied. If the as
sumption is then made that r c-1»we, the correlation
function can be expanded in powers of we2, and terms of
higher order than we2 can be neglected.26,26 Thus
For extreme "motional narrowing," that is, for
-y2Wr02«1,
G2(t) •• , the contribution to G2(t) arising from B •• ,
can be computed in a manner quite analogous to that
used in obtaining G2(th •. The calculation of G2(t)8.
follows quite closely the calculation of motionally
narrowed nuclear-nuclear dipolar interactions and ex
change-narrowed electron-electron dipolar interactions
performed by Kubo and Tomita. If the same assump
tions are made as in the calculation of G2(t)r., and if
0" •• 2 is the electronic-electronic analog of 0"1.2, For extreme narrowing, 'Y2H2rl«1,
NS(S+lh2 50" •• 2 ----[cos'YHt]--S(S+l)roltl·
3 2
(96a)
The anisotropy of the g factor and the phenomenon
of chemical reactions will not be considered here al
though both effects may be extremely important, in
fact dominant, when present. If these two effects are
neglected, G(t) for "strong exchange" is equal to the
G2(t) given in Eq. (59) plus the four contributions to
G2(t) represented in Eqs. (64), (70), (95), and (96).
For extreme motional narrowing:
NS(S+lh2
G(t)= [cos'YHt] exp[ -{A2r'+A2r,,'
3
+ (1/6)O"Ih 055(1-ro2we2)} (1/3)/(1+ 1) I t I
+{ O".h .(5/2) (1-r o2We2) }S(S+ 1) I t I], (97)
This implies that the line shape is Lorentzian with a
half-width given by the coefficient of I t I in the ex
ponent in Eq. (97).
The constant A is less than 50 Mc and may be as
small as 0.5 Mc for cases of interest.27 ] in solution
depends upon the concentration as discussed above.
Assuming that A equals 10 Mc and that] ranges from
2 A to 100 A, for S=!, A2r' varies between 10 Me and
0.2 Mc while A2rn' varies between 0 Mc and 0.002 Mc
[see Eq. (74)]. 0"1. is probably of the order of 10 Mc
while ro may well be of the order of 10-10 secl or less.
Under these conditions the dipolar interactions are
independent of exchange and (1/6)O"Ih o55=0.2 Me.
0" .. is of the order of 50 c Mc where c is the molar con
centration of unpaired electrons7 and O"./ro is thus of the
order of 1.5 c2 Mc. The factors (1/3)/ (I + 1) and
S(S+ 1) must also be considered in estimating line
widths [Eq. (97)]. These order of magnitude calcula
tions indicate that any or all of these effects may be
significant depending upon the conditions of the par
ticular problem.
Dipolar and Motional Effects: Weak Exchange
For weak exchange the identifications with the KT
theory differ from those in Eq. (82), and are given by
the relations
(98)
(98a)
(98b)
2. The cQmplex conjugate to G2'(t)/a has been added. See
reference 24. 27 B. Venkatgraman and G. K. Fraenkel, J. Chern. Phys. 23,
•• w.'= (2/3)S(S+ 1)]2. 588 (1955).
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134.153.52.226 On: Thu, 11 Dec 2014 00:04:291098 DANIEL KIVELSON
Go(t) is given by Eq. (46), Gl(t) vanishes, and there are
no cross terms between the various parts of :ref in G2(t).
The contribution to G2(t) arising from the presence of
H.<O) has already been given in Eq. (49) since HT com
mutes with all the pertinent quantities and so does not
enter into the results. It should be noted, once again,
that only the z components of spin enter into Hp in
these calculations, d. Eq. (37a).
The contribution to G2(t) arising from H.(l) is rela
tively simple to obtain if the assumption is made that
A2rc2«1, a reasonable approximation in accord with
the discussion in the preceding section. The condition
for weak exchange becomes:
(99)
An approximate expression for G2(t) J(l) is then:
N")'2S(S+1) ----- L: ei(1'H+Am)t
3(21+1) m
{S(S+1) } X (8/3) 1([J(I)J2)T cltl .
(2/+1) (100)
Note that to the order of these calculations J<I) con
tributes to the line width but does not shift the fre
quencies of the peaks. It is, of course, even more diffi
cult to evaluate J(l) on an a priori basis than it is to
obtain a value for J, and it is a difficult task to derive
the latter.
G2(t)l. and G2(t) •• can easily be obtained under the
conditions that A2Tc2«1. G2(t).. is equal to its
value in Eq. (96) except that cos,,),Ht is replaced by
cos ( ")' H + Am) t, the expression is divided by (21+ 1)
and the sum over m from -1 to 1 must be taken.
G2(t)l. is a bit more complicated. It is equal to the sum
of two terms, an intra-and an intermolecular one.
Both terms are similar to the G2(t)18 given in Eq. (95)
except that the corrections required for G2(t) •• must
also be applied here, and 1(1+ l)O"d is replaced by
1(/+l)O"d (inter) and by 3m20"d (intra) in the inter
and intramolecular terms, respectively. O"d (inter) and
O"d (intra) are defined by Eq. (88) except that in the
former j is summed over all nuclei except those in the
kth molecule, while in the latter j is summed exclusively
over the nuclei in the kth molecule.
For extreme motional narrowing, assuming the validity of the expansion in semivariants, G(t) is
NS(S+1)'Y2
G(t)= L: [expi{'YH+Am
3(21+1) m
-S(S+ 1)K(/,m)F/3A (21 +l)}tJ
X [exp{ -S(S+1)IFf2j3(21+1)}]
X [exp{ -O"d(inter)55/(/+l)/18
-0"182 (intra)55m2/6-50" 882S(S+ 1)/2
-8([J(1)]2)S(S+ 1)//3(21 + 1)} Tel t I}. (101)
This expression can be used to estimate the line shape
and width. In dilute solutions where the exchange
broadening is negligible, the lines are almost Lorentzian
in shape with a half-width .1."'1 given by the coefficient
of It I in the third exponential in Eq. (101). 0"182 (inter)
and O"d (intra) may often be of the same order of
magnitude, and the sum of the two is equal to the 0"1.2
introduced previously. In dilute solutions 0"8.2 may be
very small, as discussed above, and so the nuclear
electronic dipolar interactions may often be the domi
nant line broadening effect. The intramolecular broad
ening is seen to be proportional to m2 which means that
for I>! a dependence upon m should be observed in
the half-width of the hyperfine lines. If the exchange
term J is dominant over the other effects included in
Eq. (101), then the discussion on line shapes following
Eq. (49) is valid. The dipolar and exchange effects
could both be included if in obtaining the spectral
density the less significant of the two is expanded
before taking the Fourier transform.
An experimental program to verify the theory out
lined above and to study the relative importance of the
various terms will soon be underway. Of course, the
effect of anisotropic g factors, even in solution, and of
chemical reactions must be also considered. These
effects will be discussed in a subsequent article.
ACKNOWLEDGMENTS
The author is grateful to Professor George Fraenkel
for introducing him to the science and lore of exchange
effects during the summer that the author worked in
Professor Fraenkel's laboratory. He would also like to
acknowledge the support extended by the Research
Corporation and the National Science Foundation.
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1.1740001.pdf | Paramagnetic Resonance Absorption of Violanthrone and Violanthrene
Y. Yokozawa and I. Tatsuzaki
Citation: The Journal of Chemical Physics 22, 2087 (1954); doi: 10.1063/1.1740001
View online: http://dx.doi.org/10.1063/1.1740001
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144.32.240.69 On: Thu, 04 Dec 2014 02:08:20LETTERS TO THE EDITOR 2087
Paramagnetic Resonance Absorption of
Violanthrone and Violanthrene
Y. YOKOZAWA A:-l"D I. TATSUZAKI
The Research Institute 0/ Applied Electricity,
IIokkaido University. Japan
(Received September 20, 1954)
THE diamagnetic susceptibilities and anisotropies of con
densed polynuclear aromatic hydrocarbons have been
measured by Akamatu and ),Iatsunaga,l their results for the
violanthrone and the violanthrene being shown in Table I in
which the small diamagnetic anisotropy of the violanthrone
compared with that of the violanthrene is noteworthy.
Present investigation was undertaken to examine a view that
this small diamagnetic anisotropy was due to the cancellation by
the hidden paramagnetism involved in the violanthrone. This
hidden paramagnetism was detected by the method of microwave
paramagnetic resonance absorption using a 3.2-cm wave at
room temperature. The magnetic absorption was measured using
a rectangular reflecting cavity operating in TEol2 mode. To
eliminate the crystal detector noise at audio-frequencies, the
reflected power was balanced using a magic tee to a level at which
superheterodyne receiver and a local oscillator, followed by a
intermediate-frequency amplifier at 30 :\ic/sec, could be used.
FIG. t. Oscilloscope trace: absorption spectrum of violanthrone.
The absorption of the violanthrene is also observed (see Fig. 1).
The g values, half-widths 6.Hj, and magnetic susceptibilities are
shmm in Table II.
The paramagnetic part of the susceptibility involved in the
violanthrone ,,,as obtained from a comparison of the integrated
intensity of the absorption curve with that of a single crystal
of euso,· 5H20. The absorption intensity of the violanthrene
was very small, and the paramagnetic contribution to the total
magnetic susceptibility could be neglected. By adding these
contributions, the results are obtained: diamagnetic part of the
susceptibility -n[=264X10~G, diamagnetic anisotropy -6.K=
330X 1O~6 and averaged ... orbital radius (t~2)! = 1.5, 1A in the
violanthrone. These magnitudes are the same orders with those
of the violanthrene. Assuming this paramagnetism is originated
TARLE I. Diamagnetic susceptibilities and anisotropies
of violanthrone and yiolanthrene.
Violanthrone
Violanthrene Mole sllscept.
-x.v·IO'
204.8
273.5 Anisotropy
per mole
-"'K·IO'
141
320 Average orbital
radius Cy2)t
(A)
1.05
1.49 TABLE II. Paramagnetic resonance data.
g "'H; (Oer) Paramag.
suscep. per mole
x·IO'
Viol anthrone
Violanthrene 2.00
2.00 15
13 63
in unpaired 7r electrons, the fractional magnetic population of
these ... electrons x is obtained from the relation,
Nxg2{32S(s+n
x= 3kT ~-
where N is Avogadro's number. From this eCJuation, it is found
x~1/100.
The authors are greatly indebted to Professor Akamatu for
providing them with these organic compounds.
1 H. Akamatu and Y. Matsunaga. Bull. Chern. Soc. Japan 26, 3M (1953).
Exchange Potential in the Statistical Model
of Atoms*
C. J. NrSTERUK AND H. J. JURETSCIIKE
Polytechnic Institute of Brooklyn, Brooklyn, ~Yew York
(Received October 7, 1954)
THE statistical model of the atom which includes the free
electron exchange potential of Dirac! has the shortcoming
that it leads to electron distributions decreasing to zero discon
tinuously at a finite radius Ro from the nucleus." We want to
report some results obtained with a model based on a modifIcation
of Dirac's potential in the atomic surface region.'
The exchange potential can be interprete(1 as the potential
arising from a distribution of unit positive charge, the exchange
hole. Slater' has given a simplified expression for such a distribu
tion representing an averaged exchange hole common to electrons
of all energies. For electrons described by plane waves this
exchange hole has spherical symmetry and is always centered at
the position of the electron in CJuestion. \Vave functions proper
to atomic boundary conditions yield an exchange hole of consider
ably more complex shape. In the interior of the atom the hole is
concentrated around the electron position. As the electron
distance from the nucleus increases the hole tends to remain in
the outermost shell, at first concentrated around the nucleus
electron axis but later distributed more uniformly throughout
the shell.
This behavior of the hole suggests that in the atomic interior a
density dependent free-electron exchange potential is adequate,
while in the far outer region of the atom the exchange potential is
more nearly that due to a concentrated unit charge located at
first near the outermost shell. but approaching the nucleus as
the electron moves far away.
We have extended the variational approach of statistical theory
to include a simplified exchange potential with the above general
properties. If r is the distance of the electron from the nucleus,
then for r ~Ri the exchange potential is given by the usual
electron density-dependent expression of Dirac +e(3n/ ... )I. For
r ?,Ri it is represented by a density-independent function con
tinuous with the inner expression at Ri and approaching e/y at
large r. Ri is an additional parameter in the variational problem.
Its equilibrium value indicates the extent to which the exchange
hole follows its electron in an atom.
The differential equation for the density obtained in the
variation is identical with that set up by J ensen5 on physical
grounds. The outer boundary condition, derived formally, differs
from that assumed by Jensen. It requires that the density vanish
continuously at RD. Thus, the substitution of a density indcpend-
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144.32.240.69 On: Thu, 04 Dec 2014 02:08:202088 LETTERS TO THE EDITOR
ent exchange potential near the surface removes in a natural
manner the anomaly of the usual Thomas-Fermi-Dirac boundary
condition.
The resulting value of Ri depends on the particular function
chosen for the density-independent potential. Actually, there is
little choice, since, as Jensen has pointed out, the potential is
practically completely determined by its boundary values. We
have found that, for all reasonable potentials, Ri lies very close
to the nucleus. Thus, for Kr, Ri <0.6ao. This results because a
potential asymptotic to l/r is stronger than Dirac's potential
over most of the atom. In the interior of the atom the same
relationship is maintained because the density there is independent
of the exact form of the exchange potential.
The small value of Ri indicates that the exchange hole remains
stationary near the nucleus for all positions of its electron. There
fore the electron distributions to be expected in this model are
not very different from those obtained in the Fermi-Amaldi6
modification of Thomas-Fermi's theory. Instead of using a
reduced effective number of electrons we substitute an increased
effective nuclear charge.
The statistical approach does not lead to an asymptotic ex
change hole stationary somewhere in the surface region of the
atom. This property of the exchange hole in the more exact theory
is intimately connected with atomic shell structure, and one may
expect that a statistical model will describe atomic surface prop
erties accurately only when it also exhibits shell structure.
* This work has been supported in part by the Office of Naval Research.
I P. A. M. Dirac, Proc. Cambridge Phil. Soc. 26, 376 (1930).
2 P. Gombas, Die statistische Theorie des Atoms (Springer Verlag, Vienna,
1949) p. 80.
3 C. J. Nisteruk, M. S. thesis, Polytechnic Institute of Brooklyn (1954).
• J. C. Slater, Phys. Rev. 81,385 (1951). 'H. Jensen, Z. Physik 101, 141 (1936).
'E. Fermi and E. Amaldi, Mem. Accad. Italia 6, 117 (1934).
Different Ice Forms under Ordinary Conditions*
R. M. VANDERBERGt AND J. W. ELLIS
Department of Physics, University of California, Los Angeles, California
(Received October 1, 1954)
WHILE determining infrared birefringences of single
crystals of ice by means of channeled interference spectra,
it was observed that a group of interference maxima and minima
near 2.0).1 replaced a similar group recorded a half year earlier at
approximately 0.1).1 shorter wavelengths. The earlier results were
obtained with several crystals grown at that time, the later with
a crystal prepared approximately two months before recording
the results. A comparison of the absorption spectrum of the new
crystal with the spectrum which had been recorded of the earlier
material also showed a pronounced difference. Fortunately, some
of the earlier crystals had been preserved in a refrigerator. When
the absorption and birefringence spectra of the older material
were now reinvestigated it was found that both had changed so
that they conformed with the spectra of the later crystal. Thus
it seems that we have had at least two crystal types. We shall
designate these earlier and later grown types by A and B, respec
tively. Although it is certain that the earlier ice changed from
type A to type B during, or at some time during, the half year it
was in the refrigerator, unfortunately there is no way of knowing
whether the later ice had been grown as type B, because no study
had been made of it during the first two months of its existence.
The original type A material which had changed to type B
showed no further spectral changes during the ensuing four
months.
All of the crystals used were grown in the following manner.
A small seed crystal was cut from a large ice block and was
placed on the lower end of an aluminum rod whose upper end
projected into a refrigerating unit kept at approximately -10°C.
The seed was dipped into a container of distilled water kept at O°C by an ice jacket. Crystals grown by this method assume a
roughly hemispherical shape, exhibiting no plane faces associated
with the usual hexagonal nature of the crystal. The orientation
of the optic axis is always the same as that of the seed crystal.
Working plates of ice were cut from these larger crystals as
desired.
Near the end of our experimental program, after the existence
of types A and B had been clearly revealed, another crystal was
grown and immediately studied. Its absorption spectrum, although
more nearly like type A than type B, shvws distinct differences.
Hence we designate it type C. Its absorption spectrum was
occasionally recorded over a four months' period but no observable
change occurred. It is possible that this type C crystal was grown
more slowly than the others. These results indicate that a more
detailed investigation of crystal forms of ice could profitably be
made, with careful attention to conditions of growth.
The absorption differences among ice types A, B, and C consist
of changes in the structure of absorption bands near 2.0jl, pre
sumably associated with hydrogen bridging between water
molecules. In general the shift from A to B involves a displacement
of certain absorption maxima to longer wavelengths. Whether the
change is from greater order to disorder or vice versa in the
crystal structure seems impossible to say.
The changes involved are not associated with strain in the
crystals. We have subjected ice plates to stress and have shown
that the uniaxial form changes to biaxial without any appreciable
change in the absorption spectrum or in the dichroism which,
contrary to the findings of Plyler,l is small or lacking for all
wavelengths in the very near infrared, and with only a slight
general shift in the channeled spectrum.
Independently of the several well-known forms of ice produced
by Bridgman under extreme conditions, references to two forms
of ice found under ordinary conditions occur. Thus in the Hand
book of Chemistry and Physics' a and f3 forms are tabulated, with
hexagonal and rhombohedral symmetry, respectively. Seljakov3
believed he had shown the existence of two forms by means of
x-ray diffraction. However, Berger and Saffer< think they have
demonstrated an error in Seljakov's technique and hence seriously
question his interpretations.
* The material of this letter was taken from the Ph.D. thesis of R. M
Vanderberg. t Now at Sacramento State College, Sacramento, California.
IE. K. Plyler, J. Opt. Soc. Am. 9, 545 (1924). . . .
'Handbook of Chemistry and Physics (ChemIcal Rubber Pubhshlng
Company, New York, 1950-51), 32nd edition, p. 2225.
3 N. Seljakov, Compt. rend. acado sci. U.S.R.R. 10,293 (1936); 11, 92
(1936); 14,181 (1937). 'C. Berger and C. M. Saffer, Science 118,521 (1953).
Formation of Negative Ions in Hydrocarbon Gases*
T. L. BAILEY, J. M. MCGUIRIJ:, AND E. E. MUSCHLITZ, JR.
College of Engineering, University of Florida, Gainesville, Florida
(Received August 23, 1954)
IN connection with studies of collisions of gaseous negative
ions with neutral molecules,! negative ions produced by
electron bombardment of methane, ethane, and acetylene gases
have been investigated in a mass spectrometer. The ions observed
and their relative intensities under similar source conditions are
shown in Table I.
TABLE I. Relative negative ion intensities.
Mass Mass Electron
Gas H- 25 12-15 energy
CH, 120 12 1.5 35 ev
C,H. 73 27 0.5 70 ev
C,H, 8.5 55 0.0 70 ev
(100~1O-12 amp)
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1.1729875.pdf | Effect of Unimolecular Decay Kinetics on the Interpretation of Appearance
Potentials
William A. Chupka
Citation: J. Chem. Phys. 30, 191 (1959); doi: 10.1063/1.1729875
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Downloaded 05 Sep 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://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 30, NUMBER 1 JANUARY, 1959
Effect of Unimolecular Decay Kinetics on the Interpretation of Appearance Potentials*
WILLIAM A. CHUPKA
Argonne National Laboratory, Lemont, Illinois
(Received May 19, 1958)
The interpretation of appearance potential data on diatomic molecules should take account of possible
effects caused by predissociation, emission of light and autoionization. In the case of complex polyatomic
molecules, the kinetics of predissociation and the internal thermal energy of the molecules become especially
important. The intensities of the parent, fragment, and metastable ions produced by photoionization of
n-propylamine, n-propanol, and methyl ethyl ketone are studied as a function of photon energy. The excess
kinetic energies of the fragment ions are found to be negligibly small. The data are interpreted in terms of
Rosenstock's quasi-equilibrium theory of unimolecular decomposition and indicate that the theory is
qualitatively correct for the dissociative processes investigated. However, the theory is shown to be quan
titatively inadequate at least in the energy range near threshold. In this region the rate constant for disso
ciation varies much more rapidly with energy than the theory predicts. Some of the assumptions of the theory
are examined and compared to deductions from the data.
The meaning of appearance potential data is examined in the light of these results. The effects of both the
kinetics of dissociation and of internal thermal energy on ionization efficiency curves are significant. Most
of the methods used to determine appearance potentials tend to minimize these effects and there is probably
some cancellation of errors. A new method for the determination of appearance potentials is described. Ex
perimental methods which can yield more detailed information concerning dissociation processes of complex
molecular ions are suggested.
INTRODUCTION
THE study of the appearance potentials of various
processes resulting from electron impact on gaseous
atoms and molecules has occupied an increasing num
ber of workers in recent years. These appearance po
tentials have been interpreted to give values of ioniza
tion potentials and dissociation energies of chemical
bonds. The pioneering work of Tate,1 Hagstrum,2 and
others dealt chiefly with atoms and diatomic molecules.
The theory employed to explain the results was similar
to the theory of optical spectra. This theory has been
expanded by Morrison,3 among others4 and the simi
larity to optical spectroscopy prompted him to term
this field of study "elecfron-impact spectroscopy." It
is one purpose of this paper to point out some important
differences between the two kinds of spectroscopy which
do not seem to be fully appreciated. In particular, the
kinetic aspects of predissociation of molecular ions, es
pecially large polyatomic ions, can be very important
in the interpretation of appearance potentials. Rosen
stock et al.5 have used the concepts of kinetics of uni
molecular decomposition to develop a theory to explain
mass spectra of polyatomic molecules bombarded with
electrons of energies well above the usual range of ap
pearance potentials. In this paper, these concepts are
applied to processes occurring at the appearance po
tential and some experimental evidence is presented to
* Work performed under the auspices of the U. S. Atomic
Energy Commission.
1 H. D. Hagstrum and J. T. Tate, Phys. Rev. 59, 354 (1941).
2 H. D. Hagstrum, Revs. Modern Phys. 23, 185 (1951).
3 J. D. Morrison, Revs. Pure App!. Chern. 5, 22 (1955).
4 F. H. Field and J. L. Franklin, Electron Impact Phenomena
(Academic Press, Inc., New York, 1957).
, Rosenstock, Wallenstein, Wahrhaftig, and Eyring, Proc. Natl.
Acad. Sci. U. S. 38, 667 (1952). support this applicability. The effects of the tempera
ture of the gas on appearance potentials will also be
considered.
An analysis of these effects becomes more important
in view of recent advances in experimental techniques
which promise to yield more data of higher accuracy.
These advances include especially the use of mono
chromatic electron beams and the new technique of
photoionization to which most of the considerations of
this paper will also apply.
TECHNIQUES OF ELECTRON AND PHOTON IMPACT
Since many of the features of electron-impact spectra
are a function of the experimental arrangement, it is
necessary to consider the latter briefly. A great ma
jority of appearance potentials have been measured
using mass spectrometers of rather similar design. The
gas under investigation is bombarded by a narrow
beam of electrons produced by thermionic emission.
The resulting ions remain in the ionization chamber for
a time of the order of a few microseconds, and are then
drawn out and accelerated to energies of a few thousand
volts in about a microsecond. They then spend several
microseconds each in a field-free region, in a magnetic
field and then in another field-free region before they
are finally detected. In some arrangements, the two
field-free regions are absent.
The thermal energy distribution and other sources of
energy spread of the bombarding electrons result in the
production of an appearance potential curve (i.e., a
plot of ion intensity vs electron energy) which has no
definite "onset" but approaches the axis asymptotically.
Various methods have been used by different workers
to determine "true appearance potentials," but all in
volve assumptions which are at best unproven. This
191
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situation makes it difficult to formulate a predse
interpretation of appearance potentials. Very recently
several techniques have been developed which eliminate
the effects of the energy spread of the electrons. The
best known of these are: (1) the technique of Fox et at.,6
(2) the use of an electrostatic analyzer to produce a
monochromatic electron beam,7 and (3) the technique
of photoionization employing a vacuum ultraviolet
monochromator.s Also, the technique of Morrison,9
while not eliminating the effect of electron energy
spread, clearly displays its effect and also makes pos
sible the resolution of fine structure in appearance
potential curves. These techniques promise to provide
appearance potential data amenable to precise inter
pretation and have already led to discovery of details
in appearance potential curves unobserved by the older
techniques.
INTERPRETATION OF IONIZATION EFFICIENCY
CURVES FOR MOLECULES
A. General
The energy dependence of the cross section for forma
tion of ions varies with the type of process producing
the ions and the means of excitation. The threshold be
havior has been examined theoreticallylO and experi
mentally for most processes of interest. The results,
while varying somewhat, indicate that the cross section
at threshold for direct ionization is proportional, or
nearly so, to En-1 where n is the number of electrons
leaving the collision complex and E is the energy above
threshold. This is one fact which makes the determina
tion of thres~old energies more precise for photoioniza
tion than for ionization by electron impact. Threshold
laws for formation of un-ionized electronically excited
states, some of which can produce ions by autoionization
or by decay into positive and negative fragments, are
not as well known for electron impact. While theory
indicates that the probability of excitation should be
proportional to Ei, one experiment indicates a linear
dependence on E. t For photons, of course, discrete line,
band, or continuous absorption may occur. As regards
selection rules for the various processes, it can be ex
pected that they are less restrictive for electron impact
than for photon excitation because of the additional
electron available for transfer of spin and orbital angu
lar momentum.
In interpreting ionization efficiency curves, it is
necessary to remember that what is measured is the
6 Fox, Hickam, and Kjeldaas, Phys. Rev. 89, 555 (1953).
7 E. M. Clark, Can. J. Phys. 32, 764 (1954).
8 Hurzeler, Inghram, and Morrison, J. Chem. Phys. 28, 76
(1958).
9 J. D. Morrison, J. Chern. Phys. 21, 2090 (1953).
10 E. Wigner, Phys. Rev. 73, 1002 (1948); G. H. Wannier, Phys.
Rev. 90, 817 (1953); S. Geltman, Phys. Rev. 102, 171 (1956);
G. J. Shultz and R. E. Fox, Phys. Rev. 106, 1179 (1957). t The author is indebted to S. Geltman for an illuminating
discussion of this point. relative amounts of various ions present several micro
seconds after electron or photon impact. In general, ion
abundances will be a function of this delay. This rela
tively long delay makes the mass spectrometer a much
more sensitive instrument for the detection of predis
sodations than is the optical spectroscope. A predis
sodation is detectable in emission spectra if the pre
dissociating state has a half-life of about 10-8 sec or
less and in absorption spectra if the half-life is about
10-10 or less.n However, predissociations yielding one
or more charged particles may be detected with the
mass spectrometer if the lifetime of the predissociating
state is about 1O~5 sec or less. Of course, a weak pre
dissociation with a lifetime greater than "" 10-8 sec, in
order to be detectable, must occur from a state which
is metastable with respect to radiative transition to a
lower state. Such metastable states may be produced
readily by electron impact.
B. Diatomic and Simple Polyatomic Molecules
Much of the electron impact data on diatomic mole
cules has been successfully interpreted by application
of the Franck-Condon principle, conservation of energy
and momentum and the hypothesis that the variation
of cross section with electron energy is such that the
threshold can be measured with an accuracy of the
order of a tenth of an electron volt. However, even for
such carefully investigated molecules as CO, NO, and
O2, a few processes defy interpretation in this manner.
The technique of Morrison9 has been applied with
some success to the detection of excited ionic states.
Coupled with the use of monoenergetic electron beams,
this technique should yield very valuable results. How
ever not all peaks will correspond to excited states of
the ion. For instance, a strong predissociation leading
to fragment ions could result in a dip in the second
derivative curve as shown in Fig. 1. This would be
analogous to the observation in optical spectroscopy
of the absence in emission of bands belonging to the
predissociated state. The following peak could then be
misinterpreted as indicating another excited state of
the ion. Such a process could be identified by the
observation of a peak in the second derivative curve
for the fragment ion at the same energy as the dip in
the curve for the parent ion. Incidentally, a predis
sociation such as that indicated in Fig. 1 could yield
ions of considerable kinetic energy and yet have a
rather sharp threshold which is generally considered
to indicate a process producing fragments of zero
kinetic energy at threshold.8
The study of fragment ions produced by electron im
pact has long been used for the determination of bond
energies. In the case of diatomic molecules, it has been
11 G. Herzberg, SPectra of Diatomic Molecules (D. Van Nostrand
Company Inc., Princeton, New Jersey, 1950), second edition.
p.413.
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tacitly assumed that all the energy transferred to the
molecule by electron impact appears in the product
fragments as either kinetic energy or energy of elec
tronic excitation. In the most detailed experiments
such as those of Hagstrum,2 the kinetic energy was
measured directly and the electronic energy inferred
from a correlation of appearance potentials of several
processes with known excited states of the product ion
or atom. In spite of this evidently complete knowledge,
there are well-measured appearance potentials which
resist interpretation on this basis alone.
Besides the possibility of experimental errors, there
exists the possibility that some energy may appear in
forms other than those mentioned. For instance, it
may appear in the kinetic energy of an electron ejected
in the process of preionization either of the molecule
or of a neutral fragment thereof.t It may also appear
as radiation emitted by the excited molecule. The
latter case would seem to be more likely for higher
energy processes in which there is produced initially an
excited molecule or molecular ion which then makes an
FIG. 1. Potential
curves for a diatomic
molecule AB, show
ing the effect of pre
dissociation on the
electron or photon
impact spectrum. >to a: '" i--'-........ v-I.....-___ ---...~---. _____ _
ZI'--.......... r-1,--.-../'----....-..~
'" A+B
INTERNUCLEAR DISTANCE
allowed electronic transition (lifetime'" 10-8 sec or at
least < 10-6 sec) to a lower dissociating or predissoci
ating state. Because of the operation of the Franck
Condon principle and because the direct production of
the lower state by electron impact must also be allowed
in this case, the dissociation of the lower state usually
will have been observed directly at lower electron
energies. However, one can construct a possible set of
potential curves for which this need not be true. Such
a set is shown in Fig. 2. In any case, the value of the
threshold energy for such a process will generally not
fit into the usual simple scheme of interpretation.
Loss of energy by pure rotational transitions also
generally will not be effective. However, it is conceiv-
t The author has been informed that this suggestion was first
made by C. R. Lagergren in a thesis presented to the Department
of Physics at the University of Minnesota. FIG. 2. Potential curves for a
diatomic molecule AB, show- ~
ing loss of energy by radiation ::;
followed by dissociation. ;::: A+B
INTERNUCLEAR DISTANCE
able that a predissociation may occur after initial
production of a highly vibration ally excited molecule
followed by de-excitation to a predissociated vibra
tional level by emission of several vibrational quanta
(see Fig. 3). Generally, the emission probability would
be too small to allow emission of an energetically sig
nificant number of quanta. However, for light molecules
with large vibrational frequencies and in ionic excited
states such as lead to decomposition into positive and
negative ions, the vibrational quanta and the transition
moment may be large enough to allow such a process
to occur in 10-6 sec with sufficient loss of energy by
radiation to disrupt the usual correlation scheme.
Nevertheless, such an occurrence would probably be
very rare.
For simple poly atomic molecules, an added compli
cation is the possibility of the occurrence of appreciable
excess vibrational energy in the molecular fragments.
Also, in cases where the dissociation occurs with excess
kinetic energy but not along a line connecting the
centers of mass of the two fragments, appreciable excess
rotational energy may be present. The amount of excess
vibrational and rotational energy is not measureable at
FIG. 3. Potential curves
for a diatomic molecule
AB, showing loss of energy
by radiation of vibrational
quanta followed by predis
sociation. >-
~ v--___ -....,..~-"'---.... ___ -..,..-
'" ,~---~~'-'"'-""-----,~----'z
'" A+B
INTERNUCLEAR DISTANCE
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TABLE I. Half-lives of excited molecules as calculated by Marcus.
Active Excess rXl()6Xsec rXl()6sec
molecule energy Product (exp) (calc)
CILD* "'0.1 ev CH.D+H 1.4X1O-3 5.3XIo-'
CH.D,* ",0.1 ev CHD,+H 0.9X1O-3 16.0X1O-'
C.H6* ",0.5 ev 2CH, 6.0X1O-2 9.2XIO-2
C3Hs* ",0.5 ev CH3+ C2H5 2.0 22.0
present although in principle its effect might be noticed
by making a plot of appearance potential vs kinetic
energy as was done by Hagstrum.2 Deviation from
linearity and predicted slope could be expected in this
case unless the excess vibrational plus rotational
energy remained constant, which is unlikely.
Considering the simplicity of the interpretation used . . . . ' It IS not surprIsmg that a small number of electron im-
pact processes are apparently anomalous. Techniques
of higher resolution should clear up these anomalies
and make identification of the processes more certain.
C. Large Polyatomic Molecules
For large polyatomic molecules, it becomes imprac
tical to attempt to interpret processes occurring upon
electron impact in terms of detailed potential hyper
surfaces for the various states of the molecule and the
molecular ions formed from it. A more fruitful approach
appears to be a statistical one, such as that developed
by Rosenstock et at.5 to explain the mass spectra pro
duced by electron impact on large molecules (e.g.,
propane). This approach is based on the hypothesis
tha~ upon ver~ical ionization the molecular ion, pos
sessmg a certaIn amount of electronic and vibrational
energy, generally does not dissociate within the time of
o~e . vibratio~ but instead rapidly and randomly
dIstrIbutes thIS energy. The molecular ion may then
und:rgo unimolecular decomposition along energetically
avaIlable paths only by attaining certain configurations
with sufficient vibrational energy concentrated in the
proper modes. This hypothesis is supported by some
experimental evidence.
If this hypothesis is correct for processes occurring
near their appearance potential, two important conse
quences to be considered are the effects of the lifetime
of the parent molecular ion and the effect of tempera
ture.
The rate of decay of the parent molecular ion in a
particular mode of dissociation is expected to be a
function of the energy excess above that necessary to
cause the decay. The mass spectrometer analyzes the
products of dissociations which occur in appreciable
amounts in the time of several microseconds. Thus it . . ' IS essentIal to consider the amount of excess energy
which the molecular ion must have in order that the
dissociation be detectable by the mass spectrometer. Several authors5.12-15 have devised methods which
might be used to calculate such quantities and some
such information has also been deduced from experi
mental data.14 These various methods generally are in
very rough agreement, considering their necessary
crudity. They indicate that the excess energy necessary
to reduce the half-life to the order of a microsecond
increases with the number of internal degrees of free-.
dom of the molecule or molecular ion, as is to be ex
pected. With sufficiently complex molecules, it will
become considerably larger than the limits of error
(about ±0.1 ev) usually quoted for appearance poten
tials. As a consequence of these considerations, the
appearance-potential curve for a fragment ion should
approach the energy axis with curvature, quite apart
from any effect resulting from energy spread in the
electron beam, and the ion intensity may become
vanishingly small at energies appreciably above the
theoretical appearance potential. When the effect of
temperature is considered, the curve is expected to be
asymptotic to the energy axis. In the usual electron
impact experiments, such behavior frequently would
be masked by the effect of the thermal spread of
electron energy.
Before an estimate can be made of the magnitude of
possible errors in appearance potentials due to these
effects, it is necessary to decide on the significance of
the appearance potential of a fragment ion as usually
determined. At this point it would seem not to be far
wrong to consider the appearance potential to signify
the electron energy necessary to produce parent ions
of such excitation that they decay to produce the
fragment of interest with a half-life in the range from
10-" to 10-6 sec (or perhaps as short as 10-8 sec if
energy can be lost by an allowed radiative transition).
Then the possible error will be the excess energy re
quired in the parent ion to produce this decay rate.
The calculations of Marcus14 and of Rosenstock
et at." are particularly illuminating. Marcus used ex
perimental data and some assumptions regarding the
mechanism for deuterization of methyl radicals and
atomic cracking of ethyl and propyl radicals to calcu
late the dissociation rates of the excited molecules
produced in these reactions. The results are shown in
Table I where T is the half-life and A is a factor repre
senting the efficiency of a deactivating collision and is
usually set equal to unity. One set of theoretical values
of T calculated by Marcus is shown in the last column.
If the experimental data for propane could be applied
to the propane ion produced by electron impact, then
12 L. S. Kassel, Kinetics oj lIomogeneous Gas Reactions (Rein-
hold Publishing Corporation, New York, 1932).
13 G. E. Kimball, J. Chern. Phys. 5, 310 (1937).
14 R. A. Marcus, J. Chern. Phys. 20, 352-368 (1952).
15 N. B. Slate~, Proc. Roy. Soc. (London) A194, 112 (1948);
N. B. Slater, PhIl. Trans. A246, 47 (1953). See also A. F. Trot
man-Dickenson, Gas Kinetics (Academic Press, Inc. New York
1955). ' ,
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the experimental appearance potential would be about
0.5 ev too high, other factors such as initial thermal
energy being neglected. Actually, the dissociation of
propane into the products indicated requires appreci
ably more energy than the similar dissociation of the
propane ion and theory indicates that the excess energy
required would be smaller in the latter case. Neverthe
less, the magnitudes of the excess energies and the
rapid increase of half-life with molecular complexity
strongly suggest the importance of these factors in the
dissociation of more complex molecular ions.
A more appropriate theory which allows calculation
of these rates of decomposition is that developed by
Rosenstock et aJ.5 By making several simplifying
assumptions, the following equation is derived for the
rate of a particular decomposition,
N N-I
k=[(E-E)/E]N-1(II".;II"i), (1)
i=1 i=l
where E is the total energy less zero-point energy of the
molecular ion, E the minimum energy required for the
decomposition, N the number of oscillators in the
molecular ion, and", and" i the vibrational frequencies
of the molecular ion and of the activated complex, re
spectively. A more complex expression may be derived
for a system containing free and restricted internal
rotors.
This theory has been used by several authors,",16 to
calculate the amount of fragment ions produced in
10-6 sec as a function of internal excitation energy. It
is obvious from these calculated curves that the elec
tron-impact appearance potentials of many fragment
ions from molecules of the complexity of propane and
butane would be too high by several tenths of a volt
or more. Furthermore, if this theory is correct, any
thermal vibrational energy in the original molecule
will contribute to E. This will tend to lower those ap
pearance potentials by amounts of similar magnitude.
This situation casts doubts on the meaning of electron
impact data on such complex systems. Friedman et al.16
have considered that the apparent success of electron
impact measurements indicates that the theory of
Rosenstock et al. is not quantitatively applicable near
threshold energies. It is, therefore, important to in
vestigate the details of the process of unimolecular
decomposition of molecular ions particularly in the
energy region corresponding to lifetimes of the order
of 10-6 to 10-j) sec.
In order to get useful data in such an investigation,
it is important to minimize the energy spread of the
ionizing electrons or photons. There has been prac
tically no work done on fragmentation of large mole
cules by monoenergetic electrons. However, the recent
16 Friedman Long, and Wolfsberg, J. Chern. Phys. 27, 613
(1957). ' work by Hurzeler, Inghram, and Morrison8 on photo
ionization by monochromatic photons indicates that
this technique may be very valuable in this type of
study. These authors attempted to explain some of
their experimental ionization probability curves of both
parent and fragment ions in terms of a scheme similar
to that applied to diatomic molecules, that is, in terms
of potential surfaces and electronic transition prob
abilities, a view that had previously been elaborated
by Morrison.9 However, the shortcomings of this
interpretation were appreciated by these authors. In
fact, it will be shown that the results of Hurzeler et al.
provide excellent confirmation of certain aspects
of the theory of Rosenstock et al. If this theory is ap
plicable, one would expect the appearance of an ap
propriate metastable ion to be associated with the
appearance of each fragment ion, except where the
probability of exciting the parent molecular ion to the
appropriate energy region is too low or where a com
peting process is much more probable in the energy
region. Also, the shape and position of the ionization
efficiency curve for the metastable ion are predictable
from the theory. Furthermore, the fragment ions
would be expected in general to have kinetic energy
distributions of a type similar to a Maxwellian distribu
tion with an average energy of the order of a tenth of a
volt or so except where dissociation occurs over a po
tential "hill." More specifically, the average total
kinetic energy should be about l/n of the total excess
energy above that necessary to just cause dissociation,
where n is the number of internal (vibrational) degrees
of freedom of the molecular ion. If the diatomic-like
interpretation is applicable, average kinetic energies of
the order of a volt or so would be expected. More
specifically, the total kinetic energy should be roughly
equal to the excess energy. Also, the occurrence of
metastable ions would be infrequent and unpredictable.
The experiments now to be described were done in
an attempt to determine the validity of the theory of
Rosenstock et al. The three compounds studied,
n-propylamine, n-propyl alcohol, and methyl ethyl
ketone, were representative of the three groups of com
pounds studied by Hurzeler et al., i.e., primary amines,
alcohols, and ketones. In each case the compound
selected was the one with the largest number of atoms
so that a statistical theory would be most applicable.
The object of the investigation was to establish the
presence of certain metastable ions and to compare
the intensity and dependence on photon energy, of
these metastables as well as the parent and fragment
ions, with the predictions of the statistical theory.
EXPERIMENTAL PROCEDURE
The apparatus used for photoionization was the
same as that used by Hurzeler et al.s and is described
in detail in their article. A drawing-out potential of
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CH3CH2CH2NH2 ........ ~ ........ _ .............................. .
59 .... - ...... ~ ................. , •••
--.--;--
8.5 9.0 9.5 10.0 10.5 11.0
PHOTON ENERGY ..
FIG. 4. Photoionization efficiency curves (dashed line) and
their first derivatives (solid lines) for the production of C3H7NH2+
and CH2NH, + ions and the associated metastable ion from
n-propylamine (dashed curves).
22.5 volts was applied across the ionizing region. The
total accelerating voltage was 3000 volts. The pressure
of the sample gas was kept at about 5 X 10-6 mm or less
in order to minimize secondary reaction. The energy
spread of the photon beam used had a half-width of
about 0.15 ev. This rather large value was used in order
to get sufficient intensity of the metastable ions. The
rest of the experimental procedure was identical to
that of Hurzeler et al.
Some supplementary electron impact measurements
were made with a mass spectrometer very similar to
the one used for photoionization except that the ion
detector contained a fine wire grid of known transmis
sion before the electron multiplier. By means of this
grid, ion currents were measured both directly and
after amplification by the electron multiplier and the
gain for each ion was thus determined. The electron
beam was about 1.0 rom in diameter and all electro
static potentials were adjusted to correspond closely
to those used in the photoionization mass spectrom
eter. The pressure of sample gas was varied from about
10-6 mm to about 5.0X 10-6 mm. The ratio of parent
and metastable ion intensities was measured as a func
tion of sample gas pressure, electron energy, and focus
ing conditions in the source and collector focusing
systems.
In addition, kinetic energy measurements were made
on fragment ions produced by electron bombardment
of the three compounds investigated. The kinetic
energies were measured by a cylindrical electrostatic
analyzer using a technique previously described.17 The
energy of the bombarding electrons was 30 ev.
RESULTS AND INTERPRETATION
For all three compounds a search was made for the
metastable ion produced by decomposition of the
17 J. D. Morrison and H. E. Stanton, J. Chern. Phys. 28, 9
(1958). parent ion to produce the fragment of lowest appear
ance potential as determined by Hurzeler et al. Spe
cifically, the following unimolecular decompositions
were to be studied:
(3)
As shown by Hipple, Fox, and CondonI8 these processes
will lead to the appearance of metastable ions at gen
erally nonintegral masses m*, given by m*=m2/mo
where mo is the mass of the parent ion and m is the
mass of the charged fragment. The mass positions of
the metastable ions produced by the processes given
above are therefore 15.25 amu by process (2), 29.4 by
(3), and 25.7 by (4). These mass positions were located
by producing ions first by electron impact using 75-
volt electrons. In all cases, a small ion peak was located
at the proper position. When the electron beam was
turned off and the photon beam turned ()n at the ap
propriate wavelength, the ions were again detected in
all cases. Figures 4, 5, and 6 show the intensity of the
metastable ion as a function of the photon energy and
also the first derivative of this curve. These figures also
show similar curves obtained by Hurzeler et at. for the
9.5 10.0 10.5 11.0 11.5
PHOTON ENERGY tv
FIG. 5. Photoionization efficiency curves (dashed lines) and
their first derivatives (solid lines) for the production of C3H,OH+,
C3H,OH+, C3H,+, and CH,OH+ ions from n-propanol. These
curves are also shown for the metastable ion associated with the
reaction C3H70H+->C aH,++H,O.
18 Hipple, Fox, and Condon, Phys. Rev. 69, 347 (1946).
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parent and fragment ions. In each case, the intensity
of the parent ion was also measured at one energy in
order to determine the relative intensities of parent
and metastable ions. The measured intensities were
divided by the experimentally determined electron
multiplier gains to give corrected relative ion in
tensities. In the figures the relative intensities are cor
rect except for the indicated scale factor.
The electron impact data provided experimental
values of the electron multiplier gain for each ion. In
addition, it was found that the intensities of all parent
ions and the metastable ions from n-propylamine and
n-propanol varied linearly with pressure while the
supposed metastable ion from methyl ethyl ketone
varied roughly as the square of the pressure. This is
strong evidence that the metastable ions from n-propyl
amine and n-propanol result from true unimolecular
decomposition while the supposed metastable ion from
methyl ethyl ketone is produced by collision-induced
dissociation, a process studied by Rosenstock and
Melton.19 The ratio of metastable ions to parent ions
produced by 75-volt electrons was found to be 0.005
for n-propylamine, 0.08 for n-propanol, and about
2.6X 10-6 for methyl ethyl ketone at a source pressure
of 10-6 mm. From consideration of the respective
threshold laws for ionization, it is expected that this
ratio should be nearly the same as that obtained by
photoionization at the highest energies and, within a
factor of two, this is true for n-propylamine and
n-propanol. This ratio was independent of the pressure
for the first two compounds, roughly proportional to
the pressure for methyl ethyl ketone and essentially
independent of electron energy, except near the appear
ance potential, in all three cases. This ratio was also
independent of focusing conditions in the ion collector
but could be changed by about a factor of two by
varying potentials in the ion source, particularly the
drawing-out potential. The ratio increased as the
drawing-out potential increased, or as the time spent
by the ions in the ionization chamber decreased.
The average kinetic energies of the CH2NH2+,
C3H6+, and CHaCO+ ions produced by electron im
pact on n-propylamine, n-propanol, and methyl ethyl
ketone, respectively, were found to be essentially
identical to that of the Ne+ ion used for calibration.
The estimated error was less than 0.1 ev. Thus, there
is no evidence for the excess kinetic energies which are
to be expected if the dissociation process is similar to
that occurring in the case of diatomic molecules. This
conclusion may also be applied to the same processes
resulting from photoionization, since any state pro
duced by photoionization can also be produced in
ionization by electron impact.
In order to compare the photoionization results with
the theory, it is necessary to calculate the relative
19 H. M. Rosenstock and C. E. Melton, J. Chern. Phys. 26, 314
(1957). .......... -.................... -........ .
9.0 9.5 10.0 10.5 11.0
PHOTON ENERGY,v
FIG. 6. Photoionization efficiency curves (dashed lines) and their
first derivatives (solid lines) for the production of CH3COC2H.+
and CH3CO+ ions from methyl ethyl ketone. The same curves are
shown for an apparent metastable ion associated with the reaction
CHaCOC 2H.+ -'CH2CO+ +C2Ha.
amounts of the excited parent ions which will be de
tected by the mass spectrometer as parent, fragment,
or metastable ions. This must be done as a function of
the decay constant of the parent ion. The calculation
is complicated by the relatively large volume in which
ionization occurs and by the nonuniform electric field
gradient across the ionization region. For simplicity,
the ionization chamber was considered to be a cylinder
bounded by two plane electrodes. The equipotential
surfaces inside the chamber were estimated from curves
given by Zworykin20 for an infinitely long cylinder
bounded by one plane electrode. The time spent by a
newly formed ion in the ionization chamber was calcu
lated to range between about 5 and 1 microseconds
depending on the point of formation. For simplicity it
was assumed that equal fractions of ions spent 1, 2, 3,
4, and 5 microseconds in the ionization chamber. Ions
then spend about 1 microsecond in the electrostatic
focusing and accelerating regions of the source slit
system. Ions which dissociate in this region are spread
over a large mass range of the mass spectrum and are
thus effectively lost. The ions then spend about 5
microseconds in a field-free region before reaching the
magnetic field. Ions dissociating in this region are de
tected as so-called metastable ions. The ions then spend
about 3 microseconds in the magnetic field. Ions dis
sociating in this region are again spread over a large
mass range and are effectively lost. The ions then
spend about 5 microseconds in a field-free region before
reaching the ion detector. Ions dissociating in this
region are detected as parent ions. Thus, of total ions
produced having a dissociation rate constant of k
sect, the fractions I detected as fragment, metastable
20 V. K. Zworykin, Electron Optics and the Electron Microscope
(John Wiley and Sons, Inc., New York, 1945), p. 395.
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~I.O
o
... o
z o
t-.5 o
<l a: ... FRAGMENT
104 105
RATE CONSTANT
FIG. 7. Curves showing the fraction of ions detected as parent,
fragment, or metastable ions as a function of the rate constant for
dissociation.
or parent ions are given by
[(fragment) = 1-exp( -kx .10-6), (5)
[(metastable) = exp( -k(l +x) .10-6)
-exp(-k(6+x).1O-6), (6)
I (parent) = exp( -k(x+9) .10-6). (7)
where x is the time in microseconds spent by the ion
in the ionization chamber. Figure 7 shows the variation
of these quantities with the logarithm of the rate con
stant. To the accuracy required by this discussion, this
plot is not extremely sensitive to the exact form of the
simplifying assumptions made above.
Before a comparison can be made between the ex
perimental data and the calculated curves of Fig. 7,
the significance of the data must be examined. The
parent ions formed by photons of energy E will have
a spread of internal energies from zero (if adiabatic
ionization is attained at threshold) to (E-I.P.) where
LP. is the adiabatic ionization potential. What is
needed for comparison with theory is the number of
parent ions formed with internal energy equal to just
(E-I.P.). This number (per unit energy) is given by
dI tidE, the first derivative of the total ionization [I
with respect to energy, as shown by Hurzeler et at.
Likewise the number of parent ions of internal energy
(E-I.P.) which appear as either parent, fragment, or
metastable ions is given by dI il dE where I i is the in
tensity of the appropriate ion. This significance of the
derivative is true only if the photoionization probability
at threshold and over the energy range of the data is a
step function as experiment indicates. For comparison
with Fig. 7, each derivative curve of Figs. 4, 5, and 6
must be normalized or divided by the sum of the first
derivative curves of the figure. That is, we wish to plot
d[ iCE) dI iCE) / dI teE)
d[t(E)=~ ~ as a function of photon energy E, since this quantity is
just the fraction of total ions formed with energy
(E-I.P.) which appears as either parent, fragment, or
metastable ions and is the same quantity plotted in
Fig. 7 as a function of rate constant. The normalizt;d
plots for n-propylamine and n-propanol are shown III
Fig. 8. The case of n-propanol is complicated by the
fact that a decomposition process producing ions of
mass 59 is appreciably competitive with that producing
ions of mass 42. The dashed curve for n-propylamine
is probably due to dissociation by collision and, if so,
should be subtracted. A smaller fraction of the me
tastable ions from n-propanol may also be due to this
effect, but this correction is deemed negligible. In the
case of methyl ethyl ketone, essentially all the metas
table ions appear to be due to such collisions and this
case will be discussed later.
Comparison of Figs. 7 and 8 now enables a corre
spondence to be made between energy content of the
parent ion and the rate constant for unimolecular
decomposition and provides a possibly quantitative
test of theory. However, there are several minor diffi
culties in making this correspondence. The adiabatic
ionization potential of the parent molecule and the
energy of formation of the fragment are not accurately
known for these two molecules, and may be obtained
from the photo ionization curves of Hurzeler et at.
shown in Figs. 4 and 5 with only moderate accuracy
and reliability. This is due to the fact that the vertical
ionization potential is obviously considerably higher
than the adiabatic one and because the fragment
curves do not have sharp onsets, as indeed the statistical
theory predicts. The values for the ionization potentials
of n-propylamine and n-propanol chosen here from the
photoionization data are 8.8 ev and 10.1 ev,respectively.
The values for the appearance potentials of the two
fragment ions may be chosen in at least two ways. In
the first method, the ionization efficiency curves of
VI
Z o 1.0
.5 60 42+59
~ Ol~-L~-L~~~~-L~~~~~
o 9.5
z
!? 1.0 to
<l a: .... 5
8.5 10.0 10.5 11.0 11.5
9.0 9.5 10.0 10.5
PHOTON ENERGYe.
FIG. 8. Normalized derivative curves for parent, fragment, and
metastable ions from n-propanol and n-propylamine. The dashed
curve in the lower figure is the assumed contribution of collision
induced dissociation to the metastable ions produced from n
propylamine.
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Hurzeler et al. may be inspected and values for the
onset energy chosen in the manner of electron-impact
appearance-potential determinations. This method
gives values of about 9.4 ev for the CH2NH2+ ion from
propylamine and about 10.4 ev for the CaH6+ ion from
n-propanol. It should be noted that this method would
be expected to give correct results only if the statistical
theory were incorrect at least quantitatively.
In using the second method, we assume the correct
ness of the statistical theory and proceed to calculate E
from the rate constants obtained from Figs. 7 and 8.
Thus, the maxima in the derivative curves for Fig. 8
occur at 10.6 ev for CaH6+ and at 9.5 ev for CH2NH2+
when the collision-induced contribution is subtracted.
The most probable thermal vibrational energy of the
molecules at the estimated temperature of the ioniza
tion chamber (4000K) is about 0.08 ev for both mole
cules. The corrected maxima are thus 10.68 ev and
9.58 ev. From Fig. 7 these maxima correspond to a
rate constant of 1.7XI05 sec1. Equation (1) was then
used to calculate the rate constant for the reaction
CaH7NH2+-+CH~H2++CHaCH2. The frequency fac
tor was calculated to be 8.0X 1015 by using the fre
quencies for CaHs used by Krop£21 in calculating the
rate constant for the process CaH8+-+CHaC2H5+' The
additional frequencies of CaH7NH2+ will cancel to a
good approximation and this calculation is not very
sensitive to a change in this factor. Thus, Eq. (1)
becomes
k= 8.0X 1015[(E-E)/ EJao.5. (8)
Upon substituting the values E=9.58-8.80=0.78 ev
and k=1.7XI05 secI, the value of E is found to be
0.43 ev and the calculated appearance potential is
0.43+8.8=9.23 ev. If the expression used by Fried
man, Long, and Wolfsberg16 for similar processes is
used here, namely,
(9)
the value of E is 0.37 ev and the calculated appearance
potential is then 9.17 ev. We thus take 9.2 ev as the
calculated value of this appearance potential.
Similarly, the appearance potential of the CaH6+ ion
from CaH70H may be calculated. The expression which
Friedman et at. found to be adequate for this process is
(10)
which yields the value E=0.15 ev and a calculated ap
pearance potential of 10.25 ev.
The calculated appearance potentials appear to be
reasonable and only about 0.2 ev below the values ob-
21 A. Kropf, thesis, University of Utah (1954). '" <.> z
"" 0 z
:::>
CD
""
'" :': .... «
...J
'" Ct:
'-----;-----{. 2'-----:. 3~-.-~~---!. Sc--------L6 .. -+ .1
VIBRATIONAL ENERGY ev
FIG. 9. Vibrational energy distribution for n-butane at 400oK.
tained by inspection. However, if these values of E
and E are used in the foregoing equations to calculate
the expected half-widths of the metastable derivative
curves of Fig. 8, the calculated half-widths are much
smaller than those observed. Thus, if we calculate the
change in E required to change k from 4X 104 seci to
6X 106 sec1 (the points at half-height for the metas
table ion in Fig. 7), the values are about 0.03 ev and
0.12 ev instead of the observed 0.37 ev and 0.28 ev for
n-propylamine and n-propanol, respectively. The ob
served value for n-propylamine is that obtained when
the collision-induced contribution is subtracted. The
discrepancy between the calculated and observed
values is well outside experimental error especially for
n-propylamine.
Another discrepancy between calculation and experi
ment is shown by the maximum height of the deriva
tive curve of the metastable ion. According to the
calculated curves of Fig. 7, this maximum height should
be about 0.3, while experimentally it is very much
lower. This great discrepancy can be due to formation
of ions of a wide range of half-lives at each setting of
the monochromator, which could also explain the large
width of the curve. This formation of a wide spectrum
of half-lives can result from at least three factors: (1)
the finite resolution of the monochromator, (2) the
thermal vibrational energy distribution of the parent
molecules, and (3) inefficient randomization of the in
ternal energy of the parent molecular ion.
The energy half-width of the resolved photon beam
was about 0.15 ev and together with the calculated
theoretical widths does not quite account for the ex
perimental half-widths. This extra broadening is
probably due to one or both of the latter two factors
already mentioned.
The vibrational energy distribution for n-butane at
4000K has been calculated from the frequencies given
by Pitzer.22 The calculation was simplified by changing
the frequencies slightly until all were multiples of 115
cm-I. The frequencies used were (in cm-I) 115, 230
22 K. S. Pitzer, J. Am. Chern. Soc. 63, 2413 (1941).
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'" z o CH3CH2CH20H
MASS 29.4
FIG. 10. Comparison of calculated (dashed line) and experi
mental (solid line) normalized derivative curves for metastable
ions produced from n-propanol and n-propylamine. The alter
nately dashed and dotted line for n-propylamine is the experi
mental curve from which the collision-induced contribution has
not been subtracted.
(2), 345 (2), 1035 (13), 1495 (8), 2990 (10). The dis
tribution is shown in Fig. 9 with a smooth line drawn
through the discrete points. This distribution will also
be applicable to n-propylamine well within the ac
curacy required for this purpose.
A graphical integration of the product of the distribu
tion curves for photon energy and thermal vibrational
energy and the theoretically calculated metastable
derivative curve as shown in Fig. 7 was performed for
both n-propanol and n-propylamine. The resulting
curves are shown in Fig. 10 together with the experi
mental curves. The two curves were adjusted to co
incide at their maxima. The agreement is within the
rather large uncertainty in the experimental derivative
curves. However, it is obvious that the effects of photon
and thermal energy spread together with the experi
mental error are too great to allow any good test of the
theoretical rate equation from these curves.
Another more accurate test of the effects of tempera-
'" z o
... METASTABLE
PARENT I FRAGMENT 1.0r----....!=~2..---+_----'.-"-"'=-"-"-'----
o O~-----r--~~~-----
z
~I.O PARENT FRAGMENT
<>
METASTABLE c
0:: ... (X SOME FACTOR)
OL---~~-~-~---------
PHOTON ENERGY
FIG. 11. Illustration of the calculation of breakdown curves to
include the effect of thermal energy for the simplified case where
the breakdown curves for parent and fragment ions are step
functions and the curve for the metastable ion is a very sharp
peak. The scale factor of the metastable curve is a function of
the width of the peak. ture and of the kinetics of dissociation can be made by
examination of the derivative curves for parent and
fragment ions, especially for the simple cases of ethyl
and n-propylamine. Since these ions are of higher in
tensity than the metastable ions, they were measured
by Hurzeler et at. with higher monochromator resolu
tion, namely, about 0.04 ev and the data have less
scatter. Also, the auxiliary electron-emitting filament
was used only very sparingly in these experiments and
it is estimated that the temperature of the source
ionization chamber was in the range 300-325°K.23
If the statistical theory is quantitatively correct,
these derivative curves, where the ordinate is the frac
tion of total ion intensity, may be calculated in the
following manner. The breakdown curves for the ions
I
CH3CHZ NH2
I
X
I '\ r
CH3CHZCH2NH2
I / ~
B.5 9.0 9.5 10.0 10.5
PHOTON ENERGY IV
FIG. 12. Experimental (solid line) and calculated (dashed line)
breakdown curves including the effect of thermal energy for ethyl
and n-propylamine. The dashed curves were calculated as shown
in Fig. 10 by the use of step functions for the theoretical break
down curves. The curves calculated by the use of Rosenstock's
rate equations are so similar to the dashed curves that they are
not shown in this figure. The arrows indicate the values of E
(plus the ionization potential) chosen in the manner described in
the text.
containing no thermal internal energy are calculated
in the usual manner.Ii•16 Then the integral of the
product of this breakdown curve and the internal
energy distribution curve for 3000K is plotted as a
function of energy as illustrated in Fig. 11 for the
simplified case where the breakdown curves for parent
and fragment ions are step functions and the curve for
the metastable ion is a very sharp peak. The contribu
tion of the photon energy spread is included in a
similar manner. The vibrational energy distribution at
3000K for n-propylamine was calculated by the use of
the same data as was used previously. The calculation
for ethylamine was made by the use of vibrational
frequencies given by Pitzer24 for propane. These curves
were calculated for ethyl and n-propylamine and are
23 M. G. Inghram (private communication).
24 K. S. Pitzer, J. Chern. Phys. 12, 310 (1944).
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shown in Fig. 12. In both cases, curves were calculated
using step functions for the breakdown curves. The
agreement with the experimental curves is excellent in
both cases. For n-propylamine, the calculation was re
peated using the breakdown curve obtained by use of
Eq. (9) and Fig. 7. The agreement of this latter curve
with the experimental one is nearly as good, and
again is within experimental error. If the slightly
better agreement of the first calculated curve were
significant, it would provide evidence that the rate
constant for dissociation increases more rapidly with
Ethan Eq. (9) indicates. However, the data are not
sufficiently accurate, and the effect of thermal energy
is apparently too great to allow such a conclusion in
this case.
On the other hand, if the dissociation was of the
simple type occurring in diatomic molecules, the effect
of temperature would be negligible and the drop of the
parent peak and rise of the fragment peak would be
complete in an energy interval of about 0.08 ev. This is
clearly not in accord with the experimental curves.
A much more sensitive criterion for the amount of
broadening of the metastable derivative curves is the
height of these curves. For n-propanol, the theoretical
height is six times the experimental one which has been
corrected slightly to take account of the competing
process forming ions of mass 31. For the n-propylamine
metastable ion this factor is 23. The ratio of these
factors is consistent with the fact that the observed
broadening for n-propylamine is about 12 times the
calculated value while for n-propanol it is only about
2.3.
These considerations can be treated more quanti
tatively as follows. If energy randomization occurs so
that internal energy supplied by photoionization is
completely equivalent to thermally supplied energy,
the area under the corrected derivative curve will
remain constant, independent of the temperature of
the gas and the resolved photon energy distribution.
This area is, of course, determined by the maximum
intensity of the metastable ion. In fact, the areas under
the uncorrected derivative curves of Figs. 4, 5, and 6
are just exactly equal to this maximum intensity. This
area can be determined with fair accuracy and com
pared with areas calculated by use of Fig. 7 together
with the rate equation whose accuracy is to be tested.
For n-propylamine the area was calculated by using
both Eqs. (8) and (9). These areas were, respectively,
4.6 and 6.2 times larger than the experimentally ob
tained area shown in Fig. 10. For n-propanol, the area
was calculated by using Eq. (10) and was 4.0 times the
experimental value. These values imply that the rate
constant varies much more rapidly than Eqs. (8), (9),
and (10) indicate.
In order to see what sort of rate equation would
agree with experiment the form of Eq. (9) for propyl
amine was kept and the exponent of the energy term, i.e., the number of oscillators, was varied. It was found
that an exponent of about 6 rather than 32 gave agree
ment with the experimental value of the area. In this
calculation the value of E used was redetermined as
before. For the case of n-propanol, the form of Eq. (10)
was kept and the exponent varied in a similar manner.
Again an exponent of about 6 rather than 29 gave the
desired agreement.
The probable error of these calculations is due to
several factors. These are (1) the approximations used
in calculating the curves of Fig. 7, (2) deviations from
the assumed threshold law for photoionization, (3) the
possibility of lower transmission of the mass spec
trometer for the metastable ions, and (4) the experi
mental error in the measurement of the photoionization
data, especially that due to collision-induced dissocia
tion. The probable error due to the first factor is
estimated to be about 30%. Perhaps the most serious
possible error lies in the second factor, namely devia
tion from the threshold law. Both theory and experi
ment26.26 show that the cross section for photoionization
of a neutral atom has a finite value at the threshold
and that it will vary somewhat with frequency above
this value. For atoms with relatively diffuse bound
wave functions for the ionized electron, e.g., alkali and
alkaline earth metals, this variation will be strong as
verified by experiment. In the case of more compact
wave functions, this variation may be positive or nega
tive, but will be relatively weak. Bates26 calculated
that for the oxygen atom, the cross section will be
almost independent of frequency while for the nitrogen
atom it will decrease slowly with increasing frequency.
The scanty data compiled by Weissler26 for the mole
cules H20 and NH3 are not inconsistent with a rela
tively small variation of cross section over an energy
range of about a volt above threshold. The data of
Watanabe and Mottle27 for NH3 show vibrational fine
structure in the photoionization cross section which re
mains fairly constant with energy between vibrational
transitions. Any deviation from constancy would
generally be more apparent at the highest energy but
here the data show the greatest constancy.
The best evidence for the behavior of the ionization
cross section is the photoionization data on the mole
cules of interest themselves. The fact that the ionization
cross section for the production of the parent ion from
n-propylamine is constant to within better than 10%
over the energy range 9.6 to 10.9 ev is very strong evi
dence for the validity of the threshold law in this
region with an accuracy far better than needed in these
calculations. In the case of n-propanol the plateau
25 D. R. Bates, Monthly Notices Roy. Astron. Soc. 106, 423
(1946).
26 G. L. Weissler, "Photoionization in Gases and Photoelectric
Emission from Solids," Encyclopedia of Physics (Springer-Verlag,
Berlin, 1956), Vol. 21, p. 304.
27 K. Watanabe and J. R. Mottl, J. Chern. Phys. 26, 1773
(1957).
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1.01---=6-,,-0 ""'
en z o
II. 00
~ 1.0
o
.... o
<t a: ... 60 42
FIG. 13. The upper set of curves are the breakdown curves for
n-propanol as calculated by Friedman, Long, and W olfsberg.
The lower set of curves are the experimental breakdown curves as
calculated from the data of Hurzeler, Inghram, and Morrison.
Note that the energy scales differ by a factor of two. The arrows
on the lower set of curves indicate the values of E chosen in the
manner described in the text.
region of the ionization cross section for the production
of the parent ion was not measured over such a large
region. Nevertheless, the value of the cross section
changes by only about 10% from 10.7 to 11.0 ev.
Since the difference in apparent appearance potentials
of the parent ion and the fragment ion of mass 42 is
only about 0.3 ev, the error due to deviation from the
threshold law is small. From these considerations, it is
estimated that a generous probable error in the meas
ured area under the metastable derivative curves due
to this factor is about 10% in both cases. The probable
error due to the possibility of lower transmission for
the metastable ions is estimated to be about 20%
based on the following considerations.
It was shown that the optimum focusing condtionsi
in the ion collector were essentially the same for parent
and metastable ions and fhat the fragment has no ap
preciable excess kinetic energy and that the trans
mission for the fragment was essentially the same as
that for the parent as seen by the way the respective
derivative curves of Fig. 4 can be smoothly joined to
form a single curve representing the transition prob
ability of the ground state of the ion. The probable
error of the measurement of true metastable ion in
tensities is estimated to be about 30%. Thus a total
probable error of 50% would seem to be a reasonable
estimate and this is much less than the factor of four
or more by which the calculated and measured areas
disagree. Thus, the evidence is very strong that the
rate constant varies much more rapidly with Ethan
indicated by Eqs. (8)-(10). If the experimental finding is expressed in terms of an effective number of oscil
lators, the experiment indicates this number to be
about one-fifth of the theoretical value. However, it is
estimated that the direction of the probable error is
such that this fraction is probably slightly larger and
perhaps as large as one-third. These errors can be re
duced further by more accurate experiments and
calculations.
In the case of isolated systems, which concerns us
here, if intramolecular relaxation is slow and the
distribution in phase space is not maintained constant
for the undissociated molecules, the apparent rate
"constant" will decrease as molecules dissociate. This
apparent rate constant may vary above or below that
expected if randomization occurred. Thus, we might
expect the derivative curve for the fragment ion to
have a low-energy "tail" in addition to that caused by
thermal energy, but also the derivative curve of the
parent may have a high-energy tail which would not
be obscured by the effect of thermal energy. There is
no evidence of such a tail on the parent peak curves for
ethyl and n-propylamine in Figs. 8 and 12. However,
the data on n-propanol show some evidence of this be
havior. Figure 13 shows the breakdown curves for
n-propanol as obtained from the experimental data of
Hurzeler et al.S and as calculated theoretically by
Friedman et al.16 The curve for the parent appears to
have a high-energy tail although the scatter of experi
mental points is bad in that region. The curve for the
C3H6+ ion is even more pronounced in this behavior
and will be discussed later.
Thus, it appears that the photoionization data on
ethylamine, n-propylamine, and n-propanol can be
explained at least qualitatively in terms of the uni
molecular decay theory. In the case of n-propanol
there is some evidence for lack of complete randomiza
tion of energy. For ethylamine and n-propylamine
there is no such evidence. However, this cannot be
taken as proof of attainment of practically complete
randomization since it is possible that effects of non
randomization are obscured by the effect of thermal
energy spread. Indeed, it is possible that a considerable
part of the effect which is ascribed here to thermal
energy spread may actually result from nonrandomiza
tion. This is possible since, if randomization does not
occur, the effect of thermal energy would be decreased.
In principle, this question is easily resolvable by ex
periment. Upon cooling the gas in the ionization region
to temperatures in the range 100-200oK the average
internal energy of these molecules will be reduced to
negligible values and a good quantitative test of the
theory will be possible.
A short examination of the effects of nonrandomiza
tion is helpful. We consider a case in which the parent
molecular ions are all produced in vibronic states of
essentially identical energy, but with a distribution,
determined by the Franck-Condon principle, which is
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far from the most probable. These ions can be said to
have a single well-defined decay half-life only if the
distribution among the remaining parent ions remains
unchanged as the dissociation occurs. Usually, this will
happen only if the distribution has become random
before appreciable dissociation occurs. If appreciable
dissociation occurs in a time very short compared to
this intramolecular relaxation time, then in general
only a part of any original thermal vibrational energy
of the molecule will be effective. In the limiting case of
a molecular ion formed in a repulsive state as described
by Morrison,3 only the vibrational energy involved in
the stretching of the bond to be broken will be ef
fective.
Very little is known concerning the vibrational energy
distribution of polyatomic molecular ions immediately
after production by photoionization. If all equilibrium
internuclear distances and bond angles are nearly the
same for the molecule and the ion, then the Franck
Condon principle will require that the distribution be
similar to that in the neutral molecule. This is probably
a frequent case, especially where the resulting half
filled orbital is best described as a nonlocalized molecu
lar orbital. Nevertheless, there are many cases where
certain bond angles and distances will change much
more than others and the initial vibrational energy
distribution of the ion will be quite different from the
generally random one of the neutral molecule. In some
cases, even the symmetry of the ion can be different
from that of the parent molecule.28
For the three types of compounds investigated here,
the ionization near threshold probably corresponds to
removal of a fairly localized electron, namely, one of
the nonbonding p electrons of the oxygen29 or nitro
gen30 atom. It is reasonable to assume that the largest
change in equilibrium bond lengths and angles would
occur for the bonds to the oxygen or nitrogen atom.
The initial vibrational energy distribution would then
be a superposition of a nearly random one (similar to
that of the original neutral molecule) and one involving
strong vibrations of the oxygen or nitrogen atom and
the atoms directly bound to them. If the latter vibra
tions lead to dissociation before any appreciable ran
domization occurs, then the original random com
ponent will have little effect on the rate of dissociation.
Recent theoretical calculations on NH3 are helpful
in interpreting the ionization of the alkyl amines. It is
very likely that the ionization near threshold corre
sponds to removal of a nonbonding electron from the
nitrogen atom since the ionization thresholds C::; 8.9
ev) are considerably lower than that expected for re
moval of an electron from the alkyl group (",11-12
ev) and are much closer to the ionization potential of
28 A. D. Liehr, J. Chern. Phys. 27, 476 (1957).
29 A. D. Walsh, Trans. Faraday Soc. 42, 56 (1946); 43, 60
(1947).
30 A. B. F. Duncan, J. Chern. Phys. 27, 423 (1957). NH3 (::; 10.1 ev). Duncan30 has calculated that the first
ionization potential of ammonia, corresponding to re
moval of a nonbonding electron, is about 6.0 ev lower
than the second ionization potential. McDowell31 gives
experimental evidence for an energy difference of about
5.3 ev. Thus, the first electronically excited state of the
ethyl or propylamine ion, corresponding to removal of
the electron from the alkyl group, probably lies about
2 ev above the ground state.
While the NH3 molecule is pyramida132 with an ob
served bond angle of 106.8° and an inversion barrier of
0.26 ev, the calculations of Higuchi33 indicate that the
NH3+ ion is planar. From Higuchi's calculated variation
of orbita1 energies of the NH3 molecule with bond angle,
one can crudely estimate the energy difference between
the planar configuration of NH3+ and the configuration
in which the bond angles are equal to those of NH3
(106.8°). This difference is about 0.6 ev and may ac
count for a large part of the difference between the
vertical and adiabatic ionization potentials of NHa
which seems to be of this magnitude. This energy
would appear in the bending vibration perpendicular
to the plane of the molecular ion. This interpretation
is supported by the photoionization cross-section curve
of NH3 as measured by Watanabe and Mottl.27 The
observed spacing of vibrational levels is about 1000
cm-I which is a reasonable value for the out-of-plane
bending vibration of NH3+ since the corresponding
vibrational frequency of NHa has a value of about 950
cm-I (the mean of two components), and the nearest
other frequency is 1627 cm-I. The corresponding vibra
tions are probably excited for similar reasons in the
case of the alkyl amines. It is difficult to see how these
vibrations could lead very directly to the observed
dissociation of a bond which does not even involve the
nitrogen atom. Thus, it seems likely that appreciable
randomization of energy occurs before dissociation.
In the case of n-propanol, the vibrational energy
probably appears initially in vibrations of the OH group.
This may possibly explain the high-energy tail on the
C3H6+ curve of Fig. 13, since this initial concentration
of vibrational energy in the OH group could lead to a
high probability of loss of H20 before energy randomiza
tion could occur. This process could then compete
even at higher energies with the simple bond-rupture
processes producing the C3H60H+ and CH20H+
fragments, even though the latter processes have much
higher frequency factors.
The mechanism of energy randomization should also
be considered. Rosenstock et 01.5 assume that this
occurs chiefly by radiationless transitions resulting
from numerous crossings of potential surfaces. They
consider that the ion has a high density of electronic
3l C. A. McDowell, J. Chern. Phys. 24, 618 (1956).
32 G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand
Company, Inc., Princeton, New Jersey, 1945), p. 294.
33 J. Higuchi, J. Chern. Phys. 24, 535 (1956).
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states, with an average spacing of one millivolt for the
propane ion for example. However, it must be re
membered that this is an average spacing and that
individual spacings, particularly at the lowest energies,
will be much larger than this. Indeed, this is evidently
the case for the ketones and amines studied by Hurzeler
et at.,s for which it appears that an energy of the order
of a volt separates the ground electronic state of the
ions from the nearest excited state. Since this separa
tion is appreciably larger than the average thermal
vibrational energy of these molecules, practically no
crossing of potential surfaces will occur for ions in this
state. Yet, in the case of the amines, dissociation occurs
and does so via ions with lifetimes much longer than a
vibrational period, so that vibrational energy must
have been transferred among many modes. In this
case, it seems very likely that this transfer was ac
complished by a purely vibrational mechanism, that is,
as a result of the anharmonicity of the vibrations and
of the coupling between normal modes. This is an
interesting situation, since this process of energy trans
fer would seem to be more amenable to theoretical
calculation than would the mechanism involving cross
ing of many potential surfaces. For molecules of not
too great complexity, reasonably good force constants
and anharmonicity constants could be obtained spectro
scopically and by estimation. An attack on a similar
problem has been made by Fermi, Pasta, and Ulam.34
These authors find surprisingly little tendency toward
equipartition of the energy which is initially concen
trated in one mode of a one-dimensional chain of
particles.
It should be noted that in general the process of
intramolecular energy transfer is expected to be more
rapid for ions of higher energy of excitation for several
reasons. On the average, the density of electronic and
vibrational energy levels increases rapidly with energy
thus increasing the number of crossings of potential
surfaces. Vibrations become more anharmonic with
increasing energy thus facilitating energy transfer
among vibrational modes. At sufficiently high energies,
if lifetimes of vibronic states become comparable to
vibrational periods, it may no longer be useful to
speak in terms of normal vibrations and separable
electronic and nuclear motions. While the rate of
intramolecular energy transfer will increase with
energy, so also will the rate of dissociation, and it is
not certain which increase will be the more rapid.
Thus, it is not clear whether, as the energy of the
molecular ion increases, one can usually expect more
or less randomization of energy before appreciable
dissociation occurs.
Experimental evidence regarding the efficiency of
energy randomization is very limited. Some evidence
34 Fermi, Pasta, and Ulam, Los Alamos Scientific Report LA-
1940 (May, 1955). may be obtained from the study of normalized deriva
tive curves measured either by photon or electron
impact. When such a curve for any ion drops to zero
or even begins to decrease with increasing energy, it
should never thereafter show any sharp increase at
higher energies if energy randomization is attained
before dissociation or radiation occurs. (Very slow up
ward trends could result from deviations of the ioniza
tion cross section from threshold behavior.) No such
extreme behavior is found in the cases investigated here
although some features of the curves have been inter
preted as indicating lack of randomization. A study
covering a larger energy range would be most instructive
and a particularly enlightening case would be that of
the parent ion. If energy randomization occurs, the
ordinary ionization probability curve for the parent
ion should show no upward breaks at energies more
than a few tenths of a volt above the appearance po
tential of the lowest energy process which produces
dissociation. Accurate ionization probability curves
have been taken over a sufficiently large energy range
for some large molecules and there are fairly clear
instances of nonrandomization. For propylene,36 the
curve for the parent ion has an upward break at 13.22
ev while several dissociative processes are observed
below this energy, e.g., C3H6+ ions appear at about
11.95 ev.4 For benzene, the curve for the parent ion
has a break at about 15.5 ev35 while the process pro
ducing C6H5+ ions begins at about 14.5 ev.4 This
constitutes very good evidence that the particular ex
cited electronic states of the parent ions corresponding
to these breaks do not readily make radiationless
transitions to the lower electronic state or states which
result in dissociation.
From considerations such as mentioned in the fore
going, it seems likely that the hypothesis of energy
randomization may be satisfied in some cases and not
in others, depending on the specific molecule and even
the particular vibronic state excited in the ion. In par
ticular, electronic states of low energy may frequently
be unable to make radiationless transitions to one
another because of large energy differences aside from
other reasons such as the operation of selection rules.
Nevertheless, any dissociation of these states might still
usually occur by a process, involving long-lived ions
with vibrational energy being transferred among many
modes, similar to that of dissociation of the propylamine
parent ion in its ground electronic state.
The data on methyl ethyl ketone are shown in Fig. 6.
The curves for the parent and fragment ions obtained
by Hurzeler et at. indicate that the ground electronic
state of the parent ion does not dissociate at all while
the first excited state (which is at much higher energies)
dissociates completely. Under these circumstances,
36 R. E. Fox and W. M. Hickam, J. Chern. Phys. 22, 2059
(1954).
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where practically no parent ions of the appropriate
energy are formed, no appreciable amounts of metas
table ions are expected. Experimentally, metastable
ions of low intensity are detected and are found to have
the same energy dependence as the parent ion within
experimental error. This fact indicates strongly that
essentially all of these metastable ions actually result
from collision-induced dissociation, and this conclusion
is confirmed by the absence of true metastable ions in
the electron-impact measurements taken at lower
pressures.
From these data on methyl ethyl ketone, little can
be concluded regarding the mechanism of dissociation.
Dissociation may occur directly from the excited elec
tronic state or by a radiationless transition to a high
vibrational level of the ground electronic state. This
latter possibility while rather unlikely is not com
pletely excluded by the forms of the photoionization
curves which indicate only that no crossings of po
tential surfaces occur in the Franck-Condon region.
RELATION TO ELECTRONIC SPECTRA AND PHOTO
CHEMISTRY
The phenomena of predissociation and other radia
tionless transitions in the spectra of polyatomic mole
cules and the primary photochemical process are very
closely related to the processes discussed here. Indeed
it is found that while diffuseness in diatomic spectra
begins quite suddenly, in the case of large polyatomic
molecules it usually begins quite gradually. This be
havior is explained36 as being due to the dependence of
the rate of dissociation on the energy content of the
molecule, in a manner similar to that indicated by
various theories of unimolecular decomposition. This
behavior, in addition to other factors, usually makes it
impossible to determine accurately, from spectra alone,
the minimum energy necessary to cause dissociation.87
It is unfortunate that molecules of such complexity
as those considered here give rise to electronic spectra
in which individual lines are well nigh impossible to
resolve and in which the abundance of bands is often
so great as to give the appearance of a continuum.
Nevertheless, some information may be gained about
the processes considered here. Thus, one may look for
the occurrence of fluorescence of the excited molecule
and of structure in the spectra at wavelengths which
cause dissociation as indicated by photochemical
studies. Both of these conditions are found to occur in
the photolysis of acetone and possibly other com
pounds38 and constitute evidence for relatively long
lived parent molecules. On the other hand, the ap
pearance of true continua in these spectra would indi
cate that direct dissociation rather than predissociation
36 H. Sponer and E. Teller, Revs. Modern Phys. 13, 75 (1941).
37 Noyes, Porter, and Jolley, Chern. Revs. 56, 49 (1956).
38 K. S. Pitzer, Chern. Revs. 27, 39 (1940). occurs. True continua often do occur in the spectra
of relatively simple polyatomic molecules as would
be expected. However, it is not certain whether the
apparent continua frequently observed in the case
of very complex molecules are usually true continua
or only quasi-continua caused by lack of resolution
and by line broadening in very rich spectra.
Similar arguments can be made regarding the ap
pearance of structure in photon or electron impact
spectra. Thus, if it could be shown that the ionization
efficiency curve for any fragment has vibrational
structure, this would be decisive evidence against
direct dissociation since the lifetime of the dissociating
state would be so short as to completely smear out
such structure. Indeed, the data of Hurzeler et al.
appear to give just as much evidence for structure in
the fragment curves as in the parent curves for the
more complex molecules. However, Hurzeler et al.
conclude that this apparent structure may be instru
mental in origin. It may be noted that radiationless
transitions may smear out such structure for both
parent and fragment ions. This effect would increase
with energy for reasons mentioned previously. This
will tend to make it more difficult to detect structure
in the fragment ion curves.
IMPLICATIONS FOR APPEARANCE POTENTIALS
The question of the quantitative validity of the
statistical theory near threshold is very important to
the interpretation of appearance potentials of ion
fragments. If the theory is accurate then many appear
ance potentials of ion fragments from large molecules,
taken in the usual manner, will be too high by a large
fraction of a volt or more. Friedman et al.16 consider
the apparently general success of appearance potential
measurements as evidence for failure of the quantitative
aspects of the theory at least at low energies. Although
this evidence is good in some cases, in many others it
is not decisive for several reasons.
It is not difficult to find a wide range of values ob
tained by different investigators for the same appear
ance potential. Also, there is a large number of appear
ance potentials which can be interpreted only by
assuming the presence of excess energy in the products.
This is precisely what is predicted by the theory, the
excess energy appearing predominantly as vibrational
energy. Some of the data obtained from appearance
potentials involve comparison of two or more appear
ance potentials for similar processes in similar mole
cules. In such cases, the theory indicates that errors
would cancel to a large extent. Also, the theory indi
cates that an appreciable number of appearance po
tentials may be fairly accurate by virtue of cancellation
of the effects of two factors, either of which might
introduce a very appreciable error. These two factors
are (1), the excess energy required to produce detectable
Downloaded 05 Sep 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://jcp.aip.org/about/rights_and_permissions206
...... o c: '-? N
t-
~ ":
~
II
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t-
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.S
'" r 0
~ N
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'"
8 '" O~OO\QLf) 00 0,...-1 C""l · .... 00000
8N1.IJ--" \0 00 ............. · . . . . 00000
8-("f')U') 00 0000 · .... 00000
80-N<"') 0000 · .... 00000
~~~~~ 00000
U')lI')_OOt-
0-<"')'<1<\0 · . . . . 00000
~O_NU)
~~~~~ 00000
NLf)OI.Ot't') OO __ N
00000
o8;:!;88 · .... 00000
~N~~~ · . . . .
C",1N-.:t'\Oao
"":OO~\OO
OO'-;-N
__ Nf"')V)
N","\OooO
0000"";
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O_Nt'I')~
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00000 WILLIAM A. CHUPKA
dissociation in 10-5 sec, and (2), the temperature of the
gas in the ionizing region.
The effectiveness of temperature in lowering the
value of an appearance potential will depend to some
extent on the method used to estimate the appearance
potential from the ionization efficiency curve. It should
be in the range from somewhat less than the average
thermal internal energy to about 2 or 3 times that
value. In most cases it would probably be near the
average energy. The temperature of the ionization
chamber in electron-impact experiments is generally in
the range 400-600oK. The gas molecules usually make
the order of 10 collisions with the walls of the chamber
before escaping. The accommodation coefficient will
depend on the nature of the gas and of the wall surface.
All that can be said is that the average internal energy
of these molecules will correspond to a temperature
somewhere between the temperature of the gas inlet
system and the temperature of the walls of the ioniza
tion chamber. Table II contains a tabulation of the
average internal energy of several hydrocarbons at
various temperatures. These values were taken from
a table given by Pitzer38 after the translation and
rotational contributions were subtracted. Also given
in Table II are values of (E-~), for several values of
~, calculated using the equation
(11)
and a value of 105 sec! for k. In each case, values of
(E-~) were calculated for three values of n, namely
n= (3N -7), n= (3N -7)/3, and n= (3N -7)/5, where
N is the number of atoms in the molecule. This was
done since this work indicates that at least in some
cases It, which might be called the "effective number
of oscillators," should be roughly one-fifth (but perhaps
as high as one-third) of the total number of oscillators
in the molecule.
Where n is given the value (3N -7) and where t is
less than about 0.5 ev, it is seen that these two sources
of error tend to cancel one another to a large extent.
However, for processes with higher values of t, the
value of (E-~) becomes quite large for the more
complex molecules and it is here that the theory is
most severely tested. Unfortunately, for most processes
involving large molecules, there is a wide range of
measured values for the appearance potential of a
particular fragment. Nevertheless, there are some
carefully measured appearance potentials which are in
wide disagreement with the predictions of the un
modified statistical theory. Thus the appearance
potential of C2HS+ from butane, measured by Steven
son and Hipple,39 is consistent with appearance poten
tials of this ion from smaller molecules such as ethane.
For the production of C2Hs+ from butane, t is calculated
39 D. P. Stevenson and J. A. HippIe, J. Am. Chern. Soc. 64,
1588 (1942).
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to be about 1.5 ev by use of the heat of formation of
C2H6+ given by Field and Franklin4 and the ionization
potential of butane determined by Watanabe.40 The
value of (E-e) for n= (3N -7) and E= 1.5 ev given in
Table II is 1.86 ev. The average thermal energy at
5000K is 0.3 ev. This latter value, even multiplied by
two or three, is not enough to explain the results of
Stevenson and Hipple which indicate that (E-e) is
nearly zero.
Where n is given the value (3N -7)/3 in Table II, it
is seen that (E-e) and the average thermal energy at
5000K are quite comparable over most of the range of
values of e. In particular, for the production of C2H.+
from n-butane, these quantities are essentially identi
cal. Where n is given the value (3N -7)/5, the value
of (E-E) is practically negligible for most of the range
of values of E. In this case, the effect of temperature
is more important.
It is well known that mass spectra of large molecules
vary with temperature and the temperature coefficients
of various hydrocarbon parent and fragment ions have
been measured.41 However, these data cannot be used
in a simple manner to test the theory quantitatively.
Qualitatively, the data are reasonable in that the
temperature coefficients of the parent peaks are nega
tive and become more negative with increasing com
plexity of the molecule. For example, the temperature
coefficient of the parent peak of n-octane is about
twice that of n-butane. A more significant measure
ment would be that of the temperature coefficient of
an appearance potential. This would provide a measure
of the fraction of the thermal energy of the original
molecule which contributes to bond rupture. If the
quasi-equilibrium theory is strictly correct, this frac
tion should be near unity.
On first sight, the curves of Fig. 13 appear to be
strong evidence for quantitative failure of the theory at
threshold, since the experimental curves lie at much
lower energies than the calculated ones. However, the
value of E for production of mass 42 was chosen rather
arbitrarily and could just as well have been chosen to
give agreement with the experimental curve. In fact,
a calculation of this appearance potential by the use of
thermochemical data and the more accurate ionization
potential determined by Watanabe40 for propylene
yields a value of 10.13 ev in fair agreement with the
value of 10.25 ev obtained earlier from the experi
mental curves by use of the theory. This does not
necessarily provide support for this theory since it is
probable that there is an activation energy for this
process of the order of several tenths of a volt which
should be added to the calculated value. The value of E
for production of mass S9 was taken from the measured
appearance potential of this ion in this very same
40 K. Watanabe, J. Chern. Phys. 26. 542 (1957).
41 Reese, Dibeler, and Mohler, J. Research Natl. Bur. Stand
ards 43, 65 (1949). process. This was done in the usual manner, in which
it is implicitly assumed that the unimolecular decay
theory is quantitatively incorrect. Again a reasonable
value of € could be chosen which would give agreement
with experiment. On the other hand, the value of E
used to give the calculated curve for mass 31 is ob
tained independently by the use of thermochemical
data together with the appearance potential of the
CH20H+ ion from methanol, a process which is not
expected to be seriously affected by the kinetics of the
decomposition. The discrepancy between the calcu
lated and experimental curves for mass 31 is well
outside experimental error.
For the aforementioned reasons it seems very prob
able that at least in many cases the theory is quanti
tatively badly in error. Nevertheless, the data of
Hurzeler et at. abundantly support the hypothesis
that, in at least many cases, dissociation does not occur
immediately after ionization but rather by way of
relatively long-lived ions whose rate of dissociation
varies with energy content. In no case does a frag
mentation process have a sharp onset as would be
expected if direct dissociation occurred as it does for
diatomic molecules. The ionization efficiency curves
are all asymptotic to the energy axis and since these
curves should be similar to the first derivatives of
the ionization efficiency curves obtained by electron
impact, the latter should have even more curvature at
threshold aside from the additional curvature caused by
electron energy spread. The impossibility of obtaining
very precise bond energies from such curves without
accounting for the effects of temperature and kinetics
of decomposition seems quite clear.
The precise meaning of electron-impact appearance
potentials seems to be questionable when a comparison
is made between the best values for n-propanol tabu
lated by Friedman et al. and the relatively very accu
rate data of Hurzeler et al. The electron-impact values
for masses 60,59, and 31 are 10.7,11.35, and 11.65 ev
and seem to correspond roughly to the point at which
the derivative curve for the fragment ion drops back
to zero. This is the point at which the ionization effi
ciency curve for electron impact becomes linear if the
effect of electron energy spread is neglected. It is
possible that a constant error of about 0.2 ev may
have been introduced in the energy calibration and
subtraction of such an amount would place these ap
pearance potentials at about the maxima of the deriva
tive curves. This latter position is what might be
expected to be determined by several of the methods
of determining appearance potentials.
In the light of the results of this paper, some evalua
tion can be made of the various methods4 of determin
ing appearance potentials of fragment ions. It is quite
obvious that the linear extrapolation methods bears
little relation to the true appearance potential. The
value so determined depends on the entire shape of the
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derivative curve. That is, it depends not only upon
the process of interest, but also on the succeeding or
competing processes which cause the derivative curve
to drop to zero, at which point the ionization efficiency
curve becomes linear. This method will generally tend
to give too high values for appearance potentials.
However, in cases where the derivative curve is not
much broader than that of the process used for voltage
calibration, the method will be fairly good.
The so-called vanishing current method is quite sub
jective and therefore difficult to evaluate. It seems
likely that this method usually determines the point of
maximum curvature in the ionization efficiency curve,
that is, the maximum of the derivative curve. Or, when
the derivative curve has a very broad top, the point
determined is probably that at which this curve first
comes to about the maximum. Designation of this
point of maximum curvature as the appearance poten
tial minimizes the error caused by thermal energy to
an amount about equal to the most probable value of
the thermal internal energy. A literal application of
the vanishing current method with the use of high
sensitivity of detection and of monoenergetic electrons
would be much more affected by thermal energy. Thus,
the success of the method probably depends on the
use of low sensitivity of detection and of electrons with
a broad energy distribution such that the low-energy
tail due to molecular thermal energy is practically un
detectable. This statement also applies to the following
method.
The method of extrapolated differences will give
similar results but in an objective manner since it con
sists essentially of comparing the ionization efficiency
curve for a rare gas, for which the derivative curve is
a rather sharp peak, with that of the process in question.
When the process in question has a derivative curve
which is also a fairly narrow peak, the maximum of
this peak will be determined and the "difference line"
will be fairly straight. When the derivative curve is
broad but has a fairly steep rise on the low-energy side
as will generally be the case, the difference line will be
curved but will approach the energy axis with nonzero
slope. The energy determined will be approximately
that at which the derivative curve first reaches or
nearly reaches its maximum. When the derivative
curve has a very gradual rise on the low voltage side,
the difference line will be badly curved at low energies
and its extrapolation to the energy axis uncertain. In
this case, the value obtained will be of dubious ac
curacy and will usually be too high.
The results of this paper indicate that the appearance
potential of the fragment ion appearing at the lowest
energy should be determined in the following manner.
First, the appropriate corrected derivative curves
should be plotted as shown in Figs. 12 and 13. Then
the energy E, for which the dissociative rate constant
is about 106 seci (more or less, depending on instru-mental characteristics) is the energy at which the
curve for the fragment ion just reaches unity. If the
value of (E-e) can then be calculated or estimated in
some way, the value of e may be obtained. In the pro
cedure, the low-energy tail of the derivative curve for
the fragment ion is all ascribed to the effects of molecu
lar thermal energy, the kinetics of the decomposition
and the en~rgy distribution of the ionizing agent. The
validity of this procedure does not depend on any
particular form of the theory. Only two requirements
must be met. The threshold law for ionization must
hold over the region of interest and the rate constant
for dissociation must be a monotonically increasing
function of the internal energy (in any form) of the
parent molecular ion. However, the determination of
(E-e) may require the use of a theoretical rate equa
tion, the formulation of which may be guided by the
characteristics of the derivative curves for the parent,
fragment, and metastable ions.
The foregoing procedure is difficult to apply pre
cisely in cases where competing reactions occur and
even more difficult for fragment ions other than the
ones appearing at the lowest energy. For the latter
case, the value of E chosen in the above manner will
correspond to a rate constant much greater than 106
seci• Also, if an expression for the rate constant of
the form of Eq. (1) is adequate, the frequency factors
for the higher energy processes must be larger than
those occurring at lower energies. Since a reliable and
quantitatively accurate theory is not available, the
best procedure would seem to be the following. If the
low-energy side of the derivative curve for the frag
ment drops off about as rapidly as in the case of the
fragment appearing at the lowest energy, then E may
be selected as before. This value may be taken as at
least the upper limit to the true appearance potential
and may be fairly accurate since such a steeply rising
curve indicates a very rapid variation of rate constant
with energy. The correction (E-e) may be calculated
by the use of a modified rate equation and estimated
frequency factors. The values of the latter factors may
be suggested by the form of the low-energy tail of the
derivative curve. However, this correction will be sub
ject to considerable error until a more reliable theory
is available and these processes are better understood.
Values of E have been chosen in this manner as
indicated by the arrows in Figs. 12 and 13. It may be
noted that in many cases the derivative of the frag
ment curve never actually reaches the value unity.
This may be due to the presence of competing processes
in which case a judicious extrapolation can be made.
It may also result from lack of fulfillment of the two
requirements regarding behavior of ionization cross
section and rate constant with energy, as mentioned
previously. In this case, the value E may be chosen as
the point at which a plateau or near plateau is reached
by the derivative curve. This behavior is illustrated by
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TABLE III. Appearance potentials taken from Figs. 12 and 13 and corrected in the manner described in the text.
Ionization Energy scale
Molecule potential in ev Fragment E+I.P. in ev correction (ev) E-.· in ev E+I.P. in ev
C.H,NH. 8.9 CH.NH 2' 9.76 -0.02 0.03 9.71
C,H7NH. 8.8 CH2NH 2' 9.64 -0.02 0.08 9.54
C,H,OH 10.1 C,H.' 10.80 -0.03 0.27 10.50
C,H7OH 10.1 C,H.OH' 11.03b -0.03 ~0.25 ~1O.75
C,H70H 10.1 CH20Hi 11.500 -0.03 ~0.36 ~11.11
• The values of E were calculated using an equation of the form of Eq. (1) but with one-fourth the theoretical number of oscillators. The frequency factors
for the production of the various ions were taken to be 10" for CH,NH,+ and CH,oH+. 10' for C,H.+. and 1011 for C,H.OH+.
b Corresponds to k~10'l sec-I.
o Corresponds to k~5XI09 sec-1 if radiationless transitions to lower electronic states occurs readily_ If not, this energy may correspond to a value of k per
haps as low as 10' sec' and the value of (E-E) may be correspondingly lower and that of (E+I.P.) higher.
the curves in Fig. 13 for ions of masses 59 and 31.
Table III lists the chosen value of (E+ I.P.) where
I.P. is the ionization potential of the parent ion, the
appropriate corrections and finally (e+ I.P.) the "true"
appearance potential. The corrections to the appear
ance potentials of ions of masses 59 and 31 were made
by straightforward application of the modified rate
equation using the frequency factors and number of
oscillators given in the table. §
The calibration of the energy scale must be made
with care. The value of E, chosen in the aforemen
tioned manner, corresponds to the production by
electrons or photons of the lowest energy in the dis
tribution of molecular ions which dissociate at the
proper rate. Thus the calibrated energy scale should
refer to the lowest energy of the photon or electron
energy distribution. Since, in the photoionization work,
the energy scale refers to the center of the photon
energy distribution, the value of E must be lowered by
an amount equal to the half-width of this distribution.
In the case of ionization by electron impact the ioniza
tion potentials of rare gases are usually used as stand
ards. The uncorrected second derivative curve for a
rare gas should be a reversed electron energy distribu
tion curve,a which should be Maxwellian in the ideal
case. The edge of the sharply dropping high-energy
side should be chosen as the voltage calibration point.
Some difficulty may be encountered in cases in which
there are one or more low-lying excited states of the
ion. Where the RPD technique of Fox et al.6 is used,
and contact potentials eliminated, the low energy side
of the effective electron energy distribution should be
used to determine the energy scale.
§ It should be noted that much, if not ali, of the ions of mass 31
are produced by dissociation of parent molecular ions which appear
to be formed initially in an excited electronic state. This is indi
cated by a rise of the sum of ion derivative curves in that energy
region. It is assumed here that this state readily makes radiation
less transitions to the lower electronic state or states. This assump
tion is supported, but by no means proved, hy the fact that in
Fig. 13 the normalized experimental curve for mass 31 seems to
increase at the expense of a decrease in the curve for mass 59.
However, this may be fortuitous. A survey of appearance potentials of fragments from
large molecules, such as those tabulated by Field and
Franklin4 yields an abundance of examples which give
evidence of excess energy in the products at threshold.
In some cases, e.g., CHa+ from propane, the excess
energy is thought to exist predominantly as kinetic
energy of the fragments as indicated by some experi
ments of Kandel.42 Such kinetic energies can be under
stood in terms of the quasi-equilibrium theory only if
dissociation took place over a potential hill. In this
latter case, the ions would still have a rather narrow
energy distribution displaced from zero. The variation
of the appearance potential with kinetic energy as
found by Kandel indicates that this kinetic energy
distribution is instead rather broad and that the entire
process is very similar to that occurring in diatomic
molecules. It may be noted that the production of
CHa+ from propane is a case in which one might expect
unusual behavior as will be discussed later.
Frequently, a fragmentation process is interpreted as
yielding more than two fragments. This interpretation
is usually made in order to account for the otherwise
excessive value of the appearance potential. While this
interpretation is likely to be correct in most cases, in
others the excess energy may be in the form of vibra
tional energy as predicted by the unimolecular decay
theory. According to this theory, a simultaneous
breakup into more than two fragments is an extremely
unlikely occurrence, the usual process being successive
decomposition when sufficient energy is available. An
ion with an appearance potential which indicates that
three rather than just two fragments are being formed
should, of course, have an appearance potential higher
than that of the "parent fragment." For example, if
the process is
then the appearance potential of AB+ should be less
than that of A+. If the fragment C has a large number
42 R. J. Kandel, J. Chern. Phys. 22, 1496 (1954).
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of degrees of freedom, it may carry off a large amount
of excess energy on the average and the appearance
potential may be abnormally high.
It should be possible in most cases to establish
whether or not a fragment is produced by successive
decay as in (12). A study of derivative curves such as
the experimental curves of Fig. 13 should be suggestive
and the detection of the metastable ion produced by
the second decay of process (12) will be very good evi
dence, especially if the appearance potential of the
metastable ion can be measured even crudely.
Unfortunately, in many cases the intensity of the
metastable ion will be very low. If the theory is quali
tatively correct, the intensity of the metastable ion
will be roughly proportional to the half-width of the
metastable derivative curve and to the probability of
producing the parent ion of the proper energy. The
behavior of the first factor can be seen from Eq. (1)
from which E2 corresponding to k2=6X106 secl and
El corresponding to k1=4X104 sec I can be calculated.
The quantity (E2-E1) is then the half-width of the
metastable derivative curve as shown in Fig. 7. It is
seen that this quantity increases as n and ~ increase
and as II decreases. Thus one may expect the most
intense metastable ions to be produced by the decom
position of large molecules in such modes as to give
low-frequency factors. Such low-frequency factors have
been shown to be expected in cases such as loss of H2
and H20. This is probably the reason for the relatively
high intensity of the metastable ion observed here in
the case of n-propanol. Also, a survey of mass spectra43
indicates that the most intense metastable peaks do
correspond to such processes. The expected increase in
metastable intensity for processes of higher ~ may not
be realized in many cases because the process may have
to compete in the range of k= 105 secl with one of
lower ~ and thus possibly much higher rate constant.
This occurrence should be evident from a study of the
derivative curves. Another important type of compet
ing process is the allowed emission of radiation from
electronically excited states. It will probably be difficult
to detect the occurrence of this process experimentally.
In principle its occurrence should be detectable from
the shape of the derivative curve of the fragment ion
at the low-energy threshold.
The theory indicates certain other instances in which
the observed appearance potential of a fragment ion
might be expected to be abnormally high. These are
cases in which the process of interest must compete
with a much more likely process near the appearance
potential. This can be illustrated by considering the
process
ABCD+~AB++CD, (13)
43 Mass Spectral Data (American Petroleum Institute, Carnegie
Institute of Technology, Pittsburgh, Pennsylvania, 1955). which requires less energy than the processes
ABCD+~ABC++D, (14)
ABCD+~A++BCD, (15)
or
ABCD+~AB+CD+. (16)
If reactions (14) and (15) have frequency factors
which are only slightly higher than that of reaction
(13) then the excess energy required to make the
former reactions occur faster than the latter will be
abnormally large. The case of reaction (16) is covered
by Stevenson's Rule44 which is explained by Krauss
et al.46 in terms of the quasi-equilibrium theory. In
such cases as these, most of the A + and CD+ ions ob
served may be produced by other reactions such as
successive decomposition. It might be noted that non
randomization of internal energy may help processes
such as (14), (15), and (16) to compete with a process
such as (13).
CALCULATION OF MASS SPECTRA OF POLYATOMIC
MOLECULES
The results of this paper and the curves of Fig. 13
indicate that the theory of Rosenstock et al.6 may be
inadequate for the calculation of absolute rate constants
for dissociation. Nevertheless, for the calculation of
mass spectra as done by Rosenstock et at." and Fried
man et at.16 only relative reaction rates are important
and here the theory appears to work reasonably well.
Thus, the calculated and experimental curves of Fig. 13
differ little except in the energy scale which is essen
tially selected by these investigators to give agreement
with experimental mass spectra. This selection is ac
complished by the arbitrary selection of an energy
distribution of the parent molecular ions. A good ap
proximation to the actual energy distribution may be
obtained expelimentally as the sum of the derivative
curves of all ions corrected for variation of cross sec
tion with energy. In addition, it is not a very satis
factory situation in which appearance potential values
are used in the application of a theory which implies
that many of these values are badly in error.
CONCLUSIONS
The interpretation of appearance potentials has usu
ally neglected many complicating factors. Even for
the case of diatomic molecules, the occurrence of pre
dissociation and loss of energy by radiation or auto
ionization can lead to errors in interpretation. For
44 D. P. Stevenson, Trans. Faraday Soc. 49, 867 (1953) .
• 6 Krauss, Wahrhaftig, and Eyring, Ann. Rev. Nuclear Sci. 5,
241 (1955).
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complex molecules, experiments show conclusively that
many (if not most) appearance potentials are appreci
ably affected by either the thermal energy or the
kinetics of the dissociation of the parent molecular ion
and probably by both. The theory of Rosenstock et al.
probably does not give quantitatively accurate rate
constants at least in many instances. This failure may
be due in some instances to nonfulfillment of the
assumption of energy equilibration in the parent ion.
However, in other instances in which it is likely that
energy equilibration occurs, the experimental rate
constant seems to vary with energy much more rapidly
than calculated. This may be due to the crudeness of
the approximations used in deriving the expression for
the rate constant. If the same form of expression is
retained, experiment seems to require a much smaller
value for n, the number of oscillators in the parent ion.
Thus, appearance potentials are probably less affected
by the kinetics of dissociation than the theory predicts, although the effect is almost certainly not negligible in
many instances. Experimental techniques are now
available which can be used to determine modes of dis
sociation of molecular ions and to get quantitative
data on the reaction rates of these dissociations. Such
studies would lead to better understanding of uni
molecular decomposition in other fields such as photo
chemistry and chemical kinetics.
ACKNOWLEDGMENTS
The author gratefully acknowledges the valuable
assistance of Dr. H. Hurzeler, in taking the photo
ionization data, and of Dr. H. E. Stanton who measured
the kinetic energies of the fragment ions. He also
wishes to thank Professor M. G. Inghram, who very
kindly allowed the author the use of his photoionization
equipment, and Mr. V. Reisenleiter and Mr. G. James,
who assisted in performing some of the graphical
integrations.
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1.1721301.pdf | A Retarding Potential Method for Measuring Electrical Conductivity of
OxideCoated Cathodes
I. L. Sparks and H. R. Philipp
Citation: J. Appl. Phys. 24, 453 (1953); doi: 10.1063/1.1721301
View online: http://dx.doi.org/10.1063/1.1721301
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Downloaded 20 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 24. NUMBER 4 APRIL. 1953
A Retarding Potential Method for Measuring Electrical Conductivity
of Oxide-Coated Cathodes*
I. L. SPAllKS t ANI) H. R. Pmr.n>P
. DejJarlment of Physics, UnifJe1'sUY of Missouri, Columbia, Missouri
(Received November 10, 1952)
A. retarding potential method is developed for measuring the electrical conductivity of normal oxide
cathode coatings. The method is limited by normal current measuring devices and can not be used for
coatings which have a conductivity to thermionic emission ratio greater than 2 em/volt. Advantages of the
method are: (1) the conductivity of coatings which are in a normal state for thermionic emission may be
measured without the use of probe wires or other devices which might impair the thermionic emission of the
sample, and (2) conductivity and thermionic emission measurements may be made simultaneously on the
same coating sample. The theory of the method is discussed in detail and experimental results obtained
using this method on both BaO and (BaSr)O coatings are given.
INTRODUCTION
FROM the viewpoint of theoretical considerations,
simultaneous measurements of thermionic emission
and electrical conductivity of oxide~coated cathodes are
desirable. Investigators have encountered certain diffi
culties in the methods used for determining the electrical
conductivity. This has been especially true when
attempts were made to study thermionic emission from
the same sample. The method1 of embedding a small
probe wire in .the coating has probably been used more
widely than other methods. This is usually accomplished
by spraying a small amount of coating on a cylindrical
base metal, winding a fine platinum wire around this,
and again spraying until the wire is completely covered.
In general, a nonuniform coating surface has resulted
owing to the irregularity introduced by the probe wire.
Another disadvantage to this method is that an unusu
ally thick coating is required to completely cover the
wire. Abnormally thick coatings often give a low ther
mionic emission and are prone to develop cracks and
adhere poorly to the base metal.
Hannay, MacNair, and White2 used a ceramic base
cathode sleeve around which a pair of conductivity
leads were wound. The oxide coating was sprayed over
this base. Although both conductivity and thermionic
emission were readily measured, it was not possible to
extend these studies below the temperature range in
which an optical pyrometer could be used. The absence
of a continuous base metal prevented the use of ther
mocouples for a temperature determination.
. Loosjes and Vink's3 method of pressing the coating
between two flat base metals was satisfactory for deter
mining the conductivity but did not allow a convenient
means of measuring the thermionic emission. In order
to determine the thermiOl1'!c emission, it was necessary
to separate the two cathodes and insert an anode be
tween them after the conductivity measurements were made. This, of course, involves the hazard of damaging
the coating and of changing surface activity as was
indeed observed by them. A similar method was used
by Yount in which the thermionic emission was taken
to a ring anode. Accurate measurements of thermionic
emission were not possible as the emitting area was not
well defined.
With these difficulties in mind, it was thought de
sirable to attempt to devise a method of measuring the
conductivity which would leave the coating in its
normal state for thermionic emission. As thermionic
emission current is drawn from a cathode, a voltage
drop develops across the coating. This arises from the
fact that the coating has a finite conductivity and a
current equal to the emission current passes through it.
If this voltage drop can be measured as a function of the
emission current, the resistance of the coating can be
determined. A knowledge of the geometry of the coating
will enable one to calculate the specific conductivity.
Young and Eisenstein6 have reported on a retarding
potential method which would enable one to determine
the voltage drop across the coating. In this work they
assumed that the Schottky6 lowering of the potential
barrier at the surface of the cathode was negligible
compared to the voltage drop across the coating. They
also assumed that the method was applicable for a wide
range of anode voltages. In the present investigation a
similar method is developed without making these
assumptions. Young and Eisenstein found the slope of
the reatarding potential curve was less than the
expected value, -e/kT. A possible explanation of this is
given here.
EXPERIMENTAL TUBE
A schematic drawing of the experimental tube used in
this investigation is shown in Fig. 1. The indirectly
heated cathode, the type used in the 2C39 lighthouse
,. Supported in part by the U. S. Office of Naval Research. 4 J. R. Young, J. Appl. Phys. 23,1129 (1952).
t Now at Eastern Illinois State College, Charleston, Illinois. 6 J. R. Young and A. S. Eisenstein, Phys. Rev. 75, 347(A)
1 G. W. Mahlman, J. Appl. Phys. 20, 197 (1949). (1949).
2 Hannay, MacNair, and White, J. Appl. Phys. 20, 669 (1946). 6 F. Seitz, The Modern Theory of Solids (McGraw-Hill Book
a R. Loosjes and H. J. Vink, Philips Res. Rep. 4, 449 (1949). Company, Inc., New York, 1940), pp. 162.
453
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CATHODE
-.===~
TO PUMP
NICKEL
BAND
FIG. 1. Experimental tube.
tube, has a flat 0.5 cm2 pure nickelt button welded to the
top. A Ni-Mo thermocouple is used to determine the
temperature of the cathode base metal. Electrons
passing through the pair of one-mm holes, to mm apart
in the tantalum anode, enter the tantalum collector
through a 6-mm hole. The collector is supported by
Nonex beads and the anode by a nickel band around
the main press. In order to minimize electrical leakage,
electrical connections to the anode and collector enter
the tube through separate extended presses. A batalum
getter, shown in the small attached side tube, was
flashed immediately after seal-off. Figure 2 shows the
electrometer tube circuit with the G.E. Type 5674
electrometer tube, which is used to measure the collector
current. This circuit is capable of measuring collector currents in the range 10-9 to to-I. ampere. Collector
voltages are measured with a 0-16 volt range Leeds and
Northrup Type K-2 potentiometer.
THEORY OF THE RETARDING POTENTIAL METHOD
An energy level diagram for the cathode, anode, and
collector of the experimental tube is shown in Fig. 3.
The diagram has been simplified somewhat by the omis
sion of impurity levels of the oxide and the levels of the
filled bands, since they are not necessary for this dis
cussion. Diagrams A, B, and C represent the situations
for different values of anode and collector potentials.
In A, the anode and collector are at cathode potential.
In this case, the Fermi level of the base metal, the
JoJ",,===t
e\{,
----------------- J,.I)
METAL OXIDE METAL METAL
FIG. 3. Energy level diagram of experimental tube with
anode at different potentials.
chemical potential of the oxide coating, and the Fermi
levels of the anode and collector are at the same energy
These are designated by the symbols Jl.l, Jl.2, Jl.3, Jl.4,
respectively. If the work function of the anode, 4>A,is
greater than the work function of the cathode, 4>0, a
retarding contact potential difference (C.P.D.) equal
to (4)A -4>0) will appear between the two surfaces.
Since the anode and collect<1r of the experimental tube
were both tantalum, the work functions of these are
shown to be about equal. The application of a large
accelerating potential, VAl, to the anode displaces the
Fermi level of the anode Jl.a downward by this amount
FIG. 2. Circuit diagram. as shown in B and permits a flow of electron current
t 1001 electrolytic nickel obtained from E. M. Wise, Inter-from the cathode to the anode. The presence of a high
national Nickel Company. electric field at the cathode causes a Schottky lowering of
Downloaded 20 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_permissionsE LEe T RIC ALe 0 N Due T I V I T Y 0 FOX IDE -C 0 ATE DCA THODE S 455
its surface barrier by an amount designated as etlepcI' Due
to the finite conductivity of the oxide, a voltage drop
V iRI, will develop causing a tipping of the energy bands
in the oxide. Electrons leaving the cathode will arrive
at the anode with a minimum kinetic energy of
since this is the kinetic energy gained in passing be
tween the two surfaces. Of course, most electrons will
have energies greater than this owing to the energy
distribution of the electrons leaving the surface of the
cathode. A fraction of the electrons arriving at the anode
will pass through the one mm hole and reach the col
lector with kinetic energies which depend on the col
lector potential V c. If the potential of the collector is
positive with respect to the cathode and greater than
(epC-tlepCI), (Le., the surface barrier of the collector is
below the surface barrier of the cathode in the diagram),
all electrons passing through the hole in the anode will
reach the collector. As the collector potential is made
less and less positive, the current reaching the collector
remains constant until the surface barrier of the col
lector is at the same level as the cathode barrier.
Beyond this point, the collector surface barrier becomes
negative with respect to that of the cathode and elec
trons arriving at the anode with the minimum kinetic
energy will be repelled by the collector. As the collector
is made more negative, only the higher energy electrons
will be collected. Thus, if one makes a plot of log col
lector current, Ie, versus collector voltage, V c, the curve
should show a break at a collector voltage for which the
surface barriers of the cathode and collector are at the
same level, Fig. 4. This would occur at a point where
V R, the true cathode-collector retarding potential, has
the value zero.
If now an accelerating voltage, V A2, (VA2> VAl), is
placed on the anode, the Fermi level J.l.3 will be displaced
downward, Fig. 3-e. The surface barrier of the cathode
FIG. 4. Retarding
potential curves ex
pected from discus
sion of Fig. 3. .. ....
CD o ..J
Vc TEMP. = 8010K -Q.
Z
~
~ '4 1.\
g -II CD 46 \Q.TS 8.2XI0-7AMP
..J
9.5XI0-7 ® 92
® 275 1.6X10-6
® 445 2.1 X 10-6
-12
Vc (VOLTS)
FIG. 5. Experimentally determined retarding potential curves
for different anode potentials.
will be lowered by an amount etlepc2, (tlepC2> tlepCl)
increasing the anode current.§ A larger voltage drop
across the coating, ViR2, will occur due to this increased
anode current. A new plot of logIc versus V c will yield
a curve similar to the previous one except the saturated
collector current will now have a higher value due to the
increase in anode current, Fig. 4. Also, the break
(V R = 0) in the curve should occur for a value of V c
somewhat higher than before since the surface barrier
of the cathode is now lower than that in case B by an
amount tlE,
COMPARISON WITH EXPERIMENT
Four retarding potential curves obtained experi
mentally in the manner described above for different
values of V A are shown in Fig. 5. The shift of the curves
to the right with increasing current can be seen. The
scale is broken at V c= 3.2 volts in order to show that the
saturation collector current remains constant to 10 volts
If the collector voltage, V c, is adjusted to a particular
value VCh Fig. 3-B, such that VRl will have a negative
value, the energy difference between the collector
vacuum surface level and J.l.2 at the cathode surface will
§ In studies of this kind, it is usually assumed that the trans
mission coefficient for electrons at the surface barrier of the
cathode is equal to unity and does not change appreciably with
V A over the voltage range considered here. It is further assumed
that the energy distribution of emitted electrons does not change
with respect to 1'2 at the surface.
Downloaded 20 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_permissions456 I. L. SPARKS AND H. R. PHILIPP
be EI and give rise to a particular collector current,
I Cl. For a higher anode potential, V A2, in Fig. 3-C, the
collector voltage must be adjusted to a new value, V C2,
in order for the collector to receive the same current
ICI. When this is done, EI will have the same value as
before. From the figure it is seen that in case B,
and in case C,
E1= e(ViR1-V C2+<I».
Since EI and <I> are constant,
(V C2-V C1) = (ViR2-ViR1). (3)
(4)
(5)
In the work by Young and Eisenstein,· it was assumed
that the change in voltage drop across the coating, due
to different anode currents, was equal to the difference
in potentials at the break in the retarding potential
curves. Since the presence of any space charge near the
cathode surface affects the location of the break point,
their method is not applicable in the space charge
region. This limitation has been foreseen by Wright and
Woods7 who use a similar method but make a correction
for the space charge that is present. The method
described here is not limited by space charge since the
surface barrier of the collector is at a much higher
potential than the space charge barrier at the operating
point where conductivity determinations are made. In
the work mentioned above, it was also assumed that the
. Schottky lowering of the surface barrier of the cathode
was negligible compared with the voltage drop across
the coating. From the discussion in the previous section
and the method of taking data described below, it is
apparent that the present method is not limited by the
space charge barrier and the Schottky lowering of the
barrier at the surface of the cathode is not neglected.
The horizontal line AB in Fig. 5 represents a typical
constant collector current value, Ie. From Eq. (5) the
change in V c is equal to the change of voltage drop
across the coating corresponding to four values of
2.46
2.44
2.42
lQ
~ 2.40 o > -2.58
~
2.36
12.34 T" 802"K
2.32 OS 1.0 1.1 1.2 L3 L4 1.5 1.6 L 7 1.8 IS 1.93
lA (AMI.' x 10"1)
FIG. 6. Collector voltage versus anode current. ----
7 D. A. Wright and J. Woods, Proc. Phys.(Soc. (London), B,
65, 134 (1952). current I A through the coating. A plot of V c versus I A
should yield a straight line, the slope of which is equal
to the effective resistance of the coating, R. A knowledge
of the coating thickness t, and area A, gives the effective
specific conductivity.
u=t/(RA). (6)
Since complete retarding potential curves of the type
shown in Fig. 5 required considerable time to obtain, a
faster method was devised to determine the conductiv
ity. At a given temperature and at a particular anode
current, the saturated collector current was measured
by setting the collector at + 10 volts. Then a collector
current, one or two orders of magnitude lower than the
saturation current was selected. As the anode current
was varied by changing the anode voltage, the collector
voltage necessary to give the selected collector current
was measured. This effectively determines the points of
intersection of the line AB with the retarding potential
curves, Fig. 5. In this manner, a series of these points
could be determined in a relatively short time.
A typical plot of these intersection voltages, V c vs I A
used to evaluate (J' is shown in Fig. 6. The anode voltage
range covered to obtain the anode currents for this
curve was 46 to 455 volts. The experimental points fall
close to a straight line in the high voltage range but
deviate somewhat at the three lower currents, a situa
tion found for all curves of this type. The theory dis
cussed above assumed that the current reaching the
collector was a definite fraction of the anode current,
therefore, a constant ratio of anode current to saturated
collector current should be obtained. Table I shows the
measured ratio of anode current, I A, to saturated
collector current, I C 8, for the anode voltages used in
obtaining Fig. 6. It is seen that the ratio decreases for
the three points falling below the straight line in Fig. 6
thus explaining the reason for their deviation. The exact
cause of this effect is not known, but in the subsequent
measurement of conductivity, only anode potentials
above 180 volts were used to avoid this difficulty. From
the straight line section of this curve a conductivity of
5.7X 10-70-1 cm-1 is computed using Eq. (6).
ELECTRON ENERGY DISTRIBUTION
For the case of a diode with concentric cylindrical
geometry, Schottky8 derived an equation for thermionic
emission in retarding fields, assuming a Maxwell
Boltzmann distribution of electron velocities. This
equation reduces to a simple form for the plane parallel
geometry case:
(7)
where J R is the current density reaching the anode, Jo is
the zero field thermionic emission current density, and
V R is the true retarding potential between the surface
of the cathode and the surface of the anode. If one
plots log J R versus V R, a straight line should result, the
8 W. Schottky, Ann. Physik 44, 1011 (1914).
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slope of which is -e/kT. Hung9 investigated the emis
sion from oxide-coated cathodes in retarding fields and
found that in the high energy region the experimental
curve was a straight line and the temperature calculated
from the slope agreed well with the observed tempera
ture. Some deviation from the theoretical curve was
found in the low energy region which he attributed to
space-charge effects.
In the case of the plane parallel geometry diode, it is
apparent that only the normal component of the elec
tron velocity determines whether a given electron will
reach the anode. It is also apparent that because of the
complicated field arrangement in the experimental tube
used in this investigation, it would be quite difficult to
calculate accurately the energy distribution of electrons
arriving at the collector. It is interesting to note, how
ever, that, if one assumes that the total initiai"velocity,
rather than the normal component of velocity, deter
mines whether an electron will reach the collector, a
somewhat different expression is obtained for the col
lector current and with high anode potentials this will
probably be the case. An electron emitted from the
cathode has both normal and parallel velocity compo
nents but under the influence of the accelerating field
will arrive at the anode moving in essentially a normal
direction. Since the normal component of velocity is
large compared with the parallel components, any
electron passing through the anode will do so with
negligible parallel displacement. It is only as the
electron approaches the collector and is retarded that
the initial velocity components again influence the
direction of motion. Since the collector is a hollow cyl
inder located near the anode exit aperture all electrons
passing through the anode will be collected regardless
of their parallel velocity components. Assuming that the
electrons in the oxide with suffiicent energy to escape
have a Maxwell-Boltzmann distribution, the number
per unit volume having velocity components in the
range, dvr, dvu, and dvz is given by
N(vr, Vy, v.)dvrdvudv.
=n(27rkT/m)-t exp-[mv2/2kTJdvrdvudv. (8)
where n is the number per unit volume with all velo
cities. The velocity distribution of those escaping unit
area per unit time will be this number multiplied by
Vr with x taken as the direction normal to the surface.
To determine the number leaving unit area per unit
time having a speed between v and v+dv, the number
leaving per unit time must be integrated over a semi
spherical shell of thickness dv. That is,
v+d. ,,/2 2r f. i J: v3N(vz, vll, v.) sinO cosOdvdOd<p (9) ----
9 C. S. Hung, J. App!. Phys. 21, 37 (1950). TABLE I. Anode current-saturated collector current ratios.
VA IAllc.
455 823
410 817
365 830
320 837
275 832
227 828
183 815
137 723
92 644
46 715
where v2 sinOdvdOdcp is the volume element in v space, and
v,,=v cosO. Carrying out the integration and expressing
the result in terms of energy, the number of electrons
escaping unit area per unit time with energy. between
E and E+dE is given by
g(E)dE=CEe-ElkTdE (10)
where C=n(27rm)-l(kT)-I. The number of electrons
arriving at the collector will be the fraction, I, of these
which pass through the anode aperture with energies
greater than e V R and the current will be the electronic
charge multiplied by this number.
Io=eIcioo
Ee-ElkTdE
,VR
= eIn(27rmkT)-i(kT +e V~)e-·v RI kT. (11)
For a given temperature this may be expressed as,
Io=K(kT+eVR)e-·vRlkT. (12)
According to this equation, a plot of log I o/(kT+eV R)
vs V R should yield a straight line with slope -e/kT.
Figure 7 shows log Ie vs V c for two different tempera
tures, as would be plotted for the usual retarding po
tential case. Plotting V 0, the applied voltage, rather
than V R merely shifts the curve to the right by the
contact potential difference. These curves differ from
the usual diode retarding curves in three respects, (1)
the saturated portion is much flatter than is usually
obtained, (2) the break is sharper, and (3) the tempera
ture, calculated by assuming the slope equal to -e/kT,
is in poor agreement with the measured temperature as
shown on the figure. •
Figure 8 shows the same data plotted as log 10/
(kT +e V R) versus V R. The true retarding voltage V R
was determined by assuming V R=O at the break
points in Fig. 7. :The temperature was calculated again
assuming the slope equal to -e/kT and was found to
agree closely with the measured temperature as indi
cated on the figure.
Thus, the experimental observations seem to favor
the latter expression for the collector current. Whether
this is the exact expression or not is not important as
far as the theory of conductivity measurement is con-
Downloaded 20 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_permissions458 1. L. SPARKS AND H. R. PHILIPP
-8 I I I I I
A .
~ B
:
-9 ~ -
CURVE TMEAS. TCALC• VA
A 8500K 967"K 275Y01J'
f-8 80loK 874"1< 275VCU Q: -10
~
.5-
~
Cl)
9 -II -
-12 -
I I 1 1 I I
~ .. 5 6 7 8 9 -13
Vc (VOLTS)
FIG. 7. Retarding potential curves as would be plotted for diode
with plane parallel geometry.
cerned. In this theory the only assumption concerning
the energy distribution is that the electron density at
some level above the cathode surface level remains con
stant with respect to J.l2 as V A is changed. This seems
to be a valid assumption since the retarding potential
curves at a given temperature, Fig. S, are parallel in
the high energy region.
APPLICATION OF THE RETARDING POTENTIAL
METHOD
Results obtained in applying this method to five
different cathodes will be given. For one cathode,
results will be given with the cathode in three different
states of activation. These results are summarized in
Table II. Carbonates used in spraying the cathodes
were equal molar (BaSr)C0 3Ir and BaC03'~ The
cathodes had a coating weight of approximately 10
mg/cm2, t)le density approximately 1 g/cm3 and the
area was 0.5 cm2•
Conductivity data at various temperatures were
obtained for all cathodes except VI and VII. Three
typical curves of collector voltage, V 0, versus anode
current, I A, which were obtained for cathode II, are
given in Fig. 9. The resistance of the coating at these
" C51-2 obtained from Raytheon Manufacturing Company.
, Ultra-Pure BaCO. from Mallinckrodt Chemical Works (30
parts per million of Sr) used on cathode V and a carbonate pre
pared in this laboratory by a process involving the recrystalliza
tion of Ba(NO.h (less than 1 part per million of Sr) used on
cathodes VI and VII. Preparation carried out by Mr. Harold
John. temperatures was obtained from the slopes of the curves.
Coating conductivity values were calculated using Eq.
(6) and will be presented later.
An attempt was made to measure the coating condu-c
tivity for cathodes VI and VII using this method. Over
an approximate temperature range 550 to lOOOoK the
change in voltage drop across the coating .:l V, for these
two cathodes, was too small to be measured. The
stability of the experimental apparatus was such that a
.:l V of 0.005 volt could have been detected. Since the
coating voltage increases with increasing thermionic
emission 10, and decreases with increasing conductivity
(I, it is apparent that the ratio, (1/10, of these two
cathodes wa,s too large (i.e., giving a voltage drop too
small to be detected) in order for the conductivity to be
measured by this method. This ratio (I/lothen permits
us to calculate the range over which this method can be
used for determining the coating conductivity.
The importance attached to the ratio (1/10 in setting
the range over which this conductivity method can be
used is seen by a simple calculation. Taking the mini
mum measurable voltage change to represent essentially
the total coating voltage, as an upper limit calculation,
this voltage may be expressed as V = 10t/ (I, where t is
the coating thickness and the ratio (I/Jo=t/V. Setting
V=O.OOS volt and t=O.OI cm gives (1/10=2 cm/volt.
Thus, cathodes for which (1/10> 2 cannot be expected
to have a coating voltage greater than 0.005 volt.
Data on both the conductivity and thermionic emis
sion of (BaSr)O over the same temperature range are
found in the literature in only a few cases. Hannay,
7,....--...---.--.---,---.-----,---,
-8
-II CURVE TMEAS. TeAle.
A
B 850"1< 857"1<
801"1< 790"K
o
FIG. 8. Retarding potential curves plotted for experimental tube.
Downloaded 20 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_permissionsE LEe T RIC ALe 0 N Due T I V I T Y 0 FOX IDE - C 0 ATE DCA THO DES 459
TABLE II. Summary of results.
Cathode (:t) (~ 01 number Material (tV)
I (BaSr)O 1.25 1.18 0.39
II (BaSr)0 1.42 0.92 0.41
III (BaSr)O 1.38 1.04 0.41
IV (BaSr)O 1.34 1.42
V BaO 1.79 1.59 0.62
VI BaO 1.37
VII BaO 1.50
MacNair, and White2 show data from which the ratio
0'1 Jo can be calculated. Their value is approxiamtely
0.03 at 1000oK. Mahlman'sl data give a ratio of 0.04 at
7700K and the value for (BaSr)O reported in Table II,
in each case, is less than 2 at lOOO°K. The value for
cathode V (BaO) is also less than 2, while for VI and
VII it is evidently greater than 2. Thus, it is seen that
the method is applicable for the (BaSr)O cathodes
which are reported but is applicable only in certain
cases for BaO.
CONDUCTIVITY AND THERMIONIC EMISSION AS A
FUNCTION OF TEMPERATURE
The experimentally observed variation of the' oxide
coating conductivity with temperature is usually
expressedlO
(13)
Over the temperature range form 3000K to approxi
mately 10000K it has been found that the plot of log
0' versus liT may show two linear regions with appro
priate values of K and Q for each region. Loosjes and
Vink3 attribute their results of this type as being due to
two different conduction mechanisms in parallel. In
region I (the low temperature region), the measured
conductivity is predominately due to electron con
duction through the crystals of the coating. The
electrons pass from one crystal to the next iLt the point
of contact. In region II (the high temperature region),
the observed conductivity arises primarily from, a
conduction by the electron gas in the pores between the
crystals. Electrons thermionically emitted from the
crystals form the electron gas in the pores and may
then pass from pore to pore. A more recent explanationll
supposes that the surface and bulk conductivity of the
crystals are different; thus two parallel conduction
paths result. ,
The thermionic emission of an oxide coated cathode
is usually shown by a Richardson plot in which the
logarithm of the zero field emission current density
divided by the square of the absolute temperature,
10gJo/P, is plotted as a function of reciprocal tempera
ture, liT. For all cathodes studied this plot gave a
10 A. Eisenstein, Advances in Elecwonies (Academic Press, Inc.,
New York, 1948) Vol. 1, Pt. 1.
11 D. A. Wright, paper submitted to the Symposium on Electron
Emission, (New York City), January 30, 1951. J .(lOOO"K) .,(lOOOOK) .,/J.(lOOOOK) Break-point in conductivity (amp/em') (ohm-I em-I) (em/Vo)t) curve temp. oK
2.00XIQ-2 1.55XlO-a 7.75XIQ-2 720
1.10XIQ-2 1.35 X 10-3 1.23 X 10-1 748
7.50XIo-a 5.00XIo-' 6.67XIQ-2 742
1.26XIQ-2 1.60XIo-a 1.27XI0-1 <700
l.80XlO-' 5.23X 10-· 2.90XI0-1 750
1.36X 10-2 >2
1.29X 10-' >2
straight line. The slope of the Richardson plot is equal
to -er/>IK, where k is Boltzmann's constant. From the
slope, the apparent work function, er/>, was determined.
After the cathodes were processed and aged by draw
ing a small emission current from the cathode for
24 hours, conductivity and thermionic emission meas
urements were made as a function of temperature.
Figure 10 shows the temperature dependence of the
conductivity for cathode V. A typical curve for the
(BaSr)O cathodes (cathode II) is shown in Fig. It.
These curves are, similar to the type observed by
Young' and also by Loosjes and Vink.3 The conductiv
ity of cathode IV was measured over the range 700 to
1000 OK. Since the plot of log (f versus liT for this
cathode was a straight line, the break in the curve
evidently occurs below 700oK. Using a different method
of measuring conductivity others3.4 have found similar
straight line conductivity plots, particularly with
cathodes in a low state of activation.
The zero field emission current density, Jo, was deter-
u; ... ...
0 >
~ 2,48 T -891· K
<T -3.59 xlo-h~CM-'
2.46
2,2
2,68 T-a29·K
(7'·1.41 xI6~'CM"
2 .
2.&4
3.0 4,0 5.0 6.0
I (AMP)( 10"1
2.73r------~----...,.-----T'""""'I
2.72 T·70aoK
·5 -I -. 'cro 2.43,c10 n CM
2.71_=-___ ...L. ____ ....... _____ ""--'
1.0 1.11 2.0 2.5
I (AMPle 10')
FIG. 9. Coating voltage as a function of coating current
for cathode II.
Downloaded 20 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_permissions460 L L. SPARKS AND H. R. PHILIPP
BaO
CATHODE NO.5
8 ..J-7
0, = 0.62eV
-8.9
FIG. 10. Temperature dependence of conductivity
for cathode V (BaO).
mined by making a Schottky plotlO (log] VS V A i) and
extrapolating the straight line portion to zero anode
voltage. Conductivity and thermionic emission meas
urements were made not only on the same coating but
at exactly the same time. This ability to take simul
taneous measurements of emission and conductivity is a
distinct advantage of this method.
!fa. Typical Richardson plots for the two types of coatings
are shown in Figs. 12 (cathode V) and 13 (cathode II).
The apparent work functions of 1.79 ev and 1.42 ev,
respectively, were obtained from the slopes of the
curves. Emission data over a similar temperature range
were taken for cathodes I, II, IV, and VII. The results
for these are included in Table II. Emission data on
,
2
>-u
.... ->'a
~ u -3 \
\ \
\ (Bo Sr)O
CATHODE NO.2
~ -4
z o u
u ;;:
~ a.
en
<II 9 800 700 1000 900
-t51'=.0-~~--:1::--'--+:--+':-'-~1.t5'
FIG. 11. Temperature dependence of conductivity
for cathode II «BaSr)O). cathode VI were taken over a wide range in temperature
and anode current in order to determine whether the
Richardson plot remained a straight line over this
range. Figure 14 shows this plot to be a straight line over
the)emperature range of 368 to 1017 OK and an anode
current range of more than 12 orders of magnitude. An
apparent work function of 1.37 ev was determined from
the slope of this curve.
SUMMARY AND CONC_LUSIONS
A new method of measuring the electrical conductiv
ity has been investigated and found to give the con
ductivity of (BaSr)O and BaO coatings. This method has
the disadvantages that it cannot be used at very low
temperatures owing to the limitation set by normal
current measuring- devices, and can not be used for
-9
~ -10
o
N
!:: -12
~
9 -13
• -14 e~ ·1.7geV
A = .22 AtM?/c(b,1
OK
1000 900 800 700
1.5 1.55
Fro. 12. Richardson plot for cathode V (BaO).
coatings whose conductivity to thermionic emission
ratio is greater than 2 em/volt. Distinct advantages
are (1) the conductivity of coatings which are in a
normal state for thermionic emission may be measured
without the use of probe wires or other'devices which
might impair the thermionic emission of the sample,
and (2) conductivity and thermionic measurements
may be made simultaneously on the same coaling
sample.
A comparison of the results on BaO with results
obtained here and by others on (BaSr)O coatings
indicates that the conductivity method can be used to a
better advantage on the (BaSr)O coating since the
latter has a lower conductivity to thermionic emission
ratio. It also seems likely that this method may well
find application in measuring the resistance of the inter
face layer formed on certain cathode base metals in
which case the coating resistivity may be negligible.
Some consideration has been given to the possibility
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that the cathode voltages measured in this work arose
from an interface resistivity rather than a coating
resistivity as we assumed. This does not seem likely
since (1) a pure nickel base metal was used upon which
no chemical interface compounds are known to develop,
(2) the conditions for interface formation should have
been the same in all cathodes yet several of these showed
no measurable resistivity, (3) the magnitude and temp
erature variation of the measured conductivity are
quite similar to values reported for the oxide coating
and (4) it is reported12 that the oxide coating resistivity
always exceeds that of the interface unless special base
alloys are used.
On the cathodes for which the conductivity could be
determined, the conductivity was measured as a func-
IN
'" • N
::I u .....
A-
::I c
N I-
.....
of
II> 9 -9
-10
-II , , , , (BaSr)O
CATHODE NO.2
',9\-1.42 tV
A-0.18 AMP/CM2 °K2
FIG. 13. Richardson plot for cathode II «BaSr)O).
tion of temperature and found to yield a temperature
dependence similar to that which was obtained by
Loosjes and Vink and by Young. The slope of each of
the two portions of the curve has been explained by
Loosjes and Vink as being due to two different mechan
isms in parallel. The slopes of the conductivity curve
might also be interpreted as being due to two sets of
energy levels with different thermal activation energies.
However, this model does not seem to be plausible on a
quantitative basis since the Fermi level would be
required to shift abruptly at the temperature of the
break. In every case the Richardson plot was found to
yield only one straight line over this temperature range.
In one case, a straight line was obtained over 12 orders
of magnitude in anode current.
12 C. Biguenet, Le Vide 37, 1123 (1952). 7
8
9 BaO
CATHODE NO.6
10
N II
'" .9\-1.37eV ° N 12 A • 0.122 AMP /CM2 °K2 ::I u ...
A-13 ::I c
N 14
I-... ..: 15
III 0
,.../ 16
17
18
0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9
103/T
FIG. 14. Richardson plot for cathode VI (BaO).
In conclusion, the authors wish to express their
appreciation to Professor A. S. Eisenstein who suggested
the problem and under whose guidance the investiga
tion was completed, and to Professor G. H. Vineyard for
helpful suggestions. To the U. S. Office of Naval Re
search the authors are deeply indebted for the assistance
which helped to make this work possible.
APPENDIX
In all of the tubes used in this investigation the ratio of anode
current to saturation collector current was about 800, thus it was
nece~ary to measure collector currents at least five orders of
magnitude below the anode current to obtain the retarding poten
tial curves. Owing to the limitation set by normal current meas
uring devices this factor prevented the use of this method at very
low temperatures.
In an effort to increase the range of the retarding poten tial
method two conductivity tubes were built, each incorporating a
change in anode design. In the first tube the area of the aperture~
in the anode was increased by a factor of ten while in the second
tube the anode aperture was made larger than the cathode area
and it was covered with a tantalum coarse mesh screen. Only
sample measurements were made on these tubes but the results
indicate that the new anode designs are successful in extending the
range of the retarding potential method. Plots of collector voltage
as a function of anode current were straight lines as this method
requires. In the first tube the ratio of anode current to saturation
collector current was about 80 while in the second tube this ratio
was further reduced to 1.5. Thus measurements of conductivity
in the second tube could be made using collector currents only two
or three orders of magnitude below the anode current. In future
applications of this method it would be well to incorporate one of
the above anode designs and thus increase the range and useful
ness of this technique.
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1.1722457.pdf | Principal Electron Donors in the Oxide Cathode
R. H. Plumlee
Citation: Journal of Applied Physics 27, 659 (1956); doi: 10.1063/1.1722457
View online: http://dx.doi.org/10.1063/1.1722457
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/6?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 128.189.205.30 On: Wed, 10 Dec 2014 17:27:52LETTERS TO THE EDITOR 659
required is to replace I, Iv, iv, i.' where they occur in Eqs. (1) and
(2), by the corresponding rms values and to invoke the ac ex
tension of Jeans' theorem given by Ryder.3 This completes the
proof of the theorem.
The dual of the Shannon-Hagelbarger theorem (here stated for
the first time) asserts that the conductance G(G" G2, "', Gn) of
a two-pole network N(G" G2, "', Gn) of non-negative conduct
ances GI, G2, "', Gn is a concave downward function of G" G2,
.. " Gn, i.e., for any two sets of non-negative values G" G2,
Gn and G,', G2', "', Gn', we have
G(lCGI+G,'), lCG2+G2'), "', l(Gn+G n'»
~l[G(GI' G2, "', Gn)+G(G,', G2', "', Gn')].
It may be proved by a method strictly analogous to the one given
above, the least power theorems applicable here being due essen
tially to Black and Southwell.' [In fact, having established that
the actual power dissipation is a stationary value, these authors
complete their argument by invoking an analogy with the principle
of minimum strain energy as applied to jointed structures. A
direct proof of the minimum property for the ac case (and so, by
trivial verbal changes, for the dc case also) will be found in
Ryder.3]
1 C. E. Shannon and D. W. Hagelbarger, J. App!. Phys. 27, 42 (1956).
• James Jeans, Ekctricity and Magnetism (Cambridge University Press,
London, 1927), fifth edition, p. 322.
• Frederick L. Ryder, J. Franklin Inst. 254, 47 (1952).
• A. N. Black and R. V. Southwell, Proc. Roy. Soc. (London) AIM, 447
(1938).
High Pressure Polymorphism of Iron
P. W. BRIDGMAN
Lyman Laboratory, Harvard University, Cambridge, Massachusetts
(Received April 2, 1956)
INa recent paper' entitled "Polymorphism of Iron at High
Pressures," Bancroft, Peterson, and Minshall have discussed
the propagation of shock waves in iron. It appears that the shock
pattern is more complicated than in many materials, consisting of
three discontinuous jumps in pressure. The first is a jump from a
low value to something of the order of 10000 kg/em', the second
from 10 000 to 130 000, and the third from 130 000 to a value
varying from 165 000 to 200 000 kg/cm', depending on the experi
mental conditions. The second jump, up to 130 000 is interpreted
as due to a polymorphic transition of iron at this pressure, and it
is suggested that this is most probably the transition from the
alpha (body-centered cubic) to the gamma (face-centered cubic)
modification. This is not implausible in view of the known thermo
dynamic parameters of the transition. The transition is of the
abnormal "ice" type the high-temperature gamma modification
having a smaller volume than the alpha modification, so that in
creasing pressure decreases the transition temperature. The thermo
dynamically calculated and experimentally determined values! of
dT/dp degree in giving approximately -8.5 degrees per 1000
kg/cm! increase of pressure, so that a pressure of approximately
100 000 kg/em' would be required to depress the transition from
its normal atmospheric value at 9QO°C to room temperature. The
discrepancy between 100 000 and 130 000 is not too great in view
of the magnitude of the extrapolation.
The occurrence of a transition under shock conditions would in
any event be of much interest, because it seems to be a widely held
opinion that transitions involving change of lattice type would be
unlikely to occur in times as short as a few microseconds. This
particular transition would seem especially unlikely to occur in
such a short time because even at atmospheric pressure it is not
notably rapid or sharp, there being a hystersis of 8° under the
most favorable conditions between the occurrence of the transition
on heating and cooling. It therefore seemed of interest to me to
find whether independent evidence of the transition could be found
under static conditions at room temperature. The experiment
consisted in a measurement of electrical resistance at room tem-perature to a pressure of approximately 175000 kg/cm2• The
method was the same as that used3 in measuring the resistance
of many metals to 100 000. This limit, 100 000, of my previous
measurements was not set by any absolute limitations of the
apparatus but was primarily set by considerations of economy and
prudence in order to secure a reasonable lifetime for the apparatus.
In the present measurements two freshly figured blocks of grade
999 Carboloy (the hardest grade and presumably the grade which
would support the highest pressure on the initial application) were
pushed to destruction. Pressure was increased in steps of 4500
kg/cm! to 173000 with perfect readings. On the next step, to
177 500, there was catastrophic failure, with loud noises, complete
disintegration, and flaking off of the face of one of the blocks and
short circuiting through the silver chloride transmitting medium.
The indications for a transition were completely negative.
Resistance decreased smoothly with increasing pressure, with no
discontinuity of as much as 0.001 of the total resistance.
This negative evidence is by no means decisive, since there are
known instances (the transition of bismuth at 65 000, for example)
in which a volume discontinuity occurs with no measureable
discontinuity of resistance. But at the same time I think it in
creases the presumption that the discontinuity in the shock wave
is to be explained by something else. The whole question of what
causes such discontinuities seems to be somewhat obscure. It is
apparently recognized that such a phenomenon as reaching the
plastic limit may explain the discontinuity at 10000 mentioned
above, but the precise mechanism by which reaching the plastic
flow point may induce the discontinuity seems not to have been
worked out.
Since the pressure of 173 000 is considerably higher than any
for which I have hitherto given measurements of resistance, the
following data are now given for their own interest. The material
was highly purified iron from the General Electric Company, puri
fied by five zone meltings from iron with an original analysis of
0.004% C and 0.004% O. The relative resistances at 0: 50000,
100 000, 150000, and 175000 kg/cm2 were, respectively, 1.000,
0.907, 0.864, 0.844, and 0.838. The accuracy of these figures is not
high. Measurements on another specimen of the same material in
the conventional range to 100 000 with similar apparatus gave
for the first three values: 1.000,0.906, and 0.852.
1 Bancroft, Peterson, and Minshall, J. App!. Phys. 27, 291 (1956).
2 Francis Birch, Am. J. Sci. 238, 192 (1940). 'P. W. Bridgman, Proc. Am. Acad. Arts Sci. 81, 165 (1952).
Principal Electron Donors in the Oxide Cathode
R. H. PLUMLEE
RCA Laboratories, Radio Corporation of America, Princeton, New Jersey
(Received February 6, 1956)
THE electronic chemical potential concept' serves as the basis
of a new interpretation of the chemistry of the oxide cathode
in particular and of electronically active solids in general. Any
procedure which raises the Fermi level of a material increases its
electronic chemical potential. This corresponds chemically to a
partial reduction of the material and to making it into a stronger
reducing agent.
Through this principle, several ambiguities are apparent in the
experimental evidence on which F centers have been presumed to
be formed in typical oxide cathodes from "excess barium" and
oxygen vacancies and have been postulated to constitute the
important electron donors. For instance, chemical analyses2
(which employed cathode coating reaction with H20 to produce
H2) of excess barium content in oxide cathodes are seen to consti
tute nonspecific tests for solute barium, colloidal barium, F
centers, or other electron donor species. Any donor species in the
oxide coating or in any other material having the same low work
function would have shown the same positive reaction because it
would have shown the same strong chemical reducing property.
The conventional assumption that F centers constitute the
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principal electron donor population in the oxide cathode is con
cluded, therefore, to be unnecessary.
In addition, recent research results can be interpreted as
showing that the F -center identification of these principal electron
donors is not valid. Measurements by Timmers show that barium
dissolved at a mole fraction of 10-6 (in whatever form, whether
as atoms or as ions and F centers) in BaO behaves as a nearly
ideal solute and exerts a partial pressure five or six orders of mag
nitude larger than that measured for Ba evaporating from many
typical active cathodes.' Because of this ideal solute behavior of
Ba in BaO and the fact that a donor concentration around 10-6
mole fraction is req uired6 to account for electrical properties of
oxide cathodes, it is apparent that neither excess barium nor
F centers can be present in sufficient concentration to affect
appreciably the electronic properties of typical cathodes.
With due regard to thermochemical properties requisite of the
electron donor species and to other physical properties prescribed
by the mobile donor theory6 of the oxide cathode, a new identifica
tion of the principal donor is proposed. This species is the
OH-'e group, a hydroxide ion with an extra associated electron
which preserves charge balance in the crystal. This identification
is indirectly indicated by mass spectrometric studies in this
laboratory which detected field-dependent reactions of an opera
tive oxide cathode with various residual gases including H2 and
H20 in a high vacuum system.6
The OH-· e group is viewed as but one among many ordinary
chemical species which can be formed in crystals under proper
synthesis conditions and which can participate in electronic
processes in crystals by showing the property, "variable charge."
This property is most obviously shown by transition element
cations, but may also be shown by anions in ionic compounds and
by constituents of covalent crystals. Most of the principles govern
ing the use of variable charge species have been expounded by
. Verwey 7 and colleagues as the "controlled valency" method of
synthesis of electronically active solids.
Further details of this model will be published elsewhere. 6
1 R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics
(Cambridge University Press, London, 1939), Chap. XI.
• Wooten, Moore, and Guldner, J. Appl. Phys. 26, 937 (1955).
• Cornelis Timmer, "The density of the color centers in barium oxide as
a function of the vapor pressure of barium," thesis, Cornell University,
February, 1955, to be published in J. Appl. Phys.
• Wooten, Ruehle, and Moore, J. Appl. Phys. 26, 44 (1955). ·L. S. Nergaard, RCA Rev. 13, 464 (1952).
• R. H. Plumlee, to be published in RCA Rev. 'Verwey, Haaijman. Romeijn, and van Oosterhout, Philips Research
Repts. 5, 173 (1950).
Anomalous Polarization in Undiluted
Ceramic BaTi0 3t
HOWARD L. BLOOD, SIDNEY LEVINE, AND NORMAN H. ROBERTS
Applied Physics Laboratory, University of Washington, Seat/le, Washington
(Received March 5, 1956)
IN the course of an investigation of polarization and related
electromechanical behavior of ceramic BaTiOs, we have re
corded values of apparent remanent polarization which are in
excess of published values of spontaneous polarization in single
crystals.1 These anomalous polarization levels have been
observed in undiluted BaTiOa ceramics subjected to polariza
tion fields of long duration at temperatures above and below the
Curie transition. Values of anomalous polarization as high as
150 ,.coul/em 2 have been recorded. This polarization had a time
stability comparable to that of remanent domain polarization and
was accompanied by a volume color change from tan to gray
violet which is thought to be associated with the chemical reduc
tion of the Ti+4 ion to Ti+'.2,8
The experimental procedure for determining remanent polariza
tion consisted in heating the samples above the Curie temperature,
electronically integrating the discharge arising from the thermal
decay of the polarization,' and simultaneously monitoring .the 100
~ 10 z o
~
~ t I
cl~
u>ill: C(C(
",II: ul-
~ ~-I
~~
t;
~ -10 ...
-100 250 -----
SCHEDULE (e)
FIG. 1. Qualitative thermal behavior of electromechanical response at
zero field. Positive response is identified with domain polarization in applied
field direction. After polarization reversal samples (a) and (b) exhibit the
same qualitative behavior as those of schedule (c).
electromechanical response by means of a probe, the sensing ele
ment of which was a PbZr03-PbTi0 3 transducer.
For samples exhibiting normal ferroelectric behavior, the
integrated discharge end point coincided with the disappearance of
electromechanical activity and the thermal destruction of the
ferroelectric state. Values of remanent polarization for such
samples 'were generally less than 10 ,.coul/cm2•
For samples possessing measurable anomalous polarization,
however, the thermal behavior of domain polarization was con
siderably more complex. It is convenient to distinguish three
polarization schedules: (a) samples subjected to fields of from 20
kv/cm to 30 kv/cm for several hours at room temperature,
(b) samples polarized above 120°C at 5 to 10 kv/cm for approxi
mately one hour and then cooled through the Curie transition
under field application, and (c) samples polarized above 120°C,
as in schedule (b), and then cooled through the Curie transition
with zero applied field.6
For samples subjected to schedule (a),. electromechanical
activity corresponding to the direction of the impressed field
vanished at approximately the Curie temperature. With increasing
temperature, activity corresponding to reversed domain polariza
tion appeared, reached a maximum, and then slowly decayed to
zero coincident with the complete recovery of anomalous charge.
Similar behavior was observed for samples subjected to schedule
(b). Samples subjected to schedule (c) exhibited electromechanical
activity corresponding to a polarization direction opposite to that
of the applied field; moreover, this activity was observed to in
crease with decreasing temperature. For all schedules the range of
temperatures investigated was 25°C~T~150°C. The thermal
behavior of the electromechanical response for all three schedules is
shown in Fig. 1.
Several other characteristics of anomalously polarized samples
have been observed. If samples (a) and (b) are subjected to
thermal cycling at any time subsequent to the reversal of electro
mechanical activity, reversed domain polarization is maintained
and the thermal dependence is qualitatively the same as for
samples (c). Samples polarized above 120°C according to schedules
(b) and (c) were found to exhibit no appreciable diminution of
. activity as a result of repeated thermal cycling in the range
25°C~T~150°C. This indicated a high stability of the reversed
domain polarization attained by field application at high tem
peratures, and is correlated with the observation that the major
portion of the anomalous charge is not recovered until tempera
tures exceeding that of the initial polarization have been reached.
The range' of values for reversed domain polarization and as
sociated coupling were, respectively: 0.7-1.3 ,.coul/ em', 0.065-{).12
(radial mode).
For samples (a), the dependence of electromechanical coupling
(as obtained from resonant and antiresonant frequencies) on
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1.1722557.pdf | Study of Atomic Structure of Metal Surfaces in the Field Ion Microscope
Erwin W. Müller
Citation: J. Appl. Phys. 28, 1 (1957); doi: 10.1063/1.1722557
View online: http://dx.doi.org/10.1063/1.1722557
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of
Applied Physics
Volume 28, Number 1 January, 1957
Special Issue on Electron Physics
Study of Atomic Structure of Metal Surfaces in the Field Ion Microscope*
ERWIN W. MULLER
Field Emission Laboratory, The Pennsylvania State University, University Park, Pennsylvania
(Received July 30, 1956)
Details of the image formation in the low temperature field ion microscope are discussed. The hopping
height of the rebounding gas atom, which depends on the atom's polarizability, the tip temperature, tip
radius, and field, is significant for the resolution. Photographs of tungllten and rhenium surfaces with the
atomic lattice resolved and in different states of disorder are presented. A color printing technique, which
permits finding quickly a few displaced atoms among the many thousand that are visible, is described.
JUST 20 years ago the simple device of the field
emission microscope was introduced,1 and it was
noticed early that under certain conditiolls single
atomic or molecular objects on the tip surface became
visible as blurred diffraction disks.2 Five years ago a
way was found to operate this microscope with ions3
rather than with electrons resulting in an improvement
in resolution by a factor of about 4 down to perhaps
5 A. Recently a study of the mechanism of field ion
ization4 provided the key to a further improvement.
A resolution of better than 3 A, which is necessary
to resolve the atomic lattice of the depicted surfaces,
can now be obtained by operating the field emission
microscope, Fig. 1, at a low tip temperature and with
helium ions.6-7
So far the resolution of the field ion microscope has
been studied theoretically only under the assumption of
• This research was supported by the U. S. Air Force, through
the Office of Scientific Research of the Air Research and Develop
ment Command.
1 E. W. Milller, Physik. Z. 37, 838 (1936); Z. tech. Phys. 17,
412 (1936).
2 E. W. Miiller, Z. Physik 106, 541 (1937); ibid. 108,668 (1938);
Z. Naturforsch. Sa, 475 (1950).
3 E. W. Miiller, Z. Physik 131, 136 (1951).
4 E. W. Muller, Pittsburgh Field Emission Symposium (1954);
E. W. Muller and K. Bahadur, Phys. Rev. 102,624 (1956).
• E. W. MiilIer, Z. Naturforsch. lla, 88 (1956).
6 E. W. Miiller, J. Appl. Phys. 27,474 (1956).
7 R. H. Good, Jr., and E. W. Muller, Encyclopedia of Physics
(Handbuch der Physik) 21, 174 (1956).
1 elastic reflection of the neutral atoms at the metal
surface.4-6 The more complicated conditions prevailing
at a low temperature surface are depicted in the
schematic diagram of Fig. 2. A helium atom approaches
the tip surface with a velocity added up from gas
kinetic motion Vg ... = (2kT/m)1 and attraction of the
induced dipole Vdip=F(a/m)l(k=Boltzmann constant,
T= gas temperature, m=mass of atom, F=field at
considered place, a = polarizability). Because of in
creasing field and the image force effect the probability
of autoionization increases rapidly while the atom
approaches the surface. However, below a minimum
distance of roughly z= (Vr-rp)/F the ionization
probability goes rapidly to zero because the electron
at the ground level of the approaching atom sinks
below the Fermi level (z= width of zone of forbidden
ionization, Vr=ionization potential of atom, rp=work
function of surface). In the schematic diagram the
spatial density of ionization probability is indicated in a
qualitative manner by topographic lines. For the
operating conditions with helium and a tungsten
surface z is about 5 A. The ionization probability has
high peaks above protruding atoms or lattice steps of
the surface because of the local field enhancement, and
it is this lateral probability distribution that produces
the details in the ion image on the screen. Experi
mentally one finds that optimum conditions for reso
lution are obtained only within a small range of field
Copyright © 1957 hy the American Institute of Physics
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HIGH VOLTAGE +
LIQUID
NITROGEN
POINT EMITTER
TIN OXIDE
COATI NG TO PUMP
FIG. 1. Low temperature field ion microscope. Screen
diameter is 4 inches.
strength. While the absolute value of field, about
450 Mv/cm±15%, is quite uncertain, the applied
voltage for a given tip must be adjusted within a range
of ±1% to obtain optimum sharpness. At this optimum
field only a small fraction, perhaps 10% of the impinging
atoms will be ionized at their first approach. If the
accommodation coefficient of the surface is small, as it is
for helium on clean metal surfaces at room tempera
ture, the elastically rebounding atoms will pass, on the
average, with a large lateral velocity through the
ionization zone. Ions produced there will retain a large
tangential velccity component, and the image on the
screen of one relatively sharp dot of locally increased
ionization probability, corresponding to an underlying
protrusion, will be blurred.
FIG. 2. Motion of atoms near tip surface. At room temperature
the approaching polarized atom A will be elastically reflected (B).
At low surface temperature atom C hops through ionization zone
until it is ionized at D and is accelerated as an ion E towards the
screen. By cooling the tip surface to such a temperature as to
accommodate the impinging atoms completely to tip
temperature before re-evaporation one reduces the
average lateral velocity of the atoms and hence the ions
to about (2kTtip/m)!. This allows a potential resolution
of better than 1 A at liquid hydrogen temperature.
Actually the resolution is now limited by the distance
from the surface at which the ionization occurs. Here,
cooling provides another advantage. The rebound
atoms cannot leave the tip anymore, but are rather
being pulled back to the surface by dipole attraction.
Once an atom has touched the tip surface and given
up its kinetic energy, it can make only small hops
through the ionization zone until it is ionized and takes
off to the screen.8 The average hopping height can be
calculated as follows: the force on a dipole in the
inhomogeneous field above the tip is P= -aF(dF/dr).
The inhomogeneity of the field can be expressed4 by
the semiempirical formula F(r)=7.75Vr02/3/r4/3 (ro=tip
radius, V = applied voltage). Setting the energy in
FIG. 3. Schematic diagram of first three net planes
at 011 pole of tungsten tip.
vertical direction of the re-evaporating atoms equal to
kTtip one obtains for the hopping height
3kTtipro
h
which amounts to 5.0 A for Ttip= 22°K, ro= 1000 A,
a=2X1Q-25 cm3, and F=1.5X106 esu. Thus, with
liquid hydrogen cooling of a typical tip of radius
1000 A all the ions originating from rebounding helium
atoms are produced just at the inner border of the
ionization zone, as close as possible to the surface. If
the tip is cooled to only about 60oK, which can be
easily done with solid nitrogen, the majority of the
ions originate further away from the surface, so that
the resolution of surface details is considerably reduced.
On the other hand, a temperature that is too low, as
obtainable by liquid helium cooling, results again in a
8 E. W. Muller, Report Third International Conference on
Reactivity of Solids, Madrid (1956).
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reduced resolution as has been found experimentally.6
Then the hopping atoms stay within the zone of for
bidden ionization, diffusing away towards the shank of
the tip. The image is made by the incoming ions only,
partly at a large distance from the surface. Thus the
image is faint and more blurred. Since the field cannot
be chosen freely due to the condition of the small
ionization probability for the primarily arriving atoms,
one should choose the proper temperature for each
given tip radius in order to obtain optimum hopping
height and resolution. One could say the "focusing" of
the ion microscope should be done by adjusting the
temperature. For the larger tip radii of several thousand
A that are currently used in field electron microscopy,
liquid helium cooling is certainly advantageous. How
ever, the pictures shown here were all made with tip
radii below 1000 A radius and with liquid hydrogen
cooling.
As can be seen from Fig. 2 the most prominent detail
on an atomically smooth surface will be the steps formed
by the edges of low index net planes, in the case of
tungsten particularly the closely packed 011 and 112
planes. The individual atoms along the lattice steps
will show up whenever the sites are occupied in such
a way as to make the atom slightly protrude from the
straight edge, so that the local field strength and hence
the ionization probability above it will be enhanced.
The first three 011 net planes at the pole of the tip are
shown schematically in Fig. 3, each deeper one having
a larger diameter. Atoms in a protruding position along
the steps above which the field would be locally en
hanced have been marked more heavily in the drawing.
Comparison with actual photographs shows that these
atoms become visible. The diagram shows also that the
FIG. 4. Field ion emission pattern of a tungsten tip of radius
950 A. Picture was taken at 19400 v, 0.2 microns He pressure,
with liquid hydrogen cooling of tip (21 OK). Tip had been annealed
at 20000K and exposed to 22000 v for some field evaporation.
Dark spot in center is 011 plane, the four dark areas around it are
the 112 planes. (a)
(b)
FIG. 5. (a) Tungsten tip of radius 750 A, 17500 v, 1 micron
helium pressure, tip temperature 21°K. Tip was annealed at
2500oK, 011 plane in center, 112 planes in the four corners. (b)
Same conditions as before after application of 19500 v for field
evaporation of loosely bound atoms.
atoms need not appear in a very regular arrangement
in spite of the lattice structure of the substrate.
The high field of about 450 l\:Iv/cm necessary for the
ionization of helium limits the applicability of the
helium ion microscope to the refractory metals. The
rate of field evaporation of the metal tip can be calcu
lated in good agreement with the observation9 when
work function and ionization potential of the metal are
known. The margin between image field and evaporation
field is wide for tungsten (evaporation rate about one
monolayer per second at 570 Mv/cm at 21°K), and
fairly good for rhenium too, but tantalum and molybde
num surfaces dissolve more easily so that long time
photographic exposures are just barely possible at the
optimum resolution field. Pictures like Fig. 4 of the
same clean tungsten surface10 have been repeatedly
9 E. W. Miiller, Phys. Rev. 102, 618 (1956).
10 Of the great number of slides shown at the American Physical
Society Meeting only a few can be presented here.
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(a)
(b) FIG. 6. (a) Central
part of tungsten tip of
radius 700 A, 16500 v,
after annealing at
2200oK, with incom
plete all net plane edge.
(b) Same tip after ex
posing tip to 18 000
volts for raising central
net plane edge.
taken after one hour interval with an exposure time
of half an hour, and not one out of the more than
10000 visible atoms had changed its place. The dis
solution of the surface lattice by field evaporation can
be observed visually on molybdenum and tantalum
with very good resolution. One sees one atom after
another breaking away from the edges of the lattice
steps. In the case of rhenium and particularly tungsten
the evaporation field at liquid hydrogen temperature
is so high that all helium ions approaching the tip
ionize far above the surface. One sees then the edges of
the net planes collapsing as blurred rings.
Field evaporation or field desorption is a useful
method for manipulating the surface. Figures 5(a) and
(b) show a tungsten tip after annealing at 25000K and
after the application of a desorption field. By slowly
raising the desorption field one can gradually remove
the more protruding atoms and obtain the distribution
of binding forces over the entire surface. The stress
due to the field, about 1.5X106 Ibjin.2, also removes
dislocations.u In Fig. 6(a) the horseshoe shape of the
first 011 net plane indicates a dislocation of the plane
apparently in such a way as to have both ends of the
edge sinking into the next lower plane. Application of a
high field for a few seconds makes the edge of this
first net plane pop out to form a full circle (Fig. 6(b».
Drechsler, Pankow, and Vanselow12 observed a large
11 W. T. Read, Dislocations in Crystals (New York, 1953).
12 Drechsler, Pankow, and Vanselow, Z. physik. Chem. 4, 249
(1955). number of screw dislocations on tungsten tips by
operating the field ion microscope with hydrogen ions
at room temperature as it was originally introduced by
the present author.3 Compared with the helium micro
scope the field forces are then 6 times smaller, but of
course the resolution is only 6 to 12 A. Apparently in the
present experiments the screw dislocations are removed
or remain only as 100ps13 when the high field is applied.
The edge dislocation shown in Fig. 7 seems to have a
more complicated structure.
Typical details are shown in Figs. 8(a) to 8(e),
showing a rhenium tip of about 700 to 900 A radius.
In the annealed form fig. 8(a) several of the net plane
edges around the 1010 plane are joined to make a
double height step, and on the 1011 plane one finds again
an incompletely edged net plane. When a desorption
fIeld is applied, some of the edges of double net planes
are resolved into single steps, because of increased field
evaporation due to the local field enhancement. The
application of high desorption fields, Figs. 8( c) and 8(d),
makes the entire tip surface very uniform. Wherever
there is a local protrusion the field enhancement speeds
up th~field evaporation. The original dislocation on
the 1011 plane is still present in Fig. 8(d), although the
lattice steps are not depicted clearly enough to recognize
a spiral structure for sure. Figure 8(e) waa obtained
after the tip had been exposed to normally impinging
helium ions by operating the microscope with a negative
tip at about one twelfth of the voltage to draw a field
electron current of 10-8 amp for about 10 seconds. Ions
produced in the helium gas of 1.5 microns pressure then
bombard the tip and cause some cathode sputtering.
The result of this treatment is shown in the helium
FIG. 7. Dislocation near 011 plane on tungsten tip of radius
400 A, 11300 v, 15 microns helium pressure, tip temperature 6OoK.
13 N. F. Mott, Nature 171, 234 (1953).
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(e) FIELD ION MICROSCOPY OF ATOMIC STRUCTURE 5
(d)
(el
FIG. S. (a) Rhenium tip of about 700 A rar\ius on more pro
truding areas around 1010 plane. 17200 v, 1.5 microns helium
pressure, tip temperature 21°K. Tip was annealed at 2500oK.
(b) Same tip as before, after exposure to 20 000 v for some field
evaporation. (c) Same rhenium tip at 19200 v, after field evapora
tion at 21000 v. Tip hemisphere is now uniformly curved with
radius of about sao A. A lonely atom was left close to center of
1010 plane. Along the line between 1010 and 1120 adjacent atoms
of 2.76 A distance are resolved. (d) After 10 seconds exposure to
22000 v for more field evaporation. Picture taken at 20 500 v,
radius is increased to about 900 A. (e) Same tip as in Sed) after
bombarding with helium ions.
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ion image Fig. 8(e). At the location of the 1122 plane
a big protrusion was built up, enhancing the local field
strength so much that the ions are now produced well
above the surface, thus blurring the picture. On other
areas, including the atomically smooth 1010 plane, a
great number of individual rhenium atoms have been
scattered around.
One can easily make some quantitative measure
ments with these individual atoms. By heating the tip
to higher temperatures in the absence of a field one
can study the surface migration over a perfectly smooth
net plane. By applying gradually increased fields one
can also determine the desorption field strength. Both
experiments yield information about the binding force
to the substrate. The changes between the different
pictures that are presented here are quite considerable.
Often it is more desirable to proceed in much finer steps
in order to remove just a few or even one atom at a
time. For the purpose of finding these few changes in
location among the many thousand sites of the photo
graphs a color print technique has been worked out. A
copy of the first photograph is illuminated by green
light, and a copy of the second almost identical photo
graph is illuminated by red light. By optical means
these two pictures are brought to coincidence and the
resulting picture is then photographed on color film.
All atoms in identical positions on both photographs
appear bright yellow, the ones that are only on the
first picture appear green, and the ones that are only
on the second picture, red. If, for instance, the second
picture has been obtained after exposing the tip surface
to some field evaporation, one can say that all green
atoms on the color photograph are the ones with a lower
binding energy, and all the red ones are those that have
been brought up by the desorption field. This color print technique therefore not only makes quite spec
tacular pictures, but also allows one to see at one glance
the distribution of the loosely bound atoms over the
crystal hemisphere. Unfortunately these color photo
graphs cannot be reproduced in this journal. It may be
mentioned here that this color print technique is also
very useful for transforming small changes in current
density into easily recognizable color shades in ordinary
field electron microscope patterns of adsorption layers,
and more generally it can be employed to find the
differences between any two almost identical black and
white photographs.
The further study of metal surfaces that were
exposed to ion bombardment appears to be promising.
Producing the ions by using field electron emission
from the tip itself is not very effective because most of
the impinging ions originate near the tip and have
therefore only a few hundred volts energy on the
average. But even with these slow ions a single impact
event can be seen on the surface. On one occasion an
ion with apparently a larger energy hit the central 011
plane of the tungsten tip and produced a double spiral
of 70 A diameter. By slow field desorption at 600K the
bottom of the disturbance was reached after removal
of 10 net planes. Experiments are now being prepared
to study the impact of single fast ions shot in
tangentially to the tip surface during the observation.
ACKNOWLEDGMENTS
The author wishes to express his appreciation to Mr.
Earl C. Cooper and Mr. Russell D. Young for their
assistance with the experiments and to Professor J. G.
Aston and Mr. L. F. Shultz for the supply of liquid
hydrogen and helium.
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1.1740054.pdf | The Infrared and Raman Spectra of Formaldehyded 1 Vapor
D. W. Davidson, B. P. Stoicheff, and H. J. Bernstein
Citation: The Journal of Chemical Physics 22, 289 (1954); doi: 10.1063/1.1740054
View online: http://dx.doi.org/10.1063/1.1740054
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/22/2?ver=pdfcov
Published by the AIP Publishing
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160.36.178.25 On: Wed, 24 Dec 2014 12:43:27THE JOURNAL OF CHEMICAL PHYSICS VOLUME 22, NUMBER 2 FEBRUARY, 19$4
The Infrared and Raman Spectra of Formaldehyde-dl Vapor*
D, w, DAVIDSON,t B. P. STOICHEFF,t AND H. J. BERNSTEIN
Division of Chemistry, National Research Council, Ottawa, Ontario, Canada
(Received August 17, 1953)
The infrared spectrum of HDCO vapor has been investigated in the region 2.5}J.-25}J. and all six funda
mentals have been observed. Four of the fundamentals have also been observed in the Raman spectrum of
the vapor.
The perpendicular component of an hybrid band at 3.5}J. and the pure rotational Raman spectrum were
resolved and analyzed. The rotational constant (A -B) for the symmetric top approximation was found
to be 5.47 ±0.03 cm-I• This value, combined with the rotational constants of H2CO obtained by Dieke and
Kistiakowsky, yields approximate molecular dimensions for the ground state of formaldehyde.
THE infrared spectra of the symmetrical formal
dehydes H2CO and D2CO were obtained by
NielsenI and Ebers and Nielsen.2,3 In their work, only
the rotational structure of the three perpendicular
bands was resolved. Moreover, in both molecules the
rotational analysis of these bands was complicated by
the occurrence of strong Corio lis coupling between the
two perpendicular bending vibrations and overlapping
of the third perpendicular band by parallel bands. For
the H2CO molecule, much more accurate ground-state
rotational constants are given by the high-resolution
ultraviolet results of Dieke and Kistiakowsky4 and by
the recent microwave data of Lawrance and Strand
berg.· Since spectroscopic data on H2CO can fix only
two of the three independent geometric parameters, it
is necessary to have supplementary rotational data for
an isotopic molecule.6 In this connection the most use
ful isotopic molecule to investigate under conditions of
low resolution is the monodeuterated form.
Formaldehyde-d 1 (HDCO) was recently made avail
able and a study of the vibrational spectrum and the
rotational structure of the perpendicular bands was
undertaken. It was hoped that some of the perpendicu
lar bands in HDCO would be free of such interference
effects as observed in the infrared spectra of H2CO
and D2CO, and that a reliable value of A -B (for the
symmetric top approximation) would be forthcoming.
In the present investigation, both the infrared and
Raman spectra of HDCO vapor were obtained. Al
though all of the fundamentals were observed in the
infrared region, none of the bands showing perpendicu-
1ar structure was found to be free from Coriolis or Fermi
* Presented at the Symposium on Molecular Structure held at
The Ohio State University, Columbus, Ohio, June, 1953. t National Research Laboratories Post doctoral Research
Fellow, Division of Chemistry, 1951-1953. t National Research Laboratories Post doctoral Research
Fellow, Division of Physics, 1951-1953.
I H. H. Nielsen, Phys. Rev. 46, 117 (1934).
2 E. S. Ebers and H. H. Nielsen, J. Chern. Phys. 5, 822 (1937).
3 E. S. Ebers and H. H. Nielsen, J. Chern. Phys. 6, 311 (1938).
4 G. H. Dieke and G. B. Kistiakowsky, Phys. Rev. 45, 4 (1934).
6 R. B. Lawrance and M. W. P. Strandberg, Phys. Rev. 83, 363
(1951).
6 A few microwave lines of the H,CI30 molecule have been re
ported [R. B. Lawrance, Phys. Rev. 78, 347 (1950); "Molecular
Microwave Spectra Tables" Nat!. Bur. Standards(U. S.) Circ. 518,
(1952)]. interaction; much the same difficulty as in the sym
metrical formaldehydes was encountered in evaluating
the rotational constant A-B. However, an analysis
of the perpendicular structure of the "CH stretching"
band, which is least perturbed, was found possible.
The difficulties caused by polymerization of liquid
formaldehyde have limited the earlier investigations of
the Raman effect to aqueous solutions of H2CO.7
These difficulties were avoided, as outlined below, by a
study of the gaseous state.
The vibrational Raman spectrum of HDCO vapor
was of considerable assistance in the location of band
centers, particularly for overtone and combination
bands enhanced through Fermi resonance. Also, the
pure rotational Raman spectrum afforded a more
direct evaluation of A -B than the infrared band struc
ture, since a perturbation of the vibrational ground
level is unlikely; but the accuracy was limited by the
available resolution.
EXPERIMENTAL DETAILS
The preparation of formaldehyde-d 1 has been de
scribed by Bannard, Morse, and Leitch.8 It was made
available as the para-polymer through the courtesy of
Dr. L. C. Leitch. The mass spectrometric analysis
(93 percent HDCO) was qualitatively confirmed by the
absence of spectral evidence of H2CO or D2CO.
The infrared data were obtained with a model 12C
Perkin-Elmer spectrometer. The prism used for wave
numbers above 1900 cm-I was LiF, between 1900 and
1500 cm-1 CaF2, and below 1500 cm-I NaCl. All wave
numbers were corrected to vacuum. The infrared cell
consisted of a Pyrex tube 8 em long and 4 cm in di
ameter, with KBr windows sealed to the ends with
benolite cement. The paraformaldehyde-d 1 was placed
in the absorption cell which was then evacuated for
several hours. The cell was placed in an oven of the
type described by Bernstein9 and slowly heated to
120-140°C. Enough HDCO was added to the cell to
give a vapor pressure of ca 250 mm upon complete
vaporization.
7 K. W. F. Kohlrausch and F. Koppl, Z. physik. Chern. B24, 370
(1934).
8 Bannard, Morse, and Leitch, Can. J. Chern. 31, 351 (1953).
9 H. J. Bernstein, J. Chern. Phys. 18, 897 (1950).
289
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160.36.178.25 On: Wed, 24 Dec 2014 12:43:27290 DAVIDSON, STOICHEFF, AND BERNSTEIN
DENSITY
II
FIG. 1. Band 1/1. The equivaJent slit width is indicated.
The 115, 116 band (see the following) was subsequently
re-examined under the higher resolution of a double
pass spectrometer, with no appreciable change in the
results.
In the Raman investigation, a Pyrex tube 1.2 meters
long was used, having at the window end a diaphragm
system of black glass. One gram of the paraformal
dehyde was placed in the rear end of the Raman tube.
The tube was evacuated and sealed. The temperature
of the tube was slowly raised by means of a nichrome
heating coil wound along the length of the tube. With
the tube at a temperature of 130°C, as measured by a
thermocouple, and the window kept at about 150° by
an additional heater, the temperature of the para
formaldehyde was slowly raised to 120°C by means of
a heated oil bath. Under these conditions the vapor
pressure of the monomer was about one atmosphere.
No polymerization at the walls was observed.
Formaldehyde has absorption bands in the region
X3600, and in order to prevent photodissociation by the
intense mercury lines in this region, a solution of NaN0 2
was circulated in a Pyrex jacket surrounding the tube
(see Appendix). The Raman tube was irradiated sym
metrically along 70 cm of its length by means of four
Toronto-type mercury lamps enclosed in a MgO-coated
reflector. In this way it was possible to photograph the
rotational and vibrational Raman spectra of HDCO
excited by the Hg 4358 line. A three-prism spectrograph
was used, having a camera aperture 1/5 and a disper
sion of 130 cm-1/mm at X4358. The Raman frequencies
are based on one plate exposed for 40 hours using a
5-cm-l slit. However, the appearance of the vibrational
spectrum was confirmed by a second plate photographed
with an 1/2.5 camera in 12 hours.
FIG. 2. Band 1/2. -u-
FIG. 3. The Raman spectrum of HDCO vapor.
During the course of this investigation the Raman
spectrum of H2CO vapor was also obtained in the
manner described in the foregoing and the results are
reported in the Appendix.
VIBRATIONAL SPECTRUM
HDCO belongs to the point group Cs and has five
in-plane vibrations of type ai, and one out-of-plane
vibration of the type a". All six fundamentals are infra
red and Raman active. The a' type vibrations give rise
to hybrid bands (although parallel or perpendicular
features may predominate) and the a" type vibration
gives rise to a perpendicular band. All six fundamentals
have been observed in the infrared and four have been
observed in the Raman spectrum (see Table I). The
strong bands at 2844 and 2120 cm-1 can be attributed to
the CH and CD "stretching" vibrations and have pro
nounced perpendicular festures in absorption (see
Figs. 1 and 2). These bands show up as narrow lines in
the Raman spectrum (Fig. 3). The infrared bands at
1723 and 1400 cm-1 (Fig. 4) are almost entirely parallel
in character with a P-R peak separation of ",55 cm-1
in agreement with the value calculated by the method of
Gerhard and Dennison1o for unresolved parallel bands.
They can be atrributed to the C= 0 stretching vibration
H
"'-and C bending vibration, respectively, and appear
/
D
as narrow bands in the Raman spectrum (Fig. 3). The
remaining fundamentals 116 and 116 are strongly coupled
with one another because of Coriolis interaction and
FIG. 4. Bands 1/3 and 1/4 (1/4 is inset).
10 s. L. Gerhard and D. M. Dennison, Phys. Rev. 43, 197
(1933).
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160.36.178.25 On: Wed, 24 Dec 2014 12:43:27INFRARED AND RAMAN SPECTRA OF FORMALDEHYDE-d J VAPOR 291
exhibit irregular fine structure spacing (Fig. 5). They
were not observed in the Raman spectrum. The values
finally chosen for 1'6 and 1'6 are based on Nielsen's treat
mentll of the effect of Coriolis interaction on rotational
levels, and upon sum and product rules for isotopic
homologs (see the following); they are less certain than
the other frequencies.
Because of the low symmetry of HDCO and the close
ness of summation bands to the fundamentals 1'1 and
1'2, Fermi resonance seems likely. Indeed, both 1'1 and 1'2
exhibit irregularities suggesting the presence of over
tones or combination bands whose intensity has been
enhanced possibly by Fermi resonance. The peak at
2758 cm-I in the infrared spectrum corresponds to the
weak band observed at 2760 cm-I in the Raman spec
trum and is probably 1'3+ Vs rather than 21'4, since both
1'3 and Vs are each relatively strong, whereas 1'4 is a weak
band.
The Raman band at 2133 cm-I locates the center of
the disturbance on the high-frequency side of 1'2 and is
probably 21'6. The band at 2055 cm-I in the infrared
absorption spectrum is observed as a weak broad band
DENSITY
II
II
FIG. 5. The V5, Vo band.
,....,,2045 cm-I in the Raman spectrum and is probably
21'S. It is not clear why the band at 2045 cm-I should be
the only broad band observed in the Raman spectrum.
ROTATIONAL STRUCTURE
The perpendicular rotational structure has been
analyzed in the VI band and in the 1'5, 1'6 band. The
inertial constant A -E for the ground state was also
determined from the rotational Raman effect.
The 1'1 band (Fig. 1) has a well defined perpendicular
component, superimposed on an unresolved parallel
band contour in which the P-R peak separation is
about 5S cm-I• The principal perpendicular sub-band
frequencies are given in Table II. Although Fermi reso
nance with 1'3+ 1'5 (see the foregoing) enhances the
intensity of the perpendicular sub-bands on the low
frequency side, it appears to affect appreciably the fre
quencies of only two peaks. The remaining subbands
may be fitted by the symmetric top approximation for
the perpendicular bands (~K = ± 1, ~J = 0) with the
11 H. H. Nielsen, J. Chem. Phys. 5, 818 (1939). TABLE I. Infrared and Raman bands of HDCO vapor.
Infrared Raman
Assignment Vvac (em-I) dVvac (em-I)
VI (a') 2844.1 2846.2 (s-sharp)
V3+V5(A') 2758 2761 (w-sharp)
2v,(A') 2133 (w-sharp)
v2(a') 2120.7 2120.3 (s-sharp)
2v.(A') 2055 2045 (v.w-broad)
v3(a') 1723.4 1723.2 (v.s-sharp)
v«a') 1400.0 1397.4 (m-sharp)
v5(a') 1041
v,(a") 1074
usual formula 12 to which a term in K3 has been added:
Vlsub= 1'1+ (A' -E')±2 (A' -E')K
+ [(A I -E') -(A" -E") JK2=F4DkK3.
The notation follows Herzberg,!2 with E= (B+C)/2.
Although the term in K3 was found to be of the right
order of magnitude for a purely centrifugal effect, it
must be regarded as including a number of other in
fluences. In particular, for large K values, the asym
metry of the molecule produces a frequency shift in the
same direction as centrifugal distortion.13
From the least squares treatment of the sub-band
frequencies for K?:, 3, excluding K = 8 and 10, the fore
going equation becomes
VI sub = 2849.6± 1O.922K -0.0085K2=F0.00093K3.
TABLE II. Principal peaks in the VI band.
Calculated Calculated
Observed frequency Observed frequency
frequency K (tiK ~ +1) frequency K (tiK~ -1)
(2851) 0 (2849.7) (2843) 1 (2838.6)
2859.6 1 2860.5 2828.6 2 2827.7
2871.7 2 2871.4 2816.8 3 2816.8
2882.4 3 2882.2 2805.8 4 2805.8
2893.0 4 2893.0 2795.4 5 2794.9
2903.9 5 2903.8 2783.6 6 2783.9
2915.1 6 2914.6 2772.9 7 2773.0
2924.9 7 2925.3 2758.2
2935.1 8 2935.9 8 2762.1
2945.2 9 2946.5 2750.7 9 2751.3
2956.6 10 2957.0 2743.1
(2968.3) 11 2967.4 10 2740.3
2730.0 11 2729.6
2719.7 12 2718.9
2708.5 13 2708.4
2697.0 14 2697.7
2686.7 15 2687.2
2677.4 16 2676.5
2665.4 17 2666.0
2656.3 18 2655.7
2646.0 19 2645.4
Average deviation, excluding parenthesized frequencies, ±O.48 em-I.
J2 G. Herzberg, Infrared and Raman Spectra of Polyatomic
Molecules (D. Van Nostrand Company, Inc., New York, 1945),
p.424.
13 The actual asymmetric top levels may be obtained from the
tables of King, Hainer, and Cross, J. Chem. Phys. 11, 27 (1943);
12,210 (1944). For HDCO, the asymmetry parameter 0= (B -C)I
(A -C) ""0.032 as compared with 0.019 for H2CO and 0.054
for D2CO.
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160.36.178.25 On: Wed, 24 Dec 2014 12:43:27292 DAVIDSON, STOICHEFF, AND BERNSTEIN
The frequencies calculated from this equation are
included in Table II. The band center occurs at
VI = 2844.1 cm-1 and the rotational constant A" -E"
is found to be 5.469 cm-I.
The rotational structure of the V2 band is so strongly
perturbed that it cannot be analyzed simply, although
a K assignment, as shown in Fig. 2, is possible on the
high frequency side. This band appears to have a
stronger parallel component than VI, probably because
the least inertial axis lies nearer to the CD bond by
more than 100 than it does to the CH bond. The band
center is taken to be the frequency of the strong Q
branch (~K = 0, ~J = 0) contributed by the parallel
component.
The frequencies of the principal peaks in the V5, V6
Coriolis-coupled band (Fig. 5) are listed in Table III.
To the extent that V5 is a perpendicular band, the gen
eral effect of the Coriolis interaction on the observed
spectrum is the same as for the two perpendicular bend
ing vibrations in H2CO and D2CO. In the symmetric
top approximation as before, the sub-band frequencies
in the Vs, V6 band are given by the equation of Nielsen:11
V6+V6 _
V5.68Ub=--+eA'- (A"-B")
2
±2(AII-EII)K±[ c5~V6r+(2A'~K)2r
+[(A'-E')- (AI-E")JK2.
Here, K refers to the quantum number of the excited
state and the first ± sign to ~K = ± 1 transitions. The
second ± sign distinguishes the two states which result
from the rotational coupling between the V6 and V6
vibrational modes. The coupling parameter ~ is related
to the geometry of the molecule.
This equation can be applied in the present case only
to the outer sides of the band system, where the signs
are either both positive or both negative. When the
signs differ, the sub-band spacing is Ie sst han 2 (A -B)
TABLE III. Principal peaks in the Po, P6 band.
K Calculated Calculated
(emission frequency frequency
nomen- Observed ilK = +1 Observed IlK=-l
clature)a frequency +state frequency -state
1062.2 1055.6
1073.9 1039.7
1 1082.0 1081.1 (1021.5) 1024.6
2 1093.2 1094.3 1011.3 1011.5
3 1107.6 1108.4 998.2 997.3
4 1122.6 1123.1 983.5 982.6
5 1138.4 1138.3 968.4 967.4
6 1154.0 1153.7 951.6 951.9
7 1169.4 1169.2 935.8 936.3
8 (1183.4) 1184.9 919.4 920.5
9 (902.9) 904.6
10 (887.7) 888.7
11 (872.1) 872.7
• The K refers to the excited state.
Mean deviation (excluding parenthesized frequencies) is ±O.63 em-I. and there is an overlap of the +, -and -, + combi
nations in the center region where the structure was not
resolved. At the sides of the band, however, the spacing
is greater than 2 (A -B) and sub-bands corresponding
to individual K values may be identified. At the outer
extremity, as K gets large, the spacing approaches
2[(1+~)A-EJ=16 cm-I, from which ~""0.4. To find
the "best" K assignment the combination differences
~v8ub= V8ub(+,+ )_V8ub(_,_)
were found for various choices of K, and the left-hand
side of the following equation
plotted against K2. The slope is known approximately
from rough values of A' and ~ and the numbering scheme
that agreed with this slope was chosen.
The last equation and the combination sum
V8ub(+ +)+V8ub(_,_) V6+V6 _
, =--+eA'- (A"-B")
2 2
+[(A'-E')- (AI-E")JK2 (for K?:.2)
were each subjected to a least squares treatment in K2.
This led to
V6.68Ub= 10S2.9±10.960K
± (272.2+ 26. 70K2)!-0.0032K2,
from which the calculated frequencies of Table III are
obtained. The numerical values in this equation may
be varied somewhat and still fit the observed frequen
cies almost as well. For example, the value of A II -E"
used in the equation with ~vsub is assumed to be 5.480
cm-II4 and may be varied slightly to cause a small
change in V6-V6. The above analysis yields values for
V6 and V6 of 1040.8 and 1073.8 cm-1 (but fails to dis
tinguish between them), and ~= 0.394. These quanti
ties, especially ~, are somewhat dependent upon the
value assumed for A' (6.55 cm-I, which arises from
calculation of the molecular geometry and is probably
correct within ±0.05 em-I).
The reliability of the values of V6 and V6 resulting
from this analysis is rather less than for the other
fundamentals. This is also true of the Nielsen results for
the Corio lis-coupled bands in H2CO and particularly
in D2CO, where no rotational analysis was made.
Sum and product rules for isotopic species which
make use of the known frequencies of the other two
formaldehydes may be used to distinguish between V6
and V6 in HDCO. Both the Decius-Wilsonl5 and Bern-
14 This value was a preliminary one. Use of 5.470 em-I would
have been more consistent, but would change the parameters only
slightly.
15 J. C. Decius and E. B. Wilson, Jr., J. Chern. Phys. 19, 1409
(1951).
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160.36.178.25 On: Wed, 24 Dec 2014 12:43:27I N F R ARE DAN D RAM A N S P E C T R A 0 F FOR MAL D E H Y D E -d 1 V A PO R 293
TABLE IV. Rotational Raman shifts for HDCO.
K ~v(cm-I)a t.v/4(K+I)
4 (110.0) 5.50
5 131.4 5.48
6 152.4 5.44
7 174.8 5.46
8 196.1 5.45
9 (217.4) 5.44
10 (239.2) 5.44
a The wave number shifts are the mean for Stokes and anti-Stokes lines,
except the values in brackets which are for the Stokes' lines only.
stein-PullinI6 sum rules predict a somewhat larger value
for V6 than for V6. The Teller-Redlich product rule,
applied to the H2CO frequencies assigned by Nielsen,
gives a similar result. For this reason, the assignment
V6= 1074 cm-I and v.= 1041 cm-I is more probable
than the reverseY
The observed rotational Raman spectrum consists
of a series of widely spaced lines extending to about 200
cm-I on either side of the exciting line, superimposed
on a strong continuum. The spectrum is satisfactorily
explained by the accidentally symmetric top approxi
mation with the intense lines corresponding to transi
tions t:.J = 0, t:.K = + 2. For this case, the wave
number shifts from the exciting line are given by
t:.v=4(A"-B")(K+1) where centrifugal and asym
metry effects are neglected, since the accuracy of
measurement (± 1 cm-I) does not warrant such re
finement. Seven Stokes and four anti-Stokes lines were
used to evaluate A"-B", as indicated in Table IV.
The average value is S.46±0.04 cm-I which is in good
agreement with the infrared value resulting from the
VI analysis (S.47±0.03 cm-I).
The continuum in the neighborhood of the exciting
line is probably due to unresolved Rand S branches
(t:.J=+1, +2, t:.K=O), as well as to overexposure of
the exciting line.
GEOMETRY OF THE FORMALDEHYDE MOLECULE
Dieke and Kistiakowskr (D. and K.) evaluated the
moments of inertia (of which only two are independent)
for the ground state of H2CO from an analysis of ultra
violet bands. Their results are in excellent agreement
with the values obtained from recent microwave data.'
Two independent moments of inertia, however, provide
only two of the three relations required to determine
completely the geometry of formaldehyde, that is,
rCRo, rcoD and the HCH angle (J. D. and K. originally
assumed (J to be tetrahedral which leads to the rCRo
and rcoo values given in the first line of Table_V.
The last column of the table gives the value of (A" -B")
for HDCO calculated from these dimensions. In a simi
lar manner, several molecular models can be obtained
by assuming a value for one of the dimensions and then
16 H. J. Bernstein and A. D. E. Pullin, J. Chern. Phys. 21,
2188 (1953).
17 In the symmetrical formaldehydes, however, V6>V6. calculating the other two from the D. and K. results.
For example, the value of rco= 1.213A, obtained from
electron diffractionI8 of H2CO, leads to the results
shown in the second line of Table V. Since the HCH
group in formaldehyde is rather similar to that in
ethylene or ketene, it is not unreasonable to assume
either a value of 1.071A for rCRo (line 3 of the table) or
120° for (J (line 4). (The values of rCHo and (J0 in ethylene
are 1.071A, 119°55' 19 and in ketene20 1.07SA, 122°,
respectively.)
Since the axis of least moment of inertia in HDCO
makes an angle of only about S° with the C=O axis,
the value of (A" -B") is not very sensitive to changes
in geometry. That is, an uncertainty in the value of
(A" -B") for HDCO (assuming errors in the D. and K.
results to be negligible in comparison), leads to a rather
large range of values for rCRo, rcoo, and (JO. For the value
(A"-B")=S.47±O.03 cm-I obtained in this investiga
tion the corresponding values of the geometric param
eters are given in line 5 of Table V together with their
accompanying uncertainties. Although the geometry is
not fixed within very narrow limits, the present results
suggest that the "equilibrium" value for the CH bond
distance (1.12±O.01A) given by Lawrance and Strand
berg' is somewhat high.
We wish to express our thanks to Dr. L. C. Leitch
for supplying the paraformaldehyde-d l, and Dr. G.
Herzberg and Dr. B. R. Chinmayanandam for helpful
discussions.
APPENDIX
Four vibrational bands of H2CO were found in the
Raman spectrum of the vapor. They are identified as
follows: vI(aI)=2781.6 cm-1 (sharp, strong); v2(al)
=1742.3 cm-I (sharp, weak); v3(aI)=1499.7 cm-1
(sharp, medium); v4(b1)=2866 cm-1 (broad, weak).
The frequencies are the average measurements from
two plates, and are probably accurate to ±3 cm-I.
The accuracy of VI is lower (±S em-I) since this line is
blended with the much weaker Hg 4962 line, and the
error in the measurement of the broad band V4 is about
± 10 cm-I. The Raman frequencies are in good agree-
TABLE V. Ground state parameters of formaldehyde.
Calculateda
Assumed rCHO(A) rcoO(A) 8° (A" -B")CHDO
1. 8 = 109°27' 1.15 1.185 5.60
2. rcoo = 1.213A 1.097 118°39' 5.49,
3. rcno= 1.07lA 1.225 123°26' 5.48,
4. 80 = 120° 1.090 1.217 5.49,
5. (A" _S") =5.47 1.060 1.230 125°48'
±0.03 cm-I ±0.038 :;::0.017 :;:: 7°0'
• From the assumed parameter and two of the Dieke and Kistiakowsky
moments of inertia.
18 Stevenson, duValie, and Schomaker, J. Am. Chern. Soc. 61,
2508 (1939).
19 Reference 10, p. 439.
20 H. R. Johnson and M. W. P. Strandberg, J. Chern. Phys. 20,
687 (1952).
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160.36.178.25 On: Wed, 24 Dec 2014 12:43:27294 DAVIDSON, STOICHEFF, AND BERNSTEIN
ment with the infrared values given by Ebers and
Nielsen.3 Also, the observation of only one strong
Raman line in the 3000-cm-1 region establishes the fre
quency of VI, and confirms the earlier assignment of two
strong infrared bands at 2780 and 2973 cm-l as v] and
2V3, respectively.
The Raman spectrum of H2CO (unlike that of
HDCO) was superimposed on an intense background
which precluded the observation of all but the strongest
THE JOURNAL OF CHEMICAL PHYSICS Raman lines. This background is partly due to the
fluorescence spectrum of H2CO excited by the strong
mercury lines in the region A3650. In the initial attempt
to photograph the H2CO Raman spectrum the NaN02
filter was not used, and in this way the fluorescence
spectrum was obtained superimposed on a strong con
tinuum. It was not possible to quench the fluorescence
spectrum completely even with a 1 em thickness of
saturated NaN02 filter solution.
VOLUME 22, NUMBER 2 FEBRUARY, 1954
The Infrared Spectrum and Molecular Constants of DBr*t
FRED L. KELLER AND ALVIN H. NIELSEN
Department of Physics, The University of Tennessee, Knoxville, Tennessee
(Received September 15, 1953)
High-dispersion measurements have been made for the first time on the fundamental, first overtone,
and second overtone infrared vibrational bands of DBr, Path lengths from 40 em to 17 m were used with
pressures ranging from 90 mm to 460 mm. The isotopic splitting of the rotation lines was measured and
molecular constants determined for both isotopic species, DBr79 and DBr81. The principal constants for
DBr79 are W,= 1885.33 K, x,w,=22.73 K, y,w,= -0.0106 K, B,=4.290 8 K, 0',=0.0839 K, D,=9.6,XlO- s K,
{3= -2.2XlO-6 K. The principal constants for DBr81 are W,= 1884.75 K, x,w,=22.72 K, y,w,= -0.0106 K,
B,=4.2876 K, 0',=0.083 8 K, D,=9.5,XlO-6 K, {3= -2.2XlO-6 K.
I. INTRODUCTION
THE infrared vibration-rotation bands of HBr have
been examined by a number of investigators,
notably Imes,l who observed the fundamental (v= 1-0)
band; Plyler and Barker,2 who measured the funda
mental and first overtone (v= 2-0) bands; Naude and
Verleger,3 who photographed several lines in the third
overtone (v=4-0) band; and Thompson, Williams,
18~O 1750
(bl
FIG. 1. Records of the fundamental and first overtone
bands of DBr. (a) v= 1-0, (b) V= 2 -0. ----
* This paper was presented at the North Carolina meeting
(March, 1953) of the American Physical Society; see F. L. Keller
and A. H. Nielsen, Phys. Rev. 91, 235(A) (1953). t Submitted in partial fulfillment of the requirements for the
degree of Master of Science in Physics at The University of
Tennessee.
1 E. S. Imes, Astrophys. J. 50, 251 (1919).
2 E. K. Plyler and E. F. Barker, Phys. Rev. 44, 984 (1933).
3 S. H. Naude and H. Verleger, Proc. Phys. Soc. (London) A63,
470 (1950). and Callomon,4 who very recently re-examined the
fundamental band. Only the last tw03,4 achieved resolu
tions sufficient to permit detailed analyses to be made
of the separate contributions of the isotopic molecules
HBr79 and HBr81. A search of the literature reveals no
references to infrared measurements on the isotopic
molecules DBr79 and DBr81,
The present paper concerns recent high-dispersion
measurements which have been made on the funda
mental, first overtone, and second overtone vibration
rotation bands of DBr. The isotopic separation of the
rotational lines has been measured and molecular con
stants determined for both isotopic species, DBr79 and
DBr81. These constants are shown to be in good agree
ment with the constants for HBr recently obtained by
Thompson et al.4
II. EXPERIMENTAL DETAILS
Ail measurements on the DBr bands were made with
The University of Tennessee automatically recording,
high-dispersion, prism-grating spectrometer." As the
isotopic separation of DBr79 and DBr81 was quite small
(about 0.5 K in the fundamental band), high resolving
power was of particular importance in this investiga
tion, necessitating the use of the narrowest possible
slit widths.
A 7200 lines-per-inch echelette grating was used in
conjunction with a Golay pneumatic detector for
'Thompson, Williams, and Callomon, Acta Spectrochim. 5,
313 (1952).
6 A. H. Nielsen, J. Tenn. Acad. Sci. 22, No.4, 241 (1947).
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1.1722358.pdf | Conduction Mechanism in OxideCoated Cathodes
Eugene B. Hensley
Citation: Journal of Applied Physics 27, 286 (1956); doi: 10.1063/1.1722358
View online: http://dx.doi.org/10.1063/1.1722358
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/3?ver=pdfcov
Published by the AIP Publishing
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Conduction Mechanism in Oxide-Coated Cathodes*
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their surfaces may be pressed together or separated by a small gap. Low-field conductivity measurements
show that above approximately 700oK, the conductance of the system does not depend on physical contact
between the cathode surfaces. This result supports the theory that the high-temperature conductivity is
a property of the electron gas in the cathode pores. The ratio of conductivity to thermionic emission was
measured under conditions designed to preserve the state of activation of the cathode surface. The results
agreed with the theoretically predicted ratio and demonstrate that the higher values previously reported
were caused by a lower activation on the surface than in the interior of the cathode.
1. INTRODUCTION
PRIOR to 1949 it was generally believed that the
conductivity of oxide-coated cathodes could be
adequately accounted for on the basis of the alkaline
earth oxides begin n-type semiconductors.1,2 In 1949
Loosjes and Vink.3 proposed that at high temperatures
the conductivity of these cathodes was a property of the
electron gas in the interstices of the very porous oxide
coatings. Experiments to distinguish between these two
mechanisms are difficult and while the subsequent
literature4-8 tends to support the pore-conduction
theory, quantitative difficulties still remain.
One of the criteria that the pore conduction theory
must meet concerns the ratio of electrical conductivity
to thermionic emission. It has been shown6 that this
ratio should be of the order of 3 X 10-a cmlv for ordinary
oxide cathodes. Only a few values of this ratio in which
both measurements were made on the same cathode
hav~ appeared in the literature. Hannay, McNair, and
Whlte2 reported a value of 2SXlo-a, which is about an
order of magnitude larger than is predicted. Sparks and
Philipp9 found values ranging from 66X 10-a to 2000
X 10-a, all of which are much larger than those predicted
by theory. LoosjeslO obtained values of the same order
as the theoretical predictions by a method similar to
that which will be reported in this paper. However, the
details of his experiments were never published.
While the above experimental values of the con
ductivity to thermionic emission ratio are much larger
* This work was supported in part by the Signal Corps the
Office of Scientific Research (Air Research and Develop:nent
Command), and the Office of Naval Research. t Now with the Department of Physics, University of Missouri
Columbia, Missouri. '
1 Review papers: J. P. Blewett, J. Appl. Phys. 10, 668 830
(1939); 17, 643 (1947). A. S. Eisenstein, Advances in Elect;onics
(Academic Press, Inc., New York, 1948), Vol. 1 Part 1.
2 Hannay, McNair, and White, J. Appl. Phys: 20, 670 (1949).
3 R. Loosjes and H. J. Vink, Philips Research Repts. 4 449
(1949). '
4 J. R. Young, J. Appl. Phys. 23, 1129 (1952).
5 E. B. Hensley, J. Appl. Phys. 23, 1122 (1952).
6 R. C. Hughes and P. P. Coppola, Phys. Rev. 88 364 (1952).
7 Loosjes, Vink, and Jansen, Philips Tech. Rev. 13: 337 (1952).
8 R. Forman, Phys. Rev. 96, 1479 (1954).
I I. L. Sparks and H. R. Philipp, J. Appl. Phys. 24, 453 (1953).
10 R. Loosjes, private communication. than predicted, it was felt that this might be attributa
ble to a deactivation of the cathode surface resulting in
a lower thermionic emission from the exposed surface
than from the internal surfaces of the pores. The pres
ent investigation was initially based on the supposition
that such a deactivation could result from the evapo
ration of excess barium from the exposed surface of the
cathode. Evaporation from the internal pore walls, of
course, would not result in any net loss of barium, and
consequently the internal surfaces would not become
deactivated.
As the present investigation progressed, it became
evident that the above explanation for surface deactiva
tion was incorrect. However, a surface deactivation was
observed and was identified as related to the decay in
thermionic emission studied by Sproullll and later by
NergaardP The procedure followed in the present
investigation was such as to reduce the above effects to
a minimum.
II. EXPERIMENTAL TUBE
Figure 1 shows the experimental tube. To insure
uniform temperature and constant dimensions, the
planar cathode bases, 0.2S in. in diameter, were ma
chined from solid rods of high-purity nickel,la The wall
and end thickness was O.OSO in. A O.OOS-in. disk of a
desired cathode nickeP' was spot-welded to the flat end
of this cylinder which was then remachined to restore
the accurate geometry. The assembled cathodes were
outgassed by heating in vacuum for several hours at
lOS0°C, following which they were sprayed with
Raytheon CSl-2 spray suspension, an equimolar suspen
sion of (BaSr)CO a.
T~e lower cathode was mounted directly from the
tube press. The upper cathode was mounted on a sliding
O.060-in. molybdenum rod so that its coated surface
could be pressed into contact with that of the fixed
11 R. L. Sproull, Phys. Rev. 67, 167 (1945).
12 L. S. Nergaard, RCA Rev. 13,464 (1952).
13 The "Mond" vacuum cast nickel was obtained from the
National Research Corporation.
14 All the data presented in this paper were taken using a passive
nickel. We ~re indebted to Mr. James Cardell of the Raytheon
M:mufactunpg C:0mpany .for the following spectrographic anal
YSIS of thiS ruckel: SI-O.OO9%, Fe-0.018%, Mn-zero,
Mg-0.OI0%, Cu-0.10%, Ti-zero.
286
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cathode. A cross bar welded to this sliding rod prevented
rotation. A tungsten spring pressed the cathodes to
gether with a constant force. A rod connected to the
Monel bellows could be coupled to a loop in the sliding
molybdenum rod. Three thumb screws controlled the
bellows and pulled the two cathode surfaces apart. Their
separation, controlled to an accuracy of a few microns,
was measured by a micrometer microscope. Not shown
are Mo-Ni thermocouples which measured the cathode
temperatures and a getter contained in a side arm.
Following a vigorous out-gassing schedule, the car
bonate was converted to oxide with the cathodes
slightly separated. In all subsequent activation sched
ules, the two cathodes were pressed together.
III. de MEASUREMENTS
Conduction through the pores implies that physical
contact between the two cathodes should not be neces
sary to maintain a given level of conduction. Early
measurements on these tubes were confined to testing
this prediction.
TANTALUM CROSS
PIECES SLIDING 0.060· MOLYBDENUM ROD
TUNGSTEN SPRING
0.060· TUNGSTEN
CATHODES
FIG. 1. Schematic drawing of the experimental tube. Not shown
are the cathode heater connections and the thermocouples con
ne~te4 to each of th~ cath04es, The conductance was measured by applying 0.1 v
across the tube first in one direction and then the other.
These small voltages were used to avoid (a) nonlinear
effects discussed elsewhere3-5 and (b) possible changes
in activation brought about by electrolytic effects. The
procedure followed was to measure the conductance
with the cathodes pressed together and then with the
cathodes separated about 0.1 mm over a temperature
range of 3000K to lOOOoK.
Figure 2 shows typical results for two different states
of activation. The open points represent measurements
made with the cathodes separated; the closed points
were taken with the cathodes pressed together. The
conductance is plotted to represent the conductivity
when the cathodes are pressed together. Consequently
the open points are simply the relative conductance and
do not represent conductivity. At high temperatures no
significant difference is observed between these two
measurements as long as the separation and current
density were below that necessary to produce space
charge effects. At temperatures below the break in the
conductivity curve, the open points are observed to
continue along the same straight line that would be
expected if the conductance were a property of the
thermionic emission. On the other hand, for the more
active state the temperature dependence of the con
ductivity with the cathodes together decreases sharply
below 700 oK, as is characteristic of a well-activated
oxide-coated cathode.
Further evidence of pore conduction was observed in
the behavior of the tube during these measurements. In
'::> 0
'c:
>-t-:;
§
::>
0 z
0 0 ..
10
-. 10
-6
10
-1 10
-8
10
-. 10
1.0 15 TUBE NO.6 1ST RUN 2"" fU;
CATHODES CLOSED • ..
CATHODES OPEN 0 "
0,0.30 .•
2.0 10'
T ..
..
2.5 ..
3.0
FIG. 2. Conductance of Tube No.6 for two different states of
activation. Two separate runs were made for the higher state of
activation. The sum of the areal densities for these cathodes was
30 mg/cm! of carbonate. The resulting total thickness after
conversion WaS 0.030 em.
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..
10
·7
10
·8
10 TUBE NO. 7
CATHODES CLOSED •
CATHODES OPEN 0
O-O.4e.v.
•
4000K
2.0 2.5
10'
T 30
FIG. 3. Conductance of Tube No.7 made in the same manner as
that of Fig. 2. The sum of the areal densities for these cathodes was
17 mg/ cm2 of carbonate. The resulting total thickness after
conversion was 0.029 cm.
the high-temperature region (above the break) the
measured conductance was independent of the degree
of physical contact between the cathode surfaces.
Reproducible results were obtained whether the cath
odes were separated or together and independent of the
force pushing them together. However, in the low tem
perature region the conductance was very sensitive to
the degree of contact and might vary a hundred-fold
with respect to the applied force. Consequently all
measurements below the break in conductivity were
made without disturbing the contact. These results
indicate conduction through the oxide below the break
and through the pores above the break.
The effect of space charge on this experiment is
illustrated by the data taken near the top of the upper
curve in Fig. 2. Below each of two of the closed points
there are three open points. The upper open point
represents the conductance with the cathodes separated
about 0.05 mm. The lower points represent the con
ductance with the cathodes separated 0.1 mm and 0.2
mm, respectively. Thus, at these temperatures space
charge is observed to decrease the conductance with
increasing separation. For lower temperatures and lower
states of activation, the current density is not sufficient
to produce this effect.
The data in Fig. 2 show an abnormal amount of
curvature for the high-temperature region. A possible
explanation of this could be the existence of abnormally
large pores or cracks in the coating. For a given electron
density, the tendency of pores to become blocked by
space charge increases with increasing pore size.5 Figure 3 shows similar data taken on another tube in which
this effect is absent.
The above experiment represents easily visualized
evidence of pore conduction. However, a more quanta
tive check on the theory can be obtained by actually
measuring the ratio of conductivity to thermionic
emission. An attempt was made to measure the zero
field thermionic emission from these cathodes in the
separated position by using dc Schottky plots. However,
with only a few volts applied between the cathodes, a
pronounced decay in the emission was observed. This
was recognized as the same millisecond-decay phe
nomenon studied by Sproullll and N ergaard12 and re
ferred to in the introduction. In order to avoid these
decay effects, pulse techniques were used for measuring
the thermionic emission.
IV. PULSE MEASUREMENTS
A rather simple pulsing circuit was found to give the
most satisfactory results. A fast-acting, mechanical
relay (Western Electric type 27sB) was driven by a one
shot multivibrator. This was used to connect a 400-I.d
capacitor, charged to the desired voltage, across the
experimental tube. A small resistor in series with the
tube was used in conjunction with a DuMont type
304H oscillograph to measure the resulting current.
Because of the long-persistent screen used, a single,
1-msec pulse was sufficient to measure the current and
to note the character of decay_
N
'~161
a. :.
'" z o
iii
'" ~ w
u Z
~I(}
a: w J: ... TUBE NO. II
FIG. 4. Schottky plot of the thermionic emission at l0000K made
with single msec pulses for each point. The SUbscripts 1 and 2
indicate the lower and upper cathodes, respectively. The three
states of activation are described in the text. The sum of the areal
densities for these cathodes was 18 mg/cm2 of carbonate. The
resulting total thi~n€;!ss after ,;:ol1ver!4o!l w\tl> O,OH <;m,
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It was not the purpose of this investigation to study
the millisecond-decay phenomena in detail. However,
because of its relation to the experiments at hand,
several general observations were made. First, with the
cathode surfaces separated about 0.2 mm, it was found
that the decay could be readily observed with less than
one volt applied across the entire cathode system. This
observation is important in view of frequent suggestions
made in the past that all millisecond decays are to be
associated with poisoning effects, that is, by gases
originating from anode contaminants dissociated or
released by electron bombardment. Since 1 ev is less
than the energy usually considered to be necessary for
these effects, the experiment tends to confirm the
existence of other types of decay.
In general, it was observed that the time constant for
the decay increases with decreasing temperature, as
reported by Sproull.H The decay was observed to be
most pronounced in cathodes with low activation.
Cathodes activated to a high state of thermionic emis
sion showed little tendency to decay. This last observa
tion leads one to favor the donor depletion layer
hypothesis of N ergaard!2 over the barium dipole layer
hypothesis of Sproull.H In a semiconductor with a high
density of donors the Fermi energy will be high and a
relatively small fraction of the donors will be ionized.
Consequently, the donors that drift under the influence
of an electric field will be a small percentage of the
total, and the decay effects will be correspondingly less
pronounced.
One of the most illuminating observations was made
by applying a small but constant voltage across the
cathodes and then pulsing the tube with a voltage suffi
cient to saturate the emission. It was observed that the
magnitude of the pulsed current depended markedly on
the magnitude of the steady current. A small steady
current only slightly reduced the size of the pulse, but
larger currents could reduce the pulse to a small fraction
of its original value. Even more significant, if the steady
current was in the opposite direction to the pulsed
current, the pulsed current was increased over its
original size. Thus passing current backwards through
a cathode increases the possible emission from its sur
face. This can be accounted for by donors drifting to the
cathode surface under the influence of the electric field,
thereby reducing the work-function. This phenomenon
was first observed by Becker!· in 1929, but its implica
tions have been neglected in most subsequent considera
tions of the oxide-coated cathode.
The primary purpose of the pulse measurements was
to measure the thermionic emission in a manner that
would avoid the effects of the millisecond decay. Data
taken on Tube No. 11 represent the most complete
set of data that was obtained. The zero field emission
was measured at 10000K by using Schottky plots. Each
point in Fig. 4 represents a measurement made with a
15 J. A. Becker, Phys. Rev. 34, 1335 (1929). '::0 -3 10
-. 10
u -5 _ 10
'c:;
10000K 8000K 6000K 5000K 4000K
109 L..-,,.L,--L-''=-L--.J __ -,l. __ ----L_
1.0 1.5 2.0 2.5 3.0
103
T
FIG. 5. Conductivity plots with cathodes pressed together. The
three states of activation correspond with those shown in Fig. 4.
Q is the activation energy in ev.
single 1-msec pulse. The current was pulsed in first one
direction and then in the other. These two currents
represent the emission from cathode surfaces 1 and 2,
respectively. It will be noticed that the current from
cathode 2 was always about a factor of 3 less than that
from cathode 1. This can be completely accounted for
if the work-function of cathode 2 is about 0.1 ev greater
than that of cathode 1. Also, it is possible that the
effective areas of the two cathodes differ by a small
amount.
The thermionic emission was first measured with
the cathodes in an inactive state just after the con
version process. These Schottky plots are designated
as Ai and A2 in Fig. 4. The cathodes were then pressed
together and the conductivity plot A in Fig. 5 was
obtained with dc voltages of the order of 0.1 v.
Following these measurements, the cathodes were
activated by raising their temperature to 11500K and
by drawing currents up to 400 rna, first in one direction
and then in the other over a period of 24 hours. The
activation was very unstable during the first few hours
of treatment and exhibited asymmetrical voltage
current characteristics i but after a full day of this
treatment, a stable, relatively high state of activation
was obtained. This rather severe activation procedure
was necessary because of the passive nickeJ14 used as a
cathode base metal. Following this activation, the
conductivity was measured as a function of tempera
ture. It is shown as curve B in Fig. 5.
In order to measure the emission, it was necessary to
separate the cathodes as before. However, it was found
that the severe activation treatment had resulted in the
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TABLE I. Summary of ujJ data for Tube No. 11.
Run J u/J
mhos/em amp/em2 em/volt
A 1.2 X 10-6 4.1X1o-a 2.9XlO-a
B 2.3X1o-3 9.5X1o-' 2.4X1o-a
C 1.0X1o-a 2.1XIo-' 4.8X1o-a
two cathodes being firmly sintered together. Con
siderable force was required to achieve their separation.
Only in Tube No. 11 was a completely satisfactory
separa tion achieved; in other tubes there was a tendency
for part of one of the cathodes to break away from the
base metal. With the same techniques as those used
before, single-pulse Schottky plots represented by
curves Bl and B2 in Fig. 4 were made.
Following this measurement of the emission, the
cathodes were again pressed together and the conduc
tivity curve C in Fig. 5 was obtained. The striking dif
ference between curve C and curve B is immediately
evident. The large decrease in the low-temperature
region is attributed primarily to the poor contact be
tween the cathodes after they have been broken apart.
However, it is also evident that there has been some
deactivation of the cathode surfaces, as indicated by the
increased slopes in curve C. It should be noted that the
temperature at which the break occurs in the conduc
tivity has shifted about 200°C. This is important in
view of the suggestion frequently made that the varia
tion of this temperature as a function of the cathode
density be used as a test of the pore-conduction theory.
Clearly, the shift observed here is consistent with the
pore-conduction theory, but it also points up the
difficulties that would accompany any systematic study
of it.
Following this last measurement of the conductivity,
which required an elapsed time of about 2 hours, the
thermionic emission was again measured; it is shown
as C1 and C2 in Fig. 4. The increase in slope of these
lines is significant. As plotted in Fig. 4, the slope will
depend on the variation of the electric field at the
emitting surfaces. In a porous oxide cathode, the field in
each pore is approximately the voltage drop across the
pore divided by the distance across the pore. If the
activation of the cathode is uniform, the electric field in any of the pores is approximately the total voltage drop
through the cathode divided by the thickness of the
cathode. On the other hand, if the interface between the
two cathodes becomes deactivated, these surfaces will
limit the current, and the voltage drop will be con
centrated in this region. Consequently, the electric field
at these surfaces will be much higher for a given total
voltage drop across the cathode, and the slopes of a plot
such as Fig. 4 will be increased. We thus have further
evidence that the surfaces of the two cathodes were
partially deactivated following their initial separation
after activation.
V. DISCUSSION
From the data presented in Figs. 4 and 5 we can ob
tain the ratio of the conductivity to thermionic emis
sion at 10000K and can compare this with the value
predicted by theory,6 o/J=3.5Xlo-a cm/v. For the
thermionic emission, the average of the values for
cathodes 1 and 2 were used. The results for the cathodes
in the nonactivated state, activated state, and for the
state in which the surfaces were becoming partially
deactivated are shown in Table I. For none of these
three states does the value of q/ J differ from the theo
retical value by as much as a factor of two. For the last
case, the value is beginning to increase in agreement
with the assumption that the larger values of q/ J
previously reported are attributable to deactivation of
the cathode surface.
In addition to the data presented here, q/J for three
other tubes has been measured by similar techniques.
Even though difficulty was experienced in separating
the cathodes in these tubes, q/ J never was observed
to differ. from the theoretical value by as much as a
factor of two.
Two probable reasons can be given for the fact that
previous experiments have always resulted in larger
values for (J' / J than those predicted by theory. First,
unless the thermionic emission is measured by means of
pulses with a very low duty cycle, the donors will be
electrolyzed away from the surface, leaving it in a lower
state of activation. Second, since the surface of a
cathode is more exposed to the residual gases in the tube
than are the inner surfaces of the pores, it is probable
that some deactivation will occur at the surface because
of the oxidation of same of the excess barium.
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1.1743021.pdf | Solubility of Lithium in Doped and Undoped Silicon, Evidence for Compound
Formation
H. Reiss, C. S. Fuller, and A. J. Pietruszkiewicz
Citation: The Journal of Chemical Physics 25, 650 (1956); doi: 10.1063/1.1743021
View online: http://dx.doi.org/10.1063/1.1743021
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/25/4?ver=pdfcov
Published by the AIP Publishing
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138.251.14.35 On: Mon, 22 Dec 2014 22:48:48650 H. E. BRIDGERS AND E. D. KOLB
TABLE 1. Effective distribution coefficient of boron in germanium
crystals grown at different rates.
Growth·rate Elf. distribu Hon
Crystal no. (em/sec) XlO' coefficient
60 0.38 15
67a 0.94 11
41 1.60 5.8
38 1.70 6.9
39 3.05 4.2
42 3.30 4.8
22 3.56 4.1
72 6.05 3.5
73 6.09 3.3
40 6.19 2.6
36 6.35 2.9
44 8.38 2.4
67b 8.40 2.4
58 8.63 2.4
line with slope o/D. From the intercept a value for ko
can be obtained. The experimental data are tabulated
in Table I and plotted in Figs. 2 and 3. The experi
mental uncertainty in k is within ± 15%. The two solid
points were obtained from the same crystal. The
THE JOURNAL OF CHEMICAL PHVSICS straight line in Fig. 2 was obtained by a least squares
fit of the experimental points. Its intercept gives for
the equilibrium distribution coefficient ko= 17.4, the
uncertainty of which is estimated to be ±20%. From
the slope in Fig. 2, o/D was determined to be 56 sec/em
at 60 rpm. Using this figure and following Burton et al.,a
an estimate for the diffusion coefficient of boron in
liquid germanium was determined to be ",,3X 10-4
cm2/sec. This is included only to point out that the
value so obtained is of the expected order of magnitude.
Within the limits of experimental error the effective
distribution coefficient of boron in germanium varies
with growth rate as predicted by Eq. (1). The rapid
variation observed at low growth-rates has proved8 to
be a useful property for the formation of n-p-n struc
tures in germanium by the rate-growing2 technique.
The vacuum crystal-growing machine, which made
this work possible, was designed and constructed by
. F. G. Buhrendorf. The authors are also indebted to
Miss A. D. Mills who made the resistivity measurements.
8 H. E. Bridgers and E. D. Kolb, J. Appl. Phys. 26, 1188 (1955).
VOLUME 25, NUMBER 4 OCTOBER. 1956
Solubility of Lithium in Doped and Undoped Silicon, Evidence for Compound Formation
H. REISS, C. S. FuLLER, AND A. J. PIETR.USZKIEWICZ
Bell Telephone Laboratories, Murray Hill, New Jersey
(Received December 12, 1955)
The solubility of lithium in silicon from a lithium-silicon alloy phase has been redetermined. The original
data of Fuller and Ditzenberger appear to be in error. The solubility of lithium, from the same phase, in
boron-doped silicon has also been determined. At both low and high temperatures the solubility in the
doped crystal markedly exceeds that in the undoped one. In fact, the solubility just about equals the boron
concentration in these ranges. The low temperature disparity can be explained in terms of hole-electron
equilibria while the high temperature effect is believed due to covalent bond formation between lithium and
boron. A quantitative theory is developed which predicts the experimental results.
I. INTRODUCTION
IN a recent notel Reiss and Fuller offered a theoretical
interpretation of a curve of lithium solubility vs
temperature in silicon which had been measured earlier
by Fuller and Ditzenberger.2 The explanation invoked
the concept of hole-electron equilibria influencing the
heterogeneous process3•4 by which lithium distributes
itself between an external phase and a silicon single
crystal. The agreement between theory and what was
assumed to be the proper experimental curve was
satisfactory, especially in the occurrence and location
of a solubility maximum.
As a result the authors were stimulated to redeter-
mine the solubility curve, taking care to eliminate some
1 H. Reiss and C. S. Fuller, Phys. Rev. 97, 559 (1955).
2 C. S. Fuller and J. A. Ditzenberger, Phys. Rev. 91, 193 (1953).
• H. Reiss, J. Chern. Phys. 21, 1209 (1953).
, H. Reiss and C. S. Fuller, J. Metals (to be published). of the uncertainties in the original procedure. However,
the new curve was markedly different from the original,
and inexplicable in terms of the theory which had been
advanced. Figure 2 shows the original plot as curve A,
and the new, more reliable, data, as curve B. It should
be remarked that in both instances the external phase
was prepared by alloying pure lithium to the silicon
single crystal at the temperatures of investigation.
Provoked by this disparity, the authors then em
barked upon an investigation of the solubility of
lithium in silicon, doped with boron to the level 2 X 1018
em-a, only to come upon a new unexpected behavior,
indicated by the open circles in Fig. 2. At both high and
low temperatures the solubility departs appreciably
from that characteristic of undoped silicon and, in
fact, becomes essentially constant, at about the density
of the boron atoms, although all samples were very
slightly n-type.
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138.251.14.35 On: Mon, 22 Dec 2014 22:48:48SOLUBILITY OF LITHIUM IN SILICON 651
FIG. 1. Furnace ar
rangement for saturat
ing silicon wafers with
lithium.
/
AREA COOLED
BY DRY ICE
In Sec. III theoretical explanations of the shapes of
these new curves are presented.
II. EXPERIMENTAL PROCEDURE AND RESULTS
All measurements of impurity contents were made
by determining the electrical resistivities of silicon
crystals. There were several possible sources of error in
the original work: Lack of attainment of equilibrium,
precipitation during cooling, errors in resistivity meas
urement, errors in mobilities of holes and electrons. In
the present determinations of solubility efforts were
made to reduce these errors as much as possible.
Thinner specimens were used and at least half again
the required time (calculated from the known diffusion
constants for Li in Si) was given for equilibration.
Quenching was performed by a rapid transfer of the
specimens to an outside of the furnace tube, precooled
with dry ice. The arrangement employed is shown in
Fig. 1. This was used for determinations above 500°C
where the heating times were relatively short (less than
six hours). Heating was done on a mica support in
flowing dry, helium gas. Below 500°C, the specimens
were heated in helium at about OJ-mm pressure in
sealed quartz tubes, the latter being placed in a muffle
furnace. Temperatures were controlled to ±2°C.
Quenching was done by dropping the tubes directly
into cold water.
The specimens of silicon employed were cut from
pure silicon single crystals,5 having the following re
sistivities: a lO-ohm cm n-type, a 33-169-ohm-cm p
type and a 4S-ohm-cm n-type crystal. The first two
were grown in the usual manner with rotation (100
rpm), the last was not rotated during its growth.6 No
difference in the results among the various crystals
was noticeable. The doped specimens were cut from a
silicon single crystal to which approximately lOIS boron
atoms were added per cm3• It was grown without rota
tion and had a resistivity of 0.026 ohm-cm. In the
runs at higher temperatures the specimens had the
dimensions: t in. X! in. X 0.050 in. For the lower tem-
6 The authors are indebted to C. R. Landgren for these.
6 This was done to avoid resistivity changes caused by heat
treatment. THERMOCOUPLE
I
perature runs specimens of the same size but 0.025 in.
thick were employed.
Preparation of the specimens for saturation was as
follows: The cleaned, lapped specimen is bound with 5
mil tungsten wire between two silicon side plates so as
to form a sandwich. These side plates extended about
/6 in. beyond the edges of the specimens. The Li is
a?pl~ed to the inner surfaces of the side plates, prior to
bmdmg, as a suspension of metal filings in a 5% solution
of polystyrene in toluene. The edges of the specimen
are also liberally painted with the suspension. Upon
removal of the toluene by drying at SO-100°C, the
polystyrene plastic serves to bind the Li particles to
the silicon surface. These readily alloy with the silicon
specimen upon heating the sandwich in helium during
the high temperature runs. For the lower temperature
runs the alloying is carried out at about 500°C in
helium. The time required at this temperature is only
about 1 min and so is too short to introduce errors due
to false temperature equilibrium inasmuch as (see
below) only the inner portion of the specimens is em
ployed for the resistivity measurements.
After the alloying, equilibration and quenching steps
the sandwich is placed in water until the specimen is
freed from the side plates. It is then lapped down on
silicon carbide abrasive paper using water as a lubricant
until approximately 5 mils is removed beyond the
deepest alloy regions. The edges of the specimens are
treated similarly. After plating copper electrodes on the
ends, the resistivities are determined by means of a
two-point probe potential measurement using a field
current of 1 mao Dimensions are determined to 0.0001 in.
The errors in the measurement of resistivity are less
than 5%. The degree of agreement can be seen in
Table I which is a summary of the results of the meas
urements on all the runs. As already indicated the un
reliability in the mobilities of the carriers in the doped
specimens makes the calculated values of the concen
trations (last column, Table I) uncertain to about ±15
or 20%. These mobilities have been taken from the
work of M. Prince,1 Morin and MaitaS and Debye and
7 M. Prince, Phys. Rev. 93, 1204 (1954).
• F. J. Morin and J. P. Maita, Phys. Rev. 96, 28 (1954).
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TABLE I. Summary of results.
Not Original p
Crystal Type Rotated rotated Temperature (ohm em)
Si VI 566 P V 306°e 36.1
Si VI 1126 N V 389°e 00
Si VI 566 P V 398°e 37.4
Si VI 1126 N V 485°e 00
Si VI 1082 N V 599°e 10.4
Si VI 566 P V 649°e 33.0
Si VI 1082 N V 705°e 9.92
Si VI 566 P V 755°e 32.7
Si VI 1082 N V 802°e 10.0
Si VI 1126 N V 8500e 190.0
Si VI 1082 N V 902°e 10.2
Si VI 1126 N V 954°e 61.8
Si VI 1082 N V lO00oe 10.7
Si VI 566 P V 10600e 41.2
Si VI 566 P V 985°e 59.5
Si V 1109 P V 248°e 0.0266
Si V 1109 P V 397°e 0.0260
Si V 1109 P V 498°e 0.0265
Si V 1109 P V 6QOoe 0.0274
Si V 1109 P V 7000e 0.0262
Si V 1109 P V 8100e 0.0255
Si V 1109 P V 916°e 0.0264
Si V 1109 P V lO13°e 0.0263
Si V 1109 P V 1058°e 0.0261
Si V 1109 P V 1088°e 0.0258
Si V 1109 P V 1152°e 0.0257
Si V 1109 P V 1152°e 0.0258
Si V 1109 P V 985°e 0.0249
Kohane,9 a curve adjusted to give what we consider to
be the best values of the conductivity mobilities being
employed. The calculation of the Li concentrations in
the equilibrated doped specimens was done by a method
of successive approximation which takes account of the
scattering by both the boron and lithium impurity ions.
Figure 2 shows the solubility of Li as a function of
temperature in the pure crystals (Curve B) and the
solubility in the boron-doped crystal (Curve C). For
comparison the solubility from the previously pub
lished work is given by Curve A.
III. DISCUSSION
The high sharp maximum renders the solubility
curve, Fig. 2, for undoped silicon inexplicable on the
basis of the previous theory. C. D. Thurmond* has ex
plained it as due to the occurrence of a phase transition
in the external phase. He suggests that the lithium
silicon phase diagram has the appearance of Fig. 3
where the heavy line on the right is supposed to corre
spond to curve B of Fig. 2. Above 65QoC, lithium in the
crystal is in equilibrium either with liquid (dashed
liquidus) or a solid lithium-silicon compound (dotted
vertical line). Below 65QoC it is in equilibrium with
another solid compound (dotted-dashed vertical line).
Associated with the phase transition, involved in the
passage from one to the other compound, is a change of
sign of the heat of solution of lithium in the crystal.
This produces the sharp maximum in solubility.
The remainder of this paper will be devoted to
• P. P. Debye and T. Kohane, Phys. Rev. 97, 724 (1954). * Personal communication to the authors. Boron cou- Lithium eoO-
!' centration Final p (N) Calc.!, centration (em'/v sec) (em-a) (ohm em) (em'lv sec) (em-B)
500 <101• 0.0900 412 1.69X1017
1600 <101• 0.0283 226 9.75X1017
500 <101• 0.0260 218 1.10 X lOIS
1600 <101• 0.0153 165 2.47XlO1S
1500 <101• 0.0071 110 7.97XlO1S
500 <101• 0.0056 98 1.14X1019
1265 <101• 0.0071 110 7.97XlO1S
500 <101• 0.0117 137 3.90XlO1S
1275 <101• 0.0152 166 2.47XlO1S
1600 <101• 0.0242 210 1.23 X lOIS
1500 <1010 0.0274 224 1.02 X 1018
1550 <101• 0.0422 285 5.20X1017
1525 <101• 0.0507 310 3.97X1017
500 <101• 0.0600 337 3.09X1017
500 <101• 0.0560 325 3.43X1017
130 1.84 X lOIS 4.96 138 1.85XlOIS
130 1.90X lOIS 0.165 181 1.9 XlOlS
130 1.85 X lOIS 0.0163 113 5.2 XlOlS
132 1.75XI0ls 0.0075 93 1.07 X 1019
130 1.88XlOIS 0.0072 92 1.14 X 1019
128 1.95 X lOIS 0.0701 130 2.65XlO1S
130 1.85 X lOIS 0.158 137 2.14XlO1S
130 1.87XlOIS 1.85 183 1.87XI0ls
130 1.89XlOIS 0.468 182 1.89XlOIS
129 1.92 X lOIS 105. 181 1.92 X lOIS
129 1.94X lOIS 19.5 181 1.94 X 1018
129 1.92 X lOIS 0.652 181 1.95X lOIS
117 2.14XlO1S 00 2.14XlO18
explaining the behavior of lithium in doped silicon.
Before proceeding, it is well to reemphasize the fact
that both the undoped and doped curves retain a
certain amount of inaccuracy at densities above 1018
cm-a, not only because of uncertainties in the hole
electron mobilities which were used to convert re
sistivities to densities, but also because the specimens
have become degenerate, and not all the impurities are
ionized. The existence. of degeneracy will be ignored
throughout all of our quantitative considerations. We
assume that its effect is not great enough to obscure the
main features of our treatment, an assumption, sup
ported to some extent, by the agreement between
theory and experiment.
Turning our attention to the curve C which forms the
locus of the circles of Fig. 2 (not the drawn curve which
is theoretical), it is possible to understand its disparity
with curve B, at low temperatures, on the basis of the
hole-electron equilibrium theory4 to which we have
previously alluded. At low temperatures the presence
of boron, an acceptor, simply increases the solubility
of lithium, the donor. In fact, the circumstance that
the lithium compensates the boron almost exactly was
predicted in reference 4 in connection with the dis
cussion there of the solubility in doped material when
the external phase was formed by alloying lithium to
silicon.
The compensation at high temperatures cannot be
explained on this basis because the silicon becomes in
trinsic, and the hole-electron equilibria cannot exert
any influence. The following mechanism is proposed.
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138.251.14.35 On: Mon, 22 Dec 2014 22:48:48SOLUBILITY OF LITHIUM IN SILICON 653
At low temperatures lithium ions occupy the inter
stices of the silicon lattice as in Fig. 4. In an interstitial
position the lithium ion can come close to an oppositely
charged boron ion, but the interaction will be, at the
most, coulombic so that only an ion pair will form. A
covalent bond is unable to appear because the lithium
ion cannot get into a position where it can satisfy the
tetrahedral symmetry inherent in an Sp3 hybridization.Io
Calculationsll show that at high temperatures, at the
ion densities involved, ion pairs of the sort depicted in
Fig. 4 are completely dissociated.
Suppose, however, that as the temperature is raised
vacancies dissolve in the silicon lattice, and that one
..,
~
IJ
cr:
UJ
Q.
(f)
~ SOLUBILITY OF LI IN Si VS TEMPERATURE
~
I ,\
I \\ of! \\ II \ II \ 1\
/" VI ~\o
~ j""15'" Vr \: r"
:J
~
Z 10'8 o
~ cr: ... z
UJ IJ
Z 8
7 10'
200 ~
...... \
I I '.l.
I I ""-I I \,"
/ I \"\.,.
I "\.
L ...
/
400 600 800 1000
TEMPERATURE IN DEGREES C 1200
FIG. 2. Experimental and theoretical curves. (A) Original (in
accurate) solubility curve for lithium in undoped silicon. (B) New
(accurate) curve for lithium in undoped silicon. (C) Theoretical
curve for solubility of lithium in doped silicon. Open circles repre
sent experimental points.
such vacancy occupies a position near a boron ion, as
in Fig. 5, a slight modification of Fig. 4, in which the
dots represent electrons. It is reasonable to suppose
that the lithium ion, now able to get into a tetrahedral
position, will do so and form a covalent bond as in
Fig. 5. The lithium-boron complex so formed retains a
negative charge. If the specimen were not intrinsic at
10 L. Pauling, Nature oj Chemical Bond (Cornell University
Press, Ithaca, 1942), p. 81.
11 H. Reiss, C. S. Fuller, and F. J. Morin, Bell System Tech. J.
35, 535 (1956). 100% Ll 100% Sl
FIG. 3. Speculative phase diagram (after Thurmond) for explain
ing the sharp maximum of curve B in Fig. 2.
these high temperatures, there would still appear to
be as many net acceptors as before the addition of
lithium.
If the LiB-compound has a stability several times
RT (at these temperatures RT is of the order of 2 kcal)
the bond will be strong enough to hold the lithium atom
and the solubility of lithium will be determined, prin
cipally by the density of boron atoms. At low tempera
tures, vacancies are reabsorbed and the lithium atoms
return to their interstitial positions, at a quenched-in
density corresponding to the temperature of equilibra
tion. However, the acceptors now appear to be com
pensated since interstitial lithium behaves as a donor.t
The over-all reaction may be written in the form
(3.1)
in which D represents a vacancy and e-an electron.
The reaction to form LiB-is favored at high tempera-
FIG. 4. Lithium
ion in an interstice
in silicon near a sub
stitutional boron ion.
t Notice that this assumption is necessary in order to lend any
significance to the resistivity method for determining the quenched
in density of lithium. There is fairly good evidence that vacancies
do anneal out of silicon rather quickly.
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138.251.14.35 On: Mon, 22 Dec 2014 22:48:48654 REISS, FULLER, AND PIETRUSZKIEWICZ
FIG. 5. Formation of the LiB-complex.
ture because of the increased densities of D and e-.
Another way of stating the same fact is to assert that
the heat of reaction involves that necessary to produce
a vacancy, and this is large enough to overcome the
energy regained through the combination of the four
species shown on the left of (3.1). As a result the heat
of reaction is positive and the reaction assumes an
endothermic character which favors the product at
high temperatures.
It should be noted that the heat of formation of a
vacancy can be estimated by taking the heat of sub
limation of silicon. This leads to a figure of about 85
kcal. Furthermore, the complex LiB must represent
an acceptor state in the energy band picture of silicon,
or else the stability of the charged form LiB-would be
difficult to explain, since the crystal, at high tempera
tures, is intrinsic, and the Fermi level is located near
the center of the forbidden gap.
In the next section we give a quantitative theory
based on the mechanism just proposed together with the
hole-electron mechanism validated in reference 4. This
theory is capable of generating curve C which fits the
circles in Fig. 2 over the temperature range 250° to
1200°C.
IV. QUANTITATIVE THEORY
The theory will be based on the following set of
equilibria:
Li (external)+=±Li+
+
D + +
+ e-
n=~D-
e- (4.1) + + B-
Il
LiB-e+
Il
e+e-
The ionized lithium and boron refer to species dis
solved in single crystal silicon. It is assumed that the
impurities are completely ionized, a condition known
not to be strictly true at the high impurity densities
involved in the present data. A vacancy is denoted by
D while a vacancy which has accepted an electron is
symbolized by D-. As in reference 4, e+ stands for a
hole and e+e-for a recombined hole electron pair. As a
matter of fact (4.1) can be obtained from the set of equilibria considered in reference 4 by grafting onto it
the new vertical equilibrium involving vacancies, elec
trons and LiB-together with the equilibrium producing
D-. The following notation will be used:
D+=density of Li+,
A-=density of B-,
V = density of D,
V-=density of D-,
P=density of LiB-,
n=density of e-,
p=density of e+,
N D=P+D+-total density of lithium,
NA=P+A-=total density of boron.
The concentration of vacancies may be assumed to
follow the law
V=a exp(-€v/RT), (4.2)
where a is a constant and €v is the heat of formation of a
vacancy. The following mass action expressions apply:
D+n=K, (4.3)
np=nl, (4.4)
P
D+nA-V K*=!3 exp( -€p/RT), (4.5)
Vn/V-=K**, (4.6)
where K, K*, and K** depend on temperature, !3 and
Ep are constants, independent of temperature, Ep is the
heat of formation of LiB-, and ni is the density of
intrinsic electrons. Equations (4.3) and (4.4) have been
discussed previously in reference 4. Equation (4.5) can
be modified through the substitution of (4.2). Thus we
obtain
P --=a!3 exp( -(Ev+Ep)/RT)=7r. (4.7) D+nA-
In addition to (4.3), (4.4), (4.5), and (4.6), the
following conservation-of-charge condition applies:
(4.8)
Now the system of equations, outlined in the fore
going, can be manipulated to provide an analytical
relation between N D and N A. First we rewrite Eq. (4.6)
to define
'Y= V-/n= V/K**, (4.9)
where'Y depends only on temperature in view of (4.2).
As in reference 4 the solubility in the absence of
acceptors,
(4.10)
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138.251.14.35 On: Mon, 22 Dec 2014 22:48:48SOLUBILITY OF LITHIUM IN SILICON 655
can be shown to equal
(4.11)
Equation (4.11) can be solved for K,
K (ND°)2 +{[ (NDO)2]2 n;2(NDO)2}!, (4.12)
2(1+1') 2(1+1') 1+1'
so that K is known whenever .YDo, ni, and I' are known.
Since I' is the ratio of V-to n, it is undoubtedly small
compared to unity. For this reason it will be ignored
in (4.12) so that a knowledge of K will depend essen
tially on a knowledge of ni and N DO. The quantity ni
can be had from the data of Morin and Maita,12 while
N DO can be read from curve B of Fig. 2. The final ex
pression for N D in terms of N A is
(4.13)
If I' is ignored in comparison to unity, this becomes
(NDO)WA
ND=----
2K
(4.14)
Unfortunately, the temperature dependent constant,
11", is unknown, but if we accept (4.7) it can be deter
mined from two measurements of N D at two different
temperatures. This will be accomplished in the next
section.
V. COMPUTATIONS AND DISCUSSION
In order to determine the constants a{1 and ~v+ ~p
appearing in (4.7) we restrict ourselves to a limited
region of temperature, and then use the temperature
dependence of 7r so obtained to calculate the locus of
the circles in Fig. 2 over the entire range of measure
ment through application of (4.14). In fact we consider
specifically the temperatures 916°C and 1060°C. At
these temperatures the set of data in Table II is
obtained.
The values of N DO and N A are derived from the ex
periments described in this memorandum while ni
comes from Morin and Maita,l2 Values of 11" are ob
tained through substitution of N DO, N A, and ni in
12 F. J. Morin and J. P. Maita, Phys. Rev. 96, 28 (1954). TABLE II.
916 5.8XI0 '7 1.85 X 10'8 7.8XlO '8 3.1XlO-s7
1060 3.1X1017 1.89 X 1018 2.7Xl()19 5.8 X 1O-s7
(4.14) and subsequent solving of the expression for 11".
Insertion of these two values of 11" and their correspond
ing temperatures into (4.7) yields
a{1= 1.06X 10-34 cm6 (5.1)
and
(5.2)
With 11" available at all temperatures (by use of (5.1)
and (5.2», N D was computed over the entire range of
temperature shown in Fig. 2, using (4.14). It was
assumed that N A was equal to 1.9X 1018 cm-3 in all
samples. This figure is a good average of the boron
contents of the samples used in obtaining the data
represented by the circles. Curve C is the result of this
computation. It is seen to be in satisfactory agreement
with the experimental points.
One feature of self -consistency deserves special notice.
This is the fact that according to (4.14) N D can equal
about 2N A rather than N A (as the data requires), at
low temperatures, unless 11" becomes small enough so
that 1I"KN A/(1+lrK) in (4.14) can be ignored in com
parison with the other terms. On the other hand, this
term must still be of the order of N A at 650°C so that
the experimental data can be fitted. Thus the tempera
ture dependence of K evaluated in the neighborhood of
lOOO°C must be such as to make the above terms pass
from NAto about zero between 6500 and 400°C. It
does just this.
Another matter which deserves further experiment
may be seen from the high temperature form of (4.14).
When the specimen becomes intrinsic the first two
terms on the right should approximate t·{ DO, and we have
(5.3)
i.e., N D is a linear function of N A at anyone tempera
ture (high enough of course) with intercept N DO and
slope 1I"K/(1+1I"K). Measurement of this slope should
thus provide an independent evaluation of 1I".t
ACKNOWLEDGMENT
The authors would like to express thanks to N. B.
Hannay for helpful discussions relating to this work.
t At the time of writing preliminary measurements of this kind
have been made which show (within the not very satisfactory
accuracy thus far obtained) that N D depends linearly on N A as
required by (5.3).
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1.1722456.pdf | High Pressure Polymorphism of Iron
P. W. Bridgman
Citation: Journal of Applied Physics 27, 659 (1956); doi: 10.1063/1.1722456
View online: http://dx.doi.org/10.1063/1.1722456
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/6?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 2.233.42.114 On: Tue, 20 May 2014 05:55:51LETTERS TO THE EDITOR 659
required is to replace I, Iv, iv, i.' where they occur in Eqs. (1) and
(2), by the corresponding rms values and to invoke the ac ex
tension of Jeans' theorem given by Ryder.3 This completes the
proof of the theorem.
The dual of the Shannon-Hagelbarger theorem (here stated for
the first time) asserts that the conductance G(G" G2, "', Gn) of
a two-pole network N(G" G2, "', Gn) of non-negative conduct
ances GI, G2, "', Gn is a concave downward function of G" G2,
.. " Gn, i.e., for any two sets of non-negative values G" G2,
Gn and G,', G2', "', Gn', we have
G(lCGI+G,'), lCG2+G2'), "', l(Gn+G n'»
~l[G(GI' G2, "', Gn)+G(G,', G2', "', Gn')].
It may be proved by a method strictly analogous to the one given
above, the least power theorems applicable here being due essen
tially to Black and Southwell.' [In fact, having established that
the actual power dissipation is a stationary value, these authors
complete their argument by invoking an analogy with the principle
of minimum strain energy as applied to jointed structures. A
direct proof of the minimum property for the ac case (and so, by
trivial verbal changes, for the dc case also) will be found in
Ryder.3]
1 C. E. Shannon and D. W. Hagelbarger, J. App!. Phys. 27, 42 (1956).
• James Jeans, Ekctricity and Magnetism (Cambridge University Press,
London, 1927), fifth edition, p. 322.
• Frederick L. Ryder, J. Franklin Inst. 254, 47 (1952).
• A. N. Black and R. V. Southwell, Proc. Roy. Soc. (London) AIM, 447
(1938).
High Pressure Polymorphism of Iron
P. W. BRIDGMAN
Lyman Laboratory, Harvard University, Cambridge, Massachusetts
(Received April 2, 1956)
INa recent paper' entitled "Polymorphism of Iron at High
Pressures," Bancroft, Peterson, and Minshall have discussed
the propagation of shock waves in iron. It appears that the shock
pattern is more complicated than in many materials, consisting of
three discontinuous jumps in pressure. The first is a jump from a
low value to something of the order of 10000 kg/em', the second
from 10 000 to 130 000, and the third from 130 000 to a value
varying from 165 000 to 200 000 kg/cm', depending on the experi
mental conditions. The second jump, up to 130 000 is interpreted
as due to a polymorphic transition of iron at this pressure, and it
is suggested that this is most probably the transition from the
alpha (body-centered cubic) to the gamma (face-centered cubic)
modification. This is not implausible in view of the known thermo
dynamic parameters of the transition. The transition is of the
abnormal "ice" type the high-temperature gamma modification
having a smaller volume than the alpha modification, so that in
creasing pressure decreases the transition temperature. The thermo
dynamically calculated and experimentally determined values! of
dT/dp degree in giving approximately -8.5 degrees per 1000
kg/cm! increase of pressure, so that a pressure of approximately
100 000 kg/em' would be required to depress the transition from
its normal atmospheric value at 9QO°C to room temperature. The
discrepancy between 100 000 and 130 000 is not too great in view
of the magnitude of the extrapolation.
The occurrence of a transition under shock conditions would in
any event be of much interest, because it seems to be a widely held
opinion that transitions involving change of lattice type would be
unlikely to occur in times as short as a few microseconds. This
particular transition would seem especially unlikely to occur in
such a short time because even at atmospheric pressure it is not
notably rapid or sharp, there being a hystersis of 8° under the
most favorable conditions between the occurrence of the transition
on heating and cooling. It therefore seemed of interest to me to
find whether independent evidence of the transition could be found
under static conditions at room temperature. The experiment
consisted in a measurement of electrical resistance at room tem-perature to a pressure of approximately 175000 kg/cm2• The
method was the same as that used3 in measuring the resistance
of many metals to 100 000. This limit, 100 000, of my previous
measurements was not set by any absolute limitations of the
apparatus but was primarily set by considerations of economy and
prudence in order to secure a reasonable lifetime for the apparatus.
In the present measurements two freshly figured blocks of grade
999 Carboloy (the hardest grade and presumably the grade which
would support the highest pressure on the initial application) were
pushed to destruction. Pressure was increased in steps of 4500
kg/cm! to 173000 with perfect readings. On the next step, to
177 500, there was catastrophic failure, with loud noises, complete
disintegration, and flaking off of the face of one of the blocks and
short circuiting through the silver chloride transmitting medium.
The indications for a transition were completely negative.
Resistance decreased smoothly with increasing pressure, with no
discontinuity of as much as 0.001 of the total resistance.
This negative evidence is by no means decisive, since there are
known instances (the transition of bismuth at 65 000, for example)
in which a volume discontinuity occurs with no measureable
discontinuity of resistance. But at the same time I think it in
creases the presumption that the discontinuity in the shock wave
is to be explained by something else. The whole question of what
causes such discontinuities seems to be somewhat obscure. It is
apparently recognized that such a phenomenon as reaching the
plastic limit may explain the discontinuity at 10000 mentioned
above, but the precise mechanism by which reaching the plastic
flow point may induce the discontinuity seems not to have been
worked out.
Since the pressure of 173 000 is considerably higher than any
for which I have hitherto given measurements of resistance, the
following data are now given for their own interest. The material
was highly purified iron from the General Electric Company, puri
fied by five zone meltings from iron with an original analysis of
0.004% C and 0.004% O. The relative resistances at 0: 50000,
100 000, 150000, and 175000 kg/cm2 were, respectively, 1.000,
0.907, 0.864, 0.844, and 0.838. The accuracy of these figures is not
high. Measurements on another specimen of the same material in
the conventional range to 100 000 with similar apparatus gave
for the first three values: 1.000,0.906, and 0.852.
1 Bancroft, Peterson, and Minshall, J. App!. Phys. 27, 291 (1956).
2 Francis Birch, Am. J. Sci. 238, 192 (1940). 'P. W. Bridgman, Proc. Am. Acad. Arts Sci. 81, 165 (1952).
Principal Electron Donors in the Oxide Cathode
R. H. PLUMLEE
RCA Laboratories, Radio Corporation of America, Princeton, New Jersey
(Received February 6, 1956)
THE electronic chemical potential concept' serves as the basis
of a new interpretation of the chemistry of the oxide cathode
in particular and of electronically active solids in general. Any
procedure which raises the Fermi level of a material increases its
electronic chemical potential. This corresponds chemically to a
partial reduction of the material and to making it into a stronger
reducing agent.
Through this principle, several ambiguities are apparent in the
experimental evidence on which F centers have been presumed to
be formed in typical oxide cathodes from "excess barium" and
oxygen vacancies and have been postulated to constitute the
important electron donors. For instance, chemical analyses2
(which employed cathode coating reaction with H20 to produce
H2) of excess barium content in oxide cathodes are seen to consti
tute nonspecific tests for solute barium, colloidal barium, F
centers, or other electron donor species. Any donor species in the
oxide coating or in any other material having the same low work
function would have shown the same positive reaction because it
would have shown the same strong chemical reducing property.
The conventional assumption that F centers constitute the
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principal electron donor population in the oxide cathode is con
cluded, therefore, to be unnecessary.
In addition, recent research results can be interpreted as
showing that the F -center identification of these principal electron
donors is not valid. Measurements by Timmers show that barium
dissolved at a mole fraction of 10-6 (in whatever form, whether
as atoms or as ions and F centers) in BaO behaves as a nearly
ideal solute and exerts a partial pressure five or six orders of mag
nitude larger than that measured for Ba evaporating from many
typical active cathodes.' Because of this ideal solute behavior of
Ba in BaO and the fact that a donor concentration around 10-6
mole fraction is req uired6 to account for electrical properties of
oxide cathodes, it is apparent that neither excess barium nor
F centers can be present in sufficient concentration to affect
appreciably the electronic properties of typical cathodes.
With due regard to thermochemical properties requisite of the
electron donor species and to other physical properties prescribed
by the mobile donor theory6 of the oxide cathode, a new identifica
tion of the principal donor is proposed. This species is the
OH-'e group, a hydroxide ion with an extra associated electron
which preserves charge balance in the crystal. This identification
is indirectly indicated by mass spectrometric studies in this
laboratory which detected field-dependent reactions of an opera
tive oxide cathode with various residual gases including H2 and
H20 in a high vacuum system.6
The OH-· e group is viewed as but one among many ordinary
chemical species which can be formed in crystals under proper
synthesis conditions and which can participate in electronic
processes in crystals by showing the property, "variable charge."
This property is most obviously shown by transition element
cations, but may also be shown by anions in ionic compounds and
by constituents of covalent crystals. Most of the principles govern
ing the use of variable charge species have been expounded by
. Verwey 7 and colleagues as the "controlled valency" method of
synthesis of electronically active solids.
Further details of this model will be published elsewhere. 6
1 R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics
(Cambridge University Press, London, 1939), Chap. XI.
• Wooten, Moore, and Guldner, J. Appl. Phys. 26, 937 (1955).
• Cornelis Timmer, "The density of the color centers in barium oxide as
a function of the vapor pressure of barium," thesis, Cornell University,
February, 1955, to be published in J. Appl. Phys.
• Wooten, Ruehle, and Moore, J. Appl. Phys. 26, 44 (1955). ·L. S. Nergaard, RCA Rev. 13, 464 (1952).
• R. H. Plumlee, to be published in RCA Rev. 'Verwey, Haaijman. Romeijn, and van Oosterhout, Philips Research
Repts. 5, 173 (1950).
Anomalous Polarization in Undiluted
Ceramic BaTi0 3t
HOWARD L. BLOOD, SIDNEY LEVINE, AND NORMAN H. ROBERTS
Applied Physics Laboratory, University of Washington, Seat/le, Washington
(Received March 5, 1956)
IN the course of an investigation of polarization and related
electromechanical behavior of ceramic BaTiOs, we have re
corded values of apparent remanent polarization which are in
excess of published values of spontaneous polarization in single
crystals.1 These anomalous polarization levels have been
observed in undiluted BaTiOa ceramics subjected to polariza
tion fields of long duration at temperatures above and below the
Curie transition. Values of anomalous polarization as high as
150 ,.coul/em 2 have been recorded. This polarization had a time
stability comparable to that of remanent domain polarization and
was accompanied by a volume color change from tan to gray
violet which is thought to be associated with the chemical reduc
tion of the Ti+4 ion to Ti+'.2,8
The experimental procedure for determining remanent polariza
tion consisted in heating the samples above the Curie temperature,
electronically integrating the discharge arising from the thermal
decay of the polarization,' and simultaneously monitoring .the 100
~ 10 z o
~
~ t I
cl~
u>ill: C(C(
",II: ul-
~ ~-I
~~
t;
~ -10 ...
-100 250 -----
SCHEDULE (e)
FIG. 1. Qualitative thermal behavior of electromechanical response at
zero field. Positive response is identified with domain polarization in applied
field direction. After polarization reversal samples (a) and (b) exhibit the
same qualitative behavior as those of schedule (c).
electromechanical response by means of a probe, the sensing ele
ment of which was a PbZr03-PbTi0 3 transducer.
For samples exhibiting normal ferroelectric behavior, the
integrated discharge end point coincided with the disappearance of
electromechanical activity and the thermal destruction of the
ferroelectric state. Values of remanent polarization for such
samples 'were generally less than 10 ,.coul/cm2•
For samples possessing measurable anomalous polarization,
however, the thermal behavior of domain polarization was con
siderably more complex. It is convenient to distinguish three
polarization schedules: (a) samples subjected to fields of from 20
kv/cm to 30 kv/cm for several hours at room temperature,
(b) samples polarized above 120°C at 5 to 10 kv/cm for approxi
mately one hour and then cooled through the Curie transition
under field application, and (c) samples polarized above 120°C,
as in schedule (b), and then cooled through the Curie transition
with zero applied field.6
For samples subjected to schedule (a),. electromechanical
activity corresponding to the direction of the impressed field
vanished at approximately the Curie temperature. With increasing
temperature, activity corresponding to reversed domain polariza
tion appeared, reached a maximum, and then slowly decayed to
zero coincident with the complete recovery of anomalous charge.
Similar behavior was observed for samples subjected to schedule
(b). Samples subjected to schedule (c) exhibited electromechanical
activity corresponding to a polarization direction opposite to that
of the applied field; moreover, this activity was observed to in
crease with decreasing temperature. For all schedules the range of
temperatures investigated was 25°C~T~150°C. The thermal
behavior of the electromechanical response for all three schedules is
shown in Fig. 1.
Several other characteristics of anomalously polarized samples
have been observed. If samples (a) and (b) are subjected to
thermal cycling at any time subsequent to the reversal of electro
mechanical activity, reversed domain polarization is maintained
and the thermal dependence is qualitatively the same as for
samples (c). Samples polarized above 120°C according to schedules
(b) and (c) were found to exhibit no appreciable diminution of
. activity as a result of repeated thermal cycling in the range
25°C~T~150°C. This indicated a high stability of the reversed
domain polarization attained by field application at high tem
peratures, and is correlated with the observation that the major
portion of the anomalous charge is not recovered until tempera
tures exceeding that of the initial polarization have been reached.
The range' of values for reversed domain polarization and as
sociated coupling were, respectively: 0.7-1.3 ,.coul/ em', 0.065-{).12
(radial mode).
For samples (a), the dependence of electromechanical coupling
(as obtained from resonant and antiresonant frequencies) on
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1.1698681.pdf | Work Function in Field Emission. Chemisorption
Robert Gomer
Citation: The Journal of Chemical Physics 21, 1869 (1953); doi: 10.1063/1.1698681
View online: http://dx.doi.org/10.1063/1.1698681
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205.208.120.57 On: Mon, 08 Dec 2014 16:31:03THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 10 OCTOBER, 1953
Work Function in Field Emission. Chemisorption*
ROBERT GOMER
Institute for the Study of Metals, The University of Chicago, Chicago, Illinois
(Received April 27, 1953)
Analogs to the contact potential are calculated for the work function increase in field emission caused
by polar chemisorbates. It is found that the discreteness of the dipoles constituting the layer leads to a
work function increment smaller than the corresponding contact potential. The discrepancy becomes
more marked at low coverages and high fields. A simple Fermi-Thomas calculation for estimating the
depolarization of the electron cloud at a metal surface is given. It is probable that observed depolarizations
of oxygen and nitrogen on tungsten can be explained in terms of this factor. The effect of the high fields
used in cold emission on the work function is estimated and found to be of the order of 0.1 volt. Arguments
are presented to show that a recent explanation of the fall in heats of chemisorption with coverage may
need revision.
IT has been known for many years that adsorbates
generally cause a change in the work function of
the substrate. The work of de Boer,! Rideal,2 Bosworth,a
Mignolet,4 and others has utilized this phenomenon to
obtain information on the absorption of various sub
stances on metal substrates. The chemisorption of
oxygen, nitrogen, and hydrogen seems to increase the
work function of metals, so that a negative contact
potential results between a clean surface and one
contaminated with one of these gases. It further
appears that the contact potential varies almost
linearly with the degree of coverage of the surface.l,a
These facts are most simply and reasonably explained
by assuming that the individual adsorption complexes
(ad-atom-substrate or ad-molecule-substrate) have di
pole moments and are but weakly polarizable. The con
tact potential is then given by Voo=271"M, where M is
the dipole moment per unit area, so that V 00 is a linear
function of the coverage 8, if the dipole moment per
ada tom is constant.
The origin of the dipole moments of individual
adatom complexes is not well understood. It is probably
more or less correct that adsorbate atoms can be
considered to have a slight excess charge, which, with
its image in the metal, gives rise to a dipole of moment
M = 2dq, d being the distance of the center of the adatom
to the surface and q its effective charge. For simplicity,
the latter is considered spherically symmetric, so that
higher moments are ignored. It must be pointed out
that this model is somewhat idealized, since it is
doubtful whether surfaces can be considered smooth on
the atomic scale, or rather whether the image plane can
thus be described. Boudart5 has pointed out that
ad-atoms of slight positive charge can give rise to
* This work was supported in part by Contract AF 33(038)-6534
with the United States Air Force.
1 J. H. De Boer, Electron Emission and Adsorption Phenomena,
(Cambridge University Press, Cambridge, 1935).
2 R. C. L. Bosworth and E. K. Rideal, Physica 4, 925 (1937).
3 (a) Reference 2, this paper; (b) R. C. L. Bosworth, Proc.
Cambridge Phil. Soc. 33, 394 (1937); (c) R. C. L. Bosworth,
Trans. Roy. Soc. N.S.W. 79, 53 and 166 (1946).
4 J. C. P. Mignolet, Discussions on Heterogeneous Catalysis
(Faraday Society, 1950), p. 105.
6 M. Boudart, J. Am. Chern. Soc. 74, 3556 (1952). negative contact potentials if the adatom fits into holes
actually below the surface, so that the positive end of
the dipole points inward. It is doubtful whether this
situation exists with adsorbates other than hydrogen .
. Even in the latter instance it may be that the effect
exists only on certain loosely packed planes. The device
described below will be able to answer this question.
The method of following chemisorption by contact
potential measurements is relatively simple and fairly
clearcut. It suffers from the fundamental disadvantage
that the adsorption area must be macroscopic. This
means in practice that wires or evaporated films are
used, so that polycrystalline surfaces of unknown
structure are involved.
The advent of MUller's field emission microscope6 has
supplied a tool singularly suited for the study of individ
ual crystal surfaces under absolutely determinable
conditions. It seems close at hand to apply this device
to a study of chemisorption. Very interesting patterns
have been noted by MUller,6 Becker,7 the author, Sa and
others. If a means can be found of determining the
contact potentials of individual crystal faces of the
single crystal emitter under various conditions of
chemisorption, much valuable information can be
obtained. Drechsler and MUller have already deter
mined the work function of two crystallographic
directions in clean tungsten by field emission.9 Their
method consisted of cutting a small hole in a metal
plate, coated with fluorescent material, on which the
field emission pattern was allowed to impinge. The
portion of the beam penetrating through the hole was
then measured separately and represented the current
from a small region of the emitting crystal. By opening
the tube and rotating the tip, emission from various
directions could thus be measured and compared with
the total emission. A very similar device has recently
been built in this laboratory, consisting of a field
6 E. W. Miiller, Z. Physik 131, 136 (and previous papers).
7 J. A. Becker, Bell System Tech. J. 30, 907 (1951) and un
published work.
S (a) R. Gomer, J. Chern. Phys. 21, 293 (1953); (b) J. Chern.
Phys. 20, 1772.
9 M. Drechsler and E. W. Miiller, Z. Physik 134, 208 (1953).
1869
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205.208.120.57 On: Mon, 08 Dec 2014 16:31:031870 ROBERT GOMER
"l: ;--', ,'---",
I ,
I , f---..-->,
I ' I \
I ' I..-P:, /\ '~, I ,
I \
I \ , \
I ,
I \ P2
\
\
\
\ ,
FIG. 1. Unit cell of triangular dipole array. Dipole-dipole
distance is a [dotted line connecting dipole sites (Po)]. Equipoten
tials represented by solid lines. Points PI, P2, Pa represent points.
for which potentials have been calculated. .
emission tube with a hole in the spherical envelope
leading to a suitably constructed Faraday cage. In our
arrangement the emitter is mounted in a way permitting
its rotation (by means of magnetically operated levers
and bearings) about 2 azimuths, so that any part of
the field emission pattern may be brought to bear on
the analyzer cage.
It is apparent that this device enables one to measure
the effective work function of individual crystallo
graphic directions by use of the Fowler-Nordheim
equation/a which relates field current to applied field in
terms of a work function. Unfortunately, the evaluation
of the results is not as direct as in the case of contact
potential measurements. In the latter case one measures
the maximum potential barrier that electrons must go
over, whether this maximum occurs at the site of the
double layer (which would be the case for a completely
smeared-out dipole sheet, i.e., a condenser) or whether
it is found at a much larger distance from the emitting
surface. In our case the details of the structure of the
potential in the vicinity of the surface are important,
since field emission depends on a tunnel phenomenon
taking place within 1O-15A from the surface. Thus one
must consider that the potential will vary with position
between individual dipoles, and that it will only build
up to the final value of the contact potential in a
distance much larger than the dipole half-length d.
The chief object of this paper is, therefore, the
determination of the potential curves for the emission
of electrons from surfaces containing layers of dipoles.
The determination of these curves is carried out by
summation of individual dipole potentials. The calcula
tions are thus analogous to those of patch theory in
thermionic emission.ll From these potential curves a
10 R. H. Fowler and L. W. Nordheim, Proc. Roy. Soc. (London)
AUI, 173 (1928).
11 J. A. Becker, Revs. Modern Phys. 7, 95 (1935). graphical determination of the effective potential
barriers in field emission will then be carried out for
various fields, so that the effective work function can
be found. Similar graphical determinations have been
used by Drechsler and Mtiller9 to obtain effective work
functions for clean tungsten, taking into account
surface roughness and local field enhancement. It will
be obvious that the effective increment in work function
in field emission will always be less than the correspond
ing contact potential, since the latter builds up to its
full value at distances which increase with decreasing
coverage. The effect of the high fields used in cold
emission on the work function of the clean and gas
covered surface will be estimated in connection with
polarization effects. Finally, some interesting conclu
sions regarding chemisorption can be drawn from
the results on the structure of the electric potential
within the ad-layer.
ELECTRIC POTENTIALS DUE TO DISCRETE
DIPOLE LAYERS
For the purposes of this paper, individual dipoles will
be considered as already described. The potential P
.020,-------------------,
(I; 30 ~
15 20 25
X IN ANGSTROMS
FIG. 2. Representative potential curves for nitrogen on tungsten,
as functions of the distance x from the surface.
due to a single dipole of this kind is then
P= 14.4a(1/[y2+ (x-d)2JI-1/[y2+ (x+d)2Jl) ev, (1)
where a is the charge on the ada tom in electron charges
d the dipole half-length, x the Cartesian coordinate
parallel to the dipole axis, and y the coordinate perpen
dicular to it. The origin is taken at the center of the
dipole. P is given in electron volts if all distances are
expressed in angstroms. P given by Eq. (1) approaches
the value for the potential due to a point dipole at large
x and y.
Our task is now the summation of the contributions of
individual dipoles to the total potential at a given point.
We shall consider only regular arrays of dipoles. This
assumption is probably a good one for mobile chemi
sorbed layers, where dipole repulsion will tend to
maintain the layer with maximum dipole-dipole
distances. It may be possible to extend the present
calculations to layers of other types j however, the
present paper shows fundamental properties quite
clearly and is an obvious starting point. Calculations
were carried out for regular square and triangular
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205.208.120.57 On: Mon, 08 Dec 2014 16:31:03W 0 R K FUN C T ION IN FIE L D EM ISS ION. C HEM ISO R P T ION 1871
(hexagonal) arrays. The results proved to be almost
identical when translated into terms of coverage O.
Hence only triangular arrays will be discussed. Three
points within the triangular "unit cell" containing half
a dipole were considered, as indicated in Fig. 1. Summa
tions, using the potential of Eq. (1) were then carried
out for several values of the closest dipole-dipole
distance a, extending to distances of 4a. At distances
greater than 4a, direct summation was replaced by
integration, using the potential due to a point dipole,
41rda.X.14.4f'" ydy
P4a-oo .865a2 4a (x2+y2)!
= 209dax/ a2(x2+ 16a2) 1 ev. (2)
The total potential at a given point and a given distance
from the surface x is then the sum of the corresponding
summation and integration~ Typical curves for PI and
Pa with a= 30A are shown in Fig. 2. a is of the order of
----------------------- ~
~--x'.~ I
I
i I
2 3 4 5 6
a'
~/v.. AS A FUNCTION OF "(-aId) FOR VARIOUS X'{-X/d)
FIG. 3. Nondimensional plot of PI in terms of contact potential
V ... , as a function of distance from the surface, x, and dipole-dipole
distance a. x and a are expressed in terms of the dipole half-length
d. A curve for Po at x'=O.S is shown for comparison.
1/30 electron charges.a It is seen that the potential
does not rise as steeply as it would in the case of a
uniform dipole sheet, but builds up gradually, reaching
the value of the contact potential 211' M at large distances.
It is interesting to note that
lim P4a-<tJ= 211'M. (3)
x->'"
Perhaps it should be stated explicitly that these results
are strictly true only for infinite plane surfaces, cor
responding to an upper limit of infinity in the integral
of Eq. (2) or to a solid angle 211' in the formula 211'M.
For finite surfaces the upper limit of integration (or the
corresponding solid angle at large x) would be less,
bringing the potential back to zero at x= 00. This is
why finite crystals may have different work functions
on various faces (caused by effective dipole layers),
but the same inner chemical potential; calculations for
infinite crystals lead to different inner potentials, FIG. 4. Nondimensional plot of P3 in terms of contact potential
and dipole-dipole spacmg as function of x. Symbols as in Fig. 3.
A curve for Po at x'=2.5 is included for comparison. Note that
the crossing over of curves implies that Pa may have slight
maxima before leveling off to the value of V .. at x= <Xl.
depending on the face by which the crystal is entered.
The reason for this is obvious from the above. While it
is true that the contact potential for finite crystals will
fall off eventually, the region of interest in field emission
is so close to the surface that this effect can be ignored.
The curves shown in Fig. 2 are based on a dipole
half-length d= 2A. All subsequent calculations will
employ this value as most reasonable for actual cases.
It is possible, however, to express the potentials
nondimensionally as fractions of the contact potential
V .. and to express a and x in terms of d. The results for
PI/V .. are summarized in this way in Fig. 3.
Calculations for P2 and Pa were not extended as
far as those for Pl. Since the values of P2 lie between
the corresponding ones for PI and Pa, they are omitted.
In general the values for Pa are fairly close to those for
Pl. Thus the region enclosed by equipotentials up to
and including the points Pa can be approximated as
having a potential somewhere between PI and P3;
roughly 80 percent of the surface is thus accounted for.
For Xl = 0.5 and 2.5 calculations were also made for Po,
representing the potential at an unfilled lattice site.
Data for Po and Pa are shown in Fig. 4. It is seen that
Po is appreciably lower than the other potentials
considered.
EFFECTIVE WORK FUNCTION INCREMENTS
RESULTING FROM DIPOLE LAYERS
The results of the previous section will now be applied
to field emission. Before doing so, it may be useful to
give a very simple rationale for the Fowler-Nordheim
equation. Figure Sa shows a one-dimensional potential
energy diagram for electrons in a metal and surrounding
space, in the presence and absence of an applied external
field. The penetration coefficient of the barrier in the
presence of a field is given by
I
D=constexpC-Cm1/h)! V(V-E)dx]. (4)
o
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205.208.120.57 On: Mon, 08 Dec 2014 16:31:031872 ROBERT GOMER
(a)
"-"'---5 ~ }!----------------WN..a: SA
IMAGE POTENTIAL
It
(b)
FIG. 5. Schematic pote~tial diagrams for metal surfaces with
applied external potentials. (a) Clean metal, no image potential
assumed. x represents work function, '" the depth of the Fermi
sea. (b) Same as (a) but based on image potential. Barriers are
shown for clean metal and metal with a dipole layer of nitrogen.
The upper dotted curve represents PI alone and is drawn from an
origin at 4.5 volts on the diagram.
The exponent is thus proportional to the area under the
curve traced by the square root of the ordinate U= V -E
in Fig. 5a, since (to a good approximation8b) only
electrons very near the top of the Fermi sea contribute
to emission. The area A is nearly triangular and hence
given by
hLJ=h!/F, (5)
where F is the applied field. Thus the dependence on
work function and field of the Fowler-Nordheim
exponent seems reasonable. In this simple picture
image potential was neglected. It can readily be seen
that its effect would be to decrease the effective barrier,
which amounts to increasing the effective field. This is
the nature of the image correction later introduced
by Nordheim.I2
In order to determine the effective increase in work
function caused by a dipole layer, we proceed as follows.
For a given field a plot of the potential barrier and its
square root is constructed for the clean metal. The
contribution of the dipole layer potential to the barrier
12 L. W. Nordheim, Proc. Roy. Soc. (London) A121, 628 (1928). is then added, using the PI values corresponding to the
coverage (or dipole-dipole distance) under considera
tion. The square root areas for the clean and contam
inated cases are then determined by cutting out and
weighing. It is now assumed that the pre-exponential
parts of the penetration coefficients and of the Fowler
Nordheim equation change negligibly compared to
the exponential parts; so that
(A+aA)f x+ax=x, -A--, (5)
where X represents the work function of the clean metal,
ax the increment due to the double layer, A the area
of the square root barrier for the clean metal, and
A + aA the area of the square root barrier for the metal
in the presence of the layer. Figure 5b shows the barriers
for a representative case. The classical image potential
has been used and is blended smoothly into the Fermi
level. It is clearly not correct to relate the simple
Fowler-Nordheim equation to a potential barrier
based on image potential. The following correction is
therefore made: A third barrier is plotted, consisting
of the clean metal barrier plus a uniform linear addi
tional potential of known value (approximately equal
to the mean value of the actual layer potential under
consideration). The procedure outlined above is then
used to determine the apparent x+ ax for this case.
A correction factor given by <x+ ax) apparent/
(x+ax) can thus be found and applied to the effective
work function for the actual barrier. The effective ax
for the layer can then be compared with the correspond
ing contact potential. These results are plotted for a
range of fields and (J values in Fig. 6.
These data are based on values of Pl. A more correct
procedure would require similar calculations for P2, Pa,
and so on. The resultant values of the work function
would then have to be weighted by the corresponding
emitting areas to determine the emission current at a
given voltage. In practice, the error introduced by
using PI is small, since Pa is quite close to PI at all
except very low coverages (large a). The effect of
neglecting areas of higher work function than PI is to
give slightly low values for the effective work functions
at low coverage.
A previous investigation of the velocity distribution
of electrons in field emission8b has shown that emerging
electrons may be expected to have energies of the order
of 0.1 ev transverse to the direction of emission. This
energy is sufficient to prevent focussing effects by the
lateral potential gradients existing in the region of the
potential barrier. It can readily be shown that potential
differences between PI and points as close as lA to a
dipole site do not exceed 0.1 ev.
A clean metal work function of x=4.S ev was used.
Effective dipole moments per ad-atom were those of
nitrogen on tungsten, based on Bosworth's3 value of
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the contact potential. In order to convert from values
of a to the equivalent values of (J, a mean number of
sites/ cm2 equal to 1.2X lOIS was assumed. (J is then
given by
(J= (3.1/ a)2, (6)
i.e., the nearest neighbor distance at (J= 1 is taken as
3.1A. The contact potentials for oxygen and hydrogen
on tungsten are so close to that for nitrogen that it is
probably safe to use the percentage values given in
Fig. 6 when dealing with these gases.
Two facts emerge from Fig. 6. There is a marked
decrease in effective contact potential with increase in
field. Comparison of Figs. 3 and 6 shows that the
effective contribution to the work function can be
expressed empirically by the potential due to the layer
existing at a distance from the surface x= SA at
F=3X 107 v/cm, and 2A at F=6X 107 v/cm. Over this
range the relation is more or less linear. Second, the
decrease in effective potential becomes less important
at higher coverage, since a uniform dipole sheet is
more closely approximated.
The curves of Fig. 6 indicate that one should not
expect strictly linear Fowler-Nordheim plots in the
case of contaminated surfaces, since the effective work
function changes with field. While this is true, it is
possible to work at fields of the order of 3-4X 107
volts/cm and to stick to a very small range, so that the
variation of work function over the working range
is small.
DEPOLARIZATION EFFECTS
The calculations to this point have been expressed in
terms of the contact potentials existing at a given
coverage (J; these are average values based on measure
ments made on polycrystalline samples.
It is interesting to ask what causes the slight depolari
zations observed by Bosworth.3c Two factors must be
considered. The first is the depolarization of the
adsorbate complex itself under the influence of neighbor
ing dipoles. The existence of very strong potential
gradients in dipole arrays was recognized many years
ago by Langmuir.13 A second factor which seems to
have been overlooked is the following: Common sense
shows that the electron cloud in a metal does not
terminate sharply at the surface, since this would
lead to infinite gradients of the wave function and
hence infinite kinetic energies. The calculations of
Bardeen14 show that a spilling over of the electrons takes
place and gives rise to a double layer with the negative
end directed outward. The contribution to the work
function by this layer is of the order of 1 ev. Smoluchow
ski's14.1s refined considerations show that the details of
131. Langmuir, J. Am. Chem. Soc. 54, 2798 (1932).
14 Excellent summaries, with references, are given in C. Herring
and M. H. Nichols, Revs. Modern Phys. 21, 185-270 (1949) and
in the chapter by C. Herring in Metal Interfaces (American
Society for Metals, Cleveland, 1950).
16 R. Smoluchowski, Phys. Rev. 60, 661 (1941). .8
"1.1 .6 <I~
.5
F
WN
(a)
1.0~---------------,
t.)(/(P,la:,AS A FUNCTION OF 9 FOR VARIOUS \ALUES OF 'F'
(b)
FIG. 6. (a) Work function increment Llx in field emission in
terms of the corresponding contact potentials PIa> as functions
of the applied field for various coverages 8. (b) Llxl PI", as functions
of () for various applied fields F.
the surface, i.e., the exact boundaries of the "S poly
hedra" are subject to similar considerations, so that
electron spillover is possible from the hills to the
troughs of a surface; this produces a double layer of
opposite sign to that previously mentioned. These
facts are responsible for the differences in work function
of various faces of clean metal crystals.
It is apparent that external fields will polarize the
electron cloud at the metal surface. It will now be
shown that the observed sel£-depolarizations of chemi
sorbed electronegative layers can be rather well
explained by assuming that the effect is almost wholly
due to a depolarization of this electron cloud by the
dipoles constituting the layer. In order to calculate
the effect, we resort essentially to the Fermi-Thomas
method. We assume that the chemical potential (J.I) of
electrons in the interior of the metal is equal to that of
electrons in the external cloud. If the latter is subjected
to an external potential not experienced by electrons
in the interior, e.g., that due to an applied field or that
resulting from a chemisorbed polar layer, a reduction in
electron density p will occur in the external cloud. If
Fermi statistics are applicable, we can write
(7)
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(0)
for the ratio of densities in the cloud in the absence and
presence of an external potential P.
There is very good reason to believe that the electron
cloud at the metal surface is sufficiently dense to obey
Fermi statistics. Thus field emission experiments on
tantalum carried out between room temperature and
2°K showed no appreciable change in the work functions
or their relative values.l6 Experiments by Dyke17 and
his associates not only confirm the validity of the
exponential part of the Fowler-Nordheim equation, but
show that it is valid at very high fields; also, his patterns
for tungsten look exactly like those obtained by other
workers at lower fields. This indicates that the effect of
fields on the work function is small. All these facts can
be true only for virtual fermions.
The Thomas-Fermi method is strictly applicable
only to systems in which the density gradient of
fermions varies slowly, compared to their de Broglie
wavelength. In connection with nuclear statistics, von
WeizsackerlB has developed a correction to the Fermi
energy, which depends on this density gradient. Fig
ure 7 shows two models of the electron cloud at the
surface. A linear decrease of density is assumed in Fig.
7a and an exponential one in Fig. 7b. In terms of the
chemical potential the Weizsacker correction J.l.w turns
out to be
d [h2 [gradP]2] [d2P (dp)2] h2
J.l.w=-- = (1/p2) 2p--- --.
dp 8m p dx2 dx 8m
Application to the two cases shows that a term
J.l.w= -h2/2myo2 at p= po/4 (8)
(9a)
results for the linear decrease and a similar one but of
opposite sign
(9b)
for the exponential decay. yo represents the distance in
which the charge density drops from tpo to zero in the
linear case and the distance in which it drops from Po
to pole in the exponential one. It is seen that the
Weizsacker energies for these cases are almost identical
but of opposite sign. If yo is expressed in angstroms,
16 R. Gomer and J. K. HuIm, J. Chern. Phys. 20, 1500 (1952).
17 W. P. Dyke and J. K. Trolan, Phys. Rev. 89, 799 (1953).
18 C. F. v. Weizsacker, Z. Physik 96, 436 (1935). ( b) FIG. 7. Schematic dia
gram for the electron density
p at a metal surface. po
density in the interior. (a)
linear decrease; (b) expo
nential decrease.
the corrections to the energy in ev amount to roughly
l/Y02. Since yo is of the order of lA, no serious error
results from neglecting the Weizsacker term.
Strictly speaking, Eq. (7) should be applied at each
point of the cloud and the resultant densities used for
the determination of the new electron layer potential.
For simplicity, however, it is assumed that the mean
density in the cloud outside the metal can be taken as
ipo, and that the effect of external potentials consists
merely in a reduction of this value, leaving the effective
dipole distance unchanged. Then the ratio pp/ p repre
sents the fractional decrease in the electron layer
contribution to the work function. J.I. in Eq. (7) must
therefore be divided by 41. A value of J.l.o= 6 ev was
used for the normal chemical potential. It is further
necessary to carry out the solution self-consistently,
that is, to consider the repolarization of the electron
layer by the decrease in its self-potential. This is done
by uSIng Eq. (7) to calculate a first value of the new
electron layer potential, subtracting this from the
original value, and using this difference as the first
repolarization potential. This is subtracted from the
original external potential to obtain a new effective
external potential for use in Eq. (7). Iteration is
continued until a consistent value is reached. Calcula
tions were carried out for assumed initial double-layer
potentials of O.S and 1 ev. The results are shown
in Fig. 8.
It is possible to apply these results to Bosworth's3b,o
values of the contact potential for oxygen, nitrogen, and
.7.---------------------.
.6
..
..::. .3 ....
~ b .2 le.v.
~~-~-r_~-_rn-nl.~~-~-n.-~.o
APPLIED POTENTIAL
FIG. 8. Decrease in potential V due to electron layer at a metal
surface with applied depolarizing potential for two initial potentials
of the layer.
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hydrogen on tungsten. For this purpose Pl at x= lA
is used as the depolarizing potential. Strictly, this Pl
(at a given 0) must be based on the dipole moment per
complex at zero coverage, i.e., on the initial slope of
the V QO VS 0 plot. The results for nitrogen and oxygen
are shown in Figs. 9 and 10. The points represent sums
of the depolarization and the actual contact potentials.
The straight lines represent the initial slopes of the
V QO vs 0 plots, i.e., the contact potentials that would
result if there were no depolarization. It will be seen
that both oxygen and nitrogen can be fitted very well.
To obtain this fit, the electron cloud was assigned an
initial potential of 0.5 volt in both cases. For hydrogen
the V QO VS 0 plot shows almost no deviation from linearity
up to the highest values of 0 reached by Bosworth.3b
There is some question whether this value is 1, as believed
by Bosworth,3b or 0.7.19 In any case, there should be
very little depolarization below a contact potential of 1
volt on the basis of the present calculations.
It should be emphasized that we do not attempt to
calculate the electron layer potential a priori; the latter
FIG. 9. Contact potential
vs coverage () for nitrogen on
tungsten after Bosworth,
reference 3, (curved line).
Straight line represents ini
tial slope, solid points the
sum of the contact potential
and the calculated depolar
ization. 1.6~--------'
1.4
II)
~ o
>1.2
'!:
-'
~I.O
~
!? .8
e
WN
is taken empirically to give the best fit with experiment.
We crudely estimate the perturbation of this potential
by external fields. It should also be pointed out that
the V QO VS 0 curves taken from Bosworth 3 are not directly
determined and thus subject to quite some uncertainty.
On the whole, however, it seems reasonable to explain
the observed depolarizations of slightly polarizable
electronegative layers on the basis of electron layer
depolarization. This result will certainly not be valid
for electropositive adsorbates like cesium or barium
on tungsten.
The conclusion just reached enables one to estimate
the effect on the work function of the fields necessary
for cold emission. If the reasoning of this section is
correct, ad-atom-surface complexes like N-W are very
poorly polarizable so that experimentally produced
IV E. K. Rideal and B. M. W. Trapnell, J. chim. phys. 47, 126
(1950). 2.2,------------,
2.0
1.8
1.6
(I) ...,
~ 14
</.1.2
FIG. 10. Oxygen on i=
tungsten. Same as Fig. ~ 1.0
9.
I-
~ .8
Iz 8 .6
.4
.2 .4 .6 .8 1.0
8
WO
fields would be quite insufficient to cause polarization,
the more so as these fields (3-5X 107 v/cm) are smaller
than the inherent fields existing in the dipole layer.
Thus for both clean and contaminated surfaces the
effect on the electron cloud only need be considered.
It turns out from Fig. 8 that at fields of 3X107 v/cm a
polarization of 0.07 volt should take place, thus
increasing the work function by that amount over the
weak field work function.
ELECTROSTATIC EFFECTS ON HEATS OF
CHEMISORPTION
Experimental measurements of hea ts of chemisorption
show that initially very high differential heats drop
remarkably with increasing coverage. It has been
realized for some time that Coulomb repulsion of the
dipoles, depolarization effects, and so on are quite
insufficient to explain more than a small fraction of
this drop. Ii The latter may amount in the case of oxygen
on tungsten to 60 kcal at 0= .5. Boudart6 has recently
made the following very interesting suggestion: it is
assumed that a small integral number of electrons is
involved per ad-atom-substrate bond, and is thus
localized in a region between the adatom and the
surface; on the average these electrons will find them
selves more or less halfway between atom and surface,
i.e., at td. If a given heat of formation Ho corresponds to
each bond of this kind, the observed heat of formation
H will be smaller than Ho by the electrostatic energy of
the bonding electrons in the potential of the layer at !d.
Boudart now assumes that the potential at this point
can be taken as half the contact potential V QO at a given
coverage; since V <Xl varies linearly with 0, linear decreases
in H and their order of magnitude would thus be
explained.
If the model of the chemisorbed layer used by us is
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correct, Boudart's argument unfortunately no longer
gives correct magnitudes and should not lead to linear
changes in H. It turns out that Po at x' = 0.5 is very
much smaller than V 00/2 (see lowest curve in Fig. 3)
for all conceivable values of a'. Furthermore, it is
possible to express Po at X= lA for d= 2A (the most
reasonable actual assignment) by
Po= 18. 1 Ma-2• 85 ev (M in Debye) (10)
from which it follows that the dependence on 8 is
Po= .724MOI.43 ev. (11)
If depolarization is taken into account, the result of
Eq. (11) must be multiplied by the factor (1-0.0640.43).
This does not change the result significantly. Thus at
8= 0.5, Boudart's effect can account for only about 8
kcal in the case of oxygen, assuming two electrons per
bond. It follows that Boudart's effect can account for
the observed changes in H only if individual adatom
surface complexes are far from being the simple dipoles
by which we have represented them and have a charge
distribution which leads to effectively continuous
dipole sheets even at low coverages. It seems likely
that there is considerable deviation from spherical
charge distribution in the direction normal to the
surface. This deviation should not affect our arguments
too much. A spreading of the charge parallel to the
surface would be needed to effect quasicontinuity of the
dipole sheet. This does not seem probable for electro
negative adsorbates.
Calculations similar to those of the first section of
this paper show that dipole repUlsion energies Ep for
a triangular lattice can be expressed by
Ep= .076M28!.39 ev (M in Debye). (12) Equation (12) leads to effects of the same order of magni
tude as that of Topping,19 but is based on a dipole model
with d= 2A rather than on point dipoles. As Rideal
and Trapne1l20 point out, and as is apparent from Eq.
(12), this energy amounts to at most 2 kcal.
Electrostatic interactions of the dipoles with the
electron cloud at the surface may also be considered.
A little thought will show that this effect, while small,
must lead to an increase in H, since the only non
linear part can arise from the cumulative effect of the
dipole layer potential on the cloud. This effect, as we
have seen, is to drive the cloud back into the metal,
hence reducing the net electrostatic interaction of the
dipoles with the electron cloud.
If these arguments are correct, it would appear that
ordinary electrostatic effects are insufficient to explain
observed decreases in H. Two possibilities remain.
The first is connected with the fact that wires and films
used in heat experiments are polycrystalline and thus
may be sufficiently heterogeneous from the heat point
of view. The second possibility is that the effect exists
even on uniform surfaces and is caused by an increase
in the kinetic energy of the electron cloud, resulting
from the change in the gradient of the wave function
when the cloud is partially driven back into the metal
by the dipole layer potential. However, this effect
cannot exceed the original Coulomb interaction, unless
other quantum mechanical, essentially tunnel, effects
set in.
ACKNOWLEDGMENT
It is a pleasure to acknowledge many stimulating
discussions with Dr. Morrell H. Cohen of this Institute.
I also wish to thank Mr. Matthew Prastein for perform
ing many of the numerical computations.
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1.3067538.pdf | The nucleus
Enrico Fermi
Citation: Physics Today 5, 3, 6 (1952); doi: 10.1063/1.3067538
View online: http://dx.doi.org/10.1063/1.3067538
View Table of Contents: http://physicstoday.scitation.org/toc/pto/5/3
Published by the American Institute of PhysicsThe
NUCLEUSBy Enrico Fermi
The following article is based on the
first of six invited papers presented
during the symposium on contem-
porary physics which keynoted the
Twentieth Anniversary Meeting of
the Institute of Physics in Chicago
last October. An audience of three
thousand assembled in the Chicago
Civic Opera House to hear the ad-
dresses, four of which have ap-
peared in recent issues of this journal.
IN THE TWENTY-YEAR PERIOD since the
founding of the American Institute of Physics,
nuclear physics has been advancing perhaps as rapidly
as any other branch of our science. Twenty years ago
the neutron had not yet been discovered, and a favored
hypothesis as to the structure of the atomic nucleus
was that it consisted of protons and electrons. This
very fact may give some idea of the exponential rate
of our progress.
Perhaps, to think of another reference mark, con-
sider that it was just about forty years ago when the
discovery of the nucleus was announced by Rutherford.
In nuclear physics, as in many other branches of
physics, the past four decades have seen advances in
very many directions. These advances have occurred
both in techniques and in fundamental knowledge. Dur-
ing the period with which we are concerned, voltages
achieved in accelerating machines have been going up
in steps roughly of 10—10", 10', 10", and very soon, we
hope, 10° electron volts. The Cosmos is of course still
far ahead, and provides a formidable challenge to the
constructors of high energy accelerating machines.
Neutron sources have gone up in steps which are
more nearly (in round numbers) of the order of
one million each—from the small radium or radium-beryllium sources, to cyclotrons, to atomic reactors.
Of course quite sizeable steps have been taken in
the amount of money used for research. Large steps
have also been taken in the population growth of physi-
cists, and in the audiences that come to listen to a sym-
posium in physics—if I should judge from this audience.
Technical advances that have been less spectacular
than those mentioned previously, but I believe no less
significant, have taken place in the development of de-
tecting devices. Counters, ionization chambers, and the
more recent and very important discovery of the scin-
tillation counter should be mentioned. The latter does
automatically what Rutherford and his pupils did so
laboriously in watching the minute scintillations that
result when an alpha particle hits a crystal. The re-
fined electronic techniques used in the scintillation
counter have shortened the time of counting to the
range of 10"° seconds and less. One can thus measure
directly the time taken by particles traveling close to
the velocity of light to cross a distance of a few feet,
and consequently obtain the velocity of the particle.
The Wilson cloud chamber has led to the develop-
ment of the diffusion chamber, which promises to be
one of the fundamental tools in investigating elemen-
tary particle reactions. Photographic plates have been
PHYSICS TODAYdeveloped to a very high degree of perfection as re-
corders of tracks of particles.
Now these technical developments have resulted in
part in, and in good part have promoted, a very consid-
erable advance in the knowledge of the nucleus and of
its constituents. We have by now what seems to be
the final understanding at least of the generalities of
the nuclear structure—the nucleus built of neutrons
and protons. We have some understanding of the fea-
tures of the beta spectrum. We have discovered hun-
dreds of nuclear reactions and hundreds of new radio-
active isotopes, with the result that a new branch of
the art of nuclear science has emerged which includes
radiochemistry and all of the complex techniques in
chemistry and biology for the use of tracers.
The discovery of fission has led to the realization of
the possibility of chain reactions, soon followed by the
actual construction of nuclear reactors. This has pro-
vided the starting point for the new science of nuclear
engineering. The spectroscopy of the nucleus is ap-
proaching in complexity, although by no means in un-
derstanding, that of the atom. Charts of nuclear energy
levels with corresponding gamma-ray and other transi-
tions between them are beginning to acquire a com-
plexity that may remind one of the early atlases ofKnrico Fermi, Nobel Laureate
and professor of physics at
the University of Chicago,
came to the United States
from Italy in 1938. Professor
Fermi's contributions to the
present knowledge of nuclear
physics have been both nu-
merous and Important. He
played a prominent part in
the development of the atomic
energy program in this coun-
try, having been In charge of
the work which resulted in
the first self-sustaining- nu-
clear chain reaction produced
In the Chicaco pile in 1942.
and later having served as a
member of the wartime staff
of the Los Alamos Laboratory
in New Mexico. Professor
Fermi Is vice president of the
American Physical Society.
Wide World photo
atomic levels that were in use in the early Twenties.
Measurement of nuclear masses and moments, pri-
marily with the technique of mass spectroscopy and
radiofrequency resonances, has become an extremely
precise art. We have learned a great deal about ele-
mentary particles and, with the help of the cosmic
radiation, have discovered many new ones. Great prog-
ress has been made in the determination of beta spectra
and recently even the beta disintegration of the neu-
tron has been investigated quite thoroughly.
The mass of data resulting from these many discov-
eries presents a challenge for the understanding, and
unfortunately the business of understanding is not as
well in hand as one might wish. The present state
might be illustrated by choosing, for purposes of dis-
cussion, two of the many topics in nuclear physics that
are of current interest.
TN DISENTANGLING the problems of the atom.
A one of the major steps has been the recognition
that it is useful to speak of individual orbits of the
electrons in the atom. This, to be sure, is only an ap-
proximation, in fact a crude approximation, but still it
provides a quite invaluable starting point for the study
of complex atoms containing large numbers of electrons.
MARCH 1952When physicists became reasonably certain that the
nucleus was constructed of protons and neutrons, ques-
tions were raised concerning the orbital behavior of
these particles. Could nuclear structure be interpreted
on the general pattern of atomic structure by attribut-
ing to the various neutrons and to the various protons
within the nucleus something like individual orbits and
individual states? If so, an understanding of the nuclear
levels and the nuclear structure could possibly emerge
from the much simpler pattern of the individual states.
No definite answer has ever been given to this ques-
tion, although nuclear science has for a long time "offi-
cially" frowned on such attempts. Strong arguments
were quoted for saying that the constituents of the
nucleus are mixed so thoroughly and interact so rapidly
that there is little basis for hoping that individual orbit
considerations can lead to an understanding of nuclear
structure.
Consider one nuclcon in the nucleus travelling along
its orbit among the other nucleons. If the collision mean
free path is A. this nucleon would collide with the other
neutrons and protons in the nucleus and its orbit would
be lost after it had gone the distance of its free path.
A criterion that one might adopt in deciding whether
or not it is a sensible approach to talk of individual
orbits is to compare the mean free path with the size
of the expected orbit. If the mean free path is long,
then we may take the orbital behavior seriously. But
if the mean free path is much less than the size of the
orbit, one expects the idea of orbit to become rather
unusable. Now it is a very difficult problem to decide
the length of the mean free path, but if one takes some-
what literally the strength of the interactions between
the neutron and other components of the nucleus, one
is led to a value that seems discouragingly short.
In spite of this argument, evidence has been ac-
cumulating for the last few years, both in this coun-
try and in Germany, to the effect that orbits do exist.
The best-known feature of this evidence has been the
discovery of the so-called "magic numbers." They are
the numbers 2, 8, 20, 50, 82, 126. When a nucleus con-
tains a number of either neutrons or protons equal to
one of the magic numbers, it is particularly stable, as
if a shell of either neutrons or protons had been closed.
This and other evidence to be discussed later indi-
cate that the orbit approximation is much better than
the discussion above may have suggested. It would ap-
pear that for some reason the mean free path must be
longer than is given by a somewhat crude estimate of
its length. One possible reason for this may be the
Pauli principle, according to which collisions between
two particles may be forbidden when, after the col-
lision, one of the two particles would go to an oc-
cupied state.
Another possible explanation of the long mean free
path may haye to do with the saturation property of
the nuclear forces. It has been suggested, for example,
that the meson field responsible for these forces may
have a non-linear character and reach a saturation level
in nuclear matter due to the high density of the nucle-ons present. In spite of the fact that neither of the two
above possibilities has been worked out to the point
that it can be considered a satisfactory theory, it is
now rather generally believed that many features of
the single particle model will ultimately prove correct.
Strong additional evidence for this model is the de-
tailed explanation of the magic numbers in terms of
the assumption of a very strong spin orbit coupling.
Maria Mayer here in Chicago, and the investigators in
Germany who developed independently similar ideas,
have been able to point out very many features of the
isomeric nuclear levels which lend strong support to
these views.
There is at present no understanding of the origin of
the strong spin orbit coupling that is suggested by the
empirical evidence. Such understanding perhaps will
come only when a satisfactory theory of the nuclear
forces will have been developed. At present we must
take the existence of this coupling as an empirical fact.
In spite of our only partial understanding of the
situation, the orbit theory of nuclear structure offers a
hopeful model for at least a qualitative understanding
of nuclear structure, and already it has been possible
to fit into this picture a very great number of details.
IT IS OF COURSE IMPOSSIBLE to hope for any
deep understanding of the structure of the nucleus
without knowing a lot about the forces acting between
the elementary constituents of the nucleus—between
neutron and proton and between proton and proton
and between neutron and neutron.
The classical experimental approach to investigations
of nuclear forces has been the study of scattering. One
hurls a neutron at a proton and sees how they are de-
flected. From the features of the deflection, the angular
distribution, the energy dependence, and so on, one
hopes to deduce the force responsible for the deflection.
Early experiments by Tuve, Herb, and others, inter-
preted by Breit, gave the first knowledge of a short
range interaction between nuclear nucleons that is re-
sponsible for the fact that particles stay together. Then
came the Yukawa theory to give a great help to our
understanding of the problem by offering for the first
time a model for us to consider. The model is quite
similar in many ways to that of the electromagnetic
forces: one particle produces a field and the field acts
on another particle. In this case, however, Yukawa was
faced with the additional problem of designing a theory
that would automatically account for the short-range
character of the nuclear forces. Yukawa recognized
that a field whose quanta have zero mass (like the
photons) would have a long range, while a field whose
quanta have a finite and relatively large mass would
have a short range.
According to the Yukawa theory, a neutron will oc-
casionally convert into a proton plus a pi-meson, which
will then be reabsorbed and thrown out again and re-
absorbed and so on. The nuclear field involved in this
oscillation will extend as far from the original neutron
as the continually emitted pi-mesons can reach. And
PHYSICS TODAYhow far can they reach? The argument runs as follows:
A meson has considerable mass, and to fabricate a
meson with which to play this odd ball game requires
an amount of energy equal to the mass of the meson,
JX, multiplied by the square of the velocity of light, c.
Who pays for this amount of energy? Well, nobody;
so if nobody pays one has to borrow. Now in the bank
of energy there is a very special rule that should per-
haps occasionally be adopted by commercial banks—
namely, the larger the loan, the shorter the term. Quan-
titatively, this banking practice is represented by one
of the forms of the Heisenberg uncertainty relation.
One can borrow an amount of energy W for a time of
the order of Planck's constant // divided by W; there-
fore the time t of the loan shall be h//xc'. The meson
will be capable of moving away from its source a dis-
tance equal at most to the time / multiplied by the
velocity of light c; therefore the range of the nuclear
forces according to this mechanism is essentially h/^c
and is inversely proportional to the mass. For short-
range action, the quanta of the field that transmits the
nuclear forces must be very massive; in fact, the early
estimates of Yukawa indicated that the mass would
have to be comparable to 300 times the electron mass.
Almost on the heels of the announcement of the
Yukawa theory came the discovery of the meson in
cosmic radiation, thus giving the theory a tremendous
boost. The particle first found in the cosmic radiation,
as is well known now but was not known at the time,
is not the Yukawa meson, but is a son of the Yukawa
meson. This was discovered recently when Powell found
tracks in photographic plates that had been exposed at
high altitudes, showing the existence of two different
mesons. One of these, the so-called pi-meson, is the one
responsible for nuclear forces; the other, the mu-meson,
is a rather uninteresting offspring of the first—at least
it seems uninteresting at present.
Then, of course, came another fundamental experi-
mental result that was determined at least in part by
the Yukawa theory: if two nucleons, each of which is
surrounded by a meson field, collide with sufficient en-
ergy, some mesons are likely to be shaken loose. There
was evidence from cosmic-ray studies of the actual ex-
istence of this process, but the most spectacular ex-
perimental result in this direction was obtained at
Berkeley where Lattes and Gardner discovered that
these mesons are actually produced in the high energy
collisions in the synchrocyclotron. The discovery of an
artificial means for the production of pi-mesons has
put at the disposal of the physicists a source of this
particle that is easily controllable and extremely more
intensive than any cosmic-ray source. This is an ideal
situation for investigating the properties of these new
particles and research is going on actively in this di-
rection in many laboratories. But again, what about the
understanding?
PERHAPS, in outlining the Yukawa theory (which
*• in my opinion certainly has a considerable amount
of qualitative correctness), I should have included thewarning that there is not just one theory, but that
there are several theories, and that none of them seems
to be really the correct one. It is sometimes difficult to
say what is wrong with any particular theory because
the mathematics involved is almost prohibitively com-
plicated. But one can seldom manage to make a calcu-
lation that is really right because the theory is so com-
plicated, and if one tries, more as a rule than as an ex-
ception, one encounters divergent infinite terms which
one usually attempts to eliminate by not perfectly
orthodox procedures. Perhaps at the root of the trouble
is the fact that the theory attempts to oversimplify a
situation which may in fact be quite complicated. When
the Yukawa theory first was proposed there was a le-
gitimate hope that the particles involved, protons, neu-
trons and pi-mesons, could be legitimately considered as
elementary particles. This hope loses more and more its
foundation as new elementary particles are rapidly be-
ing discovered.
Perhaps the situation might be compared (although
comparisons are always dangerous) to that of the early
quantum theory, which provided a large amount of
qualitative insight in the atomic structure, but never-
theless failed from the quantitative point of view. Per-
haps the situation is similar; perhaps brilliant solutions
of the same type will be forthcoming.
It is difficult to say what will be the future path.
One can go back to the books on method (I doubt
whether many physicists actually do this) where it will
be learned that one must take experimental data, col-
lect experimental data, organize experimental data, be-
gin to make working hypotheses, try to correlate, and
so on, until eventually a pattern springs to life and
one has only to pick out the results. Perhaps the tradi-
tional scientific method of the textbooks may be the
best guide, in the lack of anything better.
At present, rapid progress is being made in collect-
ing data on nuclear forces, both by direct observation
from scattering experiments and by indirect study of
the mesons. Results are accumulating quite rapidly, and
while they have not yet fallen into a satisfying pattern,
perhaps they will before too long.
Some of the many Yukawa theories seem to be ex-
cluded by these experiments, and the favored one at
present is the "pseudoscalar theory with pseudovector
coupling," which in slightly plainer words means that
the meson has spin zero and behaves like a pseudo-
scalar, a symmetry property that is certainly familiar
to most physicists.
Of course, it may be that someone will come up soon
with a solution to the problem of the meson, and that
experimental results will confirm so many detailed fea-
tures of the theory that it will be clear to everybody
that it is the correct one. Such things have happened in
the past. They may happen again. However, I do not
believe that we can count on it, and I believe that we
must be prepared for a long hard pull if we want to
make sure that at the next anniversary celebration of
the American Institute of Physics we shall have the
solution to this problem.
MARCH 1952 |
1.1698840.pdf | Band Structure of Graphite
J. L. Carter and J. A. Krumhansl
Citation: J. Chem. Phys. 21, 2238 (1953); doi: 10.1063/1.1698840
View online: http://dx.doi.org/10.1063/1.1698840
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v21/i12
Published by the American Institute of Physics.
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Downloaded 20 Oct 2012 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions2238 LETTERS TO THE EDITOR
TABLE I. Quadrupole resonance frequencies. eou piing constants.
and asymmetry parameters at the liquid air temperature.
eOg
Compound VI (Me/sec) ., (Me/sec) ~ (%) (Me/sec)
CIt.I C,H,r
n-C.H71
n-C.H.I
CH,ICOOH
AsIa 264.973 ±0.01
247.115 ±0.01
250.81O±0.02
249.062 ±0.02
1298.87 ±0.1O
\297.86 ±0.1O
207.011 ±0.01 529.515 ±0.01
494.058±0.01
499.81O±0.01
497.539 ±0.03
596.12 ±0.20
592.79 ±0.20
395.777 ±0.02 2.5
1.6
5.2
3.0
4.6
6.1
18.4 1765.3
1647.0
1666.9
1658.8
1988
1978
1328.2
18.4 percent to 14.5 percent when the temperature was changed
from the liquid air temperature to the room temperature, 27°C.
The large asymmetry parameter of this molecule seems to indicate
double bonding between arsenic and iodine atoms. Above about
90°C the absorption lines split into triplets, whose frequencies at
llOoC were 206.2,208.5, and 211.7 Me/sec for the lower lines and
403.2, 407.0, and 412.5 for the higher lines.
The spectrum of stannic iodide was also measured in the tem
perature range between the liquid air temperature and 69°C. The
mean coupling constant of the doublet was changed continuously
-from 1389 Mc/sec to 1355 Mc/sec, when the temperature was
raised through this range. This spectrum was first studied by
Dehmelt4 at room temperature and recently by Livingston and
Zeldes6 at 200K and 7QoK. The obtained results agreed well with
that of Dehmelt and were consistent with those of Livingston
and Zeldes.
1 H. Kriiger, Z. Physik 130. 371 (1951); C. H. Townes and B. P. Dailey, J. Chern. Phys. 20, 35 (1952).
, H. Zeldes and R. Livingston. J. Chern. Phys. 21. 1418 (1953).
• Kojima, Tsukada, Ogawa. and Shimauchi. J. Chern. Phys. 21, 1415
(1953).
• H. G. Dehmelt. Z. Physik 130, 356 (1951).
• The melting points of methyl iodide. ethyl iodide, n-propyl iodide. and
n-butyl iodide are -66.45. -111.1, -101.3, and -103.0°C. respectively.
(Shiba "Table of Physical Constant." Iwanami. TokYo).
• R. Livingston and H. Zeldes, Phys. Rev. 90, 609 (1953).
Band Structure of Graphite
J. L. CARTER* AND J. A. KRUMHANSL
Cornell University. Ithaca. New York
(Received October 5. 1953)
SEVERAL workers have studied the electronic properties of
graphite.1 Theoretical interpretations of the results have been
based on the Wallace theory.2 Two conclusions from his theory are:
(a) the P. valence and conduction bands touch at the zone corner.
(b) energy vs K curves are symmetric about the touching energy.
From (b) one concludes that the Hall coefficient of normal
graphite should be zero, in disagreement with experiments,!
which invariably yield a sizeable negative value (electron con
duction).
This disagreement led us to examine the lattice symmetry re
strictions on band structure in greater detail. One of us3 has
applied group theoretical methods and among other results has
concluded that neither (a) nor (b) is required. Any particular
characteristic of the energy bands which is a consequence of
lattice symmetry will appear regardless of the approximation
method used for band calculations. That (b) is not required is
noted in the discussion by Coulson and Taylor,. where inclusion
of overlap integrals destroys the symmetry (b); however, near
the zone corner the effect is negligible, so that this is insufficient
to explain the anomalous electronic behaviors observed. Mrozow
skiS notes that Polder also has concluded that (a) and (b) are
accidental.
Although an extensive cellular calculation would show these
features in their proper form the labor involved seemed pro
hibitive. Rather we have attempted to modify the Wallace tight
binding calculation in a plausible manner to show how the more
general features appear in this approximation. Of the several
possible modifications of Wallace's assumptions we believe the most significant for the electronic properties near the zone corner
is the inadequacy of his Eq. (4.5). Of the four atoms in the graphite
unit cell two have nearest neighbors in adjacent planes (1 and 3)
while two do not (2 and 4); thus it is more realistic to assume
(1)
The difference is small (we estimate it from Coulomb penetration
integrals to be ;;::0.01 ev), but since the energy-level structure
varies rapidly over a few tenths of a volt near the zone corner
the effect is significant in this energy range. Appropriately modi
fying Wallace's Hermitean secular determinant to read
Hll-E, 'Ylr,
'Yt'r 5*,
Hll-E, 'Yt'r5*
'Yt'r5
-'Y05*
H22-E =0, (2)
we find near the zone corner (by neglecting 'Yl' 5 as a smalI quan
tity of second order) the four roots
E=H,,+ (Hll-~22=F'Ylr) +[ ('YQS)2+(Hl1-~22=F'Ylrrr
E=H22+(Hll-~22=F'Ylr) _ [('YoS)2+(Hll-~22=F'Y1r)'r (3)
The resulting modified band structure replacing Wallace's Figs.
lOand 11 is shown in Fig. 1, drawn with assumption that H'2-Hll
2Tkr,--------~--------~,,~
Kz
~r-~------~----~~~~~2
Ky
FIG. 1. Modified zone structure in graphite.
<2'Yl (if this is not so, the bands do not touch at alI). Similarly the
densities of states are compared in Fig. 2 with Wallace's results;
the sensitivity to the modification (1) is striking. For example,
assuming HZ2-Hll=hl the density of electron states at the
lower edge of the upper band is ""2.5 times that of the "hole"
N(E) /
/ -This model
---Wallace
E-
FIG. 2. Modified density of states.
states at the top of the lower band. One would expect similar
asymmetry in the Hall coefficient, and the resistance changes with
chemical additives.
In view of the uncertainties in relaxation times, polycrystalline
effects, etc., in current experiments we have not 'exploited this
model further. Indeed, one would not expect great precision from
the tight binding approximation. Nonetheless, the above model is
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in agreement with the more general group theoretical require
ments and is probably more representative of the actual band
structure near the zone corner than is Wallace's. Finally, it must
be noted' that as soon as the Fermi level is moved a few tenths of
a volt from the zone corner the system is essentially two dimen
sional and our modifications playa minor role. The authors would
like to acknowledge discussions of these topics with W. W. Tyler,
W. P. Eatherly, and members of the Knolls Atomic Power
Laboratory.
* Present address. General Electric Company. West Lynn. Massachusetts.
1 G. Hennig. J. Chern. Phys. 20. 1438 (1952); W. W. Tyler and A. C.
Wilson. Jr .• Phys. Rev. 89. 870 (1953).
, P. R. Wallace. Phys. Rev. 71. 622 (1947).
3 J. L. Carter. Ph.D. thesis. Cornell University. February. 1953. 'C. A. Coulson and R. Taylor. Proc. Phys. Soc. (London) A65. 815
(1952).
5 S. Mrozowski. J. Chern. Phys. 21. 492 (1953).
The Rate of Combination of Methyl Radicals
K. u. INGOLD.t I. H. S. HENDERSON.t AND F. P. LOSSING
Division of Pure Chemistry. National Research Council. Ottawa. Canada
(Received September 21. 1953)
FURTHER work on the combination of methyl radicals using
the techniques described previouslyl has shown that while
the contact time at 9 mm pressure of helium carrier gas was almost
correct, there was a considerable error in its measurement at
4.8 mm and to a lesser extent at 18.5 mm. An improved method
of measuring the contact time has shown that a factor of about
2 was involved over this pressure range, the longer contact time
corresponding to the lower pressure. The effect of pressure on the
rate of combination has therefore been re-determined over a
range from 3.4 mm to 15.0 mm of helium at 1000°C. Relative
rates are plotted in Fig. 1 against the helium pressure. It can be
seen that the pressure has a considerable influence on the rate.
Previously this effect was masked by the error in the contact time
measurements at the different pressures. The finite intercept in
the figure is due both to the first-order wall reaction which, at the
2·0 -/ ,·8 -
"6 -/ .. "4 -
:!: .. / ..
..J ,·2 -..
~ 0/ .. z ,·0 -c //0 .. .. z
0
<> 0·8 -.. /Q
=c
II: 0·6 -
0'4 r-
0·2 r-
0~~1 ___ ~1 __ ~1~1 ___ ~1 __ ~1~1 __ ~.
o 246 8 ~ ~ ~ ~
HEL'UM PRESSURE (MM I
FIG. 1. The rate of combination of methyl radicals. methyl concentrations used is negligible except at low pressures,
and to the third-body effects of the products themselves.
Since the pressure of the carrier gas is found to have an im
portant effect on the rate of combination, our results are now in
agreement with recent work on this subject.2 Moreover, since
kinetic theory predicts a negative temperature coefficient for the
combination rate in the region of pressure dependence! it is no
longer necessary to attribute this effect entirely to a variation in
the collision cross section with temperature.
The collision efficiencies reported earlierl are unaffected by this
error but refer, of course, only to the conditions stated, viz., 9 mm
of helium and the corresponding temperature. However, at con
stant temperature the rate is proportional to the helium pressure
at around 9 mm. Therefore, using the same nomenclature as
before, the values reported for the rate (i.e., the values of k,) are
really values of k,ka[MJ/k 2• In this expression the powers of
temperature involved in the collision numbers and the concentra
tion term cancel so,
In (rate)=In (constant)-(E l+Ea-E2)/RT.
El and E2 are, respectively, the activation energies of the methyl
methyl collision process and its reverse. Ea is the activation energy
of the helium-ethane* collision. For this reason it is not necessary,
when obtaining the over-all activation energy of the reaction, to
multiply the rates by a factor of 11 as was done previously. This
change lowers the activation energy (El+Ea-E2) to -1.5 kcal.
Since it is generally assumed that El and Ea are close to zero, the
negative temperature coefficient can be attributed to the in
increase of k2 with temperature. That is, the lifetime of the com
plex C2H 6* and therefore the over-all rate of reaction decrease with
increasing temperature.
A point of considerable interest is the relative deactivating
efficiency of various third bodies, i.e., the relative values of ka.
Unfortunately there are considerable experimental difficulties in
replacing helium by some other gas with larger molecules in which
a much greater efficiency of deactivation would be expected. We
are therefore indebted to Dr. R. E. Dodd for permission to use
his unpublished values of k2/k3 for acetone as the third body. By
using Gomer and Kistiakowsky's value' for kl' the value of k2/k.
for helium may be obtained from our previously published results.'
In this way a comparison can be made of the deactivating effi
ciencies of helium and acetone:
521°K
(~te
(~)A. 75.0XlO16 molecules/cc
4.0X10'6 6.4X 10'6 molecules/cc.
Since k2 is independent of the third body
(k3) He 0.059 0.085. (k3)Ac
That is, acetone is some 12-17 times as efficient as helium in re
moving excess energy from the activated complex C2Hs*.5
t National Research Council Postdoctorate Fellow .
1 K. U. Ingold and F. P. Lossing. J. Chern. Phys. 21.1135 (1953) . 'R. E. Dodd (private communication).
• S. W. Benson. J. Chern. Phys. 20. 1064 (1952).
• R. Gomer and B. G. Kistiakowsky. J. Chern. Phys. 19.85 (1951).
5 For a comparison of the efficiencies of third bodies in the iodine atom
recombination see K. E. Russell and J. Simons. Proc. Roy. Soc. (London)
A217. 271 (1953).
Erratum: The Use of Radioactive Alpha-Recoil in
the Study of Soluble Ionized Surface Layers
[J. Chern. Phys. 21. 1299-1300 (1953)J
GUNNAR ANIANSSON AND NAFTALI H. STE'GER
Division oj Physical Chemistry. Royal Institute oj Technology.
Stockholm. Sweden
10-3 CHNOa should be changed to read 10-aM HN0 3•
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1.1740413.pdf | Contributions of Vibrational Anharmonicity and RotationVibration Interaction to
Thermodynamic Functions
R. E. Pennington and K. A. Kobe
Citation: The Journal of Chemical Physics 22, 1442 (1954); doi: 10.1063/1.1740413
View online: http://dx.doi.org/10.1063/1.1740413
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/22/8?ver=pdfcov
Published by the AIP Publishing
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:461442 FUKUI, YONEZAWA, NAGATA, AND SHINGU
TABLE V. Comparison of predicted structures of addition
products with experimental results.
Conjugated molecules
butadiene
hexatriene
styrene
stilbene
anthracene
phenanthrene Predicted positions
1:2,1:4
1:2,1:4,1:6
a:",
a:(i
9:10
9:10
II Farmer. Laroria, Switz, and Thorpe, reference 12. Experimental results
1:2,1:4
1:2,1:6"
a:",
a:a'
9:10 9:10
is naturally to be preferred.) Then we can apply here
the frontier electron method analogously, using the
same classification into three types, viz., electrophilic
(E), nucleophilic (N), and radical (R).
In discussing the reactivity in polyene and some
aromatic molecules, seven cases§§§ of successive addi
tion are to be considered according to the type of re
agent, which are indicated in Fig. 10.
Taking hexatriene as an example, the pri laryattack
is predicted to occur at the terminal carbon atom, in
any case of E(l), N(l), and R(l). In the secondary attack,
§§§ Combining the three types of primary attack, viz., N(l), E(l),
and R(l>, with three types of secondary attack, viz., E(2), N(2),
and R(2), we have nine varieties of mode of addition. But the two
among these, E(1)-E(2) and N(1)-N(2) type additions, which are
not likely to happen, are left out of consideration.
THE JOURNAL OF CHEMICAL PHYSICS as one atomic 1(" orbital at the terminal carbon has dis
appeared as a result of the primary addition, the point
of attack is now controlled by the frontier electron
density of a conjugated system consisting of five carbon
atoms. From the results shown in Fig. 10, it can be
concluded that the structures of the addition products
are predicted to be 1: 2, 1: 4, or 1: 6, taking all the pos
sible cases of addition into consideration. Experimen
tally, addition product of bromine to hexatriene is
reported to be a mixture of 1: 2-and 1: 6-dibromide.I2
Similar calculations for all the possible modes of
addition are carried out as to butadiene, stilbene, sty
rene, anthracene, and phenanthrene. The results are in
a complete agreement with experiment, as is shown in
Table V·II II " It is possibly of importance in obtaining a knowledge
of the true feature of activated complexes to consider
the theoretical foundation of the fundamental postu
lates, which, however, is omitted in the present paper
and will be published elsewhere.
The authors are grateful to the Education Ministry
of the Japanese Government for a grant-in-aid.
12 Farmer, Laroria, Switz, and Thorpe, J. Chem. Soc. (London)
1927, 2937 (1927).
" " 11 The frontier electron method is also useful in the treat
ment of cationoid, anionoid, and radical polymerization, which
will be published elsewhere.
VOLUME 22, NUMBER 8 AUGUST, 1954
Contributions of Vibrational Anharmonicity and Rotation-Vibration Interaction
to Thermodynamic Functions
R. E. PENNINGTON* AND K. A. KOBE
Department of Chemical Engineering, University of Texas, Austin, Texas
(Received March 23, 1954)
Certain correction terms applying to the rigid-rotator harmonic-oscillator approximation for the thermo
dynamic functions have been worked out in a general form. Tables of the functions which appear in these
correction terms are presented. These results have been applied in the calculation of the thermodynamic
properties of nitrous oxide. A comparison of the present procedure and that of Mayer and Mayer for di
atomic molecules is given.
I. INTRODUCTION
MORE detailed knowledge of the spectra of poly
atomic molecules is gradually becoming avail
able. With the determination of the anharmonicity and
interaction constants of a molecule it becomes possible
to improve the statistically calculated thermodynamic
functions by taking these effects into account. Several
papersH have dealt with this problem in some detail.
* Present address: Bureau of Mines Petroleum Experiment
Station, Bartlesville, Oklahoma.
1 A. R. Gordon, J. Chern. Phys. 3, 259 (1935).
2 L. S. Kassel, Chern. Rev. 18, 277 (1936).
3 Stockmayer, Kavanagh, and Mickley, J. Chern. Phys. 12,
408 (1944). The calculations for determining these corrections are
rather lengthy. Therefore, approximations to a general
ized partition function and its derivatives have been
worked out, and tables of the functions which appear in
the correction terms have been compiled. These tables
were used to calculate the thermodynamic functions of
nitrous oxide at selected temperatures.
II. THE PARTITION FUNCTION
For the present purposes, it is assumed that the en
ergy levels of some molecule of interest may be repre
sented in the nomenclature of Herzberg4 in the following
4 G. Herzberg, Infrared and Raman Spectra of Polyatomic
Molecules (D. Van Nostrand Company, Inc., New York, 1945).
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:46CALCULATION OF THERMODYNAMIC FUNCTIONS 1443
manner:
T -Go= "2:, I'iVi+ "2:, XiiVi(Vi-1)
i i
+ "2:, XijViVj+ "2:, gii(li2-vi)+F.. (1)
i<i i
The term F. stands for the rotational levels and takes
different forms depending on the structure of the mole
cule: for linear molecules5
for spherical top molecules
Fv=B.J(J+ 1) (3)
for symmetric top molecules
Fv=B.J(J+1)+(A v-B.)K2 (4)
for asymmetric top molecules
In each of these representations of the rotational levels
the effective inertial quantities are used, i.e.,
B.=Bo- "2:, aiBvi
i (6)
and similarly for A. and C". In what follows, the form
B.= Bo(l-"2:, b.iVi),
i
etc., will be more convenient to employ. (7)
It is only in the case of linear molecules that a cen
trifugal distortion term has been included. Sufficient
spectroscopic data for the inclusion of such terms for the
other cases are not usually available.4 Wilson6 has pro
posed a method whereby the effects of centrifugal dis
tortion on the thermodynamic functions of nonlinear
molecules may be estimated. He has shown these effects
to be significant for such light hydrogen-containing
molecules as water and ammonia. It is to be noted also
that none of representations (1)-(5) takes into account
Coriolis splitting or Fermi resonance. The effect of
Coriolis splitting on the thermodynamic functions
should be very small.7 No generalized representation of
the energy levels which takes into account the effects
of Fermi resonance a priori is available. However,
except in the case of a close resonance in some of the
lower-lying levels, this effect should also produce a
small contribution to the thermodynamic functions.
These limitations should be kept in mind in the applica
tion of results based on Eqs. (1)-(5).
If Eq. (1) is used to represent the energy levels, a
"generalized" internal partition function may be
• Note that the term -B.l! has not been taken into the vibra
tional formula.
6 E. B. Wilson, ]. Chern. Phys. 4, 526 (1936).
7 E. B. Wilson, J. Chern. Phys. 7, 948 (1939). written as
Q=QoOQ.'QIQR(V,l)
Qoo= exp( -hcGo/kT)
Q.'= "2:, exp[ -"2:, UiV.+ "2:, Xi/1tiVi(Vi-1)
+ "2:, Xij(UiUj)fV.Vj (8)
i<i
Ql= "2:, exp[ -"2:, Giiui(ll-vi)].
I i
These expressions have been abbreviated by the short
hand
(9)
Rotational Sums
The form of QR(V,l) depends on the molecular struc
ture. For linear molecules
J=",
QR(v,l)=[exp(B.l2)] "2:, (2J+1)
J=l
Xexp[ -i3.J(J+ 1)][1-oJ(J+ 1)], (10)
with
i3.= (1-"2:, biVi)Bohc/kT= (1-"2:, biVi)i30
i i
o=D/Bo• (11)
By introducing the asymptotic Euler-Maclaurin ex
pansion first given by Mulholland8 and neglecting
second-order terms in v and I, this relation may be
reduced to
QR(v,l) = (1-"2:, b,Vi)-l(l +20/i30)QRo
i
u=symmetry number.
The expression for QRo will be satisfactory except for
large values of i30 (i.e., very low temperatures or an
extremely large value of Bo). The treatment of Mayer
and Mayer9 of the rotational partition function for
diatomic molecules is equally applicable to this case
and they give equations and tables for QRo for large i3o.
One further operation on QR(v,l) will make it more
useful when substituted back into Q. This consists of
the expansion,
(1-"2:, b.V.)-l-;:::j 1+ "2:, (bi+bnVi
i i
+:1: blvi(vi-1)+2 "2:, b,bjViVit (13)
i i<i
8 H. P. Mulholland, Proc. Cambridge Phil. Soc. 24, 280 (1928).
9 J. E. Mayer and M. G. Mayer, Statistical Mechanics (John
Wiley and Sons, Inc., New York, 1940).
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:461444 R. E. PENNINGTON AND K. A. KOBE
TABLE 1. Rotational constants in the correction terms. in which the 'i are the combinations of the rotation-
Linear Spherical vibration interaction constants given in Table I and
top 5~O only for linear molecules.
s= 2Dk/B o2hc s=O
Ti=bi+bi2 n=3b;/2+15bN8
Symmetrical top Asymmetrical top Vibrational Angular Momentum Sums
s=O s=O The quantum numbers li of the degenerate vibrations Ti= bi+a;/2+bi2+aib;/2 r;= (ai+bi+ci)/2
+3aN8 + (ai2+bi2+Ci2)/4 take on the values Vi, Vi-2, "', Vi-2Vi. The summa-+ (ai+bi+ ci)2/8 tion over the li represented by QI may be accomplished
in the following manner. For a particular li
and the neglect of purely second-order terms. Similar
L exp[ -GiiUiW-Vi)}:"'L [l-GiiuiW-vi)J expressions may be derived for the other cases.1•2
Actually, the leading term in the expansion of QRo for Ii Ii
these other cases, is, at room temperature and higher, L l=vi+l almost always a sufficient approximation. This is the Ii classical partition function for rotation and is given by
QRO=u-1[tr(kT /hc)3j ABCJ!, (14) L (liLvi) = (vi+1)(vi)(vi-1)/3 (16)
Ii
where for symmetrical top molecules C = B and for L exp[ -Giiui(li2-vi)J spherical top molecules C = A = B. A general expression li for QR(v,l) may therefore be taken as ""[Vi+ lJ[1-Giiuivi(vi-l)/3J
QR(V,l)"" (1+ L 'iVi) (1+ 25/{lo)QRO, (15) "" [Vi+ lJ exp[ -Giiuivi(vi-l)/3]. i
TABLE II. The functions nrp.
u ''I' ''I' ''I' ''I' ''I' ''I' ''I' ''I' ''I' 10'1' "'1'
0.20 4.5167 4.9834 10.0000 8.1600 9.8465 19.9874 2.0199 0.6033 0.9967 0.9033 1.7078
0.25 3.5208 3.9792 8.0000 6.1980 7.8120 15.9798 1.7604 0.6302 0.9948 0.8802 1.5087
0.30 2.8583 3.3084 6.6664 4.9022 6.4463 13.3058 1.5656 0.6575 0.9925 0.8575 1.3502
0.35 2.3862 2.8281 5.7138 3.9858 5.4619 11.3895 1.4117 0.6852 0.9898 0.8352 1.2197
0.40 2.0333 2.4669 4.9996 3.3076 4.7188 9.9522 1.2860 0.7133 0.9868 0.8133 1.1096
0.45 1.7596 2.1851 4.4437 2.7867 4.1344 8.8277 1.1804 0.7418 0.9833 0.7918 1.0151
0.50 1.5415 1.9589 3.9990 2.3762 3.6629 7.9248 1.0900 0.7708 0.9794 0.7708 0.9327
0.55 1.3638 1.7730 3.6350 2.0459 3.2738 7.1830 1.0114 0.8001 0.9752 0.7501 0.8603
0.60 1.2164 1.6176 3.3316 1.7756 2.9469 6.5620 0.9422 0.8298 0.9705 0.7298 0.7959
0.65 1.0923 1.4854 3.0748 1.5510 2.6677 6.0326 0.8806 0.8600 0.9655 0.7100 0.7382
0.70 0.9864 1.3716 2.8544 1.3623 2.4262 5.5749 0.8253 0.8905 0.9602 0.6905 0.6863
0.75 0.8953 1.2725 2.6633 1.2022 2.2155 5.1755 0.7753 0.9214 0.9544 0.6714 0.6393
0.80 0.8160 1.1854 2.4959 1.0653 2.0300 4.8235 0.7298 0.9528 0.9483 0.6528 0.5966
0.85 0.7465 1.1081 2.3481 0.9472 1.8651 4.5095 0.6882 0.9845 0.9419 0.6345 0.5576
0.90 0.6851 1.0390 2.2165 0.8449 1.7179 4.2281 0.6500 1.0166 0.9352 0.6166 0.5218
0.95 0.6306 0.9769 2.0986 0.7556 1.5854 3.9734 0.6147 1.0491 0.9281 0.5991 0.4890
1.00 0.5820 0.9207 1.9923 0.6774 1.4659 3.7420 0.5820 1.0820 0.9207 0.5820 0.4587
1.05 0.5383 0.8695 1.8959 0.6085 1.3573 3.5300 0.5516 1.1152 0.9130 0.5652 0.4307
1.10 0.4990 0.8227 1.8081 0.5477 1.2585 3.3353 0.5233 1.1489 0.9050 0.5489 0.4048
1.15 0.4634 0.7798 1.7277 0.4938 1.1682 3.1554 0.4969 1.1828 0.8967 0.5328 0.3807
1.20 0.4310 0.7401 1.6538 0.4458 1.0854 2.9884 0.4722 1.2172 0.8882 0.5172 0.3584
1.25 0.4016 0.7035 1.5856 0.4031 1.0093 2.8331 0.4490 1.2519 0.8794 0.5019 0.3376
1.30 0.3746 0.6695 1.5224 0.3649 0.9393 2.6880 0.4271 1.2870 0.8703 0.4870 0.3182
1.35 0.3500 0.6378 1.4637 0.3307 0.8746 2.5520 0.4066 1.3224 0.8610 0.4724 0.3001
1.40 0.3273 0.6082 1.4089 0.3000 0.8149 2.4244 0.3873 1.3582 0.8515 0.4582 0.2831
1.45 0.3065 0.5805 1.3577 0.2724 0.7596 2.3043 0.3690 1.3944 0.8418 0.4444 0.2673
1.50 0.2872 0.5546 1.3097 0.2475 0.7082 2.1909 0.3518 1.4308 0.8318 0.4308 0.2525
1.55 0.2694 0.5301 1.2645 0.2250 0.6606 2.0838 0.3354 1.4676 0.8217 0.4176 0.2386
1.60 0.2530 0.5071 1.2220 0.2048 0.6163 1.9823 0.3200 1.5048 0.8114 0.4048 0.2255
1.65 0.2377 0.4854 1.1817 0.1864 0.5751 1.8861 0.3053 1.5422 0.8010 0.3922 0.2133
1.70 0.2235 0.4649 1.1437 0.1699 0.5368 1.7948 0.2914 1.5800 0.7904 0.3800 0.2017
1.75 0.2103 0.4455 1.1075 0.1548 0.5010 1.7079 0.2782 1.6181 0.7796 0.3681 0.1909
1.80 0.1980 0.4270 1.0731 0.1412 0.4677 1.6254 0.2657 1.6564 0.7687 0.3565 0.1807
1.85 0.1866 0.4096 1.0404 0.1288 0.4366 1.5467 0.2538 1.6952 0.7577 0.3452 0.1711
1.90 0.1759 0.3929 1.0092 0.1175 0.4076 1.4718 0.2424 1.7342 0.7466 0.3342 0.1620
1.95 0.1659 0.3771 0.9793 0.1073 0.3806 1.4004 0.2316 1.7734 0.7354 0.3234 0.1535
2.00 0.1565 0.3620 0.9507 0.0980 0.3553 1.3324 0.2214 1.8130 0.7241 0.3130 0.1454
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:46CALCULATION OF THERMODYNAMIC FUNCTIONS 1445
TABLE II-Continued.
u 1", '", '", '", '", '", '", '", '", 10", "",
U 1", '", '", '", '", '", '", '", '", 10", "",
2.10 0.1395 0.3340 0.8970 0.0818 0.3096 1.2052 0.2022 1.8930 0.7013 0.2930 0.1306
2.20 0.1246 0.3083 0.8473 0.0683 0.2698 1.0894 0.1848 1.9741 0.6783 0.2741 0.1174
2.30 0.1114 0.2848 0.8012 0.0571 0.2349 0.9837 0.1690 2.0563 0.6552 0.2563 0.1056
2.40 0.0998 0.2633 0.7581 0.0478 0.2044 0.8874 0.1546 2.1394 0.6320 0.2394 0.0951
2.50 0.0894 0.2436 0.7178 0.0400 0.1778 0.7995 0.1414 2.2236 0.6089 0.2236 0.0856
2.60 0.0802 0.2253 0.6799 0.0335 0.1546 0.7194 0.1294 2.3086 0.5859 0.2086 0.0772
2.70 0.0720 0.2085 0.6442 0.0280 0.1342 0.6464 0.1184 2.3945 0.5631 0.1945 0.0696
2.80 0.0647 0.1930 0.6104 0.0235 0.1165 0.5801 0.1083 2.4813 0.5405 0.1813 0.0627
2.90 0.0582 0.1787 0.5785 0.0197 0.1010 0.5198 0.0992 2.5688 0.5182 0.1689 0.0566
3.00 0.0524 0.1654 0.5483 0.0165 0.0875 0.4651 0.0908 2.6572 0.4963 0.1572 0.0511
3.10 0.0472 0.1531 0.5195 0.0138 0.0758 0.4155 0.0831 2.7462 0.4747 0.1462 0.0461
3.20 0.0425 0.1418 0.4922 0.0116 0.0656 0.3707 0.0760 2.8360 0.4536 0.1360 0.0416
3.30 0.0383 0.1312 0.4662 0.0097 0.0567 0.3303 0.0696 2.9264 0.4330 0.1264 0.0376
3.40 0.0345 0.1214 0.4414 0.0081 0.0489 0.2938 0.0637 3.0174 0.4129 0.1174 0.0339
3.50 0.0311 0.1124 0.4178 0.0068 0.0422 0.2610 0.0582 3.1090 0.3933 0.1090 0.0307
3.60 0.0281 0.1040 0.3953 0.0057 0.0364 0.2315 0.0533 3.2011 0.37.43 0.1011 0.0277
3.70 0.0254 0.0962 0.3739 0.0048 0.0313 0.2050 0.0488 3.2938 0.3558 0.0938 0.0250
3.80 0.0229 0.0889 0.3535 0.0040 0.0270 0.1813 0.0446 3.3870 0.3380 0.0870 0.0226
3.90 0.0207 0.0822 0.3340 0.0033 0.0232 0.1601 0.0408 3.4806 0.3207 0.0806 0.0204
4.00 0.0187 0.0760 0.3154 0.0028 0.0199 0.1412 0.0373 3.5746 0.3041 0.0746 0.0185
4.20 0.0152 0.0649 0.2809 0.0019 0.0146 0.1094 0.0312 3.76 0.2726 0.0639 0.0151
4.40 0.0124 0.0554 0.2497 0.0014 0.0108 0.0844 0.0261 3.95 0.2436 0.0547 0.0124
4.60 0.0102 0.0472 0.2214 0.0009 0.0079 0.0647 0.0218 4.15 0.2170 0.0467 0.0101
4.80 0.0083 0.0402 0.1960 0.0007 0.0057 0.0494 0.0182 4.34 0.1928 0.0398 0.0083
5.00 0.0068 0.0341 0.1731 0.0005 0.0042 0.0375 0.0152 4.53 0.1707 0.0339 0.0068
5.20 0.0055 0.0290 0.1525 0.0003 0.0030 0.0284 0.0126 4.73 0.1508 0.0288 0.0055
5.40 0.0045 0.0246 0.1341 0.0002 0.0022 0.0214 0.0105 4.92 0.1329 0.0245 0.0045
5.60 0.0037 0.0209 0.1177 0.0002 0.0016 0.0161 0.0088 5.12 0.1168 0.Q208 0.0037
5.80 0.0030 0.0177 0.1031 0.0001 0.0011 0.0120 0.0073 5.32 0.1025 0.0176 0.0030
6.00 0.0025 0.0149 0.0901 0.0001 0.0008 0.0090 0.0061 5.51 0.0897 0.0149 0.0025
6.50 0.0015 0.0098 0.0639 0.0000 0.0004 0.0042 0.0038 6.01 0.0637 0.0098 0.0015
7.00 0.0009 0.0064 0.0448 0.0002 0.0020 0.0024 6.51 0.0448 0.0064 0.0009
7.50 0.0006 0.0042 0.0312 0.0001 0.0009 0.0015 7.00 0.0312 0.0042 0.0006
8.00 0.0003 0.0027 0.0215 0.0000 0.0003 0.0010 7.50 0.0215 0.0027 0.0003
8.50 0.0002 0.0017 0.0147 0.0000 0.0006 8.10 0.0147 0.0017 0.0002
9.00 0.0001 0.0011 0.0100 0.0004 8.50 0.0100 0.0011 0.0001
9.50 0.0001 0.0007 0.0068 0.0002 9.00 0.0068 0.0007 0.0001
10.00 0.0000 0.0004 0.0045 0.0001 9.50 0.0045 0.0004 0.0000
With this result Ql may be put in a convenient form To carry out the summation of Qv analytically it is
for substitution into the partition function. necessary to expand the small exponentials in Xii and
C;+di-l) Xij. Use is then made of the relations2
Ql=:q: exp[ -GiiUiVi(Vi-l)/3]. (17) L exp( -uv) = (1-e-u)-1
1. Vi v
Vibrational Sums L v exp( -uv) = e-u(1-e-u)-2 (19)
v
Application of the results for QR(v,l) and Ql will re-( -l)ndn
duce the over all partition function to a series of sums L vn exp( -uv) = L exp( -uv).
dun
over the vibrational quantum numbers only. This
form is If second-order terms are neglected, the algebra is not
Q= QooQRO(l + 20/i3o)Qv too difficult and the end result is that
[ C+d.-1)] InQ= InQoo+ InQRo+ InQvo+ InQc
Qv= L:n " [1+ L riz1i] InQvo= -L di In(1-e-Ui) (20)
v ~ Vi t i
Xexp[ -LUiZ1i+ L XiiUiVi(Vi-1) (18) InQc= 20/130+ L d,1' i1CPi
i
+ L Xij(UiUj)!ViVj] +t L di(di+ 1)Xii4cpi+ L didjXi/ cp/ CPj'
i<i
i<i The expression for InQvo is the harmonic oscillator re-
Xii=XJ-G;;/3= -(Xii+gi;/3)/Vi. suit. The small corrections are given in terms of the
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:461446 R. E. PENNINGTON AND K. A. KOBE
functions nrpi, which are tabulated in Eqs. (24).
Stockmayer, Kavanagh, and Mickler obtained the
last two terms in InQc without, however, taking into
account the li splitting of the degenerate levels (gii~O)
and the resulting adjustment of the "effective" Xii.
This effect may become significant at higher tempera
tures.
III. THE THERMODYNAMIC FUNCTIONS
In terms of Eq. (20) for InQ the thermodynamic func
tions may be obtained as the sum of three parts: (a)
a rigid-rotator contribution calculated with inertial
quantities for the vibrationless ground state; (b) a
harmonic oscillator contribution computed from ob
served fundamentals; and (c) a number of small cor
rections. These last are evaluated here. The corrections
are given in terms of the functions nrp and constants
characteristic of the molecule. The contributions to the
thermodynamic functions are
-Fc!RT=sT+ r:. d,1"/rpi+t r:. di(di+1)Xii4rp;
i i
+ r:. did1Xi/ rp/ rpj (21)
i<i
Ec!RT=sT+ r:. d,1"i2rpi+t r:. di(di+ 1)Xilrp;
i i
+ r:. didJXi/ rp/ rpj(8rpi+ 8rpj) (22)
i<i
Cc/R= 2sT + r:. d,1"i3rpi
i
+t r:. di(di+l)Xii6rpi
i
+ r:. d,-djXi/ rp/ rpj
i<i
For these relations the nrp and vibrational constants are
defined as follows:
lrp= (eu_1)-1
2\0= ueu(eu_1)-2
3\0= u2eu(eu+ 1) (eu_1)-3
4rp= 2u(e"-1)-2
5rp= 2u(2ue"-e"+ 1) (e"-1)-3
6\0= 4u2eu(2ueu-2eu+u+ 2) (eu_1)-4
7 rp= ui(e"-l)-l
8rp= H21teU-e"+ 1) (eu_1)-1
9\0= u2eu(eu_1)-2= (el R)HO
lO\O=u(eu-1)-1= (EI RT)HO
ll\O= -In(1-e-u)= (-FIRT)Ho
ui=hcllilkT
Xii= (-xii-gii/3)IIIi
Xij= -Xii/(lli llj)l.
The constants in the centrifugal stretching and rota-tion-vibration terms depend on the structure of the
molecule, the various forms are given in Table I.
The evaluation of the nrp for several frequencies at a
number of temperatures is a lengthy computation. To
facilitate the calculation of the correction terms a
compilation of the nrp for a wide range of the argument
u, is given in Table II. The values of the argument are
those used by Mayer and Mayer9 in their tabulation of
the harmonic oscillator functions. The last three nrp are,
respectively, the heat capacity, the internal energy, and
the free energy, functions for a harmonic oscillator.
These entries agree closely with the tables of Mayer and
Mayer and exactly with those of Johnston, Savedoff,
and Belzer.lO
IV. NITROUS OXIDE
The original analysis of the spectrum of nitrous oxide
for the anharmonicity coefficients was made by Plyler
and Barkerll and corrected by Barker,l2 A number of
calculations of various thermodynamic functions of
nitrous oxide have been published.13 Some of these
were based on the earlier-publishedll slightly erroneous
representation of the energy levels, while in others the
harmonic oscillator approximation was used. Kobe and
Pennington used the corrected assignment and a satis
factory approximation to the partition function. How
ever, the numerical differentiation used by them to
calculate the other functions produced serious error.
Thermodynamic functions have been calculated for
nitrous oxide using the tables given here and the data
TABLE III. Molal thermodynamic functions of nitrous oxide
in the ideal-gas state.
-(F<I-EoO)
T T HO-EoO so Cpo C. OK cal deg-1 cal cal deg-1 cal deg-' cal deg-1
273.16 44.211 2062.2 51.760 8.952 0.004
298.16 44.876 2290.9 52.556 9.232 0.005
300 44.925 2307.0 52.615 9.253 0.005
400 47.208 3281.7 55.412 10.207 0.011
500 49.091 4341.8 57.775 10.965 0.018
600 50.713 5470.6 59.831 11.590 0.030
700 52.149 6656.5 61.658 12.110 0.043
800 53.442 7889.6 63.304 12.542 0.058
900 54.622 9162.5 64.803 12.903 0.073
1000 55.710 10469 66.179 13.206 0.092
1100 56.721 11802 67.450 13.458 0.108
1200 57.665 13158 68.630 13.671 0.125
1300 58.551 14535 69.732 13.854 0.142
1400 59.386 15929 70.764 14.009 0.159
1500 60.177 17336 71.734 14.142 0.175
10 Johnston, Savedoff, and Belzer, Contributions to the Thermo
dynamic Functions by a Planck-Einstein Oscillator in One Degree
of Freedom (U. S. Government Printing Office, Washington, 1949).
11 E. K. Plyler and E. F. Barker, Phys. Rev. 38, 1827 (1931).
12 E. F. Barker, Phys. Rev. 41, 369 (1932).
13 W. H. Rodebush, Phys. Rev. 40, 113 (1932); R. M. Badger
and S. C. Woo, J. Am. Chern. Soc. 54, 3523 (1932); L. S. Kassel,
ibid. 56, 1838 (1934); R. W. Blue and W. F. Giaque, ibid. 57,
991 (1935); A. R. Gordon, J. Chern. Phys. 3, 259 (1935); E. Justi,
Gebiete Ingenieurw. A5, 134 (1934); K. A. Kobe and R. E.
Pennington, Petroleum Refiner 29, No.7, 129 (1950).
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:46CALCULATION OF THERMODYNAMIC FUNCTIONS 1447
of Herzberg and Herzberg.l4 These data included the
rotation-vibration interaction constants which had not
been available before. Since it is as yet not possible to
eliminate completely the effects of Fermi resonance
from the vibrational constants, the assignment of
Herzberg and Herzberg which best reproduced the
lower-lying levels was chosen for use in these latest
calculations. The resulting thermodynamic data are
presented in Table III. The correction terms exert their
greatest effect on the heat capacity, and these contri
butions are included in the tabulation.
V. DISCUSSION
The correction terms in Eg. (20) are to the first order
only. In the derivation of these results an effort was
made to carryall second-and some third-order terms.
The number of such terms is of course much greater,
and the functions involved are more complicated. After
taking a second derivative to obtain the contributions
to the heat capacity, the completely general result
becomes prohibitively complicated. An estimate of the
second-order contributions to the heat capacity of
water at 15000K indicated a value of about 0.1 percent
of the total Cpo. The water molecule was chosen specifi
cally for its very large rotation-vibration interaction.
In the case of nitrous oxide, with a fairly low-frequency
degenerate fundamental (588 cm-I) , the second-order
contributions to the heat capacity at lOOOoK are some
what less than 0.1 percent though increasing rapidly.
Since at such high temperatures the extrapolation of the
anharmonicity and interaction coefficients (to the now
important higher vibration levels) usually becomes
somewhat uncertain and the effects of Fermi resonance
become more pronounced, an error of 0.1 percent in the
most sensitive function does not seem too serious.
For the case of diatomic molecules the results given
here become particularly simple. The expression for
InQc reduces to exactly the same function as was ob
tained for this case by Mayer and Mayer.9 They,
however, give series expansions for the corrections to
the thermodynamic functions. These expansions are
proposed for use at "moderately small" values of u only.
A comparison of the correction terms for the heat
capacities of two molecules, O2 and HBr, is given in
Table IV. As should be expected, the expansion fails
badly at U= 5.0. It is also slightly smaller than the
14 G. Herzberg and L. Herzberg, J. Chern. Phys. 18, 1551
(1950). TABLE IV. Correction terms for the heat capacities of diatomic
molecules cal deg-1 mole-I.
0, HBr
u T. OK P & K' M &Mh T. OK P &K' M&Mb
10.0 223 0.004 367 0.010
5.0 446 0.010 -0.048 733 0.032 -0.068
3.0 744 0.030 +0.014 1232 0.080 +0.046
2.0 1114 0.056 0.050 1834 0.150 0.136
1.5 1489 0.084 0.080 2450 0.215 0.207
1.0 2228 0.132 0.128 3667 0.340 0.330
0.5 4456 0.271 0.267 7334 0.697 0.679
• This method.
b Mayer and Mayer (see reference 9).
function given here at lower values of u. The tempera
tures corresponding to these values of u are included in
the table.
It is to be expected that deviations from the repre
sentation of energy levels, equation (1), used here will
arise. In the case of HeN it has been found necessary
to add a cubic term, Y333va(va2-1), to obtain a satis
factory fit to the observed spectrum.I6 The inclusion
of such terms in InQc is not difficult. When expanded in
Qv this particular term would appear as
r: YaVa(va2-1) exp( -uava) = 6Ya1rp32rp3(1-e-ua)-r, (24)
1l
where Ya is yaaa/Pa, and the relations in Eg. (19) have
been applied. If the harmonic oscillator term, (t-e-u)-l,
is factored out, the resulting addition to InQc is just the
quantity 6Ylrpa2rpa. This function and its derivatives
may be evaluated conveniently from information in
Table II.
In some cases it becomes necessary or desirable to
make an empirical adjustment for anharmonicity in
very complicated molecules. Expressions which have
been used for fitting anharmonicity as determined by
comparison with calorimetric data are16
Cc=!Z6rp
Ec/T=!Z5rp
-Fc/T=!Z4rp. (25)
In these expressions (Z) serves as an adjustable de
generacy-anharmonicity constant and the arbitrary
frequency for which the nrp are evaluated provides an
additional adjustable parameter. The information in
Table II is useful in such calculations.
16 E. Lindholm, Z. Physik 108, 454 (1938).
16 McCullough, Finke, Hubbard, Good, Pennington, Messerly,
and Waddington, J. Am. Chern. Soc. (to be published).
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132.177.236.98 On: Mon, 24 Nov 2014 23:01:46 |
1.1770904.pdf | Mixing Preamplifier
F. J. Davis and P. W. Reinhardt
Citation: Review of Scientific Instruments 25, 1024 (1954); doi: 10.1063/1.1770904
View online: http://dx.doi.org/10.1063/1.1770904
View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/25/10?ver=pdfcov
Published by the AIP Publishing
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Protection of fast and sensitive preamplifiers
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HydrophonePreamplifier Optimization
J. Acoust. Soc. Am. 39, 1222 (1966); 10.1121/1.1942708
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: 155.33.120.209 On: Wed, 03 Dec 2014 14:42:321024 LETTERS TO THE EDITOR
two metals plates. Pressure can be applied by tightening the bolts
shown in the figure. We have had no trouble separating the
peUicles even two weeks later. Of course, the surfaces should be
dry and smooth during assembly. We bevel all edges with a scraper
to reduce possible air gaps in case the edges are raised or have
adhesive on them. It is helpful to have the wire markers placed
uniformly over the surface of each emulsion. We do this using
a metal plate jig with shallow holes drilled at l-cm intervals. Using
tweezers each cup-like hole is loaded with a wire marker. Then
in the dark the pellicle is placed over the jig and both turned
upside down so that the wire markers drop down on the pellicle
at the predetermined positions.
Except where stated the processing is the same as for plates of
half the pellicle thickness. Throughout the processing the peIlicles
lie in trays lined with smooth Teflon sheet. Except for the warm
development stage, all processing is at SoC where the pellicles are
quite rigid and may be handled. They should be turned over and
shifted around in the tray every few minutes at the beginning
of a new stage to assure uniform penetration from both surfaces.
During the long fixation and washing they should be turned and
shifted around every few hours. We use the same solutions and
processing times as recommended by Stiller, Shapiro, and O'Dell.2
Both surfaces should be scrubbed with wet chamois at the finish
of the stop bath. For emulsions thicker than 600 JJ. we recommend
full strength fixing solution from two to three clearing times.
Although this reduces grain size somewhat, it improves clarity.
We have processed pellicles 1000 JJ. to 2000 JJ. thick many times
using the above procedure, and have never found reduced grain
density at the surfaces.
When the washing has been completed, about 1 mm around all
the edges should be removed using a razor blade. This is not
necessary, but will reduce warping due to shrinkage in alcohol
drying. Water is displaced by ethyl alcohol in five stages where
the alcohol concentration is increased 20 percent each time. We
use about four hours per stage at about SoC. Each stage contains
6 percent glycerin. It is important to use absolute alcohol for the
final stage. Then the pellicles should be lightly pressed between
glossy surfaced cardboard until completely dry. The lateral dimen
sions will be a few percent less than original. This shrinkage can be
measured by putting marks at a known distance on the emulsion
before processing. We roll a fine-toothed wheel across the emulsion
which impresses a fine scale by pressure fogging.3 Due to the
6 percent glycerin the finished peIlicles are about as flexible as
polyethylene. They may temporarily be mounted on glass for
convenience. We usually fasten the corners down with Duco
cement and use ShilIaber oil between the glass and emulsion.
The cement can be dissolved with acetone if it is desired to have
the other surface on top.
Tracing tracks is easier when there is no intervening glass be
tween the two surfaces of in terest. This is one of the reasons for
prefering temporary to permanent mounting. The two surfaces of
interest are pressed emulsion to emulsion with Shillaber oil
between. This gives the effect of one continuous emulsion at the
interface.
The procedure for tracing a given light track is as follows.
Under low magnification, line up the nearest wire marks. This
gives alignment within 30 microns. Then under higher magnifica
tion line up a nearby heavy track which traverses the two surfaces.
This usually gives alignment better than 10 microns. The small
misalignment is due to air gaps of the order of 10 microns between
the peIlicles during exposure. In the case of a light exposure where
heavy tracks are rare, it is helpful to supply artificially produced
marker tracks. We have exposed emulsion chambers perpendicular
to protons of the proper range and intensity from the Chicago
cyclotron for this purpose. In traveling from the wire mark to
the track in question, one should stop about every millimeter to
touch up the alignment using a heavy track. The extension of the
light track in question is then narrowed down to a region about
10 microns square. In addition the extension of the light track
should have both the same azimuth and dip angles. These criteria are severe enough to rule out almost any ambiguites due to
background.
The author wishes to acknowledge Professor Enrico Fermi,
Dr. A. H. Rosenfeld, and Mr. Elliot Silverstein for the contribu
tions they have made in helping solve many of the problems which
were involved.
1 La!, Pal, and Peters, Proc. Indian Acad. Sci. 38, 277 (1953).
2 Stiller, Shapiro, and O'Dell, Rev. Sci. Instr. 25, 340 (1954).
3 Jay Orear, Rev. Sci. Instr. 25, 875 (1954).
Mixing Preamplifier
F. J. DAVIS AND P. W. REINHARDT
Health Physics Division, Oak Ridge National Laboratory,
Oak Ridge, Tennessee
(0 riginally received April 26, 1954; revised version received August 2, 1954)
IT is often desirable to mix signals from a number of photo
multiplier tubes with a minimum of signal attentuation. When
the signal outputs from two or more photomultipliers are paralleled
together, the pulse amplitude is attenuated due to the parallel
output capacitance. Under optimum conditions, i.e., where the
capacity of the connecting system is kept to a minimum, the
attenuation would be proportional to n, where n is the number
of photomultipliers in parallel.
In Fig. 1 is shown a multichannel preamplifier circuit for
paralleling a number of phototubes with a minimum signal loss.
".
FIG. 1.
The circuit is shown for only three separate inputs; more may be
used. When a pulse is applied to anyone of the preamplifier tubes
the diode in the grid circuit of that particular tube will act as a
load resistor. The diodes in the other tube grid circuits then act
as a low impedance to the grids allowing the cathodes to foJIow
the pulse with a minimum bucking action. If grid resistors are
used in Fig. 1 instead of diodes, a negative pulse from one of the
cathodes suffers degeneration from the other tubes proportional
to the number of channels, due to the large RC of their grid circuits.
The use of this circuit is of particular advantage where it is
desirable to mix signals from widely separated detectors.
If the detectors are close together and cathode followers are
not needed, the simplified circuit shown in Fig. 2 may be used.
TO 5819~
'N3~ IN3;,:J <, 50"'''' 'N34AT I ElOUTPUT Cz .01 . , I •
R. 220r
". 3.3K
FIG. 2.
In using these circuits several limitations should be taken into
consideration. With small signals, the nonlinearity due to the
nonlinear diode resistance mayor may not be below the discrim
ination level depending upon the operating conditions. Also, care
should be taken to match the counter capacitances, and diode
resistances to prevent a large signal from one cathode follower
from blocking the others.
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1.1722480.pdf | Reverse Current and Carrier Lifetime as a Function of Temperature in Silicon
Junction Diodes
E. M. Pell and G. M. Roe
Citation: Journal of Applied Physics 27, 768 (1956); doi: 10.1063/1.1722480
View online: http://dx.doi.org/10.1063/1.1722480
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/27/7?ver=pdfcov
Published by the AIP Publishing
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Reverse Current and Carrier Lifetime as a Function of Temperature in Germanium Junction Diodes
J. Appl. Phys. 26, 658 (1955); 10.1063/1.1722067
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded
to ] IP: 194.47.65.106 On: Fri, 17 Oct 2014 08:47:07JOURNAL OF APPLIED PHYSICS VOLUME 27, NUMBER 7 JULY, 1956
Reverse Current and Carrier Lifetime as a Function of Temperature
in Silicon Junction Diodes
E. M. PELL AND G. M. ROE
General Electric Research Laboratory, Schenectady, New York
(Received February 11, 1956)
Earlier measurements of the reverse current and carrier lifetime in germanium have been extended to a
series of silicon grown iunction diodes, with measurements as a function of temperature between -190°C
and 200°C. The lifetime reaches a plateau at low temperatures and can be explained in terms of the Hall
Shockley-Read recombination theory. The slope of logir vs liT, the magnitude of ir, and the slope of ir vs V
suggests that charge generation from centers about 0.5 ev deep is responsible for most of the reverse current
in these samples up to temperatures well above room temperature.
1. INTRODUCTION
THIS report is an extension of earlier work with ger
manium junctions. 1 The results and conclusions
reported in the earlier paper are supported by the
present study of silicon junctions. This latter study is
perhaps of greater interest because it suggests that the
charge generation mechanism which was important in
germanium junctions at low temperatures may often
be responsible for the reverse current of silicon diodes
at room temperature and above. Such a model would
explain the observed high reverse currents and poor
saturation characteristics of many silicon junctions.
The possibility that charge generation is important
has been suggested by earlier authors.2 The present
work, we feel, confirms their suspicions and indicates
further that the generation centers lie near the center
of the forbidden band.3
The spread of data and number of features which are
not well understood is unfortunately greater in these
experiments, perhaps because of the present immature
state of the silicon art. Because of the check afforded
by the general similarity to the behavior of germanium,
and because of the apparent importance of the sug
gested mechanism at room temperature, it is neverthe
less felt that publication is warranted.
II. EXPERIMENTAL TECHNIQUES
These followed the methods outlined in the previous
paper, with a few changes necessitated by the special
properties of silicon:
(1) The low reverse currents of silicon diodes necessi
tated greater current sensitivity in measuring i vs V.
1 E. M. Pell, J. Appl. Phys. 26, 658 (1955).
2 K. G. McKay and K. B. McMee, Phys. Rev. 91, 1079 (1953) ;
W. Shockley and W. T. Read, Jr., Phys. Rev. 87, 835 (1952).
3 Subsequent to writing this paper, the work of H. Kleinknecht
and K. Seiler [Z. Physik 139, 599 (1954)J has been called to my
attention. This work, which was roughly concurrent with that on
germanium described in reference 1, precedes the present work on
silicon and reaches essentially the same conclusions from similar
evidence. The present paper can be considered independent evi
dence for this conclusion, and in addition it contributes the
evidence of simultaneous lifetime vs temperature measurements
which permit a better comparison of theoretical and experimental
magnitudes of the reverse current. To achieve this, the breaker amplifier was replaced
with a vibrating-reed electrometer, and certain circuit
improvements were made.
(2) The reported heat-treatability of silicon4 dis
couraged the general use of alloy contacts or alloy
junctions. All but a few of the junctions studied were
produced during the growth of the ingot. For ohmic
contacts, nickel plating (in a reducing solution) was
used. The electrical properties of these contacts were
not perfect, particularly at low temperatures; but they
were sufficiently good for our purpose. The mechanical
properties of the nickel plating were outstanding and
were advantageous in obtaining a good thermal con
tact to the cryostat (the nickel-plated Si was soldered
to the cryostat with low-melting solder).
Both HF-HNOs etch and electrolytic etch in hot
concentrated NaOH were used. On each sample, either
and/or both were used until no further improvement
in the diode characteristic could be obtained.
III. RESULTS AND DISCUSSION
The results to be described are characteristic of
every single-crystal silicon grown junction diode ob
served and do not represent selected units. (Reverse
current vs temperature data and measured vs calcu
lated magnitudes of reverse current are presented for
all units; lifetime vs temperature data is presented for
one unit and described for the others; reverse current
vs voltage at low temperature is presented for one unit.)
This is important because we know of no way to
eliminate the possibility of surface effects; the con
sistency of our results, which can be appreciated only
if every pertinent result is presented, indicates that if
a surface effect is responsible, it is a reproducible
surface effect. Beyond this, the best evidence that the
observations stemmed from bulk phenomena is their
excellent agreement with results predicted by present
theory of bulk properties.
The silicon diodes were grown by two independent
sources from silicon purified by two different methods.
A few aluminum dot diodes from a third source were ex
amined; their geometry prevented quantitative checks
• C. S. Fuller et al., Phys. Rev. 96, 833(A) (1954).
768
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of reverse current vs bulk properties, but qualitatively
(ir vs l/T; ir vs V) they behaved like the other units.
It should be noted that we discuss only the reverse
characteristic; consideration of the model will show
that for forward voltages, charge generation will be
negligible [in Eq. (7) of reference 1, n«nl and P»Pl
for forward bias].
Lifetime vs 1/ T
Figure 1 shows the behavior of lifetime, as measured
by pulse injection, vs l/T in the N region of sample
No.1 of Table J.6 Although this curve is not typical
(most units exhibiting slopes which were considerably
smaller) it is reproduced because it constitutes excellent
evidence that recombination centers can be as deep as
0.5 ev in silicon. The shallower and varying slopes of
other units indicate the presence of additional shallower
centers,6 but the remarks of reference 1 indicate that
TABLE 1. Ratio of observed magnitude of reverse current to
magnitude calculated using previously published intrinsic re
sistivity data [F. j. Morin and J. P. Maita, Phys. Rev. 96, 28
(1954) ] for estimate of NvNe expc/k. The "diffusion component" is
NvNe/[1.5X 1()33T3 exp(c/k)]' where N. and Ne are the partition
functions for the valence and the conduction bands, respectively, T
is the temperature in degrees Kelvin, k is Boltzmann's constant,
and c is the temperature coefficient of the band gap, defined
by EO= Eo-cT. The "charge generation" component is N.Ne/
[1.5X1033Pexp(c/k)J'. Where the symbol ~ appears, it indi
cates that the lifetime data yielded only a lower limit to the
plateau value for deep centers.!
Charge
generation Diffusion
component component Capacitance
predominant predominant (pfds/cm'
Sample (T=2500K) (T=444°K) at 0.4 v)
1. 16 ohm-em N-8.7
ohm-em P 0.94 2.4 0.014
2. 16 ohm-em N-8.7
ohm-em P ~5 0.18 0.014
3. 8.7 ohm-em P-l
ohm-em N ~1.7 2.7 0.0053
4. 220 ohm-em N-
10wpP 1.8 0.08 0.00086
5. 18 ohm-em N-
10wpP 1.4 0.17 0.0037
6. 135 ohm-em N-100
ohm-em P ~0.1 0.02 0.00044
7. 100 ohm-em P-17
ohm-em N ~0.7 0.02 0.00058
8.32 ohm-em N-85
ohm-cmP ~2.6 0.013 0.00065
9. 11 ohm-em N-Q.05
ohm-cmP ~1.1 0.54 0.0077
Ii The lifetime measurement in this sample is characteristic of
the N region because of the higher minority carrier concentration
on the N side of the junction in this sample. When the electron
hole mobility ratio is taken into account, one sees in Table I that
for five of the samples one minority carrier was in excess of the
other by a factor of one-hundred or greater, while for the other
samples the ratio was considerably less. The effect of the second
minority carrier has been neglected in Table I. If it is taken
into account, it will reduce the ratios recorded in the table,
especially for samples 3, 6, 7, and 8, but the effect is always less
than a factor of two and thus within the eJEPerimental error.
• G. Bemski [Phys. Rev. 100, 523 (1955)J presents evidence for
a center 0.2 ev from the valence band. 100
...
'I'SECl
10
1.2 o
o
o
2.4 o
o 0
o
0° f·O .••• v
00
2.6 00
o o
o
o
o
o o 0 00 0 0 00 0 00
... ·3I'SEC TO IOOO/T'12-
2.8 3.0 3.2 3.4
1000
iliiO -
FIG. 1. Log carrier lifetime vs 1000/T; sample No. 1.
the corresponding activation energies are not safely
interpreted as recombination center depths.
In these samples trapping7 was not a problem. The
best evidence for lack of trapping is the fact that the
sample lifetime exhibited a plateau at low temperatures.
This is in accordance with the Hall-Shockley-Read
recombination theory and is contrary to what one would
expect of traps.
By comparing spark-measured bulk lifetime vs l/P
(for a sample cut from the same ingot that contained
a grown junction) with the pulse-injection lifetime
measured near the junction, it has been observed that
the silicon near the junction is not necessarily typical
of the rest of the ingot. In the particular case observed,
lifetimes were identical at room temperature, but at
lower temperatures the lifetime near the junction
dropped where the spark lifetime did not. It is con
jectured that the change in growing conditions inci
dental to introducing the junction (e.g., back melting)
introduced shallow recombination centers.
In these silicon units, lifetimes as measured near the
beginning of the decay curve were often different from
lifetimes measured further out on the decay.9 Similar
differences have been observed in germanium units,l
but in silicon the difference is generally more striking,
ratios of as much as ten being not uncommon. In silicon,
it is therefore more important to know whether the
model suggested in reference 1 is theoretically sound,
7 J. R. Haynes and j. A. Hornbeck, Phys. Rev. 100,606 (1955).
8 Measured by R. 1.. Watters.
9 This was not always true; the sample shown in Fig. 1, for
example, gave identical results whether measured near the
beginning or in the tail of the decay curve. Where a difference
was observed, the detailed temperature dependences in the two
regions would generally differ, as well as the magnitudes of T;
but both regions would exhibit a plateau at low temperature
with a rising T at higher temperatures.
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namely that the beginning of the decay curve is de
scriptive of the material near the junction and the tail
is descriptive of the material further from the junction
(but still within a few diffusionlengths of the junction),
and further, that the density of recombination centers
is a function of this distance. In the appendix, we
examine this model and show that the intuitive inter
pretation is valid. We do not intend this to be a "proof"
of the model. It is merely the most reasonable explana
tion which has occurred to us, and we have accordingly
used it in interpreting the data.lo This interpretation
affects the ratios in Table I by a factor of 1 to 3 for the
diffusion component and 1 to 10 for the charge genera
tion component, for the various diodes listed. It has
no further effect upon our conclusions, and we do not
wish to overemphasize its importance.
Reverse Current us 1/ T
Figure 2 shows the reverse current (at 0.001 volt)
vs ljT characteristics of all units tested, normalized
arbitrarily to show the variation in slopes. Junction
areas varied from 0.31 to 1.65 cm2• All units show sub
stantially the same slope at low T, and this slope
corresponds to an activation energy of 0.6 ev. For
most units, this slope persists to temperatures well
above room temperature. An activation energy of
0.6 ev is about half of what one would expect on the
basis of the diffusion model of reverse current, but it is
just what one would expect from charge generation by
0.5-ev deep centers whose presence is suggested by
Fig. 1.l When a steeper slope was evident at high
temperatures, there was insufficient data to establish
the siope, but it is presumed that the diffusion mecha
nism is becoming predominant in this region. It can be
shown that the temperature at which the two mecha
nisms contribute equally is roughly proportional to the
depth of the generation centers and independent of the
band gap, which agrees with the observed higher tem
perature transition point in silicon (0.5-ev deep centers
in Si vs 0.3-ev deep centers in Ge).
Some samples exhibited a decrease in slope over a
finite temperature region above room temperature. In
one 220 ohm-em n-type sample (No.4) this drop,
beginning around 360oK, was studied in detail and was
traced to the relatively large decrease in the magnitude
of the displacement of the Fermi levels from the
center of the band in traversing this temperature
region (the data, as presented, have not been corrected
for this change). This displacement is of importance in
calculating the width of the charge generation region,l
and for the low reverse voltage used (0.001 v) one
would expect such a drop, and of just the magnitude
10 The possibility that transition region (or any other) capaci
tance could be responsible was in each case ruled out by varying
circuit resistance to see if there was any dependence of T on Re.
Nonuniform resistivity near the junction was ruled out because
this should not affect plateau lifetimes. Other possibilities are
discussed (and rejected) in reference 1. observed. With O.S-v reverse bias, the drop disappeared.
From the foregoing, one would expect that where a
drop was not observed, it was probably because the
true slope was increasing in this region because of the
increasing importance of diffusion current.
At sufficiently high temperatures, it is possible that
mobile charge could be large enough to affect the width
of the space-charge region,!l but our estimates indicate
that this effect should be negligible below 500°K.
Magnitude of Reverse Current
Table I tabulates the ratio of observed to theoretical
reverse currents both at high temperature (4400K)
where diffusion current would be expected to pre
dominate and at low temperature (2500K) where charge
generation would be expected to predominate.
At low temperatures, generally extending to above
room temperature, the current is very nearly what one
would predict on the basis of the charge generation
hypothesis. The closest agreement was obtained with
Sample No.1, which is also the sample in which the ,
00 0
• SAMPLE .,
·1'1 ° .. 2
0 .. 3
A\, A .. 4
D *~ • .. 6 • .. 7
f " .8
(J .. 9
log
iR tit
\
f ,.. ,
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.~
IOOO/T(·K I - •
FIG. 2. Log reverse current vs 1000/T at 0.001 v reverse bias.
Mter correcting to give reverse current at saturation average
slope at low temperature gives activation energy of 0.6 ~v.
11 W. Shockley, Bell System Tech. J. 28,335 (1949).
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to ] IP: 194.47.65.106 On: Fri, 17 Oct 2014 08:47:07REVERSE CURRENT IN SILICON JUNCTION DIODES 771
plateau lifetime for O.S-ev deep centers was known with
the greatest accuracy. (Sample No. 2 is the same
sample after an unknown room-temperature effect had
changed the lifetime data; agreement is much poorer in
this case. Subsequent aging resulted in the introduction
of traps.l2) Diffusion theory would have predicted
reverse currents about three decades lower at this
temperature.
In the high temperature region, where one would
have expected diffusion current to predominate, the
results were often anomalous, the observed current in
some samples being almost two decades too low. It will
be noted in Table I that there is a strong correlation
between the discrepancy and the capacitance per unit
area; in particular, the discrepancy is within experi
mental error for units with a narrow space charge
region, while it becomes quite serious for units with a
wide space charge region (gradient-type junctions).
This is not understood.
Shape of Reverse Current vs Voltage
Figure 3 shows the log of reverse current plotted
against voltage for Sample No.1 at room temperature.
The current is observed to be proportional to Vt in
the "saturation" region. Since this was a gradient-type
junction (the capacitance was observed to vary in
versely as Vi), such a result is in agreement with the
charge-generation model. The other diodes were checked
more qualitatively, but they behaved similarly.l3
IV. CONCLUSIONS
(1) Lifetime vs reciprocal temperature generally
shows a plateau at low temperatures, in both n-and
p-type silicon, and can be interpreted in terms of the
Hall-Shockley-Read theory of recombination.
(2) The deepest center observed in lifetime measure
ments was 0.5 ev deep.
(3) At low temperatures, and usually extending
above room temperature, the r~verse current could be
interpreted in terms of charge generation, as predicted
by the Hall-Shockley-Read theory for O.S-ev deep
centers in the space-charge region. This interpretation
is supported by the slope of logir vs l/T, the magnitude
of ir, and the slope of ir vs V; and all of these magnitudes
are well outside the range predicted by the usual
diffusion-current theory.
ACKNOWLEDGMENTS
The authors wish to express their appreciation to
David Locke for the measurements and to many
12 C. B. Collins [Bull. Am. Phys. Soc., Ser. II 1,49 (1956)J has
observed, by resistivity and Hall measurements, energy levels
0.45 ev from the valence band and 0.55 ev from the conduction
band in iron-doped Si. He also has observed slow diffusion effects
(days to weeks) at room temperatures in such samples.
13 Silicon diodes in which charge generation is not predominant
at room temperature are also observed. See, for example, J. T.
Law and P. S. Meigs, J. App!. Phys. 26, 1265 (1955). 101
O.(ll iT
.... QI 1.0
v" (VQLlSl-10 100
FIG. 3. Log reverse current vs log voltage at room
temperature; sample No.1.
members of the General Physics Department, particu
larly R. N. Hall and L. Apker, for valuable discussions.
They are indebted to C. B. Collins and F. H. Horn
for the samples used in this investigation. Valuable
comments by J. L. Moll, M. Tanenbaum, and P. Weiss
are also gratefully acknowledged.
APPENDIX
Following a pulse, the carrier density satisfies the
equation,14
an i)2n (np-n) -=D-+---
at ax2 r (1)
with the boundary conditions for t>O: n=O at the
junction X= 0, and n= np for X-H.o. For the initial
condition we require that n(x,O) be the steady-state
solution of Eq. (1) with the boundary conditions n=nn
at x=O, and n=np for x---+w.
In Eq. (1) we assume D to be constant, but the life
time, r, is postulated l to be a junction of x. Define
f(x) = 1/Dr(x)
and let R{ x,f(x)} be that solution of
(lPRNx2)- f(x)R=O
which satisfies the boundary conditions,
R=l at x=O,
R=O at x---+oo. (2)
Let N(x,p) be the Laplace transform of n(x,t). N may
be written in terms of the function R.
1
N(x,p) =-[np+ (n,,-np)R{x,j(x)}
p
-nnR{x, f(x)+p/D}]. (3)
14 W. Shockley, Electrons and Holes in Semiconductors (D. Van
Nostrand Company, Inc., New York, 1950), p. 313.
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The reverse current is proportional to
Set) = [~n(x,t)] . ax x=o (4)
Since n is the Laplace inverse of N, this may be written
1[ d S(t)=2-L (nn-np)-R{x,j(x)} p dx
-nn:xR{X, f(x)+p/D} 1=0· (5)
From the properties of the Laplace transform, it can
be shown that Set) has the following limiting forms:
For t small,
[ dRJ nn [dRJ S(t)-n -- ~---n--
p dx x=o-(nDt)! n dx x=o· (6)
For t large,
(7)
where Too=limT(x).
x---+oo
The experimental lifetimes are measuredl by com
paring the oscilloscope patterns of the current with a
set of curves computed from the solution of Eq. (1)
for the special case of constant T. This solution is or,
np nn nn
S (t) -'" for t small,
(DT)! (1rDt)t (DT)'; (8)
___ e-t/T
2 (1rDt3)t for t large.
A comparison of Eqs. (6) and (8) shows that for
portions of the trace near the beginning of the decay
curve
Tmeas.'"'-'(l/D){[_dR] }-2.
dx x=o (9)
The solutions of Eq. (2) have been examined for three
cases, with f(x) chosen to have a step-function varia
tion, a linear variation, or an exponential variation
with x. In each case, and indeed, quite generally,
Eq. (9) may be written
(10)
where the quantity E! is small provided that the frac
tional change in f is small within a distance (Dr)!
from the junction. Under these conditions the apparent
lifetime near the beginning of the decay curve is a
measure of the lifetime at the junction.
For portions of the trace near the end of the decay
curve, a comparison of Eqs. (7) and (8) yields the
relation
Tmeas.=Too· (11)
However, in certain cases this result may be valid only
at points beyond the observable "end" of the decay
curve. The apparent lifetime near the end of the decay
curve is more properly interpreted as a measure of the
lifetime at a distance of several diffusion lengths, (Dr)!,
from the junction.
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1.1715374.pdf | Regenerative Beam Extraction on the Chicago Synchrocyclotron
A. V. Crewe and U. E. Kruse
Citation: Review of Scientific Instruments 27, 5 (1956); doi: 10.1063/1.1715374
View online: http://dx.doi.org/10.1063/1.1715374
View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/27/1?ver=pdfcov
Published by the AIP Publishing
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Am. J. Phys. 7, 263 (1939); 10.1119/1.1991463
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Downloaded to IP: 150.135.239.97 On: Wed, 17 Dec 2014 20:02:48LINEAR DENSITOMETER 5
>: I-+0'04-iii -:'::b-= I I 4 I pg Z
1&1 e.
<I 0 0·5 "0 "5 2'0 2'5
DENSITY
FIG. 2. Calibration curve of the circuit.
itself can be changed by varying the distance between
the lamp and the densitometer head. Obviously, if the
response of the circuit were ideally logarithmic, the
change in the output would be the same at all flux levels.
The deviation from this condition is a measure of the
change in the slope of the response characteristic."
Readings with filters of high as well as low densities
reduce the chances of cumulative errors.
THE REVIEW OF SCIENTIFIC INSTRUMENTS The deviations from a true logarithmic response are
shown on an enlarged scale in Fig. 2. The errors at the
ends are mainly due to the characteristics of the 6SK7
tubes. The density scale is linear to ±0.01 density unit
over a density range of 2.5. Since the error curve is
continuous and exhibits an extended region of inflexion,
a response linear to ±0.005 density unit can be obtained
merely by restricting the density range to 2.0.
ACKNOWLEDGMENTS
The authors wish to thank Dr. W. M. Vaidya for his
interest in this work and Dr. K. S. Krishnan, Director,
National Physical Laboratory of India, for permission
to publish this paper.
VOLUME 27. NUMBER 1 JANUARY, 1956
Regenerative Beam Extraction on the Chicago Synchrocyclotron*
A. V. CREWE AND U. E. KRUSE
The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois
(Received October 12, 1955)
The proton beam extraction system of the 450-Mev Chicago synchrocyclotron is described. The nonlinear
theory of LeCouteur has been applied and an external beam of 1011 protons per second has been obtained.
INTRODUCTION
A PEELER-REGENERATOR system of· beam ex
traction was first suggested by Tuck and Teng.!
A linear theory for this arrangement has been developed
by LeCouteur,2 and its success has been demonstrated
with the Liverpool synchrocyclotron.3 The system con
sists of two magnetic discontinuities, a region of in
creasing field (regenerator) and one of decreasing field
(peeler) 60° apart. These regions produce a radial
oscillation with exponentially increasing amplitude
without seriously increasing the amplitude of the ver
tical oscillations. Unfortunately, because of the neces
sity of working in the linear region of the magnetic
. field, about three inches of acceleration is lost. It was
hoped that some similar system could be devised which
would overcome this defect.
LeCouteur4 investigated the possibility of a nonlinear
system with a single regenerator placed only a small
distance inside the n=0.2 point. The system would use
the natural decrease of magnetic field in place of the
peeler. Calculations were performed with an idealized
* Research supported by a joint program of the Office of Naval
Research and the U. S. Atomic Energy Commission.
1 J. L. Tuck and L. C. Teng, Chicago Synchrocyclotron Prog
ress Report III (July, 1949 to July, 1950).
! K. J. LeCouteur, Proc. Phys. Soc. (London) B64, 1073 (1951);
Proc. Phys. Soc. (London) B 66, 25 (1953).
3 A. V. Crewe and K. J. LeCouteur, Rev. Sci. Instr. 26, 725
(1955); Proc. Roy. Soc. (London) A232, 242 (1955).
4 K. J. LeCouteur (to be publishedY. field shape and showed that radial instability could be
achieved without significant loss of vertical stability.
The required shape of the regenerator field depends
on two parameters of the cyclotron field, the magnitUde
of a2H/ar2 and the radial restoring force which is pres
ent at the start of the regenerator action. In particular,
LeCouteur showed that a suitable regenerator strength
would be T=0.2p+0.2p2. In this expression, T=ro/
Ho f t:.HdO, where,.o and Ho are the radius and magnetic
field measured at the start of the regenerator action,
t:.H is the deviation from the normal cyclotron field, and
finally, the integral is carried out on a circular orbit of
FIG. 1. Plan view of regenerator and channel in cyclotron.
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radius TO+p. The quantity p measures the distance
from the last unperturbed orbit to the orbit in the
regenerator, and is measured in inches. The quantity (J
is measured in radians.
A regenerator of this strength should produce radial
oscillations whose amplitudes increase by a factor of
1.3 per turn. The nodes of the oscillation are expected
to be stable and the regenerator is located about 120°
after the node. With sufficient gain in amplitude per
turn, it should be possible to make a large fraction of
the beam circulate close to the inside of a channel wall
on one revolution and enter the channel on the next.
The magnetic field in the channel would be reduced
sufficiently so that particles escape quickly from the
cyclotron. A plan view of the arrangement is shown
schematically in Fig. 1.
CONSTRUCTION OF THE REGENERATOR
Rough calculations showed that the desired re
generator field shape could be produced by two rec
tangular blocks of steel placed symmetrically above and
below the median plane. These blocks are 3X3 inches in
section, and 15 in. long with a separation of 3t in.
Because of the approximate nature of the calculations,
the regenerator blocks were constructed so that the
t~ickness could be adjusted within a small range.
FIgure 2 shows the construction of the whole regen
erator assembly. The blocks themselves are held in an
aluminum C-frame, and each one consists of a main
block with adjusting plates bolted on. Mounting plates
for correcting shims are attached to the C-frame. The
whole assembly was fitted on to guide rails clamped
on to the pole pieces of the magnet and could be moved
radially by means of the lead screw shown. This opera
tion could be performed with the tank evacuated.
The measuring equipment for the magnetic field
consisted of an electronic fluxmeter and a search coil
which could be moved in an arc of a circle centered on
the machine. The output of the fluxmeter was registered
on a pen recorder. During measurements, the search
coil was moved azimuthally in steps of 2! degrees for a
total of 30 degrees; the coil could also be moved radially
in steps of t in. from outside the machine.
The value of the field defect f !:.Hd8, was obtained by
FIG. 2. Regenerator assembly. D
TRIMMING SHIMS MAIN REGENERAT(),!
MEOIAN~ _____________ SL_OC_KS __ _
D
t ! I ! ° I 2 3 INCHES
FIG. 3. Regenerator and shim geometry.
a stepwise integration as a function of radius in the
machine. A few measurements with different disposi
tions of the adjusting plates sufficed to determine the
field shape which most nearly approximated the desired
characteristics in the steep rising portion of the regen
erator field. To trim the field inside this region, cor
recting shims were mounted on the aluminum plates.
The gap between the shims was chosen according to the
shape of the correction desired, and the thickness
adjusted to the magnitude of the correction.
The final shim disposition is shown in Fig. 3. Figure 4
shows the desired field, the field produced by the re
genera tor alone, and the field of the regenerator together
with its shims. The differences between the field ob
tained with shims and the desired field are roughly
200 gauss degrees, which was the limit of the measuring
apparatus. This was considered sufficiently accurate for
a first trial.
TESTS OF REGENERATOR ACTION
Initial tests of the effect of the regenerator on a
circulating proton beam were made with a probe carry
ing a head which is shown in Fig. S. It consists of a brass
block with a !-in. diam hole drilled i in. from the inner
edge of the block. Inside this hole is placed a small
glass ionization chamber. When the probe is placed so
that the inner edge of the brass block is in the circulating
beam of the cyclotron, a relatively small current is
obtained in the chamber. However, if it is placed in a
region where a radial gain per turn of ! in. is possible,
then particles, which on one turn circulate just inside
the block, may on the next turn pass through the ion
chamber and give rise to larger currents.
The regenerator was placed immediately adjacent to
the probe; the probe was, therefore, also at 1200 phase.
Measurements of the ionization current were taken by
scanning radially through the region of regenerative
action for various radial positions of the regenerator.
The results are shown in Fig. 6. It was evident, from
these measurements, that some regenerative action
was taking place and that a reasonable position for the
regenerator was at a radius of 76 in., ! in. inside the
n=O.2 point.
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The regenerator was fixed in this position and more
quantitative measurements of the regenerator action
were made. The beam was now studied directly, using
as a probe a flat piece of brass 3 in. high and t in. thick
on which could be mounted photographic printing
paper wrapped in aluminum foil. One minute exposures
were made at several radial positions outside the start
of the regenerator action. The photographs showed that
sufficient radial gain was being obtained to enable
particles to enter a magnetic channel and come out of
the machine. It was evident that the radial gain per
turn increased and then decreased again until only a
small number of particles of large radial gain and small
vertical amplitude persisted beyond 81 in. These
particles were presumably unstable radially and were
escaping from the machine.
30
28
",26 .. .. 24
It:
:::22 c
120 .,
~18
4
.. 16 3 ;;:14
-12
~
~ 10
u.
IU 8 o
c 6
..J
~ 4
u.
2
o
-2
-4
70 -- DESIRED FIELD
-...... -REGENERATOR WITHOUT SHIMS
X REGENERATOR WITH FINAL SHIMS
j
x )?
x I
x..x_--,,-x __ x __ x_ ,1
.11/ .. _-...... _-.... '"
RADIUS. INCHES
FIG. 4. Regenerator field defect as a function of radius. 79
These conclusions were checked by moving the probe
to investigate the oscillations at 60° phase, the proposed
phase for the entrance of the magnetic channel. At this
phase, the escaping particles are absent. It is to be
expected that the escaping particles would escape in a
wide fan, being most intense at 90° phase and very weak
beyond 180°.
Further checks were performed with a probe at 180°
phase and 77-in. radius with another probe acting as a
beam stop at 60°. No escaping particles were detected
on the probe with the beam stop at 79 in. or less, but
the particles appeared with the beam stop at 79t in. or
greater. This behavior was also verified by orbit tracing
using the technique of Parkins and Crittenden.5
Intensity measurements were performed at 60°, using
6 W. E. Parkins and E. C. Crittenden, ]. Appl. Phys. 17, 447
(1946). BRASS BLOCK PROBE
TO CENTER
- ", ~;;8'~' :L--n:=====
~/.%//
MEDIAN • .f.b~tN_E --~ rdh"', /-r.r---I
GLASS IONIZATION %'if} f------
CHAMBER '------
FIG. 5. Probe to test regenerator action.
polyethylene foil placed on the probe head in place of
the photographic paper. The foils were exposed for a
few seconds, and then counted with a calibrated counter
arrangement. It was found that a substantial portion
of the circulating beam was appearing on the foil. The
fraction was determined using the arrangement shown
in Fig. 7. It consists of a piece of copper roughly in the
shape of a C. The projection on the leading edge serves
to define the vertical extent of the beam. This frame was
then covered with polyethylene foil and exposed in the
machine as indicated in the figure. When the activity of
the foil was determined, it could be seen that about half
the activity was on the vertical portion of the frame and
the rest was on the horizontal part. There appeared to
be some blowup inside the start of the regenerator
action. This is presumably due to incomplete shimming
in this region. It was felt that this effect was not
serious enough to warrant more field measurements.
A polyethylene foil measurement in the projected
position of the magnetic channel, that is at 60° phase
and 80-in. radius, showed that we would expect 10 to
20% of the beam to enter an opening 1 in. wide and 2 in.
high.
THE MAGNETIC CHANNEL
The magnetic channel was constructed of parallel
vertical steel bars with the region of reduced magnetic
field between the bars. It was made in six sections with
a total length of 60 in. The dimensions were chosen to
obtain the greatest possible reduction in magnetic field
consistent with the deflection of a large fraction of the
beam. The length of the channel sections is determined
by the radius of curvature of the particles within the
-7
~
~ 6
~ II c
It:
~ 4
~ 3
...
~ 2
It:
It:
:::> I ()
FIG. 6. Probe currents during regenerator tests. (a) Regenerator
zero at 75.5 inches. (b) Regenerator zero at 76 inches, final setting.
(c) Regenerator zero at 76.25 inches.
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RADIUS INOHES
70 72 74 '76
I I I I I I
JI~fl~~ __ ----
'l6 :7 6 'l2
FIG. 7. Be~m patte~n on copper "C." Figures indicate ClI activity
mduced m polyethylene foil, arbitrary units.
sections. In Table I, the pertinent characteristics are
summarized.
The channels themselves have a serious perturbing
eff.ect on the circulation of particles in the machine, and
thIs effect has to be removed by restoring the magnetic
field to its original value. It was decided to perform this
correction to within! in. of the inner wall of the first
channel section. In order to make these corrections, the
field measuring equipment was modified slightly. The
search coil was mounted on a long arm pivoted at the
center of the magnet and supported on a two-wheeled
trolley. A helipot was connected to one of the wheels,
and the output of this was used on the V-drive of an
X-V recorder. The output of the electronic fluxmeter
was displayed on the X-axis. With this device, the
search coil could be moved through 150°, and the re
sulting curves could be easily integrated. It is felt that
this method is a definite improvement over the step
motion used for the shimming of the regenerator.
The shimming technique was similar to that used at
Liverpool.3 A channel section was set on its correct
radius, but placed tangentially. The search coil was then
swept past the channel in order to determine the total
field defect as a function of radius. This defect was then
corrected using standard shims of two-, four-, and six
in. gaps set parallel to the channel. The length of the
shims was kept the same as that of the channel as far as
possible. When all the channels had been individually
corrected, they were set in their final position by the
wire technique, one end of the wire being fixed at the
position of the node of the radial oscillations, and the
current and tension in the wire adjusted to the energy of
the particles at the start of the regenerator action, an
energy of 440 Mev for the Chicago cyclotron.
The channels, with their shims rigidly attached, were
mounted on an aluminum plate which could pivot at
TABLE 1. Characteristics of channel sections.
Wail Wall Channel Wail
Sections thickness height width Length material
1+2 i in. 2 in. 1 in. 7 in. each Co steel
3+4. i in. 2 in. 1 in. 7 in. each Co steel
5 3 • .. m. 2 in. 1 in. 15 in. Co steel
6 1 in. 2 in. 1i in. 15 in. Mild steel either end while the other end was adjusted by means of
the lead screw from the outside of the machine.
TESTING THE MAGNETIC CHANNEL
The whole deflection assembly was inserted in the
machine with the channel pivoted about the exit and
and the lead screw connected to the entrance. A probe
was available near the entrance, so that the beam could
be investigated as it entered the channel. The small
ionization chamber was placed inside the channel en
trance and the channel was adjusted to obtain the
greatest current in the chamber. By investigating the
beam with photographic film on the probe, it was found
that only the inner half of the entrance of the channel
was being used, there were no particles in the outer half.
Pushing the channel further in had no effect, pre
sumably because the bad field region near the channel
wall was pushing the particles away. This state of
affairs could probably be remedied by more careful
correction of the field, but there would probably be little
gain in the extracted beam; because of the curvature of
the particles in the channel sections, those. particles
which enter the outer half of the channel opening strike
the wall and are lost.
The best position of the channel entrance did not
coincide with the expected position, it was! inch further
in, and so the channel was re-aligned taking the new
position of the mouth of the channel and the position
of the expected node of the oscillations to define the
position of the wire. For the final tests, the channel was
pivoted about the entrance and the exit end was moved
by means of the lead screw. The emergent proton beam
was investigated on the outside of the machine by
means of photographic paper and polyethylene foil.
The best position of the exit end of the channel coin
cided with the position determined by means of the
wire. It was found that with the circulating current
then available, about! microampere,lOll protons per
second were emerging from the tank. This represents an
extraction efficiency of the order of 3%. The beam
leaves the tank in a fan with a spread of 7° and a vertical
height of two inches. It is hoped to focus a large fraction
of the beam by inserting a magnetic lens at the exit end
of the channel inside the tank. The beam will then pass
through a strong focusing magnet into the experimental
area.
ACKNOWLEDGMENTS
It is a pleasure to acknowledge the help and advice
of many members of the Institute, in particular, we
wish to thank Stanley Cohen, Tadao Fujii, and Robert
Swanson for their kind assistance. We are greatly in
debted to Dr. S. D. Warshaw for the loan of measuring
equipment.
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1.3060061.pdf | International Conference on the Quantum Interactions of the Free Electron:
Electron physics in America
Karl K. Darrow
Citation: Physics Today 9, 8, 23 (1956); doi: 10.1063/1.3060061
View online: http://dx.doi.org/10.1063/1.3060061
View Table of Contents: http://physicstoday.scitation.org/toc/pto/9/8
Published by the American Institute of Physics23
character was in some ways rather anomalous. He was
a mathematician, a very good mathematician, who yet
liked his theories concrete. All his life he was attracted
by the idea of tubes of force, Faraday's tubes of force,
and always tried to ascribe to them some kind of actual
physical reality. He liked something he could picture
and he entirely distrusted metaphysics. He preferred
the wave atom, the wave atom with the wave electron,
to the Bohr atom, at least as long as the waves could
be allowed to remain pictorial. He was a great experi-
mentalist who was liable to break any apparatus he
got near. He was singularly clumsy with his hands and
my mother, who was good at that kind of thing, never
dreamed of allowing him to knock a nail in.
He had most of the actual preparing of the experi-
ments done by his personal assistant Everett; my father
just took the readings, which very often took the form
of examining a photographic record, for example of
positive rays, which he would measure. But he had an
uncanny power of diagnosing the reasons why appa-
ratus, his own or other people's, would not work, and
suggesting what had to be done to make it work. He
was a man who was normally silent, but he was a wittyand amusing host at any sort of party, including the
daily teas held in his room in the Cavendish, which he
introduced. He loved flowers, wild and cultivated, and
knew a very great deal about them, but he seldom
gardened. He was fond of watching cricket, tennis, and
football, and could recall the names and achievements
of most of the leading people at Cambridge for the last
30 or 40 years in those sports. But in fact he had
played little himself. He was a man of exceptionally
wide sympathies. He could enjoy talking to almost any-
body, and had the knack of making other people talk
well about their own particular subject. He founded,
and these sympathies helped him to found, the first
school of physics, in a modern sense, at least outside
Germany, and at one time his pupils, Cavendish men,
held a very large fraction of the professorships through-
out the world. Though he had a strong sense of humour,
physics was too important to be funny, certainly too
important to be laughed at. For him the two great
qualities of a physicist, the two that really mattered,
were originality and enthusiasm; and though he rated
originality extremely high, it was enthusiasm which
stood at the top.
Electron Physics
in America
By Karl K. Darrow
The address by Dr. Darrow, a physicist at
Bell Telephone Laboratories and Secretary
" of the American Physical Society, was also
an after-dinner talk at the Electron Physics
Conference Banquet.
DR. MARTON said I was going to speak about the
history of electron physics in America. I think
you are very fortunate that he did not make this re-
quest of someone competent to fulfill it, for if he had
this person might have done it; and I can imagine
nothing less appropriate for this hour of the evening
of a busy day and particularly after so brilliant a
speech as you have just heard.
I did go so far as to try to figure out what electron
physics is, and I concluded that it is all of physics ex-
cept part of nuclear physics and the general laws of
thermodynamics and relativity, which in principle are
independent of whatever hypothesis you make aboutnature. It is somewhat devastating to reflect that the
blacksmith at his forge, the cook in her kitchen, and
the distiller in his distillery, are all practicing electron
physics; but I really see no way of making a definition
which leaves them out. So I shall not cover so vast a
field. I shall just tell about some of the figures in the
history in the United States, beginning quite a long
way back.
This year contains not only the 100th anniversary of
J. J. Thomson but the 250th of Benjamin Franklin. If
one were to omit mentioning Benjamin Franklin this
year at any speech in Philadelphia, one would be con-
sidered to have committed a crime—the crime of lese-
AUGUST 195624
Franklin, much more serious than that of lese-majestd;
and as I don't want to stay away from Philadelphia
for the rest of my days, I will avoid it. One of the most
remarkable things about Franklin, I think, was the way
in which he financed his experiments. He did not go to
the National Science Foundation, or the Atomic En-
ergy Commission, or the Office of Naval Research, or
the Office of Aerodynamic Research, or the Office of
Ordnance Research. He couldn't because they are all
in Washington—and Washington didn't exist. He didn't
get his money from an endowed university either, nor
from the taxpayers.
He came to Philadelphia as a young and penniless
boy. He started a printing business. At the age of 42
it had thrived so well that he was able to sell it for a
competence and he retired, and in his own words he
said he was going to devote himself to reading, to
study, to performing experiments, and to discussion
with ingenious people. This program he continued for
four or five years, and then he got swept up into
politics and finally into statesmanship, and that is why
he is greatly remembered now. But the work of those
few years really constitutes some sort of an epic in
the history of electrical science.
I pass over Joseph Henry just with the mention of
his name. He was a man who when he got anywhere
was likely to find that Faraday had just got there be-
fore him. But he did get to self-induction first.
I pass over a young man named Hall because I am
going to speak of him later. Next I recall to you the
very famous man who discovered the thermionic effect
and left it for others to explore. This was Thomas A.
Edison. He had developed the old-fashioned lamp with
a carbon filament of the shape of a hairpin. The inside
of the lamp grew black with sublimated carbon, and
Edison noticed that there were white lines where the
glass was shadowed from one leg of the filament by
the other leg. He thought that the evaporating carbon
atoms might be charged, so he made a tube with an
auxiliary electrode to attract them. When the auxiliary
electrode was positive it drew a current, when it was
negative it drew none; but the blackening was unaf-
fected.* At this juncture Edison turned his attention
to something else, I do not know what. He had made
enough of an impression on people's minds so that for
a while the thermionic effect was called by some the
"Edison effect", but this usage has died out. It is in-
teresting to speculate on what might have happened if
Edison had had the training of a physicist. Actually,
he had no academic training at all.
Next I introduce you to the first President of the
American Physical Society. You have heard of him as
the perfecter of the diffraction grating and as the man
who discovered the magnetic field of a convection cur-
rent, that is to say of a moving static charge. You
probably have not heard of him as the man who killed
electricity. But listen to this: "It is not uncommon for
electricians to be asked whether or not science has yet
determined the nature of electricity, and we often find
difficulty in answering the question. When it comes froma student of science, anxious and able to bear the truth,
we can now answer with certainty that electricity no
longer exists, for the name electricity as used up to the
present time signifies at once that a substance is meant,
and there is nothing more certain than that electricity
is not a substance." This is something that H. A. Row-
land published in 189S. Now, of course, one could get
all tangled up in semantic discussions as to the meaning
of the word "substance"; but from the context, which
I haven't brought along, I deduce that Rowland be-
lieved in an ether and in tubes of force in an ether, but
he thought electricity was just a name for the ends of
the tubes of force—no more significant than it would
be to have a name for the end of the rainbow where it
is supposed that a pot of gold is to be found; and he
didn't want people to put any faith or belief in the
existence of anything real or substantial at the ends of
the tubes of force. This at least is all that I can make
of it, and the coincidence of dates is such as to sug-
gest that Sir George's father might have read this and
might have set out with exemplary skill and success to
prove our Rowland wrong. But I have no evidence to
sustain this idea and unless Sir George has some, I
think we must just give it up as one of those things
that ought to be true but isn't.
Now I will go on to someone whom I do remember,
and that is Millikan. The last elementary course that
he ever gave at the University of Chicago was also the
first that I ever took; and consequently in this sense
my career begins where Millikan's teaching career
ended, though of Millikan's research career there was
still a full forty years to go. It occurred to me the
other day that I could still remember the value that
he published for the charge of the electron. Now this
is not so trivial a fact, I believe, as it appears. To me
it suggests, and I believe, that 30, 40, or 45 years ago
nine-tenths of all the physicists in this country knew
that Millikan had measured the charge of the electron
as 4.774 X 10"10, so that nine-tenths of them if they
had heard a new value given for the electron charge
would have had a standard of comparison for it, and
if the new value had been 4.25 they would have felt
there must be something queer, and that if the value
was 4.77 that it must be quite right. I doubt whether
this can be said now or can ever be said again of such
things as the value of //, the value of k, or the value of
the mass of the tau meson. My impression is that if
anyone were to give a new value for any one of these
quantities, practically all of you would have to look
up the old values to see how the new value agreed with
them. This is partly, but not exclusively, because the
numerical values of things like h/e are now given to
seven significant figures; it is also because physics has
become too much compartmented. This in turn made
me think what a towering figure Millikan was, say 30
years ago; a figure such as can hardly be imagined by
the young generation because now it is rare for a man
* This is the story as graciously provided to me by Mr. N. R.
Speiden, from the files of the laboratory of Thomas A. Edison.
PHYSICS TODAY25
to tower unless he be Enrico Fermi or Niels Bohr.
Most solid-state physicists don't know the eminent nu-
clear physicists except by hearsay and vice versa. But
30 years ago this was not yet the case. Neither was
Millikan so restricted as many of our contemporaries
have to be, for his range of research extended (not to
speak of his thesis which was on something having to
do with textiles) over the measurement of the electron
charge, over the photoelectric effect, over spectroscopy
of ionized atoms, and finally for some 25 years over
the cosmic rays; so that he was electron physicist and
nuclear physicist at once—something not easy to be-
come nowadays.
He was a man of tremendous energy, one of these
lucky people who live on the short-sleep basis and can
sleep five or six hours a night and work the remaining
hours of the night and day; and one of the few, in
fact I think of only one other, who have succeeded in
combining the career of research physicist with the
career of university president. That is a very little to
say about a very great man, but perhaps it is worth
saying. And another thing that I recall about him is
that he joined for about three years in the crusade of
J. J. Thomson to revise the terminology and make
"electron" mean the value of the unit electric charge
and "corpuscle" mean what we now call electron. It is
evident that they didn't succeed, and evident also that
any enterprise in which J. J. Thomson and R. A. Mil-
likan together failed was an enterprise in which no one
could succeed. But if that terminology had persisted,
then I suppose your father, Sir George, as discoverer
of the electron in that sense, would also have been the
discoverer of all the mesons.
Millikan was also associated, sometimes slightly,
sometimes closely, with several of the other figures who
ought to be mentioned. Davisson, for example, was one
of his early students, but only as an undergraduate and
briefly, so that it would not be reasonable to connect
Davisson's work with Millikan's. Davisson, as you
know, shared the Nobel Prize with Sir George, for the
experimental verification of what we loosely call the
wave nature of the electron. Lately I heard Sir George
relate the story of his discovery; and I was impressed
by the difference between the two. For Sir George was
acquainted with the work of Louis de Broglie, and he
was looking for what he found. With Davisson the phe-
nomenon came first and the interpretation came after.
Git was just a fortunate chance that he had taken up\
the study of the reflection of very slow electrons from
metal surfaces, for it was in the course of this study !
that he discovered that the reflected electrons grouped
themselves into clearly-defined beams. Accident played
a dramatic part. Davisson's first observations were
made upon poly crystalline masses of metal; then one
day the tube broke and the target got oxidized, and in
the course of the prolonged heating necessary to undo
the harm, the metal was changed from an aggregate of
a large number of small crystals to an aggregate of a
few large crystals. The system of beams was radically
changed. Davisson trained the incident electrons againstthe surface of a large single crystal, and the key was \
in his hand.
I think it probable that no discovery has ever been
made simultaneously in two such different ways as this
discovery; the one with very slow electrons, the other
with fast; the one with an analogue of the Laue
method, the other with the Debye-Scherrer-Hull powder
method. It was Sir George's method that had the flat-
tery of speedy imitation and application; whereas
Davisson's method has been cultivated by very few,
Farnsworth at Providence, one or two elsewhere in the
world, and otherwise remains in the state where he
left it.
Another person with whom Millikan was intimately
associated and this time definitely in the role of teacher
to pupil was the discoverer of the twin, or I guess I
should say the anti-twin, of the negative electron. This
was superficially like the discovery of the anti-proton
which has just made the headlines, but only super-
ficially, for the anti-proton was the object of a long
and tenacious search achieved finally only by new
instrumentation, whereas the positive electron just
dropped out of nowhere into Anderson's bag. This is
another instance of a discovery being made quite in-
dependently and almost simultaneously in Britain and
America, and just the hazard of chance determining the
order, and the rectitude of the Nobel Committee dis-
tributing the Prize evenly between the two. At this
point, I mention something else pertaining to the elec-
tron. This is the phenomenon loosely called paramag-
netic resonance and better named electron spin reso-
nance: the turning over of an electron in a strong
magnetic field by an applied radio frequency field. You
will find this credited everywhere to a Russian named
Zavoisky; and after naming Zavoisky, some but by no
means all writers will go on to say that the next to
publish the phenomenon were David E. Halliday of
Pittsburgh and his collaborators. But this also was a
case of independent and nearly-simultaneous discovery,
though Halliday was too modest to make his claim.
Now I turn back to E. H. Hall. Hall was a remark-
able figure and there are remarkable features about his
story. For instance, he was still a graduate student
when he sought and found an effect of such impor-
tance that within a few years it became widely known
and it took its name from him, so that such terms as
"Hall effect" and "Hall EMF" and "Hall voltage" are
now part of the everyday language of physicists. I feel
sure that there must be other such cases, but I cannot
think of any; perhaps someone else can.* Hall thus
made his discovery while he was very young, so that
he lived long to enjoy its fruits and also to experience
the ludicrous event of which L. Brillouin has told me.
He went as an honored guest to a Solvay Congress held
after World War I—the date, it seems to me, was
1924—and person after person came up to him, each
* Someone else could and did, and I have verified it at first hand.
The contributions made by E. U. Condon to the "Franck-Condon
principle" important in molecular spectroscopy were made while he
was still a graduate student.
AUGUST 195626
Robert A. Millikan C. J. Davisson Edwic H. Hall
asking, "Are you related to the old Hall?" and getting
the reply, "I am the old Hall". Evidently these in-
quirers thought that such a discovery could have been
made only by a man already middle-aged.
Another remarkable thing about Hall was this. Some-
times a man makes a discovery while looking for some-
thing else, sometimes he makes one while looking for
nothing in particular, and sometimes he makes one by
verifying some great man's theory. Hall however made
his discovery by defying a great man's theory—a very
great man's theory, that of none other than James
Clerk Maxwell. Maxwell in his Treatise on Electricity
and Magnetism said that what we now call the Hall
effect could not occur. I have indeed heard people say
that Maxwell's words can be construed otherwise, but
this is of no moment, for they were construed as I
have described by Hall himself and also by the young
Oliver Lodge, who started and then gave up an experi-
ment of which you can read the account in a speech
that Sir Oliver delivered when he was an old man—
you will find it in the section of that speech which he
entitled "How I Failed to Discover the Hall Effect".
Hall rushed in where others feared to tread, or rather,
where others thought it useless to tread: and he got
his reward.
All the stranger is it therefore, that having taken
this great and courageous step and taken it with suc-
cess, Hall did not take the next one. It is very easy
(once somebody shows you how) to derive an equation
which gives you the speed of the flowing charge, or in
more modern language the mobility of the carriers, in
terms of the Hall EMF and other measurable things.
This seems but a small step onward, and yet it was
not Hall who took it. It was another man equally
young and destined to even greater subsequent fame—
the Austrian, Ludwig Boltzmann. There is another
equation, or really the same one transformed just a
little, that enables you to go from the measured HallEMF to the density of the flowing charge. This is in-
deed a small step, but Boltzmann himself did not make
it, not at least in his first paper on the subject: I do
not know who made it first. One hates to think how
difficult it would be to analyze convincingly the be-
havior of semiconductors, were it not for the Hall
effect and for these equations that lead from it to the
density and the mobility of the flowing charges. Hall
laid the groundwork, but others found the equations.
On the other hand Hall did clearly see that the sign of
his effect gives the sign of the preponderating carriers,
and since he observed in some metals the sign appro-
priate to flowing positive charge, he has something of a
case for being regarded as the discoverer of conduc-
tion by holes.
I admit that I can scarcely claim that Hall was the
discoverer of holes. He couldn't have formed the con-
cept of holes, for this is derived from the concept of
electrons, and since Hall made his discovery before
1880 he didn't even have the concept of electrons. I
cannot claim that the first to publish the concept of
holes were Americans, nor that all of the important
discoveries in the semiconductor field were made in the
United States. Yet I think that we do not vaunt our-
selves unduly if we say that quite a big share—well
over half—of the work on semiconductors published
since World War II was done in American laboratories.
Most especially is this true of work on germanium and
silicon, those elements that almost seem to have been
designed by Nature for giving vivid demonstrations of
simple and clearcut ideas regarding conduction by elec-
trons and conduction by holes. I remember well a time
when metals were considered simple and intelligible,
semiconductors odd and mystifying; now the situation
is almost reversed, and I think that if I were trying to
lead a group of beginners into the lore of conduction,
I should commence with germanium and silicon both
pure and impure, and go over to the metals at the end.
PHYSICS TODAY27
I think that in this hypothetical case I should find it
hard to explain why for so long a time physicists as-
sumed that all of the flowing charge in a conductor
must be of one sign either positive or negative, and
did not take into account the possibility that now is
seen to be often a fact—the possibility that charges of
both signs are flowing at once. Perhaps this was due to
the discredit in which the two-fluid theory languished
for so long after the one-fluid theory was accepted.
Also I am sure that I should find it hard to produce a
simple explanation of holes. The articles that have been
written on this subject have often reminded me of
something that appeared in The New York Times some
thirty years ago, at the time of the spate of popular
books about relativity. Simeon Strunsky, then of the
staff of The Times, wrote a column about it. I no
longer remember Strunsky's exact words, but I can
paraphrase them nearly enough. In effect he said, "All
of these books have one feature in common. They are
all very lucid and fascinating until just before they get
to the point, and then all of a sudden they become un-
intelligible." To my ears this sounds sadly like the ex-
planations of holes that I have read. Nevertheless the
language of holes and electrons, the language of bands
and forbidden gaps, excitations and impurity-levels—
this has turned out to be quite a useful language for
describing vast numbers of phenomena, and indeed phe-
nomena in more than one field, and indeed phenomena
outside of physics altogether. Let me give a couple of
examples.
First, here is the example of photoconductivity in an
insulator. You have a great crowd of electrons which
are fitted together and compensate one another in such
a way that even though they are right there inside the
insulator, the outer world doesn't know anything about
them and they don't know anything about the outer
world. You may consider them as being all paired offand holding little conversations tete-a-tete, the world
forgetting, by the world forgot. They are said to form
a valence-band, also known as a filled band. Now comes
along a photon and expels one of these electrons out
of the valence-band and into the conductivity-band.
The evicted electron has to go to work, and so do all
of the other electrons have to go to work, their ac-
tivity being described by speaking of a hole. Some day
the electron will go back into the valence-band, and
things will be as they were before. The time of this re-
turn will not be decided by the exciting photon. The
photon has no control over it whatsoever. It is entirely
up to the electron to decide when to go back to the
valence-band.
Next consider a group of people all sitting together
in a dining-hall after a banquet—indeed it could be this
very group right here. They constitute a filled band, in
more senses of the adjective than one. They are paired
off or else they are grouped into clusters of not more
than eight, carrying on their conversations within their
own group. They have forgotten about the outside
world, and the outside world has forgotten about
them. But this peace is rudely shattered when the
Chairman arises and excites one of the people into the
oratory-band. Then all of the nice balancing is undone,
and everyone has to go to work, the speaker on the
one hand and the listeners on the other. Some day the
speaker will stop talking, but the time will not be de-
cided by the exciting Chairman. It will be entirely up
to the speaker to decide when to go back into the
silence-band. The speaker is all too likely to make a
mistake in judgment on this important matter, and in
fact two such mistakes have already been made this
evening. Sir George Thomson subsided too early into
the silence-band, and I have stayed out of it too long.
I can do nothing about the former error, but at least I
can refrain from compounding the latter.
International Conference on
Quantum Interactions of the Free Electron
A summary report by Harold Mendlowitz, National Bureau of Standards
THE electron has been a bona fide member of the
family of elementary particles for over a half of
a century and a great deal is now known about its
properties. As is usual in scientific endeavor, the more
one learns about something the more one finds fur-
ther questions which need to be answered. The Inter-
national Conference on the Quantum Interactions of
the Free Electron served somewhat as a pause to re-
capitulate what has been learned and understood and
to reformulate the pertinent questions that we would
like to have answered.The conference was held in commemoration of the
one hundredth anniversary of the birth of J. J. Thom-
son, the "father" of the electron. The University of
Maryland, which is celebrating its centennial and sesqui-
centennial, acted as the host institution.
A very nice feature of the conference was that there
were only nine invited comprehensive review papers,
and no short ten minute papers, in order to ensure and
facilitate adequate discussion and contributions from
those attending. For the most part this worked out as
planned, and many people were able to participate ac-
AUCUST 1956 |
1.1740610.pdf | Effects of Oxygen on PbS Films
Henry T. Minden
Citation: The Journal of Chemical Physics 23, 1948 (1955); doi: 10.1063/1.1740610
View online: http://dx.doi.org/10.1063/1.1740610
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/23/10?ver=pdfcov
Published by the AIP Publishing
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Photosensitivity in Epitaxial PbS Films
J. Appl. Phys. 39, 5086 (1968); 10.1063/1.1655928
INTERPRETATION OF HALL EFFECT DATA IN PbS POLYCRYSTALLINE FILMS
Appl. Phys. Lett. 11, 227 (1967); 10.1063/1.1755110
Reply to ``Reaction of PbS Surfaces with Oxygen''
J. Chem. Phys. 42, 4318 (1965); 10.1063/1.1695951
Reaction of PbS Surfaces with Oxygen
J. Chem. Phys. 42, 4317 (1965); 10.1063/1.1695950
Reaction of PbS Surfaces with Oxygen
J. Chem. Phys. 41, 3971 (1964); 10.1063/1.1725844
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:531948 WILLIAM T. SCOTT
It is easy to show from the material of this section that
if
](t) = it t'-!F(t-t')dt',
with ](0) = 0, as is required by this expression, then
F(t)= (1/11') it t'-!j(t')dt'.
Therefore, any pair of functions used in one way in
the integral relation we are considering can be inverted
by taking the derivative of one of them and dividing by
7r as we have indicated.
THE JOURNAL OF CHEMICAL PHYSICS Notes added in proo].-
1. Tachi and Kambara (Bull. Chern. Soc. Japan 27,
523-524 (1954), and 28, 25-31 (1955)) have carried
out a treatment somewhat similar to that of this article,
from a different point of view. In particular, they do
not consider the period of interruption to be long in
comparison to the transient decay times.
2. Vol. II of the book of Erdelyi et al.lO contains in
its section 13.1 several pairs of functions related as
](t) and F(t) above.
The author wishes to acknowledge with gratitude the
support of the Office of Naval Research. He wishes to
thank Professor D. C. Grahame for his courtesy in
making it possible for the author to participate in his
project, and for many stimulating discussions.
VOLUME 23. NUMBER 10 OCTOBER. 1955
Effects of Oxygen on PbS Films*
HENRY T. MINDEN
Chicago Midway Laboratories, Chicago, Illinois
(Received December 28, 1954)
The effect of oxygen on the conductivity of vacuum evaporated PbS films has been experimentally in
vestigated. When outgassed ~hese films are n-type semiconductors. If they are exposed to oxygen at tem
peratures below about 200°C, the conductivity decreases with increasing pressure and then increases again.
The thermoelectric power changes from negative to positive, but less than 10-3 moleO./molePbS is sorbed
by the film.
Above 200°C the conductivity merely decreases when oxygen is admitted, and the thermoelectric power
becomes small. The film gradually absorbs all the oxygen in the vacuum system and evolves a good deal of
S02. When the reaction is complete, the conductivity returns to its initial value, and the thermoelectric
power is once again negative. The film can repeatedly absorb large amounts of oxygen without there being
any permanent change in electrical properties.
It is concluded that these films are composed of two layers. Next to the substrate there is a conducting
layer that chemisorbs oxygen but does not react with it. The chemisorbed oxygen acceptors are responsible
for the observed changes in conductivity. The upper layer of the film is nonconducting and reacts to com
pletion with oxygen, possibly forming PbS0 3: PbO, and evolving S02.
INTRODUCTION
IT is well known that lead sulfide type films which
are evaporated in vacuum can be rendered photo
conductive by treatment with oxygen. The exact nature
of this treatment and its theoretical basis are still not
well understood, however.
PbS type films which have been evaporated in
vacuum seem always to be n-type semiconductors.
Levenstein and Bode! have discovered that PbTe films
can be reversibly changed from n-to p-type by mere
exposure to oxygen at room temperature. They found
that maximum photosensitivity occurred when the
resistance is greatest and the thermoelectric power is
zero. Scanlon and Humphrey2 have made a similar
* This work was supported in full by the United States Air
Force under contract number AF 33 (038)-25913.
1 D. Bode and H. Levenstein, Phys. Rev. 96, 259 (1954).
• Private communication. observation with PbSe films. Finally, in their original
work on PbS films Sosnowski and his co-workers3 re
ported the same findings, but they were quite obscure
as to the nature of the oxygen treatment.
On the basis of his work Sosnowski4 advanced the
theory that oxygen reacts with the surface of the n-type
microcrystals of the film. A surface p-region is thus
formed, and the resulting micro p-n junctions are the
sites of the photoeffect.
The chemistry proposed for these surface reactions
is quite complex. By using electron diffraction tech
niques, Wilman5 in England, and later Lark-Horovitz
and his group6 in America were able to identify a
lanarkite (PbO: PbS0 4) phase in the film. Brockway7
3 Sosnowski, Starkiewicz, and Simpson, Nature 159, 818 (1947).
• L. Sosnowski, Phys. Rev. 72, 641 (1947).
• H. Wilman, Proc. Roy. Soc. (London) 60, 117 (1948).
6 K. Lark-Horovitz et at., Phys. Rev. 79,203 (1950).
7 Private communication.
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:53EFFECTS OF OXYGEN ON PbS FILMS 1949
has recently discovered other oxidized phases in PbS
films.
Such oxidized phases as have been discovered, how
ever should not be p-type conductors, but rather,
insuiators. Moreover, insofar as can be determined,
these phases have been formed at temperatures and
oxygen pressures much higher than needed for photo
sensitization.
In the present work the kinetics and stoichiometry of
oxidation have been quantitatively investigated. The
resistance changes caused by exposure of the film to
oxygen have been correlated with the nature of the
treatment. It is believed that the results of this in
vestigation shed a new light on the nature of Sosnowski's
surface p-region, and that the general character of the
complex oxidation chemistry has been somewhat
elucidated.
EXPERIMENTAL METHODS
Evaporation Cell
The lead sulfide used in this work was synthesized
from specially purified lead and sulfur and grown into
single crystals by vapor phase deposition.8 The films
were prepared by the evaporation of small weighed
amounts of the ground up single crystals; the residual
pressure during evaporation was not mor~ tha~ 10-5
mm. Figure 1 illustrates the Pyrex glass cell m which the
films were evaporated. A weighed amount of lead sulfide
A is introduced through the sidearm B onto the quartz
evaporation head C, which is located close to the center
of the sphere G. D is a quartz to Pyrex graded seal. The
sidearm B is sealed off, and the cell is evacuated through
the tubulation E to a pressure of 10-6 mm, the ultimate
vacuum attainable by the system used. The cell is
outgassed in a furnace at 300°C, after which the lead
E
A
c.
6
.I.-----~-F
s __ ----r-- 0
Fw. 1. Evaporation cell.
8 F. Pizzarello, J. App!. Phys. 25, 805 (1953). Fw. 2. Gas handling system.
sulfide is evaporated onto G by means of the removable
heater F. In this manner a fairly uniform film is pro
duced on a hemispherical surface. The thickness T= sm
where m is the weight of PbS used and s is a constant
depending on the dimensions of the cell, the den~ity of
the PbS microcrystals, and the shape and packmg of
these microcrystals on the substrate (close-packed
spheres, cubes, etc.). Depending on the film structure
assumed, s varies by not more than about 30%.
Not shown in the figure are graphite electrodes.
These are painted along meridians 180° apart and are
separated at the north pole by a 1 mm gap. Most of the
current flows around parallels of lattitude; thus a
weighted conductivity of the whole film is measured.
Two tungsten wires are sealed through the cell wall
below the equator; each wire has a platinum strip spot
welded to the inside end. The platinum strips are in
turn sealed to the inside of the cell wall, and the graphite
electrodes are painted over the strip. Neither the plati
num nor the tungsten ever come into contact with the
PbS.
The over-all dimensions of the cell were not more than
2! X 3! inches, so it could readily be placed in a simple
furnace. The cell temperature was measured by fas
tening a calibrated thermocouple to the outside of the
cell dome by means of wet asbestos paper, which dried
hard during the outgassing.
Gas Handling System
The vacuum system is shown in Fig. 2. Pressure from
1.u to 800.u was measured by a Pirani gauge, which was
part of a special oscillator circuit.9• T~e l\;fcLeod gauge
was used for calibration only. The lOmzatlOn gauge was
used for measuring low pressures in a kinetic system .
The known volume was measured by weighing the
water the vessel contained. The volumes of the rest of
the system were measured by observing the pressure
change due to the expansion of helium from the known
volume.
Sorption Measurements
During the oxygen sorption measurements 53 and 54
(Fig. 2) were normally closed. The rest of the system
to the right of 51 was pumped out; 52 and 55 were then
9 G. von Dardel, J. Sci. Instr. 30, 114 (1953).
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:531950 HENRY T. MINDEN
closed. The trap T was usually immersed in liquid
nitrogen to freeze out condensable gas evolved by the
film during sorption. Oxygen was introduced from the
gas reservoir through the leak SI. When the desired
pressure in the manifold and gauge was reached, SI
was closed and S2 opened, admitting gas to the cell.
From the pressure changes which slowly occurred, the
amount of gas sorbed at any time could be calculated.
Blank runs were made with no film in the cell, but the
sorption was zero within the sensitivity of the method.
The minimum sorption which could be detected was
about 5 X 10---8 mole of gas; this was also the experi
mental error in the sorptions which were observed.
The gas condensed in T during the oxygen sorption
process was found to be S02 and the Pirani gauge was
calibrated for this gas as well as for oxygen. At the end
of each sorption run the system was pumped out
through Ss with T still immersed in liquid nitrogen.
(Usually there was a negligible amount of gas to pump
out.) Ss and S2 were closed and T was allowed to warm
to room temperature. From the pressure in the gauge
and manifold the amount of S02 evolved during the
oxygen sorption was determined.
Some experiments were done at low constant pres
sures in a kinetic system. S2 and ·Ss were open, and the
leak S I was partially open. The pressure was monitored
by the ionization gauge, or the Pirani gauge.
Conductivity Measurements
Conductance was measured using 60 cycle ac, partly
for convenience and partly to minimize the effect of
thermal emf's. A regulated 1 volt, low resistance source
was used and the ammeter circuit had negligible re
sistance compared with that of the cell. The device was
regularly calibrated against a standard resistance box.
Several assumptions and approximations have been
made to permit the conversion of the conductance
readings into conductivities. As was mentioned above,
the current measured by the ammeter spreads through
the whole film, not through just a small portion of it
as is usual in PbS photocells. The prime assumption is,
then, that the conductivity is indeed uniform through
out the film. Although inhomogeneities were hard to
detect, the nature of the results indicated that the
conductivity did not vary by more than a factor of 2
or 3 over the area of the film.
Since experiments with gold and with graphite elec
trodes gave identical results, it is probable that no
error is introduced by the assumption that the electrodes
contribute negligibly to the cell resistance, either in
themselves or at the electrode-film interface.
Because of slight variations in the preparation of each
cell blank, the evaporation head (e, Fig. 1) was not
always located in the center of the hemisphere G. This
introduced a possible variation in film thickness of
±0.2J.l over the area of the hemisphere. From the
subsequent discussion it will become evident, however, that the important thickness in determining the con
ductivity is not the total thickness of the film (about
O. 5J.l) , but rather the thickness of a conducting layer,
assumed to be 0.1J.l. This thickness may be low by as
much as 100% which would make the calculated con
ductivity high by a factor of 2.
The model of current flow around parallels of lattitude
is incorrect near the gap at the north pole of the cell.
The approximation is improved by adding a current
which is assumed to flow straight across the gap at the
north pole. The errors introduced by this model would
be fairly small were it easily possible to paint the
electrodes correctly and to measure the dimensions of
the gap precisely. In practice the errors so introduced
might make the calculated value of the conductivity off
by a factor of 2. In all, the calculated conductivities
ought not to be incorrect by more than a factor of five.
Thermoelectric Power
Only the sign of the thermoelectric power was de
termined. A sensitive galvanometer was used, and one
electrode of the heated cell was gently and briefly cooled
with an air stream, or alternatively, one electrode of the
cell at room temperature was heated slightly with a
flame.
EXPERIMENTAL RESULTS
General Properties of the Films
Viewed through the glass substrate, the films had a
shiny mirror-like appearance. Films thinner than about
0.1J.l also appeared shiny when viewed from the side
away from the substrate. The films used in this work
were 0.5J.l to 0.7 J.l thick. These films appeared dull on
the side away from the substrate. Moreover, after
oxidation this side of the film took on a whitish patina,
while the mirror-like substrate side remained untar
nished.
When the films were thoroughly outgassed, the con
ductance seemed to be more or less independent of film
thickness. Only one film which was 0.026J.l thick had an
appreciably lower conductance than the rest. Moreover,
the thermoelectric power was always negative. It is
TABLE 1. Conductivity of PbS films.
Weight Ndb Nto,
PbS ,,- (1017 (10-9 O'min
Film (mg) (ohm-cm)-t donors/em') mole) (ohm-cm)-t
A-Ie 0.70 7.9 0.99 0.115 0.178
A-2 17.7 19.4 2.42 0.95 0.090
A-3 14.6 13.5 1.70 0.545
A-4 17.3 43.5 5.45 2.09
A-5 9.4 14.0 1.75 0.364
A-6 16.7 323 40.4 14.9 2.23
A-7 19.3 46 5.7 2.44
A-8 15.8 44 5.5 1.93
• When the film is Dutgassed. Assuming the conducting layer is 0.1J<
thick.
b J<n is assumed to be 500 cm'/volt sec. (See S. J. Silverman and H.
Levenstein, Phys. Rev. 94, 876 (1954).)
o Film thickness was 0.0261'.
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:53EFFECTS OF OXYGEN ON PbS FILMS 1951
assumed that the contacts between the microcrystals
of the film are perfect. When the film is outgassed the
resistance is due solely to the bulk properties of the
microcrystals. The high conductivities observed tend
to support the validity of this assumption.
When outgassed, then, the films behave like very
impure electronic semiconductors. For such substances
the conductivity u= qiVaJ.ln where q is the electronic
charge, iVa is the concentration of donor impurities,
and J.ln is the electron mobility, here assumed to be 500
cm2/volt sec. From the observed value of the con
ductivity when the film is outgassed, a value of iVa can
be calculated. Table I shows u and iVa for the films
which were investigated.
Effects of Oxygen Not Accompanied by
Detectable Sorption
When oxygen is admitted to contact with the film,
the conductance decreased by one to three orders of
magnitude. There was no detectable oxygen sorption
accompanying these conductance changes. Figure 3
illustrates the typical variation of the conductivity u
with the pressure for three films at different tempera
tures. This effect showed considerable hysteresis in the
conductance in going from low to high pressures and
back down again. At low temperatures there was a
definite minimum in the conductivity; near this mini
mum there was a change from negative to positive
thermoelectric power. Above about ZOO°C the con
ductivity merely leveled off; when the conductivity
was low, the thermoelectric power was substantially
zero.
Figure 4 shows the change in conductivity with the
time when oxygen was admitted to contact with a film.
The pressure was 7J.l and the temperature 900e. Here
again the typical minimum in conductivity was ob
served. The higher the pressure (above about ZJ.l) and
the higher the temperature (up to about ZOO°C) the
more quickly was the minimum reached. Below ZJ.l or
above ZOO°C there was no minimum, as is evident from
Fig. 3. Table I shows the minimum conductivity Umin
~
V
±3~ •
I~ 21~ >-'''f--~-+---i-----+---+-----l
~" ~~
U ---a~_~·r~ _ " ___ ~ 1_
~'~ ~
u , _. , ,t ,l., 00-' -"-
PR'ESSUR[ (MILLlM[TERS)
FIG. 3. Variation of conductivity with oxygen pressure. Curve
1, film A-7, 310°C. Curve 2, film A-6, 1SSoC. Curve 3, film A-3,
200°C. o~
.. i'-"
u ± I' o '-'
I
3 .0
I .0 I '\
1\
~
\
\
I-.. V ...
3 10 .30 J 0 0 0 ., 0
TI ME (MINUTES)
FIG. 4. Variation of conductivity with time, film A-2.
Oxygen pressure, 7p.; temperature, 90°C.
observed in the presence of oxygen below ZOO°C for
three of the films investigated.
Figure 5 shows the variation of the conductance g
with temperature at several oxygen pressures. Again,
there is considerable hysteresis in the temperature cycle.
Nonetheless, the qualitative features of the curves are
readily reproducible. At higher pressures there is always
a conductance minimum in the neighborhood of room
temperature, while at low pressures the temperature
coefficient of conductance is very small and positive.
It was definitely established that the above effects
were caused by the specific action of oxygen on PbS.
Neither helium, nitrogen, nor sulfur dioxide had any
effect on the conductance, nor were any of these gases
sorbed by the film. Furthermore, no effects occurred
when oxygen was admitted to cells having no films.
Several films were vacuum evaporated onto a cooled
substrate; the pressure during evaporation was held
below 10-5 mm. The film was then heated slowly from
room temperature to 300°C under vacuum. As the
temperature rose there was a considerable amount of
outgassing. In one typical case (Fig. 6) the conductance
increased slightly and then reached a maximum while
the temperature was still below 80°C. With further
heating the conductance decreased, reached a minimum
(less than the initial value) and then rose over an order
of magnitude. The film was baked at 300°C until gas
evolution ceased and the conductance was constant;
then the film was slowly cooled to room temperature.
The conductance decreased only slightly, and there
were no reversals in the cooling curve. Levenstein and
BodelO have made a similar observation on PbTe films.
10 Private communication.
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:531952 HENRY T. MINDEN
UJ
U
Z
;'!
u
:::> o z o u 10
1.0
0.1 0~--.!:50;;---~IO;r;0--;1F,50"------02~00;;----'2;J;5°;;0---;3;l<0"0--'
TEMPERATURE (·e)
FIG. 5. Variation of conductance with temperature in the
presence of oxygen, film A-I. Oxygen pressures-Curve 1, 6X 10-6
mm; Curve 2, 2XlO-4 mm; Curve 3, lXlO-3mm; Curve 4, 2X
1O-3mm.
Sorption of Oxygen
When oxygen was admitted to contact with PbS
films at temperatures above about 200°C, there was
immediately a sudden decrease in conductance. The
thermoelectric power, which had initially been quite
negative, all but vanished. Simultaneously the pressure
in the system began to fall. If T (Fig. 2) was immersed
in liquid nitrogen, the oxygen pressure invariably fell
to zero over a period of time. The behavior of the con
ductance and the pressure as a function of time is
shown in Fig. 7. On warming the trap at the end of the
run (as described above) the condensed gas was evapo
rated. It was identified as S02 by vapor pressure meas
urements.t The above sequence of events could be
repeated at will merely by introducing more oxygen,
whether or not the S02 was pumped out. Sorption runs
could have been made almost indefinitely, were it not
for the fact that the sorption rate decreased with each
successive run, until it became excessively small.
The amount of O2 sorbed and the amount of S02
evolved was determined for each run. When the S02
was frozen out during the run, the average mole ratio
O2 sorbed to S02 evolved was 2.15±O.22. This ratio
was surprisingly independent of the film temperature,
the initial oxygen pressure, and the particular film used.
On the other hand, when the S02 was not frozen out
but rather allowed to remain in contact with the film
t The author is indebted to R. Harada for making the de
termination. during the sorption, comparatively little S02 was
evolved; the above ratio in these cases averaged about
15.
One film, having 65.9JL moles PbS, sorbed a total of
31.5JL moles O2 and evolved 12.9JL moles S02 in the
course of nine runs, during each of which the S02 was
frozen out.
The sorption rate decreased markedly with decreasing
temperature until it was almost negligible below 200°C.
Since the rate of sorption depended strongly on the
history of the film, however, it was not readily feasible
to determine the temperature coefficient of the sorption
rate.
The kinetics of oxygen sorption were investigated for
the case in which the S02 evolved was frozen out during
the run. Figure 8 is a semilogarithmic graph of pressure
versus time for the last sorption run of the film men
tioned above. Within experimental error the reaction
was first order in the oxygen pressure.
As the oxygen pressure approached zero at the end
of a run, the conductance began to increase until it
ultimately reached its initial value before the intro
duction of the oxygen (see Fig. 6); moreover, the
thermoelectric power became once again negative. In
no case was there observed a permanent change in the
conductivity or thermoelectric power.
DISCUSSION
General Structure of the Film
Although the films absorbed a great deal of oxygen,
they remained n-type semiconductors in the absence of
100r-------------r-----,
! 240
90 !
I 220 /
80 /
/ 200
70 / -/ <J) 180 0 -:r: / u
~ 60 ·0 = / 160 I UJ
UJ a:
~ 50 / ::>
f-
<l <l
f-/ a: 140 u UJ
::> <l.
~ 40 / ;:;
UJ
0 I f-
u 120
I
30 /
100
20
80
10
60
00 20 40 60 80 TIME (MINUTES) 100 120
FIG. 6. Changes in conductance during outgassing, film A-3.
Solid curve, conductance; dashed curve, temperature.
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:53EFFECTS OF OXYGEN ON PbS FILMS 1953
oxygen. Furthermore, the magnitude of the conduc
tivity was never permanently altered by the sorption of
oxygen. Finally the mirror-like layer next to the sub
strate was not tarnished by the oxidation of the films.
These facts strongly suggest that the films were not
homogeneous throughout their thickness. It is proposed
here that the mirror-like layer next to the substrate is
responsible for almost all of the electrical conductivity
of the film. Experience has shown that this mirror-like
layer is roughly O.1,u thick. On top of this substrate
layer is a thicker, nonconducting, dull layer, which
readily reacts with oxygen.
Haradall has shown that the true surface area of
evaporated PbS films varies linearly with film thickness.
This seems to be true for films as thin as 0.05,u up to
thicknesses of 5,u. For annealed films the ratio of the
true area to the apparent area (roughness factor) in
creases by a factor of 45 for every micron increase in
film thickness. It was concluded that the films are quite
porous. The calculated average size of the microcrystals
composing the film is about O.1,u in agreement with
electron microscope data.5 In the subsequent discussion
the particle size for both layers will be assumed to be
O.1,u, and the surface areas will be calculated from
Harada's work.
The Conducting Layer
The conducting layer gives rise to almost the entire
conductivity of the film. On this assumption, a free
100
80
I
I
"1\
0 \ \ / J ""--..... ~
------0 0
2 TIME (MINUTES) '0 / I
"
I 0.0
=--if!
6 oi
8
w u
2 « u
05 z o
lJ
cO
+.0
z.O
FIG. 7. Sorption of oxygen and accompanying conductance
changes, film A-8. Curve A, pressure; Curve B, conductance.
11 R. Harada, J. Chern. Phys. (to be published). ;n z o
a::
u .. 00r--------,------------,-------,
w I00r--------+-'r-------+-~
0::
:::>
If)
Vl w
a::
0:.
'oo;.-----'-----L-,--j-M-tno'(-M-j-NLUT-E-S--"-) --"'ci-no--'
FIG. 8. Sorption of oxygen showing first-order reaction
kinetics, film A-8.
electron concentration of about 1017 was calculated
from the conductivity of films when they were com
pletely outgassed (see Table I). If this was also the
free electron concentration of the nonconducting layer,
the total free electron content (Ntot Table I) of the
films prepared for this investigation (about 16 mg PbS)
was in the neighborhood of 10-9 mole.
When the oxygen gas is admitted in contact with the
film, oxygen is chemisorbed as atoms onto the micro
crystaline surfaces of the film. There the atoms act as
low energy electron acceptors. The total surface area
of the films was about 1000 cm2; the area occupied by
a single oxygen molecule is 20 N. Thus the surface of
the film can accommodate 6X 10-7 mole of oxygen
molecules close packed. This is over 100 times more
oxygen acceptors than are needed to trap out the entire
free electron content of the film. Furthermore, the
amount of oxygen just needed to trap out the free
electrons (10-9 mole) is less than l/lOth the minimum
sorption which could be detected in this investigation
(5X 10-8 mole). This is why the conductivity was
affected without there having been any detectable
sorption.
The positive evidence for oxygen chemisoprtion
rather than bulk diffusion into the microcrystals is
twofold. First, the rate at which the oxygen affects the
conductivity is fairly rapid at all temperatures. Second,
the changes in conductivity brought about by the
oxygen are readily and repeatedly reversible. What
hysteresis there is in the conductivity versus tempera
ture and pressure curves is attributed to the slowness of
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129.22.67.107 On: Sun, 23 Nov 2014 06:21:531954 HENRY T. MINDEN
(a)
(b)
(e)
FIG. 9. Band model of oxygen chemisorption. FL Fermi level;
+donors; 0-surface oxygen acceptors. (a) No oxygen present;
conduction is n-type. (b) Space-charge barrier; conduction is
"intrinsic." (c) Development of p-type conducting region near
surface.
oxygen migration along the grain boundaries between
the microcrystals of the conducting layer (see Fig. 4).
Figure 9 illustrates the band theory of the effect of
oxygen on the conductivity. The fresh film is highly
conducting and n-type (Fig. 9(a)). Chemisorbed oxygen
causes the formation of a positive space-charge region
with a compensating negative surface states' charge
(Fig. 9(b)). The potential barrier arising from the
positive space charge produces a large decrease in
electron conductivity (see Fig. 3). If enough oxygen is
chemisorbed (Fig. 9(c)), the surface region of each
particle is effectively converted to p-type material. The
n-type interior is short circuited by the surface region,
so that the conductivity of the film increases slightly
and is p-type, as was observed. If the temperature is
above 200°C, however, the intrinsic free electron con
centration is sufficiently high to prevent the Fermi level
at the surface (Fig. 9) from dropping much below the
midpoint of the energy gap with any reasonable density
of surface states. The p~region near the surface does
not develop, so that the conductivity does not increase
at the higher oxygen pressures; moreover, the thermo
electric power merely remains close to zero.
The scheme of Fig. 9 can be used to explain the
experiment illustrated in Fig. 5. If a film is cooled from
350°C in the presence of a high enough pressure of
oxygen, chemisorption will occur to an increasing extent
during the cooling. The changes in conductance should
be similar to those occurring when the oxygen pressure
is increased at a constant temperature below 200°C. If it is assumed that when PbS is evaporated onto a
cooled substrate even at pressures of 10-5 mm, the film
acquires a chemisorbed layer of oxygen, the effects of
outgassing the film can be explained (see Fig. 6). Upon
raising the temperature of the film initially, more
electrons gain enough energy to surmount the surface
potential barrier, so that the conductance increases
slightly. As the temperature is still further increased,
however, oxygen desorbs, leaving electrons which re
combine with holes in the surface p-region. This ac
counts for the observed decrease in conductivity.
Finally, when enough oxygen is desorbed, the space
charge induced barrier disappears; the conductivity
increases greatly, and is n-type, of course.
Chemical Reaction
It has been proposed here that only about 20% of a
0.5,u thick film contributes to electronic conduction.
This layer is influenced by the chemisorption of oxygen,
but is much less susceptible to reaction with oxygen
than is the nonconducting layer. Nonetheless, on several
films the amount of oxygen atoms sorbed at high tem
peratures (above 200°C) was equal to or greater than
the total number of PbS molecules in the entire film.
Clearly then, the chemical reaction cannot involve only
one oxygen atom per PbS molecule. A plausible reaction
IS
(1)
This is not the only chemical reaction which can take
place, and it is possible that it is not even a correct one,
since PbO: PbS0 4 has been previously reported as an
oxidation product of PbS films. Reaction (1) most
simply explains the results of this investigation, how
ever, and it will be used for the sake of discussion. Any
general conclusion reached on the basis of this reaction
will apply also to the formation of higher oxidation
products.
In the example quoted in the results, a film composed
of 65.9,u moles PbS had absorbed 31.5,u moles O2• For
this amount of absorbed oxygen the stoichiometry of
reaction (1) demands that m X 31.5 = 21.0,u moles of
PbS be converted into PbS0 3; hence on the assumption
of reaction (1) 21.0/65.9=32% of the film was con
verted.
On the other hand, even after this 32% of the film
had reacted, the reaction rate was becoming extremely
slow. Figure 7 is the last sorption run for the above
film. It will be noted that this run took some 2t hours,
while earlier runs took less than 15 minutes. So it is
quite possible that the outer layers of the film can be
saturated by a great deal of oxygen without the inner,
conducting layer being at all affected under the con
ditions of the experiment.
The evolution of S02 can be readily explained in
terms of the well-known decomposition
(2)
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When the S02 is frozen out, the mole ratio of O2 sorbed
to S02 evolved is very nearly 2. A combined reaction
can be proposed for this case.
It is doubtful whether this reaction has much sig
nificance, since PbS0 3: 3PbO does not seem to be a
known compound. Reaction (3) is mentioned only
because the 2: 1 ratio of O2 to S02 occurred under a
surprisingly wide range of conditions. When the S02
is not frozen out, however, its presence in contact with
the film seems to inhibit greatly reaction (2).
When the oxygen is first admitted to the cell, the
conductivity decreases, and it is almost intrinsic, as
described above. As the oxidation of the film pro
gresses, the oxygen pressure in the cell decreases. The
pressure goes so low, in fact, that at the end of the
reaction the oxygen adsorbed on the conducting layer
desorbs and reacts with the nonconducting layer. In
this manner the conductivity of the film eventually
should and does in fact rise to its original value.
Furthermore, in spite of repeated oxidation, the final
conductivity was always n-type, as would be predicted
from this model.
CONCLUSIONS
Lead sulfide films which are evaporated in vacuum
onto a smooth glass substrate are composed of two
layers. Next to the substrate there is a O.lJL thick
mirror-like layer which is responsible for almost the
entire electrical conductivity of the film. The conduc
tivity of this layer is affected by oxygen, but there is
no permanent oxidation. On top of this conducting
layer there is a nonconducting layer which is readily
oxidized. The microcrystals of the conducting layer can re
versibly chemisorb oxygen. The oxygen forms surface
acceptor states on the n-type particles, and a positive
space-charge barrier is produced. This barrier greatly
reduces the conductivity of the film. If enough oxygen
is chemisorbed, however, the space charge p-region can
be sufficiently well developed to provide a shorting p
type conduction path around the n-type interior of the
microcrystal.
When the temperature of the film is above about
200°C, in addition to being adsorbed by the conducting
layer, the oxygen reacts chemically with the upper,
nonconducting layer of the film. This reaction involves
the absorption of relatively large amounts of oxygen
and also the evolution of S02. The reaction goes so far
to completion that oxygen is desorbed from the con
ducting layer with a consequent rise in n-type conduc
tivity. Under the conditions of these experiments the
conducting layer is not affected by the oxidation process.
It is concluded that when lead sulfide films are baked
in oxygen, two relatively unrelated physico-chemical
processes occur. A nonconducting layer of the film
undergoes a gross chemical reaction with oxygen. The
chemical changes produced in this layer by oxidation
have little or no effect on the electrical properties of the
film. The observed presence of macroscopic sulfate and
oxide phasess-7 is not a relevant condition for alterations
in the electrical conductivity or for photosensitization.
On the other hand, even without gross exidation of
the film, the electrical properties are readily affected
by chemisorption of oxygen on the conducting layer of
the film. Chemisorption also occurs during the gross
oxidation process. The role of chemisorption in changing
the conductivity has been described in this paper and
will be elaborated upon in a subsequent theoretical
paper.
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1.1742023.pdf | Paramagnetic Species in GammaIrradiated Ice
Max S. Matheson and Bernard Smaller
Citation: The Journal of Chemical Physics 23, 521 (1955); doi: 10.1063/1.1742023
View online: http://dx.doi.org/10.1063/1.1742023
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/23/3?ver=pdfcov
Published by the AIP Publishing
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16THE JOURNAL OF CHEMICAL PHYSICS VOLUME 23, NUMBER 3 MARCH,1955
Paramagnetic Species in Gamma-Irradiated Ice
MAX S. MATHESON AND BERNARD SMALLER
Chemistry Division, Argonne National Laboratory, Lemont, Illinois
(Received June 25, 1954)
The paramagnetic resonance spectra of H20 and D20 ice
irradiated at 77°K have been examined. The absorbing species
have been identified as Hand OH (or D and OD), a significant
result for the radiation chemistry of aqueous solutions. The yield
of radical pairs/l00 ev is about 0.14. The hyperfine splittings of
the H doublet and D triplet area factor of 16 less than are observed
INTRODUCTION
FOR some years radiation chemists have accepted
the hypothesis that Hand OH are reactive species
produced in water subjected to ionizing radiations.
Considerable indirect evidence has been accumulated
in support of this postulate,! but no direct identification
of these species in irradiated water has been possible up
to the present because of their short lifetimes. However,
it occurred to us that in ice at sufficiently low tempera
tures these radicals or others if formed might be stable
and could be detected by the sensitive method of
paramagnetic resonance absorption. This hope has
recently been confirmed.2 Further, since thermolumines
cence has been found in irradiated ice,S we hoped to
identify the trapped electrons responsible.
EXPERIMENTAL
Materials
High-purity triply distilled H20 and D20 (99.6
percent D) were obtained from Hart4 of this labora
tory. The organic content of these materials is less
than 10-6 molar, and the inorganic impurities are less
than 5X1Q-7 molar. Matheson Company formic acid
(98-100 percent) was distilled (at 200 mm and 59°C)
through a 30-plate column at 5 percent take-off
(nD20 1.3716). Anhydrous ammonia was obtained from
Ohio Chemical and Surgical Equipment Company.
Hydrogen peroxide (90 percent) from Buffalo Electro
chemical was not further purified.
Preparation of Samples
Ice samples were prepared as follows: The Pyrex
apparatus of Fig. 1 was sealed to a vacuum line at A
with 5-10 cc of triply distilled water in B2, and the
water was thoroughly degassed by pumping on it
while it was frozen with dry ice. The water was thawed
5 times between pumpings. During pumping the
apparatus of Fig. 1 was separated from the pumps by a
liquid nitrogen-cooled trap and a tube containing gold
IF. S. Dainton, Brit. J. Radiol. 24, 428 (1951) has reviewed
some of the evidence.
2 Smaller, Yasaitis, and Matheson, Phys. Rev. 94, 202 (1954).
3 L. 1. Grossweiner and M. S. Matheson, J. Chern. Phys.
22, 1514 (1954).
• E. J. Hart, ]. Am. Chem. Soc. 73, 69 (1951). in the gas phase. This result is attributed to the effect of the
solid on the electronic state of the H or D atoms. The spectra
of H20. in H20 and D.O. in DzO irradiated at 77°K support the
identification of OH (or OD) absorption. In annealing experiments
the Hand OH disappear near 115°K. Results on solid ammonia
and solid formic acid irradiated at 77°K are also described.
foil to trap out oil and mercury. The sample was
sealed off at A when a pressure of ca 5X1Q-' mm was
attained with B2 at -78°e. Next, liquid to a depth of
12 cm was poured into each tube T and these were
sealed off at B while the liquid was frozen by dry ice
or liquid nitrogen. The tubes were thawed, placed inside
slightly larger glass test tubes, and lowered into acetone
at -15 to -25°C so as to freeze at a rate of 12 cm in
90 minutes. Clear transparent ice, usually without
cracks, resulted. The tubes were stored at -25°e.
This healed any cracks.
When ice was to be irradiated, a tube of ice was cut
into lengths in a cold box at -20°e. After warming
the lengths of Pyrex tube with a rubber gloved hand,
the ice cylinders could be pushed out of the tubes to
give samples only about 0.1 mm less in diameter
than the i.d. of the Pyrex tube. These samples were
put in 7.5 mm i.d. Pyrex test tubes which were then
suspended in liquid nitrogen for irradiation.
Samples of solid formic acid and of ice containing
sodium chloride were prepared as described above,
except that the formic acid was frozen at O°e. Ammonia
was dried with sodium' and distilled in a vacuum line
FIG. 1. Apparatus for
preparing degassed tubes
of water. T A T
• R. T. Sanderson, V acuum Manipulation of Volatile Compounds,
Oohn Wiley and Sons, Inc., New York, 1948), p. 97.
521
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16522 M. S. MATHESON AND B. SMALLER
..L • U.H.F. REGENERATIVE DETECTOR ------- - -------------------, ,
0-30MA, I
r---"'--'lfOizfOi •• ll>a '---.WIM-( !) l Bt 220V.
OHMITE I 1:1 F.e.
.0005mfd .aiml' :!l: ~#UOIO
I
I
I ,
I
I
~5~f~5CATION :
l-:;::Jh~_"--_--t+''''''1--<---"-a--=Z5",m",,ml'-- ___ i!) : GROUND
I _ .005mtd I
I ~ :!) I .V. L ____________________________ ~
FIG. 2. Quarter-wave coaxial line oscillator-detector circuit.
directly into the tubes in which it was to be frozen
The ammonia was frozen and stored in dry ice. The
samples were removed from the tubes by working
quickly at room temperature, cutting the tubes and
pushing the solid ammonia into liquid nitrogen.
The 4.44 molar H202 (aq) tubes were frozen 4 mm
per minute at -30°C and stored in dry ice, as the
solution was partially liquid at -25°C. Samples were
prepared by working rapidly in a cold box at -25°C.
The dilute peroxide samples were similarly prepared
except for four hours storage at -25°C just before
the tubes were opened. The peroxide solutions could
not be thoroughly degassed because of continuing
slow decomposition, and the frozen samples were
always cloudy.
Resonance Detection Equipment
The detection equipment centered around a quarter
wave coaxial line oscillator-detector whose circuit is
shown in Fig. 2. The output was fed to a high-gain
narrow-band intermediate-frequency amplifier using a
dual-field modulation scheme previously described.6
For recording results, a conventional low-frequency
lock-in amplifier was added with recorder output,
converting the oscillator resonance absorption signal
into roughly its second derivative. The variable
capacitor regeneration control of the oscillator was
found to perform excellently, giving quiet continuously
variable oscillation levels. The oscillator noise output
level was of the order of 1 fJ-V, while resonance signal
strengths were usually 100-1000 fJ-V. The limiting
sensitivity as determined by a 2,2-diphenyl-l-picryl
hydrazyl sample was of the order of 1014 spins, compar
ing favorably with the conventional microwave
detection systems. The entire coaxial line was mounted
in a double-wall Dewar assembly for the low-tempera
ture measurements.
RESULTS AND DISCUSSION
Paramagnetic Absorption Spectra in Irradiated
H20 and D20
When samples of pure H20 and D20 ices are irra
diated with C060 l' rays at nOK as described above,
6 B. Smaller and E. Yasaitis, Rev. Sci. lnstr. 24, 991 (1953). the absorption spectra of Fig. 3 are typical of those
measured at nOK. The various features of these
spectra are satisfactorily explained by assuming that
they are due to the existence of a free spin (i.e., unpaired
electron) located near an H or a D nucleus. Comparison
may be made of the results found with the spectra
obtained for Hand D in the gas phase.7 First, a doublet
structure is consistent with a spin of ! for the proton,
and a triplet structure is consistent with a spin of 1 for
the deuteron, as is observed (Fig. 3). The other nuclear
species, oxygen (spin 0), obviously yields no hyperfine
splitting, and the free spin concentrations are perhaps
10 OOO-foid the concentration of any impurity atoms.
The A and B peaks are identified respectively as the
strong field transitions [mI(!~!), ms(!~-!)J and
[mI(-!~-!), ms(!~-!)], while the E, F, and G
peaks are identified as [mI(1~l), ms(!~-!)J,
[mI(~O), ms(!~-!)J, and [ml(-l~-l), ms(!~
-!)]. These transitions would correlate, respectively,
with the following transitions of Nafe and Nelson using
the F and mF quantum numbers appropriate to their
weak field case (1,l~O,O) and (1,~l,-l) for the
--,------;~-,_-.,.--,----- 285 g a U S5
H20
., ..,
:::I -
r a.
E Ho <X
0 H
c
0>
(Ii
k~
Time--
-----c,----.,----.,----r-- 285 ga U ss
02°
.,
'0
.~ a.
E Ho <X
Ci H
c
0>
(Ii
Time-
FIG. 3. Paramagnetic resonance in -y-irradiated H20 and D20
ice, temperature 77°K. - - - - Field. Signal
amplitude during slow sweeping of magnetic field through
resonance.
7 J. E. Nafe and E. B. Nelson, Phys. Rev. 73, 718 (1948).
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16PARAMAGNETIC SPECIES IN GAMMA-IRRADIATED ICE 523
gaseous H atom and (!, !~!, !), (!, !~!, -!), and
(!, -!~!, -!) for the gaseous D atom. For the strong
field case A and B are expected to be of equal intensity
as observed (Fig. 3) and E, F, and G to be in the ratio
1: 1: 1. The E, F, and G peaks are not in the expected
ratio and this may be an instrumental effect occurring
only when lines are close together. The peroxylamine
disulfonate radical also gives a close tripletS and for
this case a 1: 1: 1 intensity ratio was observed for low
field modulations and approximately 1: 2: 1 for higher
modulations. The ratio of hyperfine splitting constants
calculated from these spectra is 4.0±0.4 in good
experimental agreement with the ratio of atomic
beam values of WH/WD=4.3.7 However, the splitting
constants observed (85.5 Mc/sec in irradiated H20)
are considerably less than the free atom values7 (1420
Mc/sec for H in a 15 state). For an assumed 25t state
the separation jJ of the hyperfine structure doublet
terms in absolute frequency units is given by the
Fermi9 relation,
where I is the nuclear spin in units of Ii, J.ko is the Bohr
magneton, J.kN the nuclear magnetic moment in absolute
units, and 1/;(0) is the Schroedinger wave function
evaluated at r= o. At r= 0 for a hydrogenic orbit
1/;n/(O) = (l/n")(Z3/ n3ao3),
where Z is the nuclear charge, n is the principal quantum
number, and ao is the Bohr radius. It is readily conceiv
able that the effect of the strong intermolecular dipole
field would be a promotion of the electron from the
free atom ground state to higher orbits with a sub
sequent reduction of hyperfine splitting constant. An
alternative useful classical approach may be to consider
the proton and associated electron to be embedded in a
dielectric of dielectric constant K where the nuclear
attractive potential is reduced by a factor K. Then
1/;1.2(0) is given by (l/lr)(Z3fK3a03). The value of K
required to reduce jJ from 1420 Mc/sec to 85.5 Mc/sec
is 2.55. Auty and Cole10 have found the limiting high
frequency dielectric constant to be 3.10 from -0.1
to -65°C for H20. One might well expect the effective
dielectric constant in this case to be somewhat less
than the bulk value, since close to the nucleus the
proton field is probably not shielded. Although our
accuracy does not warrant such a refinement, one may
multiply WH/WD=4.35 by (KD20/KH20)3= (3.04/3.10),3
using Auty and Cole's values and the ratio becomes 4.1.
In considering the possible importance of the inter
actions discussed above one needs to differentiate
between two types of "solutions" of H atoms. First,
if one could dissolve H atoms in liquid H20 and then
freeze the water, the structural units would take up
8 B. Smaller and E. L. Yasaitis, J. Chern. Phys. 21, 1905
(1953).
8 E. Fermi, Z. Physik 60, 320 (1930).
10 R. P. Auty and R. H. Cole, J. Chern. Phys. 20, 1309 (1952). positions consistent with the preservation of atomic
and molecular radii. The resulting "uncrowded"
paramagnetic center will show little of the promotional
effect. Second, in the case suggested by us an H atom
is produced by the dissociation of an H20 molecule
[see Eq. (3)] in an extensive crystal lattice at low
temperature. In this second case since the van der Waals
radius of a hydrogen atom is 1.2 A as against the 0.3 A
covalent radius one expects a rather crowded hydrogen
atom. The proton-electron pair interacting strongly
with the surrounding crystal molecules is clearly not
the equivalent of a ground-state gaseous hydrogen
atom, but also it is not covalently bonded to a specific
atom in its environment, so that the pair may be
designated as a strongly perturbed hydrogen atom ..
In this connection it is interesting to note that
Livingston, Zeldes, and Taylorll have recently shown
that.in irradiated H2S04 or 1: 1 molar H2S04: D20 the
hyperfine splittings are very close to those obtained
for gaseous Hand D atoms. We have confirmed this for
these substances using 350 Mc/sec and obtaining the
weak field Hand D spectra to be expected from their
results. Thus the splittings obtained for irradiated
H2S04 and irradiated H20 are quite different. When
frozen solutions of 4.4, 1.0, 0.1, and 0.001 molar H2S04
in H20 are irradiated, the resonance absorption
corresponding to that in pure H2S04 decreases in
intensity with decreasing acid concentration. Simultan
eously it is found that the relatively weaker peaks
observed in pure irradiated H20 are observable even in
4.4 molar H2S04• The magnitudes of the splittings of
the two doublets remain unchanged as the acid con
centration is varied.
The appearance in H2S04-H20 mixtures of the
resonance doublets found in the pure irradiated
components is probably to be expected, since H2S04
forms a compound with 4H20 molecules and this
compound plus water will freeze to give a eutectic of
36.8 percent H2S04 ("-'4.4M). Presumably, therefore
in the H2S04-H20 mixtures one obtains crystals of
both H2S04• 4H20 and H20.
The broadening of the resonances in irradiated ice
may be assumed to be due to neighboring proton (or
deuteron) interactions and to correspond roughly to
spacings of 2-3 A in agreement with the molecular
spacing of 2.76 A. The relative line widths (6 gauss for
irradiated H20 and 2 gauss for irradiated D20) are in
the correct ratio for the assumed magnetic species
(H and D).
Absorption Spectra of Irradiated Frozen Solutions
of H202 in H20 and D202 in D20
At T= 4 OK a second doublet structure was detected
in pure H20 ice at an intensity equal to the A, B
peaks with separation 10 gauss as compared to the
A, B peak separation of 30 gauss, while the D20 ice
11 Livingston, Zeldes, and Taylor, Phys. Rev. 94, 725 (1954).
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16524 M. S. MATHESON AND B. SMALLER
resonance spectrum showed a distortion of the F peak
attributable to a second triplet structure.
Since, as already noted, the presence of Hand OH in
irradiated water has been deduced by indirect evidence,
it is suggested that the OH free radical may be respon
sible for this second doublet. While the electronic state
for gaseous OH is 271"h the electric field splitting in ice
can be expected to quench the orbital angular momen
tum to give the residual Kramer's doublet, which
would be responsible for the resonance. A shorter
relaxation time may be expected for this state than for
the St of the hydrogen owing to orbital coupling and
hence can only be detected at very low temperatures.
In the OH radical the unpaired electron is largely
localized on the oxygen atom. The hyperfine splitting,
however, is due to the proton of the OH, which is
0.97 N2 from the 0 atom in the gaseous state. From
the C, D separation of Fig. 4, rAy [the average proton
electron distance calculated from (1/r3)AY] is 1.02 A. The
effect of dielectric in enlarging the electron orbital
would probably change the mean proton-electron
distance. However, with our present knowledge the
1.02 A is not incompatible with the postulate that we
are dealing with an OH radical.
Further support for the postulate that the C, D
peaks are due to OH is found in experiments with
'Y-irradiated frozen solutions of H202 in H20. The
spectra measured at 4°K for pure H20 and at 77°K for
0.280M H202 in H20 and 4.44M H202 in H20 all
irradiated at 77°K, are compared in Fig. 4. (Qualita
tively the spectra for H202 in H20 measured at 4° and
77°K are similar.) From data such as given in this
figure two phenomena may be noted: (1) the separation
of the C, D peaks increases with increasing H202
concentration; and (2) as the H202 concentration
increases, the C, D peaks become stronger while A and
B attributed to H weaken and disappear. The increasing
separation may be due to changing environment as
H202 is added,13 and in support of this it may be noted
that in going from pure H20 to 0.280M H202 the C, D
and A, B separations increase exactly proportionately.
The second phenomena can be correlated with the
present theory of the radiation chemistry of aqueous
systems.
In water subjected to ionizing radiation the following
reactions are assumed to occur:1
H20 (aq) + radiation----)H 20+(aq) +e, (1)
H20+(aq)->H+(aq)+OH, (2)
e+ H20 (aq)->H20-(aq)->H +OH-(aq). (3)
Equations (2) and (3) are exothermic because of the
12 G. Herzberg, Molecular Spectra and Molecular Structure. [.
Diatomic Molecules (Prentice-Hall, Inc., New York, 1939), p. 491.
13 A. G. Mitchell and W. F. K. Wynne-Jones, Discussions
Faraday Soc. No.5, 161 (1953) find H202 molecules fit into the
water structure and draw it together. Also, an OH formed from
an H202 molecule should have a different local environment than
one formed from an H20 molecule. hydration energies of the ions formed. In ice at liquid
nitrogen temperature where solvation effects are less
Eq. (2) would still be exothermic, but Eq. (3) is
estimated to be endothermic by about 1 ev.3 However,
most secondary electrons start out with more than 1 ev
of energy. Further, it is assumed for fast electrons
(from'Y rays) that radical pairs are produced in clusters
(average of 3 pairs per cluster) with the OH's near the
electron path and the H's more diffusely distributed
(perhaps 150 A radius for a cluster).14 Some H2 and
H202 are formed either directly as molecular products
t
c c:
C'
III ......--.,_-.__-.__ 285 go u 55
H20-Pure
4°K
1
H
-~~-~-~---285goU55
----''------'_--L-_-'- ___ O H202 O.28M
77°K
--.--.,.---,--.....- 285 gou 55
-L_-L __ ~_~__ °
Time-4.44M. H202
77°K
H
FIG. 4. Effect of H202 on paramagnetic resonance in irradiated
ice. ---- Field. Signal amplitude during slow
sweeping of magnetic field through resonance.
14 D. E. Lea, Actions of Radiations on Living Cells (The Mac
Millan Company, New York, 1947), p. 48.
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16PARAMAGNETIC SPECIES IN GAMMA-IRRADIATED ICE 525
or from radicals which are so close together that added
solutes do not affect the combination. For COBO 'Y rays,
however, 79 percent of the H20 decomposition gives
radicals. IS
From the above, one would expect Hand OH to be
produced in approximately equal quantities in pure
ice and the C, D peaks should be about equal in intensity
to the A, B peaks. A theoretical synthesis from four
lines of equal height and with widths and spacings as
found experimentally for A, B, C, and D gave a result~
ant pattern closely similar to that found experimentally
for pure ice (Fig. 4). Further, either of the following
reactions occurring to an increasing extent as the HZ02
concentration is increased could account for the
weakening and disappearance of the H peaks in Fig. 4,
and the simultaneous intensification of the OH ab-
sorption:
H+H202~H20+0H, (4)
e+ H202~OH-+OH. (5)
In reaction (4) the H would be formed as an immediate
neighbor of the peroxide by Eq. (3), while in Eq. (5)
the electron would react directly on the HZ02• Since
Eq. (5) is energetically more favorable than Eq. (3),
perhaps it is to be preferred over Eq. (4) at high H202
concentrations.
Other paramagnetic species which may be postulated
to occur in irradiated ice are H20+, HzO-, and electrons
trapped at imperfections. However, the HzO+, if the
unpaired electron is in an unquenched p orbit on the
oxygen,t6 would not give a resonance absorption
centering at g= 2.0. If the crystalline field splitting
results in a Kramer's doublet for the low-lying state,
then the electron orbit will be symmetrical with respect
to the two protons of HzO+. In this case a triplet
spectrum, resolved or unresolved, centering at g= 2.0
should be produced, as the proton spins may be
parallel or antiparallel. This does not accord with our
results. A similar argument may be advanced against
HzO-. Electrons trapped at imperfections such that
they cannot be assigned to any particular H20 molecule
should give a single peak at g= 2.0 of more or less
breadth, again in contradiction with our results.
Yields o(Products
By comparing the absorption obtained for a given
sample with the absorption of a 2.9-mg sample of
2,2-diphenyl-l-picrylhydrazyl the absolute concentra
tion of paramagnetic species was estimated for the
irradiated sample. (See Appendix A.) The yields of
radical pairs/lOO ev so estimated for different samples
are summarized in Table I. The measurement of
radical pairs is believed accurate to about a factor of
2. The average radical pair yield for H20 and DzO is
IJ; E. J. Hart, ]. Phys. Chern. 56, 594 (1952).
16 The lowest ionization potential of H20 in the gas phase is
for the ionization of one of the lone pair electrons on the oxygen.
R. S. Mulliken, Phys. Rev. 40, 55 (1932). TABLE 1. Radical pair yield for COM gamma irradiations.
Radical Yield
Dosage Temp. pair cone. (radical
(ev/g) Meas. (millimoles/ pairs/
Sample No. X 10-" (OK) liter) lOOev)
H2O 7 3.27 4 0.35 0.064
D.O 8 3.27 4 0.95 0.18
H2O 9 6.06 77 2.79 0.23
D20 10 5.86 77 1.78 0.18
H2O 11 7.83 4 1.84 0.14
D20 12 7.57 77 0.814 0.064
D20 12 7.57 4 0.82 0.068
H2O 21-2 7.06 4 1.26 0.109
H2O 21-1 20.0 4 2.58 0.075
H2O 24--5 8.0 77 3.04 0.23
H.0(10-;M NaCl) 15 8.0 77 2.89 0.20
H20(1O-2M NaCl) 16 8.0 77 3.08 0.23
H20 (0. 28M H2O.) 22-1 3.51 77 2.66 0.45
H.0(0.28M H2O.) 22-2 7.54 77 5.0 0.39
D20(0.194M D.02) 23-4 3.80 77 1.86 0.30
D.O(0.194M D.O.) 23-1 8.20 77 3.86 0.28
H.O(4.44M H202) 17 4.52 77 9.65 1.3
HCOOH (pure) 18 3.99 77 32.6 4.9
NHa (anhyd.) 20 15.72 91 2.36 0.086
0.14. If all radical pairs yielded Hz and H202 on
warming, then 0.07 molecule Hz (or H20z)/100 ev
would be obtained. Or, if 50 percent of the radical
pairs yielded H20, then 0.035 molecule Hz/l00 ev
would result. For tritium (3's at nOK Ghormley and
AllenI7 measured 0.27 molecule gas/lOO ev for the
initial yield. The agreement is adequate in view of the
possible error factor of 2 in our measurements. Also
molecular H2 or H202 produced at nOK would of
course not be detected in our work.
For dilute peroxide solution (0.2-0.3M) the radical
pair yield is about 2-to 3-fold higher than for pure
H20 or D20 (Table I), while for concentrated Hz02
(4.44M) the yield is still higher. The higher yields
with increasing peroxide concentration can be explained
as due to the favorable energetics of reaction (5) as
compared to reaction (3).
Samples 21-1 and 21-2 of H20 were run specifically
to test whether the yield might fall off with increasing
dosage. So that the relative figures might be significant,
the samples were prepared and handled identically
except for the amount of radiation. The irradiations of
the samples were arranged so that the irradiations for
both samples terminated at the same time, and the
resonance absorption measurements were made in
immediate succession. With the precautions used the
relative figures in Table I are believed to show a
small decrease in yield with increasing dosage. The
yields for dilute H202 in H20 and DZ02 in D20 show a
similar small decrease with dosage with less than 10
percent of the peroxide consumed.
It has been proposed that in the x-ray-excited fluores
cence of ice the emitting species is an alkali ion that
has captured an electron.s Therefore, ice containing
sodium chloride was irradiated and examined for the
17 J. A. Ghormley and A. O. Allen, Oak Ridge National Labora
tory Report ORNL-128 (September 1, 1948).
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16526 M. S. MATHESON AND B. SMALLER
100,-----------------,
..... 50 •
:I:
" W
:I:
" ..
W
Q.
W
<.) ° z .. z ° 0
'" w a: 10
r!K X 102
FIG. 5. Annealing of resonances in irradiated 0.28M H202 in H20. o sum of A, B peaks .• sum of C, D peaks.
paramagnetic resonance absorption which would be
typical of sodium (nuclear spin !). However, in ice
containing sodium chloride only the paramagnetic
species previously observed in pure ice were found
(Table I). However, such a small fraction of the sodium
ions in O.OlM NaCI are effective in fluorescence,3 that,
if this fraction existed as sodium atoms, the concentra
tion would be below the detection limit of our apparatus.
Annealing Experiments on Aqueous Systems
The relative stability of the two types of free radicals
was investigated by their temperature sensitivity.
Two methods of annealing were used, pulse and
continuous. For pulse annealing the samples were
transferred from the resonance measurement Dewar to
a test tube maintained at the annealing temperature.
After five minutes annealing, during which the tempera
ture in the test tube was followed as it came to equili
brium, the sample was transferred quickly back to the
absorption apparatus for further measurements at
liquid nitrogen or liquid helium temperature. For
continuous annealing the Dewar in which absorption
was measured was filled with cooled liquid freon-12 or
freon-13 (Matheson Company), and the change of
resonance absorption as a function of freon temperature
(next to sample) was followed. Freon-12 could be
cooled to 115°K. By careful addition of liquid nitrogen
freon 13 could be cooled as a liquid to about 78°K and
allowed to warm from that temperature. At tempera
tures below the mp (92°K) the freon-13 often tended
to from a crust of solid on top of the liquid.
In pulse annealing of H20 and D20 (samples 9 and
10) with measurement at 77°K it was observed that
the peaks of Fig. 3 disappeared rather sharply (un
changed at 108°K, essentially gone at 123°K). For
pulse annealing of H20 (sample 11, Fig. 4) measured
at 4°K, 5 minutes annealing at 118°K gave a greater drop in the A, B peaks than in the C, D. Another 5
minutes at 133°K left only a very weak doublet of
less separation than C and D which may not have
been due to the sample. These results show that the
H peaks disappear completely at 115± WOK while the
OH peaks disappear at very slightly higher temperature.
For continuous annealing (heating rate approximately
l°/min) of dilute H202 in H20 (Fig. 5) the H peaks
(A, B) disappear near 1000K in agreement with the
above results. However, almost half of the OH peaks
(C, D) anneal out at 1000K as before and the remainder
near 145°K. It is possible that the portion of the OH
peaks annealing at 1000K is due to OH originating
from H20+ [reaction (1) and (2)J and the portion
annealing at 145°K is due to OH derived from H202
[reaction (5)]. By the radiation theory discussed above
the OH's from H20+ would be in a closer cluster than
the OH's from H202. The OH's which are effectively
adjacent to each other initially might be expected to
disappear at a lower temperature than the others.
Further, neglecting the small amount of H present and
the formation of H202, the two types of OH should be
formed in equal amounts. An alternative explanation
that H reacts with OH'at 1000K and the remaining OH
with itself at 145°K cannot account for all of the OH
disappearing at 100 oK, nor can it account for the
results in 4.44M H202 below.
The results obtained by continuous annealing of
dilute (0. 194M) D202 in D20 are similar to those
described above. The weakened D peaks (E, G, Fig. 3)
and about half of the central peak disappeared at about
100oK. The remainder of the central peak (presumably
OD derived from D202) annealed near 135°K.
Likewise in the pulse annealed concentrated (4.44M)
H202 in H20 (measured at 4 OK) slightly more than
half the signal disappeared at 120oK, while the re
mainder largely disappeared at 200oK±20. This
indicates that the second species is also more stable
in the concentrated peroxide.
Results with Irradiated Solid Ammonia
Gaseous ammonia is decomposed by ultraviolet
radiation (2000-2100 A) to give Hand NH2 in the
primary process.I8 These intermediates have been
proposed also in the decomposition of ammonia in
electrical discharges.I9 Therefore, any decomposition
induced in solid ammonia at liquid nitrogen temperature
by 'Y rays might possibly yield Hand NH2 also. Solid
ammonia irradiated with C060 'Y rays exhibited the
resonance spectrum shown in Fig. 6. The outside peaks
A, E, (splitting 77 gauss) are probably due to H atoms;
if so, then the splitting of the H hyperfine doublet is
18 W. A. Noyes, Jr., and P. A. Leighton, The Photochemistry of
Gases (Reinhold Publishing Corporation, New York, 1941), p. 371.
19 G. G10ckler and S. C. Lind, The Electrochemistry of Gases and
Other Dielectrics (John Wiley and Sons, Inc., New York, 1939),
p.210.
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16PARAMAGNETIC SPECIES IN GAMMA-IRRADIATED ICE 527
less in ice than in solid ammonia. Since the high
frequency dielectric constant of solid ammonia is larger
(3.36 at 87°K)20 than for ice, one might expect the
splitting to be less in ammonia. However, in ice the
O-H···O distance is 2.76 A and in solid NHa the
N -H· .. N distance is 3.38 A with the H about 1.0 A
along the hydrogen bond in both cases.21 This longer
and weaker hydrogen bond and the lower density of
ammonia22 as compared to ice suggest that the H atom
may be less crowded in ammonia, so that the effective
dielectric constant is lower in solid NHa than in ice.
If the outer doublet is assigned to H, the inner
triplet is probably NH2. The observed splitting would
then most likely be due to the two attached hydrogens
since the nuclear moment of the nitrogen is so small.
The irradiated solid ammonia revealed a complex
annealing pattern. The A, B, D, E peaks disappeared
near 100-110oK not far from the temperatures at
which Hand OH disappeared in pure ice. However,
the C peak (due to the formation of a new and unidenti
fied species at g= 2.0) increased in intensity and
finally annealed out at about 140oK.
Results in Solid Formic Acid
Solid formic acid was irradiated at 77°K and ex
amined for paramagnetic resonances. Since irradiated
formic acid might contain OH and H, it was our hope
that resonances observed would be helpful in the
interpretation of the results in ice. The hope is not
yet realized. Two peaks of 15 gauss separation centered
at g= 2.0 were measured at 77 oK in the irradiated formic
acid. These might be OH for the following reasons: (1)
In the gas-phase photolysis of HCOOH no H atoms are
observed23 but OH radicals have been identified.24
(2) The high-frequency dielectric constant is about
2.9 at -10 to -50°CZ5 so that OH should show a
slightly larger splitting here than the 10 gauss observed
in ice. (3) In such possibilities as
O·
/
HC=O, HC =0,
and . COOH, the unpaired spin is on an atom 1.09 A or
further from the H which is responsible for the hyperfine
structure, and therefore the observed splitting seems
to be too large for such species in this medium. H is not
finally ruled out, and some charged species may be
possible. Further work is planned on these systems
"" c. P. Smyth and C. S. Hitchcock, J. Am. Chern. Soc. 56, 1084
(1934).
21 L. Pauling, Nature of the Chemical Bond (Cornell University
Press, Ithaca, 1942), pp. 168, 334.
22 L. Vegard and S. Hillesund, Chern. Abstracts 38, 4488 (1944)
find x-ray densities at -185°C, NH3=0.788, H.O=0.942 g/cm3•
23 See reference 18, p. 367 .
• 4 A. Terenin, Acta Physiochim. U. R. S. S. 3, 3181 (1935).
26 J. F. Johnson and R. H. Cole, J. Am. Chern. Soc. 73, 4536
(1951). :---,--,.----.:--- 285 gauss
t
Q)
-0
::> -C. Ho E « H
C c:
.~
({)
Time-
FIG. 6. The paramagnetic resonance absorption spectrum of
,,-irradiated solid ammonia. ----Field. Signal
amplitude during slow sweeping of magnetic field through
resonance.
involving identification of chemical products as well
as resonance spectra measurements in hope of more
positive assignment of the spectra. If the OH assign
ment is correct, the He = 0 is not observed, either
because it disappears by combination or because of
factors such as relaxation time.
On annealing, a strong single peak at g= 2.0 begins
to grow in at -103°C and reaches a maximum at
-94°C. After this peak again disappears, the two
side peaks are again observed and anneal out between
-56 and -40°C. No correlation between thermo
luminescence and resonance annealing was noted in
ice or solid ammonia, but in the solid formic acid a
single strong glow peak is observed at -99°C which
may correlate with the resonance which maximizes at
-94°C.26
APPENDIX A. ESTIMATION OF RADICAL YIELDS
For the dual modulation scheme using frequencies
FL and Fr of amplitudes aL and ar the total integrated
intensity of a signal of output height e and width ~H
can be calculated approximately from the following
formula:
1 fco HfH -f e(H)d}H.
aLar 0 0 0
For any given line shape one can approximate relative
values of the above integral by (1)
1
--emax(~H)3,
aLaI (1)
where ~H is the measured line width. The experimental
line shapes obtained for 2,2-diphenyl-l-picrylhydrazyl,
and for irradiated H20 or DzO ice samples appeared
identical, so the above approximate formula (1) was
26 The apparatus used for thermoluminescence measurements
has been described in reference 3.
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128.83.63.20 On: Thu, 27 Nov 2014 04:02:16528 M. S. MATHESON AND B. SMALLER
used to calculate radical concentrations in the irradiated
samples by comparison with the signal from a known
amount of the hydrazyl. Comparison of sample and
hydrazyl was always made at the same temperature
of measurement.
THE JOURNAL OF CHEMICAL PHYSICS ACKNOWLEDGMENT
The authors are grateful to E. L. Yasaitis for the
design and construction of the resonance detection
and auxiliary equipment, and to O. C. Simpson for
encouragement and helpful discussions.
VOLUME 23. NUMBER 3 MARCH. 1955
Study of Nuclear Resonance of the Supercooled Rotational Transition
of 2,3-Dimethylbutane*
HERBERT SEGALLt AND J. G. ASTON
College of Chemistry and Physics of the Pennsylvania State University, State College, Pennsylvania
The proton magnetic resonance lines during the transition of the supercooled variety of 2,3-dimethyl
butane to the normal state have been investigated. An increase in rotational characteristics is followed by
an exponential decrease in peak height of the resonance line. This is interpreted in terms of a transition
around Frenkel holes. An observed asymmetry in the line shape is treated mathematically on this basis.
INTRODUCTION
THIS work is part of a series of investigations of
rotation in solid solutions. Work on the system,
neohexane-cyclopentane, has already been reported.'
The investigation of rotation in solid 2,3-dimethyl
butane is the beginning of work on the system, neo
hexane-2,3-dimethylbutane. The results with 2,3-
dimethylbutane have their own significance outside of
any results of the whole phase study.
The 2,3-dimethylbutane exhibits one transition point
of the rotational type below its normal melting point
of 145.19°K,2 The ability of 2,3-dimethylbutane to
supercool below its transition temperature of 136.07°K
has been noted previously in a series of thermal studies3
and in the determination of its Raman spectrum.4.r;
The most reliable determination of its heat capacity
was reported without any indication of difficulties due
to supercooling although difficulties due to supercooling
were noted with its isomer, 3-methylpentane.2
The existence of relatively free rotation or no
rotation in the solid state can be distinguished by the
width of the proton resonance absorption line.6a•b A
wide line (greater than about 4 gauss), which indicates
no rotation, is due to the existence of internal magnetic
fields produced by neighboring protons. Relatively free
* This research was carried out under contract with the Office
of Naval Research. t Union Carbide Fellow 1953-1954.
1 Aston, Bolger, Trambarulo, and Segall, J. Chern. Phys. 22,
460 (1954).
2 D. R. Douslin and H. M. Huffman, J. Am. Chern. Soc. 68, 1704
(1946).
3 Smittenberg, Hoog, and Henkes, J. Am. Chern. Soc. 60, 17
(1938).
4 N. Sheppard and G. J. Szasz, J. Chern. Phys. 17, 86 (1949).
6 N. Sheppard and J. K. Brown, J. Chern. Phys. 19,976 (1951).
6 (a) H. S. Gutowsky and G. E. Pake, J. Chern. Phys. 18, 162
(1950); (b) Gutowsky, Kistiakowsky, Pake, and Purcell, J. Chern.
Phys. 17, 972 (1949). rotation eventually causes the internal fields to average
out to zero and produces a narrow line (less than about
4 gauss).
In the present investigation we have been able to
work with the supercooled rotating variety down to
600K and have found that it slowly loses its freedom of
rotation on standing at constant temperature as well as
by cooling. When this supercooling is removed by the
occurrence of the transition to the normal state the
substance warms up due to the heat of transition but
does not reach the transition temperature. During the
transition, the absorption line narrows from about 8
gauss to 3.5 gauss and does not broaden to the 9 gauss oj
the normal nonrotating variety until the transition is
complete. A reasonable explanation of this phenomenon
is that the transition occurs in a labile configuration
around Frenkel holes7 producing a higher but constant
spin-spin relaxation time and hence a higher peak height
as well as a narrower line due to increase rotation. The
configuration around the hole stays in this labile state
until the transition is practically complete.
EXPERIMENTAL
The nuclear magnetic resonance apparatus, perma
nent magnet, and cryostat have been described pre
viously.1 The resonance absorption occurred at 23.592
megacycles in a field of 5541 gauss. All the line shapes
were recorded after passage through a lock-in amplifier
whose output is proportional to the first derivative of
the absorption line. The width of the line was taken
as the width in gauss between the maximum and
minimum of the derivative curve. The sample of 2,3-
dimethylbutane, with less than O.I-mole percent
impurity, was prepared and purified by fractional
melting.
7 J. Frenkel, Kinetic Theory of Liquids (Clarendon Press,
Oxford, England, 1946), Chaps. I and III.
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1.1698873.pdf | A QuantumMechanical Treatment of Virial Coefficients
John E. Kilpatrick
Citation: J. Chem. Phys. 21, 274 (1953); doi: 10.1063/1.1698873
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Downloaded 27 Sep 2013 to 129.78.72.28. 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_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 2 FEBRUARY, 195,1
A Quantum-Mechanical Treatment of Virial Coefficients*
JOHN E. KILPATRICK
Department of Chemistry, The Rice Institute, Houston, Texas, and The Los Alamos Scientific Laboratory,
University of California, Los Alamos, New Mexico
(Received September 22, 1952)
The virial equation of state for gases has been developed with the quantum-mechanical grand partition
function as a basis, instead of with the usual Slater sum or the density matrix. Attention is thereby focused
on the energy levels of systems of one, two, three, etc., molecules rather than on their wave functions, An
explicit, closed solution has been found for the nth vidal coefficient in terms of the first n cluster integrals
which is valid either classically or quantum mechanically. A simple generating ftinction for the virial coeffi
cien ts has been proposed.
THE partition function for an assembly of N
moleculest is given by
ZN=L exp(-E/kT). (1)
The summation is over all possible quantum states of
the N mqlecules. E is a running symbol for the energy
of these states. Alternatively we may write
ZN=L nexp(-E/kT), (2)
E
where n is the degeneracy of the state with energy E
and the summation is over all energy states.
The grand partition function is given by
(3)
where z=exp(/I-/kT), z is the absolute activity and /I
is the usual partial molecular Gibbs free energy or
chemical potentiaL (GPF) is a function of the absolute
activity z (or the chemical potential /1-), the volume V
and the temperature T of the assembly. It is related to
the internal energy E, the number of molecules Nand
the pressure-volume product PV by the equations
E=kJ'2(a/aT) In(GPF), (4)
N =z(ajaz) In(GPF), (5)
PV=kTln(GPF). (6)
The physical meaning of Eqs. (4) and (5) can be
seen from their expansions:
L L En exp( -E/kT)ZN
E N E=----------
L L n exp( -E/kT)ZN (7)
E N
L L Nn exp( -E/kT)ZN
E N zv=---
L L n exp( -E/kT)ZN (8)
E N
* Work done under the auspices of the AEC. t Our symbol ZN is taken from the notation used by Hirsch
felder, de Boer, et al. in their forthcoming book The Properties of
Gases. Zl and ZN are the same as (pf) and (PF) as used by G. S.
Rushbrooke, Statistical Mechanics (Oxford University Press,
London, 1949). For an assembly with given values for z, V, and T, the
observed values of the internal energy and the number
of molecules present will be the average of all possible
values of E and N, each with the relative probability of
its occurrence as a weighting factor.
We now expand In(GPF) as a power series in z:
QO QO
In L ZNZN = L V gizi. (9)
N-O i-I
The coefficients, V gh are obtained by comparing corre
sponding powers of z:
Vg1 =Zl,
Vg2 =Z2-!Z 12, (lOa)
(lOb)
Vga =Za-!(Z2Z1+Z1Z2)+tZla, (lOc)
Vgm=Zm-! L L ZiZi+i L L L ZiZjZk-···. (11)
i j i j k
(i+j~m li+j+k~m
The general relation between the V g j and the Z N can be
expressed in the more compact notation:
{Lkm=i
Lmkm=j
The second summation symbol is to be interpreted as:
sum over all sets of positive or zero integers consistent
with the restrictive conditions given below.
The g's may now be considered known functions of
the ZN'S.
If we substitute the right-hand member of Eq. (9) for
In(GPF) in Eqs. (5) and (6), perform the indicated
differentiation, introduce x=N /V and clear out V,
we obtain
P=kTLgjZi,
X=Ljgi Zi. (13)
(14)
In order to eliminate z between these two equations,
we invert the latter:
(15)
274
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The Ck's may be obtained as functions of the g/s by
introducing Eq. (15) into Eq. (14) and matching
coefficients:
CI = 1/ gl, (16a)
C2= -2g2/gI3, (16b)
C 2= (8g22-3glg3)/ gl· etc. (16c)
With these values for the Ck's, Eq. (15) may be sub
stituted into Eq. (13) and z eliminated:
( g2 2g3gI -4g22 ) P=kT x--x2 r-'" .
gI2 gI4 (17)
The coefficient of xn in Eq. (17) is the nth molecular
virial coefficient.
The elimination of z between Eqs. (13) and (14) just
outlined is purely algebraic. The same result may be
achieved with considerably less effort by the use of
complex variable theory. Equation (13) is written in
the form
P=kTL:. gjzi=kTL:. Ct.kXk, (18)
1 1
and therefore
L:. gjZi= L:. Ct.kXN. (19)
1 1
We eliminate x by means of Eq. (14), multiply by z-n-1dz
and integrate around the origin:
gn = _l_!z-n-lL:.Ct.k(L:.jgjZi)kdZ. (20)
2ri
Therefore gn is equal to the coefficient of zn in the ex
pansion of L:.Ct.k(L:.jgjZi)k. It is advantageous to replace
jgi by the new symbol Pi-We obtain the family of
equations
Pl=hCt.l,
tp2= PzCt.l+PI2Ct.2, (21a)
(21b) These equations give Ct.n, the nth molecular virial coeffi
cient, in terms of the Pi and therefore of the gi' The gj
are known functions of the partition functions for n
molecules, n-1 molecules, "', two molecules and one
molecule. The physical significance of the gj will be
discussed later. '
A more complicated treatment of Eqs. (13) and (14)
leads to an explicit solution for the nthvirial coefficient:
(n> 1)
The sets of kj, the powers of the various pj products,
are merely the various ways (independent of order)
that 2n-2 can be partitioned into n-1 parts. The proof
of this expression will be found in the appendix.
As an example of the use of Eq. (24) we shall calcu
late one of the terms of Ct.s. The step by step solution of
Eq. (22) as far as Ct.s is tedious, and the direct algebraic
solution would be very laborious.
All of the ways that 14 can be partitioned into 7 parts
are given below:
8 1 1 1 1 1 1 5222111
4322111
721 1 1 1 1 3332111
6311111
541 1 1 1 1 4 2 222 1 1
3322211
622 1 1 1 1
532 1 1 1 1 322 2 2 2 1
442 1 1 1 1
433 1 1 1 1 2 2 2 2 2 2 2
tp3= P3Ct.l+2P2PICt.2+pI3Ct.a, (21c) It is evident there are 15 terms in Ct.s. The partition
or in general, 5 2 2 2 1 1 1 corresponds to the Pk product P6P23P13•
The complete term is Pn p.r.
-=L:. i!Ct.i L:. II -,-, n ... 1 (r,) _I,.! (22) ( _ )8-3-1(7)(1O!) P6P23h3 105P6P23h3
{L:.,. =i
~ "..~n
These equations can be solved for Ct.l, Ct.2, Ct.3, in turn:
Ct.2= (-tpz)! P12,
Ct.3= (-jP3PI+P22)/PI4,
Ct.4= (-!P4PI2+3pap2PI-5/2P23)/ P16,
Ct.. = (-tp.PI3+4P4P2PI2+ 2Pa2P12 (23a)
(23b)
(23c)
(23d)
-12pap22h+ 7 P24)/ PIS. (23e) 1 !3!
Written directly as a function of the g,'s, this term be-
comes
105 (5)(23)g6g23gN gll4.
CLASSICAL FORM OF EQUATIONS
It is evident upon comparing Eqs. (13) and (14) with
those of Rushbrookel and of Mayer and Mayer2 that
our gj is the quantum-mechanical equivalent of Rush
brooke's g;, and that gi= gl ibj, where bj is the cluster
integral introduced by Mayer. If the classical partition
1 See reference t.
2 J. E. Mayer and M. G. Mayer, Statistical Mechanics (John
Wiley and Sons, Inc., New York, 1940).
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function is introduced by ZN free particle in a container of volume V:
X-3N
ZN= N! f··· f exp( -U(r)/kT)d(r) ,
(V) . (25) ZI = (2s+ 1) £, £, t exp(r2+s2+12) ( Jz2)
r=1 8=1 t=! 8mkTVf
where X2= h2/27rmkT, (r) is the set of N position vectors,
and U(r) is the potential energy of the N molecules, our
set of equations (10), (11), and (12) at once reduces to
the Ursell-Kahn3 expressions for the cluster integrals.
If the assumption of additivity of potential energy
between pairs of molecules is introduced, we get Mayer's
expressions for the cluster integrals.
The substitutions necessary to obtain the second
virial coefficient in classical form are quite easy. Upon
substituting Eq. (25) in Eqs. (lOa) and (lOb) we have
X-6
Vg2=2"! f f exp( -U(r)/kT)d(r) -!X-6V2. (27)
If the relative potential energy of two molecules de
pends only upon their separation, we can perform five
of the indicated six integrations and write
V g2= 27rX-6V foo (e-U(r)/kT -1)r2dr. (28)
r=0
Now, since gl=h and 2g2=P2, Eq. (23b) yields
a2= 27r .{'" (1-e-U(r)/kT)r2dr (29)
in the usual classical form.
Expressing Eq. (24) in terms of the classical cluster
integrals bj introduces no complications. Since pj= jbjPIi,
b1 cancels out identically, even though its value were
not unity:
THE SECOND VIRIAL COEFFICmNT IN
QUANTUM-MECHANICAL FORM
The partition function for one molecule of a mono
atomic gas (with nuclear spin s) can easily be evaluated
by summing over the translational energy states of a
3 Boris Kahn, thesis, Utrecht, 1938. = (2S+1)X-3V. (31)
The last expression is of course identical with the corre
sponding classical partition function except for the
nuclear spin degeneracy factor. It is a very accurate
representation of the triple sum except for very high
gas densities and for temperatures far below 10K. We
have then for gl (or PI)
(32)
It is advantageous to replace the denominators of
Eqs. (23b) , etc., with this expression but not to replace
the ZI'S that will appear in the numerators. For the
second virial coefficient, we get
V(2s+1)2 (33)
This expression has been given by de Boer4 for gases
with zero nuclear spin.
The precise way in which the nuclear spin of any gas,
real or ideal, affects its second virial coefficient can
easily be deduced from Eq. (33). We need to introduce
several symbols: Z18, Z82(BE)' and Z82(FD). The super
script refers to the nuclear spin, and the parenthetical
BE or FD refers to whether the summation is over even
or odd states (with respect to exchange of identical
atoms). Naturally no such subscript is necessary for ZI
since the states of only one atom are involved. Every
state of one atom is degenerate by the factor 2s+ 1 re
suiting from the nuclear spin. Therefore,ZI 8= (2s+ 1)ZI0,
everything else being the same. A state for two (identi
cal) atoms may be even or odd, aside from the sym
metry resulting from nuclear spin. For a spin of s, there
are (s+1)(2s+1) even and s(2s+1) odd nuclear spin
wave functions. It follows that
Z82(BE) = (s+ 1)(2s+ l)Z02(BE)+s(2s+ 1)z02(FD), (34a)
Z'2(FD) = (s+ 1) (2s+ l)z02(FD)+s(2s+ 1)z02(BE). (34b)
When these relations are inserted into Eq. (33), we get
(35a)
4 de Boer, Van Kranendonk, and Compaan, Physica XVI, 545
(1950).
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and
+_S_[ X6(Z02(BEl-!Z012)J. (35b)
2s+1 V
The physical interpretation of these two equations is
s+l s
(a2)' (BEl = --(a2)0 (BEl+--(a2)0 (FD), (36a)
2s+1 2s+1
s+l s
(a2)8(FD)=--(a2)0(FDl+--(0!2)0(BEl' (36b)
2s+1 2s+1
These two equations have been previously derived but
by a different argument. So far as we are aware, this
direct deduction from Eq. (33) is new. (a2)0(FD) prob
ably is not the second virial coefficient of any actual gas,
since zero spin and Fermi-Dirac statistics are inconsis
tent. It is really only a convenient abbreviation for a
certain sum.
The relations between the second virial coefficients of
ideal Bose-Einstein, Fermi-Dirac, and (corrected)
Maxwell-Boltzman gases also follow readily from Eq.
(33). We have evaluated a2 in these three cases directly
(including nuclear spin), but the argument is greatly
shortened if use is made of the general relation of
Eqs. (36a) and (36b). We need only then to consider the
case of zero spin.
The argument is perhaps most readily visualized as
follows. We arrange the energies of all of the possible
translational energy states of one atom in a row,
reading from left to right, and in a column, reading
from top to bottom, in the same (and completely arbi
trary) order in both cases. The energy of every state of
two atoms will be found by taking every possible sum
of one number from the row and one from the column.
We form a square table (infinite in extent to the right
and downward) by entering the Boltzman factors for all
of these energies in the appropriate places. Z2 for an
ideal MB (distinguishable atoms) gas is the sum of all
of the entries in the table, i.e., Z2=Z12. For a CMB
gas, Z2= (1/2 !)Z12 or the sum of all terms above the
diagonal plus half the sum of the diagonal terms. It
follows at once that Z2-!Z12=O and 0!2=O. For a BE
gas, Z2 is the sum of all the terms above the diagonal
and all those on the diagonal. Z2-!Zl, is not zero; it is
exactly half the sum of these diagonal terms:
Therefore, the second virial coefficient of an ideal zero
spin, BE gas is
(38) Z2 for a FD gas is just the terms above the diagonal and
none of those on the diagonal. The second virial coeffi
cient is equal and opposite to that of the BE gas.
Including nuclear spin, according to Eqs. (36a) and
(36b), we have
2-5/2X3
(a2)'{BE}= =F--.
FD 2s+1 (39)
As usual, the presence of a spin greater than zero serves
to reduce the quantum deviation from classical sta
tistics. It is sometimes stated that ideal BE and FD
gases have equal and opposite second virial coefficients.
This is of course true if the two gases have the same
nuclear spin, but apparently FD statistics are always
associated with half integral spin and BE statistics with
integral spin.
APPENDIX
We multiply Eq. (19) by x-n-1dx, eliminate x from
the left-hand member by Eq. (14) and integrate around
the common origin of x and z:
x CE gjZi)(L: 12gzz1-1)dz.
1 1
Therefore, an is just the coefficient of zn in the expan
sion of
Upon replacing jgj by Ph as before, and performing the
indicated operations, we obtain
00 00 oc (-)i(n+i) 1 kp 'Pk P.r'z(8-1lr, L: L: L: _J_zi+k-1 L: II ,
i=O i=l k=l n lP1n+Hi j (r) 8=2 ,.1
{ L: r8=i.
,=2
The coefficient of zn is
{L: r8 =i
~ sr8=n+i+l- j-k.
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This expression is one form of the general solution but
is unduly complicated. Its principal disadvantages are
that it has about three too many summation signs and
that a given p. product arises in several ways. We can
group all P. products of the same sort together in the
following way. Let k.'=,. for s2':2; k."=o.; and
k.''' = O.k. Both k." and k.''' are defined for all s in
cluding s= 1. In addition, let k.= k.'+k."+ks''' for all s.
We have not as yet defined k/. The product
is the form of the general term except for the factor in
Pl. From the first restrictive condition we have
L: k.=i+2-k l"-k/",
2
and from the second,
L: sk.=n+i+1-k l" -kl""
2
We now consider the factor in Pl. The largest value of i
possible is n-1, since for no larger i is it possible to
divide n-1 +i into i parts, each two or larger. We will
therefore write the general term with the denominator
P12n-2 and write the remaining factor (if any) in PI
in the numerator in the form Plkl. This statement
defines kl (and therefore kl'). It follows then that
-n-1-i+k/'+k l'''=kl-(2n-2),
i=n-3-k l'.
The two restrictive conditions now simplify:
L: k. = n-1,
8=1
L: sk.=2n-2.
_I
The last remaining task is to determine the numerical
coefficient of the general term. A close inspection shows that this coefficient is
(-)n-3-k1(2n-kl-3) '{II _1 }
n' .-2 k.'
X { (2n-kl -1)(2n-k l -2)
-(2n-kl-2) ~2 (S+~)k.
+ L: L: (~+~)k.kt+ L: k.(k. -1) },
.-2 I>. t s .=2
Of the four terms in the braces, the first represents all
terms in which j=k=1, that is, kl"=kllll=1 and all
other k." and k.''' =0. In that case, the,. are just the k8•
The second term represents all terms with either j or k
equal to unity. One of the '. is then just one less than
the corresponding k •. The third term represents the
case when j,e1, k,e1 and j,ek. The fourth term is
similar, except that j=k. In the last two terms,
kl" = kllll = 0 so kl' = kl. In the first and second terms,
kl=k/+1 and kl=k/+2, respectively. The index i, of
course, is replaced by n-3-kl'.
With the aid of the two restrictive equations the com
plicated expression in the braces reduces to just n-1.
The general expression for an is therefore
( -)n-3-k1(n -1)(2n -kl -3) , p/s
an= L: hkl II -,
(k.) n 'P12n-2 _2 k.'
r~ k. = n-1
1 (n> 1).
l~ sk.=2n-2
After completing this proof, we observed that this
final expression for an is obtained directly as the coeffi
cient of z2n in the expansion of
(L: p;zi)-n+l/n.
I
This expression can probably be obtained directly from
Eqs. (14) and (19), but we have not pursued the matter.
In any event, this expression may prove useful as a
generating function for the virial coefficients.
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1.1715521.pdf | Hydrogen Bubble Chamber Used for LowEnergy Meson Scattering
D. E. Nagle, R. H. Hildebrand, and R. J. Plano
Citation: Review of Scientific Instruments 27, 203 (1956); doi: 10.1063/1.1715521
View online: http://dx.doi.org/10.1063/1.1715521
View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/27/4?ver=pdfcov
Published by the AIP Publishing
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Rev. Sci. Instrum. 28, 645 (1957); 10.1063/1.1715961
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Downloaded to IP: 132.248.9.8 On: Mon, 22 Dec 2014 04:14:00HIGH TRANSIENT MAGNETIC FIELDS 203
mechanical properties, but is less desirable from the
point of view of eddy currents.
IV. ACKNOWLEDGMENTS
The authors are indebted for valuable technical
advice to G. M. Moore and F. Cameron of Raytheon
THE REVIEW OF SCIENTIFIC INSTRUMENTS Manufacturing Company. We also acknowledge grate
fully the assistance of R. L. Smith, A. M. Koehler,
P. C. Bondi, and S. Engelsberg in several phases of this
development. Semiconductor material for the solid
state investigations has been generously supplied by
Clevite Transistor Products.
VOLUME 27, NUMBER 4 APRIL, 1956
Hydrogen Bubble Chamber Used for Low-Energy Meson Scattering*
D. E. NAGLE, R. H. HILDEBRAND, AND R, J. PLANot
The Enruo Fermi [nstitutejor Nuclear Studies and Department oj Physus,
The University of Chicago, Chuago, Illinois
(Received January 30, 1956)
A.2.5X2,5X10-cm hydrogen bubble chamber has been developed for experiments on the scattering of
particles from the 450 Mev synchrocyclotron. Seventy-five thousand pictures have been taken at the rate
of one every two seconds and are being scanned at the rate of two thousand per day. The average track
length per picture is about one gram per square centimeter. The characteristics of the bubble chamber are
described and examples of the pictures are shown.
1. INTRODUCTION
IN this paper we describe a Glaser bubble chamber,!
operating with liquid hydrogen,2-4 which has been
used to study low energy pion-proton scattering. It is
valuable to study these events at such low energies that
counter techniques are difficult because of the short
range of the pions, while cloud chamber and emulsion
studies are difficult because of the low cross sections. A
bubble chamber for this type of study must allow rapid
scanning and accurate measurement of angles and
ranges. Since it is used with the synchrocyclotron in a
search for rare events, it should recycle rapidly and
should operate reliably for hundreds of thousands of
expansions.
The present 2.5X2.5XlO-cm chamber is designed to
meet these specifications. Its cycling time is two
seconds. With it, we have taken seventy-five thousand
pictures of low-energy meson tracks. Fifty thousand of
these pictures have been scanned and preliminary
results will soon be published.
II. GENERAL FEATURES OF THE APPARATUS
The central feature of this apparatus is the all-glass,
square-cross-section bubble chamber. The four flat
transparent walls allow 90° stereo-photography, which
in turn assures maximum accuracy, rapid scanning of
* Research supported by a joint program of the Office of Naval
Research and the U. S. Atomic Energy Commission. t Now at Columbia University, New York, New York
1 D. A. Glaser, Phys. Rev. 91, 762 (1953); Donald A. Giaser and
David C. Rahm, Phys. Rev. 97, 474 (1955).
'R. H. Hildebrand and D. E. Nagle, Phys. Rev. 92, 517 (1953).
a J. G. Wood, Phys. Rev. 94, 731 (1954).
4 D. Parmentier and A. J. Schwemin, Rev. Sci. lnstr. 26 954
(1955). ' pictures, and maximum use of the chamber volume.
The completely smooth inner surface minimizes spon
taneous boiling at the walls.
The principal parts of the bubble chamber apparatus
are shown schematically in Fig. 1. There are three
distinct hydrogen systems: The first is the reservoir
which contains liquid hydrogen boiling continuously at
one atmosphere. It serves as a source of cooling for the
others. The second, consisting of the jacket and its
attached condenser, serves as a temperature controlling
bath for the bubble chamber. The bubble chamber itself
and the metal bellows connected to it comprise the third
circuit. All are enclosed by an aluminum shield kept at
liquid nitrogen temperature in order to reduce the flux
of thermal radiation. The whole apparatus is vacuum
jacketed and continuously pumped.
The temperature of the chamber is maintained by the
lWi!Il
FIG. 1. Schematic diagram of apparatus.
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Downloaded to IP: 132.248.9.8 On: Mon, 22 Dec 2014 04:14:00204 NAGLE, HILDEBRAND, AND PLANO
bath of liquid hydrogen in the jacket which surrounds it.
The bath temperature is determined by the balance be
tween heat lost through the heat leak to the reservoir
and heat gained from the surroundings and from the
heater. Good thermal contact with the end of the heat
leak is achieved by means of a reflux condenser of large
surface area. The heater is used for temperature control,
and also provides a simple method for momentarily
stopping the boiling in the jacket at the instant the
picture is taken (see Sec. IV).
The pressure in the bubble chamber is controlled by
a metal bellows pushing directly on the liquid hydrogen.
The bellows unit is coupled by a push rod to a second
bellows unit at room temperature which is driven by
compressed air.
III. BUBBLE CHAMBER AND JACKET
The bubble chamber is a hollow glass prism of inside
dimensions 2.SX 2.SX 10 cm and wall thickness 4.S mm.
It is oriented so that the mesons travel parallel to its
long axis. It is made from square Pyrex tubing4
& whose
inner surfaces are accurately plane and whose outer
surfaces have been ground flat and polished. The tubing
is sealed off at one end and a Kovar-to-Pyrex graded
seal is attached at the other end. Lines are etched on
the outside surfaces to provide a reference system for
scanning.
The jacket is also made of Pyrex tubing of the same
type, but of inside dimension 4-cm square, and length
about 1S cm. This is also ground and polished, sealed at
one end, and attached to a S.7-cm diameter Kovar cup
at the other end.
The bottom of the Kovar cup is removable to allow
the bubble chamber to be assembled inside the jacket.
The two portions of the Kovar cup are soft soldered
together.
The assembled chamber and jacket are shown ill
Fig. 2.
IV. JACKET TEMPERATURE CONTROL
AND MEASUREMENT
The jacket surrounding the bubble chamber specifies
the temperature of the bubble chamber walls, and hence
the average temperature of the bubble chamber. The
jacket temperature control depends on the following
mechanism: During operation, the liquid hydrogen in
the jacket is maintained at a level somewhat above the
highest point of the bubble chamber. The liquid in the
jacket normally boils because of heat from thermal
radiation, and from the jacket heater. The resulting
vapor liquifies in the condenser and drops back into the
jacket, a process of heat transfer so efficient that the
condenser temperature and the jacket temperature are
closely the same.
Control of the temperature is achieved by empirically
choosing the heat leak (a copper bar 2 in. long with a
fa Obtained from the Fischer & Porter Company, Hatboro,
Pennsylvania. FIG. 2. Assembled bubble chamber and jacket.
H;-sq in. cross section) to have approximately the
correct thermal conductance, and then adjusting the
heater power to produce the desired temperature. The
heater is turned on and off by the pressure "pickup" in
the jacket system which is shown in Fig. 1. Thus a drop
in temperature reduces the pressure causing the switch
to turn on the heater. The pressure pickup is simply a
diaphragm between the jacket system and a reference
system. A difference of 0.01 atmos between the jacket
and reference pressures is sufficient to cause the dia
phragm to deflect, thus making or breaking the heater
control circuit.
During the time the chamber is waiting for a meson,
the jacket must be free of bubbles which would obscure
the track. This is done by momentarily disarming the
pressure switch and keeping the heater on, at a high
current, for about 200 msec before the picture is to be
taken. The boiling around the heater, which is hidden
from the camera, is then too rapid for the condenser to
cope with, so the pressure rises rapidly, and the bubbles
disappear in the portion of the jacket around the bubble
chamber. After the picture is taken, the pressure switch
is given control again, the heater is turned off, and the
pressure falls. The use of the heater to stop jacket
boiling does not materially increase the average heater
power delivered to the whole system. The power dis
sipated by the heater accounts for about 20% of the
total evaporation of liquid hydrogen.
The jacket pressure is read on a bourdon gauge, and
the temperature of the jacket estimated from the known
vapor pressure data for hydrogen. 5
Typical operating values are given in Table I.
The condenser surface is a spiral of O.OOS-in. copper
sheet. The turns of the spiral are separated by a O.OlS-in.
spacer strip. The whole assembly makes a roll 1t in.
long and 2 in. in diameter. This spiral is soldered into
a copper can to form the condenser.
V. PRESSURE CONTROL SYSTEM
The pressure in the bubble chamber is controlled by
the motion of the lower metal bellows. The lower and
i Wooley, Scott, and Brickwedde, J. Research, Nat!. Bur.
Standards 41, 379 (1948).
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upper metal bellows are mechanically coupled by a
hollow push rod, sliding inside of a guide tube. Com
pressed air is admitted to the upper bellows by a
solenoid valve,6 forcing the push rod down and com
pressing the liquid hydrogen in the lower bellows.
On the "expand" part of the operating cycle, the sole
noid valve closes off the compressed air line and releases
the air in the upper bellows to the atmosphere. As the
push rod moves up, the pressure in the bubble ~h~mber
falls rapidly, and the chamber becomes sensItlVe to
ionizing particles.
The bellows7 are Ii-in. inside diameter two-ply
stainless steel units with an effective area of 3.45 sq in.
They are assembled so as to allow a maximum stroke
• • •• 3 • of ! in. The actual motIOn durmg operatIOn IS 16 m.
The displacement is, thus, about 0.74 cu in. The volume
of the chamber system is 9.4 in.3 of which there is 5.3
in.8 at 27°K, 3.7 in.3 near 20oK, and 0.4 in.3 filled with
vapor. Most of the vapor is at room t~mperature. ..
The design of the bellows system IS such as to mInI
mize the hazard due to bellows failure. As shown in
Fig. 1, the compressed air and the liquid hydrogen are
in contact with the outside of the upper and lower
bellows respectively, while the push rod runs thro~gh
the inside. The inside system composed of the gUlde
tube and the two bellows is evacuated at the beginning
of a run by opening a valve leading to the main vacuum
system. The valve is then closed leaving the bellows
system isolated. A failure of either bellows is registered
by.the vacuum gauge mounted below the upp~r bell.ows
(see Fig. 1). No danger results from such a faIlure smce
the main vacuum system remains undamaged.
A rod attached to the upper bellows extends into a
Lucite tube at the top of the assembly so that the posi
tion and motion of the bellows can be easily seen.
VI. EXTERNAL PLUMBING
The external plumbing, which is shown in Fig. 1, must
insure the purity of the hydrogen gas entering the
system and must provide for pressure control and
limitation.
All the hydrogen gas entering the system is purified
by passing through a charcoal trap immersed in liquid
nitrogen. A sintered glass filter beyond the charcoal
trap removes any dust which may be present.
TABLE I. Operating conditions for bubble chamber.
Heater power (average)
Heater power (during pulse
to stop jacket boiling)
Jacket pressure (average)
Pressure rise due to heater
pulse
Bubble chamber temperature 6 watts
50 watts
75 psi
10 psi
27°K
• Obtained from the Flexonics Corporation, Elgin, Illinois.
7 Obtained from the Bellows Manufacturing Company, Chicago,
Illinois. TABLE II. Time sequence of operations.
Starting time Duration
Operation mUJlsec" millisec b
Clock pulse 0 short
Heater pulse to stop
jacket boiling 0 250
Compressed air 300 70
release"
Pulse to cyclotron 385 short
Meson pulse 400±10 0.001
Flash (400±10)+3.5 «1
Next clock pulse 2000 short
• Column two refers to the delay time between the operation In that row
and the initiai clock puise. .
b Coiumn three refers to the duration of the operatIOn. .
• There is a delay of about SO msec between the operatIOn of the com
pressed air valve and the change of bubbl~ chamb~r pressure. At the end
of the release operation. the compressed ~Ir IS reapph~d. but.the same delay
leaves the bubble chamber sensitive untIl after the pIcture IS taken.
Before a run while the bubble chamber is still warm,
hydrogen is admitted to the chamber and jacket and
then pumped out again. This process is repeated several
times to flush out all other gases. The system is then left
full of hydrogen under pressure while the nitrogen shield
and hydrogen reservior are filled.
The bubble chamber and jacket cool slowly by con
duction and radiation to the reservoir. After about one
hour, liquid hydrogen begins to cond~ns~. When the
bubble chamber and jacket are full of lIqUld hydrogen,
the valves are closed isolating them from the filling line
and from each other. The heaters are then turned on and
the apparatus is ready for operation.
Each of the three isolated pressurized systems,
namely, the charcoal trap, the j~cket, a~d the cham?er
is protected by a safety valve, smce accIdental heat~ng
of any of these systems could cause a large pressure nse.
Valves bypassing the safety valves are used for normal
release of gas at the end of a run.
The volume of the external (room temperature)
chamber and jacket system is kept to a minimum so as
to reduce heat loss due to the flow of gas in and out of
the internal (cold) apparatus.
vn. OPERATION OF THE BUBBLE CHAMBER
After the apparatus has been aligned with the desired
beam of the cyclotron and the chamber has been filled
with liquid hydrogen at the proper temperature, a clock
pulse starts the following cycle of operation, summarized
in Table II.
First the heater is pulsed for a period of 250 msec.
By the 'end of this period, all boiling stops in the region
around the bubble chamber (see Sec. IV).
About 50 msec after the jacket heater is turned off,
the chamber is expanded by releasing the pressure in
the upper bellows. When this pressure has dropped to
1.5 atmos (after about 70 msec) the cyclotron rf
(normally off) is operated for one frequency modulation
cycle.
The beam arrives at the apparatus about 15 msec
after the cyclotron is pulsed. Its arrival is detected by
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Downloaded to IP: 132.248.9.8 On: Mon, 22 Dec 2014 04:14:00206 NAGLE, HILDEBRAND, AND PLANO
FIG. 3. Two views of a 7r-p scattering event. The pion scatter
ing angle is 102°0' ± 1 °30'. The short horizontal lines are spaced
1 cm apart.
a pair of scintillation counters in coincidence. The
second counter is placed as close to the bubble chamber
as possible, so that even for low energy beams, very few
particles are lost by scattering before entering the
chamber.
The coincidence pulse is delayed about 3 msec to
allow time for the bubbles to grow. At the end of this
delay, the pulse fires the flash lamps and the picture is
taken.
The coincidence pulse also starts the "reset" opera
tions of advancing the film and the registers which num
ber each picture. The chamber remains under com
pression until the next clock pulse starts another cycle.
The period is determined by the time required to clear
the chamber of old bubbles.
In test runs, it has been found that the chamber will
remain sensitive for intervals of about t sec after being
expanded because of its clean, all-glass construction.
This feature is valuable when working in very weak
beams. If no particle emerges in the first rf pulse, the
cyclotron may be pulsed again and again until one does
emerge and fires the coincidence counters.
VIII. PHOTOGRAPHY AND REPRO]ECTION
The photographic system is shown schematically in
Fig. 1. The ground glass screens provide bright back
grounds against which the bubbles appear as dark spots.
Lines etched on the outside walls of the bubble chamber
are visible in the pictures, thus establishing a coordinate
system to which the bubble tracks may be referred.
These features are illustrated by the sample pictures
shown in Figs. 3 and 4.
The flash tubes are Amgo Type H-D-1 excited by
the discharge of an 80-I.d condenser charged to 250 v.
The duration of the flash is about 20 J.Lsec.
The pictures are taken by a pair of Bell and Howell
Eyemo-K 35-mm motion picture cameras, converted to
single frame operation. The lenses are 21-cm Zeiss
Tessars, stopped down to f-30. The object distance was
105 cm. The film was Kodak Linograph Ortho.
In order to scan the pictures, they are reprojected
three and a half times life size on a flat table. The two views appear side by side and are measured independ
ently. Angles and lengths are measured with a Bruning
Drafting Machine. The angles are read to 5 min of arc
and the magnified lengths to 0.2 mm. These are recorded
as the initial data.
From these measurements, the true angles and lengths
in the plane of the event are rapidly and accurately
determined with an analyzing instrument which uses the
familiar properties of stereo-graphic projection. This
instrument is described by Pless and Plano.8
Pictures containing more than five tracks are difficult
to scan. The beam is therefore limited to give an average
of two tracks per picture.
IX. BUBBLE TRACKS
The scanning speed and the accuracy of the measure
ments are influenced by the size and the number of
bubbles along the tracks.
The size of the bubbles, as they are photographed, is
controlled by adjusting the delay between the time the
FIG. 4. Two views of a 7r-P.-e decay.
particle traverses the chamber (as detected by the
counters) and the time the light is flashed. The bubble
diameter used in most of our work is about 0.2 mm,
which is larger than the resolution of the photographic
equipment (about 0.1 mm) but smaller than the
average distance between bubbles (about 0.4 mm).
The number of bubbles per unit track length depends
on the operating conditions and on the rate of energy
loss of the particle. Under our conditions, a 20-Mev pi
meson track has about 30 bubbles per centimeter, while
a high-energy electron track has about 15 bubbles per
centimeter. This difference is usually sufficient to dis
tinguish electrons from mesons, even though the mesons
have energies extending all the way from 10 to 30 Mev.
Thus, Fig. 5 is a histogram showing the bubble densities
of 20 pion and 20 electron tracks. In making this histo
gram, each pion was identified as a component of a 7r-p
scattering event or 7r-J.L-e decay, while each electron
was identified as a component of a 7r-J.L-e decay or
Dalitz pair associated with 7r-capture. Three centimeter
lengths of track were counted to obtain the bubble
8 I. Pless and R. Plano (submitted to Rev. Sci. Instr.).
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densities. Statistical fluctuations in bubble-spacing
along the 3-cm lengths are sufficient to explain much of
the spread in the two groups of points.
The 'II"-p.-e decay in Fig. 4 illustrates the difference
in the appearance of electron and slow-meson tracks.
X. BUBBLE CHAMBER CHARACTERISTICS
One measure of the usefulness of the apparatus is
the number of scattering events which can be found for a
given expenditure of liquid hydrogen, cyclotron time,
and scanning time. This information, which is sum
marized in Table III, may be interpreted as follows:
For a process with a cross section of 10-26 cm2, each
event requires 0.6 I of liquid hydrogen, 0.15 hr of
cyclotron time, and three quarters of an hour of scan
ning time. Although we hope to reduce these figures by
further improvement of the apparatus, it is apparent
o Tr MESON TRACKS
U)8 G ~ ELECTRON TRACKS
~6 f-
LL.
°4 0:: w III
~2 z
FIG. 5. Histogram showing bubble densities of twenty r-meson
and twenty electron tracks. Three centimeter track lengths were
counted.
that processes with cross sections of this magnitude may
already be studied.
A second factor affecting the usefulness of the device
is the accuracy of the measurements. In the event shown
in Fig. 3, the meson scattering angle is measured as
102°0' ± 1 °30'. The accuracy of this measurement is
limited by mUltiple scattering. The proton range in the
same event is 2.08±0.1S mm from which we obtain the
incident pion energy of 14.3±1.0 Mev. In this case, the TABLE III. Data on most recent operation of chamber.
Cyclotron time used"
Total liquid hydrogen consumption
Number of (pairs of) pictures taken
Average number of (10 em) tracks
per pictureb
H;ydrogen traversed by each track
Time to scan 1000 pictures· 45 hours
200 liters
56000
1.5
0.6 g/cm2
4 hours
• This figure includes only time during which there was liquid hydrogen
in the apparatus. It does not indude setup time, repair time, or time to
measure the properties of the beam.
b This figure includes only tracks which traverse the entire length of the
chamber and which remain far enough from the walls so that the events
will be measurable.
• This figure includes the time to find. measure. analyze. and record all
events in which a track is deflected more than five degrees. In this experi
ment. about one such event was recorded for every fifty pictures.
error depends primarily on the bubble size and density.
Finally, we must examine the efficiency for detecting
events. In the present experiment, we consider only
those events for which the recoil proton has a measur
able range so that we can determine the energy of the
pion, and so that we can distinguish scattering events
from the more frequent '11"-J.L decays. Under the condi
tions of this experiment, this means that the scattering
angle must be greater than 50°. In one strip of film
containing 2000 pictures, we found a total of 17 tracks
with deflections greater than 30°. These events included
'11"-J.L decays and 'II" scatterings above and below the
minimum angle. When these pictures were rescanned by
a different scanner, the same 17 deflections were found.
On the basis of this test, we feel that the efficiency for
seeing the relatively conspicuous scattering events be
yond 50° must be greater than 90% and may
approach 100%
XI. ACKNOWLEDGMENTS
Weare indebted to Professor Earl Long and Professor
Lothar :Meyer for many helpful suggestions about the
design of the apparatus. We also wish to thank Dr.
Irwin Pless for help during the construction and
operation of the equipment and Mr. Konrad Benford
for assistance in the design and construction of the
electronic circuits. Finally, we must thank Mr. Christian
van Hespin for the construction of the jacket and
chamber in his glass shop.
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1.1722126.pdf | Slowing Down Distribution to Indium Resonance of U235 Fission Neutrons
from a Point Fission Source in Two Aluminum Light Water Mixtures
L. D. Roberts, J. E. Hill, and T. E. Fitch
Citation: Journal of Applied Physics 26, 1018 (1955); doi: 10.1063/1.1722126
View online: http://dx.doi.org/10.1063/1.1722126
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/26/8?ver=pdfcov
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Slowing Down Distribution of U235 Fission Neutrons from a Point Source in Light Water
J. Appl. Phys. 26, 1013 (1955); 10.1063/1.1722125
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] IP: 75.102.73.105 On: Sat, 22 Nov 2014 04:25:19JOURNAL OF APPLIED PHYSICS VOLUME 26. NUMBER 8 AUGUST. 1955
Slowing Down Distribution to Indium Resonance of Um Fission Neutrons from a Point
Fission Source in Two Aluminum Light Water Mixtures
L. D. ROBERTS, J. E. HILL,* AND T. E. FITCH
Oak Ridge National Laboratory, Oak Ridge, Tennessee
(Received February 8, 1955)
The mean square slowing down length, f2, has been measured for two aluminum light water mixtures,
the aluminum-to-water volume ratios being 1: 1 and 1: 2 by volume. The values of j" obtained are 460.7
em2 and 297.4 em2, respectively.
INTRODUCTION
SINCE aluminum is an important structural material
in water-moderated reactors, it was considered
important to determine the mean square slowing down
length, f2, in light water aluminum mixtures. We have
measured this quantity for two aluminum-water
mixtures, the volume ratios being 1: 1 and 1: 2 for the
two samples. The values of f2 obtained for the 1: 1
mixture was 460.7 cm2 and for two volumes of water to
one volume of aluminum, 1'2= 297.4 cm2•
APPARATUS AND MEASUREMENT PROCEDURE
A 2-S aluminum tank 5 ft by 5 ft by 6 ft high, resting
on the thermal column at the top of the Oak Ridge
National Laboratory graphite reactor was filled with
a stacked grid structure of 2-S aluminum plates of
thickness 0.250 in.±0.003 in. To obtain the two
mixtures studied, these plates were separated by one
inch diameter aluminum spacers of thickness 0.250 in.
±0.003 in. for the 1: 1 mixture and of 0.500 in.±0.006
in. thickness for the 1: 2 mixture. The plates lay in
horizontal planes parallel to the top of the pile and
perpendicular to the central vertical axis of the tank
along which the measuring foils were located. After
the grid structure had been assembled in the tank, the
latter was filled with water, particular care being taken
to insure the removal of air bubbles.
In all of the measurements reported here, indium
foils 4 cm by 6.35 cm by about 0.10 g/cm2 thick were
used. They were of the same thickness to within 0.5
percent. The foils were enclosed in cadmium boxes with
a wall thickness of 0.320 cm or in thin aluminum covers,
and they lay in a plane parallel to the plates of the
aforementioned aluminum grid structure. The alumi
num covers were used for r> 52.7 cm for the 1: 1
mixture and r> 46.8 cm for the 1: 2 mixture. See the
previous paper (referred to as paper I) for a discussion
of this point. A cylindrical structure of aluminum disks
0.250 in. thick and 4 in. in diameter, spaced the same as
the aluminum grid for a given mixture, was used to
support the cadmium and aluminum covered foils, and
this structure was so ananged that it could be easily
withdrawn from the tank to insert and remove the foils.
Figure 1 shows the top of the water tank and the upper
* Now with The Rand Corporation, Santa Monica, California. surface of the aluminum grid with the above cylindrical
structure partially withdrawn. This structure was
quite precisely made so that all of the values of r, the
distance from the foil to the neutron source, were good
to at least 0.03 cm. The source of the fission neutrons
which was one of those used for the "age in water"
measurement, paper I, was a disk of U235_Al alloy 5.08
cm in diameter and 0.2 cm thick. The alloy was of
eutectic composition, or 18% uranium which was 96%
U235. The source was mounted on the aforementioned
cylindrical structure on the lower end and adjacent to
the bottom of the tank for values of r greater than
22.86 cm for the 1: 1 mixture and 16.31 cm for the 1: 2
mixture. For measurements in the region O<r~ 22.86
cm for the 1: 1 mixture, the source was located at 20
cm from the boundary of the AI-H20 cube; and for the
1: 2 mixture, with values O<r~ 16.31 cm, the source
FIG. 1. A view of the top of the water tank showing the upper
surface of the aluminum grid with the foil supporting structure
partially withdrawn.
1018
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6.0
!I.O
3.0
o
, (em) • 2 -I H20. AI MIXTURE
o I 'I H20. AI MIXTURE
FIG. 2. Plot of the experimental data for the two aluminum water mixtures.
was located 13.3 cm from the boundary. It is assumed
that this experimental arrangement gives results
corresponding to an infinite cube of the mixture. A
discussion of this point is given in paper I.
The foils were counted according to the procedure
described in detail in paper I. It should be emphasized,
however, that all of the activities here reported corre
spond to the average value of the activity on the front
of the foil with the activity on the back. Also, that
geometry corrections have been made for foil size and
source size so that the activities A. quoted are propor
tional to slowing down densities.
EXPERIMENTAL RESULTS, CALCULATIONS,
AND DISCUSSION
Table I gives the experimental data as the function
loglOA.r2 vs r, where A. is the measured saturated
activity of the foil (corrected as stated in the foregoing),
and r is the distance from the center of the source to
the center of the foil. These results are plotted on the
graph, Fig. 2, and it is seen that the data fall on reason
ably smooth curves. These curves were drawn on quite
a large scale and values of logloAsr2 read off every two
cm. The smoothed values of Asr2 thus obtained are
given in Table II. These experimental functions were
extrapolated to infinity using A.(extrapolated) = k(e-rIA)/r2• For the 1: 1 mixture the values, k= 6.886
X106 counts/min and A=8.975 cm were used, and for
the 1: 2 mixture we used k = L 923 X 106 counts/min
and A = 8.489 cm. These values were obtained by fitting
A. (extrapolated) to the experimental curves at large r.
TABLE I. Experimental data.
1 volume water! 1
volume of aluminum
T(em) !oglOA.T'
3.31 5.1830
6.85 5.7101
10.99 5.9070
14.45 5.9269
18.67 5.8724
22.10 5.7649
26.35 5.5929
26.89 5.5764
29.75 5.4409
30.19 5.4348
34.39 5.1977
37.69 5.0449
41.86 4.8383
45.19 4.6748
49.39 4.4798
52.69 4.2786
53.72 4.2415
61.22 3.9047
68.72 3.5161
76.22 3.1479 2 volumes water: 1
volume of aluminum
T (em) !oglOA.r'
2.98 5.321
6.78 5.806
10.95 5.906
14.40 5.874
16.31 5.759
18.21 5.640
22.02 5.500
23.93 5.368
29.64 5.114
31.55 4.937
35.13 4.739
39.17 4.493
42.75 4.238
50.37 3.767
57.99 3.320
65.61 2.930
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TABLE II. Interpolated data from loglQA.r2 curves.
1 volume aluminum: 1 1 volume aluminum: 2
volume water volumes water
T A.r' A.r' A.r2 A.r'
0 0.0000 0.0000 0 0.0000 0,0000
2 0.6700X105 0.00267 X 108 2 0,8800XI05 0,3520XI07
4 2.104 0.03373 4 3.3200 0.5312
6 4.179 0.1504 6 3.6664 2.0399
8 6.124 0.3919 8 7.3152 4.6817
10 7.586 0.7586 10 8.0360 8.0360
12 8.356 1.2033 12 7.8782 11.344
14 8.492 1.6644 14 7.1501 14.014
16 8.260 2.1146 16 6.0698 15.538
18 7.709 2.4978 18 5.0414 16.334
20 6,823 2.7292 20 4.0460 16.184
22 5.848 2.8304 22 3.1978 15.477
24 4.920 2.8339 24 2.4687 14.219
26 4.064 2.7472 26 1.8969 12.823
28 3.319 2.6021 28 1.4433 11.315
30 2.685 2.4165 30 1.1025 9.9225
32 2.123 2.1740 32 0.8419 8.6210
34 1.686 1.9490 34 0.6381 7.3764
36 1.340 1.7366 36 0.4838 6.2700
38 1.064 1.5364 38 0.3636 5.2504
40 0.8453 1.3525 40 0.2730 4.3680
42 0.6714 1.1843 42 0.2027 3.5756
44 0.5333 1.0325 44 0.1496 2.9863
46 0.4236 0.8963 46 0.1108 2.3434
48 0.3365 0.7753 48 0.0818 1.8847
50 0.2673 0.6683 50 0.0604 1.5100
52 0.2123 0.5741 52 0.0456 1.2330
54 0.1687 0.4919 54 0.0347 1.0119
56 0.1346 0.4221 56 0.0265 0.8310
58 0.1086 0.3653 58 0.0206 0.6930
60 0.08630 0.3107 60 0.0162 0.5832
62 0.06902 0.2653 62 0.0128 0.4920
64 0.05521 0.2261 64 0.0102 0.4178
66 0.04416 0.1924 66 0.0080 0.3485
68 0.03540 0.1637
70 0.02831 0.1387
72 0.02265 0.1174
74 0.01811 0.09917
76 0.01452 0.08387
78 0.01162 0.07069
80 0.009290 0.05945
The value of 1'2, the quantity which we sought to obtain,
is given by the integral
~<X>A.y4dr
1'2=
f"'Asr2dr
0
.. /~AL~
~ 90
~ 80 ·s
;; 70 THEORETICAL CURVE ;, eo ..
~ 50
20
.0
°0~------~0~5------~1.0~------~1.~$--------~
VOLUME OF AI VOLUME OF H20
FIG. 3. Comparison of a theoretical function for the
age with the experimental values. TABLE III.
One volume of Al to Qne volume H.O
measured J:80A."2d"",, 183.1XI05
extrapolated 1.'" A.r2dr=0.08X105 sa
measured .£80 A rd"= 83.68XlOs
extrapolated 1.'" A.r4dr=0,67X 108
80
f2=460.7 ern2
r=fl/6= 76.8 em'
One volume of Al to two volumes H.o
f66 measured Jo A,r2dr=137.2X105
extrapolated fOO A.r2dr=0.07XI05
J66
f6S measured Jo A.r4tir=4O.44XI08
extrapolated f"'A,rdr= 0.39X 108 J ••
fl= 297.4 ern2
r=i'2/6=49.6 ern2
The following Table III gives our values for the
function, f2 to indium resonance.
It is seen that only 1% of our 1'2 values is due to
extrapolated area. From a consideration of total count
and of the measurement of r at the experimental
points, it would seem that the values of 1'2 and T given
in Table III are good to the order of 1%. Here T is the
neutron age defined as 1'2/6.
Table IV gives our age values from fission energies to
indium resonance enhanced by an estimated age
increment from indium resonance energies to thermal
energies.
The graph, Fig. 3, gives a theoretical functionl for
T l)S the ratio vol. Al/voL H20 together with our experi~
mental points. It is seen that the experimental point
for pure water falls close to the theoretical function,
but that the values of the age T for the two aluminum
TABLE IV.
Theoretical
Experimental age indium Total age to
Vol. All age to indium resonance thermal
vol.H.O resonance to thermal energies
0 30.8 em2 1 crn2 31.8 crull
0.5 49.6 em' 2 em' 51.6 em2
1.0 76.8 em2 3 em' 79.8 em2
1 Given in the Oak Ridge National Laboratory Report Mon
P-219 by Weinberg, Soodak, Dismuke, and Arnette.
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water mixtures fall progressively further from this
curve. Even though fair agreement with experiment is
obtained for pure water, the theoretical calculation
procedure is not really adequate. This is brought out
by the fact that various attempts to improve the
calculation for pure water have led to poorer rather than better agreement.2 It is thus of interest to note the pro
gressively increasing disagreement between experiment
and this first-order theory for aluminum water mixtures.
2 This calculation has been made recently at a number of
laboratories; see, for example, J. Certaine and R. Aronson,
Rept. NDA 15C-40.
JOURNAL OF APPLIED PHYSICS VOLUME 26. NUMBER 8 AUGUST, 1955
Rectification Properties of Metal Semiconductor Contacts
E. H. BORNEMAN, R. F. SCHWARZ, AND J. J. STICKLER
Philco Corporation, Philadelphia, Pennsylvania
(Received December 22,1954; revised manuscript received March 17,1955)
Metal semiconductor contacts of a number of different metals
were made on n-and p-type germanium using jet etching and
plating techniques. Current voltage curves taken on 12 of these
metals on 5 ohm-cm n-type germanium showed rectification which
follows the diode equation J =Jo(eqYlkT -1). No correlation was
found between the reverse saturation current densities of these
diodes and such properties of the metals as work function, electro
motive force, etc. For those metal contacts possessing the lowest
saturation current densities, calculations indicated the current
crossing the contaCt was to a large percent hole current and that
the magnitude of the hole current was controlled primarily by
the geometry of the diode. All metals plated on 5 ohm-em p-type
INTRODUCTION
As reported previously by Bradley et al.,l when
indium is electroplated onto freshly etched n
type germanium, rectifying contacts are obtained. Used
as diodes, or as emitters and collectors in transistors,
the electrical behavior of these contacts is very similar
to indium fused p-n junctions even though no diffusion
of impurities has taken place. This discovery instigated
the following study, a survey of the rectifying properties
of various metals electroplated on both n-and p-type
germanium.
The rectification of the indium contacts has been
attributed to a surface barrier to electrons (see Fig.
1). For such a model, theory predicts that the electron
current at the contact I" should be of the form
I"=I,,.(eQv/kT-l), (1)
where V is the applied voltage, q is the value of the
electronic charge, k is Boltzmann's constant, and Tis the
absolute temperature. The electronic saturation current,
I nB, should be of the form
(2)
where A is a constant and <Po is the potential difference
in electron volts between the Fermi level and the bottom
of the conduction band at the germanium-metal inter
face. Thus, <Po is a measure of the height of the surface
barrier. The hole current at the metal contact, I p,
1 W. E. Bradley, Proc. Inst. Radio Engrs. 41, 1702 (1953). germanium produced ohmic contacts of resistivity comparable
to the spreading resistance expected for the diode geometry used.
For indium diodes, a study of rectification versus resistivity
indicated that the barrier produced on both n-and p-type ger
manium with plated contacts is one to electron flow rather than
hole flow. When the assumption of only hole current crossing the
barrier was made, it was shown that the I-V curves calculated
from the diode theory, for different resistivities of germanium,
were in qualitative agreement with the measured curves. Curves
of zero voltage conductance versus temperature for different
resistivities of germanium were also found to be in good agreement
with those calculated on the assumption of all hole current.
should also be of the form
(3)
where Ips, the hole saturation current, should be limited
by the ability of the holes to diffuse in the bulk of the
germanium. Since the hole current is limited by this
diffusion process in both the p-n junction and the sur
face barrier model, the value of I p8 can be calculated in
the same way for both. Its actual value depends on the
recombination rate of the holes in the bulk of the
germanium and at the surface, as well as on the geom
etry of the contact. For a circular contact of radius a on
a semi-infinite slab of n-type germanium, the hole
saturation current would be
(4)
if the bulk and surface recombination are negligible.
Here iJ,p is the mobility of the holes, and peq is the
equilibrium value of the hole concentration in the bulk.
cond uct Ion band
_ Lor.!!L l!'y'eL __
_ .. tal valenee band
FIG. 1. Diagram of the electron barrier at
germanium-metal interface.
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1.3061226.pdf | Recent advances in high-energy physics
Henry Pierre Noyes
Citation: Physics Today 6, 5, 14 (1953); doi: 10.1063/1.3061226
View online: http://dx.doi.org/10.1063/1.3061226
View Table of Contents: http://physicstoday.scitation.org/toc/pto/6/5
Published by the American Institute of PhysicsRecent Advances in
High-Energy Physics
By H. P. Noyes
' I ^HE PAST YEAR has seen what may turn out to
*• be the start of great progress in our understanding
of meson problems. This was brought into focus at the
Third Annual Rochester Conference on High-Energy
Physics, held December 18-20, 1952, and the topics
discussed there form a convenient framework for pre-
senting recent work in this field. The series of Roches-
ter conferences, organized by R. E. Marshak and made
possible by the support of a group of Rochester indus-
tries together, in the present instance, with that of the
National Science Foundation, has consisted of informal
sessions in which the latest experimental and theoretical
results have been mulled over by representatives of
most of the American and some of the foreign groups
directly concerned with high-energy physics.
The starting point for the discussions last December
was a survey of the experimental evidence for the hy-
pothesis of the "charge independence" of nuclear forces.
This hypothesis was originally suggested by the experi-
mentally observed approximate equality of neutron-
proton and proton-proton (symbolized below by n-p
and p-p) scattering in the singlet state and eventually
formulated in terms of a fictitious "isotopic spin space"
or charge space. That is, just as elementary particles
have an internal angular momentum or "spin" and re-
actions obey certain selection rules because the total
angular momentum, /, is conserved, it is assumed that
they also have an intrinsic "isotopic spin" (whose z
component plus 1/2 is their electric charge) and that
in reactions the total isotopic spin, 7, is conserved (in
addition to the usual conservation of charge), also
giving selection rules. This hypothesis was brought to
the fore at the Second Annual Rochester Conference
when it was pointed out that the 7r-meson scattering
in hydrogen observed by the Chicago group in the
one hundred Mev region could be most simply inter-
preted by assuming a "resonance" in the / = 3/2, J =
3/2 state of the pion-nudeon system and the con-
servation of isotopic spin.
But the experimental basis for the charge-independ-
ence postulate is still very shaky. In fact, as E. P.
Wigner remarked, "never has there been as much
theoretical thinking done on a subject the experimen-
tal foundation of which is as inadequate as this one".
Thus, while there is a considerable body of evidence
for the equality of n-n and p-p nuclear forces (which
is sometimes referred to as "charge symmetry"), and
while the correspondences between energy levels in
light nuclei are suggestive of charge independence, the
direct proof of the equality of n-p and p-p forces isstill in doubt. II is known that the scattering lengths
(essentially the strength of the nuclear potential) differ
slightly for the two systems, but the comparison is sub-
ject to corrections involving the variation of the forces
with distance and the interaction of the magnetic mo-
ments of the particles, which corrections cannot be
made with sufficient precision without a more detailed
knowledge than we possess about nuclear forces. Hence
the single direct datum on which the charge independ-
ence hypothesis rests is the approximate equality of «-/>
and p-p effective ranges of interaction, and this is sub-
ject to about a twenty percent experimental error. In
addition, charge independence requires that the n-p dif-
ferential scattering cross section at 90° be greater than
one-quarter of the corresponding p-p cross section; but
again the experiments at high energy are not suffi-
ciently precise to decide whether or not this condition
is violated.
Charge independence has many detailed consequences
for the production and scattering of 7r-mesons and nu-
cleons, but unfortunately the verification of most of
these relations would require the use of polarized beams
and targets, which is beyond the reach of present tech-
nique. However, one special case that can be checked
is the prediction that of two processes leading to the
production of a n-meson and a deuteron, the process
p + p —> TT* + d should have precisely twice the cross
section for the process « + p —> TT° + d for the same
energy and angle of emission of the pion. It is encour-
aging that experiments at Chicago have shown that the
angular distributions coincide within experimental error;
the ratio of two has yet to be demonstrated. During the
past year there has been considerable improvement in
the precision of the data on pion-nucleon scattering and
the photoproduction of pions, but this will bear only
indirectly on charge independence, and then only if it
can be shown that "charge independent" meson theories
are capable of explaining it. In particular, the extension
of the energy range of the measurements of the photo-
production of 7r° mesons from hydrogen to 450 Mev at
Caltech has shown a drop by a factor of four from the
maximum cross section at 315 Mev, indicating that this
is a rather special energy for the pion-nucleon system.
npHEORETICAL PROGRESS made in the last year
-*• was summarized at the Third Annual Conference
by J. R. Oppenheimer in the phrase "all the classic
arguments that the pseudoscalar meson theory with
pseudoscalar coupling—PS (PS) theory—is in disagree-
ment with experiment have been shown to be incor-
rect." This is very important since it had already been
shown that 7r-mesons have spin zero and odd parity
with respect to nudeons, and the PS (PS) theory is,
so far as we know at present, the only theory with these
properties that is capable in principle of giving finite
and unambiguous answers. The "pseudoscalar" char-
acter of the meson means essentially that if the nudeon
Henry Pierre Noyes, assistant professor of physics at the University
of Rochester, received his PhD for work in theoretical physics at the
University of California at Berkeley in 1950.
14 PHYSICS TODAY15
velocity is nonrelativistic, the meson can only be emit-
ted directly in states of odd parity and hence of odd
angular momentum, that is, p-states at low energy,
while the elementary interaction in states of even an-
gular momentum (in particular in s-states) must in-
volve nucleon pairs (which have odd parity by the
Pauli principle) and hence high energies or an intrin-
sically strong interaction. "Pseudoscalar coupling" de-
scribes the specific meson-nucleon interaction Hamil-
tonian which has the property of including both types
of interaction.
The difficulties with this theory came from trying to
treat the interaction as intrinsically weak in the sense
that only a couple of terms in a perturbation theory ex-
pansion need be included. The first great advance was
made by Levy, who showed that even though the inter-
action between two nucleons carried by this meson field
is very strong at short distances, it is repulsive and
hence can be simulated phenomenologically by an infi-
nitely strong repulsion at a radius chosen to give agree-
ment with experiment, while for low nucleon velocities
the region outside this "core" could be represented by
the potential derived from a few terms in a perturba-
tion theory expansion. He was then able to fit all the
low energy- properties of the n-p system approximately
by taking the dimensionless strength of the interaction
G2/47r to be about 10 and the core radius to be about
0.4 X 10"13 cm. What was not realized immediately was
that certain "radiative corrections" which enter as a
multiplicative constant in certain terms of the potential
are not affected by the nucleon velocities and involve
arbitrarily high powers of the interaction parameter
even outside the "core". Thus this theory actually con-
tains a third arbitrary parameter which is in principle
calculable, but is at present completely unknown. There
is some reason to believe that it is much smaller than
the value used in Levy's calculations, but fortunately it
has been shown that the results are very insensitive to
the value of this parameter provided compensating ad-
justments are made in the interaction strength and core
radius. The structure of the core, which has been ig-
nored by this treatment, must rapidly become of de-
cisive importance at higher energies, even perhaps at
20 or 30 Mev, and the quantitative predictions of the
theory for p-p scattering in this region are still in doubt,
but a not unreasonable theoretical program has led to
a better over-all charge independent model for the nu-
cleon-nucleon system than the previous phenomenologi-
cal approaches. Further, when the three-body forces
analogous to this potential are included, approximately
the correct density and nucleon energy are obtained for
infinite nuclear matter, whereas previous models led to
collapse to very high densities.
It also became evident at the conference that calcula-
tions of the pion-nucleon scattering have been greatly
improved by abandoning the straightforward perturba-
tion theory expansion. The essential method used here
(as well as in Levy's work) was first suggested by
Tamm and Dancoff. The basic idea is that even when
the interaction is weak, a single meson can be absorbedand re-emitted many times in the course of a single
scattering, and, since the energies of these successive
intermediate states can be very close to the initial en-
ergy of the system, "resonance" effects can lead to
results quite different from the perturbation theory
approximation of including only one absorption and
rc-emission.
The simplest application of this point of view to the
meson-nuclcon system requires the inclusion of all
states with zero, one, or two mesons present in the field
at a time. G. F. Chew obtained an approximation to
this situation by ignoring the relativistic properties of
the nucleon and simulating nucleon recoil by simply
cutting off integrals over momenta for values greater
than the rest mass of the nucleon. His results repro-
duce the observed "resonance" in the / = 3/2, J = 3/2
state and fit the scattering in the other /(-states approxi-
mately. It is clear from our previous discussion of
angular momentum and parity that s-state scattering is
ignored by treating the nucleon as nonrelativistic. When
this approach is improved by taking the relativistic
properties of the nucleon seriously, as has been done
by F. J. Dyson and the group at Cornell, it is found
that the quantitative behavior of the system does, in
fact, depend very critically on the high momentum
parts of the wave function, but it is still possible to
obtain agreement with experiment for the p-state scat-
tering using an interaction strength G2/4TT of about 14
(the only parameter in the theory). Another interesting
feature of these equations is that they predict that the
resonance will be very closely followed by an anti-
resonance much as has been observed in the photo-
production of 7r°'s (cf. above), although the exact re-
lation between scattering and photoproduction is not
clear.
Unfortunately, since the problem of formulating this
approximation in a covariant way has not yet been
solved, it is impossible to separate out divergences
which occur in the / = 1/2 states in any satisfactory
way. The 5-state interaction in this approximation cor-
responds to a strong repulsion at the Compton wave-
length of the proton; this "hard sphere" scattering
varies with energy essentially as the first power of the
meson momentum and has about the correct magnitude
at 135 Mev. However, the experimental energy- depend-
ence is completely different, falling rapidly toward zero
at 60 Mev and rising again at 40 Mev. If, as is now
being investigated experimentally, the 5-phase shift ac-
tually has opposite signs in the regions above and be-
low 60 Mev, this would indicate an attractive region
surrounding the "hard core" predicted by theory; in
any case the experiments are extremely interesting be-
cause they provide evidence that pion and nucleon
interact strongly at distances much greater than the
proton Compton wavelength. Presumably this is a re-
sult of the interaction of the meson with the virtual
meson cloud which envelops the nucleon out to dis-
tances of the order of the meson Compton wavelength.
In fact, the theory does predict a very strong meson-
meson interaction, but so far attempts to include it in
MAY 1953the theory in a quantitative way have not been suc-
cessful. Thus, although we are obviously still a long
way from a quantitative theory of either the nudeon-
nuclcon or pion-nudeon system, the theoretical predic-
tions have finally begun to bear some resemblance to
what is found experimentally, which was hardly even
remotely true a year ago.
But in order to keep this advance in its proper per-
spective, it must be remembered that only the vaguest
guesses have been offered to explain the host of un-
stable particles heavier than 71- or /i-mesons whose
numbers keep on increasing. Since they are apparently
produced with relative ease in very-high-energy nu-
cleon-nucleon encounters, they must possess strong
interactions with this system, and our present theoreti-
cal ideas would lead us to expect that they would de-
cay to lighter particles via this interaction in times
shorter by a factor of 1O10 than are actually observed.
The bright side of the picture has been the rapid ex-
perimental isolation of many of the new particles and
the addition of a considerable amount of quantitative
data. It is only possible, at present, to study these par-
ticles in the cosmic radiation, but it became apparent
at the conference that a great deal has been learned
from these studies. In fact considerably more is known
about some of the particles than was known about -n~
and ju.-mesons prior to the artificial production of pions
by accelerators, and it is to be hoped that once the
new particles have been produced by billion-volt ac-
celerators, progress will be correspondingly rapid.
"PXTENSIVE EXPERIMENTATION has been done
-*—' at a number of laboratories on the charged and
neutral F-partides first found by Rochester and Butler
in 1948. It is well established that the type called Vi
decays into a proton and a Tr-meson, but in order to
prove that no neutral particle is also emitted, it is nec-
essary to show that the plane defined by the charged
daughters also contains the line of flight of the neutral
parent. Cloud chamber studies at MIT, using several
lead plates inside the chamber, reveal the high energy
event in which the V°i originates in several cases and
show conclusively that only the two charged particles
are emitted in the decay; the decay energy, Q, is 37
Mev and the lifetime is about 3 X 1CT10 seconds. A
second neutral F-particle with the decay scheme V°2 —>
TT* + ir~ has also been observed, but the Q is only very
roughly estimated as 200 Mev, while the lifetime is
probably not very different from that for Vi. Experi-
ments at Caltech were also cited as giving evidence for
FV-^P + TT" with a Q of about 75 Mev, V\ -» p +
(?)°, and F°a—> K~ + •** where the negative particle
is definitely heavier than a if meson and could be as
heavy as a negative proton.
The alternative method used for the study of the
decay of heavy particles is the study of their tracks in
photographic emulsions, developed primarily at Bristol.
Unstable charged particles whose masses probably fall
in the range between 1000 and 1500 electron masses,
and which are collectively denoted as /sT-particles, havebeen split into two types. The first to be discovered,
the K meson, has a ^-meson daughter whose energy is
not unique and ranges up to a momentum of at least
280 Mev/c. Since at least two neutral particles must
be emitted in addition to the ^-meson, and the decay
appears quite analogous to the /J-decay of the /A-meson,
the tentative decay scheme is written as * —* p + 2v,
where v may or may not prove to be the same neutrino
as is "found" in /J-decay. The second type of JiT-particle
was at first classed with the K-meson, but has since been
shown to have a 71-meson daughter with the unique mo-
mentum 212 Mev/c; in the decay scheme x~*7r +
(?)°, the neutral particle could be as massive as Vi
(i.e. —850me). A third type of charged particle, also
observed first in emulsions, decays into three charged
particles of mesonic mass, the preferred decay scheme
being T ~~* 3-n-, although T ~> w + 2/x cannot yet be com-
pletely excluded; the decay energy Q = 74 ± 2.5 Mev
in the former case. More work is needed to establish
definitely the existence of another particle which ap-
pears to decay according to £ ± —* tr1 + (?ir°) with
a Q-value less than 6 Mev.
Work with a multiplate cloud chamber has shown
that many particles which stop in the plates and give
rise to a meson (S-particles) or decay in the gas of the
chamber (V—-particles) are identical with the ^-mesons
observed in emulsions in that they give rise to a 7r-meson
with the same unique decay energy; that some of the
other examples observed are K-mesons is also quite
likely. T-mesons have also been observed in cloud cham-
bers. It has recently been shown that the neutral pion
exhibits an alternative mode of decay TT° —> e* + e~ + y
which occurs with a frequency of I/SO of the usual
mode 7r° ~~* 2y; this is in good agreement with a pre-
vious theoretical prediction. Since the electron pair can
be directly observed, its energy and distance from the
point of origin of the w° give a much better indication
of the 7r° lifetime than was previously obtainable; the
result is that the TT° meson decays in about 5 X 10"15
seconds. Thus, although there is no reason to believe
that we have as yet seen anything like all the unstable
particles, it is encouraging to see how much progress
has been made toward establishing the decay schemes
and energies of those already encountered.
Clearly, in so brief an account it has been impossible
to touch on more than a few highlights of the Third An-
nual Rochester Conference on High-Energy Physics or
even to give proper credit for the various contributions.
The chairmen of the various sessions were C. D. Ander-
son, H. A. Bethe, E. Fermi, J. R. Oppenheimer, B.
Rossi, and E. P. Wigner. Foreign representatives in-
cluded E. Amaldi (Italy), C. J. Bakker (Holland), L.
Le Prince-Ringuet (France), and D. H. Perkins (Eng-
land). A comprehensive report of the proceedings has
been prepared by three members of the Rochester staff:
H. P. Noyes, M. Camac, and W. D. Walker, to which
the interested reader is referred for details; the Pro-
ceedings are available through Interscience Publishers
(New York) at a nominal cost.
PHYSICS TODAY |
1.1700760.pdf | The Vibrational Spectra of Molecules and Complex Ions in Crystals. VI. Carbon
Dioxide
W. E. Osberg and D. F. Hornig
Citation: The Journal of Chemical Physics 20, 1345 (1952); doi: 10.1063/1.1700760
View online: http://dx.doi.org/10.1063/1.1700760
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/20/9?ver=pdfcov
Published by the AIP Publishing
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130.239.116.185 On: Tue, 22 Apr 2014 11:38:24THE JOURNAL
OF
CHEMICAL PHYSICS
VOLUME 20, NUMBER 9 SEPTEMBER, 1952
The Vibrational Spectra of Molecules and Complex Ions in Crystals. VI. Carbon Dioxide*
W. E. OSBERGt AND D. F. HORNIG
Metcalf Chemical Laboratories, Brown University, Providence, Rhode Island
(Received April 14, 1952)
A previously reported discrepancy between the predicted and observed infrared spectrum of crystalline
carbon dioxide is shown to arise from the presence of two peaks due to C130Z, a combination band involving
a lattice frequency near 110 cm-1 and two reflection peaks. The infrared spectrum was studied at -190°C
and aside from the previous features shows one component from V3 and two from V2, shifted very little from
the gas frequencies. The difficulties encountered in interpreting the spectrum of this simple crystal occur
quite generally in the spectra of more complicated substances.
INTRODUCTION
IT has been pointed out that the reported infrared
spectrum of crystalline carbon dioxide is incom
patible with the structure determined by x-ray diffrac
tion studies. 1 Such a discrepancy in so simple a
molecular crystal seemed worthy of further investi
gation.
According to the x-ray structure determinations,
carbon dioxide forms a face-centered cubic lattice of
symmetry Th6 with four molecules per unit cell.2-4
The molecules all lie on sites of symmetry Cai(=Sa).
This site symmetry alone is sufficient to establish that
the exclusion rule between infrared and Raman spectra
should hold, as indeed it does. Only the Fermi doublet
arising from the symmetric stretching vibration (VI) has
been observed in the Raman spectrum," while the anti
symmetric stretching vibration (Va) and the bending
vibration (1'2) were found only in the infrared spectrum.6
The relation between the vibrations of the isolated
molecule and of the active vibrations arising from the
* Based on a thesis presented by W. E. Osberg in partial fulfill
ment of the requirements for the degree of Doctor of Philosophy,
Brown University, 1951. This work was supported by the ONR. t Present address: Hercules Powder Company, Wilmington,
Delaware.
1 D. F. Hornig, Disc. Faraday Soc. No.9, 115 (1950).
2 J. de Smedt and W. H. Keesom, Proc. Amsterdam Acad. 27,
839 (1924); Z. Krist. 62,312 (1926).
3 H. Mark and E. Pohland, Z. Krist. 61, 293 (1925); 64, 113
(1926).
4 J. C. McLennan and J. O. Wilhelm, Trans. Roy. Soc. Can.
Sec. III (3) 19,51 (1925).
6 J. C. McLennan and H. D. Smith, Can. J. Research 7, 551
(1932).
6 W. Dahlke, Z. Physik 102, 360 (1936). coupled motions of the molecules in the crystal is just
the relation between the site symmetry and the space
group.7
This relation is illustrated for the present case in
Fig. 1. It is seen that VI should give rise to two com
ponents in the crystal. They may both be Raman
active. In fact this vibration is split by Fermi resonance
with 21'2, just as in the gas, and no further splitting
has been observed. The antisymmetric vibration Va
should give rise to two components (species Au and Fu
of Th) of which only the second should be infrared active
in the crystal. However, two peaks have been reported
by Dahlke.6 Finally, the bending vibration 1'2 should
yield three components in the crystal, one of species E ..
and two of species F ". Only the latter pair should be
active, but three peaks were observed. Consequently,
we have reinvestigated the infrared spectrum of crystal
line carbon dioxide in order to determine the origin of
these differences.
EXPERIMENTAL RESULTS
Commercial carbon dioxidet whose purity was stated
to be 99.S percent was used for the work. The maximum
amounts of impurity were stated to be 0.34 percent
nitrogen, 0.09 percent oxygen, and 0.07 percent water
vapor. The infrared spectrum of the gas disclosed no
impurity which might affect the results of this investi-
, gation.
The films were prepared by subliming the CO2,
which had previously been condensed in a cold trap,
7 D. F. Hornig, J. Chern. Phys. 16, 1063 (1948).
:j: Pure Carbonic Company, Boston, Massachusetts.
1345
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130.239.116.185 On: Tue, 22 Apr 2014 11:38:241346 W. E. OSBERG AND D. F. HORNIG
Vibrational
Mode
V, eli Th
A __ Ag
9 Au
Eg
Eu
Fg
Fu
FIG. 1. Relation between the local symmetry of CO2 in the crystal
and the symmetry of the unit cell containing 4 molecules.
onto an NaCI backing plate which was mounted in a
low temperature transmission type cell maintained at
a temperature of -190°C. The resulting films were
quite transparent to the eye, except for the thickest
ones, and showed little scattering in the infrared region.
The spectra were obtained at -190°C on a double
beam infrared spectrophotometer,S using CaF2, NaCl,
and KBr prisms. They are shown in Figs. 2 to 5 and
the results tabulated in Table I. The frequencies given
are believed to be accurate to ±5 cm-I at 3700 cm-r,
±3 cm-I at 2350 cm-r, and ±1 cm-I at 650 cm-I•
The spectrum of crystalline carbon dioxide obtained in
the present work agrees well with that of Dahlke in the
vicinity of 650 cm-I and 3600 cm-I• However, it is
quite different in the neighborhood of 2300 cm-r, and
the absorption maximum which was reported at 2288
cm-I actually occurs at 2344 cm-I• Since the spec
trometer was calibrated on the atmospheric carbon
dioxide band at 2349 cm-I before and after each run,
it does not seem possible that the present results are
in error. In addition, the study of a thick film revealed
an absorption band at 637 cm-I which had not been
reported previously.
IOO.~~.~~~~~\~(~,~ __ ~\,~.=_.=-_/~~~~~~o
80 ,., ... /.-..... , L,\' / .... "'",..'
60
~
jjj40 I
t-20 I
10
52500 2450 " \ 1/ ,"i .1[' ][, I " .1.
I , Ii ! !
I , I' t 1 i " :1 ji . I,' .j I \,i !. iIi !! i \ I I! · I, ' I I.
· I I ! : I ! I I I I'
· " I I I
i \ 1 iii i I,
i :! . II ! I,
I " -11-
2400 2350 2300 22110
FREQUENCY IN CM-I -0.2
-0.4
os
-1.0
-1.2
2200
FIG. 2. The infrared spectrum of crystalline CO2 at -190°C in
the region of the fundamental Va.
8 Hornig, Hyde, and Adcock, J. Opt. Soc. Am. 40, 497 (1950). z 100 ----, 0
~ 80 , , I ,
(/) ,
\
~60 \ -0.2 \ , § z , ct I 0: I I- ,
-04 t-tlo-- 40 ,
\ l- I
Z '\ I
ILl '.) (.) -fl- o: 06 ILl
Q. 20 700 675 650 625
FREQUENCY IN CM-1
FIG. 3. The infrared spectrum of crystalline CO2 at -190°C in
the region of the fundamental V2 taken with NaCI prism.
DISCUSSION
It is to be expected that at this low temperature
(-190°C) all of the fundamental vibrations should
yield sharp lines in the infrared spectrum.7 If all of the
sharp lines in the spectrum of the thin film are assigned
to the fundamentals, the pattern of lines coincides
exactly with that expected theoretically. Only one line
which can be ascribed to the asymmetric stretching
vibration 1'3 is observed (Fig. 2), and in contrast to
the earlier work this line has a frequency only 5 cm-I
different from that found in the gas. Similarly, two
sharp lines due to the bending vibration 1'2 are observed.
The mean observed width of these three lines at half
the maximum absorption coefficient is only 5 cm-I;
most of the width of 1'3 and of the less intense component
of 1'2 certainly originates in the finite resolution of the
spectrometer, but the more intense component of 1'2
appears to have a natural width of approximately
3 cm-I. Consequently, there appears to be no genuine
discrepancy.
In addition to these three lines there are two sharp
but weak bands which can be ascribed unambiguously
to CI302• Since the vibration frequencies of the CI302
molecule differ from those of the surrounding molecules,
the CI302 spectrum should be simple, showing no struc
ture caused by intermolecular coupling. This is indeed
the case and only one peak due to 1'2 is found in CI302
in contrast to CI202 which shows two. This effect has
40
20~~700~--~6~~~--~650~--~'~25~
FREQUENCY IN CM-I 2 r-g
04.t+
0.6
FIG. 4. The infrared spectrum of crystalline C02 at -190°C in
the region of the fundamental V2 taken with a KBr prism.
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130.239.116.185 On: Tue, 22 Apr 2014 11:38:24SPECTRA OF CO2 CRYSTAL 1347
been observed previously in solid solutions of HCI in
DCl9 and of naphthalene in anthracene.Io
Aside from the peaks near 3600 cm-1 which are
readily interpreted in terms of the large number of
components of the two combinations lying in this
vicinity (all of which may resonate) and those at
667 cm-I and 2379 em-I which we believe are caused
by reflection, all that remains is the relatively broad
peak at 2454 cm-I and a region of extremely weak
absorption, scarcely above the noise level in the thickest
film, between 2235 and 2270 em-I. It seems highly
probable that the former arises from a combination
of Va with the torsional oscillation frequencies of the
molecules in the lattice with a maximum density of
frequencies near 110 em-I. It is not impossible, how
ever, that the combination involves translational lattice
frequencies since the u-g selection rule applies only to
limiting modesll but the selection rules for limiting
modes appear to be valid empirically, no definite
TABLE 1. Infrared absorption frequencies of crystalline
carbon dioxide at -190°C.
Gas Crystal A. Line width
(em-') (em-1) (em-1) (cm-1) Assignment
3748"
~::} 3716 3712 -4 VI+V3 and
3639' 2V2+V3
3609 3610 +1
2454 35 V3+V Torsion
2379' Reflection
2349 2344 -5 7 V3
2284 2280 -4 7 Va (CI302)
667 Reflection r60 : } 667 -10 V2
653
642b 637 6 V2 (Cl302)
• Shoulder.
b Calculated.
violations having yet been found. The absorption near
2260 em-I may arise from the corresponding difference
bands.
THE EFFECT OF REFLECTION
The effect of reflection from the vacuum-sample and
sample-backing interfaces, plus the reflection from the
same interfaces traversed in the opposite direction,
9 D. F. Hornig and G. L. Hiebert, J. Chern. Phys. 20, 918
(1952).
10 G. C. Pimentel, J. Chern. Phys. 19, 1536 (1951).
II H. Winston and R. S. Halford, J. Chern. Phys. 17, 607
(1949). FIG. 5. The infra
red spectrum of
crystalline CO. at
-190°C in the vi
cinity of "'+"3 and
2"2+1'3-
appear in the spectra of crystals as spurious absorption.
Because of the rapid variation of the index of refraction
near an absorption band the reflectivity of each inter
face varies widely in the neighborhood of an absorption
band and may shift the apparent peak position or
produce a spurious peak nearby, usually on the high
frequency side. In a recent investigation of the reflection
spectra of the ammonium salts, Bovey found reflection
peaks shifted as much as 30 cm-I to the high frequency
side of the absorption maximum.12
However, they have also been observed shifted to
low frequencies.13
A very characteristic feature of reflection maxima
which frequently is sufficient to identify them is their
very great intensity increase when the thickness of thin
films is increased. This effect arises because the de
structive interference of the beams reflected from the
front and back surfaces of a vanishingly thin film
changes to constructive interference when the film
thickness is AI 4.
Such behavior is clearly apparent in the peaks at
667 cm-1 and 2379 em-I, both of which occur on the
high frequency shoulders of the true absorption and
both of which increase in intensity much more rapidly
than the true absorption, as is evident from a com
parison of the spectra of thin and thick films (Figs. 2
and 3).
CONCLUSIONS
The spectrum of crystalline carbon dioxide agrees
with the theoretical expectations. It is complicated by
the presence of peaks due to C1302, reflection and com
bination between internal and lattice vibrations. These
complications are of very general occurrence and may
lead to difficulties in the interpretation of the infrared
spectra of crystals of more complicated molecules.
12 L. F. Bovey, J. Chern. Phys. 18, 1684 (1950).
13 C. Schaeffer, Z. Physik 75, 687 (1932).
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130.239.116.185 On: Tue, 22 Apr 2014 11:38:24 |
1.3067418.pdf | The isotope effect in superconductivity
E. Maxwell
Citation: Physics Today 5, 12, 14 (1952); doi: 10.1063/1.3067418
View online: http://dx.doi.org/10.1063/1.3067418
View Table of Contents: http://physicstoday.scitation.org/toc/pto/5/12
Published by the American Institute of Physics14
The following article, based on a paper
given at the May meeting of the
American Physical Society in Wash-
ington, describes a recently discovered
phenomenon in the behavior of super-
conducting elements. By E. Maxwell
the ISOTOPE EFFECT iJYER SINCE Kamerlingh-Onnes discovered in 1911
that the electrical resistance of mercury abruptly
vanished at a temperature just a few degrees above the
absolute zero, the phenomenon of superconductivity has
presented an intriguing challenge to physicists. Subse-
quent experiments by Onnes and others definitely estab-
lished that the resistivity of a superconductor, if at all
finite, must be immeasurably small, and less than 10~20
ohm cm. Consequently the superconductor is assumed
to have zero resistance. Onnes soon found that the su-
perconducting property was not peculiar to mercury but
was characteristic of a sizable group of metals. Twenty-
one of the metallic elements are known to be supercon-
ductors as are innumerable alloys and intermetallic com-
pounds. The known superconductors are exhibited in
Fig. 1 and are seen to fall into definite areas of the
periodic table. None of the monovalent metals are ob-
served to be superconducting, at least down to tem-
peratures of the order of a few tenths of a degree
absolute.
Onnes discovered rather early in his researches that
the superconducting property was destroyed if the metal
were subjected to a sufficiently strong magnetic field.
When the field was removed, however, superconduc-
tivity was restored. This critical or threshold field in-
creases as the temperature is lowered and has an ap-
proximately parabolic temperature dependence. Some
typical threshold field curves are exhibited in Fig. 2.
The threshold field curve is an important characteristic
of a superconductor and is quite analogous to the vapor-
pressure-temperature curve for a liquid. It does, in
fact, define the phase diagram for a superconductor.
For temperatures and magnetic fields corresponding to
points under the curve the superconducting state is the
stable phase, while for higher temperatures and fields
the normal or nonsuperconducting phase appears. The
intersection of the threshold curve with the tempera-
ture axis is the so-called transition temperature or
E Maxwell has served as a physicist in the Cryogenics Section of
the National Bureau of Standards since 1948. Prior to that he was
a member of the Radiation Laboratory staff at MIT.more properly it is the zero-field transition temperature.
It was not at first clear that the superconducting
state represented a true equilibrium state or that the
threshold curve was a true phase boundary in the ther-
modynamical sense. In fact there appeared to be good
reason to expect that the disturbance of superconduc-
tivity by a magnetic field would inherently be an irre-
versible phenomenon.
For a long time it was supposed that the electro-
dynamic behavior of a superconductor was simply that
of a perfect conductor. On this basis it was expected
that the magnetic induction, B, inside a superconductor,
should be invariant with time, i.e., £ = 0. If this were
true the thermodynamical state of a superconductor
could not be specified by merely the temperature and
magnetic field but would depend on the manner in
which that field and temperature had been reached.
Consider, for example, that such a conductor is cooled
below the transition temperature and some magnetic
field, H, less than the critical field, is then applied.
Currents will then be induced in the surface of the con-
ductor in such manner as to screen the field from the
interior and maintain B — 0. (These are "transient"
currents of infinite duration which suffer no damping
because of the infinite conductivity.) Next consider an
alternate process in which the conductor is cooled be-
low its transition temperature while in the field H. Since
ZJ = 0 for a perfect conductor, the magnetic induction
B inside the conductor must then remain invariant and
"frozen-in", in contrast to the previous case. In fact if
the external field were removed the conductor would be
left with a "frozen-in" magnetic moment. The state of
the conductor for a given field and temperature would
therefore be dependent on its previous history. Further,
if the "perfect conductivity" is destroyed by a magnetic
field the transition would be reversible in the case where
B = 0 inside the conductor, but irreversible if there
were a "frozen-in" field. In the latter case the decay of
the surface currents associated with the "frozen-in"
field would result in Joule heating.
This picture of a superconductor was accepted im-
PHYSICS TODAYBe
Mg
Co
Sr
Bo
RoSc
Y
La
AcTi
-Zr
HfNb
Ta:
PaCr
Mo
WMn
MaFe
JRtk
Os:Co
Rh
IrNi
Pd
PtCu
Ag
AuZn
pd
HcjB
Al
Go
In
TIC
SI
Ge
Sn:
Pb^N
P
As
Sb
Bi0
S
Se
Te
PoF
Cl
Br
IHe
Ne
A
Kr
Xe
Rn800
STED! -OER FIELD ICAL CRIT
Fig. 1. The superconductors (shaded areas).15
I 2 3 4 5 6 7°K
TEMP °K
Fig. 2. Threshold field curves for
some typical superconductors.
SUPERCONDUCTIVITYplicitly for many years. When in 1933 Keesom and van
den Emde discovered that a superconductor suffered a
discontinuous jump in specific heat at the transition
temperature, it was shown that this could be accounted
for if the destruction of superconductivity by a mag-
netic field were a reversible transition in accord with
the second law of thermodynamics. On the basis of
the perfect conductor theory this seemed inconsistent.
Shortly thereafter, however, Meissner and Ochsenfeld
made the startling discovery that when a superconduc-
tor is cooled in a magnetic field the field is expelled as
soon as the metal becomes superconducting. This led
to the far reaching conclusion that a superconductor
must be characterized ideally by the condition 5 = 0
and not merely B = 0. Consequently it would make no
difference in the final state of a superconductor if it
were cooled in a finite field or in zero field. B would be
zero in both cases, there would be no "frozen-in" field,
and therefore no irreversibility of the transition.
The existence of the Meissner effect insures the ther-
modynamic reversibility of the transition and removes
the ambiguity. A thermodynamic treatment of the prob-
lem, based on the premise that B = 0 in the supercon-
ducting state, was given by Gorter and Casimir in
1934.1 In 1935 F. and H. London proposed a phenome-
nological theory to describe the electrodynamics of a
superconductor.2 We shall not describe it here except to
say that it not only contains the Meissner effect as a
bulk property, but in addition predicts small but finite
penetration of the magnetic field into a superconductor
to depths of the order of 10~5 cm. These finite penetra-
tions have been verified experimentally and have been
studied in an important series of researches, chiefly at
Cambridge.
One of the interesting postwar developments, relating
to the microscopic nature of superconductivity, was an-
nounced in March 1950 at a conference on low-tem-
perature physics held in Atlanta, Georgia, under the
iC J Gorter and H. Casimir, Physica 1, 30S (1934).
2 F. London and H. London, Physica 2, 341 (1935); Superfluids
v. 1, F. London, John Wiley and Sons, Inc., New York (1950).sponsorship of the Office of Naval Research. At this
conference the discovery of a new phenomenon in su-
perconductivity, an isotope effect, was announced inde-
pendently by groups working at Rutgers University and
at the National Bureau of Standards. At both of these
laboratories experiments had been performed on sepa-
rated isotopes of mercury with the result that the tran-
sition temperatures were observed to vary with isotopic
mass. These results came as a surprise to many. An
isotope effect had previously been looked for in lead by
Kamerlingh-Onnes (1922) and by Justi (1941) but none
had been found. We know now that the effect was too
small to have been detected with their limited resolv-
ing power.
Precisely what this effect is can best be seen in terms
of the threshold field curve. The solid curve of Fig. 3
is the magnetic threshold field curve, HC(T), for natu-
ral tin (average atomic mass 118.7). The dashed curves
in Fig. 3 are the threshold fields observed for lighter
and heavier samples, enriched in isotopes 113 and 124
respectively. (Because the separation is not complete
the average masses are actually 113.6 and 123.0.) For
the lighter sample the threshold field curve is shifted to
higher temperatures and fields, while for the heavier
sample it is shifted to lower temperatures and fields.
The effect, first discovered in experiments with sepa-
rated isotopes of mercury made available by the Atomic
Energy Commission, was small, of the order of 0.01 °K
per mass unit, so that special care and precision were
required to resolve the shifts accurately. Working in
the Low Temperature Laboratory at the National Bu-
reau of Standards, Maxwell observed that the threshold
curve for a sample of very pure Hg193 was displaced to
higher temperatures as compared with natural mercury
(M = 200.6). The Hg19S, which had been produced by
the transmutation of gold, was part of the supply used
by Meggers to establish the spectroscopic standard of
length. Independently, and at about the same time,
Reynolds, Serin, Wright, and Nesbitt, working at Rut-
gers University, carried out experiments along similar
lines using three samples of mercury which had been
DECEMBER 1952electromagnetically concentrated in isotopes 199, 202,
and 204, respectively, and likewise observed the iso-
tope shift. The investigation was rapidly extended to
other superconductors by these and other workers, and
the effect has now been confirmed in tin, lead,' and
thallium, in addition to mercury. In all cases the criti-
cal field curves for the lighter isotopes are shifted to-
ward higher temperatures and fields. It is now pre-
sumed that the isotope effect is a general property of
all superconductors.
Let us digress briefly to describe the technique of the
experiment. In the superconducting phase, correspond-
ing to the region under the threshold curves of Fig. 3,
the metal is completely diamagnetic, that is, the mag-
netic induction, B, is zero inside the superconductor.
In the normal state, however, B is equal to H (neglect-
ing the small normal diamagnetism which may be pres-
ent). In going from the superconducting to the normal
state the magnetic susceptibility changes sharply from
zero to — 1. 4,r. The threshold field is determined by
observing the flux penetration into the sample at con-
stant temperature as a function of the applied field. A
simplified sketch of a typical apparatus is given in Fig.
5. The flux penetration into the superconductor is de-
tected by noting the change of induction in the pickup
coil which is registered as a sharp kick on the ballistic
galvanometer. In practice various refinements are em-
ployed. A null method is used in place of a deflection
method, several samples are observed simultaneously,
and the bath temperature is accurately controlled. At a
number of fixed temperatures the critical fields of each
of several isotopes are successively measured. From
these measurements a set of curves, such as those in
Fig. 3, are obtained.
Some time after the original observations of the ef-
fect had been made it was shown by the Rutgers group
that the mercury data could be correlated by the re-
lation MtTc = const, where Tc is the zero-field tran-
sition temperature. It is shown in standard treatments
of lattice dynamics that the Debye characteristic tem-
perature, 0, is proportional to the square root of the
ratio of the force constant to the atomic mass. Assum-
ing that the force constant is invariant with isotopic
mass it follows that the transition temperature is pro-
portional to the Debye 9 and consequently suggests a
connection between lattice properties and superconduc-
tivity. As a matter of fact, theoretical treatments of
superconductivity, based on interaction between lattice
vibrations and electrons, were then in the making, and
were published a few months later by Frohlich and by
Bardeen. Frohlich had in fact developed his theory,
which implicitly contained a mass dependence, prior to
learning of the isotope effect.
Since our treatment is concerned primarily with the
phenomenology of the isotope effect we shall not at-
tempt to give any detailed discussion of the basic theo-
retical problems or of the differences in the various
points of view. A review of the lattice vibration theories
of superconductivity has been given by Bardeen." Both
3J- Bardeen, Revs. Modern Phys. 23, 261 (1951).Frohlich and Bardeen, though using different models,
find interaction mechanisms between electrons and lat-
tice vibrations such that a new electronic state of
slightly lower energy is set up. This state is identified
with the superconducting state, and the difference in
free energies of the normal and superconducting states
at absolute zero is found to be proportional to 1/M.
From very general thermodynamic arguments, however,
we know that the free energy difference between the
normal and superconducting states at the absolute zero
is simply H^/Sn per unit volume, where Ho is the
critical field at T — 0. Consequently, if we equate the
expressions for the free energy difference obtained from
the lattice vibration theories and from the thermody-
namic treatment, it would follow that Ml-H0 = const. It
is experimentally observed that the critical field curves
for the different isotopes are geometrically similar fig-
ures, so that the ratio H(j/Tc is invariant with isotopic
mass and therefore we can extend the prediction to say
that MiT0 = const, as had been shown for mercury
isotopes.
How does the half-power law work out in the case
of the other superconductors? Table I exhibits the re-
Table 1. The Exponent e in M'Te = const
Element
Hg
Sn
Sn
Sn
Sn
Pb
Tl* Source
0-504 Reynolds, Serin and Nesbitt
0.5OS±.O19 Maxwell
0.462±.014 Lock, Pippard and Shoenberg
0.46 ±.02 Serin, Reynolds and Lohman
0-50 Olsen, Bar and Mendelssohn
0.73 ±.05 Olsen
0.50 ±.05 Maxwell
suits for mercury, tin, lead, and thallium. Except in the
case of lead, the exponents are all consistent with, or at
least close to, J. In the case of tin there are some small
disagreements among different investigators. Whether or
not these small departures from J are real, or are
caused by small secondary effects, such as strain or im-
300
200
UJ
o
X
Fig. 3. Threshold field
curves for natural tin
CM =118.7) and for two
isotopically enriched sam-
ples. These curves estab-
lish the phase diagram
for a superconducting ele-
ment. They are all geo-
metrically similar figures.100NORMAL
SUPERCONDUCTING
PHYSICS TODAY17
purity, is not yet clear. It has been indicated that vari-
ous approximations in the Frohlich and Bardeen treat-
ments could conceivably introduce small deviations from
the J power. Whether or not they contain enough flexi-
bility to account for the 0.73 power, reported by Olsen
for lead, is a matter of conjecture. This point should of
course be further investigated.
The critical field measurements on isotopes contain
much more than the mass-temperature dependence and
it is worthwhile to explore the phenomenology in greater
detail. The threshold field curves of Fig. 3 are all simi-
lar figures to approximately one part in a thousand.
They transform one into another by a uniform expan-
sion or contraction of the scale, a property first ob-
served by Lock, Pippard, and Shoenberg and verified by
other experimenters. In the experiments carried out at
NBS on tin, the results of which are given in Fig. 3, it
was observed that the curves could all be well repre-
sented by a universal equation
/*=/(') (1)
where h = S/Bo, t = T/Tc, and where the same func-
tion f(t) holds for all isotopes. This equation, together
with the additional observation that H0/Tc is the same
for all isotopes, expresses the similarity property.
The similarity property has some interesting conse-
quences for the entropy characteristics of the isotopes.
It has long been recognized that the superconducting
state is characterized by a greater degree of order, hence
less entropy, than the normal state. The entropy dif-
ference is related in a very direct way to the threshold
field curve. A standard thermodynamic treatment of the
phase transition shows that the excess of entropy of the
normal over the superconducting state is given by
VmHodh(2)
where Vm is the atomic volume and Sn and S3 the en-
tropies of the normal and superconducting states, re-
spectively. Because the entropy of the lattice vibrations
appears in both Sn and Ss, Eq. (2) essentially expresses
the difference in the electronic entropies of the twostates. Because Vm and H,,/T,. are universal constants
and dh/dt is a universal function for the family of iso-
topes, it will be clear from (2) that AS, considered as
a function of T, possesses the similarity property previ-
ously described. In Fig. 6 we have the AS curves corre-
sponding to the threshold field curves of Fig. 3. At very
low temperatures these curves are all linear and coinci-
dent. The linear term is, in fact, identified with the elec-
tronic entropy yT of the electron gas in the normal
metal and is the same for all the isotopes. In the
theory of metals it is shown that y is proportional to
the density of electronic states at the surface of the
Fermi distribution at the absolute zero. These experi-
ments show that changing the isotopic mass makes no
difference in y and consequently in the density of states
in the normal metal.
By subtracting out the linear term in AS we can see
how the electronic entropy of the superconducting state
alone, Ss(cil), changes with isotopic mass. It will of
course exhibit the same similitude property that AS
does. It turns out that Ss(cl) is in fact given by
SS(e\) = yTF{t). (3)
F(t), a function of the reduced temperature, t = T/Tc,
is the same function for all the isotopes, y is also inde-
pendent of isotopic mass. The recognition of this simili-
tude feature of the entropy is an interesting by-product
of the isotope investigations.4 It is very consistent with
an early macroscopic concept of the superconducting
state, one not apparently concerned with the isotope
effect at all, the so-called two-fluid model proposed by
Gorter and Casimir0 in 1934. In this model they visual-
ized the electron gas in a metal, which is in the super-
conducting state, as a mixture of a normal fraction, x,
with which is associated the usual entropy, and a super-
conducting fraction, 1 — x, of zero entropy. The in-
ternal parameter * is a function of temperature and at
any temperature it must adjust itself so that the free
energy is a minimum under conditions of thermody-
1 E. Maxwell, Phys. Rev. 87, 1126 (1952).
= C. J. Gorter and H. Casimir, Physik. Z. 35, 963 (1934).
Fig. 4. Expulsion of the magnetic field from a superconductor.
A superconductor, which is cooled in a magnetic field (left)
from above the transition temperature, suddenly thrusts out
the field (right) when it enters the superconducting state.
This property, discovered by Meissner and Ochsenfeld, in
1934 is one of the fundamental characteristics of a super-
conductor.
V)
DECEMBER 1952Fig. 5. Simplified drawing
of apparatus used to de-
termine threshold fields.
The temperature is stabi-
lized by holding the vapor
pressure of the helium
bath constant. The Hclm-
holtz coils supply the
magnetic field. If the
field is raised above the
critical value the mag-
netic induction penetrates
into the specimen, induc-
ing a voltage in the pick-
up coil which is regis-
tered by the ballistic
galvanometer.TO MANOMETER
TO VACUUM PUW
LIQUID HELIUN
PICK-UP COIL
HELMHOLTZ COILS18
Fig, 6. Entropy difference be-
tween normal and superconduct-
ing states for tin samples of
different mean atomic niass.
These curves are geometrically
similar figures.
namic equilibrium. As the temperature is lowered more
and more of the normal fraction "condenses" to form
part of the superconducting fraction. The model is set
up so that x varies from zero to unity as the tempera-
ture goes from the absolute zero to the transition tem-
perature TB. Quite naturally therefore * is a function
of the reduced temperature T/Tc.
The total electronic entropy, according to the two-
fluid model, is taken to be yT, the ordinary linear term
for an electron gas, multiplied by a function of x, the
normal fraction. An expression of this sort will auto-
matically exhibit the similitude feature if the function
contains no hidden mass dependent parameters. In the
original formulation of the two-fluid model the elec-
tronic entropy was taken to be
Ss(ei) = yTxa, (4)
a form chosen because it was consistent with early ex-
primental data, a is an adjustment parameter, charac-
teristic of each superconductor and empirically found
to be of the order of one-half. Inasmuch as a is essen-
tially an electronic parameter it is plausible to assume
that it would be independent of isotopic mass as is y.
With this assumption Eq. (4), therefore, clearly ex-
hibits the similitude property experimentally observed.
This is a direct result of the fact that the internal
parameter x depends on the reduced temperature, T/Tc,
and is an intrinsic feature of the two-fluid model. The
equation for the critical field curve which follows from
(4) can be derived in a straightforward way. As would
be expected, the reduced field, h, is a function of the
reduced temperature, t, and the parameter, a, and con-
sequently also exhibits the similitude feature.
In concluding this brief survey we note that the
phenomenology of the isotope effect gives us an inter-
esting insight into both the microscopic and macro-
scopic pictures of superconductivity. From the micro-
scopic point of view it suggests an intimate connection
between the dynamical properties of the crystal lattice
and superconductivity, and the general trend of agree-
ment with the "half-power law" of the lattice vibration
theories reinforces this notion. On the macroscopic side
it exhibits the similitude property inherent in the two-
fluid model of a superconductor.Les Atmospheres Stellaires (in French). By Daniel
Barbier. 238 pp. Flammarion, Editeur, Paris, France,
1952. 625 francs.
This excellent book is really somewhat more than the
title suggests, and in fact covers quite a broad part of
the domain of astrophysics. It deals in some detail with
classifications of stars and spectral types, radiative equi-
librium and hydrostatic effects, the absorption coefficient
and the continuous spectrum, and the contours of the
absorption lines in stellar spectra. It also discusses such
incidental topics as molecular spectra, bright (emission)
line spectra, and gaseous nebulae, and presents a good
summary of students' work done in this field.
The book, well illustrated with figures and diagrams,
is written on the graduate student level, or for physi-
cists desiring to familiarize themselves with recent de-
velopments in this neighboring field. It assumes the fa-
miliar forms of mathematics and atomic physics; how-
ever, since it is not intended as a treatise to instruct
the experts, the references are given by name and oc-
casional date only. It will make excellent supplementary
reading, not only in astrophysics, but also as an ex-
ample of the applications of the principles of atomic
physics.
The style is clear and lucid and the book is strongly
recommended to all interested in this subject. As usual
with French editions, the volume is uncut and has pa-
per covers.
Serge A. Korff
New York University
Electrons and Holes in Semiconductors. With Ap-
plications to Transistor Electronics. By William Shock-
ley. 592 pp. D. Van Nostrand Company, Inc., New
York, 1950. $9.75.
"I have said it thrice: What I tell you three times is
true," said the Bellman. Following this sound pedagogi-
cal precept, Shockley has organized his excellent text
into three parts of increasing mathematical complexity
or increasing level of abstraction: Part I, Introduction
to Transistor Electronics; Part II, Descriptive Theory
of Semiconductors; Part III, Quantum-Mechanical
Foundations. The structure is discussed in the preface:
"In Part I, only the simplest theoretical concepts are
introduced and the main emphasis is laid upon inter-
pretation in terms of experimental results. This mate-
rial is intended to be accessible to electrical engineers
or undergraduate physicists with no knowledge of quan-
PHYSICS TODAY |
1.1721810.pdf | On the Nature of Radiation Damage in Metals
John A. Brinkman
Citation: J. Appl. Phys. 25, 961 (1954); doi: 10.1063/1.1721810
View online: http://dx.doi.org/10.1063/1.1721810
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v25/i8
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
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Downloaded 11 May 2013 to 128.135.12.127. 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_permissionsTHERMIONIC EMISSION AND ELECTRON DIFFRACTION 961
work function of between 1.2 and 1.3 ev, both values
comparing favorably with the reported3 emission
values for thick, sprayed oxide cathodes. It is now
worth considering whether the emission mechanisms
operating in these two physically different structures
are related. We may also consider whether the emission
from a thin BaO film on the cathode base metal plays
any essential role in the normal operation of thick,
sprayed oxide cathodes.
If the conduction of electrons from the base metal to
the external vacuum surface in sprayed cathodes takes
place by means of crystal conduction12•13 and is followed
by thermionic emission from the vacuum surface, it is
difficult to assign any essential role to a thin oxide film
present at the base metal-coating interface. However,
the similarity in emission from a film composed of
crystallites 20 monolayers thick and from the micron
size crystals which comprise the surface of sprayed
oxide coatings is not too surprising. These small
crystallites have a fairly well-developed crystal struc
ture, as evidenced by the electron diffraction, and thus
may be expected to exhibit electronic properties similar
to the larger crystals.
If, on the other hand, at normal operating tempera
tures an electron gas within the pores14 of the sprayed
coating conducts electrons through the coating, an
emission mechanism at the base metal-coating inter
face is desired. A thin film of the oxide on the base
metal could supply the necessary emission to provide
an adequate electron gas density. This density would be
12 Hannay, MacNair, and White, J. App!. Phys. 20, 669 (1949).
13 D. A. Wright, Phys. Rev. 82, 574 (1951).
14 R. Loosjes and H. J. Vink, Philips Research Repts. 4, 449
(1949). maintained in the pores by absorption and re-emission
from the pore walls. Since the density of the electron
gas at the vacuum surface of the cathode is primarily a
function of the absorption and re-emission from the
oxide crystals in that region, the thin oxide film at the
base metal serves only to maintain an adequate density
of the electron gas at the interface. It is difficult to be
lieve that during cathode processing at least a few
monolayers of the oxide have not been evaporated
onto the base metal from the adjacent oxide crystals.
This pore conduction hypothesis has been given added
impetus by the recent work of Hensley1s and Young,16
Although in this cathode model the oxide film at the
base metal serves an important role, the emission
characteristics of the cathode are determined by the
oxide near the surface. Therefore, the similarity in
cathode emission with that of a thin film must be
interpreted as in the previous paragraph.
This paper summarizes the results of a study of the
thermionic emitting surface BaO on pure nickel.
Somewhat similar results were obtained when BaO
films were evaporated onto a 4.7-percent W -Ni alloy.
In general, the results were less reproducible with the
alloy base metal, but the levels of attainable emission
were the same as on the pure nickel. No evidence of an
interface compound was found from the diffraction
patterns. It is anticipated that the sealed-off, glass
electron diffraction tube developed in this study will
find application in the examination of other surfaces
which must be handled under very high vacuum
conditions.
to E. B. Hensley, J. App!. Phys. 23, 1122 (1952).
16 J. R. Young, J. App!. Phys. 23, 1129 (1952).
JOURNAL OF APPLIED PHYSICS VOLUME 25, NUMBER 8 AUGUST, 1954
On the Nature of Radiation Damage in Metals
JOHN A. BRINKMAN
North American Aviation, Inc., Downey, California
(Received October 5, 1953)
The nature of the permanent damage retained in metals from irradiation has been investigated in some
what greater detail than has been done in the past. The usual assumption has been that the damage in all
metals consists chiefly of interstitial-vacancy pairs. The model presented in this paper reduces to this picture
for the light elements but introduces a new concept in the case of damage in the heavy metals, called a
displacement spike. Calculations are made from which one can estimate the relationship between the density
of interstitial-vacancy pairs and the temperature of the associated thermal spike. An assumption regarding
the extent to which interstitial-vacancy pairs persist throughout the duration of the thermal spike has been
made, based upon these calculations. The number of interstitial-vacancy pairs predicted in the heavy ele
ments is considerably smaller than that predicted by the former model. A mechanism is proposed by which
small dislocation loops can be produced in the heavier metals by irradiation.
This article is based upon studies conducted for the U. S. Atomic Energy Commission under Contract
AT-1l-1-GEN-S.
INTRODUCTION
CHARGED particle or neutron irradiation is known
to produce lattice changes in metals which can be
retained as permanent damage as long as the metals are held at sufficiently low temperature. The nature of
these lattice distortions has been the subject of con
siderable theoretical study, in both the open and classi
fied literature, by such workers as F. Seitz, H. Brooks,
Downloaded 11 May 2013 to 128.135.12.127. 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_permissions962 JOHN A. BRINKMAN
J. D. Ozeroff, H. M. James, F. W. Brown, and M. M.
Mills. Seitz has described the development and present
status of the theory concerning such radiation induced
lattice imperfections in solids. I For metals, according
to the presently accepted picture, the damage is com
prised of two aspects (1) Frenkel defects, or interstitial
vacancy pairs and (2) effects resulting from thermal
spikes.
The assumption has frequently been made' that
essentially every atom which receives an energy greater
than a certain threshold necessary to displace it from
its lattice site will persist as a permanent interstitial
atom, while its lattice site will remain as a vacancy.
The number of Frenkel defects which should be pro
duced in metals by irradiation has been calculated by
Seitz on the basis of this assumption.2 Thermal spikes
have been assumed capable of producing effects of
similar nature to those resulting from heating and
rapidly quenching the metal. The disordering of ordered
AuCua by neutron irradiation observed by Siege13 has
been cited as an example of such an effect.1,2 The
Frenkel defects produced by a given primary knock-on
atom will all lie within the limits of the associated
thermal spike. The above assumption regarding the per
sistence of the radiation induced interstitials and
vacancies therefore must include the more basic assump
tion that the time duration of the thermal spike is too
short to permit appreciable annealing of the Frenkel
defects.
The present paper represents an attempt by the
author to construct a model of radiation damage in
metals based upon somewhat different assumptions.
While a detailed theoretical investigation of these
assumptions, as well as those made by other workers,
would be desirable, the author can at present only give
qualitative arguments for them and appeal to experi
mental work to differentiate between the two models.
THERMAl SPIKE
• ---INTERSTITIAL ATOMS
o ---VACANT LATTICE SITES
-·-...-rH OF PRIMARY KNOCK-ON
_0 .... ---PATH 0' SUBSEouENT IWOCK-<*
INTEAs(CTIONS OF IACKGROUHO UHES
RE~SENT NORMAL. LAtTICE SITES
FIG. 1. Schematic representation of radiation damage model
in two-dimensional square lattice.
1 F. Seitz, Phys. Today 5, No.6, 6 (1952).
2 F. Seitz, Discussions Faraday Soc. 5, 271 (1949).
3 S. Siegel, Phys. Rev. 75, 1823 (1949). RADIATION DAMAGE MODEL
In this section, the atomistic picture of radiation
damage which will be developed later is described in a
qualitative manner along with the assumptions upon
which it is based. These are contrasted with the assump
tions made by others, and the resulting models are
compared.
The present paper deals only with the damage pro
duced in a metal by knock-on atoms and the dependence
of this damage on the energy of the knock-ons. This
treatment is therefore independent of the type of
bombarding particle used. To apply the general
concepts developed in the present paper to any specific
type of irradiation, one must calculate the energy
spectrum of the primary knock-ons resulting from such
irradiation. By this means, one should obtain a more
accurate picture of the total damage than by simply
assuming that each primary knock-on possesses a cer
tain average energy as has usually been done in the
past,2,4 because the nature as well as the amount of
damage is found to vary with this energy.
Concerning the damage from a knock-on atom of
very high energy, the picture is represented in Fig. 1.
Here the metal crystal is represented schematically in
two dimensions as a square lattice, the intersections of
the background lines representing the lattice sites.
Interstitial atoms and vacancies are represented as
indicated, as are the paths of the primary knock-on
and subsequent secondaries, tertiaries, etc.
In the present paper (1) the persistence of radiation
induced interstitials and vacancies and (2) the pro
duction of appreciable atomic interchange by thermal
spikes are assumed to be two mutually exclusive effects
of radiation damage. This is in contradistinction to the
usual assumption,' that the interstitials and vacancies
persist through the duration of the thermal spikes, even
though the spikes may disorder an ordered alloy. The
assumption that these two effects are mutually ex
clusive leads to two separate regions along the path
of a high-energy primary knock-on, each retaining a
different form of damage, as shown in Fig. 1. The high
energy region to the left of point A will retain as inter
stitial-vacancy pairs all of the displaced atoms produced
here, and there will be no appreciable atomic inter
change among the remainder of the atoms. On the
other hand, the low-energy region to the right will
retain essentially none of the interstitial-vacancy pairs
produced, but the normal atoms will not retain their
respective sites. Thus, if this region were initially part
of an ordered superlattice alloy, these atoms will
become disordered.
The author was led to make the above assumption by
the results of the calculations of the next section. It is
found that, for many metals, the thermal spikes reach
temperatures well above the normal melting point of
the material. Seitz' has estimated that the spikes reach
4 J. D. Ozeroff, KAPL-205, 1949.
Downloaded 11 May 2013 to 128.135.12.127. 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_permissionsRADIATION DAMAGE IN METALS 963
temperatures of the order of 104 OK for periods of the
order of 10-11 sec. The present calculations indicate
that, in regions which reach temperatures above the
normal melting point of the material, the concentration
of interstitial atoms and vacant lattice sites is at least
several percent of the concentration of normal atoms
and lattice sites. Thus, the average separation between
interstitials and vacancies in such regions should be
only two or three interatomic distances or less. This
should give rise to large local strains in the material.
If a true melting point exists for the material which
is heated to high temperatures by a thermal spike, it
will be at a somewhat higher temperature than the
normal melting point at atmospheric pressure, because
the material will be held under high pressure by the
surrounding undisturbed lattice. However, the tem
perature of the melting point should not be changed by
more than about a factor of 2. Thus temperatures of
the order of 104 oK are still large compared with the
melting point, implying that it is not correct to think
of the material as a solid while it is at high temperature.
Because of the high pressures and temperatures, it is
uncertain whether this material can be more appro
priately referred to as a liquid or as a dense gas. In
subsequent discussion, the words "melting" and
"liquid" will be used to describe this material, but it
should be remembered that their meanings are not neces
sarily conventional.
The large local strains in the material associated with
the production of interstitial-vacancy pairs cannot
persist indefinitely in the liquid, since the lattice sites
surrounding an interstitial atom in a solid no longer
exist. Upon melting, an atom, which in the solid was
an interstitial, is no longer defined as an interstitial.
The region including a few atoms around the location
of this atom will contain an abnormally high concen
tration of atoms, due to the presence of the "extra"
atom. Likewise, the region around a vacancy will, after
melting, have an abnormally low concentration of
atoms. Thus, immediately after melting, the former
interstitials and vacancies will be considered to be
replaced by local "density fluctuations." If the quench
ing is extremely rapid, it is expected that these density
fluctuations will give rise to an equal number of inter
stitials and vacancies again upon resolidification. If the
liquid state is maintained, these density fluctuations
should relax as a result of the associated local strains.
If this relaxation time is longer than the time the
material remains melted, the radiation-induced, inter
stitial-vacancy pairs will persist during the melting and
resolidification, while if it is much shorter, they will not.
The time for relaxation of these strains has been
estimated by the author to be of the order of 10-12 sec,
as the frequency of oscillation of any atom should be
of the order of 1013 secI, and a chain of two or three
atoms must participate in the relaxation of a given
"density gradient." As this is appreciably shorter than
the time the spike remains melted, the interstitials and TABLE 1.
Z Element To(A) E,,(ev) Z Element TO (A) E,,(ev)
11 Na 3.708 180 56 Ba 4.34 860
12 Mg 3.190 550 57 Ta 3.73 4200
13 Al 2.856 1200 58 Ce 3.64 5900
19 K 4.618 140 59 Pr 3.633 6000
20 Ca 3.93 420 60 Nd 3.62 6500
21 Sc 3.205 2500 63 Eu 3.960 1500
22 Ti 2.91 5000 64 Gd 3.554 8300
23 V 2.627 9600 65 Tb 3.508 10000
24 Cr 2.493 15000 66 Dy 3.499 10000
26 Fe 2.476 20000 67 Ho 3.480 11 000
27 Co 2.501 20000 68 Er 3.459 12000
28 Ni 2.486 23000 69 Tm 3.446 13000
29 Cu 2.551 23000 70 Yb 3.866 3700
30 Zn 2.659 19000 71 Tu 3.439 13000
37 Rb 4.87 150 72 Hi 3.14 33000
38 Sr 4.30 610 73 Ta 2.854 73000
39 Y 3.59 3000 74 W 2.734 110000
40 Zr 3.16 9000 75 Re 2.734 105000
41 Cb 2.853 25000 76 Os 2.670 150000
42 Mo 2.720 36000 ~~ II Ir 2.709 120000
44 Ru 2.644 51000 78 Pt 2.769 110000
45 Rh 2.685 47000 79 Au 2.878 80000
46 Pd 2.745 43000 81 Te 3.401 16000
47 Ag 2.882 31000 82 Pb 3.493 14000
48 Cd 2.972 21 COO 90 Th 3.59 9000
vacancies are assumed not to persist in a region which
has been heated above the melting temperature.
From considerations of ordinary diffusion data, it
seems that atomic interchange should not occur appre
ciably during the short existence of the thermal spike
unless the temperature is well above the melting tem
perature. This leads to the assumption cited earlier
regarding two mutually exclusive types of damage. The
transition energy possessed by the primary knock-on at
point A in Fig. 1, at which the rate of energy loss
becomes large enough to anneal the interstitial-vacancy
pairs and produce appreciable atomic interchange, has
been calculated for most metals, the values being given
in Table I.
The region to the right of point A is assumed to have
undergone melting and resolidification. It is believed
that the atomic interchange occurs during the time the
material is melted, predominantly as a result of a certain
amount of random motion of the atoms, possibly re
sembling turbulence in the flow of a liquid, initiated by
the relaxation of the local strains when the density
fluctuations relax. The stored energy released upon
relaxation of these strains will be sufficient to raise the
temperature even higher, thereby maintaining the liquid
state for a brief period after most of the density fluc
tuations have disappeared. During this period, the
turbulent motion initiated by the relaxation of the
strains can presumably continue to a sufficient degree
that, upon resolidification, most of the atoms will
occupy new lattice sites. Thus, essentially all of the
atoms in this region will be "displaced atoms," in the
sense that each will be displaced to a new lattice site.
A region of crystal which has undergone melting and
resolidification in such a manner will therefore be
called a "displacement spike."
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The resolidification of a displacement spike should
occur predominantly on the parent lattice, as it will
form an ideal nucleus for crystallization. Thus, the
crystal structure of the material should not be destroyed
by displacement spikes. A few small microcrystals of
entirely new orientation may possibly be formed,
however, somewhat as indicated schematically in Fig. 1.
The displacement spikes proposed here should not
be confused with the thermal spikes described by Seitz,!
as they differ in the following aspects. By their defini
tion, displacement spikes cannot sustain radiation
induced, interstitial-vacancy pairs, while it has been
assumed that thermal spikes would not anneal the
associated interstitial-vacancy pairs. The entire volume
of a displacement spike is required to have been melted
and resolidified, thus giving it well-defined boundaries,
while this is not true for a thermal spike.
Interstitial-vacancy pairs are still produced, accord
ing to the present model, in the region to the left of
point A in Fig. 1, along with an associated thermal spike.
This part of the presently proposed picture is therefore
very similar to the model described by Seitz,! with the
exception that the thermal spike does not reach a tem
perature at which atomic interchange can occur. The
main difference between the model proposed here and
the former one, therefore, is the displacement spike
concept.
CALCULATIONS
It is desired to calculate the density of interstitial
vacancy pairs produced in the region of the thermal
spike along the path of a primary knock-on atom. A
"displacement collision" will be defined as a collision
of the primary knock-on with a normal lattice atom in
which sufficient energy is transferred to the normal
atom to separate it from its lattice site, creating an
interstitial-vacancy pair. One must then calculate the
mean free path of the primary knock-on between suc
cessive displacement collisions.
Basic to a calculation of the mean-free path between
displacement collisions is a calculation of the cross
section for scatter of an atom by an identical atom
with an energy transfer greater than a given minimum
EO. This in turn involves a knowledge of the interaction
potential energy between two identical atoms VCr).
This will be calculated, assuming that the potential of
each atom is essentially it screened Coulomb potential
of the form
cf>(r) = (Zq/r)e-r/a, (1)
where q represents the charge of a proton and Z is the
atomic number of the atom. From Schiff," one finds
that, for moderately heavy atoms, the radius a of the
atomic electron cloud is of the order of magnitude where C is a proportionality factor of the order of unity
and
ao= h2/mq2=0.5282X 10-8 cm
is the radius of the first Bohr orbit of hydrogen m
representing the electronic mass.
Ozeroff4 has made a calculation of VCr) based on the
same model, but his result is apparently in error. The
detailed calculations are therefore presented in Ap
pendix A, giving the expression
(3)
This approaches the expected Coulomb repulsion as r
approaches O. At r= 2a, it changes sign, becoming a
weak attractive potential at large distances, with a
minimum at r=a(l+v3). This is consistent with the
model used, in which an atom consists of a nucleus sur
rounded by the rigid charge distribution p, given in
Appendix A. It may not be consistent with the true
physical picture, however, because effects such as the
redistribution of charge during a collision and exchange
interactions have been neglected. The closed shell re
pulsion between ions is probably the largest effect
neglected. Certainly, for r somewhat less than 2a, the
above expression is expected to be approximately
correct, as the shielded Coulomb repulsion is expected
to dominate at short distances.
The potential function, Eq. (1), which has been used,
is best suited for heavy un-ionized atoms. For the
lighter metals, the quantitative results of such a poten
tial may be inaccurate, because the manner in which
the screened Coulomb field drops off does not approx
imate the exponential form as closely as in the case of
the heavier elements. However, the range and strength
of the interaction should still be correct as to order of
magnitude; and, therefore, it should be possible to
draw good qualitative conclusions by use of it, which
should be of the correct order of magnitude, even for
the very light metals, as long as they are not ionized.
A good approximation to the upper limit of kinetic
energy at which a moving atom can be considered
un-ionized is
E= (M/m)E ioIl,
where M is the mass of the atom, m is the electronic
mass, and Eion is the ionization energy of the atom. If
it is assumed that an ionization energy of 5 ev is suf
ficient to ionize any atom to a point such that the above
expression for V (r) no longer gives good results, then
the following table gives the kinetic energies below
which the present treatment is valid.
Atom
Be9
Al27
Cu63 Energy, ev
80000
250000
570000
or
a=CaoZ-1/3, In most types of irradiation, the energy transferred
(2) to the primary knock-on will be low enough so that the
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complete history of the primary knock-on and all suc
cessive knock-ons can be treated by use of Eq. (3).
The accuracy of this potential is justified for atomic
separations less than about 1.5a. At distances between
about Sa and lOa, the closed shell repulsion is expected
to be the dominant interaction. Thus, in this range,
the potential is assumed to be an exponential repulsion
of the Born-Mayer type,6 Ae-'Br, with constants ad
justed to fit observed compressibility data.7 In the
interval1.5a<r<5a, the potential will be left unspeci
fied; the assumption will be made, however, that it is
a monotonically decreasing function of r in this range
connecting smoothly with the assumed expressions at
both ends.
Having arrived at a potential function, a satisfactory
method for calculating the scattering cross section must
now be chosen. The general condition which must be
satisfied in order that either the Born approximation or
the classical treatment may be valid is
X«a, (4)
where
X=h/Mv
is the de Broglie wavelength of the moving particle and
a represents the order of the dimensions of the scattering
field. This condition is fulfilled for atoms with energies
greater than 100 ev when the scattering field is given
by Eq. (3). Hence, in all collisions involved in the
present problem, either the Born approximation or the
classical treatment, or a combination of both, can be
used according to the conditions described by Williams.8
The classical treatment can be applied when
Vb/liv»l, (5)
where b represents the impact parameter in collisions
in which the energy transfer is small compared with the
energy of the moving particle; V represents the strength
of the potential evaluated at this distance, and v is the
velocity of the moving particle. In potentials and
regions of potentials where the opposite condition is
fulfilled, the Born approximation can be applied.
Collisions of interest in the present problem involve
energy transfers of between 1 and 25 ev, where the
energy of the moving particle is between 1000 and
100000 ev. Thus, b in Eq. (5) can be considered as
truly representing the impact parameter. The rela
tionship between b and ~ will be determined later, and
it will then be shown that Eq. (5) is fulfilled for values
of ~o between 1 and 25 ev as long as Z> 10. The clas
sical approximation will therefore be used, and it must
be remembered that the results will only be good for
metals whose atomic number exceeds 10.
6 L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Com
pany, Inc., New York, 1949), p. 168.
B M. Born and J. E. Mayer, Z. Physik 75, 1 (1932).
7 P. W. Bridgman, The Physics oj High Pressure (G. Bell and
Sons, Ltd., London, 1949), p. 160.
8 E. J. Williams, Revs. Modern Phys. 17, 217 (1945). In Appendix B, the classical treatment of the problem
is carried through, assuming that Eq. (3) represents
the correct potential over the entire range of r. On this
assumption, the expression for the energy transfer is
found to be
Zl4/3 ER2
~=4--F(b/a),
C2 E (6)
where ER represents the Rydberg energy 13.52 ev and
F(x) == {K1(x)-(x/2)Ko(x)F, (7)
where K denotes the modified Bessel function of the
second kind.
The cross section for scattering with an energy trans
fer greater than ~ for such collisions will then be
f1.=7rb Z=7raZ{F-l( C2Ee ) }2,
4ERzZ14/3 (8)
where F-l(X) is the inverse of the function defined in
Eq. (7).
The function F(x) is shown as the solid line in Fig. 2.
According to Eq. (6) this function is proportional to
the energy transfer when the impact parameter is ax.
It is noticed that at x= 2.4, the curve drops to 0, so
that a particle with an impact parameter equal to
2.4a should not be deflected. The reason for this is
apparent from the potential used, which has its mini
mum at 2.73a. Thus, an impact parameter of 2.4a
apparently corresponds to the path which a particle
must follow if it is acted on equally by the attractive
and repulsive forces, giving no net deflection.
100
0\ SOLID L.INE ---TIfE FUNCTION, F(ll,
DEFINED IN £(1.111
IIfOlIWt UHf--- THE CORRECTl:O FI_' FUNCTION
I \
.,
'1---
~ \
\
. I--1--' -\ 1\
. t--f---~ 10
10
10
10
10 '51--I--i\.
1\
1\ 10
10 -71--I--\
-8 10 012345678910
X
FIG. 2. Function F(x) as defined in Eq. (7) and corrected
F(x) function.
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Consider now the effects of having neglected the
closed shell repulsion. This interaction is approximated
by a simple exponential form Ae-Br, which is just the
negative of the form approached by Eq. (3) for r»2a.
The energy transfer in such a collision will be inde
pendent of the sign of the potential and will depend
only on the magnitude. The magnitude of the closed
shell repulsion has been calculated for several of the
inert gases,9 and it can be estimated from experimental
compressibility data7 for metals. In general, the evalu
ation of A and B from compressibility data yields
values comparable with those of the attractive term in
Eq. (3). Thus, the total interaction at distances greater
than about 7 a is of the same order of magnitude as
given by Eq. (3) and is positive rather than negative.
The energy transfer for large impact parameter should
therefore be of the same magnitude as that calculated
here. Thus, for large x, although the sign of the potential
is wrong, the F(x) curve in Fig. 2 should be approxi
mately correct.
At distances somewhat less than 2a, Eq. (3) is
expected to represent the true interaction, as the
Coulomb interactions should predominate here. The
F(x) curve in Fig. 2 should therefore be correct up to
about x= 1.5.
As the true potential is a monotonically decreasing
function both at large and small distances, rather than
a function whose slope changes sign, as is Eq. (3), there
seems to be no further reason for supposing that it has
a minimum in the neighborhood of 2a. It will therefore
be assumed that VCr) is a monotonically decreasing
function of r throughout the range 0<r<10a. This
will then eliminate the zero in the F(x) curve and
demand that F(x) be a monotonically decreasing func
tion of x. It is therefore assumed that the true F(x)
curve can be obtained by bridging the gap smoothly,
giving the broken line in Fig. 2.
he
00 3 L
p",!~o:213 o. i,~~ / 2 p
00 /
00 , /'
---/
10-2 3 5 B,er 2 , 5 Bier' Z 3 5 a 10° ;::: " :; ~i"d
FIG. 3. The mean free path of a knock-on atom between dis
placement collisions 1 as a function of its energy E in cases where
E»to.
9 A few examples of such calculations are J. C. Slater, Phys. Rev.
32,349 (1928); P. Rosen, J. Chem. Phys. 18, 1182 (1950); W. E.
Bleick and J. Mayer, J. Chem. Phys. 2, 252 (1934); M. Kunimune,
Progr. Theoret. Phys. 5, 412 (1950). It would be possible to estimate an F(x) curve for
each metal, basing the right-hand end on the compres
sibility data for the particular metal. This process is
tedious and seems unwarranted as a result of the uncer
tainty of other approximations which must be made.
Rather, the corrected F(x) curve in Fig. 2 will be
assumed as an average and will be applied to all metals.
Therefore, throughout the remainder of the present
paper, and in the use of Eqs. (6) and (8), F(x) will be
taken to be defined by the broken curve in Fig. 2,
rather than by Eq. (7) which gives the solid curve.
For purposes of determining whether or not Eq. (5)
is fulfilled, the form of the assumed potential at large
distances can be taken approximately as the magnitude
of the attractive part of Eq. (3),
Z2q2
V (r)-+--e-r/a•
2a (9)
Using this and Eq. (6), it can now be seen that Eq. (5)
is fulfilled if foE> 1000 (ev)2 and Z> 10.
The mean free path between displacement collisions
is given by
(10)
where 0"0 is the cross section for a displacement col
lision and No is the density of atoms. It will be assumed
that, for both close-packed and body-centered cubic
metals, No is given by
No= 1.4/r03, (11)
where ro is the interatomic distance. This is accurate
only for close-packed metals, but is less than 8 percent
in error for the body-centered cubic structure. Equation
(8) will be used for 0"0' One can then pick any metal
with a given Z and ro, assume a value for the displace
ment energy fO and plot 1 vs E.
A more general curve can be obtained, however, for
all metals by plotting the quantities P and Q, which are
defined by the expressions
la02
P=--,
r03Z2/3
EfO Q=-.
Z14/3
Then, from Eq. (8) it is seen that if one sets
F(x) = (C'l/4ER2)Q,
the cross section for displacement is
7rC'la02
u,o=7ra2x2=--x2•
Z2/3 (12)
(13)
(14)
(15)
Therefore, combining Eqs. (10), (11), (12), and (15),
P= 1/1.47r(;2x2• (16)
The value of C will be taken as 2.09, as used by Ozeroff4
to agree with the Thomas-Fermi atom model. Then,
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0.010 PvsE
0.009 ~ ,.,;
0.008 _ J'002
P=ro3Z2/3
0.007
0006
P
0005
0.004
0.003
0.002
0.001
o 50 100 150 200 250 300 350 400 450
E(ev)
FIG. 4. The mean free path of a knock-on atom between dis
placement collisions 1 as a function of its energy E in cases where
E",.o.
from Eqs. (14) and (16) and the F(x) curve, one can
plot P vs Q, giving the curve in Fig. 3. For any given
metal, one knows Z and '0; and, by estimating a value
for EO, one obtains the 1 vs E curve for a particular
metal from Eqs. (12) and (13) as P and Q are directly
proportional to 1 and E, respectively.
The impulse approximation was used to obtain the P
vs Q plot in Fig. 3. It is therefore possible to set a lower
limit on the values of Q for which this curve is valid, by
applying the validity criterion for the impulse approxi
mation, namely,
Eo<<E.
This gives
(17)
To obtain a rough idea of the form of the 1 vs E curve
for lower values of E, it will be assumed that the scat
tering caJ? be treated as hard sphere scattering, the
cross section for scattering with an energy transfer
greater than EO being given approximately by
O'Eo=7I'bo2[1.-(Eo/E)], (18)
where bo represents the distance of closest approach of
the atoms in a head-on collision.
To obtain an expression for bo, the form of the poten
tial at large distances will be taken as that given in
Eq. (9). Setting this equal to E, one can solve for bo,
giving Combining Eqs. (10), (11), (12), (18), and (19), it
is found that in this low-energy limit, P is given by
In this case, it is not possible to choose a single coor
dinate proportional to E which will give a single curve
representing all metals. Thus, a separate curve must be
plotted for each set of values for Z and Eo. Figure 4 has
been plotted showing the family of 1 vs E curves which
one obtains by choosing EO as 25 ev. The curves from
Figs. 3 and 4 have been combined to give the 1 vs E
curve for copper as a typical example in Fig. 5. While
the two curves do not join exactly, the agreement
between them is satisfactory for present purposes.
The displacement energy EO assumed to be 25 ev, is
in general of the order of, or less than, 100 times the
energy per atom necessary to melt the material within
a displacement spike. From this fact, one can see from
Eq. (8) and Fig. 2 that the number of displaced atoms
in any region should be about one-fourth the number of
atoms which are heated to the melting temperature.
Thus, in regions in which the lattice is heated to suf
ficiently high temperatures that melting may occur,
the density of displaced atoms according to this theory
should be about 25 percent of the total atom density.
While this figure may be somewhat in error as a result
of the rather crude approximations made in the pre
ceding calculations, it seems certain that the number
should be at least 5 percent, supporting the assumption
that radiation induced Frenkel defects cannot persist
in a region which has been melted and resolidified.
A criterion must now be set up to determine whether
or not such melted regions are produced, and what the
transition energy Etr of the knock-on is when such
production starts. There are three factors which con
tribute to the heating of a displacement spike: (1) the
nondisplacement elastic collisions of the knock-on atoms
with the normal atoms of the lattice; (2) the transfer of
i.nE FOR COPPER
002" 5,.
0020
4,.
3,.
2ro -
0008
psOOO45
or £~ro 0004
10 la' 1O' la' 1O' 10·
E(e'l}
bo= Cao In(Z7/3 ER).
Zl/3 C E (19) FIG. 5. The mean free path of a copper knock-on atom between
displacement colIisions 1 as a function of its energy E.
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energy to the lattice from the electrons which have been
excited by the knock-on atoms; and (3) the release of en
ergy as a result of the annealing of the Frenkel defects.
Calculations made by BrookslO show that the energy im
parted to the electrons by a knock-on atom is dispersed
among a large number of electrons before it can produce
an appreciable temperature rise in the atomic lattice;
hence Process (2) is negligible. Process (3) occurs as a
result of Process (1); therefore, the heating as a result
of Process (1) alone must be great enough to initiate
the action which is then continued by Process (3). It
will therefore be assumed that the lattice heating re
sulting only from nondisplacement elastic collisions of
the primary knock-on with lattice atoms must be suf
ficient to melt a continuous cylindrical region along its
path, from two to four interatomic distances in diam
eter, in order to initiate the production of a displace
ment spike. Thus, the value of Etr will be taken as the
energy of the primary knock-on at which the rate of
energy loss to nondisplacement elastic collisions is just
large enough to heat to melting temperature from 4 to
12 lattice atoms per interatomic distance along its path.
This criterion can be simplified somewhat by con
sidering the previously estimated ratio of the relative
numbers of displaced and melted atoms. One can
therefore estimate that the value of Etr is the energy
of the primary knock-on at which the mean free path
between successive displacement collisions becomes of
the order of one interatomic distance. This will be
taken as the criterion for the determination of Etr• It is
therefore seen that Etr for copper will be given by the
second intersection of the curve in Fig. 5 with the
horizontal broken line, which represents a mean-free
path between displacement collisions equal to one
interatomic distance.
Similarly, values of Etr for all metals can be obtained
from Fig. 3 by calculating the value of P corresponding
to one interatomic distance in the given metal, and
determining Etr from the corresponding value of Q,
given by the curve. Table I has been calculated by this
method, assuming EO to be 25 ev for each metal.
THE NATURE OF DISPLACEMENT SPIKES
From Table I, one deduces that, if the radiation
damage model proposed in this paper is correct, the
damage produced in the heavier metals by most of the
common kinds of irradiation should consist primarily
of displacement spikes. Only in a few of the lightest
metals should the production of interstitial-vacancy
pairs be appreciable. One likely means of experimentally
differentiating between this model and the former one
should be a determination of the number of interstitial
vacancy pairs produced in some of the heavier metals
by pile neutron irradiation. The present model predicts
few or none, while other workers have assumed this to
be the primary damage resulting from such irradiation.
Another difference is that the former model assumes
10 H. Brooks (private communication). that the damage produced in both the light and heavy
elements is of essentially the same nature, the primary
effect being the production of interstitial-vacancy pairs.
The presently proposed model, however, predicts that
the damage in the heavy elements will be of a different
nature than in the light elements, consisting chiefly of
displacement spikes. In this connection, some specu
lation on the nature of displacement spikes and on the
differences expected to be observed in the physical
effects resulting from displacement spikes and from
interstitial-vacancy pairs seems appropriate.
It is easy to imagine processes by which dislocation
loops can be formed within a displacement spike during
its resolidification. The formation of small microcrystals
of new orientation, as illustrated in Fig. 1, should be a
more difficult process and, if it occurs, should be much
less frequent than the production of dislocation loops.
The reason for this is that the boundary of such a small
microcrystals should consist of an array of these dis
location loops. Thus it seems that small dislocation
loops should be one of the primary products of displace
ment spikes.
As a result of the small size of these dislocation loops,
they will be in strong tension; and, as a result, they
should anneal easily, simply by collapsing. The activa
tion energy for annealing should increase with the size
of the loop. Thus, the annealing of this type of damage
can be expected to have a variable activation energy,
giving an annealing process which occurs over a rather
wide range of temperature. It should be possible to
observe annealing beginning at very low temperatures,
but which will not run to completion until the tem
perature is raised considerably higher.
This is in contrast to the annealing which one should
observe for a process involving only a single activation
energy. Here, if a temperature is found at which an
nealing will begin, the process should run to completion
at this temperature. From measurements of the an
nealing of radiation damage, it may therefore be possible
to separate the two models, because the annealing of
interstitials and vacancies should be characterized by a
small number of discrete activation energies rather than
a continuous range.
The size of displacement spikes can probably be
estimated crudely by dividing the energy available for
production of the spike, which is just the energy of the
primary knock-on at the time spike production starts,
by an average energy per atom when the spike is in
the melted condition. The energy per atom necessary
to melt typical metals at atmospheric pressure is
between 0.1 ev and 0.2 ev. The metal within displace
ment spikes will be held at high pressure by the sur
rounding lattice, raising the melting point somewhat,
to a value which should still be less than 0.5 ev. To
account in a rough manner for heat losses to the sur
rounding nonmelted lattice and to the electronic system,
the estimate will arbitrarily be raised to the order of
1.0 ev per atom as the amount of available energy dis-
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sipated by each atom of the spike. Using this figure,
one obtains an average size for the displacement spikes
produced by 2-Mev neutrons in copper of about 2X 1()4
atoms. If the region is assumed spherical, this represents
a sphere about 75A in diameter. This then also repre
sents an approximate upper limit on the size of dis
location loops which can be produced.
It was argued earlier that a density of interstitial
vacancy pairs giving an average vacancy-interstitial
separation of only two or three interatomic distances
would not persist throughout the associated thermal
spike, because the relaxation ti.me of the pairs was prob
ably short relative to the melted period of the spike. It
is possible, however, that a lower concentration of
interstitial-vacancy pairs could persist during the tem
perature pulse. In particular, one might think that an
average vacancy-interstitial separation of 8 to 10
interatomic distances might have a relaxation time
somewhat longer than the time the spike remains
melted. Relaxation of the interstitial-vacancy pairs
therefore may be thought of as progressing until the
concentration of atoms in interstitial positions is
decreased to the order of 10-3, at which time resolidi
fication occurs, freezing in the remaining interstitial
vacancy pairs. Hence, it is possible that a few in
terstitial-vacancy pairs will be produced in each dis
placement spike, the maximum concentration being
estimated to be of the order of 10-3• For pile neutron
irradiation on copper, this corresponds to about 20
interstitial-vacancy pairs per primary interaction, a con
siderably smaller number than one would obtain from
calculations based on the earlier model, of the type
made by Seitz.2 In the lighter elements, Be, C, Na, Mg,
and AI, the two models should predict about the same
type of damage, as the size of the displacement spikes
in such cases should be negligible. Attempts to experi
mentally differentiate between the two models should
therefore be carried out on the heavier metals.
CONCLUSIONS
It would be desirable to measure a direct effect of
irradiation on some physical property of the heavier
metals which could be definitely assigned to either
interstitials and vacancies or to dislocation loops. Such
an effect, however, may be difficult to find, as both
should increase the electrical resistivity, both are ex
pected to produce increases in hardness, and other
effects are probably common to both. Two more
indirect methods have been described in the last section
by which the accuracy of the present model may be
checked. These are (1) an analysis of the dependence of
the nature of the damage on atomic number and (2) an
analysis of the annealing of property changes produced
by the damage. It is suggested that such experiments
be carried out on metals irradiated with pile neutrons
at temperatures as low as possible. The average size of
displacement spikes produced by pile neutrons should
be considerably larger than those produced by cyclotron irradiation. Thus the production of dislocation loops
and other displacement spike effects should be more
pronounced in neutron irradiation. In order to retain
as much of the damage as possible, the metals should be
held at temperatures as low as can be maintained during
irradiation, because, according to the present model,
displacement spike effects may begin to anneal at tem
peratures below that of liquid nitrogen.
ACKNOWLEDGMENTS
The author wishes to express his gratitude to the
following persons for many valuable suggestions which
have been incorporated in this paper: D. B. Bowen,
C. E. Dixon, J. S. Lomont, 1\1. 1\1. Mills, and F. Seitz.
APPENDIX A
Consider a system of two atoms. The calculation is
simplified considerably by imagining the atoms to be
nonidentical at first. The separation distance will be
denoted by CR, the first being located at coordinates RI
and the second at R2• The respective potentials will be
and as exp(-I r-Rll/al)
4>1 =Zlq------
Ir-Rd
exp(-lr-R21/a2)
4>1 =Z2q'------
Ir-R21
1
p= __ \T24>,
47r
the associated charge distributions are
PI ___ Zlq{exp(-lr-RII/al) } 47r0(3) (r-Rl) ,
47r a121 r-RII
P2 ___ Z2q{eXp(-lr-R21/a2) } 47r0(3)(r-R 2) ,
47r a221 r-R21
where 0(3) is the three-dimensional 0 function repre
senting the nuclear point charge. The electrostatic
energy of the system is given by
w=~i (4)1+4>2) (Pl+P2)d(3)r.
2 ail space
The cross terms will give the interaction energy
The two terms in this integral must be equal as a con
sequence of Green's reciprocation theorem,!l and there
fore
v = I 4>IP2d(3)r.
all space
11 W. R. Smythe, Static and Dynamic Electricity (McGraw-Hili
Book Company, Inc., New York, 1950), p. 34.
Downloaded 11 May 2013 to 128.135.12.127. 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 JOHN A. BRINKMAN
An equivalent result should be obtained if cf>2P1 is
integrated instead of tP1P2. Thus,
v 1 --=-----
CR 47l"a22
f exp(-lr-R 11/a1-lr-R 21/a2)
X L d(3)r
all space I r-R111 r-R21
exp ( -CR/ a2) 1
i exp( -I r-R11/a1-1 r-R21/a2) X d(3)r,
all space Ir-R11Ir-R21
and therefore
exp ( -CR/ a1) -exp ( -CR/ a2)
= 47l"a12a22 .
CR(a12-a22)
for a1:;6 a2.
The atoms can now be considered identical again,
setting
a1=a2=a, Zl=Z2=Z.
V(CR) then takes on an indeterminate form, which when
evaluated becomes
Z2q2 (CR) V(CR)=- exp( -CR/a) 1--.
CR 2a
APPENDIX B
As Eo is generally small relative to the energy of the
moving particle, the classical treatment can be sim
plified by use of the "impulse approximation," in which
the moving atom is considered undisturbed by the
stationary atom. The impulse given to the stationary
particle is given by
[= f+'" FJ.dt=2i+"'(M)lFedx,
-'" 0 2E
where F J. is the component of the force exerted on the
stationary atom by the moving atom perpendicular to
the path of the moving atom, and is given by
FJ. =Fr sine, Fr= -av/ar.
Here, e is the angle between the line joining the two
atoms and the path of the moving atom, and V is
given by Eq. (3). This gives
~=b(2M)\ d1+ d2+ d3}, Z2q2 E where
f'" 1 e-r/a
£f3= - dr,
b r2 (r2-b2)i
and b is the impact parameter.
-2a2 fJ 1 is just the Laplace transform of the function
for O<r<b
for b<r< 00,
which is equal to Ko(bj a), where K denotes the modified
Bessel function of the second kind. The integrals 92
and 93 can then be obtained by integrating 91 with
respect to II a. Thus,
1 '"
92=-f Ko(xb)dx,
a 1/a
fJ3= f'" f'" Ko(xb)dxdy.
1/a 11
By use of the formula
f'" dy f'" f(x)dx= -f'" (a-x)f(x)dx.
a y a
93 can be shown to be equal to
Thus 93= -f'" [(1ja)-xJ Ko(xb)dx.
1/0
and
The energy transfer is
[2 Z14/3 ER2
E=-=4--F(bja),
2M C2 E
where C is defined in Eq. (2), and
ER=q2/2ao= 13.52 ev=Rydberg energy,
and
F(x) == {K1(x)-(x/2)Ko(x)}2.
Downloaded 11 May 2013 to 128.135.12.127. 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 |
1.3061038.pdf | Second Sound Propagation in liquid helium II
John R. Pellam
Citation: Physics Today 6, 10, 4 (1953); doi: 10.1063/1.3061038
View online: http://dx.doi.org/10.1063/1.3061038
View Table of Contents: http://physicstoday.scitation.org/toc/pto/6/10
Published by the American Institute of PhysicsSecond Sound Propagation/~\NE OF THE STRANGEST anomalies thus far
v-/ exhibited by matter has been the special thermal
wave property of liquid helium II, known as second
sound. Characteristic of no other substance than helium
II—that weird form adopted by liquid helium below
the A-temperature of 2.19°K—second sound is essen-
tially an undamped thermal wave propagation. Such
thermal waves display all the usual properties of wave
phenomena, including resonance and reflection charac-
teristics. This property of heat flow conforming to a
wave equation, rather than to the classical diffusive
heat flow equation, results in such seemingly paradoxi-
cal situations as heat flowing uphill against thermal
gradients. Anomalous even in name, second sound never
activates microphones and is generated by heat im-
pulses rather than by mechanical impulses. Finally, its
behaviour provides perhaps the most effective means
for investigating and understanding the true nature of
this so-called quantum liquid, helium II, and the asso-
ciated quantum hydrodynamics.
Let us commence our discussion by considering a
case of one-dimensional thermal propagation, in which
heat pulses are introduced at one end face of a cylindri-
cal enclosure containing helium II. Such heat pulses
may be generated electrically and then detected, after
a characteristic time delay in transit, by the tempera-
ture sensitive opposite end functioning as a bolometer.
Using timing techniques analogous to radar, the results
can be presented oscillographically as illustrated by the
photograph of Fig. 1, where the horizontal time scale
provides a direct measure of this delay time in terms of
the number of calibrated marker pips (and thus the
wave velocity).
The oscillogram of Fig. 1 illustrates rather simply
the true wave nature of second sound propagation. That
is, following the primary signal representing the directly
arriving thermal wave packet, there appears another
signal corresponding to roughly three times the initial
delay time. This of course represents the heat pulses
which have been reflected back from the receiver sur-
face to the transmitter, and return. Fig. 1 thus demon-
strates pictorially the reflectivity property for thermal
waves in liquid helium II, which of course was inherent
in the original thermal standing wave experiments of
Peshkov.1
Thermodynamics of Pulses
We have introduced the subject of these heat pulses
because their study reveals a great deal about the gen-
eral behavior of second sound. At the risk of boring
the reader with a few equations, some of the mathe-
matical relationships for second sound propagation will
be formulated on the basis of such pulses. In this man-
Fig. 1. Oscillogram of Second Sound Pulses. Direct pulse
arriving after 9 delay marker intervals is followed by re-
flected (triple transit) pulse at 27 delay marker intervals.ner we obtain the mathematical results without refer-
ence to the wave equation, other than to accept the
shape-preserving feature of its solutions.
Let us assume that a sudden one-dimensional heat
pulse enters liquid helium II initially at ambient tem-
perature T. If the heat current density H is constant
during the pulse duration, a resultant square wave re-
gion of excess temperature T will progress through the
liquid at a constant wave velocity v2 characteristic of
the temperature T. This pulse is represented by (A) of
Fig. 2.
Since the heat delivery H (erg sec^crrr2) across any
hypothetical normal plane (a-a) must represent the
heat transported by the region of excess temperature
r proceeding at velocity v2 through liquid of density p
and specific heat capacity C (erg gnr'deg-1), we have
H/r=pCVi. (1)
Here we have divided through by r in order to put the
expression in the form of characteristic thermal admit-
tance. Relationship (1) provides the basis for treating
second sound propagation analogously to electrical or
acoustical systems, which can always be done, and ex-
presses the dependence of heat current on temperature
variations (from the ambient) rather than on tem-
perature gradient.
"Cold pulses" may also be propagated through liquid
helium II. Thus in (B) of Fig. 2 we see such a cold
pulse represented by a region within which the tempera-
ture is below the ambient T by amount T. Instead of
coinciding with the direction of wave propagation, the
heat flows in this case in the reverse direction, toward
the source of cold impulses, as indicated. At the pulse
front (b) heat current H pours from the undisturbed
region ahead into the pulse region cooler by amount T.
Since reversibility is inherent to the wave equation
[solutions here being of the form T(x± v2t)], this
constitutes a reversible heat flow between a source at
temperature T and a heat receiver at temperature T — T.
The reverse process occurs at the rear b' of the pulse
where heat actually flows from the cold interior up to-
ward the ambient reservoir temperature following the
pulse. Accordingly we may regard the cold pulse (B)
as a self-contained thermodynamic unit constituting a
reversible heat engine at the pulse front coupled to, and
thus driving, a reversible refrigeration unit at the rear.
This coupling is provided by the shape-sustaining prop-
erty of the thermal pulses.
Such a system of heat flow into and out of a colder
region is consistent with the second law of thermody-
namics provided an appropriate amount of mechanical
energy appears, and is in turn consumed, as the pulse
passes. This requires the entirely new concept of a
mechanical energy content in a region (to first order)
of zero net mass flow and zero pressure fluctuations!
Such a radical form of energy cannot be rationalized in
John R. Pellam, chief of the Cryogenic Physics Section in the Heat
and Power Division of the National Bureau of Standards, is a member
of the American Physical Society.
PHYSICS TODAYn LIQUID HELIUM II By John R. Pellam
1
b b
(B)Fig. 2. Thermal Pulses. Temperature distributions (one-
dimensional) in various thermal pulses progressing through
liquid helium II; in each case shown here, the pulse is
moving to the right. (A) Square-wave heat pulse, (B)
square-wave cold pulse, and (C) saw-tooth heat pulse,
showing uphill heal Mow.
(C)
classical materials and, as we shall sec, requires a new-
concept of liquid structure for visualization. We will
not attempt for the moment to visualize the exact form
taken by such energy, deferring this to the discussion
of the two-fluid model. However we can at this point
deduce from thermodynamic considerations the amount
of this energy, and provide direct experimental verifica-
tion of its existence.
Referring to (B) in Fig. 2, the fraction of heat flow
H converted to such mechanical energy at the pulse
front (b"> must equal the usual ratio of temperature
difference to absolute temperature. This rate of con-
version constitutes a complex packet of mechanical en-
ergy flow y (erg sec"1cm"2) transported with the pulse
7,'// = T/T; or 7 = rff/r (2)
(even though for cold pulses heat flows counter to the
pulse propagation, this mechanical energy progresses
with the pulse). Similarly at the rear of the pulse the
reverse process occurs, corresponding to a refrigerator
returning its working substance to the higher tempera-
ture T. In complete analogy with other wave propaga-
tions, the quantity y representing energy flow is essen-
tially the Poynting vector for heat waves! On this basis
it is not difficult to see for example that within a saw-
toothed shaped heat pulse, such as (C) in Fig. 2, the
uphill heat flow is not a violation of the second law of
thermodynamics, but rather a result of it. (Direct ap-
plication of the second law to second sound waves was
first made by Gogate and Pathak.2)
The parallelisms to other types of wave propagation
are manifest in many ways. For example, one of the
thermodynamic requirements of such thermal wave
propagation is that the peculiar mechanical energy of
such a travelling second sound wave be divided equally
between a kinetic energy form (depending on H-) and
a potential energy form (depending on T2) ; this is in
complete analogy with ordinary mechanical wave propa-
gation. Such equidivision of thermal wave energy may
be deduced easily by a modification of Rayleigh's early
method for the equivalent acoustical case. That is, one
examines the juxtaposition of identical square-wave
heat pulses approaching from opposite directions and
reconciles the quadratic dependence of y on H [from
(1) and (2)] with the conservation of mechanical
energy.
If we divide energy flow y of expression (2) by wave
velocity v2, we obtain simply the mechanical energy
density. Then considering the equidivision of this en-
ergy between the two forms (i.e. one-half kinetic) wehave for the kinetic energy density KE
(in units of erg cm"3). Although this result is thermo-
dynamically independent of any fluid model, we shall
see shortly that the two-fluid model provides an excel-
lent basis for visualizing both the kinetic and potential
energy forms existing within second sound waves.
The Two-Fluid Hypothesis
Thus far we have deliberately avoided the two-fluid
hypothesis in our discussion of second sound waves in
liquid helium II. This was for the purpose of present-
ing the concept of second sound propagation on as
purely a thermodynamic basis as possible. In this way
we have been able to recognize some of its properties
as quite general, and not dependent on any particular
fluid model.
We must now introduce the two-fluid concept, not
only as a means of visualizing such quantities as the
kinetic and potential energy densities, but also for de-
riving an expression for the wave velocity of second
sound. The two-fluid hypothesis was originally pro-
posed by F. London 3 as a Bose-Einstein condensation
phenomenon. This model enabled Tisza 4 to predict the
existence of second sound and to foretell correctly some
of its properties. Some time later Landau,5 employing
a somewhat different two-fluid hypothesis, independently
predicted second sound, and deduced in fact the correct
velocity behavior for temperatures all the way down to
a few tenths of a degree above absolute zero.
The two-fluid hypotheses presuppose liquid helium II
to be made up of two component liquids occupying the
same space at the same time. One of these, the so-called
normal fluid component, is responsible for all of the
entropy of liquid helium II and also its viscosity; the
superfluid, on the other hand, is considered totally
devoid of both entropy and viscosity. The absence of
viscosity in superfluid is so complete, in fact, that this
component can actually flow through the normal fluid
component without friction or interference! This situa-
tion may be handled most conveniently by ascribing
separate flow fields to the two component fluids, so
that momentum pnvn is associated with normal fluid
flow and pHv8 with superfluid flow; where pn and vn
refer to density and particle velocity respectively for
normal fluid, and p9 and vs refer to superfluid.
We can thus write, in terms of p and v for the liquid
as a whole,
P=Pn+p,,
OCTOBER 1953and, for second sound waves,
v. = - I - (4)
The density equation simply expresses the composite
density p as the sum of the component densities. Equa-
tion (4) states the condition of zero net momentum as-
sociated with second sound waves (recalling that micro-
phones are not affected), and specifics the remarkable
condition that within these thermal waves the two fluid
components are actually flowing directly through each
other in opposite directions! This "internal counter-
flow" occurs along the direction of wave propagation
(as does also the heat flow) so that in this sense second
sound may be considered "longitudinal".
We may now specify the quantities discussed earlier
more definitely in terms of this two-fluid system. For
example, the kinetic energy density KE may be written
directly in terms of the component particle velocities
and densities
KE=ipnvn* + lP,v.\ (5)
Furthermore, since the entropy of liquid helium II re-
sides entirely within the normal fluid component, vn
may be related directly to heat current density H. Re-
ferring once again to (A) of Fig. 2, the entropy flow
H/T supported within the pulse by this normal fluid
component is given by the London expression
H/T = pSv, (6)
where 5 is the entropy (in erg gm-Meg"1) of liquid
helium II. For this we have visualized the flow of heat
as a mass transport process associated with the motion
of normal fluid component, but completely unaffected by
the counterflowing and "thermally empty" superfluid.
Finally we can write for the "mechanical" expression
for kinetic energy density KE
1 PP n
KE = -
As we shall see, this relationship can be equated to our
earlier "thermodynamic" expression (3) for this same
quantity, to give an expression for second sound ve-
locity. Before going on to this, however, we shall con-
sider an experiment bearing on the above subject.
We have already noted that the existence of this
kinetic energy density within such thermal waves can
be verified by a direct mechanical test. Thus far we
have treated second sound propagation as a purely
thermal phenomenon, both in excitation and detection.
We have in fact emphasized that ordinary acoustical
devices, such as vibrating sources and microphones, are
ineffective for dealing with these waves. Nonetheless, as
we have also seen in the foregoing, a mechanical energy
content is fundamental to the existence of such waves,
and can be observed mechanically by appropriate
methods.
Direct observation of this kinetic energy is made pos-Fig. 3a. Thermal Raylcigh Disk Experiment. Thin disk suspended by
sensitive fiber within resonant second sound field swings slightly
crosswise to propagation axis from equilibrium 45° orientation. Heat-
flow distribution // indicated by dotted line; internal counterflow of
normal fluid component (n) and superfluid (s) around disk develops
torque-producing "internal stress" in liquid.
sible by a device invented by Lord Rayleigh for measur-
ing acoustic energy. It is noteworthy that this Rayleigh
disk, developed in the century before microphones, is
capable of detecting the internal counterflow of quantum
hydrodynamics to which modern microphones are deaf.
Rayleigh ° suspended a small disk within the sound field
of a resonant acoustic cavity and oriented at an angle
TT/4 to the axis of wave propagation. For conditions of
resonance this sensitive disk was deflected slightly by
a torque tending to swing it cross-wise to the direction
of wave-motion. Konig showed by integrating the Ber-
noulli pressure over the surface of such a disk of radius
a that this torque was given by (4a?/3)pV2, in terms
of undisturbed fluid velocity v.
For the purpose of illustrating this mechanism we can
refer to Fig. 3a. Although this diagram represents the
present application to second sound, it enables us to
visualize the process for Rayleigh's classical application
also. Thus during the portion of the cycle during which
the particle flow is from left to right, a stagnation point
is formed where particles encounter the disk on the far
left side (1). Similarly a stagnation point occurs on the
near right side (2). At the same time, however, unim-
peded streamline flow takes place tangentially past the
corresponding opposite sides of the disk. The resultant
Bernoulli pressure difference across the thickness of the
disk provides a torque as shown. It is easily seen that
an identical pressure distribution is set up during the
opposite half of the cycle. The quadratic nature of the
Konig expression requires this condition so that the
disk acts as an acoustic detector.
The special application 7 to thermal waves in liquid
helium II stems from this quadratic dependence on
particle flow. Thus we can see from Fig. 3a that each
of the two counterflowing fluid components should exert
its contribution to torque independently of the other.
That is, both terms of the kinetic energy expression (5)
exert their separate influences on the disk, giving
Torque =-a3pnvn2+-a3pavsi
4,a PP-
3 P-I{H/pST)\ (8)
Thus in terms of the two-fluid concept the resultant
torque produced by second sound on the Rayleigh disk
is visualized as the sum of the torques exerted by nor-
mal fluid and superfluid separately as they stream past
the disk in opposite directions! And this occurs in the
complete absence of any detectable acoustic-type pres-
sure fluctuations or momenta, in an otherwise perfectly
quiescent medium.
Experimental confirmation of the Konig formulation
extended to thermal waves is shown in Fig. 3b, where
PHYSICS TODAY190
100Fig. 3d. Torque on Disk versus Tem-
perature. Torque ratio (T/<//*>AV.)
versus temperature 7"(°K) for ther-
mal Rayleigh disk in liquid helium
II. Circles represent experimental
values, and solid line gives the theo-
retical value, equation (8). Dotted
lines represent separate contributions
of the component fluids as indicated.
A- POINT
1.25 1.5 1.75 20
TEMPERATURE (°K)2 25
Torque'(ff1-Av >s plotted versus temperature T. It
may be observed that the torque exerted by superfluid
is greatest near the A-point. where superfluid is scarce;
and similarly, the effect of normal fluid is greatest at
the lowest temperatures where its concentration is low.
These effects are the direct result of the zero momen-
tum condition (4), requiring the minority component to
travel faster, plus the quadratic dependence of torque
on particle velocity (S) more than off-setting the de-
creased density.
Wave Velocity of Second Sound
As in other forms of wave propagation, the velocity
of second sound is the most readily measurable quantity
associated with the phenomenon. It is also perhaps the
most physically significant. We already have the expres-
sions from which this wave velocity Vn may be writ-
ten; combining equations (3) and (7) for the thermo-
dynamic and mechanical expressions for mechanical
energy KE, we have the Tisza-Landau equation
(9)Pn C
That is. second sound velocity v2 is the quantity relat-
ing these two energy expressions.
We note that equation (9) is essentially a thermo-
dynamic expression, giving us a great deal of insight to
the behavior of second sound. For example, near the
A-point where superfluid disappears (p — pn —>0) the
wave velocity drops to zero. At temperatures in the
1°K-2°K range, the value of (p — pn)/pn may be deter-
mined by an independent mechanical measurement, the
Andronikashvilli experiment, thus affording a check of
(9). Finally at temperatures below 1°K, where v2 can
still be measured directly but pn cannot, expression (9)
provides an indirect evaluation of pn. We next consider
Andronikashvilli's direct measurement of pn.
Andronikashvillis suspended a set of closely-spaced
disks in liquid helium II on a torsion fiber, as shown in
Fig. 4, and observed the dependence upon temperature
of the angular rotation period of the system. Now it is
well known by experiment that the exceptional heat
flow properties of liquid helium II are suppressed in
narrow channels (presumably a close correlation be-
tween entropy and viscosity). Andronikashvilli spacedSuspension
Disks (edge -view)
Fig. 4. Andronikashvilli Ex-
periment. Parallel disks sus-
f)emled for rotational oscil-
ation in liquid helium II.
these plates so close together that the heat content, and
thus the normal fluid, would necessarily be carried with
the disks during their angular oscillations. At the same
time, the completely non-viscous superfluid component
would ignore the motion of the disks and remain sta-
tionary. Accordingly, by observing the period of this
torsion pendulum he was able to measure the effective
mass pn associated with the normal fluid component of
the helium (subtracting of course the background mo-
ment of the torsion pendulum itself).
Actually what is really involved in applying these re-
sults for pn to equation (9) is to express second sound
velocity in terms of Andronikashvilli's observed entropy
moment, viz.
h-I
C(10)
Here / represents the effective moment of inertia of
the system attributable to normal fluid density, and /„
the moment at the A-point (i.e. where the liquid is en-
tirely normal fluid). The known correctness of expres-
sion (10) in the 1°K-2°K range thus merely expresses
a consistent relationship between two different types of
thermo-mechanical experiments. The role of the two-
fluid concept here has really been to provide a vehicle
for relating such experiments, and formulation (10) is
the truly basic one.
The over-all second sound behavior is illustrated in
Fig. 5 where wave velocity (m/s) is plotted vs tem-
perature (°K) from the A-point down to a few hun-
dredths of a degree Kelvin above absolute zero. The
solid curve (Peshkov-Pellam-Herlin) in the region above
1°K shows the velocity behavior in the upper tempera-
ture range where Tisza's and Landau's results agree
[given by (9) and/or (10)], and illustrates the rapid
decrease to zero near the A-point as the liquid becomes
all normal fluid.
The results in the lower half of the temperature
range confirm the qualitative correctness of Landau's
early prediction 5 that second sound velocity would in-
OCTOBER 1953
-Fig. 5. Second Sound Wave Velocity. Second
sound velocity (m/s) versus temperature T
from the X-point (2.19°K) down to a few
hundredths of a degree above absolute zero.
In the upper temperature range (1°K-2°K
roughly) the curve (Peshkov-Pellam-Herlin)
agrees favorably with both Landau's and Tisza's
predictions. In the range below 1°K the veloc-
ity rises as predicted by Landau. The dotted
curve represents Atkins's and Osborne's data;
the solid turve below 1°K represents moreu
o200
100\
—
Atkins
Osbo
1\
\
\
\
\
ne »
1V
\\\
I^NBSdou
1v,/V5
1Vo
|VC
1X -
Ipoint. ^jo"1
0~
0
o's
o"6
o"y
i iy
i i i/
i L ./
opc '
), I 1y^VSiOp,.3.3
t«.^-poinf
0.3 0.4 0.5 0.6 07 0.8 0.9 10 1.25
TEMPERATURE T ("Kelvin!
fig. 6. Normal fluid Concentration. LOR-IOR plot of pti p for liquid
helium II vs tcmpi-rature T. Above 1.2°K data are Andronikashvilli's
direct measurements; below 1.2°K results are deduced indirectly from
velocity measurements, using Eq. (6). Below about one-half degree Kelvin,
pn p obeys the T4 behavior predicted by Landau.
crease drastically at temperatures just below 1°K. Meas-
urements at these extreme low temperatures had to
await the application to the problem of cooling by
adiabatic demagnetization. First steps in this direction
were taken at the National Bureau of Standards in
1949, when a sample of liquid helium II was cooled
sufficiently to observe a doubling in wave velocity. The
pulse technique was used and a velocity increase from
IS.4 m/s to 34 m/s measured at temperatures well be-
low 1°K, thus strongly favoring Landau's treatment.
Some time later Atkins and Osborne 9 extended such in-
vestigations down to much lower temperatures, observ-
ing a gross velocity increase to values apparently taper-
ing off at about ISO m/s, leaving little doubt about the
over-all correctness of Landau's predictions. The dotted
line of Fig. 5 gives these results and provides the gen-
eral shape of the velocity curve.
The solid line below 1°K in Fig. 5 represents second
sound velocity measured relatively recently 10 at NBS
Cde Klerk, Hudson, and Pellam) under conditions more
closely approaching temperature equilibrium (i.e. warm-
up times of one-half hour). At roughly one-half degree
Kelvin there is a partial levelling-off in the neighbor-
hood of Landau's predicted upper limit v^y/3, where
vx is the first (ordinary) sound velocity. However, it is
also evident that at still lower temperatures the velocity-
continues to rise, apparently to an eventual value nearer
to i', than the "Landau velocity" vx/\/3. Thus although
Landau was essentially correct in his prediction of
sharply increased velocity below 1°K, his treatment
clearly requires some refinements to account for the
continued velocity increase.
As mentioned earlier, at temperatures below 1°K
where an Andronikashvilli measurement would be futile
(pjp is less than 1 percent below 1°K), the second
sound velocity determinations lead to an indirect evalua-
tion of normal fluid concentration. Using these velocity
results in conjunction with equation (9), pjp is given
in Fig. 6 down to about 0.1°K. These determinations
are plotted versus temperature on a log-log basis toillustrate the approach below about one-half degree
Kelvin to the T* behavior predicted by Landau (to be
discussed shortly). Note the extremely low normal fluid
concentrations which can be determined by these second
sound measurements—pjp is but one part in 100 mil-
lion at 0.TK1
The Phonon Gas Concept
Landau's correct anticipation of the increase in second
sound velocity below 1°K was based primarily upon his
special consideration of the phonon gas behavior at
temperatures approaching absolute zero. Phonons may
be regarded as quantized sound excitations, conforming
in many ways to the behavior of photons. Thus an as-
semblage of phonons may be pictured as a (sound)
radiation gas and as such displays the same T* total
heat content of black body radiation, with the attendant
Debye T3 behavior of entropy and specific heat. There
is an important difference, however, in regard to in-
teractions between phonons. Whereas photons do not
interact directly, maintaining equilibrium distribution
rather through interaction with the container wall,
phonons suffer direct collisions with each other. Thus
phonons may be regarded on a "particle" basis, and
the influence of effects attributable to "mean-free-path
lengths" between collisions may become important.
Perhaps the basic viewpoint most contributing to
Landau's success in predicting second sound behavior
near absolute zero was his recognition of the "radiation
mass" of these phonon excitations as normal fluid
density. This was consistent with Landau's insistence
on treating the partition of the liquid in terms of
thermal energy, associating all thermal excitations of
any kind with a normal fluid, rather than to regard
specific groups of atoms as superfluid and other specific
groups as normal fluid, as did Tisza. Landau essentially
deduced the moment of inertia associated with a hypo-
thetical Andronikashvilli experiment near absolute zero
by computing the net momentum associated with the
phonon gas carried between hypothetical disks. This in-
volved an integration over the classical sound momen-
tum p = e/v1 for phonons of energy t (and first sound
velocity i>j) obeying a Bose-Einstein energy distribu-
tion. We shall not go into the details of this rather
involved integration here, but the resulting effective
densitv ratio due to the phonon gas becomes
(11)
where Eph is the energy (per gram). Although we refer
to this quantity as the ratio of normal fluid density to
liquid helium density, we tacitly understand that it
really represents the ratio ///„ of an imaginary An-
dronikashvilli experiment.
Relationship (11) for Pn/P (or I/Io), plus the de-
pendence of Eph on the fourth power of temperature,
are sufficient for evaluating equation (9)—or (10)—for
second sound velocity. Thus Cpll = 4Eph/T and Sph =
4Epj3T (where the symbols Cph and Sph represent
PHYSICS TODAYphonon specific heat and entropy, respectively), and
substitution into (9) leads directly to the well-known
Landau velocity.
(12)
While appearing somewhat remarkable at first en-
counter, this Landau velocity is actually almost a fun-
damental requirement of thermal propagation in a
phonon gas. For with constant phonon velocity vt
(phonons all must travel at the velocity of sound!)
the average individual messenger velocity along any
particular axis becomes «i/V3, and it is indeed diffi-
cult to see how a signal could be transmitted at any
other velocity! The situation is easier to visualize than
the propagation of ordinary sound in air, where the
Newton velocity related to mean particle speed re-
quires the Lagrange correction for variation in molecu-
lar velocities. (A simplified derivation of the Landau
velocity based on the phonon gas picture has been given
by Ward and Wilks,11 following a treatment by de Hoff-
mann and Teller i: for sound propagation in a relativ-
istic gas.)
Role of Particle Statistics
Thus far we have discussed the entire subject of
liquid helium II without direct reference to the role
played by the fundamental particle statistics. Although
we have relied upon the two-fluid model often during
the foregoing, we have not considered the question of
why there should be two fluid components in liquid
helium II. In fact, even in discussing Landau's essen-
tially correct computations of second sound velocity it
did not appear necessary to introduce the subject of
particle statistics. This seems somewhat surprising, par-
ticularly in view of the marked differences known to
exist between the liquid properties of helium 4 and the
rarer isotope helium 3.
In London's original proposal of the two-fluid model
of liquid helium II as an example of a Bose-Einstein
condensation, he intimately related the properties to
the even-particle nature of helium 4. And by the same
token, the possibility of direct experimental verification
of this straight-forward hypothesis was provided in
terms of the properties of liquid helium 3; for the odd-
particle helium 3 should display Fermi-Dirac behavior,
and thus no A-point nor superfluidity properties. The
subsequent liquefaction of helium 3 and verification 13
that no transition of the helium I-helium II type oc-
curred, at least down to a few tenths degree Kelvin,
substantially supported the London hypothesis.
Accordingly we may justifiably ask at this point how
Landau made his correct predictions regarding second
sound behavior on the basis of a theory apparently in-
dependent of the Bose-Einstein condensation hypothe-
sis. The answer is probably that, by properly associat-
ing normal fluid density with all thermal excitation in
liquid helium II, the generally correct behavior can be
associated with any two-fluid model, given sufficient
flexibility in arbitrary parameters. In Landau's case, heappears arbitrarily to have chosen a two-fluid model in
which phonons contribute to the excited state at the
lowest temperatures, and in which he ascribed the
rapidly augmented energy content of the excited state
above one-half degree Kelvin to rotons. By assigning a
convenient energy gap A to these rotons, and introduc-
ing other arbitrary parameters, he was able to fit the
corresponding entropy expressions to known values, and
thus produce a mathematically valid result.
The success of the Landau treatment need not de-
tract at all from the validity of London's condensation
hypothesis. Tisza's calculation notwithstanding, there is
every reason to believe that the entire problem can be
treated on the basis of the Bose-Einstein hypothesis.
Actually in his original derivation London mentioned
the existence of Debye waves in the Bose-Einstein
liquid, and the picture presented earlier of thermal
signals transmitted by phonon collisions should apply
with the same resultant velocity i^/x/3 near absolute
zero; the only requirement is the truism that phonons
travel with the velocity of sound!
The London hypothesis has the special advantage of
not requiring any new concepts, such as rotons, or other
devices to explain the drastic increase of specific heat
with temperature between one-half degree Kelvin and
the A-point, because this behavior is inherent in a con-
densation model. Perhaps the weakest feature of the
condensation model to date has been the arbitrary ap-
plication of an essentially gas model to a liquid state.
While the basic condensation property of an even-
particle substance should persist as well for the liquid
state, other considerations must be introduced to make
the situation more physically realistic.
Of these probably the most important concerns the
zero-point energy of liquid helium which is credited
with preserving the liquid state down to absolute zero
for either helium 3 or helium 4. A more detailed
quantum-mechanical treatment of liquid helium II
should take into account not only the effects of the
particle statistics but also such background effects as-
sociated with the liquid state itself as this zero-point
energy. It could reasonably be expected that an analysis
of this nature, with the corresponding modifications in
distribution function, ought to result in the same nu-
merical results as the Landau treatment but with de-
termined, rather than adjusted, constants.
References
1. V. Peshkov, /. Exptl. and Theor. Phys. (USSR) 16, 1000 (1946).
2. D. Gogate and P. Pathak, Proc. Phys. Soc. 59, 457 (1947).
3. F. London, Nature 141, 643 (1933); Phys. Rev. 54, 947 (193S).
4. Tisza, J. de Phys. et Rad. 1, 165, 350 (1940); Phys. Rev. 72,
838 (1947).
5. L. Landau, /. Exptl. and Theor. Phys. (USSR) 11, 592 (1941).
6. Lord Rayleigh, Phil. Mag. 14, 186 (1882).
7. J. Pellam and P. Morse, Phys. Rev. 78, 474 (1950); J. Pcllam
and W. Hanson Phys. Rev. 85, 216 (1952).
8. E. Andronikashvilh, J. Exptl. and Theor. Phys. (USSR) 18, 424
(1948).
9. K. Atkins and D. Osbornc, Phil. Mag. 41, 1078 (1950).
10. D. dc Klerk, R. Hudson and J. Pellam, Phys. Rev. 89, 326, 662
(1953); and a paper to be published.
11. J. Ward and J. Wilks, Phil. Mag. 42, 314 (1951).
12. F. de Hoffmann and E. Teller, Phys. Rev. 80, 692 (1950).
13. D. Osborne, B. Weinstock, and B. Abraham Phys. Rev 70 9SS
(1949).
J. Daunt and C. Heer, Phys. Rev. 79, 46 (1950).
OCTOBER 1953 |
1.1748046.pdf | RBranch Heads of Some CO2 Infrared Bands in the CO + O2 Flame Spectrum
W. S. Benedict, Robert C. Herman, and Shirleigh Silverman
Citation: The Journal of Chemical Physics 19, 1325 (1951); doi: 10.1063/1.1748046
View online: http://dx.doi.org/10.1063/1.1748046
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/19/10?ver=pdfcov
Published by the AIP Publishing
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Linemixing and durationofcollision effects in the ν3 Rbranch band head of CO2
AIP Conf. Proc. 328, 306 (1995); 10.1063/1.47457
The vibrational infrared spectrum of CoO
J. Chem. Phys. 71, 474 (1979); 10.1063/1.438093
Erratum: On ``Determination of dissociation energies for some alkaline earth (hydro) oxides in CO/N2O
flames''
J. Chem. Phys. 60, 1698 (1974); 10.1063/1.1681260
Determination of dissociation energies for some alkaline earth (hydro) oxides in CO/N2O flames
J. Chem. Phys. 59, 2572 (1973); 10.1063/1.1680373
Photographic InfraRed Emission Bands of O2 from the CO – O2 Flame
J. Chem. Phys. 17, 220 (1949); 10.1063/1.1747228
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131.111.185.72 On: Fri, 05 Dec 2014 11:52:37LETTERS TO THE EDITOR 1325
. ..
o.o:.-,L._~.L-. __ ----1-------- .. l-,.o-------'
MOLARITY 0# HCI.
FIG. 2. Optical density at tB versus HCl concentration.
A. 0.00203 M Na,S,03. B. 0.00520 M Na,S,o,. C. 0.0121 M Na,S,o •.
Our results indicate that even for the dilute range, the optical
density at tB is not constant. The variation could be interpreted as
due to a change in the critical supersaturation concentration of
sulfur with reactant concentration. However, it seems unlikely
that this could account for the tremendous range of optical densi
ties observed at tB. It is more likely that molecular species other
than sulfur which absorb at 3000A 0 are preferentially formed at
the higher acid concentrations.
The polythionates have been identified among the acid decom
position products of thiosulfate.5 Lorenze and Samuel6 have
measured the absorption spectra of the polythionates. They give
for the molar extinction coefficient of tetrathionate and penta
thionate at 3000A 0, 40, and 320, respectively. On the other hand,
La Mer and Kenyona found no appreciable absorption by tetra
thionate at 3000A 0 and it was on the basis of this that they con
cluded that the transmission of their solutions during the homo
geneous reaction was only a function of sulfur concentration.
Messrs. Robert Penn and Carleton Hommel of this laboratory
have redetermined the absorption spectra of tetrathionate and
pentathionate and confirm the results of Lorenze and Samuel.
This indicates that the optical density at 3000A 0 measures both
polythionates and molecular sulfur.
This interpretation is consistent with the minimum in tB. Both
polythionates and sulfur are among the acid decomposition
products of thiosulfate. The formation of polythionate is favored
at higher acid concentrations so that although the rate of decom
position increases monotonically with acid concentration (as
attested by the high optical densities atrained) the net rate of
formation of sulfur becomes low, hence the large values of lB.
, H. Reiss and V. K. La Mer. J. Chern. Phys. 18. 1 (1950).
, Bassett and Durant. J. Chern. Soc. 129. 1401 (1927).
3 V. K. La Mer and A. S. Kenyon. J. Colloid Sci. 2. 257 (1947).
• E. M. Zaiser and V. K. La Mer. J. Colloid Sci. 3. 571 (1948).
'Janickis. Z. anorg. u. allgern. Chern. 234. 193 (1937) .
• Lorenze and Samuel. Z. physik. Chern. (B) 14. 219 (1931).
The Polarization of Rayleigh Scattering As an Aid
to the Determination of Molecular
Orientation in Solids
E. G. Cox
Chemistry Department. University of Leeds. Leeds. England
(Received August 14. 1951)
ATTENTIONI has been drawn by D. H. Rank to the value
of the polarization of Rayleigh scattering p as an aid to the
determination of molecular configuration in liquids. It may be of
interest to point out that this constant can also be used to deter
mine the approximate orientation of molecules in crystals. In the
case of a non-associated molecule having two of its three principal
polarizabi\ities equal or apprOltimately so, the values of p and R, the molecular refraction, suffice to determine the numerical values
of the polarizabilities; knowledge of the space-group, density, and
refractive indices of the substance in the crystalline state then
enables the approximate molecular orientation to be deduced.
The results are not so accurate as those based on measurement of
magnetic anisotropy, but the method is likely to be convenient
for relatively simple substances of low melting point on which
magnetic measurements are less easily made and for which the
values of p are more likely to be available. A simple example is
provided by benzene where it was found possible in this way
to confirm the molecular orientation deduced by x-ray methods;'
details of the calculation were not published at the time but are
quoted by Hartshorne and Stuart. a
, D. H. Rank. J. Chern. Phys. 19.511 (1951). 'E. G. Cox. Proc. Roy. Soc. (London) AI3S. 491 (1932).
3 Crystals and the Polarizing Microscope (Edward Arnold and Company.
London. 1950). p. lSI.
R-Branch Heads of Some CO2 Infrared Bands in
the CO+0 2 Flame Spectrum
W. S. BENEDICT
National Bureau of Standards. Washington. D. C.
AND
ROBERT C. HERMAN AND SHIRLEIGH SILVERMAN
Applied Physics Laboratory.* The Johns Hopkins University.
Silver Spring. Maryland
(Received August 13. 1951)
SOME time ago the authors observed the presence of several
small peaks on the short wavelength side of the CO2 funda
mental V3 in the emission spectrum of the CO+02 flame. These
observations were made with a spectral slit width of ~8 cm-I.
Recently the spectrum was re-examined with a Perkin-Elmer
spectrometer using a LiF prism, a sensitive Golay detector, and
slit widths of 40 microns which give a resolving power of ~2.2
em-I. At this resolution these peaks were recognized as the
R-branch heads of some of the vibration-rotation bands of the
CO2 molecule. The emission spectrum is shown in Fig. 1 together
with the atmospheric transmission band of CO2 at ~4.2.u. A
comparison of the observed and calculated positions of the
R-branch heads found is given in Table I. The calculated values
were obtained from well-known formulasl•2 using a set of constants
which were obtained in the course of a recent effort to fit all avail
able data. These constants which differ slightly from previously
published values!. are given in cm-I as follows: VI = 1342.9,
V2= 667.3, va=2349.3, Xu =3.06, X,,=0.67, Xaa= 22.5,
X12= -3.20, X13=20.50, X23=11.75, Xll= -1.17, b'=51.69
(1-0.039 Va), <>1=0.00044, <>2= -0.00072, <>3=0.00307,
Bo=0.3904, and Do= 1.8X 10-7• The calculated value of the
rotational quantum number at the R-branch heads of all the bands
is J(head) = 127. The existence of moderately prominent heads is
to be expected since in the CO+02 flame employed the rotational
temperature has been determined from the CO overtone bands
to be ~2700oK.a
TABLE La
" (R-branch head) (cm -,)
Band Transition ",(cm-') Observed Calculated
a 001 --+000 2349.3 2396.7 2397.1
b 01'1 ~ 01'0 2337.6 2385.3 2385.4
c {IOI }--+eOO} 2327.4 2377.1 2.H5.3
d 02'1 0200 2326.3 2374.1 2373.9
02'1 --+ 02'0 2325.8 2373.6
e 002 --+ 001 2324.2 2.372.4 2372.0
• Wave numbers are referred to vacuum (see Fig. I).
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131.111.185.72 On: Fri, 05 Dec 2014 11:52:371326 LETTERS TO THE EDITOR
z o
I-
t3
....J
lJ...
W o
2300 2350 2400
WAVELENGTH IN CM-1
FIG. 1. (Al Infrared emission spectrum of the CO +0. flame showing the
R-branch heads of several vibration-rotation bands of the CO. molecule.
(Bl The CO, atmospheric absorption band is shown for comparison.
It is clear that many additional bands from higher vibrational
levels must appear in this spectral region. However, they are
mostly at lower frequencies and overlap with each other as well as
with the CO fundamental so that even high resolving power would
be of little avail in unraveling these bands. It would be of interest
to obtain the relative intensities of the bands reported in order to
estimate a vibrational temperature. Unfortunately, the difficulties
of overlap as well as atmospheric and self-absorption would seem
to make this type of determination too inaccurate to be of much
value.
* This work was supported in part by the U. S. Navy Bureau of Ordnance.
1 D. M. Dennison. Revs. Modern Phys. 12. 175 (1940). 'G. Herzberg. Infrared and Raman Spectra (D. Van Nostrand Company,
Inc., New York. 1945) .
• Plyler, Benedict, and Silverman (submitted for publication).
Compressibilities of Concentrated Metal
Ammonia Solutions*
ROBERT H. MAYBURyt AND LOWELL V. COULTER
Department of Chemistry, Boston University, Boston. Massachusetts
(Received August 14, 1951)
CONCENTRATED solutions of the alkali and alkaline earth
metals in ammonia exhibit such characteristically metallic
properties as high electrical conductivity,l reflectivity to light,2
and degenerate paramagnetism.3 In addition, these solutions,
upon preparation, undergo an anomalous volume expansion with
an accompanying decrease of density< indicating unusual orienta
tion of the molecules and atoms in the solution. Further indication
of this has been revealed by compressibility studies which are re
ported in part at this time. Adiabatic compressibilities of concentrated sodium, lithium,
calcium, and potassium iodide solutions have been determined
between -33°C and -70°C from sound velocity measurements
carried out by a modified electronic pulse technique.6 Two x-cut
quartz crystals were immersed face to face in the solution. The
additional transit time of a sound signal sent from one crystal to
the other upon increasing their separation by a measured amount
was determined electronically. The Browning P-4 Synchroscope
was used as the timing and presentation unit. Combination of the
measured velocity with the corresponding density of the solution
yields the adiabatic compressibility according to the expression:
fJ= Ifc2p.
The results presented in Fig. 1 are striking in the case of the
metal ammonia solutions in that the compressibility rapidly
increases as the concentration increases. The generally observed
behavior in the case of electrolyte solutions is a decreasing com
pressibility with increasing concentration, reflecting the operation
of electrostrictive forces as is observed in the case of KI in am
monia. It appears that something other than electrolytic nature
must be assigned to the metal ammonia solutions as a result of the
observed compressibility behavior.
To account for the properties of the metal ammonia solutions, a
model is proposed which regards these as an expanded metal6 in
w z
>-90
~~80
" ..,
"-70 o
~ " §60
;;; o SODIUM
~ LITHIUM
" CALCIUM
() POTASSIUM
IODIDE
~ 50 -.JPURE LIQUID
~O -~~~r---~
LOG,. MOLES N~PER GRAM ATOM METAl.
(OR PER MOLE SALT)
FIG. 1. Compressibilities of metal and KI ammonia solutions.
which the lattice unit is the cation solvated by six ammonia mole
cules having an outward orientation3 of the hydrogens. This com
plex is reasonable in view of the strength of the ion-dipole bond as
reflected in the existence of isolable solids as Ca(NH3)6 having
metallic properties.7 A degree of randomness, of course, exists in
the lattice of the actual solution.
By analogy with metals the cohesive energy of the system arises
from the binding together of the complexes by the metal valence
electrons in the role of weak resonating covalent bonds.s That the
metal electrons must be considered in such a role follows from the
observed degenerate paramagnetism3 implying Fermi statistical
behavior and from the high electrical conductivity indicating
broad conduction bands. The repulsive energy originates in the
mutual electrostatic repulsion of the complexes since their ex
terior surfaces are composed of the hydrogen ends of ammonia
dipoles.
The volume expansion observed suggests that a large equilib
rium inter-complex distance is established in concentrated solu
tions by the mutual repulsion of the complexes; as a consequence,
the magnitude of the attractive and repulsive forces at equilibrium
is small. This conclusion provides a ready explanation of the ob
served high compressibility. The complexes are located in very
shallow potential wells and as a result experience only small
potential energy changes when the sound wave effects small dis
placements about the equilibrium position [i.e. (d2V /a"')r=ro is
very small and since 1/{J a: (d2V /ar2)r=ro, {J is large as observed]'
The individuality of the curves "and the calcium point are ac
countable on the basis of the differences in the respective cohesive
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131.111.185.72 On: Fri, 05 Dec 2014 11:52:37 |
1.1748045.pdf | The Polarization of Rayleigh Scattering As an Aid to the Determination of
Molecular Orientation in Solids
E. G. Cox
Citation: The Journal of Chemical Physics 19, 1325 (1951); doi: 10.1063/1.1748045
View online: http://dx.doi.org/10.1063/1.1748045
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/19/10?ver=pdfcov
Published by the AIP Publishing
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141.209.144.159 On: Wed, 10 Dec 2014 18:45:06LETTERS TO THE EDITOR 1325
. ..
o.o:.-,L._~.L-. __ ----1-------- .. l-,.o-------'
MOLARITY 0# HCI.
FIG. 2. Optical density at tB versus HCl concentration.
A. 0.00203 M Na,S,03. B. 0.00520 M Na,S,o,. C. 0.0121 M Na,S,o •.
Our results indicate that even for the dilute range, the optical
density at tB is not constant. The variation could be interpreted as
due to a change in the critical supersaturation concentration of
sulfur with reactant concentration. However, it seems unlikely
that this could account for the tremendous range of optical densi
ties observed at tB. It is more likely that molecular species other
than sulfur which absorb at 3000A 0 are preferentially formed at
the higher acid concentrations.
The polythionates have been identified among the acid decom
position products of thiosulfate.5 Lorenze and Samuel6 have
measured the absorption spectra of the polythionates. They give
for the molar extinction coefficient of tetrathionate and penta
thionate at 3000A 0, 40, and 320, respectively. On the other hand,
La Mer and Kenyona found no appreciable absorption by tetra
thionate at 3000A 0 and it was on the basis of this that they con
cluded that the transmission of their solutions during the homo
geneous reaction was only a function of sulfur concentration.
Messrs. Robert Penn and Carleton Hommel of this laboratory
have redetermined the absorption spectra of tetrathionate and
pentathionate and confirm the results of Lorenze and Samuel.
This indicates that the optical density at 3000A 0 measures both
polythionates and molecular sulfur.
This interpretation is consistent with the minimum in tB. Both
polythionates and sulfur are among the acid decomposition
products of thiosulfate. The formation of polythionate is favored
at higher acid concentrations so that although the rate of decom
position increases monotonically with acid concentration (as
attested by the high optical densities atrained) the net rate of
formation of sulfur becomes low, hence the large values of lB.
, H. Reiss and V. K. La Mer. J. Chern. Phys. 18. 1 (1950).
, Bassett and Durant. J. Chern. Soc. 129. 1401 (1927).
3 V. K. La Mer and A. S. Kenyon. J. Colloid Sci. 2. 257 (1947).
• E. M. Zaiser and V. K. La Mer. J. Colloid Sci. 3. 571 (1948).
'Janickis. Z. anorg. u. allgern. Chern. 234. 193 (1937) .
• Lorenze and Samuel. Z. physik. Chern. (B) 14. 219 (1931).
The Polarization of Rayleigh Scattering As an Aid
to the Determination of Molecular
Orientation in Solids
E. G. Cox
Chemistry Department. University of Leeds. Leeds. England
(Received August 14. 1951)
ATTENTIONI has been drawn by D. H. Rank to the value
of the polarization of Rayleigh scattering p as an aid to the
determination of molecular configuration in liquids. It may be of
interest to point out that this constant can also be used to deter
mine the approximate orientation of molecules in crystals. In the
case of a non-associated molecule having two of its three principal
polarizabi\ities equal or apprOltimately so, the values of p and R, the molecular refraction, suffice to determine the numerical values
of the polarizabilities; knowledge of the space-group, density, and
refractive indices of the substance in the crystalline state then
enables the approximate molecular orientation to be deduced.
The results are not so accurate as those based on measurement of
magnetic anisotropy, but the method is likely to be convenient
for relatively simple substances of low melting point on which
magnetic measurements are less easily made and for which the
values of p are more likely to be available. A simple example is
provided by benzene where it was found possible in this way
to confirm the molecular orientation deduced by x-ray methods;'
details of the calculation were not published at the time but are
quoted by Hartshorne and Stuart. a
, D. H. Rank. J. Chern. Phys. 19.511 (1951). 'E. G. Cox. Proc. Roy. Soc. (London) AI3S. 491 (1932).
3 Crystals and the Polarizing Microscope (Edward Arnold and Company.
London. 1950). p. lSI.
R-Branch Heads of Some CO2 Infrared Bands in
the CO+0 2 Flame Spectrum
W. S. BENEDICT
National Bureau of Standards. Washington. D. C.
AND
ROBERT C. HERMAN AND SHIRLEIGH SILVERMAN
Applied Physics Laboratory.* The Johns Hopkins University.
Silver Spring. Maryland
(Received August 13. 1951)
SOME time ago the authors observed the presence of several
small peaks on the short wavelength side of the CO2 funda
mental V3 in the emission spectrum of the CO+02 flame. These
observations were made with a spectral slit width of ~8 cm-I.
Recently the spectrum was re-examined with a Perkin-Elmer
spectrometer using a LiF prism, a sensitive Golay detector, and
slit widths of 40 microns which give a resolving power of ~2.2
em-I. At this resolution these peaks were recognized as the
R-branch heads of some of the vibration-rotation bands of the
CO2 molecule. The emission spectrum is shown in Fig. 1 together
with the atmospheric transmission band of CO2 at ~4.2.u. A
comparison of the observed and calculated positions of the
R-branch heads found is given in Table I. The calculated values
were obtained from well-known formulasl•2 using a set of constants
which were obtained in the course of a recent effort to fit all avail
able data. These constants which differ slightly from previously
published values!. are given in cm-I as follows: VI = 1342.9,
V2= 667.3, va=2349.3, Xu =3.06, X,,=0.67, Xaa= 22.5,
X12= -3.20, X13=20.50, X23=11.75, Xll= -1.17, b'=51.69
(1-0.039 Va), <>1=0.00044, <>2= -0.00072, <>3=0.00307,
Bo=0.3904, and Do= 1.8X 10-7• The calculated value of the
rotational quantum number at the R-branch heads of all the bands
is J(head) = 127. The existence of moderately prominent heads is
to be expected since in the CO+02 flame employed the rotational
temperature has been determined from the CO overtone bands
to be ~2700oK.a
TABLE La
" (R-branch head) (cm -,)
Band Transition ",(cm-') Observed Calculated
a 001 --+000 2349.3 2396.7 2397.1
b 01'1 ~ 01'0 2337.6 2385.3 2385.4
c {IOI }--+eOO} 2327.4 2377.1 2.H5.3
d 02'1 0200 2326.3 2374.1 2373.9
02'1 --+ 02'0 2325.8 2373.6
e 002 --+ 001 2324.2 2.372.4 2372.0
• Wave numbers are referred to vacuum (see Fig. I).
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z o
I-
t3
....J
lJ...
W o
2300 2350 2400
WAVELENGTH IN CM-1
FIG. 1. (Al Infrared emission spectrum of the CO +0. flame showing the
R-branch heads of several vibration-rotation bands of the CO. molecule.
(Bl The CO, atmospheric absorption band is shown for comparison.
It is clear that many additional bands from higher vibrational
levels must appear in this spectral region. However, they are
mostly at lower frequencies and overlap with each other as well as
with the CO fundamental so that even high resolving power would
be of little avail in unraveling these bands. It would be of interest
to obtain the relative intensities of the bands reported in order to
estimate a vibrational temperature. Unfortunately, the difficulties
of overlap as well as atmospheric and self-absorption would seem
to make this type of determination too inaccurate to be of much
value.
* This work was supported in part by the U. S. Navy Bureau of Ordnance.
1 D. M. Dennison. Revs. Modern Phys. 12. 175 (1940). 'G. Herzberg. Infrared and Raman Spectra (D. Van Nostrand Company,
Inc., New York. 1945) .
• Plyler, Benedict, and Silverman (submitted for publication).
Compressibilities of Concentrated Metal
Ammonia Solutions*
ROBERT H. MAYBURyt AND LOWELL V. COULTER
Department of Chemistry, Boston University, Boston. Massachusetts
(Received August 14, 1951)
CONCENTRATED solutions of the alkali and alkaline earth
metals in ammonia exhibit such characteristically metallic
properties as high electrical conductivity,l reflectivity to light,2
and degenerate paramagnetism.3 In addition, these solutions,
upon preparation, undergo an anomalous volume expansion with
an accompanying decrease of density< indicating unusual orienta
tion of the molecules and atoms in the solution. Further indication
of this has been revealed by compressibility studies which are re
ported in part at this time. Adiabatic compressibilities of concentrated sodium, lithium,
calcium, and potassium iodide solutions have been determined
between -33°C and -70°C from sound velocity measurements
carried out by a modified electronic pulse technique.6 Two x-cut
quartz crystals were immersed face to face in the solution. The
additional transit time of a sound signal sent from one crystal to
the other upon increasing their separation by a measured amount
was determined electronically. The Browning P-4 Synchroscope
was used as the timing and presentation unit. Combination of the
measured velocity with the corresponding density of the solution
yields the adiabatic compressibility according to the expression:
fJ= Ifc2p.
The results presented in Fig. 1 are striking in the case of the
metal ammonia solutions in that the compressibility rapidly
increases as the concentration increases. The generally observed
behavior in the case of electrolyte solutions is a decreasing com
pressibility with increasing concentration, reflecting the operation
of electrostrictive forces as is observed in the case of KI in am
monia. It appears that something other than electrolytic nature
must be assigned to the metal ammonia solutions as a result of the
observed compressibility behavior.
To account for the properties of the metal ammonia solutions, a
model is proposed which regards these as an expanded metal6 in
w z
>-90
~~80
" ..,
"-70 o
~ " §60
;;; o SODIUM
~ LITHIUM
" CALCIUM
() POTASSIUM
IODIDE
~ 50 -.JPURE LIQUID
~O -~~~r---~
LOG,. MOLES N~PER GRAM ATOM METAl.
(OR PER MOLE SALT)
FIG. 1. Compressibilities of metal and KI ammonia solutions.
which the lattice unit is the cation solvated by six ammonia mole
cules having an outward orientation3 of the hydrogens. This com
plex is reasonable in view of the strength of the ion-dipole bond as
reflected in the existence of isolable solids as Ca(NH3)6 having
metallic properties.7 A degree of randomness, of course, exists in
the lattice of the actual solution.
By analogy with metals the cohesive energy of the system arises
from the binding together of the complexes by the metal valence
electrons in the role of weak resonating covalent bonds.s That the
metal electrons must be considered in such a role follows from the
observed degenerate paramagnetism3 implying Fermi statistical
behavior and from the high electrical conductivity indicating
broad conduction bands. The repulsive energy originates in the
mutual electrostatic repulsion of the complexes since their ex
terior surfaces are composed of the hydrogen ends of ammonia
dipoles.
The volume expansion observed suggests that a large equilib
rium inter-complex distance is established in concentrated solu
tions by the mutual repulsion of the complexes; as a consequence,
the magnitude of the attractive and repulsive forces at equilibrium
is small. This conclusion provides a ready explanation of the ob
served high compressibility. The complexes are located in very
shallow potential wells and as a result experience only small
potential energy changes when the sound wave effects small dis
placements about the equilibrium position [i.e. (d2V /a"')r=ro is
very small and since 1/{J a: (d2V /ar2)r=ro, {J is large as observed]'
The individuality of the curves "and the calcium point are ac
countable on the basis of the differences in the respective cohesive
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1.1748407.pdf | The Properties of the Interstitial Compounds of Graphite. I. The Electronic
Structure of Graphite Bisulfate
Gerhart Hennig
Citation: The Journal of Chemical Physics 19, 922 (1951); doi: 10.1063/1.1748407
View online: http://dx.doi.org/10.1063/1.1748407
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example N 2. The magnitude of this inner-shell effect
on the bond energy, as judged by the Swi values,12 is
perhaps not large, but far from negligible,13
The phenomenon of forced hybridization leads to a
curious paradox. In a molecule such as N2, to which
Fig. 1 is approximately applicable, the 2(jg, 2(ju, and
311'11 MO's (see Eqs. (1)) are each filled with two elec
trons. If these MO's are approximated by LeAO forms,
and we then ask how much 2s and how much 2p(j
population the above distribution corresponds to for
the two separate atoms, we may obtain what at first seem
like strange results. If we should assume no hybridiza
tion and then ignore the requirements of orthogonality,
there woutd be two electrons each in (jg2s, (ju2s, and
(jg2p; and, since each LeAO MO belongs equally to
both atoms, we would say that this corresponds to two
2s and one 2p(j electron on each atom. On requiring
LeAO orthogonality, it would still be allowable to
assume that 2(jg is pure (jg2s and 2(ju is pure (ju2s, but
3(jg would then have to be a hybrid containing a con
siderable amount of (jg2s mixed into (jg2p. If now we
count each LeAO MO as divided equally between the
two atoms, we obtain more than two 2s electrons on
each atom. Why does this not violate the Pauli prin
ciple? It might be argued that there is a violation,
resulting from unjustified initial assumptions. But this
can be disproved, since, (1), with both the shells 2(jg and
3(jg filled, as here, the total wave function when anti
symmetrized can be shown to be independent of how
12 Compare Figs. 1 and 2 with the p=4.0 figure in Fig. 1 of
reference 3.
13 This discussion is relevant to the question of Van Vle{:k's
"nightmare of inner shells" (J. H. Van Vleck and A. Sherman,
Revs. Modern Phys. 7,185 (1935», and to the discussion of inner
shell-outer-shell repUlsions in reference 4 and by Pitzer in an
earlier paper. . .
THE JOURNAL OF CHEMICAL PHYSICS 2(jg is chosen with respect to degree of hybridization,!1
and, (2), the form of 2(ju is not subject to any direct
restrictions depending on the numbers of electrons in
2(jg and 3(jg.
The answer to the paradox undoubtedly lies in the
implications of the fact that the pure forms (jg2s and
(jg2p are not orthogonal (see Eq. (14)), or of the related
fact that 2s of one atom is not orthogonal to 2p(j of the
other. The importance of the paradox is as a warning
that caution is needed in any attempt to conduct a
census of 2s and 2p(j electrons in the separate atoms of a
stable molecule on the basis of the LeAO MO method.
With regard to the specific question raised above
for the configuration 2(jg22(ju23(jg2 of N2, it is to be noted
that if 2(ju were pure (ju2s, then, regardless!1 of what
forms are used for 2(jg and 3(jg, the best answer would
appear to be that there are precisely two 2s and one
2p(j electrons on each atom. However, if (as is in prin
ciple possible, and in fact to be expected, for the best
-energy-minimizing-wave function of N2) the 2(ju
corresponds to some degree of 2s-2p(j hybridization,
then the number of 2s electrons per atom is less than
two and the number of 2p(j electrons correspondingly
greater than one.
A more thorough analysis of the matters touched on
in this Section will be postponed to a later paper.14
ACKNOWLEDGMENT
The author is very greatly indebted to Mr. Tracy J.
Kinyon for his cooperation in the preparation of the
tables and graphs. Mr. Kinyon carried out all the ex
tensive numerical computations for the tables.
14 Reference should be made to P. O. Liiwdin, J. Chern. Phys. 18,
365 (1950) for a discussion of certain closely related problems and
methods.
VOLUME 19, NUMBER 7 JULY, 1951
The Properties of the Interstitial Compounds of Graphite. I. The Electronic
Structure of Graphite Bisulfate
GERHART HENNIG
Argonne National Laboratory, Chicago, Illinois
(Received April 4, 1951)
1. The electrical resistance, its temperature dependence, and the Hall coefficients of graphite bisulfate
compounds have been determined at various oxidation stages. The measurements have indicated that the
oxidation removes electrons from a nearly full conduction band.
2. Reduction of the lamellar bisulfate compounds produces residue compounds which retain about a third
of the bisulfate ions and half the sulfuric acid originally present in the lamellar compounds. The formula of
the residue compounds is approximated by Cn• HSO.· 4H2SO •.
3. The model of a hypothetical graphite which has lost electrons from its conduction band, is approxi
mated closely by these residue compounds, since the impurities are distributed in a state of high disorder.
The lamellar compounds, on the other hand, may distort the band structure of graphite because they form a
superlattice.
INTRODUCTION
GRAPHITE is a semiconductor in which the empty
and full conduction bands are separated by either
a small1 or vanishing 2 energy gap. Those properties of
1 S. Mrozowski, Phys. Rev. 78, 644 (1950).
I P. R. Wallace, Phys. Rev. 71, 622 (1947). graphite which depend on the conduction electrons
should therefore be very sensitive to small amounts of
impurities which change the number of electrons in the
conduction bands. A strong dependence of properties
on the purity of graphite has, in fact, been known for a
long time. It was therefore decided to investigate the
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41GRAPHITE BISULFATE 923
relations between the electrical properties and the band
structure experimentally by adding known amounts of
electron donors or acceptors to graphite.
Nearly all chemical reactions of graphite produce
either lamellar compounds or residue compounds.
Lamellar Compounds
In these substances, planes of carbon atoms alternate
in a definite periodic sequence with single planes of the
reactant. The period of alternation becomes smaller as
the activity of the reactant is increased, but, with the
exception of the ferric chloride compound, the concen
tration of the reactant remains constant in those planes
which it has already invaded. A review of these sub
stances has been published by Riley. 3 Numerous
additional lamellar compounds, some of which will be
described in subsequent papers, have been observed in
this laboratory. It has always been assumed that all the
lamellar compounds except the graphite salts were
unionized. This will be shown in subsequent papers to
be incorrect. The graphite salts, however, are known
to be ionized,4 i.e., the reactants are intercalated as
negative ions, and the graphite planes share the· positive
charge.
Residue Compounds
Most of the reactant in the lamellar compounds can
be removed by essentially reversing the procedure by
which the lamellar compounds were formed. In all
cases examined, however, a small amount 'of re~ctant
remains which cannot be removed without resortmg to
fairly drastic procedures. The amount of reactant re
maining is dependent upon the amount of reactant
present in the original lamellar compound. Compounds
of this type, in which the reactant is held very strongl!'
in the graphite and cannot be removed by cathodIc
reduction or chemical washing, have been termed
"residue" compounds. These substances have not been
described previously. In residue compounds, the re
actant is distributed more irregularly through the
graphite than in lamellar compounds. It is likely that the
reactant in the residue compounds becomes trapped at
imperfections and twinning planes of the lattice. This
irregular structure of the residue compounds has been
established by x-ray diffraction measurements which
will however be reported in a subsequent paper.
I~ both la~ellar graphite salts and in the correspond
ing residue compounds, the reactant is present as n.e?a
tive ions, and the graphite planes share the pOSItIve
charges. Such a charge distribution is typical of elec
tron acceptor impurities. However, in a study of
graphite, the residue compounds are more imp0.rtant
than the lamellar compounds, since they do not dIstort
the band structure of graphite as much as do the lamel
lar compounds. The electrical properties of both
3 H. L. Riley, Fuel 24, 1, 43 (1945).
4 W. Riidorff and U. Hofman, Z. Anorg. Allgem. Chern. 238, 1
(1938). w
~o.e
:; o·!il-----+--.:~~+---i---__t---I
~ 04
w 03 S o.e g 0.1
°0~~---±,0-~,~,--+,,0~~"--'3~0--~"._~40--~~"0
OXIDATION STATE. 10-4 'QlliWClleftt5/grcII" 010111)
FIG. 1. Effect of oxidation in sulfuric acid on the
electrical resistance of graphite.
lamellar and residue graphite salts have been investi
gated. The results of the investigation on the graphite
bisulfates are reported in this paper.
Graphite bisulfates are prepared by the oxidation of
graphite in concentrated sulfuric a:id. Numero~s
oxidizing agents, as well as an electrIC current, wIll
cause this oxidation. The fully oxidized compound
has the composition4 C24+·HSOC·2H 2S04• In the
present investigations, various lamellar and. resi.due
graphite bisulfates were prepared by elec~rolY~Ic OXIda
tion and reduction in concentrated sulfurIC aCId.
The electrical resistance, its temperature dependence,
and the Hall coefficient were measured for various
graphite bisulfates. The resistance and its temperature
dependence are obviously important prop~rties of the
conduction electrons. The Hall coeffiCIent IS a measure
of the number of conduction electrons. A negative
coefficient is obtained if the conduction electrons move
near the bottom of an empty band; a positive coefficient
is obtained if the electrons move near the top of an
almost full band ..
EXPERIMENTAL
Materials
The graphite used was Acheson graphite similar in
properties to the National Car~on Company'~ Spec~ro
scopic Electrodes except that It showed a hIgh amso
tropy. All except specially designated samI;>les :vere cut
so that the current direction and the Hall dIrectIOn were
perpendicular to the axis of extrusion, i.e., the direction
of lowest resistivity.
A few samples of natural graphite from Ticonderoga
were obtained from Ward's Natural Science Establish
ment, Rochester, New York. The samples were thin
plates about 1 cm long. They were purified. by a.1ternate
washing with hydrochloric and hydroflUOrIC aCIds. T?e
platelets cleaved very easily, but showed many strIa
tions on the basal planes.
The sulfuric and nitric acids used were Baker
Analyzed Chemicals.
Electrical Resistance and Oxidation State
Samples of graphite were moun~ed in sU:h a w~y t~at
their resistance could be determmed durmg OXIdatIOn
in sulfuric acid. The samples weighed about 0.4 g and
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41924 GERHART HENNIG
0.o..----.----,--,--.--,--,-----,---,---,-,
'; D..
$0 .•
:1 0.7 I't.ductioll
= o.e ::::1.: .~!I, • O.,I------+~ __ ___;'.q,...==-::"-t---__+---___j
ii
:: 0.4
~ O.l 5 O.t
:t 0 .•
0~Q-~-~.0~~.~.-~.0~-+. •• -~>~0-~>.~-+..0~~4~.-~.0
10.4 EQUIVAlENTS OF OXIDIZING CHARGl/aRAfill ATOM
FIG. 2. The electrical resistance of graphite during
oxidation and reduction in sulfuric acid.
their dimensions were 2 mmX2 mmX40 mm. Several
millimeters from each end, a very small hole was drilled
through the sample to accommodate platinum potential
leads. The current leads were sturdy platinum wires,
which fitted into recesses at the ends of the sample.
Resistance measurements were reproducible to 0.5
percent. The whole assembly was lowered into a tube
containing concentrated sulfuric acid. The tube com
municated through a sintered glass disk with a cathode
compartment which contained concentrated sulfuric
acid and a platinum electrode. The graphite was oxi
dized by a current which varied for different experi
ments from 2 to 10 milliamperes. The oxidation state
in equivalents of electrons per gram atom of carbon
was calculated from the formula:
tI(60) (12)/96500w,
where t is the time in minutes, I is the current in am
peres, and w is the weight of the sample in grams.
The resistance decreased as the graphite became
oxidized. In a few cases this decrease was delayed, pre
sumably because of traces of oxidizable impurities
present in the acid or in the sample. In such cases the
zero time was chosen to coincide with the first decrease
in the resistance. The choice also eliminated the small
error in the measurements of the resistance which was
caused by that fraction of the oxidizing current which
flowed through the graphite. Each resistance measure
ment was corrected for a relatively small potential
recorded by the potentiometer immediately after the
resistance measuring current had been discontinued.
This potential, which is probably caused by tempera
ture fluctuations and strains in the graphite, was usually
quite small, but occasionally rose to several hundred
microvolts.
.. O...---.----..---,----,---.--..---,r---,.-....,..--,
;: 0.' ......... ..,.--, ., •• 1 ... " .• '19 .... I
~ 0 ••
~ 0.1 ""... ~ '::5
...... ~ .. 15
= 0.' ~':" .. : o.,\----__+---''''-.::--+----t---__+---___j
.0.4 .......... ,1..
0.'
0..
0.. J-..... _ ...... -... --.. _----...
... ....... ~ .. --..
o.oo~~-__7;;.o~~.~. -~.;;-o -:!:.,--;,!::o --; .. ~-±.0~~4~.-~.O
OIUOATION STATE PO'· nul .. OI."'I/"OIII .tollll
FIG. 3. Effect of reoxidation on the electrical resistance
of graphite bisulfate "residue" compounds. The resistance data obtained for twelve separate
runs on twelve different samples are plotted in Fig. 1.
All resistances are reported relative to the resistance of
the unoxidized sample in sulfuric acid .
One parallel cut and six perpendicular cut samples
were run at five milliamperes in concentrated sulfuric
acid, and one parallel cut sample was run at 10 milli
amperes in concentrated acid. The other samples were
all perpendicular cut samples and differed from each
other by the fact that two were run at two milli
amperes and ten milliamperes, respectively, one :was
oxidized in 13 molar sulfuric acid, one was oxidized
while a stream of helium was passed through the acid,
and one was oxidized while a stream of sulfur dioxide
was passed through the acid. The agreement, seen in
Fig. 1, between such a number of runs under different
conditions demonstrates that the current efficiency of
the oxidation is very probably close to unity. If it were
not unity it would have changed with the current
TABLE 1. Properties of the graphite bisulfate residue compounds.
Oxidation state 10'X Oxidation state of corresponding fractional H,SO./HSO.-of lamellar residue weight gain ratio in
Sample compound compound in residue residue
no. (10-' equivalent/g atom) compound compound
17 17.8 5.3 2.15 4.0
19 10.55 3.55 1.79 5.2
21 29.6 9.35 3.40 3.5
27 43.5 9.75 4.29 4.4
29 19.75 6.5 2.94 4.6
35 11.2 4.0 1.96 5.1
53 25.4 7.5 3.19 4.3
57 31.5 9.1 3.71 4.1
63a 31.1 9.2
63b 40.6 11.15
63c 42.7 11.70 4.88 4.2
69 20.85 6.65 2.66 4.0
71 32.5 9.85 3.46 3.4
73a 33.7 9.95
97 37.75 11.15 4.37 3.8
101 35.6 12.00 4.59 3.7
105a 36.65 10.2
density. Furthermore if it were less than unity due to a
concurrent oxidation of sulfur dioxide, for example, the
presence of excess sulfur dioxide should have decreased
the current efficiency considerably.
At any oxidation level, the lamellar graphite bisulfate
compound could be decomposed by a current which
made graphite the cathode. The progress of this reduc
tion was followed again by resistance measurements.
Fig. 2 represents two typical runs. At the beginning of
the reduction, the rate at which the resistance increased
was less than the rate at which it had decreased during
oxidation, but later the rate was higher than the corre
sponding rate during oxidation. After approximately
two-thirds of the missing electrons had been restored,
the resistance quite suddenly reached a constant value
unaffected by further passage of current. At this point
the material had the characteristics of a residue com
pound. The constant resistance ultimately reached in
the reduction is dependent only on the degree of oxida-
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41GRAPHITE BISULFATE 925
tion of the original lamellar compound. Thus, for every
lamellar compound there is a corresponding residue
compound.
The current efficiency during the reduction of the
lamellar compound was not always unity as evidenced
by a lack of reproducibility. Therefore a reliable de
termination of the number of electrons trapped in the
residue compound was not possible from measurements
made during the reduction. Fortunately the reoxidation
of the residue compound proceeded at unit current
efficiency. In fact, the rate of the resistance change
during reoxidation was identical with the rate during
the original oxidation when compared at equal resis
tances. This indicates that the resistance of a lamellar
compound and a residue compound are identical if they
are oxidized to the same oxidation state, and that the
number of electrons missing from the graphite in the
residue compound is the same as the number of elec
trons lost in the formation of a lamellar compound hav
ing the same resistance. This fact was used in drawing
Fig. 3. The dashed line in Fig. 3, which is the resistance
vs oxidation state curve for lamellar compounds, also
applies to residue compounds, as was just shown. In
this way the oxidation states for residue compounds
were determined, for which only the resistance was
known. The experimental points represent the resis
tances of residue compounds as they were gradually
oxidized.
The chemical composition of residue compounds was
determined in the following manner: The material was
washed in running water for 24 hours, dried and
weighed. Rewashing for several days in running water,
or in boiling sodium hydroxide, affected the weight
only slightly. It was assumed that the excess weight
above the original weight of graphite was due to trapped
bisulfate ions and sulfuric acid only. Since the number
of bisulfate ions should be equal to the number of
electrons lost during oxidation, the weight of these ions
could be calculated from the oxidation state. The rest
of the weight gain was assumed due to sulfuric acid.
The number of molecules of sulfuric acid associated with
each bisulfate ion is listed in Table I, column 5, and is
roughly four molecules per ion. The second column of
Table I lists the number of equivalents of electrons
missing from the corresponding lamellar compound
from which the residue compound was produced by
reduction. The entries in the third column were de
termined from the measured resistance of the residue
compound, by referring to Fig. 1.
It was found that the resistance of artificial graphite
which had been oxidized more than 0.005 equivalent
per gram atom began to increase gradually. At the same
time the samples became bent and distorted. A few
samples were sealed into glass tubes to provide a rigid
support. The tubes were perforated in six places to
admit platinum leads and acid. The resistance ratio of
one of these samples passed through a minimum of _.-,.-±::=--+--
/~ ..• . ..
" ? I.ol.----+------..I(,.j..<~--+_--__+-__I '.0
~o., 0.';
~ 0 8 0 I'ID,,,'OI O,opllit. 0.8 !
; 0.1 • A.II'ltial G.Clphlt. 0.7 •
0.6 101
w.-<"'----+------f-.----+----f---Io .•
0.'
0.'
0.'
0.'
0.00 300 400 ~ 600 100 '00 100 0.0
IO-4EQUI\lAL[NTS OF OllOlllNO CHARGE JURAN ATOM
FIG. 4. The electrical resistance of graphite during oxidation in
concentrated sulfuric acid, and the corresponding electromotive
force of the cell Hg I Hg2S041 H2S04/ CnHS04• 2H2SO,.
0.106, corresponding roughly to 0.007 equivalents of
oxidation, but increased again beyond this value.
It was anticipated that natural graphite of large
crystal size would not distort as badly as fine-grained
artificial graphite. Accordingly, a plate of natural
graphite (# 37) was set in plaster of Paris so that it was
supported on three sides for reinforcement. The plate
was about 3 mm wide, 10 mm long, and 0.2 mm thick.
Platinum wires bent into clamps provided current leads.
Fine platinum wires were threaded through holes in the
plate for potential leads. The resistance of this plate is
shown in Fig. 4. The resistance decreased 50-fold before
it began to fluctuate. It is reasonable to conclude that
the resistance would continue to decrease even further
if more rigid and more perfect samples were available.
A plate of natural graphite (# 119) mounted without
the plaster of Paris backing was oxidized 0.00286
equivalents/g atom. Its resistance ratio dropped to
0.262. On reduction, a residue compound was formed,
but its exact resistance was uncertain. The resistance
0: ......
~ ..
" Z
4 I-..
::
0: .. > ;::
~
'" 0.'
0.0 L-___ -'-...L-___ L-.L-. __ ...J
-000 -100 -80 o o.
fEMPERATURE ("C)
FIG. 5. The electrical resistance of graphite bisulfates
as a function of temperature.
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41926 GERHART HENNIG
Porenti
~.ad
Graphite
Sample al
--Current
~.ad.
I ~
f
1"-
J Wolfrom
/ROd~
V I-Platinum Clip --
PI
V
==~ -
~\
\ 1.5
... 4rnm .... Qtlnum
Wire
mm 3 em
FIG. 6. Graphite sample used for Hall coefficient and
resistance measurements in sulfuric acid.
kept increasing very slightly, as long as a reducing
current was passed. The sample was therefore washed in
running water, and its resistance ratio rose from 0.656
to 0.762 after three days of washing. Obviously, then,
the residue compounds of natural graphite are not as
stable as those formed from artificial graphite, probably
because the imperfections in natural graphite are not
very efficient traps.
The Electrode Potential
The electrode potentials of artificial and natural
graphite against a standard mercurous sulfate electrode
were measured at various oxidation states. Graphite
itself was sometimes slightly positive and sometimes
slightly negative but became consistently negative after
a very small reducing current had passed. The lamellar
compounds were always strong oxidizing agents and
produced large potentials against the standard elec
trode. During the reduction of lamellar compoun~s, the
potential decreased, passed through zero close to the
end of the reduction and became negative. The residue
compound itself was always a weaker oxidizing agent
than the mercurous sulfate electrode. The potentials
for a sample of artificial and a sample of natural graph
ite have been plotted in the upper part of Fig. 4. The
current efficiency of this oxidation was incidentally not
always unity because of the gradual disintegration of the material. Detailed measurements of the electrode
potentials and their temperature coefficients will be
published later.
The Temperature Dependence of the Resistance
A resistance sample was placed in sulfuric acid con
taining a few drops of nitric acid as an oxidizing agent
and the resistance was measured as the sample oxidized.
The sample was withdrawn periodically, washed with
sulfuric acid, and, still wet with acid, dipped into liquid
nitrogen. The resistance was redetermined. The sample
was next transferred to a dry ice-petroleum ether
mixture and the resistance remeasured. Finally the
petroleum ether was washed off with sulfuric acid, the
sample was reoxidized further, and the above measure
ments repeated. The resistances have been plotted in
Fig. 5 as a function of temperature. The resistance of
various residue compounds was also determined at
these temperatures on samples prepared by electrolytic
oxidation followed by reduction. These samples were
washed and dried before measurement. The results of
these measurements are also plotted in Fig. 5.
TABLE II. The effect of temperature on the electrical
resistance of a natural graphite bisulfate compound.
Relative resistance at
Sample 25°C -75°C -195°C Average tempera
ture coefficient
between -195
and 25°C
(t>.RX103)
(t>.T)(R,,)
Unreacted 1 0.896 0.602 + 1.8
Bisulfate 0.756 0.620 0.388 +2.2
residue compound
The temperature coefficient of the natural graphite
sample described previously (# 119) was also deter
mined, both on the unreacted sample and on its washed
and dried residue compound. The resultant data are
shown in Table II.
The Hall Coefficient
The resistance and Hall coefficient were measured
on a sample which was cut and mounted as shown in
Fig. 6. The current leads were rigid platinum clips,
while all potential leads consisted of platinum wire.
Stray potentials, and particularly the Ettingshausen
potential, were reduced considerably by leaving stubs
of graphite attached to the sample at the desired points
and making contact to these stubs. All platinum leads
were gold soldered to wolfram rods which were sealed
through a ground glass cap. The whole assembly fitted
into a flattened glass tube of 1 cm o.d. which was
mounted between the round pole pieces, 5 cm in diam
eter, of an electromagnet.
The magnet was calibrated between 4 and 15 kilo
gauss with a search coil and ballistic galvanometer. The
Hall measurements were always made at 14 kilogauss
and a current of two amperes. The field was reversed at
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41GRAPHITE BISULFATE 927
least sevep times and the average value of the differences
in the induced potentials was determined. The Hall
coefficient (A H) was calculated from the following
formula:
(AE)d(109)
2IH
where AE is the average potential difference (volts);
d is the thickness of sample (cm); I is the current (amp);
H is the magnetic field (gauss); and 109 is the conversion
factor from practical units to emu. To insure that the
Hall coefficient referred to the same number of carbon
atoms even after a chemical reaction, any changes in
the dimensions of the sample during the reaction were
ignored, and the coefficient was calculated from the
thickness of the sample before reaction. The precision of
the measurements was ca ±O.OO5 emu.
The following experimental procedure was used. A
sample was immersed completely in concentrated sul
furic acid and its properties were measured. It was
oxidized briefly by a current to a platinum electrode.
Its properties were then remeasured. Subsequent reduc
tion converted it to a residue compound. This cycle of
oxidation and reduction was repeated so that the sample
was alternately measured as a lamellar and as a residue
compound. A separate set of experiments was performed
in which the sample was oxidized in dilute sulfuric acid
by a few drops of nitric acid added to the sulfuric acid.
It was found that little oxidation occurred in 12 molar
or more dilute acid, but a very small increase in the
acid concentration near 12 molar changed the final
oxidation state of graphite considerably.
The Hall coefficients obtained after chemical or
electrical oxidation have been plotted in Fig. 7 against
the corresponding resistance. Two quite distinct curves
were obtained for the lamellar and the residue com
pounds.
DISCUSSION
Reaction Mechanisms
It is probable that the cause for the formation of
lamellar graphite bisulfates with the bisulfate planes
occurring with a definite periodicit y5 is an electrostatic
phenomenon. A layer plane filled with bisulfate ions
constitutes such a concentration of charge that, in spite
of the dielectric action of sulfuric acid, the concentra
tion of bisulfate ions at the graphite surface will be low
near this plane. Therefore, the next layer of bisulfate
ions enters as far away as possible, namely, exactly
midway between planes already filled, thus giving rise
to the stepwise or periodic reaction. Such a mechanism
would not apply to those lamellar compounds which
are not ionic. Most lamellar compounds are formed in
this stepwise fashion, and it seeems reasonable that such
lamellar compounds should also be ionic in character.
6 W. RUdorff, Z. physik. Chern. B45, 42 (1940). That this was found to be true will be shown in sub
sequent papers.
The conversion of lamellar to the corresponding
residual graphite bisulfate compounds may involve a
transitory reduction of the bisulfate ions. When Fig. 2
was discussed earlier, the peculiar delay in the resistance
increase was mentioned. The lamellar compounds act on
reduction as if the electrons were at first "stored away"
without entering the graphite conduction band, but
electrons can only be "stored away" by reducing the
bisulfate or sulfuric acid. Later during the reduction
when the ions and molecules are ejected from the
graphite they release these electrons to the graphite.
This mechanism would also explain why the current
efficiency during reduction is often not unity. If the
reduced ions escape too rapidly they may fail to return
some of the ".borrowed" electrons to the graphite and
may, in the presence of even a weak oxidizing agent,
release the electrons to this agent. It must be empha
sized that this mechanism is highly tentative and has
been postulated only to explain the behavior of the
resistance during the reduction of lamellar compounds.
Two alternate explanations are possible to explain
why a residue compound is formed at all in preference
to complete expulsion of interstitials. Isolated ions
may simply be trapped with the graphite planes col
lapse. Once trapped, the isolated ions and molecules are
certainly unable to move because the energy required
to separate two carbon planes is enormously large, due
to the number of carbon atoms involved per ion. As an
alternate explanation of residue compounds, it may be
postulated that the bisulfate ions and acid molecules in
, :
...
~
" ::: ..
0.
"
.J
.J .. 0.5
0.'
0.3
0.2
0.'
0
-0.1
-0.21-----1----+\-1------1
% -0.3
-0.' 1-----I--------1r+-I------1
-0.5
-0 .• 1-----1-------\\1-----1
-0.7
0!-:lO.L, -,10.::-2 o:':.3:-:0f-: .• -::0~.s-:lo.':-. -:':0.7:--0:': .• :-:of-: .• -,~.0------l
RELATivE RESISTANCE (R/Ro'
FIG. 7. Hall coefficient of graphite bisulfates as a
fllnction of o~idation ptatc::.
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41928 GERHART HENNIG
1
.. 0.3 z
OJ ;; 0.2
§ 0./
0 0 "
~ -0.1
oJ c
:z. ·0.2 /
/
/
/ , ", I ,
I ,
I " I ,
I '
- - --Calc\llation -- Ellpe,iment
-0.' f-f-----+-----f-------+-----j
-k " 0: ,
~ 0.7 I " = 0.6 \
; 0.15 f+' -\T----><"<:-~----r_---_+-___l
OJ \ \
a 0.4 \ ,
\ ,
~ 0.3 \ ', .... -J~ __
;: ,
~ 0.2 ' ....
a 0.1 --='29';' - 2:,·
---200;-- - - - -____ -_-___ -:. :~2~.
°0~------~,~5----~3~0------~.~.--~
OXIDATION STATE (10-4 equIYGlent,/orom atom)
FIG. 8. Calculated and experimental resistances and
Hall coefficients of oxidized graphite.
a layer plane have also invaded adjacent cracks and
imperfections. On reduction they are trapped in these
positions. Evidence in favor of this second view will be
advanced in subsequent papers.
Comparison of Compounds
One of the purposes of this investigation was the
preparation of graphite containing acceptor impurities.
Their distribution should be uniform, but statistical,
to prevent the formation of a superlattice, which would
change the band structure of the parent compound.
It is likely that lamellar bisulfates are not ideally
suited as acceptor impurities. The bisulfate ions are
concentrated in widely separated layers, where they
are probably arranged in planar lattices. The residue
compounds are certainly more dilute and very probably
more disordered than the lamellar compounds and
should, therefore, have more nearly the same band
structure as graphite itself, perhaps differing from it
only in the effective number of electrons. The experi
mental results confirm that either the band structure is
different for lamellar and residue compounds or that
the lamellar compounds are not as homogeneous as the
residue compounds. Either of these explanations could
account for the large difference in the Hall coefficients
of the compounds (Fig. 7).
The Oxidation of Graphite
The oxidation of graphite has been shown to diminish
its resistance and change the sign of the Hall coefficient.
These changes are in agreement with the commonly accepted theory that the electrons in graphite fill one
conduction band completely except for a small number
of electrons which are excited into a nearly empty band.
The excited electrons and the "missing electrons" in
the lower band constitute the current carriers. Removal
of electrons by oxidation increases the number of un
paired electrons in the lower band but decreases the
number of excited electrons in the upper band by a
smaller amount. Therefore, the net number of current
carriers increases, and the resistance decreases. Since the
carriers are predominantly in the lower band, the Hall
coefficient is positive. The number of positive carriers
depends only slightly on temperature and therefore the
temperature coefficient of resistivity becomes small
and even changes sign, because eventually the tempera
ture fluctuations of the lattice increase the resistances
more than the temperature excitation of electrons into
the empty band can compensate for. The temperature
coefficient of natural graphite is already positive,6
because of the smaller number of permanent scattering
centers,2 and increases further on oxidation.
A quantitative comparison of the experimental
results with Wallace's2 band theory of graphite was
attempted. For the purpose of this comparison it was
assumed that the intercalation of ions and molecules
into graphite reduces the number of electrons in the
graphite lattice without distorting the electron bands
and without increasing the effective number of scatter
ing centers appreciably. Furthermore, it was assumed
that the distribution of electrons in the graphite
lattice is uniform on a microscopic scale and does not
change in the vicinity of an intercalated negative ion
layer.
With these restricting assumptions the resistance
and Hall coefficient of oxidized graphite were calculated.
Wallace has derived equations for the energy and con
ductivity of electrons both in a two-dimensional and a
three-dimensional model of graphite. The calculations
have been extended to a two-dimensional model of
oxidized graphite only.
The following symbols have been used by Wallace:
Ee is the energy at the comers of the Brillouin zone, E
E is the energy of an electron, 1'0 is an exchange
energy of magnitude 0.9 ev,Jo is the Fermi distribution,
N(E)dE/N a is the density per atom of electronic
energy states between E and E+dE, du is a surface
element in the surface of-constant energy, p is the
resistivity, 1/ T is the probability per unit time of
scattering an electron wave, a and c are fundamental
lattice displacements in graphite. We define the addi
tional terms: m is the number of electrons per atom
removed from the graphite lattice by oxidation, .1. is
the difference between Fermi energy and Ee, and AH
is the Hall coefficient.
The Fermi energy of electrons coincides in graphite
with the energy at the comers of the Brillouin zone.
6 D, E. RQberts, Phil. Mag. 26, 159 (1913),
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41GRAPHITE BISULFATE 929
Oxidation, i.e., removal of m electrons, lowers the
Fermi energy by an amount A, which is a function of
the number of electrons removed and of the tempera
ture. This relation between A and m can be calculated
by equating the number of empty states created (mI2)
to the total number of states available before oxidation
minus the available electrons after oxidation:
Ee N(E) 00 N(E)
ml2 = i --dE-J --!(E)dE
-00 Na -00 Na
fEe Ec-E foo E-Ee - dE- dE
-00 1+e(E-E e+<1l/kT Ee l+e(E-Ee+<1l/kT
-2k2J'2(e-<1/kT -ie-2<1/kT +te-S<1/kT -••• ).
From this equation, A in units of ev can be computed
at a given temperature for various values of m.
For a given value of A, the resistivity can be calcu
lated from Wallace's equation (3.12):
87re2r foo d fo - IE-Eel-=-dE
h2c --00 dE
The resistance relative to unoxidized graphite at a
given temperature is then
RIRt= -2kT In2/(A-2kT In[1 +e<1/kTJ).
The Hall coefficient was calculated from the equations:7
7 H. Jones and C. Zener, Proc. Roy. Soc. (London) A14S, 269
(1934). N = ffdfo( aE)2 du dE " dE ak" I gradE I
ffd!O( aE)2 du
Nil = dE akll I gradE I dE.
Substitution of Wallace's relations
Ec-E= +v.h'Yoal (ke-k) I for E<Ec
E-Ee=v.h'Yoal(k-ke)/ forE>Ee
results in
This must be multiplied by (3X 1010) to convert to emu.
Calculated and experimental values of the Hall
coefficient at room temperature and of the resistance
at three temperatures have been plotted in Fig. 8
against the oxidation level m. The resistances have
been plotted as RI Rt, where Rt is the resistance of
unoxidized graphite at the temperature stated. The
experimental values were obtained from Figs. 1 and 5.
Comparison of the calculated and the measured
resistances shows fairly good agreement at room tem
perature, but progressively larger deviations at lower
temperatures. The calculated Hall coefficient differs
considerably from the experimental one obtained for
the residue compounds. They differ, however, by a
nearly constant value of m, as if a certain fraction of the
positive carriers behaved in reality as negative carriers.
Qualitatively, the calculated values agree with the
general trend of the electrical properties of graphite
observed during oxidation, but quantitatively, the
measured and calculated properties differ considerably.
The difference may be due in part to the choice of the
two-dimensional model of graphite for these calcula
tions. However, the assumptions which were made
earlier about the nature of oxidized graphite may also
be inadequate. These assumptions .will be tested in
subsequent papers which deal with the effects of other
acc.eptor impurities, and of donor impurities, on
graphite.
ACKNOWLEDGMENT
It is a pleasure to thank O. C. Simpson and J. R.
Gilbreath for numerous discussions and for their in
valuable help in editing this paper.
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128.197.26.12 On: Thu, 31 Oct 2013 19:40:41 |
1.1747680.pdf | On the Magnetic Properties of Liquid He3
L. Goldstein and M. Goldstein
Citation: The Journal of Chemical Physics 18, 538 (1950); doi: 10.1063/1.1747680
View online: http://dx.doi.org/10.1063/1.1747680
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/18/4?ver=pdfcov
Published by the AIP Publishing
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128.138.73.68 On: Mon, 22 Dec 2014 03:20:42THE JOURNAL OF CHEMICAL PHYSICS VOLUME 18. NUMBER 4 APRIL. 1950
On the Magnetic Properties of Liquid He3 *. t
L. GOLDSTEIN AND M. GOLDSTEIN
Los Alamos Scientific Laboratory, Los Alamos, New Mexico
(Received November 7, 1949)
The object of this paper is to investigate the possibility that liquid He3 might have a tendency toward
nuclear ferromagnetism in the approximation where this fluid is considered to be an antisymmetrical collec
tion of atoms having an angular momentum of h/2 and a finite nuclear magnetic moment. In the rather
crude approximation where the motion of the individual atoms is described with the help of plane de Broglie
waves, the exchange energy, originating in the interatomic mutual potential energy of He3 atoms, favors
the parallel alignment of all spins at the temperature of absolute zero. The total energy of the fluid is, how
ever, smaller in the antiferromagnetic configuration because its kinetic energy is smaller than in the ferro
magnetic configuration. To the approximation of these studies then, liquid Hea does not show nuclear
ferromagnetism. Its nuclear paramagnetism is then discussed.
1. INTRODUCTION
ONE of the interesting problems raised by a collec
tion of particles having an angular momentum of
hl2 and a finite magnetic moment concerns its magnetic
behavior. Antisymmetrical statistics resulting from first
principles lead for such a collection of completely free
particles to the well-known balancing of their spins
and magnetic moments. At the temperature of absolute
zero in such a system the total angular momentum and
magnetic moment vanish. This, however, is not neces
sarily true if the particles of the collection exert forces
upon each other. Indeed, as first shown by Bloch,! for
a so-called free electron gas, in which the potential
energy of any electron is constant or vanishes on the
average, the Coulomb repulsion between the electrons
leads to a quantum mechanical exchange energy for
electrons of parallel spin. This exchange energy is
negative and favors ferromagnetism. However, this
energy is to be regarded as a correction to the first
approximation energy (e.g., the kinetic energy) of the
electrons, which is positive. Actually the preceding
method of obtaining the total energy of the free electron
gas may be said to be equivalent to an approximation
method which proceeds according to the increasing
powers of the coupling parameter, e'2/hvj, where VI is
the electron velocity at the top of the Fermi distribu
tion associated with the antiferromagnetic configura
tion.2 Ferromagnetism cannot occur in this model unless
the coupling parameter is larger than unity. This latter
condition is however equivalent to that of the break
down in the validity of the approximation method.
Within the region of validity of this method then, the
free electron gas cannot exhibit ferromagnetism.
A collection of Rea atoms at liquid Rea densities may
be regarded as approximating what one might call an
antisymmetric assembly of loosely bound particles. It
* This paper has been reported on at the Cambridge meeting
of the American Physical Society, June 16-18, 1949. t This document is based on work performed at Los Alamos
Scientific Laboratory of the University of California under Gov
ernment Contract W-7405-Eng-36.
1 F. Bloch, Zeits. f. Physik 57, 545 (1929).
2 L. Goldstein, J. de phys. et rad. 7, 141 (1936). appears that a study of the magnetic properties of
liquid Hea using the admittedly crude model of de
scribing, in a first approximation, the motion of in
dividual atoms by plane de Broglie waves and com
puting the total energy of such a system by including
the mutual potential energy of He atoms, is of interest.
It is realized that there will be objections to this limit
ing gas model. It may, however, be expected that as
long as the quantum mechanical exchange energy cor
rections are of reasonable magnitude in comparison
with the average classical potential energy or the
average kinetic energy, such an approximation method
would be justified.
2. THE TOTAL ENERGY OF LIQUID Hea
In the present studies it was assumed that liquid
Hea stays liquid down to the absolute zero temperature,
a situation often conjectured in connection with liquid
Re4. Since the latter fluid does not solidify down to the
lowest temperature at which it has been observed, unless
subjected to external pressure, it is to be expected that
this conjecture is better justified in the case of liquid
Hea which is considerably more volatile and less densea
than liquid He4. The present investigation refers essen
tially to the temperature of absolute zero, where the
density of the liquid was taken to be somewhat larger
than the highest density observed at Los Alamos,
namely p(OOK) was assumed to be 0.08 g/cc. This
density determines the maximum kinetic energy or
linear momentum at the top of the Fermi distributions
in both the ferromagnetic (f) and antiferromagnetic (a)
configurations. One has, Po denoting the liquid density
at the temperature of absolute zero, for the maximum
momentum p, and kinetic energy Ej,
Pr=h(3po/47rM)1,
Er= p?/2M = (h2/2M) (3po/47rM)J, (1)
M being the mass of a Rea atom (5X 10-24 g), and
Pa = P ,/2\ Ea = E,/2J• (2)
a See Sydoriak, Grilly, and Hammel, Phys. Rev. 75, 303 (1949)
and Grilly, Hammel, and Sydoriak, ibid. 75, 1103 (1949).
538
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128.138.73.68 On: Mon, 22 Dec 2014 03:20:42MAGNETIC PROPERTIES OF He3 539
The ratios of 21 and 21 between the moments p,/p"
and kinetic energies E,/Ea in the (J) and (a) configura
tions result from the inflated momentum or energy
spheres in the former as compared with the latter.
Indeed in the (a) configuration two atoms of opposite
spin are accommodated per free particle state, while
in the (J) configuration there being only one atom per
state, the volume of the momentum sphere has in
creased by a factor two with respect to that in the (a)
configuration.
The average kinetic energy per atom is then
(3)
as in a completely degenerate collection of antisym
metric particles, since in our approximation the kinetic
energy of an atom in liquid Hea is that of a free particle
within an ideal Fermi-Dirac fluid.
The total wave function of the system in the ferro
magnetic configuration, normalized to unity in the
volume V, is, in the present approximation, the de
terminant:
Nt
1/IAIr,12,"IN)=(NlV N)-!L: (-)PP
P=l
where the summation extends over all the N! permuta
tions of the permutation operator P acting on the radius
vectors Ij of the Hea atoms of wave vector kj(=pi/h)
in the system formed by N atoms. In the antiferro
magnetic configuration the wave function is a product
of two determinants associated with the (N /2) atoms
in the two spin states respectively. Let the I'S denote
the radius vectors of the atoms in one spin state and
the R's those of the second group of eN /2) atoms in
the other spin state, the approximate wave function is
(N /2) I (N /2) !
X L: L: (-)P+PpPexp{i[(k11 1+'" p=l P=l
Here, the permutation operators p and P operate on
the I'S and R's respectively, the K's being the wave
vectors associated with the atoms R in one of the spin
states. It is, of course, fully realized that the extreme
ideal gas wave functions may be worse approximations
here than in the case of the electron gas in the theory of
metals. While in the latter case a relatively good justifi
cation can be given in some types of metals for de
scribing the less tightly bound electrons as forming an
antisymmetric assembly of practically free particles,
no such justification is attempted in the present prob
lem of liquid Hea. We should like to invoke here the
experimental results of the Argonne workers4 on the
flow properties of liquid Hea. According to these results
4 Osborne, Weinstock, and Abraham, Phys. Rev. 75,988 (1949). the viscosity coefficient of this liquid is of the same
order of magnitude as that of liquid He4I, i.e., some
10-6 c.g.s. unit.6 This in turn is of the same order of
magnitude as, or somewhat larger than, the viscosity
coefficient of He4 vapor at liquid He temperatures.
This argument should merely be considered as an in
direct indication of the gas-like behavior of liquid Hea
where the atoms have a small but finite binding energy. 6
One finds with the wave functions (5) or (6), the fol
lowing potential energy of two Hea atoms:
Ep=~f<I>(r)dVldV2-~8(81- 82)f<I>(r)
V2 V2
X exp[ i(k1-k2) I JdVldv2 = Ep, c+ E.,( I k1-k21 ). (6)
Here, Ep, c is simply the classical potential energy of
two stationary Hea atoms averaged over their positions
in the volume V of the fluid, with <I>(r) denoting the
mutual potential energy of two stationary atoms at a
distance r. This was taken to be given by either one of
the following two expressions due respectively to Slater
and Kirkwood7 and Margenau:8
<I> S-K(r) = A e-ar -Br6,
<I>M(r) = Ae-ar-Blr6-B2r8. (7)
(8)
In these formulas, the distance r = I Il-I21 of the two
atoms is expressed in angstrom units, the constants
having the following numerical values:
A
77 B
0.149 Bl
0.139
while a is equal to 4.60A-l. B2
0.37X 10-11 erg,
Ez is the quantum mechanical exchange energy of
two Hea atoms whose distance in wave vector space is
I k1-k21 while the 8-function simply indicates that in
the (a) configuration only atoms of parallel spin direc
tion give rise to exchange energy (81 = 82). The co
ordinate integrations in (7) extend over the volume V.
With the origin of the coordinate system placed at
one of the atoms, one obtains
(10)
The integrals (9) and (10) diverge on account of the
peculiar behavior of the interatomic potential energies
(8), so that the integrations have to be cut off at some
5 See the monograph of W. H. Keesom, Helium (Elsevier Pub
lishing Company, Inc., Amsterdam, 1942).
G It is to be noted that the equality in order of magnitude of
the liquid and vapor viscosity coefficients also holds for hydrogen
but the difference in viscosities becomes increasingly large with
heavy elements.
7 J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931).
8 H. Margenau, Phys. Rev, 56, 1000 (1939).
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finite small distance a. Physically, only those lower
limits are acceptable which lead to a large enough nega
tive average potential energy, since the atoms are
evidently bound in the liquid. One obtains thus, at once,
(11)
It is found that the preceding averages become nega
tive at values of a, such that a2::2.14A in the S-K case
and a2:: 2.02A in the M case.
The exchange energy of a pair of Re3 atoms becomes,
after an elementary calculation, with the S -K poten
tial energy,
471" 1 e-aa [( (32-1) Ex(k)R= ---A-- aa+-- sinka
V k3 1 +{32 {32+ 1
+(ka+~)coSka], (13)
1+{32
due to the repulsive part (R) of the potential energy
or the term Ae-aa. The attractive part, (A), leads to:
where
fx sinx
(3=a/k; Si(x)= -dx.
o x
With the Margenau potential energy, the repulsive
exchange energy is of course, the same as with the
Slater-Kirkwood potential energy, while the attractive
exchange term has a form similar to (14). The oscillat
ing character of these exchange energies is of course
already evident in the original formula (10).
The total energy of the liquid, in the ferromagnetic
configuration becomes:
ET.,=!NE,+!N(N -1)Ep.c(a)
+n:kILk2Ex( / kl-k2/)' (15)
The first term on the right-hand side is the free particle
kinetic energy in a system of N atoms, with Ef being
given by Eq. (1); the second term is the total classical
potential energy of the system, there being N(N -1)/2
pairs of energy Ep.c(a) per pair of atoms. Finally the
last term represents the total exchange energy. In the
latter, one has to sum over all possible k-values of both
atoms up to k/. Similarly, in the antiferromagnetic configuration one
obtains
ET.a=!NEa+!N(N -1)Ep.c(a)
+ 2(!LkILk2Ex( / kl-k2/». (16)
In this expression, the maximum kinetic energy Ea is
given by Eq. (2); the classical potential energy term is
the same as in the (f) configuration, since this quantity
does not depend on the spin configuration. In the last
exchange energy term, the summations over the k's
extend only up to ka( = 2-lk,), the radius of the con
tracted wave vector sphere associated with (N /2)
parallel spins, in either direction. The total exchange
energy is of course the sum of the exchange energies in
either spin direction, and this is indicated by the factor
2 in front of the single spin direction exchange term.
We now tum to the evaluation of these exchange
energies.
3. THE FREE PARTICLE EXCHANGE ENERGY
IN LIQUID He3
The summations appearing in the total exchange
energy expressions in Eqs. (15) and (16) may be re
placed by integrations over the distribution of free
particle levels, with a level density,
dn(k,O) V --k2dk
271" sinOdO (271")3 '
associated with the solid angle 271" sin8d8 and wave
vector band (k,k+dk), so that:
where the origin of the coordinate system in k-space
was chosen to coincide with one of the particles, say
atom 1, so that an integration over the k-space of this
atom leads just to N, while the polar axis coinciding
with the direction of (kl-k2), the integrand in the
second integral becomes independent of the polar angle
8, leading to a factor 47r of the total solid angle. It is
now seen that by substituting ExCk), as given by Eqs.
(13) and (14), one obtains a series of definite integrals
all expressible in terms of impractical infinite series at
worst. Instead of following this procedure it appeared
more useful to express the total exchange energy in
another form. With the definition (10) one finds
! Lk1Lk2Ex( / kl-k2/ )
with dkl denoting the volume element in k1-space.
Choosing a polar coordinate system whose axis coin
cides with k2' applying the Gegenbauer addition the-
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128.138.73.68 On: Mon, 22 Dec 2014 03:20:42MAGNETIC PROPERTIES OF He, 541
orem9 to sinlk1-k2Ir/(lk 1-k2Ir), integrating over
kb replacing the summation over k2 by an integral
with the level density (V(21r)3)dk 2, and integrating
again one obtains finally
(17)
where JJ(z) is the Bessel function of order !, k, and ka
denote the maximum wave vectors in the two con
figurations, respectively, e.g.,
k,=p,/h, ka=2-ikf•
Replacing, in Eq. (17), <I>(r) by the Coulomb energy
t?/r acting between two electrons, one obtains at once
the total exchange energies in the two configurations of
the free electron gas mode1.10
In the present case of the He-He interaction the
total exchange energy has no simple analytical ex
pression. It is, however, easy to see that the exchange
energy, for a chosen cut-off length a, is in general
larger, the smaller the radius of the momentum or wave
vector sphere of the system, a result similar to the
simpler case of electrostatic coupling, where this is
always true. The complication which exists in the case
of helium results from a superposition of effects due to
the attractive and repulsive components of the poten
tial energy <I>(r).
Since a simple analytical expression of the exchange
energy is lacking here, a discussion of the limitations in
the validity of these calculations is not as straight
forward as in the electron gas problem. A further com
plication arises here from the somewhat arbitrary
choice of the cut-off length a. It appears, however, that
the limitations on the validity of the exchange energy
in the He3 case should be somewhat different from those
of the electron problem. In the latter because the aver
age classical potential energy vanishes or becomes con
stant, the exchange energy can only be compared with
.a
I T,f
-2.oh2.o~J.21--J2\'2 ----,2,l,.3-.J..4.--~2ii;;.5=~42.r:..n-n______3.o
alAI
FIG_ 1. Average potential, exchange, and total ene,rgies per
atom in liquid Hea. (Slater-Kirkwood interactlOn_) ----
9 G_ N. Watson, Theory of Bessel Fttnclions (Cambridge Uni
versity Press, London, 1922), p_ 363 or 366_
10 See reference 1 or F. Seitz, Modern Theory of Solids (McGraw
Hill Book Company, Inc., New York, 1940), p. 341. the kinetic energy or first approximation energy of the
electrons. In our case, where the plane de Broglie wave
approximation of the one atom wave function is postu
lated at the start also, the exchange energy may be
compared to both the average potential and kinetic
energies. Since the atoms are bound in the liquid, neces
sarily the potential energy is larger than the kinetic
energy per atom. The comparison of the approximate
exchange energy to the potential energy rather than to
the smaller kinetic energy would then lead in the present
calculations to a wider validity range in the wave vec
tors or particle densities than in the electron case.
The exchange energies have been computed here by
evaluating numerically the integrals in (17).* In all the
cases studied here with the smaller Slater-Kirkwood
potential energy, the exchange energy turned out to
be a small fraction of both the kinetic and potential
energies for the same physically significant cut-off
lengths, e.g., a~ 2.2A. As expected, the exchange
energy is negative for the small cut-off values, since
there, the repulsive portion of the potential energy is
more important than the attractive part. For cut-off
lengths a which are physically significant the exchange
energy starts by being negative, it then increases with
increasing a, at constant k" becomes positive and reaches
a maximum corresponding to that value of a, beyond
which the potential energy <I>(r) is negative, e_g_, for a
equal to the root of <I>(r). The positive exchange energy
is associated with the predominance of the attractive
part of the potential energy.
At the temperature of absolute zero, with the as
sumed liquid He3 density, the exchange energies have
been investigated with both types of potential energies
and the results are given in Figs. 1 and 2. ** It is seen
that with both types of potential energies, the exchange
energy alone favors the ferromagnetic configuration
since one has always Ex.,<Ex.a. However, in the whole
physically significant region of the cut-off lengths, as
a function of which the exchange and average potential
E-:~
o"
""Q
>
~
"' -I.
-I
-2
-2. m I----:;::'-
V
~
1\\ ~I
\ ~ ----"---
~
"" i'----
1 -:; /
Ex•o E(,
~Tf _J----t::= -----------V
7 ~
ET,a
-------I/Vlal V V
alAI
FIG. 2. Average potential, exchange and total energies per
atom in liquid Hea_ (Margenau interaction_) ----* We should like to thank here Mr. D. W. Sweeney for his co-
operation in this work_ _
** One finds that the kinetic energy is Ea = 4.05X 10-16 erg/
atom.
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128.138.73.68 On: Mon, 22 Dec 2014 03:20:42542 L. GOLDSTEIN AND M. GOLDSTEIN
TABLE 1. Paramagnetic, (xT/p), and total, (Xto';p), mass
susceptibilities of liquid Hes.
T (xTlp) -(xtotlp) OK 10-'1 e.g.s. 10-'1 e.g.s.
0* 0.508 5.80
0.245* 0.507 5.80
0.490* 0.504 5.81
1.2 0.478 5.83
1.5 0.464 5.85
2.0 0.431 5.88
2.5 0.404 5.91
2.8 0.391 5.92
3.1 0.384 5.93
3.34 Q.400 5.91
* These values correspond to the same liquid He. extrapolated density
of 0.08 glee.
energies have been investigated, the total energy ET, a
of the antiferromagnetic configuration is smaller than
that of the ferromagnetic configuration ET, f. Hence,
in the present, rather crude, model of liquid Hea with
a density value assumed at the temperature of absolute
zero, the liquid would not exhibit nuclear ferromag
netism.
It is not entirely without interest to notice that with
the larger Margenau potential energy there is, in a
small but physically acceptable cut-off length region,
some evidence for the somewhat increased stability
of the ferromagnetic configuration. Clearly, it is not
justifiable to attach any particular importance to this
result beyond noticing its existence. The main reason
for this lies in the fact that for these smaller cut-off
lengths the exchange energies are about equal to the
potential and kinetic energies. The validity of the ex
change energy at these cut-off lengths cannot, however,
be justified. This situation is similar to the one en
countered in the electron case, insofar as the electron
gas model leads also to ferromagnetism in the extrapola
tion of the approximation method to exchange energy
regions where the method ceases to be valid.
We should like to add here that the model could be
improved along lines suggested by Wigner's workll in
the electron case. This improvement would correspond
to the introduction of correlation between Hea atoms of
opposite spin in the antiferromagnetic configuration.
It is likely that one would thus be led to a further
strengthening of the conclusions reached above on the
larger stability of the vanishing total spin configuration
of the system.
4. ON THE NUCLEAR PARAMAGNETIC
SUSCEPTIBILITY OF LIQUID Hea
Since our crude model of liquid Hea leads to a stable
non-ferromagnetic configuration, the finite nuclear mag
netic moment of the Hea atoms should manifest itself
through a small paramagnetism of the liquid. The main
interest of this paramagnetism of this liquid lies in
that its experimental study might be of importance in
11 E. P. Wigner, Phys. Rev. 46,1002 (1934) and Trans. Faraday
Soc. 34, 678 (1938). showing, together with other thermodynamic proper
ties, the possible intervention of antisymmetric sta
tistics in the behavior of this fluid.
In the limit of the ideal antisymmetric fluid model,
the paramagnetic susceptibility is given by12
np.2 [-F'(a)]
XT=-
kt F(a) (18)
where n is the atomic concentration, the spin per atom
being h/2, p. is the nuclear magnetic moment of
Hea (p.= -1.07X 10-23 c.g.s.),1a F(a) is the statistical
function determined by the temperature and concentra
tion of the fluid according to
nharm
F(a)
(2s+ 1) (211'mkT)!
=j(To/T)!; F'(a)=dF/da,
with m denoting the mass of a Hea atom (S.008X 10-24
g), and To the degeneration temperature at the con
centration n,
To= (h2/2mk) (3n/ 41!'(2s+ 1))I.
At the limit of very low temperatures, e.g., T«To,
3 np.2
lim XT=--,
T«To 2 kTo
XT reduces to the temperature independent Pauli para
magnetic susceptibility. The resultant magnetic sus
ceptibility of liquid Hea is the sum of its diamagnetic
and paramagnetic susceptibilities. The latter is given
by the Langevin formula
Z
XD= -n(ro/6) L T;2.
i=I (19)
Here, ro denotes the classical electron radius (e2/mc2),
and T;2 is the mean square of the distance of the
i'th atomic electron to the nucleus, the sum being ex
tended over all the Z electrons of the atom. The re
sultant or total susceptibility, per unit mass, is then
Xtot= (XT-XD)/ P
XA,D(He4) p.2 [-F'(a)] ---+ (20) 3· mHea(kT) F(a)
since the atom-gram diamagnetic susceptibility of Hea
should be practically the same as that of He4.
Using the approximate Los Alamos liquid Hea densi
ties together with the experimental value of XA,D(He4)
(= -1.90X 10-6 c,g.s.),14 we have collected in Table I,
12 L. Brillouin, Les Stalistiques Quantiques (Les Presses Uni
versi taires, Paris, 1930).
1S H. A. Anderson and A. Novick, Phys. Rev. 73,919 (1948).
14 A. P. Wills and L. G. Hector, Phys. Rev. 23, 209 (1924);
2A, 418 (1924); G. G. Havens, Phys. Rev. 43, 992 (1933).
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128.138.73.68 On: Mon, 22 Dec 2014 03:20:42PERFLUOROCYCLOBUTANE SPECTRA 543
a series of liquid Hea susceptibilities per unit mass.IS
Since the diamagnetic mass susceptibility of Hea is
(-6.31X1O-7 c.g.s.), it is seen that its paramagnetic
mass susceptibility is but a small fraction, 6-8 percent
of the former.The paramagnetic mass susceptibility is
practically temperature independent in this ideal anti
symmetric fluid model, in contrast with the Curie
paramagnetism.
It is, of course, realized that the experimental in
vestigation of the very feeble magnetic properties of
liquid Hea would encounter serious difficulties. One of
these is the problem of the establishment of statistical
equilibrium in an external magnetic field necessary for
the nuclear paramagnetic susceptibility to manifest
itself in a physically acceptable way. It appears that an
extension of the experimental methods of Bloch, Bloem-
16 The ratios (-F'(a)/F(a» have been obtained with the help
of the tables of J. McDougall and E. C. Stoner, Phil. Trans.
Roy. Soc. London A237, 67 (1938).
THE JOURNAL OF CHEMICAL PHYSICS bergen, Purcell, and Poundl6 to the investigation of
liquid Hea could yield important information on the
statistical behavior of a rather elementary monatomic
system in the liquid phase. It is, indeed, clear that the
understanding of the laws governing the elementary
processes of energy exchange between the nuclear spin
system, and the "rest" of this liquid could be of great
help in furthering the knowledge of similar phenomena
in more complicated systems. We should like then to
conclude by saying that the experimental investigation
of the magnetic properties of liquid Hea could yield,
together with its other thermal properties, information
regarding the possible intervention of antisymmetrical
statistics in the general behavior of this fluid. Simul
taneously, the nuclear paramagnetism of this liquid
offers interesting possibilities in the study of approach
to statistical equilibrium, in an external magnetic field
and, at very low temperatures, of a unique monatomic
system.
16 F. Bloch, Phys. Rev. 70, 460 (1946); Bloembergen, Purcell,
and Pound, Phys. Rev. 73,679 (1948).
VOLUME 18. NUMBER 4 APRIL. 1950
Vibration Spectra and Normal Coordinate Treatment of Perfiuorocyc1obutane*t
HOWARD H. CLAASSEN
Department of Physics, University of Oklahoma, Norman, Oklahoma
(Received November 28, 1949)
The Raman spectra of gaseous and liquid cyclic C,Fg at room temperature have been obtained and polar
ization measurements made for the liquid state. The infra-red absorption spectrum of the gas between 2 and
23.1).1 obtained by Dr. D. C. Smith of the Naval Research Laborotory is also reported here. A normal coordi
nate treatment, based on the assumption that the molecular symmetry is D'h, has been made and applied
to assign the 23 fundamental vibration frequencies and to evaluate a set of force constants. The observed
spectra have been interpreted in detail.
INTRODUCTION
ALTHOUGH considerable interest has recently been
shown in the spectroscopic properties of fluoro
carbons, few papersl•2 have discussed potential functions
for fluorinated compounds other than methane deriva
tives. Values reported for C-F bond stretching force
constants have ranged from 9.15 to 3.80X lOS dynes/cm.
Perfluorocyc1obutane is an example of a relatively
complicated molecule which, because of its high sym
metry, is amenable to a vibrational analysis. If a
fairly simple potential function is assumed, force con
stants can be calculated for this molecule. The Raman
spectrum of liquid perfluorocyc1obutane has been
studied by Edgell,S who also reported four infra-red
* This work has been supported by the Office of Naval Research
under contract N7-onr-398, Task Order 1. t From a dissertation submitted to the Faculty of the Graduate
College of the University of Oklahoma in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
1 E. L. Pace, J. Chern. Phys. 16, 74 (1948).
• W. F. Edgell and W. E. Byrd, J. Chern. Phys. 17, 740 (1949).
3 W. F. Edgell, J. Am. Chern. Soc. 69, 660 (1947). absorption maxima and made an assignment of funda
mentals.
As a part of a larger project on the spectroscopic
properties of fluorocarbons and fluorinated hydrocar
bons the vibration spectra of perfluorocyclobutane have
been investigated. This paper reports the spectra, their
interpretation and a normal coordinate analysis.
EXPERIMENTAL
The sample of perfluorocyc1obutane was prepared
and purified in the Jackson Laboratory of E. I. du
Pont de Nemours and Company. No information was
available about its purity. Since it has been possible
to interpret satisfactorily all but a very few of the
faintest infra-red and Raman bands, the purity is
probably high.
The infra-red absorption from 2 to 23.1/-1 was studied
by Dr. D. C. Smith of the Naval Research Laboratory
by means of a prism spectrometer of high resolution.4
4 Nielsen, Crawford, and Smith, J. Opt. Soc. Am. 37, 296 (1947).
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128.138.73.68 On: Mon, 22 Dec 2014 03:20:42 |
1.1698913.pdf | Addendum: Directed Valence as a Property of Determinant Wave Functions
Howard K. Zimmerman , and Pierre Van Rysselberghe
Citation: The Journal of Chemical Physics 21, 381 (1953); doi: 10.1063/1.1698913
View online: https://doi.org/10.1063/1.1698913
View Table of Contents: http://aip.scitation.org/toc/jcp/21/2
Published by the American Institute of PhysicsLETTERS TO THE EDITOR 381
t 110
.!!'
~90
~ (3ii plane)
FIG. 2. N8
In order to have a standard for comparison of intensities of
extra spots at different temperatures, a small quantity of alu
minum powder was dusted over the crystal. The variation of
the intensities of aluminum lines over the small range of tem
perature studied was neglected, however. The intensities were
compared by means of a standard wedge, prepared according to
the method of Robinson. The ratio of the (111) aluminum line
to the maximUl}l intensities of the extra spots were found by
matching each of them against the standard wedge. It appears
from the curve that intensities of extra spots increase very
rapidly with the temperatures, and the variation of structure
factor amplitude of two planes are different (see Figs. 1 and 2).
Erratum: Theory of Absorption Spectra
of Carotenoids
[J. Chern. Phys. 20, 1661 (1952»)
GENTARO ARAKI
Faculty of Engineering. Kyoto University, Yosida, Kyoto, Japan
IN the previous letterl we attempted an explanation of the
relation between absorption spectra and molecular lengths of
carotenoids, making use of Tomonaga's method for electron gas
with arbitrary couplings. We took into account the electron-spin
for enumerating the number of electrons (occupying levels up to
the Fermi maximum) only. If we take into account the spin
degrees for wave functions we have, instead of Eq. (2) in the
previous letter,l the following equation:
8= (L/7r)'(4/ A)(N _2)-1.
The empirical value of A thus becomes twice as large, A =709.5.
The rest needs no change.
1 G. Araki and T. Murai, J. Chern. Phys. 20, 1661 (1952).
Addendum: Directed Valence as a Property of
Determinant Wave Functions
[J. Chern. Phys. 17, 598 (1949»)
HOWARD K. ZIMMERMAN, JR., Department of Chemistry, Agricultural and
Mechanical College of Texas, College Station, Texas
AND
PIERRE VAN RYSSELBERGHE, Department of Chemistry.
University of Oregon, Eugene, Oregon
IN his valuable recent review on "Quantum Theory, Theory of
Molecular Structure and Valence" Professor Coulson! states
that the idea of deriving directed valence properties from atomic
wave functions (as written, for instance, under the form of
determinants) "seems to have originated with Artmann,2 but it has
been "rediscovered" by Zimmerman and Van Rysselberghe3• " ."
We wish to put on record that the tetrahedral valences of carbon
were derived in this manner by one of us (P.V.R.) in 1933, that the
calculations and results were communicated orally and by letter to
several persons interested in the field, and that a communication to the editor of the Journal oj the American Chemical Society
giving a condensed presentation of this fundamental point was not
published for reasons which, coupled with other preoccupations,
resulted in our abandoning further work of this type. The problem
was resumed in 1946 and led to our submitting a papers to The
Journal of Chemical Physics in July, 1948, publication following in
July, 1949. In this paper we present the derivation of the tetra
hedral valences of carbon in a manner identical with that of the
intended communication of 1933, and we give reasons for offering
our treatment of the whole problem of directed valences as an
alternative to that of Artmann whose work had come to our
attention through the abstract' published in September, 1947.
1 C. A. Coulson. Ann. Rev. Phys. Chern. 3. 1 (1952). see p. 8.
• K. Artmann. Z. Naturforsch. 1. 426 (1946). 'H. K. Zimmerman. Jr .• and P. Van Rysselberghe. J. Chern. Phys. 17.
598 (1949).
• Chern. Abstracts 41. 5785 (1947).
The Dissociation Energy of Fluorine*
PAUL W. GILLES AND JOHN L. MARGRAVEt
Department of Chemistry. University of Kansas. Lawrence. Kansas
(Received December 16. 1952)
RECENT spectroscopic datal on CIF imply a value for the
dissociation energy of fluorine in the range 3~ kcal/mol.
Such a low dissociation energy would mean that considerable
dissociation of diatomic fluorine into atoms must occur at rela
tively low temperatures. Doescher2 and Wise' have reported
experimental results that indicate a value for Do(F2) between 36
and 39 kcal/mol.
The experiments reported here were carried out in 1950 on a
sample of fluorine obtained from the Pennsylvania Salt Manu
facturing Company, The pressure exerted by this sample of
fluorine, when contained in a closed system of copper which had
been previously treated with fluorine, was measured as a function
of temperature over the range 300-860oK with a Bourdon type
Dura gauge in two runs on different days. Between the two runs
a slight leak into the system occurred so that about six percent
of the gas was air in the second run.
When corrections are made (1) for the presence of this air in
the second run on the basis that it did not react and (2) for the
cooler zones of the system, the pressure calculated on the basis
of no dissociation of an ideal gas agreed at all temperatures
below 8000K with the experimentally observed pressure with
standard deviations of ±0.02 inches of Hg for six points on the
first day and of ±0.03 for six points on the second day.
Three measurements at temperatures above 8000K showed
differences between observed and calculated pressures consider
ably greater than any found for the twelve lower temperature
measurements. If it is assumed that these differences are caused
by the partial dissociation of F 2 into atoms, one may calculate at
each temperature the degree of dissociation, the dissociation
equilibrium constant, and, by using the available data for the
free energy functions of F and F 2,' the dissociation energy of F 2.
Because of the corrections necessary for different temperatures in
different parts of the system, the degree of dissociation IX and
the equilibrium constant K are not simply related to the pressure
difference.
The data and results are shown in Table I, in which the calcu
lated and observed pressures in the second run have been cor
rected for the air leak. The uncertainties listed are obtained by
assigning to each pressure an uncertainty of ±0.03 inch of Hg.
TABLE 1. Degree of dissociation. dissociation equilibrium constant.
and dissociation energy of fluorine.
Pobs Peale K D.(F,)
Run TOK (inches of Hg) a (10-' atmos) (kcal)
1 815 1.48 1.40 0.07±0.04 0.975±1.10 33.4±2.0
2 810 1.52 1.38 0.11±0.05 2.49 ±1.80 31.6±1.6
2 860 1.72 1.46 0.21±0.05 10.6 ±5.4 31.2 ±0.8 382 LETTERS TO THE EDITOR
The weighted average of the values in the last column gives
for Do(F2) a value of 31.5±0.9 kcal/mol. Using the free energy
functions4 one calculates for the same quantity 36.5±1.0 from
the data of Doescher 2 obtained in similar experiments at higher
temperatures in a nickel container, and 39±1 from the graph of
Wise.s It appears that the best value is 36±3 kcal/mol.
The electron affinity of fluorine may be related to the dissocia
tion energy through a Born-Haber cycle. Studies by lonov and
Dukelskii,5 in which positive and negative ion currents were
observed during evaporation of alkali metal halides from a
tungsten filament, allow calculation of the electron affinities of
the halogen atoms if the proper work function for the tungsten
surface is known. These experimenters found values for the
electron affinities of chlorine, bromine, and iodine in good agree
ment with those given by other workers, when potassium halides
were used and the work function for a clean tungsten surface was
assumed. A similar treatment of their data on KF indicates a
value of 83±3 kcal/mol for the electron affinity of F.
Metlay and Kimba1l6 have studied the relative currents of
negative ions and electrons emitted from a hot tungsten filament
in the presence of fluorine. Although originally misinterpreted,
the experimental data yield an average value for the electron
affinity of 82±3,1·8 in good agreement with the result of lonov
and Dukelskii.
If one uses the value 82±3 for the electron affinity of F in a
Born-Haber cycle along with thermochemical data from the
National Bureau of Standards Table "Selected Values of Chemical
Thermodynamic Properties" and crystal energies of the alkali
metal fluorides computed after Pauling,9 he finds Do(F2)=31±4
kcal/mol, in agreement with the experimental value.
The authors are pleased to acknowledge the support of the
U. S. Atomic Energy Commission in this work.
* Abstracted in part from a thesis presented by John L. Margrave in
partial satisfaction of the requirements for the degree of Doctor of
Philosophy at the University of Kansas. December 28. 1950. t Present address: Department of Chemistry. University of Wisconsin.
Madison, Wisconsin.
1 A. L. Wahrhaftig. J. Chern. Phys. 10.248 (1942); H. Schmitz and H. J.
Schumacher. Z. Naturforsch. 2a. 359 (1947).
2 R. N. Doescher. J. Chern. Phys. 19. 1070 (1951); 20. 330 (1952).
• H. Wise. J. Chern. Phys. 20. 927 (1952).
• R. M. Potocki and C. W. Beckett. National Bureau of Standards
Report 1294. December 1. 1951; L. Haar and C. W. Beckett. ibid. Report
1435. February 1. 1952. 'N. lonov. Compt. rend. acado sci. U.R.S.S. 28. 512 (1940); N. ronov
and V. Dukelskii. FIZ. Zhur. 10. 1248 (1940). 'M. Metlay and C. Kimball. J. Chern. Phys. 16. 779 (1948).
7 J. L. Margrave. thesis. University of Kansas (1950).
• R. B. Bernstein and M. Metlay. J. Chern. Phys. 19. 1612 (1951).
• L. Pauling. The Nature of the Chemical Bond (Cornell University Press.
Ithaca. New York. 1940). p. 340.
Partition Functions for Relating Entropy to
Disorder in the Melting of Pure Metals
JOHN F. LEE
Mechanical Engineering Department. North Carolina State College.
Raleigh. North Carolina
(Received December 11. 1952)
THE partition function due to Lennard-Jones and Devonshirel
has been modified to avoid the controversial "communal
entropy" following the suggestions of Ono.2 Application has been
made to the body-centered cubic lattice characteristic of some
liquid metal coolants such as sodium.
The mean energy of an atom at a distance r from the center of
its cell due to the nearest neighboring atom at a distance a from
the same center is
u(r) = ~ J" u{(r2+a 2-2ar cosO)!} sinOdO. (1)
The energy of interaction of two spherical atoms separated by the
distance r is
u(r) =4eo{ (ro/r) 12_ (ro/r)·}. (2)
The energy u(O) may be obtained by substituting Eq. (2) in
Eq. (1) with the limit r ...... O. Then the mean energy of the central
atom is as follows, the number of nearest neighbors Ii being 8 for a body-centered cubic lattice:
_ {(VO)4 (r)2 (VO)2 (r)2} zu(r)-u(O)=zeo -; I ~ -2 -; m ~ . (3)
Letting y= (r/a) 2 for convenience, the functions ley) and m(y) are
defined.
ley) = (1+12y+25.2 y2+12y3+y<)(1-y)-IO-1,
m(y)= (Hy)(1-y)-4-1.
The "free volume" is defined
v(O) = 27ra3g,
and
g= J.Y y!exp{-:~[(;rl(~r-2(;rmGr]}dy, (4)
where the upper limit of y= (3/47rV3')! for a body-centered cubic
lattice; or Eq. (4) may be expressed
v(O) = J exp{ -[zu(r) -u(O)J/kT}dr. (5)
The integral extends over the cell.
The partition function for a single atom is
(27rmkT)! f= ~ v(O){ -zu(0)/2kT}. (6)
The partition function for the whole assembly is·found to be too
low by a factor of eN in the limit of the lower densities, this
factor being in essence the "communal entropy." The partition
function for the whole assembly is therefore
F=fNeN. (7)
The "communal entropy" is avoided by regarding the existence
of vacant sites which the preceding development ignores. If
xi=Ni/N represents the ratio of the vacant sites to the total
number of sites then ZXi is the number of vacant sites and z(1-x.)
is the number of neighboring occupied sites. The energy at the
center of the cell is z(l-xi)u(O), and the energy of the assembly is
u=~(N -~ Xi)U(O)+~ (l-xi)ui. 2 i i
The partition function for the assembly is
(27rmkT)3NI2 J J F= ~ ~ dr,,,· drN exp(u/kT).
When we substitute from Eq. (8),
_ (27rmkT)3NI2 [-zeN -~i Xi)U(O)]
F -h2 ~ exp 2kT
X J exp{ -(l-xi)u;/kT}dri,
the generalized free volume being
V(Xi) = J exp{ -(l-x.;)ui/kT}d ri. (8)
(9)
(10)
(11)
When Xi=O, the neighboring sites are all occupied, and Eqs. (5)
and (11) are identical. If Xi= 1, the neighboring sites are all
vacant. It is clear that. some simple relationship must be found
between vex) and x. Assuming lnv(x) to be linear in x, it can be
shown that
lnv(X) =x InvI*+(l-x) lnvo*. (12)
Following the suggestion of Ono modified for a body-centered
cubic lattice,
'00*= '0(0) = 2tra2g= 27rV3'r03(v/vo)g,
'0,*='0(1) =a3/V3' =r03(V/VO).
The solution now may be obtained using the methods of Fowler
and Guggenheim."
1 Lennard-Jones and Devonshire. Proc. Roy. Soc. (London) A163. 53
(1937); A165. 1 (1938); AIM. 317 (1939); A170. 464 (1939).
'Ono. Memoirs of the Faculty of Engineering. Kyushu University. Japan
10. 190 (1947). • R. H. Fowler and E. H. Guggenheim. Statistical Thermodynamics
(Cambridge University Press. Cambridge. 1949), pp. 576-581. |
1.1698912.pdf | Erratum: Theory of Absorption Spectra of Carotenoids
Gentaro Araki
Citation: The Journal of Chemical Physics 21, 381 (1953); doi: 10.1063/1.1698912
View online: https://doi.org/10.1063/1.1698912
View Table of Contents: http://aip.scitation.org/toc/jcp/21/2
Published by the American Institute of PhysicsLETTERS TO THE EDITOR 381
t 110
.!!'
~90
~ (3ii plane)
FIG. 2. N8
In order to have a standard for comparison of intensities of
extra spots at different temperatures, a small quantity of alu
minum powder was dusted over the crystal. The variation of
the intensities of aluminum lines over the small range of tem
perature studied was neglected, however. The intensities were
compared by means of a standard wedge, prepared according to
the method of Robinson. The ratio of the (111) aluminum line
to the maximUl}l intensities of the extra spots were found by
matching each of them against the standard wedge. It appears
from the curve that intensities of extra spots increase very
rapidly with the temperatures, and the variation of structure
factor amplitude of two planes are different (see Figs. 1 and 2).
Erratum: Theory of Absorption Spectra
of Carotenoids
[J. Chern. Phys. 20, 1661 (1952»)
GENTARO ARAKI
Faculty of Engineering. Kyoto University, Yosida, Kyoto, Japan
IN the previous letterl we attempted an explanation of the
relation between absorption spectra and molecular lengths of
carotenoids, making use of Tomonaga's method for electron gas
with arbitrary couplings. We took into account the electron-spin
for enumerating the number of electrons (occupying levels up to
the Fermi maximum) only. If we take into account the spin
degrees for wave functions we have, instead of Eq. (2) in the
previous letter,l the following equation:
8= (L/7r)'(4/ A)(N _2)-1.
The empirical value of A thus becomes twice as large, A =709.5.
The rest needs no change.
1 G. Araki and T. Murai, J. Chern. Phys. 20, 1661 (1952).
Addendum: Directed Valence as a Property of
Determinant Wave Functions
[J. Chern. Phys. 17, 598 (1949»)
HOWARD K. ZIMMERMAN, JR., Department of Chemistry, Agricultural and
Mechanical College of Texas, College Station, Texas
AND
PIERRE VAN RYSSELBERGHE, Department of Chemistry.
University of Oregon, Eugene, Oregon
IN his valuable recent review on "Quantum Theory, Theory of
Molecular Structure and Valence" Professor Coulson! states
that the idea of deriving directed valence properties from atomic
wave functions (as written, for instance, under the form of
determinants) "seems to have originated with Artmann,2 but it has
been "rediscovered" by Zimmerman and Van Rysselberghe3• " ."
We wish to put on record that the tetrahedral valences of carbon
were derived in this manner by one of us (P.V.R.) in 1933, that the
calculations and results were communicated orally and by letter to
several persons interested in the field, and that a communication to the editor of the Journal oj the American Chemical Society
giving a condensed presentation of this fundamental point was not
published for reasons which, coupled with other preoccupations,
resulted in our abandoning further work of this type. The problem
was resumed in 1946 and led to our submitting a papers to The
Journal of Chemical Physics in July, 1948, publication following in
July, 1949. In this paper we present the derivation of the tetra
hedral valences of carbon in a manner identical with that of the
intended communication of 1933, and we give reasons for offering
our treatment of the whole problem of directed valences as an
alternative to that of Artmann whose work had come to our
attention through the abstract' published in September, 1947.
1 C. A. Coulson. Ann. Rev. Phys. Chern. 3. 1 (1952). see p. 8.
• K. Artmann. Z. Naturforsch. 1. 426 (1946). 'H. K. Zimmerman. Jr .• and P. Van Rysselberghe. J. Chern. Phys. 17.
598 (1949).
• Chern. Abstracts 41. 5785 (1947).
The Dissociation Energy of Fluorine*
PAUL W. GILLES AND JOHN L. MARGRAVEt
Department of Chemistry. University of Kansas. Lawrence. Kansas
(Received December 16. 1952)
RECENT spectroscopic datal on CIF imply a value for the
dissociation energy of fluorine in the range 3~ kcal/mol.
Such a low dissociation energy would mean that considerable
dissociation of diatomic fluorine into atoms must occur at rela
tively low temperatures. Doescher2 and Wise' have reported
experimental results that indicate a value for Do(F2) between 36
and 39 kcal/mol.
The experiments reported here were carried out in 1950 on a
sample of fluorine obtained from the Pennsylvania Salt Manu
facturing Company, The pressure exerted by this sample of
fluorine, when contained in a closed system of copper which had
been previously treated with fluorine, was measured as a function
of temperature over the range 300-860oK with a Bourdon type
Dura gauge in two runs on different days. Between the two runs
a slight leak into the system occurred so that about six percent
of the gas was air in the second run.
When corrections are made (1) for the presence of this air in
the second run on the basis that it did not react and (2) for the
cooler zones of the system, the pressure calculated on the basis
of no dissociation of an ideal gas agreed at all temperatures
below 8000K with the experimentally observed pressure with
standard deviations of ±0.02 inches of Hg for six points on the
first day and of ±0.03 for six points on the second day.
Three measurements at temperatures above 8000K showed
differences between observed and calculated pressures consider
ably greater than any found for the twelve lower temperature
measurements. If it is assumed that these differences are caused
by the partial dissociation of F 2 into atoms, one may calculate at
each temperature the degree of dissociation, the dissociation
equilibrium constant, and, by using the available data for the
free energy functions of F and F 2,' the dissociation energy of F 2.
Because of the corrections necessary for different temperatures in
different parts of the system, the degree of dissociation IX and
the equilibrium constant K are not simply related to the pressure
difference.
The data and results are shown in Table I, in which the calcu
lated and observed pressures in the second run have been cor
rected for the air leak. The uncertainties listed are obtained by
assigning to each pressure an uncertainty of ±0.03 inch of Hg.
TABLE 1. Degree of dissociation. dissociation equilibrium constant.
and dissociation energy of fluorine.
Pobs Peale K D.(F,)
Run TOK (inches of Hg) a (10-' atmos) (kcal)
1 815 1.48 1.40 0.07±0.04 0.975±1.10 33.4±2.0
2 810 1.52 1.38 0.11±0.05 2.49 ±1.80 31.6±1.6
2 860 1.72 1.46 0.21±0.05 10.6 ±5.4 31.2 ±0.8 382 LETTERS TO THE EDITOR
The weighted average of the values in the last column gives
for Do(F2) a value of 31.5±0.9 kcal/mol. Using the free energy
functions4 one calculates for the same quantity 36.5±1.0 from
the data of Doescher 2 obtained in similar experiments at higher
temperatures in a nickel container, and 39±1 from the graph of
Wise.s It appears that the best value is 36±3 kcal/mol.
The electron affinity of fluorine may be related to the dissocia
tion energy through a Born-Haber cycle. Studies by lonov and
Dukelskii,5 in which positive and negative ion currents were
observed during evaporation of alkali metal halides from a
tungsten filament, allow calculation of the electron affinities of
the halogen atoms if the proper work function for the tungsten
surface is known. These experimenters found values for the
electron affinities of chlorine, bromine, and iodine in good agree
ment with those given by other workers, when potassium halides
were used and the work function for a clean tungsten surface was
assumed. A similar treatment of their data on KF indicates a
value of 83±3 kcal/mol for the electron affinity of F.
Metlay and Kimba1l6 have studied the relative currents of
negative ions and electrons emitted from a hot tungsten filament
in the presence of fluorine. Although originally misinterpreted,
the experimental data yield an average value for the electron
affinity of 82±3,1·8 in good agreement with the result of lonov
and Dukelskii.
If one uses the value 82±3 for the electron affinity of F in a
Born-Haber cycle along with thermochemical data from the
National Bureau of Standards Table "Selected Values of Chemical
Thermodynamic Properties" and crystal energies of the alkali
metal fluorides computed after Pauling,9 he finds Do(F2)=31±4
kcal/mol, in agreement with the experimental value.
The authors are pleased to acknowledge the support of the
U. S. Atomic Energy Commission in this work.
* Abstracted in part from a thesis presented by John L. Margrave in
partial satisfaction of the requirements for the degree of Doctor of
Philosophy at the University of Kansas. December 28. 1950. t Present address: Department of Chemistry. University of Wisconsin.
Madison, Wisconsin.
1 A. L. Wahrhaftig. J. Chern. Phys. 10.248 (1942); H. Schmitz and H. J.
Schumacher. Z. Naturforsch. 2a. 359 (1947).
2 R. N. Doescher. J. Chern. Phys. 19. 1070 (1951); 20. 330 (1952).
• H. Wise. J. Chern. Phys. 20. 927 (1952).
• R. M. Potocki and C. W. Beckett. National Bureau of Standards
Report 1294. December 1. 1951; L. Haar and C. W. Beckett. ibid. Report
1435. February 1. 1952. 'N. lonov. Compt. rend. acado sci. U.R.S.S. 28. 512 (1940); N. ronov
and V. Dukelskii. FIZ. Zhur. 10. 1248 (1940). 'M. Metlay and C. Kimball. J. Chern. Phys. 16. 779 (1948).
7 J. L. Margrave. thesis. University of Kansas (1950).
• R. B. Bernstein and M. Metlay. J. Chern. Phys. 19. 1612 (1951).
• L. Pauling. The Nature of the Chemical Bond (Cornell University Press.
Ithaca. New York. 1940). p. 340.
Partition Functions for Relating Entropy to
Disorder in the Melting of Pure Metals
JOHN F. LEE
Mechanical Engineering Department. North Carolina State College.
Raleigh. North Carolina
(Received December 11. 1952)
THE partition function due to Lennard-Jones and Devonshirel
has been modified to avoid the controversial "communal
entropy" following the suggestions of Ono.2 Application has been
made to the body-centered cubic lattice characteristic of some
liquid metal coolants such as sodium.
The mean energy of an atom at a distance r from the center of
its cell due to the nearest neighboring atom at a distance a from
the same center is
u(r) = ~ J" u{(r2+a 2-2ar cosO)!} sinOdO. (1)
The energy of interaction of two spherical atoms separated by the
distance r is
u(r) =4eo{ (ro/r) 12_ (ro/r)·}. (2)
The energy u(O) may be obtained by substituting Eq. (2) in
Eq. (1) with the limit r ...... O. Then the mean energy of the central
atom is as follows, the number of nearest neighbors Ii being 8 for a body-centered cubic lattice:
_ {(VO)4 (r)2 (VO)2 (r)2} zu(r)-u(O)=zeo -; I ~ -2 -; m ~ . (3)
Letting y= (r/a) 2 for convenience, the functions ley) and m(y) are
defined.
ley) = (1+12y+25.2 y2+12y3+y<)(1-y)-IO-1,
m(y)= (Hy)(1-y)-4-1.
The "free volume" is defined
v(O) = 27ra3g,
and
g= J.Y y!exp{-:~[(;rl(~r-2(;rmGr]}dy, (4)
where the upper limit of y= (3/47rV3')! for a body-centered cubic
lattice; or Eq. (4) may be expressed
v(O) = J exp{ -[zu(r) -u(O)J/kT}dr. (5)
The integral extends over the cell.
The partition function for a single atom is
(27rmkT)! f= ~ v(O){ -zu(0)/2kT}. (6)
The partition function for the whole assembly is·found to be too
low by a factor of eN in the limit of the lower densities, this
factor being in essence the "communal entropy." The partition
function for the whole assembly is therefore
F=fNeN. (7)
The "communal entropy" is avoided by regarding the existence
of vacant sites which the preceding development ignores. If
xi=Ni/N represents the ratio of the vacant sites to the total
number of sites then ZXi is the number of vacant sites and z(1-x.)
is the number of neighboring occupied sites. The energy at the
center of the cell is z(l-xi)u(O), and the energy of the assembly is
u=~(N -~ Xi)U(O)+~ (l-xi)ui. 2 i i
The partition function for the assembly is
(27rmkT)3NI2 J J F= ~ ~ dr,,,· drN exp(u/kT).
When we substitute from Eq. (8),
_ (27rmkT)3NI2 [-zeN -~i Xi)U(O)]
F -h2 ~ exp 2kT
X J exp{ -(l-xi)u;/kT}dri,
the generalized free volume being
V(Xi) = J exp{ -(l-x.;)ui/kT}d ri. (8)
(9)
(10)
(11)
When Xi=O, the neighboring sites are all occupied, and Eqs. (5)
and (11) are identical. If Xi= 1, the neighboring sites are all
vacant. It is clear that. some simple relationship must be found
between vex) and x. Assuming lnv(x) to be linear in x, it can be
shown that
lnv(X) =x InvI*+(l-x) lnvo*. (12)
Following the suggestion of Ono modified for a body-centered
cubic lattice,
'00*= '0(0) = 2tra2g= 27rV3'r03(v/vo)g,
'0,*='0(1) =a3/V3' =r03(V/VO).
The solution now may be obtained using the methods of Fowler
and Guggenheim."
1 Lennard-Jones and Devonshire. Proc. Roy. Soc. (London) A163. 53
(1937); A165. 1 (1938); AIM. 317 (1939); A170. 464 (1939).
'Ono. Memoirs of the Faculty of Engineering. Kyushu University. Japan
10. 190 (1947). • R. H. Fowler and E. H. Guggenheim. Statistical Thermodynamics
(Cambridge University Press. Cambridge. 1949), pp. 576-581. |
1.1700076.pdf | Heat Conduction in Simple Metals
M. L. Storm
Citation: Journal of Applied Physics 22, 940 (1951); doi: 10.1063/1.1700076
View online: http://dx.doi.org/10.1063/1.1700076
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/22/7?ver=pdfcov
Published by the AIP Publishing
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[This article is 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 03:34:52JOURNAL OF APPLIED PHYSICS VOLUME 22. NUMBER 7 JULY. 1951
Heat Conduction in Simple Metals*
M. L. STORMt
Research Division, New York University, New York, New York
(Received January 12, 1951)
The partial differential equation of heat conduction is a nonlinear equation when the temperature de
pendence of the thermal parameters (i.e., the thermal conductivity, K, and S, the product of the density and
the specific heat at constant pressure) is taken into account. It is shown that a mathematical condition for
the transformation to linear form of the one-dimensional, nonlinear, partial differential equation of heat
conduction is the constancy of [1/(KS)!J(d/dT)log(S/K)!. This discovery is the motivation for an investiga
tion of the relations between the thermal parameters of simple metals on the bases of the theory of solids
and available experimental data. It is found that KS is essentially constant, its variation with temperature
being much less than that of either K or S considered separately. It is also shown, as a result, that the
condition for the above-mentioned transformation is valid for simple metals. Applications of the trans
formed equation to the solution of problems in heat conduction are considered.
1. INTRODUCTION
THE differential equation of heat conduction in an
isotropic solid through which heat is flowing, but
in which no heat is being generated, is
V' (KVT)= S(aT/at). (1)
In the above, T is the temperature of the solid at time
t and position (x, y, z), K is the thermal conductivity,
and S= pCp, where p is the density and Cp the specific
heat at constant pressure. The two quantities K and S
are termed the "thermal parameters." In the cgs sys
tem the units of K are caVcm sec DC, and the units
of S are caVcm3 DC.
Equation (1) is nonlinear, since the thermal pa
rameters are functions of temperature. In the usual
mathematical treatment of heat conduction, it is as
sumed that the thermal parameters are constant and
solutions of the resulting linear equation have been
thoroughly investigated. However, in the case of metals,
this approximation holds for limited ranges of tempera
ture only, and discrepancies between the measured and
calculated temperatures are usually attributed to the
neglect of the variation of the thermal parameters. In
particular, the usual assumption of constant thermal
parameters is inadequate for heat conduction problems
in such devices as jet engines and rockets where large
temperature ranges and rapid rates of heating are
encountered.
Previous investigators who allowed the thermal
parameters to vary had more success in handling the
steady state heat conduction equation than in solving
the problem of nonsteady state heat conduction.t
In the former case, many problems can be handled by
* This paper is part of a dissertation presented for the degree
of Doctor of Philosophy at New York University. The work was
done with the support of the ONR, Department of the Navy,
and the Office of Air Research, Department of the Air Force,
under Contract N6ori-ll, Task Order 2, as part of Project Squid. t Now at the Naval Ordnance Laboratory, White Oak, Mary
land. t A detailed survey of past literature on this subject will be
found in a doctoral thesis by M. Storm, "Heat Conduction in
Simple Metals;" New York University (1950). the methods of Van Dusen1 and Ellion.2 In the latter
case, the least restrictive solutions were obtained in
problems where the variation of the thermal parameters
was small, thus allowing approximate solutions to be
obtained.3
It is too much to hope for an analytic solution of
Eq. (1), subject to arbitrary boundary conditions, when
the thermal parameters are represented as general
functions of temperature. Whereas Van Dusen1 suc
ceeded in transforming Eq. (1) to a form for which
solutions could be obtained for noncrystalline, poorly
conducting solids, in this investigation we shall limit
ourselves to heat conduction in simple metals4 and
consider the one-dimensional form of the nonlinear
equation
(a/ax) [K(aT /ax)] = scaT/at). (2)
Future considerations will show that our treatment of
Eq. (2) for simple metals is restricted to the tempera
ture range in which the thermal parameters can be
represented approximately by the following linear func
tions of temperature:
K=Ko(l-a[T-To]), and S=So(l+a[T-T o]). (3)
However, a straightforward substitution of (3) into (2)
does not lead to any simplification of the mathematical
problem of solving the nonlinear equation; this indi
cates that a different mode of attack must be adopted.
II. TRANSFORMATION OF THE ONE-DIMENSIONAL,
NONLINEAR HEAT CONDUCTION EQUATION
The equation to be solved is
(a/ax)[KcaT/ax)]=S(aT/at), (2)
1 M. S. Van Dusen, J. Research Natl. Bur. Standards 4, 753
(1930).
2 E. Ellion, Bell Aircraft Corp., Report No. B.A.C.-21, No
vember, 1948.
B M. R. Hopkins, Proc. Phys. Soc. (London) 50, 703 (1938).
4 F. Seitz, Modern Theory of Solids (McGraw-Hill Book Com
pany, Inc., New York, 1940), says that monoatomic metals can be
subdivided into two groups, depending upon whether or not the
d shells are filled. If the d shells are completely filled or com
pletely empty, the properties of the metal are usually simpler
than if they are not, and these metals are called "simple metals."
In the alternative case, the metals are called "transition metals."
940
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to ] IP: 147.143.2.5 On: Mon, 22 Dec 2014 03:34:52HEAT CONDUCTION IN SIMPLE METALS 941
where K=K(T), S=S(T), and T= T(x, t). Introduce
a new variable Q, where
Q= iT [K(X)5(X)]!dX,
To (4)
To being an arbitrary reference temperature. Equation
(2) then becomes
(K/5)~(a/ax)[(K/ 5)!(aQ/ax)] = aQ/at. (5)
Now let
where Q(x, t)=Q*[X(x, t), t],
X= 1 ""(S/K)!dx.
o (6)
(7)
Using Eqs. (6) and (7), transform from the variables
x and t to the variables X and t in the following manner:
(K/5)!(aQ/ax)
= (K/5)t(aQ*/aX)(ax/ax)=aQ*;'ax, (8)
and
a2
Q* =~[(K)! aQ]= (K)l ~[(K)! aQ]= aQ (9)
aX2 ax 5 ax 5 ax 5 ax at
after using Eqs. (7) and (5). Also,
~g= aQ*[fx{~(~)!}dX]+aQ*,
at ax 0 at K at (10)
but
:/:~~) 1=LdT(:) IJ::'
= L~ (: YJ:~ = [d~ (: y J:2~~ (11)
after using (4) and (9). Substituting (11) into (10) and
using the fact that dx= (K/ S)tdX, we get
aQ = aQ*[ rX[~(~)tJa2Q*(K)ldXJ+aQ*
at ax )'J dQ K aX2 5 at
_ aQ*[£X{ d (5)!}a2Q* ] aQ* --- -log --dX +-. (12) ax 0 dQ K aX2 at
Upon equating (12) and (9), Eq. (5) is put in the fol
lowing form where the thermal parameters have been
gathered into one term:
a2Q* _ aQ* aQ*[JX{ d (5 )t}a2Q* ] ---+- -log --dX. (13)
aX2 at ax 0 dQ K aX2
Now assume that
(d/dQ) log (5/ K)t= [1/(K5)!J(d/ dT)10g(5/ K)t= A, (14) where A is a constant. The validity of this relation for
simple metals will be investigated theoretically in the
next section. Substitution of (14) into (13) yields
a2Q* /ax2= aQ* /at+ A (aQ* /aX)2
-A (aQ*/ax)[aQ*/axJx=o. (15)
Let the flux of heat into the metal through the face
at x=o be denoted by j, where j can be a function of
time. The boundary condition there is
-j= K(aT/ax) I x=O= (K/5)!(aQ/axl %=0
= (aQ*/ax)I x=o. (16)
Substitution of (16) into (15) yields
a2Q* /ax2=aQ*/at+ A (aQ* /aX)2+Aj(aQ*/aX). (17)
If the face of the metal was situated at x=b, with the
boundary condition there being K(aT/ax) I x~b= -j,
then Eq. (7) can be redefined as
and after carrying out the mathematics Eq. (17)
would be obtained as before.
Equation (17) is still nonlinear. Consider the further
transformation
Q*= -(1/ A)logr.
Equation (17) then becomes
a2r/ax2=ar/at+Ajcar/aX). (18)
(19)
This is the final transformed form of the heat flow
equation. It should be noted that the applicability of
this equation is limited to those problems in which the
flux of heat at one surface of the solid is known, since
j appears in the final equation.
Since the validity of Eq. (14) for simple metals is
by no means obvious, it is necessary to investigate the
relations between the thermal parameters on the basis
of theory and available data. It is easily seen that the
most general forms for K and S which satisfy Eq.
(14) are
K=Kog(T)ex p[ -A(Ko5o)tl~g(T)dT J. (20)
and
(21)
where the subscript zero means that the function is to
be evaluated at T= To, and the otherwise arbitrary
function geT) satisfies the relation g(To) = 1.
III. INVESTIGATION OF THE RELATIONS BETWEEN
THE THERMAL PARAMETERS OF SIMPLE METALS
A. Introduction
The starting point for the investigation will be the
formula for thermal conductivity due to the heat cur-
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rent carried by the electrons in a metal:6
K = 7r2k2TnjL/3Jmv caIj em sec DC, (22)
where k is Boltzmann's constant, m the mass of an elec~
tron, v and L are, respectively, the velocity and mean
free path of the conduction electrons evaluated at the
Fermi level, nf is the effective number of free electrons
per unit volume, and] equals 4.18X107 ergs/cal. For
monovalent metals, nf is of the order of the number of
atoms per unit volume but is much less than this for
metals of higher valency. The investigation will be re
stricted to good conducting metals for which the con
tribution of the lattice to the thermal conductivity can
be neglected.
It turns out that, for conditions best satisfied by the
monovalent metals, the mean free path is inversely
proportional to the probability per unit time, P, that
an electron makes a transition to a state lying in an
area of the Fermi distribution. 6 The differences between
the various methods of calculating the electron scatter
ing lie in the different assumptions that have been made
concerning the interaction between the lattice and the
electrons. In this paper we shall follow the treatment
given by Mott and Jones,6 as the results will then be in
the form most suitable for a discussion of the relations
between the thermal parameters.
An Einstein model is used to describe the lattice
vibrations. Each atom is treated as vibrating inde
pendently of all the others, and the scattering by each
atom is calculated separately. The metal is considered
to be free from imperfections such as impurities and
lattice defects, and only scattering due to the thermal
vibrations of the lattice is considered. It turns out that
P is proportional to N(X2)Av, where N is the number of
atoms per unit volume, and (X2)Av is the mean square
amplitude of the atomic oscillations; hence, L is in
versely proportional to N(X2)AV.
In seeking a relation for K and S, we are dealing with
two different mechanisms; one is the thermal con
ductivity, which is mainly of electronic origin, and the
other is the product of the specific heat, at constant
pressure, and the density, which is mainly of atomic
origin. The connecting link between the two mecha
nisms is the mean free path, which appears in Eq. (22)
for the thermal conductivity, since it is inversely pro
portional to the mean square amplitude of the atomic
oscillations. Thus, it will be sufficient for the purposes
of this investigation to represent the mean free path by
L= 1/ BS(X2)Av, (23)
where B, which is dimensionless and differs for the
different metals, is in the first approximation inde
pendent of temperature.
6.Frohlich, . Elektronentheorie der M etalle (Verlag, Julius
Spnnger, Berlm, Germany, 1936), Chapter III. '
6.Mot,t and Jones, Properties of Metals and Alloys (Oxford
Umverslty Press, New York, 1936), Chapter VII; also see refer
ence 4, Chapter XV. The procedure followed will be to express the mean
square amplitude of the atomic oscillations in terms of
the thermal energy of the body and then approximately
in terms of the atomic heat of the body. To be consistent
with Eq. (23), which holds for temperatures greater
than the Einstein temperature, this calculation will be
carried out for a similar temperature range. However,
the atomic heat must of necessity be that at constant
volume, for it is only for this atomic heat that theo
retical expressions are available. In order to convert
from C v to C p, it is necessary to use the thermodynamic
relation
(24)
where C p and C v are the molar heats at constant pres
sure and constant volume, T is the absolute tempera
ture, V the molar volume, Ct. the coefficient of volume
expansion, and i3 is the compressibility. This means that
the model of the metal crystal must possess a coefficient
of volume expansion, and, for this to be true, the model
must be composed of anharmonic oscillators. Debye7
was the first to point out that a model of a solid in
which the atoms obey Hooke's law is too idealized in
that it has a zero coefficient of volume expansion. In
order to represent the actual behavior of a solid body,
Debye replaced Hooke's law of force by an expression
involving terms of the second order in the displacement.
The atoms then execute unsymmetrical oscillations
and a displacement of their rest positions with increas
ing energy of vibration occurs, so that the body increases
in volume. To obtain an equation of state for such a
solid, Debye first considered the case of a single an
harmonic oscillator which was originally at rest. The
oscillator was then stretched, by means of an external
force, to a new equilibrium position where it was
allowed to carry out oscillations. Debye showed that
the asymmetric oscillator in its new position behaved
approximately like a harmonic oscillator whose fre
quency of oscillation was a function of the displacement
of the rest position. Proceeding similarly for a solid
body which is composed of many anharmonic oscilla
tors, Debye considered the solid in a first approxima
tion as one composed of harmonic oscillators vibrating
about displaced equilibrium positions with frequencies
dependent on the magnitUde of the original imposed
extensions. Thus, in calculating the free energy of the
solid, he allowed the Debye characteristic temperature
to be a function of volume. It should be noted that
more accurate representations of experimental specific
heat curves are obtained when temperature-dependent
Debye characteristic temperatures are used. Moreover,
when the Debye temperature, and hence the maximum
frequency of vibration, is expressed in terms of the
elastic constants of the body, a variation of these
constants and hence of the frequency with volume oc
curs for actual solids.
7 P .. I?~?ye, Vortriige iiber die Kinetische Theorie der M aterie und
J!,lektnzttat (B. G. Teubner, Leipzig, Germany, 1914).
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In addition, Griineisen8 developed an equation of
state for metals on the assumption of central-force
interactions between the atoms, and he found that
(25)
is a constant which depends on the exponents of the
attractive and repulsive terms in the central-force law
of interaction. He also found that for atoms oscillating
with a monochromatic frequency 11, that 'Y is also given
by the relation
'Y= -(d log 11/ d log V) = -(d loge E/ d log V), (26)
where the Einstein characteristic temperature 19 E
equals hll/k. This result of Griineisen is in essential
agreement with that found by Debye.
Finally, combination of Eqs. (24) and (25) yields
Cp/Cv= l+'YQvT. (27)
Eucken9 used Eq. (27) to compare the measured and
calculated values of lOO(Cp/Cv-l) for monatomic
metals. He concluded that it is a useful formula for a
majority of metals. Though based on the assumption
of a central-force law of interaction between atoms, we
shall find Eqs. (26) and (27) as well as Eqs. (22) and
(23) to be of great use in the investigation of the rela
tions between the thermal parameters.
B. Derivation of an Expression for the
Product KS
Consider a model that is composed of anharmonic
oscillators which, for simplicity, all have the same fre
quency of vibration v. As an approximation we will
follow Debye's procedure and consider an anharmonic
oscillator as behaving like a harmonic oscillator about
a displaced equilibrium position with a f~equency de
pending on this displacement. However, (X2)A' will be
calculated for an undisplaced harmonic oscillator. The
dependence of frequency on volume is taken to be that
of Eq. (26), and the relation between the atomic heats
is taken to be that of Eq. (27). Smirnov1o used the same
general procedure of treating anharmonic oscillators
when he calculated the influence of the anharmonic
part of the thermal oscillations on the electrical re
sistance of a metal.
Hence, the first step is to express (X2)A' in terms of
the thermal energy of the body and then approximately
in terms of the atomic heat at constant volume, sub
ject to the above considerations. The oscillation of a
simple harmonic oscillator of mass M and frequency v
is described by
x = Xo sin (21rvt+ 0)
so that
(28)
8 Griineisen, Handbuch der Physik, Vol. X, contains a survey of
this work.
g Eucken, Handbuch der Experimental Physik, Vol. VIII.
lOSmirnov, Physik. Z. Sowjetunion 5, 599 (1934). and its total energy E is
E= 47r2M Jfl()[l) A,. (29)
According to the quantum theory, the internal en
ergy per mole is
and
Therefore, we have
U/CvT=(1+(e E2/6:f2)+···). (32)
If W is the atomic weight, then for a one-dimensional
oscillator of mass M, the energy is MU /3W. Equating
this to the energy of the oscillator given in (29) and
solving for the reciprocal of ()[l)Av yields
l/()[l)Av= 12rll2W /U
= 121r2Wk2eE2/h2CvT(1 + (19 il/6T2) + ... ). (33)
Substitution of (33), (23), and the relation Cv
= W JCp/(l+'YQvT), which is obtained from (27), into
(22) yields
KS= (4?r4Jt1(rll/N») eE2[1+'YcxvT] p. (34)
h2.J2MB [1+ (eE2/6:f2)] v
C. Investigation of the Temperature
Dependence of KS
Because of the presence of the factor of propor
tionality B in (34), it is not possible to carry out an
absolute determination of the magnitude KS. Hence,
we will calculate the temperature variation of KS/H,
where H is a constant which includes all temperature
independent factors. The effective number of free elec
trons per atom, nf/N, will be considered temperature
independent.
For mod-erate or high temperatures, the Einstein
atomic heat function approximates fairly well to the
Debye function and gives a fair representation of the
atomic heats in this region. If the first two terms of the
high temperature expansions of the Einstein and Debye
atomic heat functions are equated, it turns out that
19 E = 0.77 19 D. On the other hand, if in this temperature
region the Einstein temperature is taken to be propor
tional to the mean frequency of the Debye frequency
spectrum, it turns out that eE=!B D, which essentially
agrees with the above result.
In this section, it is convenient for calculational pur
poses to evaluate eE by means of the relation
(35)
However, the theory presented is still a monochromatic
theory, and the relation between 19 E (or 19 D) and volume
is still given by Eq. (26). The variation of eE with
temperature is obtained in the following manner:
(d 10geE/dT)
= (d loge E/ d log V) (d log V / dT) = -'YQv. (36)
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TABLE L The variation of KS/H with temperature.
TieD ••
Metal 3 4 6
Cu 0.88 0.90 0.87 0.83
Ag 0.89 0.91 0.88 0.84 0.78
Cd 0.88 0.90 0.86
Zn 0.86 0.87
Al 0.85 0.84
Ph 0.90 0.94 0.93 0.90 0.88 0.84
Integrating (36) and expanding the solution to first
order terms (for most metals 'Yav",1Q-4, since 'Y,.....,2 and
av"'SOX 10-6), one obtains, after introducing 0 D via
(35),
(37)
where 0D•O is the value of the Debye characteristic
temperature at the absolute zero of temperature.
The variation of density with temperature is given by
p=PD[1-a v(T-273)J, (38)
where Po is the density at zero degrees centigrade.
The only factor whose temperature dependence re
mains to be considered is v, the velocity of the electrons
at the Fermi level. For free electrons v is obtained
from the relation
The Fermi statistics is a constant volume statistics,
but, since we are allowing thermal expansion in our
model, we will make the further approximation that
n, the number of electrons per unit volume which is
proportional to the density, can vary with temperature
due to the variation with temperature of the density.
As will be seen, this assumption will not affect the re
sults of the calculation of the temperature dependence
of KS to any appreciable extent. Thus v"-'pt, since
n"'p or
p/v"'pl. (39)
Upon substituting Eqs. (39), (38), (37), and (35) into
(34) and collecting all the temperature-independent
factors into the constant H whose magnitude is of no
immediate importance, we get
The values of -y, eD•D, and av, vary from metal to
metal. The right-hand side of (40) will be calculated
numerically for several metals, which include among
them a wide range of values for 'Y, 0D•D, and avo The
results of the calculation for KS/H, carried out for
each metal from 0D.D to a temperature roughly equal
to the melting point temperature, are listed in Table I.
The variation of the coefficient of volume expansion
with temperature was neglected in performing the calculation. The values used for 'Y and av were taken
from Mott and Jones;6 the values of 0D.C were taken
from Seitz;4 and the values for the melting point tem
peratures were taken from the Handbook of Chemistry
and Physics.
Considering all the approximations and assumptions
made in arriving at Eq. (40), it is stretching the results
of the theory too far to believe that the above calcula
tion will predict the actual detailed variation of KS.
However, the calculation does show that the product
KS is essentially a constant over most of the tempera
ture range considered. The average deviation from the
mean divided by the mean, expressed in percent, has
the value of 1 percent for Cd, 2 percent for Cu, 2
percent for Pb, and 4 percent for Ag. Hence, we con
clude that a relationship between the thermal pa
rameters, in this temperature range, is
KS"-'constant. (41)
We shall later see that Eq. (41) is also borne out by
experimental data where the variation of KS with
temperature is much less than that of either K or 5
alone; and use of experimental values for the magnitude
of the constant KS will allow an estimate to be made of
the constant of proportionality B.
Hume-Rotheryll combined the empirical Griineisen
relation that R",C pT with the Wiedemann-Franz law
that K/O"T= constant, where 0"= 1/ R is the electrical
conductivity, to obtain the relation between the ther
mal conductivity of electronic origin and the atomic
heat at constant pressure;
KCp=constant. (42)
Since 5 = pCp and the variation of density is much less
than that of, K and Cp, it is seen that (41) based on
theoretical considerations is essentially the same as
(42) which is obtained empirically. However, Eq. (41)
differs in functional form from the relation obtained
by Bidwell,12
K/ pCv= kl/T +k2, where kl and k2 are constants.
D. Investigation of the Temperature Dependence
of [lj(KS)!J(djdT) log(S/ K)!
The first step is to calculate (d/ dT) log (5/ K)t, where
the factor Sj K is obtained by dividing 52= p2Cp2 by K5
which is given by (34). It is seen that the taking of a
logarithmic derivative will cause all the constant terms,
including the factor of proportionality B, to vanish
from the final result, leaving an answer which can be
compared with experimental data. Hence, after sub
stituting (39), and using (27) and (31) to write cp
=(3R/WJ)(1-(0 E2/12P)+''')(1+-ya vT), we get,
1\ Hume-Rothery, The Metallic State (Clarendon Press, Oxford,
1931), p. 79.
12 C. C. Bidwell, Phys. Rev. 58, 561 (1940). For a discussion and
critique of Bidwell's result see R. W. Powell, ]. Appl. Phys. 19,
995 (1948).
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after differentiating out all the constant terms and
neglecting powers of eh/T higher than the second,
~ 109(~)t =~ ~ logP413[1+'YavT]. (43)
dT K 2 dT GE2
The dependence of density on temperature is given by
(38), and (d/dT) logGE is given by (36). Equation (43)
then becomes approximately
(d/dT) log(S/K)l='Yav+av(h-i), (44)
a positive constant. This result will be compared with
experimental data in the next section.
If the relation KS = constant is substituted in Eqs.
(20) and (21), it is seen that geT) = 1 and the varia
tions of K and S with temperature, given by
K = Ko exp[ -A (KoSo) l(T -To) ] and
S=Soexp[A(KoSo)l(T-To)], (45)
satisfy Eq. (14).
When (41) and (44) are written in the form KS
= KoSo and (d/dT) 10g(S/ K)t=A(KoSo)t, where A (KoSo)l
",,10-4 from (44), and solved simultaneously, it turns
out that the variations of K and S with temperature
are, naturally enough, given by Eqs. (45). Since the
coefficient A(KoSo)t is small, the exponentials in (45)
can be expanded to nrst-order terms yielding
K = Ko(1-A (KoSo)![T -To]), and
S=So(1+A(K oSo)![T-To]), (46)
which is in agreement with the usual linear form in
which the thermal parameters of simple metals are
represented empirically in the temperature rapge we
are considering.
In summary, the conditions for the constancy of
1/(KS)l(d/dT) 10g(S/ K)! are that the temperature
variations of K and S be given by Eqs. (20) and (21).
However, for the case of simple metals, sufficient con
ditions for the validity of Eq. (14) are the constancy
of the product KS and the exponential behavior of
K and S given by (45); the latter equations are equiva
lent to a linear variation of K and S with temperature
in the range considered.
IV. COMPARISON OF THEORY WITH AVAILABLE DATA
A. Introduction
The results of calculations made to investigate the
variation with temperature of the quantities KS and
logS / K are presented in graphical form. On one graph,
K, S, and KS are plotted to the same scale as a function
of temperature, and on the other, logS/K is plotted as
a function of temperature; the linearity of the latter
is a measure of the constancy of (d/dT) 10gS/K. The
plot of logS / K versus T is considered since the slope
of this line can be compared with the theoretical result
given by (44); and the division of this slope by 2(KS)!
yields the value of [1/(KS)iJ(d/dT) 10g(S/K)i. It is desirable to have data extending from at least
the Debye temperature to a temperature roughly equal
to the melting point; and while such data are available
for the calculation of S, the situation is quite different
when the thermal conductivity is considered. Although
some data are available for each metal, the data for
many are too sparse, and hence unusable for purposes
of checking the variation of the relations between the
thermal parameters with temperature. In many cases
it was necessary to piece together the data taken by
various investigators. An additional problem arising
out of this procedure was that the various sets of data
sometimes did not join together too smoothly. Aside
from experimental errors, this was due to the fact
that values of thermal conductivity are sensitive to the
purity of the metal used, and different investigators
usually used specimens of varying purity. It is to be
noted that the thermal conductivity data are less
accurate than the specinc heat data.
B. Examination of Data for Some Simple Metals
Sufficient data were available in the literature to
investigate the behavior of copper, silver, sodium,
cadmium, zinc, aluminum, and lead. Plots of S, K, KS,
and logS/K versus temperature are shown in Figs. 1
through 7. The straight line drawing in the KS plot
represents the mean value of the data in the tempera
ture range indicated.
The thermal conductivity of copper at -50°C and
-100°C was measured by Lees,13 the data from 0-600oe
were calculated from an interpolation formula in the
International Critical Tables which is based on measure
ments made by Schofield,14 and the remainder of the
data were taken from a graph in a paper by Hering.15
The thermal conductivity data for silver, cadmium,
and zinc from -170°C to ooe were measured by Lees,13
and the rest of the data were measured by Bailey.16
Both silver and copper are monovalent metals, and one
would expect the thermal conductivity of silver to
behave in the same manner as that of copper. This is
true till a temperature of 400°C at which K is a mini
mum; but above this temperature, K increases with in
creasing temperature. No other measurements of the
thermal conductivity of silver were found in this
temperature range. The data for zinc check fairly well
with data taken by Konno17 and by Van Dusen and
Shelton. IS
The thermal conductivity of sodium was measured
by Bidwell.l9
The thermal conductivity of aluminum from -160°C
13 Lees, Trans. Roy. Soc. (London) Al08, 381 (1908).
14 Schofield, Proc. Roy. Soc. (London) AI07, 206 (1925).
15 Hering, Trans. Am. rnst. Elec. Engrs. 29, 285 (1910).
16 Bailey, Proc. Roy. Soc. (London) A134, 57 (1931).
17 S. Konno, Phil. Mag. 40, 542 (1920).
18 Van Dusen and Shelton, J. Research Natl. Bur. Standards
12, 429 (1934).
19 C. C. Bidwell, Phys. Rev. 28, 584 (1926).
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P 0.7
11, 0, ...
~rt'f 0
is 5
00
'" '" 0 .E .S A ALUMINUM
Melting Point is 659·C
KS
X
:.: (f) 0.2
0.1 eo,.
~--~--~0----~--~200~--~~~0~~4~00~--~~0--~ ·200 .100
0 B-
06
0
0.4
0
:.:: 0.2 In
'" 01
(>
...J 0
·01
.0.2
·0.
-OA
·200 ·100
T i' rr.
11
siC.
.S .S
" ..
-0
-Q,4
·0'
·0.6
-0.7
·eoo -100 Temperature in -c
FIG. 1.
SILVER
Mellin, PoiII' i, 960 '0 4
K$ •
J< "x
o 100 200 300 400 500 600
Temp.,utur. i" -c
/ 0 0
/
~" . I I I
100 200 300 400 500 600
remperuture in °C
FIG. 2. to ODe was measured by Lees.13 The data used from
ODe to 6000e were measured by KonnoP Konno's
measurements are in agreement with those taken by
Bidwell and Hogan20 for 99.2 percent aluminum.
The thermal conductivity of lead from -2000e to
ooe was measured by Bidwell and Lewis;21 and the rest
of the data were measured by Van Dusen and Shelton,18
whose measurements agree with those of KonnoP
Examination of the graphs reveals that the relation
KS:~constant can be regarded as valid for simple
metals. Empirically, the reason for this lies in the fact
that for those metals where S increases monotonically
SODIUM
Melling Point i. 97.5·C
0.4
0.3
"t~ ill 1'1 02
E E " " 88
,S of:
'" ",
'" 0.1
6D,.
~~O~O---~2~OO~~4~O~O--~O~--~~~O~
o
-0.1
-0.2
-0.3 /
/
o ,0
o ,0 / \ .s
/ \ I / \J
(Ii .0.4 I
go
.J -0.5 /
I
-0.6 /
-0.7 l
/
.0.8 0 ~D,O
-W~O---~2~OO~~~~O--~-2~OO~---~WO~
Temperature in "C
FIG. 3.
and K decreases monotonically with increasing tempera
ture, the individual temperature variations cancel in
the final product and the temperature variation of KS
is less than the variation in K or S. Although eD,o is
not an absolute criterion for the lower limit of the
temperature range in which the relation is valid, it is
seen that, in accordance with the considerations of
Sec. III, the thermal parameters are linear functions of
temperature for temperatures roughly greater than e D,O.
Surprisingly enough, even for sodium and cadmium,
20 C. C. Bidwell and C. L. Hogan, J. Appl. Phys.18, 776 (1947).
21 Bidwell and Lewis, Phys. Rev. 33, 249 (1929).
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where discontinuities occur in the thermal conductivity,
the variation of KS is much less than that of K or S
alone.
Although not shown here, the values of Kcp were
calculated and compared with KS. Because of the small
variation of density with temperature, it was not found
possible to decide on the basis of available data whether
the relation KS"-'constant or KcpC'-:'constant was more
correct. The products Kcv and Kpcv were also calculated
for aluminum, copper, silver, and lead, the values of Cv
being obtained from the Debye atomic heat function.
In each case, their products were less constant than
CADMIUM
Melling Poinl is 321°C
0.5
0-0_0 K -0_° ............ /0-0-°_0 "8 "80.2
'" (/)
0.1 X X
80,0 x x x KS
It
~~0~0--~'1~0~--~0----~10~0~~2~0~0~~~0
Temperature in °c
09f-
0.8f-
'" u; 0.7t-
! I
0.5t- ~D,O I
-200 -00 o 100 200 300
Temperature in °c
FIG. 4.
either KS or Kcp. This is to be expected empirically,
because Cv is more nearly constant than Cp at higher
temperatures, and the product Kcv will primarily
possess the temperature variation of K, the temperature
variations of the individual factors not canceling as
well as in the product Kcp.
Examination of the plots of logS/ K versus tempera
ture reveals that if no discontinuity appears in the
data, then logS / K does become a linear function of
temperature for temperatures roughly greater than
eD,o. On the other hand, the linear region is by no
means obvious in the plots for sodium and cadmium.
In Table II, the measured values of (d/ dT) 10g(S / K)t llNC
Melting Point is 419·C P 081~----~~~~~--------~~-' _ ,. A. s.
~ P 0.1 A..-"'A_'_O-'_O-O-O-
,. "I 0.6 A..-'"
5 5 0 t{tf'
'8 '804
.~ .5 o.
~ CJ) o..o--o-o-o-o-o-o_o_o_o-.!..o
0.2 x x x X A K A xks eo. OJ
.200~--~~oo~~Ho~--~OO~--~200~--~~0--~4trOO
Temperature in ·C
12,-----------------------------,
1.1
10
~ 09
gO!
-.J 0.7
06 B.
O!
-200~--~.OO~~~0~--~O~0--~2~O~0--~~~--~~
Temperature in ·C
FIG. S.
are compared with the theoretical values calculated
from Eq. (44), and the mean values of KS calculated
from the data, together with the average deviation
from the mean, are also listed.
Considering the approximations made in obtaining
Eq. (44), the agreement between theory and experi
ment is much better than might be expected.
lOne can estimate B, the factor of proportionality
introduced into the expression for the mean free path,
if the value of KS obtained from the data is used in
COPPER
Melting Point i-s 1083 coc 10,------------------'--:.---'-----'-=---------------,
I>
-; ~ 0.9
'81
g ! OB
" <II 0
02
01
.ol
-0.2 0 100 200 eoo
Te",perotLlte ift 00.
FIG. 6. 1100 000
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LEAD
0~~ __ ~M~~~li~n~PO~ln~l=is~3~27~~~-,
A. 0'.
-"'6 OJ2 -....
B U 010 0-......
• Si .5 0---'0_
0
~ q, 008 --o--o~o
006
0.04 .s )I l! ); I
002 60,0
~200 -100 a 100 2;00 500
Ternperalure in ·C
,~--------------------,
,1 e. ,.
" "" ,.
V;
8' ...J ,2
J.I
FIG. 7.
FIGS. 1-7. The variation of the thermal parameters and the
combinations KS and logS/K for simple metals.
Eq. (34). The calculation may be carried out simply
for monovalent metals for which the electrons can be
considered as behaving approximately like free elec
trons. It turns out that for silver Br-v6, and for copper
Br-v7. Now, the form given by Eq. (23) is equivalent
to writing the mean free path in terms of a scattering
cross section B(X2)AV, and if for (X2)AV we substitute
Xo2/2 (as given by (28)), then the scattering cross
section is 1/2BX02. Since B/2 is approximately 'll', the
atoms act more nearly like spheres of radius Xo in
these cases.
C. Examination of Data for Fused Quartz
The thermal conductivity of fused quartz was meas
ured by Seeman,22 the specific heat was measured by
Moser,23 and the variation of density with temperature
TABLE II.
Theoretical Experimental
value of value of
Temperature
xl~~i5k)1 (d/dT)
Metal range Xlog(S/K)!
Cu l00-1000°C 1.1 X 10-4 lAX 10-4
Ag -50--300°C 1.7 X 10-4 3.0XlO-4
Zn -50--350°C 2.1X10-4 3.4XlO-4
Al 0-600°C 1.7 X 10-4 4.5XlO-4
Pb -50-300°C 3.0X 10--4 5.0X 10--4
22 Seeman, Phys. Rev. 31, 119 (1928).
23 Moser, Physik Z. 37, 737 (1936). Mean Average
KS deviation
0.765 0.004
0.553 0.014
0.175 0.002
0.276 0.008
0.028 0.001 was calculated from values of the liner expansion co
efficient measured by Souder and Hidnert.24
In the range from o-700oe, K increases linearly
from a value of 0.0027 to 0.0054; and S increases from
0.374 to 0.612. The product KS increases from 0.001
to 0.003, an increase of 200 percent. Hence, the relation
KS constant, where the variation in KS is much less
than that of K or S individually, certainly does not
hold for fused quartz .
A plot of logS / K versus temperature shows that
10gS/K increases from 4.93 to 4.97 in the range from 0
to 200oe, but decreases steadily in the remainder of
the range, reaching a value of 4.73 at 700°C. Although
the plot is fairly linear in the latter part of the range,
the slope is negative, in distinct contradiction to the
result of the theory for the simple metals, as given by
(44), and in contradiction with the data for the simple
metals.
Thus, investigation of the data for an insulator,
where the electrons do not contribute to the thermal
conductivity, shows that the relations derived previ
ously are not valid for all substances, conductors, and
insulators. This justifies our looking to the theory of
conduction by electrons in the investigation of the rela
tions between the thermal parameters of simple metals.
D. Examination of the Data for Iron and
0.80 Percent Carbon Steel
The theory presented does not apply to transition
metals or alloys. This is because the approximation of
the mean free path used applies best to the monovalent
metals, but does not apply to the transition metals or
alloys where the scattering of the electrons is a more
complicated process. However, examination of Figs. 8
and 9 shows that the relations between the thermal
parameters that are valid for the simple metals also
hold empirically for iron and 0.80 percent carbon steel,
till the temperature at which the magnetic phase
change occurs.
The thermal conductivity data used for iron were
measured by Powell,25 and all data for 0.80 percent
carbon steel were taken from a paper in the Iron and
Steel Institute.26
However, all transition metals and alloys do not
behave in a similar manner. The thermal conductivity
of platinum, as given in Landolt-Bornstein, 1936, in
creases linearly from 0.167 to 0.215 in the range from
200e to 1020oe; and the thermal conductivity of
brass16 increases from 0.175 to 0.354 in the range of
-170°C to 450oe. Thus, the relations between the
thermal parameters do not hold for these metals.
24 Souder and Hidnert, Natl. Bur. Standards (U.S.), Sci. Techno!.
Papers 21, 1 (1926--1927).
26 Powell, Proc. Phys. Soc. (London) 46, 659 (1934).
26 Iron and Steel Institute, Special Report No. 24, 1939; Second
Report of Alloy Steel Research Comm., Sec. 9.
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IRON
Melting Point is 1535°C
0.8,.. CARBON STEEL
100 200 ~ 400 roo 600 100
KCI!, Temperature in ·C
'"
0 loa 200 300 400 500 600 700 30 B Temperature in "c
28 e, 3.0
2.
U
:.: 2.4
in ~ 2
'" 0 '" ..J 2.2 0 .J 2.4
2.0
U
2.00 100 200 lIOO 400 !\OO soa 100
00 300 400 500 600 700 Temperature in °C Tempera1ure in "C
FIG. 8. FIG. 9.
Fres. 8 and 9. The variation of the thermal parameters and the combinations KS and logS/ K for iron and 0.80 percent carbon steel.
V. CASE OF A SEMI-INFINITE METAL WITH A
CONSTANT HEAT FLUX
A. Formulation and Solution of Problem
Consider a semi-infinite metal with a constant flux
of heat into its surface. The boundary condition at
x=O is given by (16), and the other condition used will
be limx .... ooT(x, t) = O. The initial temperature will be
chosen as zero; and To, the arbitrary temperature which
appears in (4), may be set equal to zero without any
loss in generality. Then, after using Eqs. (4), (6), (7),
and (18), the transformed boundary and initial condi
tions become
limr(X,t)=l, (dr/dX) =Aj, r(X,O)=1. (47)
X ... oo r X=o
Introduce the following dimensionless variables:
1}=AjX, r=Aj(t)l, x=Aj(So/Ko)!x. (48)
It turns out that the solution of (19) for r, subject to
conditions (47), is
.1= 1 + eV
[r2+ 1 +1}Jerjc(~+~)
2 21' 2
-!erjc(;r -;) -(:)1 exp[ -(;T -;)] (49) In addition, the expression for X in terms of 1} and r
turns out to be
eV
[ 11] ( 1} 1') X =--r r+-erfc -+-
2 T 21' 2
Equation (50) is obtained from (7), which can be re
written as
x= (;0°) 'l\dX
after using the relation (K/S)!= (Ko/So)!r; the latter
being obtained from (45), (4), (6), (18), and the fact
that KS is constant. The subscript zero refers to the
value of the function at T=O.
Equations (49) and (50) represent the solution of the
problem . .I and X can be calculated as functions of 1} and
T, and then r can be expressed in terms of X and 1'. The
result of doing this for a partial range of values of the
dimensionless variables is depicted in Fig. 10. Q can
then be calculated as a function of X and r by means
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T·0.09
0.98
0.9
'T·0.27
0.1
0.04 0.08 0.12 016 0.20 0.24 X
S versus 'X for various values of "t
FIG. to. Dimensionless representation of the solution for a semi·
infinite solid with constant heat flux at one end.
of (18); and once Q is known, T can be obtained from
(4). Since KS is constant, the conversion from Q to
T is simple.
B. Calculation of a Numerical Example and
Comparison with the Results of the Usual
Linearized Theory
The calculation will be performed for a specimen
of 0.80 percent carbon steel. In order to do this it is
necessary to know,the magnitudes of (KS)!, A, j, and
(So/Ko)!. From Fig. 9A the mean value of KS is 0.108,
and from Fig. 9B we find that (d/dT) log(S/K)!=5.9
X 10-4 for the temperature range of Q-600oe. Thus the
magnitude of A is 18X 10-4• The value of (So/ Ko)! is
2.74. For the heat flux into the metal at x=O, choose
a value of j=500 cal/cm2 sec.§
Since Aj and (So/ Ko)! are known, r, which is given
as a function of X and T in Fig. 10, can be obtained as a
function of x and t. The results of the calculation for T
are shown as the solid lines in Fig. 11. In carrying out
the calculation, temperatures greater than 6000e were
not considered as, for 0.80 percent carbon steel, the
relations used hold best in the range from Q-600°C.
In the usual mathematical treatment where the
thermal parameters are considered constant, the dif
ferential equation to be solved is
(51)
subject to K(iJT/iJx)I~=-j, and lim..,-+OOT(x,t)=O,
the initial temperature, being zero. The solution of this
§ This roughly corresponds to propellant gas in a rocket at a
tern perature of about 3000°C and a heat transfer coefficient of
abou t 0.18 cal/cm2 sec. °C. problem is
T=j/(KS)!{2(t/rr)! exp( -Sx2/4Kt)
-x(S/K)!erjc(S/K)!(x/2(t)!)}. (52)
However, since K and S are really functions of tem
perature, there is no unique method of calculating the
temperature distribution for there is no way of deciding
which values of the thermal parameters are to be used.
For example, in the case of 0.80 percent carbon steel,
the following is the variation of S / K with temperature:
Toe
o
300
600 S/K
7.52
10.73
15.29
Thus, in calculating the temperature distribution from
(52) in order to compare the results with the solution
of the nonlinear equation, three calculations were
made corresponding to the above listed values of S/K
at 0, 300, and 600°C. The associated values of KS
were taken from Fig. 9A. The results of the three calcu
lations are shown in Fig. 11 together with a plot of the
solution of the nonlinear equation.
It is ipteresting to note that for this particular
problem, the distribution obtained from the nonlinear
equation is almost entirely bracketed by solutions of the
420.
180. Temperature Distributicn in a Semi-infinite
Slab cf 0.8 'Yo Carbcn Steel
Sclution o.f non-linear equation-
Solution cf linear equaticn with K and S
\ ~ constant at: 0 °C ----
\ 300OC--
\~~ 600"C ----
.. " \ ",
'. " \ ", \ ,
\~ \,
~,~~ '. ~ "-,,-
\~,~ \"" " """-
, :\' '-, ~ "-
\, ~" '"-,, t=C09 sec '."''' ' \, ~ "-,,-'",
',,-"" "-
120. '\~ "" "~'"
\~~ ~ "'" .......... .."',.
",,~~ -""""",_ ............. t=o..C4 sec.
'-..'-~ .............
'--......~~ 60.
, t=o.·o.l sec
0. 0..0.2 0..0.4 0.0.6 0.0.8 0..10. 0..12
Distance in cm
FIG. 11. A camparison of the temperature distribution ob
tained by solution of the nonlinear equation and the linearized
equation.
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linear equation calculated when the thermal parameters
are taken to be those at 0 and 300°C. This shows that
in this temperature range, the distribution obtained
from the linear equation best fits the temperature dis
tribution obtained from the nonlinear equation when
the constant values for K and S are taken to be those
at a temperature which is about one-fourth of the
maximum temperature considered.
VI. A FURTHER APPLICATION FOR THE
TRANSFORMATION
If the relation (K/S)!= (Ko/So)!eAQ is substituted
into Eq. (5), the latter becomes
(Ko/ So)te-AQ(a/ax)
X [(Ko/ So)te-AQ(aQ/ax)]=aQ/at. (53)
Upon making the substitution y= (Ko/ So)!e-,tQ, Eq.
(53) becomes
i(a2y/ ax2) = ay/at, (54)
and the boundary condition (16) becomes
ay/ax/ x==o=Aj. (55)
Although at first glance (54) appears different from
the nonlinear equation already considered, the two
must be similar, and (54) should be put in a linear form
by similar methods. Accordingly, make the change of
variables
and
Then we have (' dx
X= Jo y(x, t)
y(x, t) = Y[X(x, t), tJ.
ay/ax= 1!y(ay/aX), (56)
(57) and
also
ay = ay + ay[ fX (-ay/at) dXJ= ay _ ay[ fX a2y
dX),
at at ax Jo i at ax Jo ax2
or
ay/at=ay /at-(aY /ax) (ay/ax)+Aj(a y/ax). (59)
Substitution of (58) and (59) into (54) yields
a2y /ax2=ay/at+Aj(ay/aX),
and (55) becomes
l/y(ay/aX)! x=o=Aj. (60)
(61)
Equations (60) and (61) are identical with the corre
sponding equations for S already considered. Thus, it is
seen that the nonlinear partial differential equation
i( a2y / ax2) = ay / at can be transformed to a linear form
if ay/ax/ x==o is known. As was mentioned in Sec. II,
this can also be done if ay / ax /_b is known.
Finally, it is clear that the diffusion equation,
(a/ax)[D(ac/ax)]=ac/at, (62)
where D is the coefficient of diffusion, and c the con
centration, can be handled by the mathematics pre
sented in this paper for those substances for which the
coefficient of diffusion can be represented by the form
(D)!= (Do)!e-ac•
In conclusion, the author wishes to thank Dr. George
Hudson, Dr. Hartmut Kallman, and Dr. Fritz Reiche,
for their constant help and encouragement. The author
also wishes to acknowledge his indebtedness to the late
Dr. J. K. L. MacDonald, who first suggested the problem
to him and helped him overcome the initial mathe
matical difficulties.
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1.1698915.pdf | The Dissociation Energy of Fluorine
Paul W. Gilles and John L. Margrave
Citation: The Journal of Chemical Physics 21, 381 (1953); doi: 10.1063/1.1698915
View online: http://dx.doi.org/10.1063/1.1698915
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Published by the AIP Publishing
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155.33.16.124 On: Sat, 22 Nov 2014 21:01:28LETTERS TO THE EDITOR 381
t 110
.!!'
~90
~ (3ii plane)
FIG. 2. N8
In order to have a standard for comparison of intensities of
extra spots at different temperatures, a small quantity of alu
minum powder was dusted over the crystal. The variation of
the intensities of aluminum lines over the small range of tem
perature studied was neglected, however. The intensities were
compared by means of a standard wedge, prepared according to
the method of Robinson. The ratio of the (111) aluminum line
to the maximUl}l intensities of the extra spots were found by
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from the curve that intensities of extra spots increase very
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factor amplitude of two planes are different (see Figs. 1 and 2).
Erratum: Theory of Absorption Spectra
of Carotenoids
[J. Chern. Phys. 20, 1661 (1952»)
GENTARO ARAKI
Faculty of Engineering. Kyoto University, Yosida, Kyoto, Japan
IN the previous letterl we attempted an explanation of the
relation between absorption spectra and molecular lengths of
carotenoids, making use of Tomonaga's method for electron gas
with arbitrary couplings. We took into account the electron-spin
for enumerating the number of electrons (occupying levels up to
the Fermi maximum) only. If we take into account the spin
degrees for wave functions we have, instead of Eq. (2) in the
previous letter,l the following equation:
8= (L/7r)'(4/ A)(N _2)-1.
The empirical value of A thus becomes twice as large, A =709.5.
The rest needs no change.
1 G. Araki and T. Murai, J. Chern. Phys. 20, 1661 (1952).
Addendum: Directed Valence as a Property of
Determinant Wave Functions
[J. Chern. Phys. 17, 598 (1949»)
HOWARD K. ZIMMERMAN, JR., Department of Chemistry, Agricultural and
Mechanical College of Texas, College Station, Texas
AND
PIERRE VAN RYSSELBERGHE, Department of Chemistry.
University of Oregon, Eugene, Oregon
IN his valuable recent review on "Quantum Theory, Theory of
Molecular Structure and Valence" Professor Coulson! states
that the idea of deriving directed valence properties from atomic
wave functions (as written, for instance, under the form of
determinants) "seems to have originated with Artmann,2 but it has
been "rediscovered" by Zimmerman and Van Rysselberghe3• " ."
We wish to put on record that the tetrahedral valences of carbon
were derived in this manner by one of us (P.V.R.) in 1933, that the
calculations and results were communicated orally and by letter to
several persons interested in the field, and that a communication to the editor of the Journal oj the American Chemical Society
giving a condensed presentation of this fundamental point was not
published for reasons which, coupled with other preoccupations,
resulted in our abandoning further work of this type. The problem
was resumed in 1946 and led to our submitting a papers to The
Journal of Chemical Physics in July, 1948, publication following in
July, 1949. In this paper we present the derivation of the tetra
hedral valences of carbon in a manner identical with that of the
intended communication of 1933, and we give reasons for offering
our treatment of the whole problem of directed valences as an
alternative to that of Artmann whose work had come to our
attention through the abstract' published in September, 1947.
1 C. A. Coulson. Ann. Rev. Phys. Chern. 3. 1 (1952). see p. 8.
• K. Artmann. Z. Naturforsch. 1. 426 (1946). 'H. K. Zimmerman. Jr .• and P. Van Rysselberghe. J. Chern. Phys. 17.
598 (1949).
• Chern. Abstracts 41. 5785 (1947).
The Dissociation Energy of Fluorine*
PAUL W. GILLES AND JOHN L. MARGRAVEt
Department of Chemistry. University of Kansas. Lawrence. Kansas
(Received December 16. 1952)
RECENT spectroscopic datal on CIF imply a value for the
dissociation energy of fluorine in the range 3~ kcal/mol.
Such a low dissociation energy would mean that considerable
dissociation of diatomic fluorine into atoms must occur at rela
tively low temperatures. Doescher2 and Wise' have reported
experimental results that indicate a value for Do(F2) between 36
and 39 kcal/mol.
The experiments reported here were carried out in 1950 on a
sample of fluorine obtained from the Pennsylvania Salt Manu
facturing Company, The pressure exerted by this sample of
fluorine, when contained in a closed system of copper which had
been previously treated with fluorine, was measured as a function
of temperature over the range 300-860oK with a Bourdon type
Dura gauge in two runs on different days. Between the two runs
a slight leak into the system occurred so that about six percent
of the gas was air in the second run.
When corrections are made (1) for the presence of this air in
the second run on the basis that it did not react and (2) for the
cooler zones of the system, the pressure calculated on the basis
of no dissociation of an ideal gas agreed at all temperatures
below 8000K with the experimentally observed pressure with
standard deviations of ±0.02 inches of Hg for six points on the
first day and of ±0.03 for six points on the second day.
Three measurements at temperatures above 8000K showed
differences between observed and calculated pressures consider
ably greater than any found for the twelve lower temperature
measurements. If it is assumed that these differences are caused
by the partial dissociation of F 2 into atoms, one may calculate at
each temperature the degree of dissociation, the dissociation
equilibrium constant, and, by using the available data for the
free energy functions of F and F 2,' the dissociation energy of F 2.
Because of the corrections necessary for different temperatures in
different parts of the system, the degree of dissociation IX and
the equilibrium constant K are not simply related to the pressure
difference.
The data and results are shown in Table I, in which the calcu
lated and observed pressures in the second run have been cor
rected for the air leak. The uncertainties listed are obtained by
assigning to each pressure an uncertainty of ±0.03 inch of Hg.
TABLE 1. Degree of dissociation. dissociation equilibrium constant.
and dissociation energy of fluorine.
Pobs Peale K D.(F,)
Run TOK (inches of Hg) a (10-' atmos) (kcal)
1 815 1.48 1.40 0.07±0.04 0.975±1.10 33.4±2.0
2 810 1.52 1.38 0.11±0.05 2.49 ±1.80 31.6±1.6
2 860 1.72 1.46 0.21±0.05 10.6 ±5.4 31.2 ±0.8
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155.33.16.124 On: Sat, 22 Nov 2014 21:01:28382 LETTERS TO THE EDITOR
The weighted average of the values in the last column gives
for Do(F2) a value of 31.5±0.9 kcal/mol. Using the free energy
functions4 one calculates for the same quantity 36.5±1.0 from
the data of Doescher 2 obtained in similar experiments at higher
temperatures in a nickel container, and 39±1 from the graph of
Wise.s It appears that the best value is 36±3 kcal/mol.
The electron affinity of fluorine may be related to the dissocia
tion energy through a Born-Haber cycle. Studies by lonov and
Dukelskii,5 in which positive and negative ion currents were
observed during evaporation of alkali metal halides from a
tungsten filament, allow calculation of the electron affinities of
the halogen atoms if the proper work function for the tungsten
surface is known. These experimenters found values for the
electron affinities of chlorine, bromine, and iodine in good agree
ment with those given by other workers, when potassium halides
were used and the work function for a clean tungsten surface was
assumed. A similar treatment of their data on KF indicates a
value of 83±3 kcal/mol for the electron affinity of F.
Metlay and Kimba1l6 have studied the relative currents of
negative ions and electrons emitted from a hot tungsten filament
in the presence of fluorine. Although originally misinterpreted,
the experimental data yield an average value for the electron
affinity of 82±3,1·8 in good agreement with the result of lonov
and Dukelskii.
If one uses the value 82±3 for the electron affinity of F in a
Born-Haber cycle along with thermochemical data from the
National Bureau of Standards Table "Selected Values of Chemical
Thermodynamic Properties" and crystal energies of the alkali
metal fluorides computed after Pauling,9 he finds Do(F2)=31±4
kcal/mol, in agreement with the experimental value.
The authors are pleased to acknowledge the support of the
U. S. Atomic Energy Commission in this work.
* Abstracted in part from a thesis presented by John L. Margrave in
partial satisfaction of the requirements for the degree of Doctor of
Philosophy at the University of Kansas. December 28. 1950. t Present address: Department of Chemistry. University of Wisconsin.
Madison, Wisconsin.
1 A. L. Wahrhaftig. J. Chern. Phys. 10.248 (1942); H. Schmitz and H. J.
Schumacher. Z. Naturforsch. 2a. 359 (1947).
2 R. N. Doescher. J. Chern. Phys. 19. 1070 (1951); 20. 330 (1952).
• H. Wise. J. Chern. Phys. 20. 927 (1952).
• R. M. Potocki and C. W. Beckett. National Bureau of Standards
Report 1294. December 1. 1951; L. Haar and C. W. Beckett. ibid. Report
1435. February 1. 1952. 'N. lonov. Compt. rend. acado sci. U.R.S.S. 28. 512 (1940); N. ronov
and V. Dukelskii. FIZ. Zhur. 10. 1248 (1940). 'M. Metlay and C. Kimball. J. Chern. Phys. 16. 779 (1948).
7 J. L. Margrave. thesis. University of Kansas (1950).
• R. B. Bernstein and M. Metlay. J. Chern. Phys. 19. 1612 (1951).
• L. Pauling. The Nature of the Chemical Bond (Cornell University Press.
Ithaca. New York. 1940). p. 340.
Partition Functions for Relating Entropy to
Disorder in the Melting of Pure Metals
JOHN F. LEE
Mechanical Engineering Department. North Carolina State College.
Raleigh. North Carolina
(Received December 11. 1952)
THE partition function due to Lennard-Jones and Devonshirel
has been modified to avoid the controversial "communal
entropy" following the suggestions of Ono.2 Application has been
made to the body-centered cubic lattice characteristic of some
liquid metal coolants such as sodium.
The mean energy of an atom at a distance r from the center of
its cell due to the nearest neighboring atom at a distance a from
the same center is
u(r) = ~ J" u{(r2+a 2-2ar cosO)!} sinOdO. (1)
The energy of interaction of two spherical atoms separated by the
distance r is
u(r) =4eo{ (ro/r) 12_ (ro/r)·}. (2)
The energy u(O) may be obtained by substituting Eq. (2) in
Eq. (1) with the limit r ...... O. Then the mean energy of the central
atom is as follows, the number of nearest neighbors Ii being 8 for a body-centered cubic lattice:
_ {(VO)4 (r)2 (VO)2 (r)2} zu(r)-u(O)=zeo -; I ~ -2 -; m ~ . (3)
Letting y= (r/a) 2 for convenience, the functions ley) and m(y) are
defined.
ley) = (1+12y+25.2 y2+12y3+y<)(1-y)-IO-1,
m(y)= (Hy)(1-y)-4-1.
The "free volume" is defined
v(O) = 27ra3g,
and
g= J.Y y!exp{-:~[(;rl(~r-2(;rmGr]}dy, (4)
where the upper limit of y= (3/47rV3')! for a body-centered cubic
lattice; or Eq. (4) may be expressed
v(O) = J exp{ -[zu(r) -u(O)J/kT}dr. (5)
The integral extends over the cell.
The partition function for a single atom is
(27rmkT)! f= ~ v(O){ -zu(0)/2kT}. (6)
The partition function for the whole assembly is·found to be too
low by a factor of eN in the limit of the lower densities, this
factor being in essence the "communal entropy." The partition
function for the whole assembly is therefore
F=fNeN. (7)
The "communal entropy" is avoided by regarding the existence
of vacant sites which the preceding development ignores. If
xi=Ni/N represents the ratio of the vacant sites to the total
number of sites then ZXi is the number of vacant sites and z(1-x.)
is the number of neighboring occupied sites. The energy at the
center of the cell is z(l-xi)u(O), and the energy of the assembly is
u=~(N -~ Xi)U(O)+~ (l-xi)ui. 2 i i
The partition function for the assembly is
(27rmkT)3NI2 J J F= ~ ~ dr,,,· drN exp(u/kT).
When we substitute from Eq. (8),
_ (27rmkT)3NI2 [-zeN -~i Xi)U(O)]
F -h2 ~ exp 2kT
X J exp{ -(l-xi)u;/kT}dri,
the generalized free volume being
V(Xi) = J exp{ -(l-x.;)ui/kT}d ri. (8)
(9)
(10)
(11)
When Xi=O, the neighboring sites are all occupied, and Eqs. (5)
and (11) are identical. If Xi= 1, the neighboring sites are all
vacant. It is clear that. some simple relationship must be found
between vex) and x. Assuming lnv(x) to be linear in x, it can be
shown that
lnv(X) =x InvI*+(l-x) lnvo*. (12)
Following the suggestion of Ono modified for a body-centered
cubic lattice,
'00*= '0(0) = 2tra2g= 27rV3'r03(v/vo)g,
'0,*='0(1) =a3/V3' =r03(V/VO).
The solution now may be obtained using the methods of Fowler
and Guggenheim."
1 Lennard-Jones and Devonshire. Proc. Roy. Soc. (London) A163. 53
(1937); A165. 1 (1938); AIM. 317 (1939); A170. 464 (1939).
'Ono. Memoirs of the Faculty of Engineering. Kyushu University. Japan
10. 190 (1947). • R. H. Fowler and E. H. Guggenheim. Statistical Thermodynamics
(Cambridge University Press. Cambridge. 1949), pp. 576-581.
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1.1721645.pdf | Cathode Effects in the Dielectric Breakdown of Liquids
J. K. Bragg, A. H. Sharbaugh, and R. W. Crowe
Citation: J. Appl. Phys. 25, 382 (1954); doi: 10.1063/1.1721645
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Downloaded 11 Mar 2013 to 131.211.208.19. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJOURNAL OF APPLIED PHYSICS VOLUME 25, NUMBER 3 MARCH, 1954
Cathode Effects in tJ?e Dielectric Breakdown of Liquids
J. K. BRAGG, A. H. SHARBAUGH, AND R. W. CROWE
General. Electric Research Laboratory, Schenectady, New York
(Received August 14, 1953)
The apparent (measured) electric strength of a liquid dielectric depends on the nature of the cathode used.
The ;field emission of electrons from the metallic cathode is assumed to be responsible for this effect; it may
provide space charge which distorts the electric field in the dielectric. The nature of the cathode determines
the range of electric fields over which the emission becomes appreciable.
In t~is paper we ~iscuss in detail the emission of charge and the formation of space charge. Because of
theoretical and expenmental uncertainties connected with supposedly uniform metallic surfaces, an electrode
consisting of an electrolyte solution is also considered. Although the experimental results demonstrate that
ions are emitted from such cathodes under the influence of strong electric fields, the role in providing space
charge is similar to that of metal electrodes which emit electrons.
For the former electrode, the considerations regarding space-charge formation predict a definite de
pendence of apparent electric strength on electrolyte concentration and provide a way of testing the basis
of the theory. Certain experiments reported here confirm the predictions of the theory.
I. INTRODUCTION
THE influence of the nature of the cathode metal
upon the breakdown of dielectrics has been ob
served by many investigators. von Hippel and Algerl
:first attributed this effect, as observed in the study of
alkali-halide crystals, to field emission of electrons from
the cathode. Evidence that the origin of the effect in
liquids is also field emission was found by Salvage,2 who
succeeded in showing a rough correlation between his
experimental results and the vacuum work function of
the metal used as cathode. Because of the sensitivity
of the electron-emission characteristics of a metal to
the physical and chemical condition of its surface
however, such correlations are at best qualitative, and
may be difficult to reproduce.
We have tried to put the suggestion of von Hippel
in a quantitative form, and have designed a new kind
of breakdown experiment to give a reliable test of the
resulting theory. The experimen1 involves the use of a
cathode consisting of an aqueous solution of electrolyte;
the emission of ions from this cathode is a reproducible
phenomenon whose dependence upon electrolyte con
centration can be predicted. The theory depends on a
distortion of the electric field which may be caused by
the emission of charge from the cathode into the
dielectric.
For example, if an electric field of 106 v/cm is im
pressed on a dielectric, field emission of electrons from
a metal cathode may occur as shown by LePage and
DuBridge.3 This emission may provide a considerable
density of negative charge in the gap between elec
trodes. The amount of such charge which can be trans
ported across the gap to the anode is limited by the
properties of the dielectric (the mobility it allows the
charges, and the dielectric constant) and by the geom
etry of the gap. The current that flows is, therefore, dependent on cathode properties at low fields, but when
the field is high enough to cause strong emission, the
current becomes independent of the nature of the
cathode, and is prescribed by the properties of the gap
through which the charge must be transported.
In the latter case, a nonuniform distribution of space
charge arises, and the electric field in the gap becomes
inhomogeneous. It is to this feature that the cathode
effects are ascribed; that is, under these "space-charge
limited" conditions, the electric field in a region of the
dielectric is enhanced to a magnitude considerably
greater than the average (measured) field. The value of
the average field at which this occurs is however , ,
dependent on the nature (emission characteristics) of
the cathode.
The connection of this with breakdown strength
measurements must be made with the help of a some
what arbitrary assumption. We first suppose that the
intrinsic electric strength of a dielectric is a meaningful
property. We define it in terms of the magnitude of the
homogeneous electric field, existing in a region of the
dielectric, necessary to disrupt this region or establish
a conducting path across it, initiating processes from
outside the region being excluded. The region con
sidered must be large enough so that the property so
defined is independent of the size of the region.
T~e intrinsic electric strength, defined in this way,
obvlOusly depends only on the nature of the dielectric' . ' what IS usually measured does not. Its relation to
cathode effects is the subject of our basic assumption.
We assume that a dielectric will break down whenever
the average electric field over a region of dimension 0
in the direction of the field becomes equal to the in
trinsic electric strength, the magnitude of 0 being just
that for which the homogeneous electric field necessary
for disruption of the region becomes independent of 0.
It should be noticed that this assumption is inde
pendent of the nature of the breakdown process, and
~ A. von Hippel and R. S. Alger, Phys. Rev. 76, 127 (1949). 11 hi'
3 B. Salvage, Proc. Inst. Elec. Engrs. 98, 227 (1951). ate conc USI0ns we draw from it will apply whatever
w. R. LePage and L. A. DuBridge, Phys. Rev. 58, 61 (1940). the breakdown mechanism. However, the understand-
382
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ing of breakdown mechanisms and the techniques of
measurement are not yet sufficiently advanced to make
this hypothesis operationally useful. We will conse
quently use a somewhat indefinite idea; we ask simply
that the electric field become equal to the intrinsic
strength somewhere in the dielectric.
In the remainder of this paper, we will develop the
consequences of the assumptions set forth above and
report the extent to which these ideas are supported by
experiment.
II. MECHANISMS OF CHARGE EMISSION
A. Field Emission
Evidence that the high field-emission current from
metal cathodes into dielectrics is essentially field emis
sion has been presented by LePage and DuBridge.3 They
measured the emission current into toluene as a func
tion of field strength and temperature. Figure 1 shows
some of their results, plots of the logarithm of the cur
rent against reciprocal temperature for various field
strengths. It will be observed that the magnitudes of
the slopes decrease as the field strength is increased.
This suggests the gradual transition from Schottky
emission, which has a pronounced exponential tempera
ture dependence, to field emission, which is independent
of temperature. The electric fields with which we deal
are greater than any listed in this diagram.
Field emission into the vacuum is described fairly
well by the Fowler-Nordheim equation,4
j= f(x)P exp( -bxi/E). (1)
Here b is a constant, X is the vacuum work function,
and E is the electric field. The quantity bx! is of the
order of 108 if E is expressed in volts per centimeter, so
that this expression describes a current which increases
very strongly in the region around 107 v/cm. f(x) is
usually of such a magnitude that appreciable emission
appears at about 107 v/cm. The applicability of the
law depends on the presence of uniform surfaces; in
general, the presence of points and patches must be
taken into account.
If the emission is into a dielectric, the situation is not
so clear. If the dielectric may be regarded as structure
less, one may derive the equation3
j= f*(X*)P exp(-b*x*!/E). (2)
The constants 1* and b* now involve the dielectric
constant E of the dielectric, and x* is the work function
for the cathode-dielectric system, i.e., the difference
between the Fermi level of electrons in the cathode
and the electrostatic potential of an electron removed
to infinity in the dielectric. It can be shown that
X*<X/E, but beyond this, x* can be specified only by
measurement.
• R. H. Fowler and L. Nordheim, Proc. Roy. Soc. (London)
A1l9, 173 (1928). N 10-12 ,.
o
....
en
'" 0:
'" .. ,.
~ 10-14 ..
in
~
...
'" '" 0: '5 10-16
OJ 25000 VOLTs/eM
~ ~ 400 VOLTS/ eM
FIG. 1. Currents in toluene at various field strengths as
function of temperature (after LePage and DuBridge).
In actual practice, the equation just given probably
does not apply at all. If the dielectric is crystalline so
that its electron states are distributed in bands, then a
different kind of barrier to emission arises, whose shape
depends on the concentration and nature of impurities
in the crystal.s Something similar may occur in amor
phous solids and liquids; furthermore, the mean free
path may be so short in these substances that the whole
basis for the derivation of the electron emission laws
is invalidated.
Nevertheless, the field-emission current should de
pend on the field in an exponential way, and the
equation
j= a exp( -b/ E),
with empirical constants, should be satisfactory.
B. Ion Emission
Experimental evidence to be described in this paper
indicates that, under proper conditions, an emission
current of ions may be obtained. The experiments in
volve the use of aqueous electrolyte solutions as
cathodes in the study of the breakdown of liquids
immiscible with water. We wish to discuss here the
nature of the emission, dealing mainly with a simple
picture of the process which has to be modified to agree
with experiment, but which must be understood in order
to interpret the actual results.
In Fig. 2 we have sketched the "motive" (electro
static potential plus an effective potential caused by
image forces) of a negative charge in a system consisting
of aqueous electrolyte cathode, a dielectric liquid, and
a metal anode, in the presence of an electric field. The
slight decrease in the motive just to the left of the
electrolyte-dielectric interface is due to the distortion
of the ionic distribution by the electric field; in contrast
to what occurs in metallic conductors, the field must
extend an appreciable distance into the cathode. The
• See, for example, H. C. Torrey and C. A. Whitmer, Crystal
Rectifiers (McGraw-Hill Book Company, Inc., New York, 1948).
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-''/10-+---...11
+ +
+
+ +
+ --- ....... -0
ELECTROLYTE DIELECTRIC ANOOE
o Xm x~ A
MOTIVE OF NEGATIVE CHARGE IN ELECTROLYT E
DIELECTRIC SYSTEM
FIG. 2. Motive of negative charge in electrolyte
dielectric system.
rise to the right of the interface is due to an assumed
increase in energy of the ions as they move from the
water layer to the dielectric layer, and the shape of the
curve in this region is prescribed by the image force
between the ion and the electrolyte.
Calculation shows that for as Iowa concentration of
ions as 1o-4N, the drop in potential in the electrolyte
is less than 0.25 v out of 10 kv impressed across 0.01
cm of dielectric. This drop will be ignored henceforth.
The difference in energy of a univalent ion in water
and in benzene may be of the order of 2 ev. To illustrate,
we adopt the very simple view that the energy of an
ion in a dielectric liquid is just that required tQ charge
the ion in the medium6
(3)
(e= ionic charge, E= dielectric constant, ro= ionic ra
dius) ; then the energy difference is
exo-~(~-~)~~. E2»EI. (4)
-2ro EI E2 ~ 2Elro'
Although this is obviously very much oversimplified,
we may obtain a certain degree of consistency by using
ionic radii as determined from heats of solution with
the aid of this formula.6 This procedure is still not
quantitative, because we have to use, for the ion in the
dielectric, the radius determined from aqueous solutions.
For a typical ion, CI-(ro= 2.13A), we find exo= 1.4 ev
for the water-benzene transfer.
There is the possibility that the ion may enter the
dielectric still associated with several water molecules.
If the resulting droplet has a radius r, the energy differ
ence for the transfer is
e2
ex = -+4?rr20',
2Elr (5)
where 0' is the water-benzene interfacial free energy per
unit area. The value of r which makes this expression
a minimum is
rt= (e2/167rEO')t::::::3.8A,
6 M. Born, Z. Physik I, 45 (1920). a number about double the ionic radius. Despite the
uncertainty in using an interfacial free energy for a
droplet so small, this makes it probable that a few
molecules of water are carried alvng with the ion. Ex
perimental evidence that this is so will be presented
later. The energy difference thus obtained is about
ext = 1.2 ev.
The quantity exo (or ext) plays the role of a work
function for the ion. The motive for the ion is of the
same form as that for an electron in a dielectric opposite
a metal cathode, as evaluated by LePage and Du
Bridge.3 The height of the maximum in the motive
above the value in the electrolyte is
eiE!
eXm=ext---.
Ei (6)
The concentration of ions at Xm (see Fig. 2), provided
they are supplied sufficiently rapidly from the cath
ode, is
(7)
here no is the concentration of ions in the electrolyte.
This is similar to the Schottky correction for the effect
of a finite collecting field on the barrier to thermionic
emission into a vacuum.
A difference arises when we try to calculate the emis
sion current. In the present case the ions cross the
barrier by diffusion. There is no way of taking this into
account precisely, so an approximate assumption which
is exceedingly simple will be made. We suppose the ions
move across the barrier with the drift velocity J.lE they
would have in the presence of the field E alone. Then
we have for the current density
j= -J.leEno exp( -ext/kT) exp(eiEl).
EikT (8)
This approximation is exact if applied to the sim
plified problem of the barrier shown in Fig. 3, which
can be solved for the motion of ions in the combined
electric and concentration gradients. The result is that
the concentration at the top of the barrier, multiplied
by the drift velocity of the charges in the electric field
alone, gives the current across the barrier.
ABSENCE OF FIELD
FIG. 3. Potential barrier for simplified emission problem.
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With sufficient labor one can obtain arbitrarily ac
curate numerical solutions for a barrier of any com
plexity, but the qualitative conclusion we want to
establish here does not demand such refinements, nor
does our physical knowledge of the situation justify
them.
Equation (8) predicts a negligible emission current
at any reasonable field strength. The reason is that the
Schottky correction, eiEl/ elkT, lowers the barrier by
only 0.25 v for a field of 106 v/cm. This implies that the
barrier layer is about 25A thick; a barrier layer of
100A, on the other hand, gives a correction to ext of
1 v and an emission current of the order of 0.5 amp.
We have obtained evidence of emission currents of
almost this order of magnitude, but only after one or
more preparatory electric pulses. Apparently the func
tion of the preparatory pulses is to blur the Schottky
layer, thus providing enhanced emission. We will
simply make the obvious assumption that such an
emission current depends exponentially on the elec
tric field.
III. SPACE-CHARGE EQUATIONS
A. Simple Space-Charge Equations
The charge and field distributions in the gap (as
sumed to consist of two infinite plane conductors
separated by a distance a centimeters) are governed by
the following relationships:
e(d2cp/ dx2) = -41rp;
j=-pp,(dcp/dx), O~x~a. (9)
In these equations e is the dielectric constant, cp the
electrostatic potential, p, the charge mobility, j the
current density, p the charge density, and x the coordi
nate (which varies from ° to a across the gap). We
consider only negative charge, so that p, and pare
negative. For the moment we assume that the mobility
is independent of the electric field. The consequences
of any failure of this assumption will be assessed
shortly.
The problem is made definite by the statement of
certain boundary conditions:
cp=o at x=O,
cp= cpo at x= a, (10)
-(dcp/dx)=Ee at x=o.
The first two conditions always obtain. The simplest
assumption about the third is that Ee=O, and the solu
tions corresponding to this boundary condition are the
following :7
cp= cpox!/ ai,
p= (3/321l")(ecpo/a ix1),
j= (geJ..l,cp02/321l"a3). (11)
7 See, for example, J. D. Cobine, Gaseous Conductors (McGraw
Hill Book Company, Inc., New York, 1941), p. 128. 'Yo
1J
'Ya o
2.25
1.50 0.20
2.14
1.48 TABLE I.
0.40
1.84
1.41 0.60
1.35
1.31 0.80
0.70
1.16 1.00
0.00
1.00
These are analogs of the Langmuir-Childs equations,
valid, however, for liquids, solids, and high-pressure
gaseous dielectrics. A consequence of the first of these
is that the electric field at the anode, Ea, is the average
field Eo= cpo/a increased by a factor 3/2.
B. The Self-Consistent Solution
One of the results of the simple model is that a finite
current flows through the gap. The solution is thus
obviously not self-consistent, since the assumption
Ee=O implies that the field-induced emission current is
zero. The boundary condition must be so adjusted that
the current flowing in the gap, as derived from the
space-charge equations, is equal to the emission current
of the cathode given by its field-emission law.
We must, therefore, solve Eqs. (9) under the general
boundary condition -(dcp/dx)_o=Ee. We will then
regard Ee as a function of j in the solutions. The inte
grations, together with appropriate determination of the
constants arising, give now
(12)
This and succeeding results reduce to the simple space
charge equations if we set Ee=O.
The current is derived by putting cp equal to cpo when
x equals a. The implicit equation for j is then
(13)
In terms of convenient dimensionless quantities 'Yo
= EclEo and 1]= 81raj/eJ..l,E o2, this equation becomes
(14)
Table I gives solutions of this cubic equation for 1], for
various values of 'Ye. In addition to the reduced current
1], the table lists values of the reduced anode field
'Ya=Ea/Eo obtained by differentiating Eq. (12) with
respect to x and then setting x equal to a. As previously,
if the cathode field is made to be zero, the anode field
is three-halves of the average field. On the other hand,
if 'Ye is 1.0, then 'Ya is 1.0 also.
Equation (14) gives a relation between the reduced
current and the reduced cathode field which arises from
consideration of the transport of charge through the
dielectric. There is another equation connecting them,
namely the cathode emission law
(15)
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For a given average field Eo, Eqs. (14) and (15) can be
solved simultaneously to give numerical values of the
reduced current and cathode field, and hence the re
duced anode field.
In Fig. 4 we have drawn a typical cathode-emission
characteristic, and the space-charge-limited current as
a function of field strength, to illustrate the meaning
of these results. The assumed cathode characteristic
has the form given by Eq. (15) and is plotted against
Eo. In the region Eo<E h emission is so feeble that
'Y.~ 1, and the field is essentially undistorted. Note that
our self-consistent solution follows the cathode charac
teristic in this region. At some field El the reduced
emission current has grown to, say, 0.2, and Table I
shows that appreciable deviations of 'Ya and 'Y. occur.
As Eo is further increased, the strong tendency of the
emission current to increase rapidly forces the system
over into space-charge-limited conditions under which
the reduced current in the gap is independent of the
field (the actual current varies as the square of Eo).
Comparison of these results with those of the pre
ceding section shows that the simple treatment of the
problem gives values of the anode field which are
asymptotically correct at low and high fields. Conse
quently, the discussion of the breakdown process to be
given later will be developed on the basis of the simple
space-charge laws.
C. Dependence of the Mobility on the
Electric Field
In the discussion just given we have assumed that
the mobility of the neg!l.tive charges is independent of
the electric field. This assumption is undoubtedly de
fective in the field strength region treated here. Never
theless, there is no information about the field de
pendence of electron drift velocities in liquids under
0 I,' '" 0 ...
It' 1,0
"' Q
i 0,5 ..
Q
"' U 0 a
"' II:
o u
~ I
~ I
~ I
~ I
j!; I ______ ~_I ____ -_____ _
I
I
I
I
I
E,
AVERAGE ELECTRIC FIELD. Eo
(ARBITRARY UNITS)
FIG. 4. 10 high fields. Shockley8 has used the work of Seitz9 on the
mobility of electrons in crystalline, nonpolar insulators
to show that in this case the mobility ought to decrease
with electric field at high fields, eventually varying as
B-1. On the other hand, it is possible that an electron
in a nonpolar liquid may become trapped or "solvated,"
and thus behave like a molecular ion, in which case the
mobility would be expected to increase with field at
high fields (see below). It can be proved that if the
mobility varies as g-i, the anode field under space
charge-limited conditions is 5/3 times the average field.
For ionic charge carriers the matter is somewhat
clearer. It is certain that the drift velocity of such
charges will increase strongly with field in the high field
region. Not so certain is the functional form, but a
crude approximation to the current, often used, is
j=ap sinh «(3d",/dx) ; a, (3)0. (16)
This current deviates from an ohmic law if (3(d",/dx) is
not small compared to unity. If ",o/a (the average field)
is large compared to 1/(3, it can be shown that the field
in the gap is essentially undistorted by space charge.
Because of uncertainties in both emission and con
duction laws, it is not worth while to develop inter
mediate cases. We may conclude, however, that in
practice the extent of distortion of the electric field in
the gap due to ionic space charge is expressed by a
factor lying between 1 and 1.5, multiplying the average
field. On the other hand, an electronic space charge
may produce a distortion of field expressed by a factor
lying between 1 and 2.
IV. DEPENDENCE OF MEASURED ELECTRIC
STRENGTH ON CATHODE CHARACTERISTICS
In the previous sections, we have described possible
kinds of field-dependent charge emission, and the space
charge effects which they may produce. We have still
to show the connection between these and measure
ments of electric strength. In the detailed discussion to
follow, we will use principally the results of the simple
space-charge calculation. The modification that may be
made by a more refined treatment will be indicated
afterward.
In Fig. 5 we have plotted some current densities
versus electric field strength. The solid lines represent
the emission characteristics of a hypothetical series of
cathodes as a function of the field strength at the cath
ode surfaces; the emission described by these curves is
supposed to increase exponentially with electric field.
The dotted curve is the space-charge-limited current
plotted against the average field in the gap.
Consider first the cathode curve marked (I). If such
a cathode is used, an increase in the applied electric
field will bring about an increase in the emission cu.rrent,
following along the solid curve until it intersects the
8 W. Shockley, Phys. Rev. 82, 330 (1951).
t F. Seitz, Phys. Rev. 76, 1376 (1949).
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dotted one. Beyond this point, the potential emission
current of the cathode is larger than can be collected.
The actual current, therefore, follows the dotted line
thenceforth; the emission is "space-charge-limited,"
and the field in the gap is no longer uniform. The
simple space-charge law states that, in this situation,
the maximum field in the gap (that at the anode) is
three-halves the average field. Thus, when we have
progressed to the right in Fig. 5 until the average field
is two-thirds the intrinsic strength, breakdown will
occur. This is true of any cathode whose characteristic
intersects the dotted curve at a field less than (2/3)E i,
where Ei is the intrinsic strength of the dielectric.
Next we consider the cathode curve marked (III).
The emission current from this cathode, at the field Ei,
is still below the space-charge-limited current. Conse
quently, the field on the cathode and throughout the
gap is Ei, and breakdown occurs at the average field Ei.
Finally, when the curve (II) intersects the dotted
line and the field becomes distorted, a field greater
than E. is produced at the anode, so that breakdown
occurs when the average field corresponds to the point
of intersection. We can describe the entire situation in
FIG. 5.
the following approximate way: any cathode which
begins to emit strongly at fields below (2/3)E i gives
breakdown at (2/3)E i. A cathode which begins to emit
at a field between (2/3)E i and E. gives breakdown at
that field, while a cathode whose emission is negligible
below E .. gives breakdown at Ei• The resulting plot of
breakdown strength versus cathode nature (effective
work function of a metal increasing to the left, or the
concentration of an aqueous electrolyte cathode in
creasing to the right) is shown in Fig. 6, and is charac
terized by two cathode-independent regions (the upper
and lower flat portions) and a transition region.
The equation of the transition curve is obtained by
equating emission current and space-charge-limited
current; if we treat the latter as constant compared to
the rapidly changing emission current, we obtain, in
the case of ion emission,
no exp(aE9) = constant,
or
E9= l/a(constant-Iog no). (17)
Here (J is a constant near unity (if the emission were of
the Schottky type, it would be one-half). Thus, the
breakdown field should vary approximately linearly I ]I m
GATHODE CHAIIA()TERI8TI()
FIG. 6.
with the logarithm of the concentration in the inter
mediate portion of the graph of Fig. 6.
One modification of the results of this section, intro
duced by more detailed space-charge considerations,
should be mentioned. The relation which the upper
and lower flat portions of the curve of Fig. 6 bear to
each other depends on the law relating the drift ve
locity of the charge carrier to the electric field. In
Sec. III, we saw that for electrons this relationship
is not understood, so that no predictions can be made
about the ratio of the two flat portions if the cathodes
used are metals.
On the other hand, if the cathode emits ions, we can
say with confidence that the drift velocity varies at
least as strongly as the first power of the field. So,
according to the results of Sec. III, the ratio of the
fields given by the upper and lower flat portions must
lie between one and three-halves.
V. EXPERIMENTAL METHODS
In practice, we are usually concerned with metallic
cathodes, but any detailed theory of their effects is
difficult to develop and impossible to verify experi
mentally at present. We have developed a theory for
the ion-emitting cathode because it runs parallel to
that for an electron emitter, and at the same time, is
susceptible to a detailed experimental study. We will
now report the results of our experiments with such
ion-emitting cathodes.
A. Experimental Techniques
The experiments were carried out with a rectangular
pulse generator which delivers single, well-defined,
rectangular voltage pulses of variable amplitude and
duration. Amplitudes up to 15 kv may be obtained,
with durations ranging from 0.25 to 54 ILSec. The use of
such pulses should essentially eliminate erroneous re
sults due to local, prebreakdown heating of the spark
gap. This effect, giving rise to what is known as "thermal
breakdown," has often been observed in systems to
which de and ac voltages were applied. This is obviously
undesirable if one is attempting to study the funda
mental mechanism of breakdown.
The construction details of the breakdown cell are
shown in Fig. 7. The general design is similar to that
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FIG. 7. Assembly of breakdown cell.
used by Bahre/o except that one electrode is made
movable through the use of a bellows and micrometer
drive arrangement. The cell is equipped with optical
windows to make it possible to monitor the breakdown
during an experiment, and to adjust the gap spacing
by optical means. The ball and socket arrangement for
the upper electrode is provided so that a single solid
electrode may be used for a number of experiments, a
fresh part of its surface being used for each one. For
the present series of experiments, the cell was used in
the vertical position, and a glass cup filled with aqueous
electrolyte substituted for the lower electrode.
Benzene was chosen as the dielectric for the measure
ments because of its ready availability in a relatively
pure condition, and because of its immiscibility with
water. The material used was Baker and Adamson (cp)
thiophene-free benzene; no attempt was made to purify
it further. The benzene was filtered through an ultrafine
(0.9-1.4j.1 pore size) fritted glass filter to remove solid
particles.
After assembly of the dielectric-electrode system, the
travelling microscope (32X) was focused on the gap
through the optical windows of the cell (Fig. 8). The
gap was illuminated with the microscope illuminator.
The electrode spacing of 0.0084±0.0001 cm was set
using the interference colors which appeared when the
electrode separation was very small; we used a particu
lar red color which corresponded to a separation of
10 W. Bahre, Arch. Elektrotech. 31, 141 (1937). 0.0008±0.0001 cm. From this point, the spacing was
increased by 0.0076 cm, as read from the micrometer,
to its final value of 0.0084 cm.
Two techniques were used for making the electric
strength measurements. The first, hereafter referred to
as the "many-pulse" technique, consisted of the appli
cation of single, constant-length, rectangular pulses of
gradually increasing amplitude, until disruption of the
dielectric occurred. In the second method, only one
pulse, near breakdown, was applied to the cathode
dielectric-anode system. If breakdown did not occur, a
pulse of slightly higher voltage was applied to the next,
completely new system. If breakdown resulted, a pulse
of slightly lower voltage was applied to the next system.
In the present investigation, this method served to
demonstrate an effect to be discussed in later sections.
B. Correction for Deformation of the
Electrode Surface
. When a voltage is applied to a breakdown gap having
a liquid electrode, it causes a deformation of the elec
trode surface whose extent depends on magnitude and
duration of the voltage. A method of correction for this
deformation has been described;l1 it involves the use of
voltage pulses of several different lengths. For systems
involving the mercury cathode, it was found that the
decrease in electrode separation resulting from such
deformation is linear with pulse length over a consider
able range of pulse lengths in excess of one microsecond,
giving rise to a linear dependence of electric strength
on pulse length in this region. A linear extrapolation of
the electric strength to zero time gives a value approxi
mately corrected for electrode deformation.
The apparent electric strength of benzene was meas
ured as a function of pulse length, using aqueous lithium
chloride solutions as cathodes; the results are given in
Fig. 9. The curves are not far from linear for pulse
lengths from 0.7 to 4.7 microseconds, but their slopes
indicate a more pronounced deformation than was
realized with the mercury cathode. (The rapid rise
below 0.7 j.lsec is a property of the dielectric itself, not
of importance here.) Since the nature of the deformation
in the pulse-length region below one microsecond is not
known, a linear extrapolation of the curve to zero time
may yield a result which is somewhat in error, especially
because electrode distortion is large. It should also be
mentioned that the electric strengths are those of
benzene saturated with water, as will always be the case
when an aqueous solution is used as an electrode.
~STEEL ELECTRODE I MICROSCOPE
GAP I /ILLUMINATDR I \.V *
TRAVELlNG.-T I
MICROSCOPE LIQUID OPTICAL
ELECTRODE WINDOWS
FIG. 8. Assemhly for setting gap.
11 Sharbaugh, Crowe, and Bragg, J. Appl. Phys. 24,814 (1953).
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VI. EXPERIMENTAL RESULTS AND DISCUSSION
A. The Many-Pulse Technique
The principal purpose of the present study is the
verification of the theory of cathode effects which we
have described. According to the theory, a plot of the
electric strength versus the logarithm of the ion con
centration should consist of three parts: (1) a level por
tion at high concentrations, (2) a linear increase over
an intermediate concentration range, and (3) another
level portion at low concentrations.
This effect has been demonstrated using lithium
chloride solutions as cathodes (the high solubility of
lithium chloride in water makes it possible to cover a
wide range of concentrations). The measurements were
made with voltage pulses of length 1.8 microseconds,
and corrected to zero time as described in the preceding
section. The data in Fig. 9 show that the value obtained
in this manner exceeds by about 15 percent that meas
ured with a pulse length of 1.8 microseconds.
The electric strengths measured in this study are
plotted in Fig. 10 against the logarithm of the ionic
concentration of the cathode. The hollow circles give
the data obtained by the many-pulse technique. The
two level parts of the curve, together with the connect
ing linear portion, are clearly evident from these data.
The significance of the single-pulse results, represented
by the filled circles in Fig. 10, will be discussed in a
later section.
If our interpretation of the results in Fig. 10 is cor
rect, the value of the field strength corresponding to the
upper flat portion of the curve has a special significance.
It is the electric strength measured in the absence of
appreciable space charges, and corresponds to the
intrinsic strength of the dielectric. Unfortunately, for
reasons which we have already given, the result has no
absolute significance; this will be the case until an
extrapolation procedure free from arbitrary assumptions
is developed. Nevertheless, the relative values reported
~ u
~
~
:t: ....
'" z .... 0: .... en
0
0: .... 0 .... ..J .... b ~ ~
1.5 \ 0.01 N LiCI STEEL \ • 1.0 N LiCI STEEL " lb BENZENE, .00B4 CM GAP
\ \
,&-_--~ 1.0
~..... ---... ---4 --..... -...... -~ -....... .... ... -------0.5
0~0--~----~--~3~--+4--~~~
PULSE LENGTH (,. SEC.)
FIG. 9. Dependence of the electric strength of
benzene upon pulse length. 1.5
_ 1.4
~
!:! > ! 13
:r t;
~ 1.2
>., • SINGLE PULSE RESULTS
o MULTI PULSE RESULTS
CATHODE, li CI (Aq)
ANODE, STEEL
BENZENE
.0084 CM GAP
------.-~--..- ...
<.> VALUES OBTAINED BY EXTRAPOLATING
~ I I ~~~~~~I\~T~i:;T~M~S ~~~~E ~~~N;rH
~ REPRESENTS AVERAGE OF AT LEAST
'" SIX RUNS; STANDARD DEVIATIONS
1.0 ALWAYS LESS THAN 4 %
'6
LOG,. CATHODE NORMAL! TY
FIG. 10. Dependence of the corrected electric strength of
benzene upon cathode normality.
here are quite precise, and serve to support the basic
ideas involved in the theory of cathode effects.
B. The Reversal of Electrode Polarity
The field emission of electrons from negative ions in
aqueous solution has been observed under certain cir
cumstances.12 It is, therefore, necessary to identify the
charge carriers causing the effect reported in the previ
ous section.
The mechanism by which charge emission exerts its
influence upon the measured electric strength of a
dielectric applies equally well whether the emission is of
negative charges from the cathode or of positive charges
from the anode. It has been shown repeatedly that the
measurement of electric strengths of liquids and solids
is not influenced by the nature of the anode when metal
electrodes are used. While electrons may be emitted
from a metallic cathode, apparently positive "holes"
are not drawn from a metallic anode in appreciable
numbers by the electric field (the emission of a hole
into a liquid dielectric is the extraction of an electron
from a molecule of the liquid by the anode, leaving a
positive ion).
We have, therefore, carried out experiments in which
the aqueous electrolyte solution was used as the anode.
The results are compared in Fig. 11 with analogous data
obtained with the electrolyte as cathode. Except for an
apparent small translation of the anode curve, the
concentration effect is observed as before. Because of the
close correspondence between the two curves, we must
assume that emission from the electrolyte surface is
possible in both cases. The symmetry of the behavior
itself indicates that the charge carrier in both cases is
an ion. This conclusion may be supported by simple
energy considerations .
Consider the over-all process,
Cl-(Aq)->CI(Aq)+e(benzene).
This may be synthesized from a series of steps, the first
of which is the removal of the chloride ion to the
12 A. Guntherschulze and H. Betz, Elektrolyt-kondensatoren (M.
Krayn, Berlin, 1937).
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1.8
::E u
~
~ 1.6
J: l-
e.!>
Z
UJ 1.4 II:: l-I/)
u
ii: 1.2 I-u
UJ
..J
UJ CATHODE ANODE
o STEEL LiCI (Aq)
• LiCI (Aq) STEEL
BENZENE, 0.25 X 10-6 SEC. PULSE
o .0084 eM GAP
LOG,o ELECTRODE NORMALITY
FIG. 11. The effect of reversal of electrode polarity upon
the electric strength of benzene.
vacuum:
Cl-(Aq)--tCI-(v)
CI-(v)--tCI (v)+e(v)
CI (v)--tCI (Aq)
e (V )--te (benzene) 3.0 electron volts,
3.8 electron volts,
- u electron volts,
.-W electron volts.
The positive terms constitute work done on the system.
The total energy change is
(6.8-W -u) electron volts.
The quantity u ought to be negligible (if the process is
sufficiently irreversible, it will not appear at all). To
assess the result, one still ought to have some informa
tion about the quantity W j unfortunately, there is none.
However, it is sufficient to observe that the predicted
electron emission will be less than that from a metal
whose vacuum work function is 6.8 volts (provided no
specific interaction occurs at the metal-benzene inter
face). This is because the ion concentration is always
less than the concentration of quasi-free electrons in a
metal, and because the water-benzene interface has no
points such as enhance emission from a metal. The
emission of charge actually observed is orders of mag
nitude greater than that from metals whose vacuum
work functions range from 4 to 5 electron volts.
C. The Nature of the Ion
In a separate series of experiments we have investi
gated the behavior of cathodes containing various nega
tive ions. These experiments were performed with the
0.25-microsecond pulse, using the many-pulse tech
nique, and the results (not corrected for deformation)
are reported in Fig. 12. It is clear that the nature of
the negative ion has little or no influence on the effect
studied here.
There are two conclusions which may be drawn froll!
this result. As an observation supplementary to the
discussion of the preceding section, we may say that if
electron emission were to occur, the effect of the widely
different electron ionization energies of the hydroxyl
ion and chloride ion should have been observed (this difference is about 2 electron volts for the gaseous ions).
Secondly, as we have had occasion to comment, it is
likely that several water molecules accompany the ion
in its transfer to the benzene. This will have the result
that the effective ion sizes are about the same, a con
clusion in harmony with the experimental results.
D. The Single-Pulse Technique
When aqueous electrolyte cathodes of cqncentration
lower than 10-2 normal were used in the breakdown
studies of benzene, the application of pulses near
breakdown invariably resulted in the appearance of a
cloud of small particles (1 to 5 microns in diameter) in
the gap. Since these particles could not easily be re
moved between pulses, erroneous breakdown values
often resulted. It was, therefore, necessary to use the
single-pulse technique with systems involving these low
cathode concentrations.
An extension of this technique to systems in which
cathodes of higher concentration were used has demon
strated another interesting effect. The results, plotted
in Fig. 10, show that the dependence of the electric
strength on concentration has disappeared. The single
pulse strength obtained with any cathode whose con
centration is greater than 10-3 normal is equal to the
value obtained from the upper flat portion of the many
pulse curve. Evidently a single pulse is incapable of
inducing ion emission. The function of the pulses pre
ceding breakdown, when the many-pulse technique is
used, is then to "blur" the Schottky barrier, which is
initially too narrow to allow appreciable emission. Al
though it is impossible to say exactly what the pre
breakdown pulses do, it is probable that they create
something like an emulsified layer about 100A thick at
the interface. In the presence of the electric field, the
barrier presented by this layer may be lowered as much
as one volt, making intense emission possible. In fact,
a group of experiments involving a "double-pulse" tech
nique showed that a single pulse, preceding the break
down pulse itself, was sufficient to restore the concen
tration effect almost completely.
The portion of the single-pulse curve in Fig. 10 which
lies below 10-3 normal in concentration shows a second
increase in apparent electric strength. This is not a
property of the breakdown system at all, but is caused
by the effect of the high electrical resistance of these
solutions on the transmission of the short pulses to the
breakdown gap.
The appearance of the small, solid particles mentioned
earlier deserves further comment. They appeared during
pulses near breakdown when very dilute cathodes were
used. Under such conditions, the electric field in the gap
is homogeneous, so that the entire dielectric, rather
than a small region, is subjected to a high field. It is
possible that the particles are bits of carbon produced
by prebreakdown discharges, and that they show up
under the present circumstances because these dis-
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charges occur throughout the dielectric rather than in a
small region near the anode.
E. The Motion of the Ions
Inasmuch as the duration of the electric field is very
short in these experiments, it serious question arises as
to whether the ions can really move as rapidly as is
required for the validity of our t?eory. This is to sa~,
the ions must be capable of movmg across an appreCl
able fraction of the gap in a time of the order of a
microsecond.
We have no direct knowledge of the mobility in
benzene of the ions with which we deal, but we can infer
a reasonable value from some measurements which have
been made of the transport of ions of another sort in
liquid hydrocarbons.13 When. a liq~id h.ydrocarb~n .is
bombarded with x-rays, a vanety of Ions IS formed mIt.
The mobilities of these ions may be obta~ed from a
measurement of the time interval between their pro
duction and their collection on an electrode spaced a
known distance away. The ions themselves are both
positive and negative, and presumably consist of singly
or multiply ionized hydrocarbon molecules or frag
ments thereof. We have, of course, no information as
to the relative sizes of these ions and the ions we use in
our experiments. It is probably safe to infer that the
mobility in benzene of the inorganic ions plus attached
water molecules is comparable with the mobility of
ions formed from fairly large hydrocarbon molecules,
e.g., n hexane. In this case, the mobilities of the several
kinds of carriers were grouped about a value of 5X 10--4
cm2/v sec.IS
Using this mobility, we find that an ion may travel
5X 10--4 centimeter in a microsecond when the electric
field is 106 v/cm. This distance is about 6 percent of the
gap spacing used in our experiments, whereas we require
a travel of at least one-third of the gap in order to
provide the required distortion of the electric field.
There are two factors, ignored so far, which alleviate
this situation. The first is that more than one pulse is
necessary to provide the concentration effect. Second
and more important, in this estimate we have used a
mobility measured at low electric fields. As pointed out
earlier, the mobility of ions is expected to increase with
electric field at high fields. Evidence that this is so is
131. Adamczewski, Ann. phys. 8,309 (1937). i"
Co)
~ !.
:z: t; z
lIJ
0::
Iii
Co)
a: I-
Co)
lIJ
...J
lIJ 1.8 CATHODE
o HCI
x NoOH
1.6 • LiCI
STEEL ANODE
BENZENE, 0.25 X 106 SEC. PULSE
1.4 .0084 CM GAP
1.2
1.~1!;3--~:-----~--+--~-----::!
LOG10 CATHODE NORMALITY
FIG. 12. The effect of the nature of the ion upon
the electric strength of benzene.
provided by the fact that the ratio of the. upper and
lower flat portions of the curve of electnc strength
versus cathode concentration is 1.25 instead of the 1.5
ratio which would result from ohmic behavior. This has
a secondary effect which also lessens the requirement of
rapid motion of the ions. There is less field distortion;
consequently, the distance the ions must travel in order
to produce the required distortion is somewhat less.
Altogether, it seems that the requirements put on the
rapidity of ion motion are not excessive.
To support this picture we have substituted a heavy
mineral oil for benzene and repeated many of the experi
ments described in the preceding sections. Stokes's
law, which has an approximate validity when applied
to the motion of ions through a viscous medium under
the influence of an electric field, shows that the mobility
of an ion ought to vary inversely with the viscosity of
the medium through which it travels. The viscosity of
the mineral oil in question is more than a hundred times
that of benzene, so that the motion of ions should be
inappreciable during a microsecond.
Our experiments on mineral oil show this to be so.
There was no observable dependence of the breakdown
strength upon cathode concentration, whether the
many-pulse or single-pulse technique was used.
ACKNOWLEDGMENT
The authors wish to express their gratitude to Dr.
M. H. Hebb for his numerous helpful suggestions and
criticisms.
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1.1713991.pdf | Gas Velocity Probe for Moving Ionized Gases
Charles Cason
Citation: Journal of Applied Physics 36, 342 (1965); doi: 10.1063/1.1713991
View online: http://dx.doi.org/10.1063/1.1713991
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Published by the AIP Publishing
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] IP: 128.123.44.23 On: Mon, 22 Dec 2014 16:11:38JOURNAL OF APPLIED PHYSICS VOLUME 36, NUMBER 2 FEBRUARY 1965
Gas Velocity Probe for Moving Ionized Gases
CHARLES CASON
U. S. Army Missile Command, Redstone Arsenal, Alabama
(Received 5 June 1964; in final form 16 September 1964)
A dire.ct electrical method for measuring the velocity of a flowing plasma is based upon the polarization
voltage mduced when a plasma flows transversely to an applied magnetic field. A method has been de
veloped to allow the estimation of certain background signals and the determination of allowable measure
ment currents arising from plasma and probe properties. Applications for this technique of ionized gas
velocity measurements are indicated.
It was ~ound that an ac. magnetic field of the order of 5 G would give a sufficient signal to determine the
gas velOCity of a plasma Jet ll;s compared to a dc field of the order of 100 G. Gas stream velocities produced
by a low-power argon plasma Jet were found to vary from 1000 to 3000 m/sec depending slightly on the mass
Bow rate and predominantly on the ambient pressure.
INTRODUCTION
METHODS for measuring gas velocity in a flowing
plasma may be categorized as average-type
measurements and local measurements. Betchov and
Fuhs1 measured the gas velocity of a plasma jet with
a pair of pickup coils, one located upstream from the
other one. "Noise" signals from each coil were dis
played on a double-gun oscilloscope and the record
photographed. Gas velocity was estimated by the time
delay in the signal between the upstream and the down
stream coils. Gourdine2 also used coils, but he developed
an rf method in which a moving plasma distorted the
magnetic field within the coils and this variation was
related to the gas velocity. Fuhs3 also used a macrnetic
disturbance technique and measured the prod~ct of
~lectrical conductivity and gas velocity in plasma
Jets but could not separate this product by his method.
Freeman' used photomultiplier tubes with lenses as the
sensors. Except for the sensor used, this method was
essentially the same as Betchov's and Fuhs'. Methods
which use detectors to observe disturbances at different
axial positions to relate time delays to velocity will
give, at best, average velocities for wide separations,
or inaccurate velocities for very close spacings of
detectors. This property is comparable in a way to an
"uncertainty principle."
Disadvantages of the above methods are that the
data must be photographed point by point and analyzed
at a later time and also that good spatial resolution is
not achieved. Streak camera photography using a focal
plane shutter camera as used by Freeman4 will produce
the same results. The chief advantage of these "average"
measurements is that the plasma flow is undisturbed by
the measuring equipment.
Probes immersed in a plasma have been successfully
used to measure many plasma properties. Jahn and
I R. Betchov and A. E. Fuhs, TDR-169(3153)TR-l Aerospace
Corporation (1962) (unpublished). '
2 M. C. Gourdine, PLR-71 , Plasmadyne Corporation (1960)
(unpublished).
3 A. E. Fuhs, Am. lnst. Aeron. Astronautics J. 2, 667 (1964).
4 M. P. Freeman. S. U. Li, and W. V. Jaskowsky J. Appl.
Phys. 33, 2845 (1962). ' Grosse5 used paired electrostatic probes to measure
characteristics of shock velocities. As before, the
arrangement of the probes was axial or time separated.
Baker and Hammel6 suggested a new way of measuring
plasma gas velocities. They investigated the properties
of a plasma streaming transverse to a magnetic field.
In general the behavior of a moving plasma in the
presence of a magnetic field rarely follows the simple
classical theory due to a multitude of plasma processes.
However, Baker and Hammel demonstrated that when
a plasma streams through a transverse magnetic field
B with a velocity v, an orthogonal electric field E is
present. According to simple classical theory a polariza
tion electric field, v x B, would be generated in the
plasma to allow it to pass through the magnetic field.
The va~ue of Ej B which they observed was in agree
ment WIth the plasma velocity as determined by mag
netic probes.
Clayden and Coleman7 applied the above method to
an arc-heated low-density wind tunnel. They inserted
a pair of symmetric probes mounted 1 cm apart along
the flow radius normal both to the applied dc magnetic
field and the gas flow to detect the electric polarization
field present. A recording galvanometer was used to
simultaneously measure the voltage across the probes
and the current to the coil. Their analysis indicated a
linear v.ariation in polarization voltage with applied dc
magnetIC field. Their study did not estimate the sen
sitivity or linearity of the equipment and they did not
report any special ~xperimental difficulties in applying
thIS apparatus to wmd tunnel research.
THEORY
Important parameters in the design of velocity probe
experiments are: (1) impedance requirements on the
detector, (2) thermionic emission effects, and (3) mini
mum detectable gas velocity. A review of standard
: R. G. Jahn and F. A. Grosse, Phys. Fluids 2,469 (1959).
D. A. Baker and J. E. Hammel, Phys. Rev Letters 8 157 (1962). . ,
7 W. A. Clayden and P. L. Coleman, Memo (b) 57/63, Royal
Armament Research and Development Establishment Fort
Halstead, Kent, England (1963) (unpublished). '
342
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] IP: 128.123.44.23 On: Mon, 22 Dec 2014 16:11:38GAS VELOCITY PROBE FOR MOVING IONIZED GASES 343
electrostatic probe theory with modifications to allow
for polarization voltage, measurement currents, and
probe thermionic emission will yield this information.
The theory for electrostatic probes was devised by
Langmuir and Mott-Smith8 for a single probe floating
in a plasma. It was modified by Johnson and Malter9 to
apply to two equal area probes. Chen10 has considered
the effect of probe temperature on Langmuir probe
analysis. Figure 1 (a) depicts an idealized potential
diagram across a plasma with a v x B potential added.
Subscripts land 2 are probe index numbers and <I>
denotes the surface work function. The potentials of
the two probes with respect to the immediate vicinity
plasma are VIand V 2 and the electric polarization
voltage of the plasma due to a magnetic field is VB.
Vo denotes the output voltage as would be measured by
a voltmeter. The output voltage then has components
due to plasma properties, electrode work functions, and
applied external magnetic fields.
When an electron is thermionically emitted it has
overcome a barrier of <I> above the Fermi level and
leaves with a kinetic energy proportional to the surface
temperature. Energy is added as the electron falls
through the probe sheath. For probes perpendicular to
the B field and to the stream velocity, the electron then
loses kinetic energy proportional to the polarization
voltage of the plasma relative to electrons back at its
source probe. It next loses kinetic energy proportional
to the other probe's potential when collected by it.
Then relative to its source electrons, its energy is then
further reduced proportional to the work function.
Arrows in the figure indicate the path this electron
would take as its potential is changed. Collection and
emission rates of electrons are assumed not to disturb
the plasma. Charges then must be generated or absorbed
by the plasma as rapidly as they are drained away or
emitted by the electrodes.
Figure l(b) shows an idealized plasma and instru
ment system. In this figure, ii and ie refer to the mag
nitude of the random ion and electron current densities
near electrodes of areas A 1 and A 2. The primed terms
correspond to those plasma charges actually collected.
Langmuirll has shown that ion currents may be pro
duced at a hot electrode when it is in an alkali vapor.
This method for ion production will be neglected, even
for hot probes, since the application is for air, nitrogen,
and inert gas plasmas. Thermionic electron current
density is represented by Ith while conventional circuit
current is labeled I. The measurement impedance is
called R and the observed voltage is then Vo.
Complete current expressions are obtained by apply-
81. Langmuir and H. Mott-Smith, Jr., Gen. Elec. Rev. 27, 449,
538, 616, 762, 810 (1924).
9 E. O. Johnson and L. Malter, Phys. Rev. 80, 58 (1950).
10 C. J. Chen, J. Appl. Phys. 35, 1130 (1964).
11 1. Langmuir and K. Kingdon, Phys. Rev. 21, 380 (1923). (a)
R
" (b) v.
FIG. 1. (a) General potential diagram for the polarization probes,
Contributions due to probe work functions, induced electric
polarizations, and plasma sheath potentials are shown. Arrows
indicate the path an electron would take moving from probe 1 to
probe 2. (b) Gas velocity probe showing the currents which pass
through the plasma sheath. The voltage measured is Vo in both
diagrams.
ing Kirchhoff's law at each of the two electrodes.
I = in' A1-i.1' A 1+Ith1A 1,
1= -ii2' A2+i./ A2-Ith~2. (1)
(2)
The primed terms are the electrons and ions collected
by each probe. Langmuir8 used the Boltzmann dis
tribution function for the x component of velocity to
determine collection currents. The effect of applied
magnetic fields and induced electric fields on the velocity
distribution function are neglected because they are
required to be very weak so the flow will be undis
turbed. When this is done for Eqs. (1) and (2) the
potential on each probe is calculated to be
V 1= -Te11n(in/i e1)-Telln[1-(I -IthANiilA1], (3)
V2= -T.dn(ii2/i e2)-Te2ln[1+ (I+ Ith~2)/ii2A2J, (4)
when Te is in electron volts. The random electron
current density terms are a convenient grouping of
constants from the integration while in and ii2 are the
saturated ion current densities to each probe. Chen10
found that thermionic emission effects will modify
Langmuir-type probe current-voltage characteristfc
curves to the extent that saturation ion current may
appear to be increased by an order of magnitude. From
the picture presented here these effects may be cal
culated, including the changes made on the wall poten
tial. For I=O the wall or floating potential VI, on each
electrode is obtained, Equation (5), written without
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] IP: 128.123.44.23 On: Mon, 22 Dec 2014 16:11:38344 CHARLES CASON
subscripts, is valid at either electrode.
1/1= -Te In(ii/'ie)-T e In (1 +Ith/i i). (5)
Equation (5) has one additional term added to the
usual electrostatic probe theory for the probe floating
or wall potential. It may be seen from this equation
that if Ith is sufficiently large VI may even be positive.
This condi tion is
(6)
Generally V f is negative except for rarefied plasma
flows and hot electrodes with low work functions.
The potential as measured with a voltmeter can be
deduced from Fig. lea) and is,
Vo=<t>1-(h+V 2-V1+VB• (7)
Effects of absorbed atoms which would modify the
work function are neglected in this study since each
electrode is assumed to be hot. When Eqs. (3) and (4)
are substituted into Eq. (7) one obtains for the dc
case,
VO=<t>1-<t>2+ V B+ Te1ln(iiI/iel) -Te2ln(ii2/ie2)
+ Te1ln[1-(1-IthlAl)/iilA1]
-Te2ln[1+(I+Ith2A2)/ii2A2]. (8)
Equation (8) shows the effects on the measured voltage
due to (1) thermionic emission, (2) plasma thermal
gradients, (3) fluctuations of velocity, charge density,
and temperature, and (4) magnetic fields. Problem
areas may be divided into those associated with dc and
ac measurements.
If an ac magnetic field of Eo sinwt is applied, the
measurement voltage may be tuned to the angular
frequency w which is used. From Eq. (8) the dc terms
not affected by drawing current may be dropped thereby
leaving the following:
Vo= VB sinwt+T e1ln[1-(I-IthlA l)jiilA1]
-Te2ln[1+(I+Ith2A2)/ii~2]. (9)
The required minimum impedance of the measure
ment circuit may be estimated by comparing the terms
containing I which represent the voltage adjustment
due to measurement current used. The last two terms
on the right of Eqs. (8) or (9) give the error voltage due
to the plasma-probe current drain to the measuring
instrument. For a 1% drop in the voltage, the sum of
these two terms are set equal to 0.01 VB. The maxi
mum allowable current Ia is then determined. For
-1< (Ia+1thA)/iiA<1, the logarithm is expanded as
a series and the first term only is retained to give,
V BX 10-2"", Te1(I.-Ith1A1/i.lAl)
+ Te2(Ia+lth2A 2/ii2A 2)' (10)
If the electrodes are assumed to have equal area and
be in identical plasma environments this gives I a'
(11) Nobata12 has investigated the effects of a strong
magnetic field, i.e., the electron cyclotron frequency
exceeds the collision frequency of electrons and gas
molecules, on probe current. He found that for a mag
netic field of 530 G aligned in the direction of a probe in
a low-pressure neon plasma, the ratio of the saturation
electron current to the saturation ion current dropped
by the order of lo to that at zero magnetic field.
Following the results of Nobata one should be aware
that I a may be overestimated for certain geometries
if a strong magnetic field is used but it would be un
affected in the case of a weak magnetic field.
The physical orientation of the probes has a further
restriction other than merely being aligned to sample
the polarization voltage. This restriction is caused by
the plasma self-loading caused by gradients in the
product of v x B. Assume that a gradient in velocity
due to a boundary formed by a wall is present. A curl
v )( B will exist when a velocity gradient component is
normal to a constant magnetic field. Circulation cur
rents J induced by curl (v x B) will change the electric
field in the plasma to E' which is reduced from v )( B by
an amount J/u. This current may also interact with the
field B to produce a Lorentz force which will tend to
accelerate the slower portions of the boundary layer
and slow down the faster portions of the boundary layer
and the portion of the uniform velocity flow just out
side of the boundary layer. This tends to create a
sharper gradient near the surface and effectively in
creases the depth of the boundary layer. The induced
electrical field which must be probed is not affected by
self-loading in the plane where gradients in velocity
are perpendicular to the applied B field. At other posi
tions the induced electric field is reduced due to the
circulation currents arising from plasma self-loading.
EXPERIMENTAL PROCEDURE AND DATA
Steady-State Induced Fields
In the first experiments a dc magnetic field was
placed across a plasma jet perpendicular to the flow.
Clayden and Coleman7 used an iron core electromagnet
greater then 100 G but found hysteresis in their field
due to the iron. Therefore, an air core Helmholtz coil
was used in this study to produce a magnetic field
linear with current and uniform over the volume con
taining the probes.
The magnet consisted of 8 turns of copper tubing
wound on a radius of 16.5 cm. The magnetic field was
calculated by the Helmholtz equation to be 0.436 G/ A.
The induced magnetic field was measured to be 0.443
G/ A (which suggests that a partial turn was generated
from a return lead). Since agreement is within 2%, the
calculated value was used in the analysis. Field strengths
of 85 to 117 G were used in these experiments.
Source of the plasma gas was a small de plasma
generator. The cathode was water-cooled tungsten and
12 K. Nobata, Japan J. Appl. Phys. 2, 719 (1963).
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IP: 128.123.44.23 On: Mon, 22 Dec 2014 16:11:38GAS VELOCITY PROBE FOR MOVING IONIZED GASES 345
the anode nozzle was water-cooled brass with a i-in.
(0.32-cm)-diam throat. Power for the plasma generator
came from a bank of 10, 12-V truck batteries in series.
The arc was initiated by means of an in-line Tesla
coil arrangement. The minimum power delivered to the
electrodes was 325 Wand the maximum was 1350 W.
Argon gas was used in all cases.
The probe assembly used in the dc experiments was
made of tantalum foil 0.2 mm thick. The two probes
each of 1 cm2 area had small separate quartz tubes as
support insulators which were aligned with the applied
magnetic field. The separate probe supports prevented
a zone of stagnant plasma from developing which could
short any signal present. Separation between these
probes was 1.25 cm. Polarization electric field strength
was probed in the plane where gradients in velocity
would be perpendicular to the applied magnetic field
as required.
In each experiment the vacuum reservior 3360-m3
volume was pumped down to a pressure less than 10-3
Torr. Then the test gas was injected at a constant pre
determined flow rate. Next the arc was initiated by the
Tesla coil and then the magnetic field was applied.
Polarization voltage, a measure of velocity, was
graphed on an X-V plotter as a function of the output
voltage of a thermocouple vacuum gauge attached to
the vacuum reservior. It was at this point that an un
desirable feature of the dc system became evident. The
sampling probes became white hot and began to
thermionially emit electrons. One probe was in the
center of the flow and one was near the side thus pro
ducing important temperature differences on each
probe's surface.
By alternately turning the magnet on and off a
"zero" base line for measured probe voltage could be
drawn on the X-V plotter in addition to the total
signal. Figure 2 (a) is a typical data plot. The free stream
gas pressure is plotted on the abscissa and probe voltage
is on the ordinate; in this case the gas flow was argon
at 0.35 g/sec. Figure 2(b) is a plot of the gas velocity
for several flow rates for a constant power setting.
Uncertainty in the reading was ± 170 m/sec due to
velocity fluctuations in the flow of the plasma. Stray
dc effects in the data, such as plasma thermal gradients,
thermionic emission rate variations at each probe, etc.,
are shown by several terms of Eq. (8) and appear on
the "base line" in Fig. 2(a). These effects can lead to
very large errors unless they are taken into account or
properly eliminated. Isolation amplifiers must be used
to measure the polarization voltage because of probable
interaction with the arc power supply through ground
loops. The response time of the de system was 0.15 sec.
The magnet had a noticeable influence on the plasma.
When it was turned on the plasma jet could be seen to
deflect at a shallow angle; it would then return to its
original direction when the magnet was turned off. The
same experimental procedure was then followed except 003 "
0.6
1/1
~ g
0.4
02~--~--~~--~~~~--~--~~ .003 .017 .052 .07S .103, .142 .220
>... g
iill
>
1/1 < PRESSURE IN TORR
(d)
FLOW RATE
'0 .4110 ..,.&1'<: •• 215
4 .~~s •• !tE.O
~O~--~~~~--~~~~--~~~~ .003 .017 .052 .07S .103 .142 .220
PRESSURE IN TORR
{b)
FIG. 2. Data and results from dc experiments: (a) Typical data
from an X-V plotter showing output voltage vs free stream pres
sure. Bottom trace is background dc "noise" while top trace is
dc "noise" plus VB. The argon flow rate was 0.35 g!sec and B was
117 G. (b) Reduced data for several flow rates of argon. Gas
velocity is plotted as a function of indicated free stream pressure.
The magnetic field used in the experiments ranged between 85 and
117 G.
with a weak ac magnetic field in order to avoid non
meaningful dc effects and the effects of moving the
plasma about.
ac Induced Fields
A new coil was made from 0.12-cm-diam wire.
Fifty-one turns were wound on a radius of 8.25 cm.
In this case the reduced magnetic induction is 5.5 G/ A.
Magnet current was supplied from a commercial high
fidelity amplifier accepting a selected frequency from
a signal generator. Probe voltage was fed through
capacitors to an audio interstage transformer with a
60-kn impedance and a 3.5-kn dc resistance. This
transformer was used to isolate the amplifier from the
probes to avoid ground loops through the arc power
supply. An amplifier tuned to 2 kc/sec was used for
the voltage measurement. It has a sensitivity of 1 p.V
and a gain of 104• Sensitivity to gas velocity is deter
mined by the amplifier but this is reduced when "noise"
is present.
The probe assembly used in the ac experiment was
made of tungsten wire 0.064 cm diameter with a
separation of 1.04 cm between centers. The probe
housing was a two-hole aluminum oxide thermocouple
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.018
.016
.014
........ 012
> '-.01
1.0 1.4
MAGNET CURRENT (A)
. (eI)
~~~
> 0 . CI .... 'po. "no
~~ o . 1.0 1.5
MAGNET CURRENT (A)
(b)
FIG. 3. Rectified rms signals from ac experiments: (a) Recording
of probe signal as a function of magnet current. Bottom trace is a
pickup signal with no plasma while top trace is a typical record
with an argon plasma. The free stream pressure increased from
0.075 to 0.095 Torr during the run. The rectifiers used were non
linear in response. (b) Probe signal recordings to test linearity of
induced electrical polarization. Top curve is a 5-sec time sweep
with no applied magnetic field to the plasma. The detector was
tuned to 2 kc/sec in all cases. Middle curve is observed signal as a
function of 666.6-cps magnetic field to test for 3rd harmonic con
tent. Lower curve is observed signal as a function of 1000-cps
magnetic field to test for 2nd harmonic content.
insulator tube which had been ground down along one
hole so that it opened 1.03 cm shorter than the end one.
Each wire was inserted and bent perpendicular to the
tube. The wire from the top hole was 0.87 cm in length
while the wire from the shorter hole was 0.68 cm. This
arrangement placed the leading edge of each wire at the
same axial distance downstream from the plasma jet noz
zle. The separating insulator between the end probe and
the short-length probe was seen to produce a small zone of
stagnant plasma. It appeared that most of this gas was
located to the rear of the short-length probe even
though it was seen to be in contact with the long probe.
An improvement on the experimental arrangement
used here would, as in the dc case, have no separating
insulator which would allow a stagnant zone of plasma
to be present near the probes which could short out any
signal developed. Gas velocities indicated by this
assembly were similar to those observed by the dc
probe.
An X-Y plotter was used to record rms polarization
voltage versus rms magnet current. The data plot in
Fig. 3(a) has curvature due to the nonlinear response of the rectifier filter in the plotter for the signal voltage
used. Signals were filtered to have a response time of
0.01 sec. Only the first part of the data from this run
is shown. Two "error" voltages are noted. One is the
smooth curve due to a pickup voltage with no plasma
present which is seen at the bottom of the figure. This
would result in a 660-m/sec excess in gas velocity. The
other, a background signal for the plasma with no
magnetic field, is evident from the Fig. 3(b) time sweep
(note displaced zeros for each curve). The level of the
background 2-kc/sec signal field is in the order of 2 to
5 mY. The other two curves reflect an attempt to deter
mine linearity of the response of the plasma polarization
to the ac field. Linearity was tested by applying a
1-kc/sec magnetic field to the plasma and measuring
response at 2 kc/sec (2nd harmonic) and again at
666.6 cycles when measured at 2 kc/sec (3rd harmonic).
Changes in background level would be produced by
changes in higher harmonic polarization fields if the
response was nonlinear. No changes were obvious as
compared to the normal background drift seen in Fig.
3(b). It appears that 2 kc/sec was not the best choice in
detection frequency because of an apparent fluctuation
present in the plasma but was required due to the con
struction of the amplifier.
3000
..
~2800
i
~26
o
o
20000~--~1----~2--~3----4~--~5~--~6--
MAGNET FIELD STRENGTH IN GAUSS
(d)
OIRECTION OF
GAS FLOW
(b)
FIG. 4. (a) Reduced data from Fig. 3 (a). The plot of gas velocity
vs magnetic field strength indicates the presence of a pickup error
assumed proportional to B. A least-squares fit to a straight line is
made to find the B=O intercept. (b) Station for boundary layer
velocity profile measurement. This probe orientation is free from
plasma self-loading effects.
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Upon reducing the data a background "error voltage"
was taken to be a constant 0.003 V. Provision was not
made to monitor its fluctuations as seen from Fig. 3(b);
therefore, scatter should result from this source of
error. Gas velocities were calculated and plotted in
Fig. 4(a). Error signals proportional to the magnet
current were estimated by a least-squares fit to the
equation.
p=mB+po, (12)
where m is the proportionality constant due to an
error, Po the true gas velocity including the 660-m/sec
pickup error, and p the gas velocity indicated by adding
in an error assumed proportional to the magnet current.
The results of the data reduction showed the gas
velocity to be 2160 m/sec with a rms deviation of
± 137 m/sec. Substraction of the "pickup" signal
equivalent of 660 m/sec gives a gas velocity of 1500
m/sec.
The average gas velocity may be crudely estimated
from mass flow rate and electric power. For 100%
efficiency the enthalpy for 0.19 g of argon per second
heated at 500 W is 2.63X 104 Caljmole. For pressure
equal to or greater than 0.01 atm this enthalpy would
result in a stagnation temperature of approximately
SOOOOK.13 From the relation !mp2=!kT a velocity of
1750 m/sec is estimated.
An alternate way of making the same type of crude
estimate is from the power equation for a flowing gas;
that is,
(13)
For the example cited, Eq. (13) indicates a gas velocity
of 2200 m/sec. These two results, although rough,
nicely overestimate the results obtained from both the
dc and ac data except for very low free stream pressures.
This was expected since the plasma generator does not
have 100% efficiency.
An attempt was made to use a photo tube system to
measure gas velocity for an independent comparison.
This system was found to exhibit similar difficulties as
reported by Freeman4 and was eventually abandoned.
At best, these measurements indicate a gas velocity of
about 2000 m/sec (±SOO m/sec). Freeman and co
workers also found that several disturbances such as
those produced by temperature and total pressure
fluctuations may propagate in a subsonic plasma
jet at different velocities giving rise to additional
uncertain ties.
CONCLUSIONS
Comparison between the velocity results obtained by
the dc method as shown in Fig. 2eb) and the ac result
13 F. Bosnjakovic, W. Springe, K. S. Knoche, and P. Burgholte,
Papers Presented at the ASME Symposium on Thermal Properties,
Purdue, 23-26 February 1959 (McGraw-Hill Book Company, Inc.,
New York, 1959), pp. 465-472. of 1500 m/sec from 0.075-0.090 Torr is satisfactory. The
plasma jet was found to exhibit rapid fluctuations in
gas velocity as seen in both the dc and ac data. Gas
velocity was found to vary considerably upon free
stream conditions and moderately with gas flow rate.
Also magnetic fields in the order of 100 G were found
to slightly disturb the gas flow. For probes in a hot
flow where a dc field is used it was found best to alter
nate the field between "on" and "off" to determine and
subtract extraneous dc effects. When ac measurements
are employed a field of the order of 5 G is all that is
necessary since background "error voltages" may be
eliminated at this level. It is expected that flows other
than plasma jets would exhibit less fluctuations at the
frequency of the tuned voltage amplifier. The lowest
saturation ion current to the probe obtained to date in
the plasma jet was 1.3 rnA while the minimum current
required by the measurement equipment is of the order
of microamps. From Eq. (11), 5X 10-3 V /cm, and
Te=O.173 V, it was found that fa requires a minimum
dc resistance of 2.5 kf!.
APPLICATION
It is thought useful to mention two specific applica
tions where important measurements of gas velocity
may be performed by the current method. Other than
obvious wind tunnel use when the working fluid is
partially ionized, one may also consider the ionized
boundary layer flow on high-speed nose cones and
rocket and jet engine exhaust flows. To date it is not
known to the author that precise measurements have
been made of the gas velocity about re-entry nose cones
or of the spatial distribution of the flow field or rocket
exhausts. Figure 4(b) illustrates a possible system
proposed for hypervelocity nose cone flight tests which
utilize rf telemetry. All equipment in the instrument
package can be transistorized and will essentially con
tain the following: an oscillator, power amplifier, tuned
amplifier, and tuned calibration voltage source. A
switching network would scan the measured variables;
viz., magnet current, polarization voltage output, cali
bration signal, and also a zero point for an index point.
The output could directly feed a telemetry set. Measure
ments of the gas velocities in the boundary layer can
be directly related to heat transfer rates. At present,
boundary layer velocity profiles must be estimated to
solve heat transfer problems. A series of experiments on
the measurement of boundary layer velocities is planned
for the AMICOM 8000-kW plasma facility in conjunc
tion with a re-entry simulation program.
ACKNOWLEDGMENT
Appreciation is expressed to Dr. J. F. Perkins of the
U. S. Army Missile Command for his helpful comments
and illuminating discussions on the subject.
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1.1735291.pdf | Magnetic and Electrical Properties of ReactorIrradiated Silicon
E. Sonder
Citation: Journal of Applied Physics 30, 1186 (1959); doi: 10.1063/1.1735291
View online: http://dx.doi.org/10.1063/1.1735291
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
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IP: 141.217.58.222 On: Wed, 26 Nov 2014 01:06:021186 ]. A. KRUMHANSL
is a gauge variable which has the major effect of trans
forming the origin of the vector potential to the mth cell.
The Hamiltonian in this representation is then solved to
second order in terms involving H, for the case of a
uniform magnetic field.
The contributions to the resulting energy involve the
following:
(a) Pauli spin terms.
(b) An energy which is determined for the ,uth band
from the energy operator E", with argument p-(eA/c),
i.e., the effective mass Hamiltonian which gives the
cyclotron susceptibility.
(c) Normal core terms of the form (e2/6mc2)(r2).
(d) First-order Zeeman terms of the "in cell" variety;
these are quenched in most cases.
(e) A paramagnetic correction which indeed has the
form
(J).,k 1M· H III,k')2
E= L
k=k',,,,r!v E",(k)-Ev(k') (4)
where M is the magnetic moment operator, H the mag
netic field, and the matrix element is taken over the unit
cell between the cell periodic parts of the Bloch func
tions of same k between bands. This contribution applies
to all occupied states, with matrix elements only to higher unoccupied states; it IS therefore always a
paramagnetic contribution.
(f) Additional second-order terms which arise from
nonorthogonality matrix elements of exp[i(Gn-G m)]
between Wannier functions in different cells and differ
ent bands. The omission of these can be objected to, and
only further careful study can determine their im
portance.
If (4) is evaluated, using matrix elements determined
from k· p analysis of the effective masses in Ge, the
paramagnetic susceptibility so calculated is in good
agreement with that predicted from the phenomeno
logical approach. Such a contribution would also be
significant in metals, so that it is to be hoped that the
correspondence between these terms and those in exact
theories of the susceptibility can be established.
CONCLUSIONS
Enough is known about the various contributions to
the susceptibility in semiconductors to plan and inter
pret many of the electronic properties of doped and
irradiated semiconductors. Perhaps the most fertile
field of investigation which these experiments can lead
to is the more rigorous development of the theory of
solids in a magnetic field.
JOURNAL OF APPLIED PHYSICS VOLUME 30, l\iUMBER 8 AUGUST. 1959
Magnetic and Electrical Properties of Reactor-Irradiated Silicon
E. SONDER
Solid State Divisions, Oak Ridge National Laboratory,* Oak Ridge, Tennessee
Magnetic susceptibility measurements above 3°K and Hall effect and resistivity determinations between
50 and 3000K are reported for n-type silicon samples irradiated with increasingly higher doses of fission
neutrons. The paramagnetism due to electronic states in the forbidden gap shows an initial decrease after
short irradiation but a reversal, increase, and final saturation at a value less than that originally contributed
to the paramagnetism by the filled donors after longer irradiation.
The Hall coefficient shows evidence of a distribution of irradiation-produced energy levels in the neighbor
hood of 0.3 ev below the conduction band. The mobility goes through an initial sharp decrease with irradia
tion but recovers partially after longer irradiations. The results are discussed in terms of several models of
radiation damage. It is concluded that a simple model based on uniformly dispersed interstitials and vacan
cies is not adequate to explain the results and that interactions between centers, and nonuniform distribution
of damage will probably have to be taken into consideration.
INTRODUCTION AND BACKGROUND
ONE of the desirable goals of irradiation-effect
studies in semiconductors is the formulation of a
workable model of the actual damage site. Most of the
work! in the past has been concerned with the deter-
* Oak Ridge National Laboratory is operated by the Union
Carbide Corporation for the U. S. Atomic Energy Commission.
1 For a review of radiation effects in semiconductors see, for
instance, J. H. Crawford, Jr., and J. W. Cleland, Progress in
Semiconductors (Pergamon Press, Inc., London, 1956), Vol. 2,
p. 67; or "Energy levels produced in semiconductors by high
energy radiation," Tech. Memo. No. 4 (July, 1958) (Batelle
Memorial Institute, the Radiation Effects Information Center). mination of electronic energy levels and with the com
parison of these with predictions of the James-Lark
Horovitz2 model, in which a uniform distribution of
irradiation-induced vacancies and interstitial atoms is
assumed. However, this model is only one of many that
would be consistent with various electronic level
schemes. In fact, recent careful analysis of Hall mobility
and lifetime measurements3 of electron-irradiated
silicon has indicated that pairing of imperfections may
2 H. M. James and K. Lark-Horovitz, Z. physik Chern. (Leipzig)
198, 107 (1951).
3 G. K. Wertheim, Phys. Rev. 110, 1272 (1958).
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well be important. Furthermore, it has become evident,
both for silicon4 and other semiconductors,! that differ
ent types of residual damage may be obtained with
different irradiating particles, i.e., neutrons, deuterons,
or electrons. As a result, it is becoming accepted that
more complicated models of residual damage at room
temperature must be devised to explain irradiation
effects. Some of these complications involve pairing and
clustering of defects, introduction of large disordered
regions, and possible effects of charge segregations and
electrostatic fields around the damage. This is especially
true for the case of neutron irradiation, where a very
large amount of energy (many times the displacement
energy) may be transferred during the primary collision.
It is at this stage where Hall and resistivity measure
ments alone are no longer adequate, and information in
addition to energy-level positions is necessary to point
the way. The present work is an attempt to obtain a
fairly systematic picture of both Hall mobility and
magnetic changes of a semiconductor material under
reactor irradiation. It is hoped that these measurements,
combined with simultaneous determination of the usual
electrical properties, will help to point out qualitatively
which of the many types of possible complications
should be included in future, more sophisticated models
of radiation damage in semiconductors.
THEORY
1. Magnetic Susceptibility
The gram susceptibility or simply the susceptibility,
X, of a material is the ratio of the magnetic moment
per gram to the magnetic field, or x=MlpH, where M is
the magnetization (magnetic moment per unit volume),
p is the density, and H is the field strength. It is con
venient to consider the total susceptibility to be made
up of three parts: the contribution of the pure lattice,
xe, that of the conduction electrons or holes in the
conduction or valence bands, respectively, XC) and that
of the unpaired electrons in states in the forbidden gap,
Xi. The term Xc can be made to vanish by the simple
expedient of limiting measurements to temperatures
low enough such that there are no electrons or holes in
the conducting bands. In the present experiment this
limits the useful susceptibility data to T<SOoK. If the
assumption is made that irradiation causes no large
changes in the lattice contribution,5 then all changes
measured will be changes in Xi, the susceptibility of
states in the forbidden gap. The contribution of the
imperfection and impurity states may be further
subdivided into paramagnetic and diamagnetic con
tributions of the various types of centers present. The
configuration of one of these, the simple donor center, is
fairly well understood; its contribution to the sus-
• G. K. Wertheim, Phys. Rev. 111, 1500 (1958).
5 The room-temperature value of the susceptibility was meas
ured after each irradiation of the n-type samples. It was found that
within experimental error there was no change in the lattice
contribution, Xl, at room temperature. ceptibility is given by
xi(donor)=nd/J2/kTp- (diamagnetic term) (1)
where the first term is the contribution of nd spin
t electrons, (3 is the Bohr magneton, k is the Boltzman
constant, and T is the absolute temperature. The
second term, which is the contribution to the diamag
netism of the electrons in their orbit is, in the range of
interest to us, a factor of 10 smaller than the first term.
Experiment6 confirms the above expression for n-type
silicon with low enough donor concentration (less than
SX 1017).
A temperature dependence similar to that of Eq. (1)
might be expected for any isolated trapped electron
state. We shall thus, in order to concern ourselves with
the damage qualitatively, consider Xi to have the form
of Eq. (1). Then from the liT dependence an estimate
can be obtained of the number of unpaired electrons.7
A large temperature-independent diamagnetism or
strong interactions between magnetic centers would be
evidenced by curvature in plots of Xi versus liT.
2. Electrical Properties
In the following we shall confine our attention to
n-type silicon. From results of Hall coefficient measure
ments the number of conduction carriers, n, may be
obtained; from this the behavior of the Fermi level can
in turn be calculated. This latter is of primary interest
here, since position of the Fermi level will give informa
tion on occupation of various levels and since it is the
relation between occupation of levels and the magnetic
properties that might be significant for the under
standing of the behavior of irradiation-induced defects.
For the purposes of the present paper, a value of 1 has
been used for the mobility ratio, r, in the Hall expres
sion, n=rIRe, where R is the Hall coefficient and e is the
electronic charge. The Fermi level has been calculated
from the expression
where m* is the effective mass, (mlmt2)!, h is the
Planck constant, and Ec-Ef is the energy difference
between the Fermi level and the bottom of the con
duction band.
The Hall mobility, JL, which is the product of the Hall
coefficient and the conductivity, will herein be used
only as a qualitative guide. This is necessary since the
transport equations have been solved only for very few
idealized cases and since the situation in irradiated
material is much too complicated to permit quantitative
comparison with theory.
6 E. Sonder and D. K. Stevens, Phys. Rev. 110, 1027 (1958).
7 It should be pointed out that two assumptions are made here;
namely, one-electron states (s=!) and quenching of orbital
moments. (g=2) and (j=s) in the more general expression
x =n (gfJ)2j(j+ 1)/3pkT.
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(0) (b)
FERMI LEVEL
-eo -eo -eo -eo -€I--€I--€I-
'NTSETRASTTE'TS'AL ..... -& -e- INTERSTITIAL
STATES .... ~ ~ .e-
..... -e--e- VACANCY
VACMJCY STATES
STATES t
lei (dl
RISE IN BAND EDGES AND TRAP StATES
DUE TO CONCENTRATION OF CHARGE
-& .....
~ -e-..e-
o€/-
~
DISTRIBUTION
OF STATES
..... VALENC
BAND ----FERMI -& -e- ..,..
LEVEL
~..e-~
o€/--e-Z "\& .e-
FIG. 1. Diagram of trap-level distribution for various models of
the neutron damage. (a) and (b) show the type of distribution
expected for the James-Lark-Horovitz model of uniformly
distributed single vacancies and interstitials. In (al both ionization
states of the interstitial are above the higher vacancy states. In
(b) the lower interstitial state is below the upper vacancy state. In
(cl is shown a more gene tal distribution of levels that might come
about by interactions of centers at various small distances from
each other. In (d) nonuniformity of the introduced traps actually
causes a variation of the band edges and otherwise discrete states.
3. Models of Radiation Damage
To form a basis for later discussion of the data, we
shall describe here some possible models and the
susceptibility and mobility changes with neutron
irradiation that these models might predict.
A . Uniformly Dispersed, N oninteracting Vacancies
and Interstitials (the James-Lark-Horovitz Model)
In this model two filled states, with an energy falling
in or near the forbidden gap, are postulated for the
interstitial site; two empty states are postulated for the
vacancy. If both vacancy states lie at an energy below
the lower interstitial state [see Fig. lea)], then, in the
absence of extrinsic carriers, two electrons from the
interstitial will drop to the empty vacancy sites, causing
electrons to remain paired.8 Thus, no change in para-
8 It might seem that Hundts' rule should apply and that the
triplet state of the two-electron systems (which would be a
paramagnetic state) would be energetically favored. This would
imply a monotonic increase of paramagnetism with irradiation, a
result which was not observed in these measurements. Moreover,
calculations for vacancies in a similar structure (diamond)
[C. A. Coulson and M. J. Kearsley, Proc. Roy. Soc. (London)
A241, 433 (1957)J indicate that configurational interaction
depresses the single state below the triplet. magnetism should be expected in essentially intrinsic
materiaL
In n-type material the electrons from the donors
would drop to the lower interstitial level and would
thus retain their paramagnetism after long irradiation.
However, it should be noted that at a low irradiation
dosage (when the number of interstitials is less than
that of the original donors) some of the electrons
originating from donors will fill the upper interstitial
level, as well as the lower one, and will thus be non
magnetic due to pairing.9
If only one of the interstitial levels is above the higher
vacancy level [see Fig. 1 (b)], then one might expect
that for each Frenkel defect pair introduced unpaired
electrons are created at the interstitial as well as at the
vacancy. A linear increase in the paramagnetism with
irradiation dose might then be expected in pure mate
rial. In n-type material behavior similar to that
described above for the model pictured in Fig. lea)
would be expected as a result of low irradiation doses.
However, whereas in the case pictured in Fig. 1 (a) the
paramagnetism would saturate after long irradiation, in
the case pictured in 1 (b) it would continue increasing
indefinitely.
For single-vacancy and interstitial sites, the diamag
netism to be expected would be less than that for donor
centers.lO
B. Interacting Vacancies and I nterstitials
If these centers produce fairly well-defined energy
levels, it might be expected that, depending upon
whether the uppermost filled level is due to a paired or
an unpaired electron, the donor paramagnetism is
either removed or retained, respectively, in a manner
very similar to the specific model of vacancies and
interstitials assumed in Sec. A.
If the interactions are such as to produce distributions
of levels [Fig. l(c)], it is difficult to make any predic
tions without using specific assumptions about the level
distribution. However, if the distribution covers a large
part of the forbidden gap, the position of the Fermi level
of the material in question might tend to have a weaker
effect upon the paramagnetism than it would in material
of well-defined levels, where the relative position of the
Fermi level and a given irradiation-produced level
would be all important.
C. Clusters
If the damage is such as to create clusters of great
density of defects while the remainder of the material
9 This would not happen if the upper interstitial level were
higher in energy than the donor state. However, since the tem
porary disappearance of paramagnetism was observed in these
measurements, this particular variation of the James-Lark
Horovitz model seems improbable.
10 On the basis of the hydrogenic model the diamagnetic
susceptibility varies as the inverse third power of the binding
energy of the electron to the charge center. [See, for instance,
reference 6, Eq. (6).J Thus, the more tightly bound electrons in
deep states will contribute a smaller diamagnetism than will donor
electrons.
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remains relatively undamaged, a nonuniform spatial
distribution of electron traps might be expected. Space
charge effects due to these nonuniformities must then
be taken into consideration.ll A small region of damage
might be able to exhaust donors from a larger surround
ing region, causing disappearance of the donor para
magnetism but little or no contribution to the mag
netism of the traps in a cluster, due to the high spatial
density of electrons and the resulting strong inter
ations. As the irradiation is continued and more
damage is introduced, the exhausted volume surround
ing each damaged region would begin to overlap the
exhaustion region of a neighboring damage area. As a
result, the additional damage introduced would be able
to cause less and less additional donor paramagnetism
to disappear. Meanwhile, the increased volume of the
actual damaged material would make it possible for the
trapped electrons to be more widely separated. This
could conceivably lead to a reappearance of para
magnetism. For even greater irradiation doses the
whole specimen might become filled with damaged
material. The behavior of the magnetic susceptibility
would then depend upon the density of states in the
"imperfection band" and upon the occupancy of the
states. A similar situation, that of an impurity band, has
been considered by Mooser.12 According to those
results, a temperature dependence anywhere between
(T)-1 and (T)O might be expected.
The different models discussed also predict respec
tively different behavior of the Hall mobility as a
function of irradiation. Of course, for any of the models
a decrease of mobility, especially in the low-tempera
ture, defect-scattering range, is to be expected.
For the James-Lark-Horovitz model (A) theories of
charged-point scattering13 should apply so that at low
temperature the mobility,}J., should vary as a low power
(1.5) of the temperature and should be proportional to
the square of the charge and to the density of centers.
For (B) a variety of modes of behavior, including
TABLE 1. List of samples and irradiation history.
Pre irradiation net
donor density Irradiation
Hall plate Susceptibility Doping Time Total flux
Sample (em-') sample (em-') agent (hr) neutrons/em'
Mll 1000 ohm-em 0
p-type floating 154 2.2 X 1017
zone-grown
329B 2.7 Xl0'7 2.1 Xl0'7 Arsenic 0
4.24 6.3 Xl0" 329 1.65 Xl0'7 2.1 XlO'7 Arsenic 0
16.2 2.3Xl0'6
39.1 5.7 XlO'6
59.6 8.6 XlO"
1262 5.8 XlO'7 6.4 XlO'7 Arsenic 0
74.5 1.1 XI017
148.5 2.IXIOI7
219 3.IXIOI7
286 4.1 XlO'7
11 B. R. Gossick and J. H. Crawford, Bull. Am. Phys. Soc. Ser.
II, 3, 400 (1958).
12 E. Mooser, Phys. Rev. 100, 1589 (1955).
13 P. Debye and E. M. Conwell, Phys. Rev. 93, 693 (1954). TEMPERATURE (OK)
20 10 70 5.0 4.0
(I( 10-7) ~-~-~-n---'----,-----'--",,--,----, 3.0 2.5
0.,5 • 329, PREIRRADIATED
• 329, 2.3 X lO'6 nvl
& 329, 5.7 X tot6 nvl
o 329, 8.6 X to'6 nvl
03298, PREIRRADIATED __
6 3298, 6.3 X lO'S nvl
o ~ __ L-__ ~_il-_-L_-L_~_~ ____ ~~
o O.lO 0.20 0.30 OAO
'IT (OK)-t
FIG. 2. Change in the magnetic susceptibility, plotted as a function
of reciprocal temperature for specimens 329 and 329B.
effects similar to those seen in heavily doped and
compensated material,14 should be possible. Strong
temperature dependence for some conditions and weak
temperature dependence for slightly different conditions
would not be surprising. Such effects would be even
more probable in the case of imperfection clusters (C).
The electric fields resulting from a nonuniform dis
tribution of defects would act as very effective scatter
ing sources at low temperature, leading to a temperature
dependence of the mobility that might be appreciably
faster than the 1.5 power expected for point-charge
scattering. A reversal of the irradiation-produced
mobility decrease might also be expected in this case
for large irradiation doses. This would occur when
damaged material begins to fill the volume of the
crystal, thus removing the strong space charges and
nonuniformities which cause such large diminution of
mobility.
EXPERIMElU
A number of samples of n-type silicon were irradiated
at room temperature with varying doses of fission
spectrum neutrons in hole 51 of the Oak Ridge National
Laboratory graphite reactor. Table I lists the samples
and their irradiations. Susceptibility specimens and
Hall plates were irradiated together in all cases except
that of Mn, a high-purity, zone-grown comparison
specimen, where only a susceptibility cube was used. It
might be noted that samples 329 and 1262 are both
n type but differ in donor concentration by a factor of
three. Susceptibility cube 329B came from the same
portion of the ingot as did 329. Unfortunately, the Hall
plates accompanying 329B and 329, respectively, came
from adjacent portions of the ingot on the higher and
lower concentration side, respectively, so that these
latter differ slightly in original donor concentration.
14 E. M. Conwell, Phys. Rev. 103, 51 (1956).
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IP: 141.217.58.222 On: Wed, 26 Nov 2014 01:06:021190 E. SONDER
lXI0-7)
0.3
>-I-:::;
iIi ;::
Q.
W
~ 0.2
:::>
VI
u ;:: w z
;;j
::;;
<t 0.1 a:
If.
0 TEMPERATURE (OK)
20 10 7 4
.,
o 1262, PREIRRADIATED
r--+--~--I-""""~- 0 1262,1.1 X 10'7 nvl
to 1262.2.1 X 10'7 nvl
r--H'----+.-c'Of----f-.- V 1262, 3.1 X 10'7 nv/
.. 1262, 4.1 X lo'7nvl
• MW PREIRRADIATED
f---+V=--+"~=-J---+--. M ... 2.2 Xl0<7nvl
0 0.10 0.30 0.40
FIG. 3. Change in the magnetic susceptibility, plotted as a function
of reciprocal temperature for specimens Mil and 1262.
Magnetic susceptibility as well as Hall and resistivity
measurements were made following each irradiation.
The magnetic susceptibility was measured over the
range 3.soK-300oK with the emphasis on the low
temperatures, where all electrons are in donor or trap
sites. The equipment and technique has been described
previously.6,ls Determination of temperature in the
range 4 oK to 200K was a problem due to the fact that
the susceptibility cubes must hang free and cannot
therefore be in direct contact with a thermometer.
However, a reliability of O.l°K was obtained in the
temperature by the expedient of using a ruby sus
ceptibility specimenl6 to calibrate two germanium
thermometers,17 which were located in the wall of the
specimen chamber and were held in thermal contact
with the specimen by the use of 100 JJ. of helium ex
change gas.
The Hall and resistivity measurements were made by
standard dc techniques, using 7.5 koe of magnetic field.
Contacts were soldered to nickel-plated "dog ears"
before the irradiations. Thus, it was unnecessary to
heat the samples subsequent to neutron irradiation.ls
RESULTS
The results of the magnetic susceptibility measure
ments are shown in Figs. 2 and 3. Since the term Xi is the
one of interest here, the lattice contribution has already
been subtracted out. Although there is a small tem
perature dependence of the lattice susceptibility above
16 Stevens, Cleland, Crawford, and Schweinler, Phys. Rev. 100,
1084 (1955); D. K. Stevens, Oak Ridge National Laboratory
Report, ORNL 1599 (unpublished).
16 The temperature behavior of ruby has been measured
accurately by other means [J. G. Daunt and K. Brugger, Z.
physik. Chern. (Frankfurt) 16, 203 (1958)].
17 Kunzler, Geballe, and Hull, Rev. Sci. Instr. 28, 96 (1957).
18 One of the Hall samples, 1262, was heated to 100°C for about ! hr while the Hall measurements were in progress after the first
irradia~ion. There was evidence of a slight amount (~10%) of
annealmg. sooK,19 it is fairly clear, both on theoretical grounds20
and by comparison with actual measurements on
germanium,21 that the lattice susceptibility tends toward
a temperature-independent behavior near absolute
zero. However, measurements of the supposedly pure
sample, Mn, did yield a small paramagnetic term at
low temperature. This has been attributed to surface
states, dissolved oxygen, or possibly to other impurities
that do not contribute donor or acceptor states and
has thus been included as a contribution to the impurity
term, Xi, rather than to the lattice susceptibility, X •• In
the subtraction of the lattice susceptibility from the
values measured in the n-type sample, therefore, no such
paramagnetic term has been included. In any case this
paramagnetic contribution to the susceptibility of the
"pure" sample is so small that its inclusion or neglection
is totally unimportant compared to the much larger
changes in magnetic properties of n-type material, with
which we are concerned in this paper.
A look at the curves for specimens 329 and 329B
(Fig. 2) shows that before irradiation a fairly steep
slope, consistent with a donor concentration of 2X 1017,
is observed. There is an immediate decrease of sus
ceptibility upon irradiation, as shown by the results of
lightly irradiated 329B. Consecutive doses of fission
spectrum neutrons cause a continuation of this decrease
of slope and then cause it to rise again and remain
constant at a value significantly less than that of the
unirradiated sample. Identical behavior occurs, how
ever, at higher irradiation levels for a specimen that is a
factor of three less pure (specimen 1262, Fig. 3). These
results are combined in Fig. 4, where the number of
magnetic centers is shown as a function of irradiation.
Figure 4 is intended only as a guide due to the fact that
it is based on using the first term of Eq. (1). This is
correct to within about 10% for the case of donor
centers but may not be as good for other types of
defects, where strong diamagnetism, interactions be
tween centers, or effects due to orbital magnetic
moments might conceivably be present.
I\. \. \. 1262
\. ~
1\ -.~. './
M" .
3 4 (x 10")
IRRADIArION (nv/)
FIG. 4. The number of magnetic centers as a function of irradiation.
19 D. K. Stevens and]. H. Crawford, ] L, Bull. Am. Phys. Soc.
Ser. II, 1, 117 (1956).
20 ]. A. Krumhansl and H. Brooks, Bull. Am. Phys. Soc. Ser. II
1, 117 (1956). '
21 R. Bowers, Phys. Rev. 108, 683 (1957); also, unpublished
results of the present author.
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The results of the electrical measurements of the
companion Hall plates are shown in Figs. 5-7. The
behavior of the Hall coefficient shown for the two
samples, 329 and 1262 in Figs. 5 and 6, respectively, is
similar to that found previously in neutron-irradiated
material,4 even though our doping levels, as well as
irradiations, were 100 to 200 times greater. There was
no evidence of a discrete level. From the results of
specimen 329B we can obtain an estimate of initial
introduction rate of electron traps. The number, 5
traps/cm3/neutron/cm2, is in surprisingly good agree
ment with the introduction rate of 5.6 obtained in the
much purer material by Wertheim. This agreement is
almost surely fortuitous in view of the fact that ex
posures to neutrons were conducted in different reactors
with the possibility of different flux spectra.
The Hall mobility is shown as a function of tempera
ture in Fig. 7. Three results are immediately evident.
First, there is a very sharp decrease of the mobility
throughout the temperature range but mainly at the
lower temperatures. Second, the slopes of some of the
2
I08c=~--~r-+-~---+--4---~-4--~--d
5
2
5
2
w3 c=~---+-+-~6oL--+---+
5
2
102~-+~-+ __ +-~~
5
2
o 2 4 6 8 10 12 14 16 18 20
fOOOlr {'K)-I
FIG. 5. Hall coefficient of specimens 329 and 329B
as a function of reciprocal temperature. I
Z
W 5
U 2 ;:;:
~ 104 I=-!---¥-
U
..J 5 -' «
I
QO
5
2
10 o 2 4 6 8 W 12 14 16 18 20
lOOOIT ('Kf'
FIG. 6. Hall coefficient of specimen 1262 as a
function of reciprocal temperature.
curves are much greater than the 1.5 power that one
would expect on an assumption of point-charge
scattering centers. Third, for both specimens the
mobility begins to increase again at moderately high
irradiation doses. It might be pointed out that the
minimum in the mobility seems to occur at somewhat
higher doses than the minimum in the number of
magnetic centers as obtained from the slope of the Xi
versus l/T curves.
DISCUSSION
In order to consider some of the implications of
these data, it is first useful to see how the Fermi
level in the irradiated material varies with irradiation
and temperature.
The Fermi level calculated from Eq. (2) is shown in
Fig. 8. The significant result here is that for moderately
heavy irradiations the Fermi energy becomes tempera
ture independent. However, it keeps dropping with
radiation dosage, indicating that no discrete level exists
closer than 0.35 ev to the conduction band. In fact, the
implication is that there is a continuum of energy levels,
at least between 0.3 and 0.36 ev below the conduction
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IP: 141.217.58.222 On: Wed, 26 Nov 2014 01:06:021192 E. SONDER
4000
I I
I 1
"~ e. 'K[ "'''''''t ---
329 S .1."", .-'~r \ ! I
!
I
II I
I I
.... 2000
1000 "~:tt·! .,'",
I."" ........ -, " .262 lJNIRRADIATED
800 t-(329Sl 6.3 'W'5nvt "- '\.
I .:-. ".
GOO "" -, ~ ....
i "
u 400 ., ..
g
"'E
~ 200 )-
t-:::;
iii
0
::E
-' -' 400 4 :r i 1
I ....-1-f---(329) 2.3, X lOi6 nr I
~ /' ~t~~
/" 'O~"./ I
~'I ~ It ('>'1: 'l
1I & /" ,,~l
• !\. • ["".
-
I !
I -r,",,-
! ...........
~ r-
...., ." '\ .
'\
I
~.l x 10'7 nvt L...
IT xi017 nv; ~
tf;;. I' V .~ .'
/ 1.£1 !-
FIG. 7. Temperature
variation of Hall mobility
for samples 1262, 329B,
and 329 after various
irradiations with fission
spectrum neutrons.
80 f---' ... i,/ "'. =IE fJ f--t--3.1' 10<7 nvl I .' L
60 • I -3
I j ' ........... 1 I
/ / I I
/ •
40 I II
I
i---..
!t: w'" o 8 V , V /
I) j 2.1 X to'7 T'
e' I 20 ~!l
i;t t¥ II )
329; 3298
II 1
I 1262
I 10
40 60 80 100 200 40040 60 80 100 200 400 600
TEMPERATURE !OK)
band edge. One additional fact might be pointed out
here; this is that the depression of the Fermi level is
much faster in the purer specimen than in the less pure
one. This is to be expected if the introduction rate of
deep traps is comparable in the two specimens, since it
is the ratio of the concentration of vacant, deep-lying
states to that of donor states that pulls the Fermi level
down.
It should be pointed out here that the calculation of
the Fermi level from Hall data is strictly correct only
for a model that has uniformly dispersed point-trapping
centers. Large damage regions and consequent non
uniformities would tend to make the meaning of the
Hall coefficient ambiguous. It is conceivable thatll a
region of high damage would exhaust, at least to some
extent, a larger region surrounding it, thus decreasing
the volume of crystal through which current can flow.
Juretschke, Landauer, and Swanson22 have calculated
22 Juretschke, Landauer, and Swanson, J. Appl. Phys. 27, 838
(1956). a correction to the Hall coefficient in material containing
spherical insulating regions. For these the Hall coeffi
cient should be corrected by a factor of (l-}e)/ (1-~:).
where E is the relative volume of insulating material,
This shows that as long as the sample is not predomi
nantly "insulating," the Fermi level which essentially
varies as the log of the Hall coefficient, still has at least
qualitative meaning.
It is probably evident by now that the choice of the
model may in itself determine the interpretation of the
experimental data. As a result, it might be misleading to
try to fit the conclusions to a particular, favorite model.
Rather, we shall take the various results of the measure
ments and discuss each in turn in terms of the models
outlined earlier in this paper. Table II will show a.
summary, indicating general agreement with a model by
the word "yes" and disagreement by "no."
1, 2. Initial decrease and subsequent increase in the
number of magnetic centers: For Model A a minimum
in the number of magnetic centers is to be expected
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IP: 141.217.58.222 On: Wed, 26 Nov 2014 01:06:02MAG NET I CAN DEL E C T RIC ALP R 0 PER TIE S 0 FDA MAG ED S i 1193
when the number of defects is just large enough to
cause all electrons to be removed from the donor
centers and to be trapped in pairs at the interstitials.
This would imply an actual vanishing of the para
magnetism, which, according to our data, does not seem
to occur, but cannot be ruled out.
For the more general model, B, it is conceivable that
a fraction of the uppermost irradiation-induced levels
are due to unpaired electrons, while others are due to
paired electrons similar to those postulated for model A.
Such a situation could easily lead to a minimum, but
nonzero, value of the number of magnetic centers at
low irradiation doses, with an increase in the number
after longer irradiation.
On the basis of the cluster model, C, initial dis
appearance is easily accounted for by the interactions
of the electrons in the spatially small damage sites.
Further irradiation, which increases the volume of
damaged material, makes it possible for the trapped
electrons to be further separated, thus decreasing their
interactions and consequent loss of paramagnetism.
3. Saturation of paramagnetism at less than original
value after long irradiation: On the basis of model A a
return of the paramagnetism and saturation at the pre
irradiation value should occur after long irradiation if
TABLE II. Summary of comparison of experimental results
with predictions of various models.
A
(Simple
Lark-
Result of experiment Horovitz)
1. Initial decrease of the Yes
susceptibility with ir
radiation dosage
2. Increase in susceptibility No
for larger irradiations
with no evidence that
the susceptibility has
gone through the van-
ishing point .
3. Saturation of the num-No
ber of magnetic cen-
ters at high irradiation
doses at a value less
the original number
of donors
4. Lack of evidence of dis-No
crete levels in Hall
versus liT curve
5. Temperature-independ- No
Fermi level at high ir
radiation doses; con
tinuous dropping of
Fermi level with irra
diation dose
6. Steep slope at low tem-No
perature in the tem
perature dependence
of the mobility
7. Reversal of the mobility No
and an increase for
large irradiation dose
8. Correlation between sus-Yes
ceptibility behavior
and Fermi level Agreement with model
B
(Pairing or other
method for C
distributing levels) (Clusters)
Yes Yes
Yes Yes
Yes Yes
(but requires
diamagnetism,
interactions,
or banding)
Yes Yes
Yes Yes
No Yes
Yes Yes
Yes No o
0.1
-;: 0.2
~
w g
w
~ at
<l
'" Z o ;:: g 0.2
a z 8
:> g 03
u. a w a::
::>
~OA
:>
(f)
<l
>
"' a:: 0.1 w z w
~ a::
~ 0.2
0.3
0.4
o -r--. r-dADIATED
SAMPLE 329B ,15 ...............
6.3x10 nvi t---
1-""::: r:::::::::: ~ r--..UNIRFADIATED
SAMPLE 329 ~ t--'6 2.3x to n('
-
5.7, lD'jnvl ---. "-
B.6"oi6nVI
. --.:::: ~TE~ ~ r---t---
SAMPLE 1262 ~ 1.1,10 r i
I 2.1 x IOt7 nvl
I
3. t x lO17nvt
4.tx1l0t7nvt
I
50 100 150 200 250 300 350
TEMPERATURE ('K)
FIG. 8. Behavior of the Fermi level in n-type silicon after irradia
tion with various amounts of fission-spectrum neutrons.
both interstitial levels were above the upper vacancy
level; an additional increase which does not saturate
should be observed both in pure-and n-type material if
the lower interstitial level were below the upper
vacancy level. Neither of these types of behavior is
observed.
On the basis of Model B various ratios of magnetic
to nonmagnetic states can be postulated for any
amount of filling of irradiation-induced states (i.e.,
position of the Fermi level). However, it is a little
surprising that the number of magnetic centers stops
changing, while the Fermi level continues to drop with
additional irradiation. The model is general enough,
though, to be consistent with the results obtained.
The cluster model easily accounts for a return of less
than the original number of magnetic centers. On the
basis of this model, the absence of continued change in
the number of magnetic centers after heavy irradiation
might mean that the total damaged volume has reached
its final value, i.e., the sample is filled with damaged
material.
4, S. Lack of evidence of discrete levels in the be
havior of the Hall data and the Fermi level: This result
also excludes the simple (A) model but can be explained
by either of the others considered. Interacting irradia
tion-produced centers (B) would create a level dis
tribution if the distances between the interacting
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IP: 141.217.58.222 On: Wed, 26 Nov 2014 01:06:021194 E. SONDER
centers were not all equal. Nonuniform damage (C)
would create space charges, which would lower or lift
otherwise "discrete" levels with respect to the local
Fermi level. Since the energy difference between the
Fermi level and the edge of the conduction band
depends upon the electrostatic potential associated with
nonhomogeneous distribution of defects, there would
be seemingly continuous variations of levels [see
Fig.l(d)].
6. Steep slope of the temperature dependence of the
Hall mobility: It is difficult to account for a slope much
greater than 1.5 in a graph of log mobility versus log
temperature on the basis of point-charge scattering.l3
Similar steep slopes have been observed over small
ranges of temperature and impurity content in samples
where impurity banding effects were becoming impor
tant. For the present case, however, the similarity in
behavior of the two samples which underwent irradia
tion doses differing by a factor of three tends to cause a
little hesitancy in ascribing the behavior to an "imper
fection band."
Steep slopes similar to those observed here have been
observed in the case of n-type germanium. The germa
nium results have been discussed at some length23 and
it seems reasonable that space-charge effects (model C
in the present case) could cause steep slopes in the
temperature dependence of both germanium and silicon.
7. Mobility increase after long irradiation: In order
to explain the result on the basis of point-charge
scattering (model A or B), it must be postulated that
heavy irradiation causes many of the centers to become
either less strongly charged or better shielded. This is
inconsistent with the James-Lark-Horovitz model,
according to which continued irradiation would cause
additional charged centers to be introduced but would
not cause any increase in free carriers to improve the
shielding. On the basis of the more general level,
model B, it is conceivable, however, that pairing of
oppositely charged centers is such that as the Fermi
level drops and the occupation of the centers changes
the net charge of some of the paired centers will de
crease, thus causing a higher mobility to be observed.
The behavior is also consistent with model C, accord
ing to which lowering of the Fermi level in the un
damaged portions of the sample to where it approaches
the Fermi level in the damage dusters would decrease
the space charges surrounding the damaged areas. This
23 Cleland, Crawford, and Pigg, Phys. Rev. 98, 1742 (1955). would decrease the charge scattering, thus increasing
the mobility.
8. Correlation between susceptibility behavior and
Fermi level: As is the case with the mobility, the re
versal of the decrease of number of magnetic centers and
of the point of saturation is correlated with the Fermi
level rather than with the total amount of irradiation.
This behavior is difficult to explain on the basis of the
cluster model (C) since, on the basis of that model, the
return and saturation of the susceptibility would be a
result of overlap of damage regions, which should occur
for the same irradiation dosage in samples of differing
original donor concentration. If model B were assumed,
the susceptibility behavior would essentially be deter
mined by the number of electrons filling a given center.
Correlation between the susceptibility and Fermi level
would then be expected.
A look at Table II will show that the simple model
(A) is inadequate to account for the magnetic and
electrical changes observed after neutron irradiation.
Also, it seems that neither a model based on simple
pairs or small groups of imperfections (B) or one based
on only large damage clusters (C) is quite adequate in
itself. This is not really surprising for as complicated a
process as room-temperature, neutron irradiation of a
semiconductor. It should be recalled that the neutron
transfers enough energy to make possible large clusters
of imperfection but that at room temperature at least
some annealing takes place. lVforeover, in a reactor a
distribution of neutrons of various energies is incident
upon the sample, causing damage centers of various
sizes and shapes to be introduced. Thus, it seems that
the complexity is such that parts of models Band C
must probably be used to account fully for the irradia
tion effects observed in a semiconductor like silicon.
This means that there probably are space-charge
effects and large damaged regions present but that the
effects of filling of states in the forbidden gap is still
all-important.
ACKNOWLEDGMENTS
The help of L. C. Templeton in obtaining some of the
experimental data is gratefully acknowledged. Discus
sions with J. H. Crawford, Jr., H. C. Schweinler, R. A.
Weeks, and other members of the Solid State Division,
Oak Ridge National Laboratory, have been very
helpful and are much appreciated.
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IP: 141.217.58.222 On: Wed, 26 Nov 2014 01:06:02 |
1.1726398.pdf | Investigation of TripletState Energy Transfer and Triplet—Triplet Annihilation in
Organic Single Crystals by Magnetic Resonance and Emission Spectra: Diphenyl
Host
Noboru Hirota
Citation: The Journal of Chemical Physics 43, 3354 (1965); doi: 10.1063/1.1726398
View online: http://dx.doi.org/10.1063/1.1726398
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Published by the AIP Publishing
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130.239.20.174 On: Sat, 22 Nov 2014 17:54:163354 S. J. LADNER AND R. S. BECKER
molecular relaxation time and (2) some steric influence
of the solvent molecules. The internal degradation
process must include the crossing of the triplet-state
molecule to an isoenergetic vibrational level of the
ground state, irrespective of how the excess vibrational
energy is then lost. This loss of vibrational energy mayor
may not occur in the manner described by Porter.12.26-28
More detailed knowledge is needed before further
clarification of the viscosity influence on the radiative
as well as the internal degradation process is possible.
CONCLUSION
It is concluded that a viscosity change can influence
the rate constants for at least two unimolecular de
activation processes, namely, the radiative and non
radiative decay of the triplet state. Although it has not
been shown that such effects:will occur in all cases, it
has been demonstrated for a metalloporphyrin and an
28 G. Porter, Proc. Chern. Soc. 1959, 291.
THE JOURNAL OF CHEMICAL PHYSICS aromatic ketone. The results of this study have pro
vided information concerning the changes in the
internal degradation process quite similar to that
obtained in the case of the aromatic hydrocarbons and
their halogen derivatives studied by Porter, Livingston,
and others. Further, although the directional change
of kl as a function of viscosity may be the same for
different molecular species, this is not true for ko•
ACKNOWLEDGMENTS
One of us (R. S. B.) gratefully acknowledges partial
support for this project by the Robert A. Welch
Foundation and the Council for Tobacco Research. In
addition, S.J.L. wishes to thank the National Science
Foundation and the National Aeronautics and Space
Administration for Fellowship Grants. Thanks are due
Professor J. N. Pitts, University of California, at
Riverside, for the donation of a sample of p-phenyl
benzophenone.
VOLUME 43, NUMBER 9 1 NOVEMBER 1965
Investigation of Triplet-State Energy Transfer and Triplet-Triplet Annihilation in
Organic Single Crystals by Magnetic Resonance and Emission Spectra: Diphenyl Host*
N OBORU HIROTA t
The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois
(Received 14 June 1965)
Temperature and concentration dependence of the decays of the paramagnetic resonance signals from
various guest molecules in their triplet states were studied extensively in diphenyl and diphenyl-d lO host.
Temperature-dependent and -independent delayed fluorescence spectra of the various guest molecules and
their lifetimes were also studied. The observations are discussed in terms of three types of triplet-state
energy transfer and triplet-triplet annihilation. The rates of transfer and activation energies were determined
for the temperature-dependent part of the triplet-state energy transfer. The rates are discussed in connection
with the triplet exciton interaction in single diphenyl crystals.
1. INTRODUCTION
TRIPLET-energy transfer and triplet-triplet annihi
lation processes in organic single crystals have been
studied recently by several authors.1-4 Nieman and
Robinson 2 have demonstrated the occurrence of rapid
triplet-energy transfer among various isotopically sub-
* This work was supported by the U.S. Atomic Energy Com
mission and the National Science Foundation. The frequency
counting equipment used in this investigation was granted from
the Advanced Research Project Agency. t Present address: Department of Chemistry, State University
of New York at Stony Brook, Stony Brook, Long Island, New
York.
1 M. A. El-Sayed, M. T. Walk, and G. W. Robinson, Mol.
Phys. 5,205 (1962).
2 G. C. Nieman and G. W. Robinson, J. Chern. Phys. 37,
2150 (1963).
3 R. W. Brandon, R. E. Gerkin, and C. A. Hutchison Jr., J.
Chern. Phys. 37, 447 (1962).
4 R. M.: Hochstrasser, J. Chern. Phys. 39, 3153 (1963); 40,
1038 (1964). stituted benzenes in benzene-d s at around 4 OK. Stern
licht, Nieman, and Robinson5 gave a theory which ac
counts for their experimental results. Brandon, Gerkin,
and Hutchison3 showed by magnetic resonance studies
that triplet-state-energy transfers from phenanthrene
to naphthalene in single crystals of diphenyl at 77°K.
The presence of this transfer was shown using a filter
which eliminates the excitation of naphthalene and of
diphenyl. From the accurate measurements of the zero
field parameters of phenanthrene and naphthalene, they
concluded that there is no complex formation between,
or juxtaposition of, phenanthrene and naphthalene
molecules. Thus triplet energy must transfer through
diphenyl host molecules. Such a transfer was later
found to occur at temperatures as low as 4°K.6 The
5 H. Stemlicht, G. C. Nieman, and G. W. Robinson, J. Chern.
Phys. 38,1326 (1963).
6 Experiments made by A. Forman (private communication
with Dr. A. Forman).
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130.239.20.174 On: Sat, 22 Nov 2014 17:54:16TRIPLET STATE IN ORGANIC SINGLE CRYSTALS 3355
rate of transfer was estimated to be much greater than
the phosphorescence decay rate in this transfer. Hirota
and Hutchison 7 later found both a different type of
triplet-energy transfer and also triplet-triplet annihila
tion in the same system. This transfer process is
strongly temperature dependent and significant only
at higher temperatures in dilute crystals. Another type
of triplet-energy transfer and annihilation was also
shown to take place at 77°K in the crystals with high
guest concentration, which proceeds with rates compa
rable to the phosphorescence decay rates. Therefore
several different mechanisms of energy transfer and
annihilation seem to be operative in these systems
depending on the temperature and the concentration
of guest molecules. Studies of the delayed optical emis
sion spectra in the same system further confirmed the
presence of the transfer and annihilation processes.
Since the decay of the phosphorescent state is very
sensitive to the transfer and annihilation processes, we
have examined carefully the decays of the paramagnetic
resonance signals due to triplet molecules over a wide
range of guest concentrations and temperatures in
order to study the rates and the mechanisms involved.
We have also studied delayed optical emission spectra
in order to understand the nature of the triplet-triplet
annihilation. We report here the results of these experi
mental investigations. The present work is a continu
ation and extension of the previously published work7
by Hirota and Hutchison on the transfer and annihila
tion in diphenyl crystals containing phenanthrene and
naphthalene.
2. SYSTEMS
The available values of the lowest triplet-state ener
gies of the molecules studied in this work are sum
marized in Table 1. The source of the information is
indicated below the table.
3. EXPERIMENTAL PROCEDURES
All the data on the triplet decays were obtained
from the paramagnetic resonance signals due to ~M =
±1 transitions from the oriented guest molecules ex
cept for the data obtained for 2-naphthylamine phos
phorescence. Experimental details of the crystal grow
ing, magnetic resonance measurements, and delayed
emission measurements are the same as described in
the previous paper7 and are not presented here. All
the magnetic resonance measurements were made at
the microwave frequency, (9.60±0.05) X 109 cycle secl.
Most of the lifetime measurements in diphenyl crystals
were made on crystals mounted on the brass wall of
the resonant cavity with their cleavage planes against
the wall. Absorption peaks with H nearly along the y
axis of a molecule were used to do most of the magnetic
7 N. Hirota and C. A. Hutchison Jr., J. Chern. Phys. 42, 2869
(1965) . TABLE I. Lowest triplet states.
Lowest
Molecule triplet Source
Diphenyl 23 010 crn-1 a
22 800 b
23 000 c
Diphenyl-d 1o 23 120 a
23 100 c
Phenanthrene 21 370 d
Phenanthrene-d lO 21 410 d
Naphthalene 21 100 d
Naphthalene-d s 21 190 d
21 280 e
2,3-Dirnethylnaphthalene 21 230 d
2-Naphthylarnine 20 490 d
I-Methylphenanthrene 21 280 d
Durene 26900
• Determined in the present work. The given fignre is the maximum" for
phosphorescence in dibenzy\ host.
b Determined by G. N. Lewis and M. Kasha in E.P.A. [J. Am. Chern. Soc.
66,2100 (1944)].
o Given by Ermolaev, see Ref. 9.
d Determined in the present work. The given figure is the maximum" for
phosphorescence in diphenyl host.
e Determined in the present work. The given figure is the maximum" for
phosphorescence in durene host.
f Determined by D. Olness and H. Sponer, J. Chern. Phys. 38, 1779 (1963).
experiments because the intensities of these peaks are
greater than those of the x and z peaks. The y axis is
defined here in the same way as in the Refs. 8 and 9.
All the concentrations given in this paper were deter
mined by uv spectral analysis with a Cary Model 14
spectrophotometer.
4. CHEMICALS
All deuterated compounds used in this work origi
nated from Merck Sharp & Dohme of Canada Ltd.
Diphenyl, naphthalene, and durene were obtained from
Eastman organic chemicals. 2,3-Dimethylnaphthalene,
2-naphthylamine, and 1-methylphenanthrene were ob
tained from Aldrich Chemical Company. All chemicals
used in this work were zone refined at least 50 passes
prior to being used for crystal growing.
5. EXPERIMENTAL RESULTS
Treatment of all the data obtained in the Sec. 3 are
made in the same way as described in the previous
paper.7 The main observations are summarized in the
following.
s C. A. Hutchison Jr., and B. W. Mangum, J. Chern. Phys. 34,
908 (1961).
gR. W. Brandon, R. E. Gerkin, and C. A. Hutchison Jr., Chern.
Phys. 41, 3713 (1964).
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130.239.20.174 On: Sat, 22 Nov 2014 17:54:163356 NOBORU HIROTA
18
I I J
16
..
14
12
u 10 ..
II!.
....
8
6
4
2
0
70 90 110 130 150 170 190
T,K
FIG. 1. Temperature dependence of the l/e times in two-component systems. -D.-, 1.45% phenanthrene-d 1o .in d~phenyl; -0-
0.22% phenanthrene-dlo in diphenyl; -$-, 0.06% phenanthrene-dlO in diphenyl; -X-, 1.0% phenanthrene III dlphenyl; -0-
0.07% naphthalene in diphenyl; -_-, 0.06% naphthalene-da in diphenyl; -()-, 0.5% 2-naphthylamine in diphenyl; -e-, 0.25%
phenanthrene-d 1o in diphenyl-d1o.
5.1. Two-Component Biphenyl Systems
In Fig. 1, the temperature dependence of the decays
of the magnetic resonance signals from triplet mole
cules is given for various two-component systems in
which diphenyl is host. The times (hereafter referred
to 1/e times) in which the signal intensities are re
duced to 1/ e of the initial values are plotted against
temperatures. Each point in the figure is the average
of more than two measurements at each temperature.
Concentration dependence of the decay is also seen
from the decay curves in three different crystals with
different concentrations of phenanthrene-dlO' These
curves for different crystals were obtained with simi
lar intensity of light and the same heat filter was used
for each crystal. Standard deviations in the measure
ments of 1/ e times are not greater than approximately
0.4 sec for the curves with 1/e times from 5 to 16 sec,
0.2 sec in the curves with 5-to 2-sec lie time and less
than 0.2 sec in others.
Figure 2 gives the temperature and concentration
dependence of the delayed emission spectra of phenan
threne-d lO, naphthalene-ds, and other two-component
systems at 77 oK. Delayed fluorescence at 77°K was
found to be strongly concentration dependent. Figure
3 shows the decays of the delayed fluorescence of the
diphenyl crystals containing (1) 3.15% phenanthrene-dlo, (2) 0.06% phenanthrene-d lo, (3) 0.05% naphtha
lene-ds, (4) 0.25% 2-naphthylamine at various tem
peratures. In each figure the logarithm of the intensity
was plotted against time. The initial intensity is normal
ized to 35. These figures together with Figs. 3 and 5 of
the previous paper7 show the essential features of the
decays of the triplet-state signals and the delayed
fluorescence in the two-component systems. The main
observations in the two-component systems are sum
marized as follows.
5.1.1. Crystals with Low Concentration
In the crystals of low guest concentration decays of
the triplet signals are exponential at 77°K. The rates
of the decays of the phosphorescence signals start to
increase at higher temperatures as shown in Fig. 1
and deviate from exponential decay. The qualitative
characteristics of the decays of phosphorescence and
delayed fluorescence at higher temperatures are the
same as described for phenanthrene-d lo in diphenyl in
the previous paper.7
5.1.2. Crystals with High Concentration
In the crystals of high guest concentration (higher
than 1.5%) the decay rates of the triplet signals are
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130.239.20.174 On: Sat, 22 Nov 2014 17:54:16TRIPLET STATE IN ORGANIC SINGLE CRYSTALS 3357
(1) (2) (3)
300
A ,mp. A,mp.
300
A ,mp. A,mp. A,mp.
FIG. 2. Temperature and concentration dependence of the delayed fluorescence spectra. (A) Temperature dependence of the
delayed emission spectra of 0.05% naphthalene-d s in diphenyl: (1) 78°K, (2) 124°K, (3) 134°K. (B) Concentration dependence of
the delayed fluorescence spectra of phenanthrene-d lO at 77°K: (1) 3.2%, (2) 1.5%, (3) 0.25%.
greater as much as 5% than in the crystals of low
concentration at 77 oK. Considerable delayed fluores
cence appeared at 77°K in the crystals of high guest
concentration and the intensity increases as the guest
concentration increases. This delayed fluorescence was
found to be almost temperature-independent from 77°
to 90°K.
>-'iii 10 5.2. Three-Component Diphenyl Host Systems
We define the donor as the component whose decay
rate is greater in the three-component systems than in
the two-component systems and the acceptor as the
component for which the opposite holds. In Fig. 4
the decay rates are plotted vs temperature for various
(A) (B)
FIG. 3. Decays of the delayed fluorescence spectra ~ 7
at different temperatures. (A) -e-, 3.15% phen- £:
anthrene-d lD in diphenyl; delayed fluorescence decay, 5,--,-- ,,:.---c;-r
(1) 78°K, (2) 117°K, (3) 1UOK, (4) 123°K.
(B) -e-, 0.06% phenanthrene-diD in diphenyl; 31-----t---f--C--t---t--t----t---I
delayed fluorescence decay, (1) 104°K, (3) 109°K,
(4) 114°K, (5) 119°K, (6) 127°K. -0-, 0.25% 0
phenanthrene-d lo in diphenyl; delayed fluorescence
decay, (2) 103°K. (C) -e-, 0.05% naphthalene
ds in diphenyl; delayed fluorescence decay, 5 6 7
(1) 1WK, (2) 122°K, (3) 132°K, (4) 142°K,
(5) 150oK, (6) 156°K. (D) -e-, 0.25% 2-naph- 50.--.-----,--.-,---.----,-----,
thylamine in diphenyl; phosphorescence decay,
(1) 113°K, (2) 167°K, (3) 184°K, (4) 190oK. -@-, 0.25% 2-naphthylamine in diphenyl; de
layed fluorescence decay, (5) 170oK, (6) 177°K,
(7) 186°K, (8) 197°K. --X --, in Figs. 2 and 3
are decays predicted from Eq. (IV).
o
(C) o 2 3 4 5 9
Time,Sec
50,---,---,--,--,
(D)
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u • NOBORU HIROTA
I 6"1---~---I--
14~----+-----~-----
12~-----~~------~~--------r--
10~--~------------~--------~-----9-+++ + +
+++
I -t-.-i---f---t--
--t----t-
.. 8 :1-------1------..:
6
,
/
I-------I-----~-- ® , 4 i // ...... 0 ...
~," --.... U ..... oft .' ... ' -8-.n::.._---«--.~=.= 8___ -"II _
. ~---.
2
10 eo 90 100 110 120 130
T~K
FIG. 4. Decay rates vs temperature in three-component systems.
Solid lines represent donor decays; broken lines represent acceptor decays. ..............
140 150
Donor decays Decay observed Crystal -0--0-
-.&.--e--x--61--ct--.--b,.-phenanthrene-diD 0.37% phenanthrene-d lO and 0.15% 2,3-dimethylnaphthalene in diphenyl
phenanthrene-d lO 0.45% phenanthrene-diD and 0.07% naphthalene in diphenyl
phenanthrene-diD 0.1 % phenanthrene-d ID and 0.29% 2-naphthylamine in diphenyl
phenanthrene-diD 0.1 % phenanthrene-diD and 0.47% 2-naphthylamine in diphenyl
phenanthrene-d lO 0.19% phenanthrene-d lO and 0.39% 2,3-dimethylnaphtahlene in diphenyl
phenanthrene-diD 0.25% phenanthrene and 0.05% naphthalene in diphenyl
phenanthrene-diD 0.42% phenanthrene-dlO and 0.05% naphthalene in diphenyl-dlO
naphthalene-d 8 0.05% naphthalene-d 8 and 0.68% I-methylphenanthrene in diphenyl
naphthalene-d 8 0.05% naphthalene-d8 and 0.58% I-methylphenanthrene in diphenyl-dlO
Acceptor decays
Same symbols are used to indicate the same crystals.
---0---
---0----x---2,3-dimethylnaphthalene ---ct--- naphthalene
naphthalene --.-. 1.methylphenanthrene
2,3.dimethylnaphthalene --b,.- 1.methylphenanthrene
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three-component systems. The decay rates for the
donors were obtained by a least-squares fit of the
intensity against time with the relation lnIl1o= -kt
in the region where the decays are approximately ex
ponential. Standard deviations in 11k were not greater
than 0.2 sec for all the least-squares fits which were
made. For the acceptors lie times were plotted vs
temperature. Each point given in Fig. 4 at a particular
temperature is the average of more than two measure
ments. The uncertainty to be attached to each point
is estimated to be not greater than approximately
0.3 sec.
In Fig. 5 intensities of the triplet-state signals from
phenanthrene were plotted against time in various
three-component systems with different acceptor con
centrations in order to show how the decay of phenan
threne-d lO is affected by the presence of acceptor. It is
seen that in the crystals of low acceptor concentration
the decays of phenanthrene are nonexponential and the
>-:= 10
<J)
~ 7
C
5
31--+--+-t--
o 2 4 6 8 10 12 14
Time. Sec.
FIG. 5. Effect of the acceptor on the decay of the donor. Decay
of phenanthrene-dlo in different crystals at 106°K; (1) 0.22%
phenanthrene-dw in biphenyl, (2) 0.37% phenanthrene-dl o+
0.15% 2,3-dimethylnaphthalene, (3) 0.17% phenanthrene-d lo+
0.50%1-methylphenanthrene, (4) 0.10% phenanthrene-d w+
0.29% 2-naphthylamine.
decay rates are small, but in the crystals of high ac
ceptor concentration the decays are close to exponential
decay and the rates are larger. These decays of the
donor and acceptor phosphorescence signals in three
component systems when compared with their decays
in two-component systems show clearly that temper
ature-dependent triplet-energy transfer is taking place.
Figure 6-(1) shows the decays of the magnetic reso
nance signals in three-component systems of phenan
threne-d lO and phenanthrene keeping their ratio con
stant (1: 1) and changing the total concentration.
Since the spin Hamiltonian parameters D and E for
the protonated and deuterated phenanthrene are al
most the same, the signal due to one species cannot
be distinguished from that due to the other because
the signals overlap. The intensities of the signals were
measured by observing the value of the maximum or
minimum of the first derivative of the absorption vs
time. It is clearly seen that the decay rates increase
as the concentration increases.
In Fig. 6-(2) the decays of phenanthrene-d lO in the
mixed crystals of phenanthrene-d 1o and 1-methylphe-. Time , Sec Time ,Sec
(A) (B)
FIG. 6. Concentration dependence of the decay at 77°K.
(A) Phenanthrene+phenanthrene-d lo (1: 1); (1) 0.31%,
(2) 0.75%, (3) 1.35%, (4) 2.54%, (5) phenanthrene-dw decay,
(6) phenanthrene decay, (7) rapid transfer decay. (B) Phenan
threne-dw+1-methylphenanthrene; (1) phenanthrene-dlo 0.22%,
1-methylphenanthrene 0.1S%, (2) phenanthrene-dlo 1.11%, 1-
methylphenanthrene 0.S1 %.
nanthrene are shown for two different concentrations.
It is shown that the decay rate of phenanthrene-d 1o in
the crystal of high concentration of 1-methylphenan
threne is much larger than in the crystal of low con
centration and deviates from the exponential decay of
phenanthrene-d 1o and 1-methylphenanthrene, indicating
the transfer of energy. The ratio of the intensities
changes in favor of 1-methylphenanthrene in the crys
tal of high guest concentration, which indicates the
quenching of the phenanthrene-d lO triplet by the en
ergy transfer.
The filter experiments initiated by Brandon, Gerkin,
and Hutchison3 were repeated in naphthalene and phe
nanthrene-d1osystems. Guest concentrations were varied
and the heat filter described in the previous paper7 was
used. By the insertion of a naphthalene filter the phe-
300 (1)
500
A,mfL
A,mfL p (2)
600 300
A,mfL
600
A,mfL
FIG. 7. Miscellaneous delayed emission spectra. (I) Two-com
ponent systems; (1) "'1% 2-naphthylamine in naphthalene at
SOcK, (2) "'1% 2-naphthylamine in diphenyl at 190oK.
(II) Three-component systems; (1) 0.25% phenanthrene-dlo,
0.06% naphthalene at 150°K., (2) 0.06% phenanthrene-dlo,
0.49% dimethylnaphthalene at 145°K.
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nanthrene signal intensities were reduced to about 25%
to 35% of the intensities without filter. The ratio
R= (1/10),'/(1/1 0)1' was measured to be from 0.8 to
1.2 when the phenanthrene concentration was in the
range from 0.05% to 1.5%. Here 1 is the intensity of
the signal with filter and 10 is without filter and Sub
scripts nand p indicate naphthalene and phenanthrene,
respectively. Even in the crystal with 0.05% phenan
threne and 0.05% naphthalene, a considerable amount
of transfer of energy was still observed.
6. DISCUSSIONS
The experimental results described in the previous
section are discussed in terms of different types of
triplet-triplet energy transfer and annihilation in the
following section.
6.1. Temperature-Dependent Energy Transfer and
Annihilation
6.1.1. Effect of the Temperature on the Transfer Rates
A number of investigations of the energy transfer
have been made in glasses and the mechanism of the
transfer has been discussed in terms of the direct ex
change interaction between donor and acceptor. lo.n
Transfer due to this mechanism should not be very
temperature-dependent until softening of the glass
takes place at higher temperature. A different theory
of the mechanism of the triplet-state energy transfer
has recently been proposed by Sternlicht, Nieman, and
Robinson5 which describes the transfer via virtual tri
plet states of the host crystal. Transfer due to this
mechanism should depend very much on the separa
tion of triplet-state energies (!lE) between guest and
host and therefore on the temperature. At higher tem
peratures higher vibrational states of the lowest triplet
state are occupied and a larger transfer rate would be
expected because of small !lE in higher vibrational
states. The effect of temperature on the excitation
transfer has been discussed previously,5.l2 particularly
in systems with small!lE, such as benzene in benzene-dB.
We consider the transfer from three regions of guest
triplet states in the following discussion. Here !lE is
the difference in the triplet-state energy between guest
and host,j{3{3 is the triplet exciton interaction inc~uding
vibrational factor between guest and host. x IS the
vibrational energy of a particular triplet vibronic state.
(a) !lE-x»!f3{3. The formula for the trap-to-trap
transfer rate constant, k, was given by Sternlicht, Nie
man, and Robinson 5 to be
k= (4/h) [(f{3{3) 2 (fp'{3') N-l/ !lEN]. (I) -----
10 V. L. Ermolaev, Soviet Phys.-Usp. 6, 333 (1963) CUsp.
Fiz. Nauk 80, 3 (1963) J. Many examples of rigid-glass experi
ment can be found in this article.
11 V. L. Ermolaev and A. Terenin, Izv. Akad. Nauk SSSR 26,
21 (1962).
12 G. W. Robinson and R. P. Frosch, J. Chern. Phys. 37, 1962
(1962) ; 38, 1187 (1963). This formula is strictly true only at OaK. At temper
atures higher than OaK contributions from higher vi
brational states must be taken into account and the
rate constant will be given by the average over all
possible contributions. The trap-to-trap rate constant,
k, is then given by
k= jk(X)p(x) exp( -k~)dX / jp(x) exp( -k~)dX,
(II)
where k(x) is the rate constant for the transfer from a
particular vibronic state whose vibrational energy is x
higher than the zeroth vibrational state. k(x) is given
by a formula similar to the Eq. (I) and p(x) is the
density of states. However, the contribution from
higher vibrational states in this region cannot be very
important in dilute mixed-crystal systems. Since
k(x) exp( -x/kT) is smaller than k(O) until x ap
proaches !lE, contribution to the transfer from this
region is nearly temperature-independent and small
when the concentration of guest is low and k(O) is
already small.
(b) !lE-x> 0, but not much greater than !f3{3. The
transfer rate from this region would be large and ap
proaches !f3{3/h as !lE-x approaches f{3{3. Previous for
mula (I) based on the perturbation theory, however,
cannot be applied when !lE-x is close to !f3{3.
(c) !lE-xSO. In this region the excitation energy
is transferred from the excited vibrational states of
the guest triplets to the corresponding triplet states
of the nearest host molecules. Then the triplet excita
tion energy will be transferred by a diffusion process
until it reaches another guest. Once the excitation
energy is transferred to the host the process of transfer
will be similar to that in pure crystals. Similar situ
ations in pure crystals have been discussed by several
other authors.13•l4
Although the transfer rate from the region (b) is
not known, we may neglect this contribution compared
with the transfer from the region (c), because!f3{3 is
much smaller than kT around 1000K and the amount
of the transfer from the region (b) may be only a small
fraction of the transfer from the region ( c). Thus
temperature-dependent energy transfer should have a
sharp activation process whose activation energy is
approximately !lE. However, this argument may not
be correct in crystals with small!lE, such as benzene
in benzene-dB, because kT approaches !f3{3.
6.1.2. Kinetics of the Phosphorescence Decay and
Delayed Fluorescence
Predictions from the following kinetic model are in
satisfactorily agreement with the observations on the
13 R. G. Kepler, J. C. Caris, P. Avakian, and E. Abramson,
Phys. Rev. Letters 10, 400 (1963).
14 J. Jortner, S. Choi, J. L. Katz, and S. A. Rice, Phys. Rev.
Letters 11, 323 (1963).
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temperature-dependent energy transfer and annihila
tion. Although most of the kinetic processes have been
discussed in the previous paper,7 we describe them
briefly for completeness:
lk2
GIT~GIT*'
lk~ 1 (1)
1(2) they also involve two triplets. We use hereafter the
rate constant k40 to indicate all annihilation processes
which involve HT and GT. Various possible paths of
triplet-triplet annihilation have been recently discussed
theoretically.6.14 In the discussion of two-component
systems we consider the processes which involve only
GIS and GIT.
The kinetic equations for the decays of the triplet
signals in such cases have been obtained in the previous
paper,7 assuming that the T-S transition (1) and
the annihilation processes (4) are much slower than
the processes (2) and (3) and thus that an instantane
ous equilibrium among GIT, G1T*, and HT is established.
The rate of the decay is given by
2k2
G2T"'---:-G 2T *,
2k~ 2(2) d[GIT J/dt"'_lkl[G IT J
lkg
GIT*+ Hs"'---:-G 1S+ H T,
lky,
2kg
G2T*+Hs"'---:-G2S+HT,
2ka 1(3)
2(3)
Here H means host molecule, Gl means first guest
molecule, G2 means second guest molecule, subscripts
Sand T mean singlet state and triplet state, respec
tively, of molecules, Superscript * means a vibrationally
or electronically excited state, and k's are rate con
stants in an appropriate rate equation. The differences
between triplet state energy of HT and GIT and G2T are
given by !lEI and !lE2, respectively. !lEl<!lE2 is as
sumed here.
The process (1) is the transition from the lowest
triplet state of a guest GT to its ground singlet state
Hs. (We call this T-S transition.) (2) is the thermal
excitation and de-excitation of a triplet state GT• (3) is
the energy transfer from a thermally excited triplet
state of a guest GT to a host Hs and its reversed process,
transfer of the triplet energy from a host triplet HT
to a guest Gs. (4) is the triplet-triplet annihilation
which produces an excited singlet state (Gs* or Hs*)
and a ground singlet state (Gs or Hs). There may be
other types of annihilation processes, such as
k6
HT+ Gr-4HT *+Gs
or
(5)
These processes, however, cannot be distinguished
from the process (4) on kinetic bases only because -(lk401kg[HsJ/lka[GlsJ}[GlTJ2 exp( -!lEI/kT) (III)
under the condition lka[GlsJI»lU[G IT J.
This condition will be approximately correct when
the concentration of the triplet guest molecules is low
relative to that of the unexcited guest molecules. If
this condition is reversed, the rate-determining step
will be the transfer of the energy from the guest to
host, since the triplet energy once transferred to the
host will eventually be annihilated by the other triplet
before it is trapped by another guest. In this case the
annihilation rate will be larger, but no simple formula
tion is possible.
When the decay of the guest Gl can be described
by Eq. (III), the delayed fluorescence intensity is
given by the formula
with { [GIT Jt-O exp( -lklt) }2
100 , 1+ (IKj1k 1)[1-exp( -lklt)]
1K = lk40 lkg[HsJ exp-(!lEl/kT) .
lkg-[GISJ
When lKj1k1«1 is satisfied,
I 00 [GIT J21=0 exp( -21klt). (IV)
(V)
Here, I is the intensity of the delayed fluorescence.
The delayed fluorescence lifetime is thus approximately
half of the phosphorescence lifetime at the highest tem
perature at which the delayed fluorescence is observable.
The experimental results will deviate from this predic
tion when the concentration of the triplet is high rela
tive to the concentration of the guest in its ground
state. In the three-component systems we have to
consider all the processes, but the process
2kg
G2T*+ Hs-4G2S+ HT
can be neglected compared with the other processes
when !lE2 is much deeper trap than !lEI' This condi
tion is satisfied in such systems as phenanthrene-d lo-
naphthalene and phenanthrene-d lO-2-naphthylamine.
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We consider three cases under this assumption. G1
represents donor and G2 represents acceptor, respec
tively.
(a) Low acceptor concentrationj2klf[G2s]«lklf[G18].
The rate of the decay of the donor is given by
d[GIT ]/d~-kl[GIT]- Pka[Hs][GlT J/lkjf[G18]}
Xexp( -AEJkT) Pk40[GIT]+2k3[G28]}' (VI)
The rate of the decay of the acceptor is given by
dt 2k [G ]+ Ika[Hs][GIT] x (_ AEl)
I 2T Ikjf[GIS] e p kT
X {2ka-[G2s]_2k40[G2T]}' (VII)
(b) High acceptor concentration; 2kjf[G2s]»
Ikjf[GIS]. The rate of the decay of the donor is given
by
d[Gldit~_lkl[GIT]-lka[Hs] exp( -AEI/kT) [GIT].
(VIII)
The rate of the decay of the acceptor is given by
In this case triplet-state energy transferred to the host
is eventually caught by the acceptor. The rate-deter
mining step will be the transfer of the energy from an
excited vibrational state of the guest to the host as
suming the Boltzmann distribution for the thermal
excitation.
The rate at which the guest triplet-state energy is
transferred to the host is of special interest here. The
transition is from a guest triplet molecule with the
vibrational energy AE+a to a surrounding diphenyl.
(a is the same order of magnitude or smaller than kT.)
The rate of the energy transfer from one molecule
to another has been discussed by several authorsI2.14.15
using time-dependent perturbation theory. The trans
fer rate of the triplet energy from guest to host Ika[Hs]
can be calculated from
Here I/;'s are electronic and vibrational wavefunctions
for guest and host molecules. Summation represents
the sum over all guest-host interactions.
Robinson and Froschl2 derived a formula for the
excitation transfer rate in a slightly different way:
their formula is given by
Ika[Hs]= I)2/1i2) Tvib(ffJfJ)2. (XI)
It was pointed out by them that this formula is equiv
alent to what is derived from Eq. (X) except a factor
11 D. L. Dexter, J. Chern. Phys. 21,836 (1953). of 7C". Here hfJ is the triplet exciton interaction and Tvib
is the vibrational relaxation time. Taking the average
of the exciton interaction over nearest neighbors, we
get Iks[HsJ= (2/fI,2)mvibUfJP)2, where n is the number
of nearest neighbors and JfJp is the average triplet
exciton interaction.
(c) Intermediate acceptor concentration. In the case
of intermediate acceptor concentration, the triplet
energy transferred to the host goes partly to the
acceptor and partly back to the donor or is annihilated
by the other triplet molecules, depending on the re
spective concentrations of the donor and acceptor.
Temperature dependence of the rate, however, would
be the same as in the other cases. As the concentration
of the acceptor increases compared with that of the
donor, the decay of the donor will approach more
closely exponential behavior.
6.1.3. Interpretations of the Experimental Results
6.1.3.1. Qualitative analysis of the observations in two
component systems. In the crystals of low guest con
centration (concentration is less than 1 %), there is no
significant annihilation and transfer of triplet-state
energy at nOK except the type of transfer discussed in
6.2.2. Phosphorescence decay and delayed fluorescence
in these systems can be analyzed by the kinetic scheme
described in the preceding sections.
Magnetic resonance signal: The analysis of the mag
netic resonance signals of phenanthrene-dlo has been
given in detail in the previous paper.7 The agreement
between the experimental results and the predictions
was satisfactory. Naphthalene-d s in diphenyl also be
haves in a similar way, but the quantitative agreement
between the actual decay and the prediction of the
model is not so good as in phenanthrene-dlo, Other
two-component systems given in Fig. 1 also seem to
behave in a similar fashion, but no quantitative studies
were made.
Delayed fluorescence spectra: From the decays of
the delayed fluorescence shown in Fig. 3, it is seen
that the decays of the delayed fluorescence approach
the exponential decay with a lifetime equal to one
half of the phosphorescence lifetime at the highest
temperature where the delayed fluorescence was ob
served. The initial decay of the delayed fluorescence
of phenanthrene-dlo and naphthalene-ds, however, devi
ates considerably from exponential decay and decay
rates are much larger than the predicted decay rate.
This deviation is probably due to the fact that the
concentrations of the triplet molecules are high and
the condition lklf[ GISJ»lk40[ GIT] is not satisfied ini
tially. This conjecture is also supported by the fact
that the buildup time of the triplet-state signal is con
siderably shorter than the decay time in dilute crystals
of phenanthrene-dlo and napthalene-d s. In the crystals
of higher concentration annihilation of the triplet may
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take place because a small fraction of guest molecules
are relatively close to each other owing to the statisti
cal distribution of guest molecules. Weak delayed fluo
rescence is seen in 0.5% phenanthrene-dlo crystals at
77°K. This is another cause of the deviation from the
prediction in the crystals of higher concentration.
The decay curves of the delayed fluorescence at
higher temperatures behave as predicted by Eq. (IV).
Some of the predicted curves are also shown in Fig. 3.
Similar delayed fluorescence spectra were observed in
many other systems, such as I-methylphenanthrene,
2,3-dimethylnaphtalene and 2-naphthylamine in di
phenyl and 2-naphthylamine in naphthalene. They can
be explained on the basis of the same model.
The characteristic temperatures T. at which the
decay rates start to increase with deviation from ex
ponential decay and, at which crystals of 1 % guest
concentration show strong delayed fluorescence are
approximately 125°K for phenanthrene, 1300K for
I-methylphenanthrene, 145°K for naphthalene, and
185°K for 2-naphthylamine. Triplet-energy difference
between guest and host also increases in this order. If
the electronic interactions and the vibrational over
lapping factors are same for various systems the ratio
between their characteristic temperature T. and the
energy difference !!.E must be same for various com
binations at the same concentration. This is found to
be approximately correct for the above combinations.
Temperature dependence of the decay in two-compo
nent systems with large triplet-energy difference (!!.E)
was also studied in the systems, such as naphthalene
durene and naphthalene-tetrachlorobenzene. Although
the decay rates are temperature dependent, no delayed
fluorescence was observed in these systems at higher
temperatures. Detailed results will be reported else
where.
6.1.3.2. Qualitative analysis of the observations in
three-component systems. Magnetic resonance signal:
The decay behavior of the triplet signals of the donor
and acceptor can be explained by the formulas given
in the preceding section. [Eqs. (VI)-(IX)] Although
the assumptions made to obtain the two limiting cases
are not strictly correct in any of these actual cases,
the following examples are considered to be close to the
two limiting cases.
(a) Low acceptor concentration.
0.45% phenanthrene-d lO+0.07% naphthalene
0.39% phenanthrene-d lO+0.18% 2,3-dimethylnaph
thalene
In these cases the decays are in good agreement with
what is expected from Eqs. (VI) and (VII). Decays
of the donors are not exponential but approach expo
nential behavior when the concentration of the triplet
state molecule is low. Phenanthrene-dlo and naphtha-lene systems were already discussed in great detail in
the previous paper.7
(b) High acceptor concentration.
0.05% phenanthrene-d lo+0.47% 2-naphthylamine
0.05% naphthalene-d s+0.58% 1-methylphenanthrene
In these cases the assumption, 2kJ[~]»lkJ[GlS],
seems to be correct. Both of the decays of the donors
and acceptors are almost exponential and the acceptor
decay rates are not affected much by the presence of
the donor because the donor concentration is relatively
low. The triplet-state energy level of naphthalene-ds is
slightly lower than that of I-methylphenanthrene and
the assumption, lk3[G1T*]»2k3[G 2T*] is no longer cor
rect. Triplet-state energy is probably transferred from
both naphthalene-ds and 1-methylphenanthrene to di
phenyl host. Nevertheless, lkll[GIS]<<(2kJ[G2S] is still
correct and the triplet-state energy transferred to the
host is mostly trapped by 1-methylphenanthrene.
Therefore, the analysis given for the limiting case of
the high acceptor concentration would be approxi
mately applicable until the decay rate of naphthalene
approaches to that of I-methylphenanthrene. The
decay of naphthalene-ds in the presence of I-methyl
phenanthrene is affected by the reversed transfer of
energy from 1-methylphenanthrene to naphthalene at
higher temperatures where the lifetimes of both species
are similar. This is seen in Fig. 8. This temperature
region was avoided for the determination of activation
energies.
( c) In termedia te cases.
The other cases would be considered as intermedi
ate. Decays of the donors are almost exponential.
0.17% phenanthrene-d lO+0.50% methylphenanthrene
0.19% phenanthrene-d lO+0.39% dimethylnaphtha-
lene
0.10% phenanthrene-d lo+0.29% naphthylamine
Delayed fluorescence: The delayed fluorescence ob
served in phenanthrene-naphthalene diphenyl system
is only phenanthrene fluorescence in the crystal of
0.06% naphthalene+0.25% phenanthrene. Even in the
crystal of 0.06% phenanthrene-dlo and 0.49% 2,3-
dimethylnaphthalene, delayed fluorescence from phe
nanthrene is about three times stronger than that of
methylnaphthalene at 140 oK. These results show con
siderable singlet excitation transfer in these crystals in
the period of 10-8 sec.
6.1.3.3. Evaluation of the activation energies. The
activation energies were obtained for several com
binations of donor and acceptor in diphenyl and
diphenyl-d lO• These values were obtained by the same
procedure as described in the previous paper7 by the
least-squares fit of In(k-kl) vs liT. k is the observed
decay rate in three-component systems and kl is the
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0.51---1--
0.3 :-....
0.1 ~\
-,---
\
• 0.07 'A-
: 0.05 1---+---", --+--\ t ~
0.03 1---+---- ~ -1----
~ , FIG. 8. Temperature dependence
ofthe transfer rates. In (k-kl) vs liT.
Same symbols as in Fig. 4 are used to
represent different crystals, except
that -0-indicates phenanthrene
dlo and I-methylphenanthrene crystal.
0.01 ~--+------+------~-----~--------~
0.0071---+------+-------~------r-------~
0.005~----~----------~----------~----------~----------~
e 9 10
;, 30-1
T 10. K
T -,;S transition rate. Obtained activation energies are
tablulated in the Table II.
The observed activation energies are seen to be con
sistent with the model already described. The tem
peratures at which these measurements were made are
relatively high (90°-120°) and kT/hc at these temper
atures is about 60-80 cm-1. Since vibrational factors
vary with energies and kT is relatively large, activation
energy thus determined may differ from the actual
f!l.E/hc by 50 cm-1•
6.1.3.4. Pre-exponential factor and transfer rate. From
the data on the limiting case of the low acceptor
concentration we can estimate the order of 2k~, assum
ing lk3= lk~. 2k~ thus estimated is 1 X 10-13 molecule- 1
cm-3 sec1 for phenanthrene16 and naphthalene in di
phenyl and diphenyl-d lO, and 3 X 10-14 molecule- 1 cm-3
sec1 for 2, 3-dimethylnaphthalene in diphenyl. Since
we do not know the exact concentration of the triplet
molecule, it is not possible to estimate lk40 accurately
but it is considered to be the same order of magnitude
(,,-,10-13 molecule- 1 cm-3 sec1) as the transfer rate,
2k~, from the fact that annihilation process competes
with the transfer process in dilute acceptor crystals
and from the rough estimate of the triplet concen
tration.I6
lk3[Hs] can be estimated from the decay rate in the
16 k. given in the previous paper7 is in error by factor of to.
Estimated k. in the previous paper is also about one order of
magnitUde larger than estimated here. This is mainly due to the
fact that the activation energy determined for annihilation process
is 1520 cm-I instead of 1420 cm-I from the transfer. II 12
limiting case of high acceptor concentration. From
f!l.E= 1425 cm-1 and k=0.25 secI at 1000K for phe
nanthrene-d lO and 2-naphthylamine we obtain Ik3[Hs] =
2X10s secl• In the same way we obtain 3X10s sec1
and 6X lOS secI for naphthalene-ds in diphenyl and
diphenyl-d lO, respectively. Since we have 50 cm-I un
certainties in f!l.E/hc, this number may be off by factor
of 3. From Eq. (XI) we can estimate Jf3~ assuming
Tvib"'1O-I3 sec and taking n=6. Jf3~ is 0.07 cm-I for
Ik3[H.]=2X10 s secI, 0.09 cm-I for Ik3[H.]=3XlO s
secI, and 0.11 cm-I for Ik3[H.]=6X10 s secl•
The estimated rates of transfer and annihilation are
two-orders-of-magnitude smaller than the transfer and
annihilation rates observed in pure anthracene crys
tals.I3 Although the triplet exciton interaction in the
present system is between vibrationally excited triplet
states of phenanthrene and the ground state of bi
phenyl, the numbers given above are much smaller
than the estimated triplet exciton interaction in ben
zeneI7 and anthracene}3.14 Vibrational factor if3 here
involves the vibrational state with energy 1500 cm-I.
The shape of the phosphorescence spectra of phenan
threne, however, seems to indicate that ff3 may not be
seriously different from that for zeroth vibrational
state. Recent theoretical calculations by Jortner, Rice,
Katz, and Chops in fact predict that the triplet exciton
interaction in diphenyl is smaller than in naphthalene
17 G. C. Nieman and G. W. Robinson, J. Chem. Phys. 39,
1298 (1963).
18 J. Jortner, S. A. Rice, J. L. Katz, and S. Choi, J. Chem. Phys.
42, 309 (1965).
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130.239.20.174 On: Sat, 22 Nov 2014 17:54:16TRIPLET STATE IN ORGANIC SINGLE CRYSTALS 3365
TABLE II. Rates and activation energies.
1 2 3 4
Host (B) Acceptor (fh) Ika CBs] (secl)
Diphenyl phenanthrene-d lO naphthalene
Diphenyl phenanthrene-dlo 2-naphthylamine 2X10s
phenanthrene-dlo l-rnethylphenanthrene 5
2ka (molecule-I
• crn-a.secl)
1XlO-la 6
D.E/he from
rates (crn-I)
1411±22
1412±32
1412±36
1481±47
1391±31 7
till/he
spectroscopica
(crn-I)
1440
1440
1440 Diphenyl
Diphenyl
Diphenyl-d lo
Diphenyl
Diphenyl-d lo phenanthrene-d 1O 2, 3-dimethylnaphthalene 3X1O-14
1X10-13 ...... 1440b
phenanthrene-d lo naphthalene
naphthalene-ds l-rnethylphenanthrene
naphthalene-d s I-methylphenanthrene 3X10s
6X10s 1534±40
1671±66
1895±39 1550
1690
1800
a 6.Elhc was estimated assuming that diphenyl triplet energy level is 22 900 em-I and diphenyl-dlo triplet level is 23 010 em-I.
b Estimated on the basis of measurements at only four different temperatures.
::nd anthracene. Assuming ir--0.l, our estimate of
fJ"-'1 cm-I is in good order-of-magnitude agreement
with their calculation.
6.2. Triplet-Triplet Energy Transfer and
Annihilation at Low Temperatures
6.2.1. Triplet-Triplet Energy Transfer and Annihilation
in the Crystals of High Guest Concentration
. ~xperimental results given in Figs. 2-(B), 3, and 6
mdlcate the occurrence of triplet-triplet annihilation
in two-component systems at 77°K when the guest
concentration is approximately 1.5% or higher. Figure
6-(1) shows that in 0.35% and 0.75% crystals, the de
cay of the phosphorescence signal can be reproduced
very well by the superposition of two exponential decays
of phenanthrene-d lO and phenanthrene with lifetimes of
10 and 3 sec and the initial intensities approximately
4 to 1. In the crystals of higher-guest-concentration
decays were found to be much faster as shown in Fig. 6.
Triplet-triplet annihilation in the crystals of high guest
concentration can shorten the decay, but this large
difference cannot be accounted for by only this reason.
The shortening would be accounted for by the increased
energy transfer between deuterated and protonated
phenanthrene in high guest concentrations. If the rapid
triplet-energy transfer from phenanthrene-dlO to phe
nanthrene and vice versa is taking place at a much
greater rate than the T-+5 transition rate, the decay
should be exponential with the rate constant,
k 0.323+0.1 exp( -t:.E/kT)
1 +exp( -!:tE/kT) .
Here, !:tE is the energy difference between phenan
threne-d io and phenanthrene. Taking t:.E=40 cm-I and T= 77 oK, k is given to be 0.25 seci. This decay is
shown by a dotted line in Fig. 6.
The existence of the triplet-state energy transfer
and triplet-triplet annihilation is well known in rigid
glass,1O·1l·I9 when the concentration of the guest is as
high as 0.1M. The main mechanism of the triplet
energy transfer in rigid glass would be due to the direct
exchange interaction between donor and acceptor as
previously discussed,lO.l1 because the energy difference
(!:tE) between host and guest is so large that the
transfer due to the mechanism discussed by Sternlicht,
Nieman, and Robinson would be small. In diphenyl
systems t:.E is relatively small and the transfer due to
this mechanism could be significant.
The observation that the low-temperature transfer
sets in at concentration about 1.5% or higher is con
sistent with the small value of h(3 as estimated in the
preceeding section. Guest concentration 1.5% or higher
is approaching the concentration at which considerable
energy transfer was observed in glasses and the trans
fer by the same mechanism is expected to occur in the
present systems. Therefore present results are not
enough to decide definitely which of the two mecha
nisms is the important one in the present case.
The rate of the transfer of energy due to Sternlicht,
Nieman, and Robinson's mechanism can be given by
Eq. (I) in 6.1.1. If we assume h(3=0.1,,-,0.2 cm-I as
estimated in the preceding section, the transfer rate
calculated using Eq. (I) is faster than T -+5 transition
rate only in case of N ~ 2. Although the accurate num
ber of host molecules between guest molecules is diffi
cult to know and the rate of transfer depends highly
on the mutual orientation of the guest molecules, this
indicates that high guest concentration is required for
the mechanism of Sternlicht, Nieman, and Robinson's
IUT. Azurni and S. P. McGlynn, J. Chern. Phys. 38, 2773
(1963) j 39, 1186 (1963).
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130.239.20.174 On: Sat, 22 Nov 2014 17:54:163366 NOBORU HIROTA
to be the important mechanism. Recently Siegel and
Judeikis20 reported the experiments which support the
view that the transfer due to the direct exchange inter
action between donor and acceptor is important in the
mixed system of diphenyl and ether. Their results and
the present results may indicate that the transfer due
to the direct exchange interaction is still the main
mechanism in diphenyl host.
6.2.2. Fast Triplet Energy Transfer
The existence of the fast triplet-state energy transfer
from phenanthrene to naphthalene was demonstrated
at low temperatures and in crystals of low guest con
centration by filter experiments. This transfer, there
fore, cannot be explained by the mechanisms discussed
in the preceding sections.
The transfer described so far takes place after the
excitation energy degrades to the lowest triplet states
of the guest molecule. On the other hand, in the filter
experiment only the excitation of the acceptor and
host is eliminated, but the possibility of energy trans
fer before the completion of degradation is not elimi
nated. Thus this transfer seems to occur through ex
cited singlet or triplet states of phenanthrene.
It was found that there is no strong dependence of
this transfer on the concentration of phenanthrene.
This might be due to the following fact. In the crystals
of high phenanthrene concentration direct excitation
of naphthalene is reduced because of the stronger ab
sorption of phenanthrene even without naphthalene
filter. Then the excitation of naphthalene may mainly
come from the transfer and the ratio,
is close to unity. In crystals of low phenanthrene con-
20 S. Siegel and H. Judeikis, J. Chern. Phys. 41, 684 (1964). centration naphthalene is more directly excited but a
larger fraction of the excitation energy which is trans
ferred to diphenyl host may be distributed to naphtha
lene, if we assume that the energy is transferred from
phenanthrene to diphenyl and then to naphthalene.
Thus the ratio R may not be very sensitive to the
phenanthrene concen tra tion.
The experiments made so far are not enough to
elucidate the nature of this type of energy transfer.
The transfer of energy may take place through higher
excited triplet states (higher vibrational states or an
other triplet state, if any) by the mechanism discussed
in 6.1.1. This mechanism, however, has the difficulty
that the vibrational relaxation is supposed to be much
faster than the guest-host triplet transfer rate discussed
in the last section. Another possibility is the inter
molecular spin-forbidden energy transfer. A small ad
mixture of singlet character in the higher vibrational
triplet states of diphenyl may allow the transfer of
energy from the lowest excited singlet state of phenan
threne to the excited triplet state of diphenyl. An
example of a similar spin-forbidden intermolecular en
ergy transfer was discussed very recently by Bennet
et al.21 Further detailed studies are required in order to
elucidate this type of transfer.
ACKNOWLEDGMENTS
The author thanks Professor Clyde A. Hutchison Jr.
to whom he is greatly indebted for continuous encour
agement, helpful discussions, and critical readings of
the manuscript. Thanks are also due to Professor D. S.
McClure for his generous assistance in the optical meas
urements, Dr. Arthur Forman and Dr. R. P. Frosch
for helpful discussions, and Clark E. Davoust and
Warren E. Geiger for the construction of apparatus.
21 R. G. Bennet, R. P. Schwenker, and R. E. Kellog, J. Chern.
Phys. 41,3040 (1964).
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1.3057953.pdf | Quantum Chemistry and the Solid State
Herbert A. Pohl
Citation: Physics Today 15, 12, 90 (1962); doi: 10.1063/1.3057953
View online: http://dx.doi.org/10.1063/1.3057953
View Table of Contents: http://physicstoday.scitation.org/toc/pto/15/12
Published by the American Institute of PhysicsMEETINGS
Quantum Chemistry and the Solid State
Like the ever-widening ripples from a pebble dropped
into a pond, the clarifying concepts of quantum me-
chanics spread wider through the years into all prob-
lems of science and life. This was particularly evident
at the recent conference sponsored by the Quantum
Chemistry Institute at Uppsala University and held
at the tiny Swedish resort town of Rattvik in pictur-
esque Dalecarlia, August 27 through September 1. The
Quantum Chemistry Institute, with the stimulating
guidance of Per-Olov Lowdin, specializes in attacking
those problems in chemistry and solid state which may
be formulated at the outset in terms of the Schrodinger
equation.
The symposium dealt with a wide range of problems,
ranging from the four-body problem for the H2 mole-
cule, considerations of density matrices in many-body
theory, solid-state theory, and ligand-neld theory, to
recent work in "quantum biology", including suggestive
considerations of protonic tunneling as affecting gene,
DNA, RNA, and protein synthesis.
It was clearly apparent in the discussions that the
means of application, and even to some extent the
quantum theory itself in its furthest details and in its
time dependency, is still under test. Much of the effort
reported upon at the symposium dealt with the means
available now to circumvent the considerable mathe-
matical and computational difficulties which now beset
the quantum chemist. W. Kolos of the Institute for
Nuclear Research, Polish Academy of Science, Warsaw,
described a successful and precise calculation of the
H^ molecule as a four-body problem, including nuclear
motion and involving eighty terms. This was regarded
by many attending as something of a mile-marker in
testing and applying quantum theory.
Headway in attacking problems with the Schrodinger
equation was disclosed on several fronts. J. Coleman,
P. 0. Lowdin, and Fukashi Sasaki described advances
in the density matrix approach in many-body theory,
whereas Norman Bazley and David W. Fox gave new
methods for determining lower bounds to the energy
levels of atomic and molecular systems. The electron-
electron interaction (correlation) problem was also dis-
cussed in terms of the alternant molecular orbital
scheme (different orbitals for different spins) by R.
Pauncz for hydrocarbons, by George Dermit for dia-
mond, and by J. W. Moskowitz for the interesting
hypothetical molecule, annular Ho.
A statistical theoretical study along the lines of the
Fermi-Thomas approach was described for atoms by
Rezso Gaspar. The evaluation of zeta-function expan-
sions for molecular integrals was described by Mos-
kowitz. Remarks on linked-cluster expansions were pre-
sented by Lowdin. An interesting extension of density-
matrix theory in a Hiickel-type approximation wasmade and applied to conjugated hydrocarbons and
benzenoid hydrocarbons containing heteroatoms by H.
Looyenga of TNO, Delft, Holland.
On hearing the grave and complex computational
difficulties facing present-day quantum chemists, and
of their need to deal with many high-order determinants
of complex integrals, etc., one is repeatedly struck with
the hope that a way out will be found. Perhaps it will
be analogous to the invention of nonunit numerators for
fractions which so greatly eased the problems faced by
early Egyptians who had conceived only the use of
sums of fractions of numerator, 1, to express a given
fractional value.
Bernard and Alberte Pullman, in masterful presenta-
tions, described the considerable progress made in ac-
counting for the relative reactivity and natural selection
of many molecules of biological importance. Particular
success has been had in the interpretation of the role
of enzyme constituents important in redox reactions, in
calculating stability to ultraviolet radiation, in evalu-
ating the role of functional molecular portions (as op-
posed to whole molecules) in carcinogen action, and in
the evaluation of hydrogen bonding through the amino
acid residues as potential pathways for electron trans-
fer. Low7din presented an interesting and potentially
fruitful notion of protonic tunneling between the doubly
hydrogen-bonded base pairs of the double-stranded
DNA molecule. If such a process did occur, it was
pointed out, then inversion of pairing and other in-
formation misstorage could occur. This then has direct
implication in the problems of mutations, evolution,
aging, and tumor inception.
Rembrance was given the perennial problem of phase
determination in electron and x-ray diffraction deter-
minations, by K. Hedberg.
New areas for quantum chemistry considerations
were seen in (1 ) the discussion by Ronald Hoffmann
of the many new polyhedric organic and inorganic mole-
cules of cage-like structure; (2) in the development by
Jan Linderberg of the Naziere-Pines many-electron ap-
proach to the treatment of the dielectric constant of a
solid and the consequent estimation of London inter-
molecular force terms; (3) in the discussion by H. A.
Pohl of (a) the nature of carrier transport vis-a-vis
molecular overlap in molecular solids with special ref-
erence to conductivity and to piezo-resistivity, (b) the
existing gap in the theory of carrier mobility in solids
in the transition ranges between that well described
by wave-packet "drifing", and that describable by
uhopping" processes (i.e., between about 500 and 0.01
cm2/volt sec), (c) the much-needed extension of theory
using random coordinate spacings to the problem of
electronic transport processes in amorphous solids and
liquids, (d) the problem of the near identity of the
9Q PHYSICS TODAY91
!
SARMOUR RESEARCH FOUNDATION
Establishes a New Section for
BASIC & APPLIED RESEARCH
in the
PHYSICS OF FLUIDS
Positions are available now for Scientists to pursue investigations in the following areas
turbulence
planetary atmospheres
surface tension
cryogenic & super fluids
vibrational relaxation
Future studies are projected in the areas of the Dynamics of Biological Mate-
rials, Molecular Bond Cleavage and Short-Lived Microbubble Dynamics.
While Armour engages in both basic and applied research sponsored by
industry and government, individual investigations of merit are supported
by the Foundation itself. More than half of all programs in the Physics
Division are the result of staff-generated ideas.
Research and development are also carried on at ARF in Chemistry; Metals
and Ceramics; Fluid Dynamics and Propulsion; Mechanics; and Electronics.
Interdisciplinary cooperation throughout the Foundation is a customary and
valued function of the ARF staff.
Physicists with advanced degree and a background in the dynamics of fluids,
their electromagnetic interactions or molecular phenomena are invited to
inquire about these new positions. Write in confidence to Mr. Ron C. Seipp.
ARMOUR RESEARCH FOUNDATIONOF ILLINOIS INSTITUTE OF TECHNOLOGY
TECHNOLOGY CENTER • 10 West 35th St., CHICAGO 16, ILLINOIS
An Equal Opportunity Employerdynamics of conducting liquids
cohesive force
adhesive force
viscosity & dynamic rigidity
non-newtwiian liquids
DECEMBER 196292
ELECTRONIC ENGINEERS
PHYSICISTS
METALLURGISTS
An Invitation to Join
a
NEW
RESEARCH
DEPARTMENTThis group is now forming at Bell Aerosystems
Company to perform a variety of investigations
in the aerospace field. Current studies are on
advanced high-performance chemical propellants,
nuclear propulsion systems and electrical pro-
pulsion devices in the very low-thrust ranges.
Other planned projects include energy conversion
for new sources of electrical power for space
equipment, space dynamics, solid state physical
materials, and the effects of radioactivity in the
Van Allen Belt on rocket engine components and
other materials for space applications.
Available to staff members are the most modern
research tools, including an IBM 7090 computer,
and extensive test facilities. In addition, re-
searchers at Bell benefit from the knowledge and
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Inquiries are invited from Scientists and En-
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BELL AEROSYSTEMSco.DIVISION OF BELL AEROSPACE CORPORATION -A fOXtrOtl! COMPANY
P.O. BOX #1 BUFFALO 5, NEW YORK
An Equal Opportunity Employeractivation energy of conduction to the lowest triplet
energy in molecular solids of organic nature, (4) Cole-
man's laudatory reference to the equation of Wentzel
for many particles which is relativistically invariant;
(5) Lowdin's challenging discussion of the reaction-rate
problem in terms of the wave-mechanical evolution
operator for the time-dependent Schrodinger equation.
Lowdin urged a fresh consideration of the evolution
operator in treating kinetic problems and expressed
confidence that it would become a powerful tool.
The attending scientists, from many nations, united
in expressing their deep appreciation for the hospitality
extended them by their Swedish hosts, and for the
stimulating approaches in quantum chemistry presented
at the symposium.
Herbert A. Pohl
Polytechnic Institute of Brooklyn
Calorimetry Conference
The seventeenth annual Calorimetry Conference was
held August 22-24 at the University of California in
Berkeley. Hosts for the occasion were the Inorganic
Materials Research Division of the Lawrence Radiation
Laboratory and the College of Chemistry. Local ar-
rangements were made by a committee consisting of
N. E. Phillips (chairman), R. Hultgren, D. N. Lyon
and I. Pratt.
In keeping with the traditions of previous confer-
ences, a wide variety of calorimetric topics was dis-
cussed, ranging from techniques, through results, to
interpretation. Thirty-seven papers were presented, the
principal one being that given as the Huffman Memorial
Lecture by E. F. Westrum, Jr. (University of Michi-
gan). Professor Westrum, whose topic was the thermo-
dynamics of globular molecules, offered a lucid dis-
cussion of the problems of understanding the behavior
of the so-called plastic crystals. Therm odynamic
measurements on these substances can yield valuable
information about transitions, and about rotation of
molecules and molecular groups in the solid. Much of
the available experimental information has been ob-
tained by Professor Westrum and his students.
Invited papers were given by M. L. McGlashan
(University of Reading), D. Patterson (University of
Montreal), and A. M. Karo and A. W. Searcy (Uni-
versity of California). Each of the papers served to
keynote a particular part of the program. For example,
McGlashan's discussion of the calorimetric determina-
tion of the change of enthalpy of vapors with pressure
and Patterson's application of the Prigogine theory to
the explanation of heats of mixing of polymer solutions
introduced a series of contributions on measurements
of heats of mixing, solution, and dilution.
Karo described the information about lattice vibra-
tions which is obtainable from an understanding of the
thermodynamic properties of crystalline solids, illus-
trating his theme with examples of alkali-halide crys-
tals. In particular, he showed that accurate experi-
mental heat capacities are sufficient to distinguish be-
PHYSICS TODAY |
1.1736056.pdf | Surface Electrical Changes Caused by the Adsorption of Hydrogen and
Oxygen on Silicon
J. T. Law
Citation: Journal of Applied Physics 32, 600 (1961); doi: 10.1063/1.1736056
View online: http://dx.doi.org/10.1063/1.1736056
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/4?ver=pdfcov
Published by the AIP Publishing
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] IP: 130.216.129.208 On: Mon, 24 Nov 2014 22:24:19JOURNAL OF APPLIED PHYSICS VOLUME 32, :\UMBER 4 APRIL, 1961
Surface Electrical Changes Caused by the Adsorption of Hydrogen and
Oxygen on Silicon
J. T. LAW'
Bell Telephone Laboratories, Murray Hill, New Jersey
(Received September 9, 1960; in final form December 8, 1960)
Measuremen~s of conductance, lifetime, change in contact potential with light, and contact potential
have been carne~ out on bombardment-cleaned silicon surfaces and during the adsorption of molecular
oxygen and atomic hydrogen. In the c~se o~ oxy.gen adsorption, the work function increased linearly with
coverag.e. A change of 0.35 ev was obtamed m gomg from 0=0 to 0= 1. Very small changes in the transport
properties were observed. Hydrogen atoms produced an initial decrease in work function of 0.1 ev for
~overages below 0=0.35: From 0=0.35 to 0=1.0 the work function was increased by 0.3 ev. The changes
m the transport properties were substantial and indicated a downward movement of the energy bands at
the s~rface by about 0.08 ev. In the clean condition, the valence band edge was 0.12 -0.14 ev below the
~erml level at the surface compared to 0.36 ev in the interior. The effect of hydrogen adsorption is discussed
m terms of the adsorption data previously obtained on this system.
INTRODUCTION
I~ a previous paper,! some details were given of an
lOn-bombardment annealing treatment which suf
ficed to produce a clean boron-free silicon surface. At
that time a quasi-cylindrical geometry was used which
had certain disadvantages from the point of view of
precise electrical measurements. The principal problem
arose in measuring (LlCPh, the change in contact
potential with light, since the cylindrical grid surround
ing the sample "saw" not only the central cleaned and
annealed region of the sample, but also the cleaned but
not annealed heavier end blocks. In the present work
this problem has been eliminated by changing to ~
parallel plate system where the sensing plate could be
moved away from the sample during the cleaning
process.
In an attempt to elucidate the electronic structure
of the silicon surface, we measured changes of the
following quantities induced by adsorption: (i) surface
c~?d~cta.nce, (ii~ c~ange in contact potential with light,
(m) hfetIme of mJected carriers, and (iv) the contact
potential difference between the silicon and a reference
electrode. From these quantities one can determine
how the ionic double layer and the space' charge are
e~ch affected by ~dsorption and, with certain assump
tions, say somethmg about the electronic structure of
t~e clean sur~ace and the changes induced by adsorp
tIOn. Data WIll be presented for the two cases of (i)
molecular oxygen, and (ii) atomic hydrogen adsorption
both at 300oK. Some fragmentary data obtained a~
higher temperatures will also be given.
EXPERIMENTAL DATA
. !he sar.nples were cut from single-crystal floating-zone
SIlIcon WIth a p-type body resistivity of 150 ohm-em
and had the geometry shown at (A) in Fig. 1. The thin
central section had the dimensions 1.91XO.22XO.05
• Present .addre.ss: Semiconductor Products Division, Motorola
Inc., Phoemx, Anzona.
1 J. T. Law, J. Phys. Chern. Solids, 14, 9 (1960). cm. Before mounting in the tube, the sample ends were
sandblasted and rhodium plated, after which the rest
of the silicon was etched in a mixture of HF and HN0 3•
It was then thoroughly washed in deionized water and
mounted on the stem in molybdenum clips. Small
tungsten heaters were attached to the two outside clips
(B and B') for raising the sample temperature over the
range 300o-460oK. To prevent sputtering of the lead
assembly, the heaters and leads were enclosed in a
molybdenum can C with the sample protruding through
a small slit of dimensions 2.0XO.45 cm.
The rest of the tube consisted of a bombardment
stem, a vibrating plate, and sliders to shield the window
and reference electrode during ion bombardment. After
suitable baking and degassing, vacuums of about
5X 10-10 mm Hg were obtained. With the present elec
trode arrangement, ion current densities of 100 J.l.A cm-2
to the sample were obtained. This current removed
1 X 10-4 em of silicon in about 5X 103 sec. Following the
ion bombardment, the sample was annealed at 13800K
for 2 hr, a time previously! demonstrated to be sufficient
to remove the bombardment damage detectable by
conductivity changes.
Spectroscopically pure oxygen was admitted through
a bakeable valve at pressures from 1 X 10-8 to 1 X 10-4
mm Hg. The source of atomic hydrogen was a hot
~ungsten filament in an ambient of molecular hydrogen
m .the pressure range lX 10-LIX 10-4 mm Hg. In using
thIS method of atomization, there is a danger of con
tamination resulting from the formation of carbon mon
oxide and water vapor at the hot filament but in a . ' prevIous paper2 we gave a detailed account of the
evidence which, we believe, shows that only atomic
hydrogen is adsorbed in our system .
The various electrical properties of the silicon were
then. s~udied in the clean condition and during the
admISSIOn of gas.
(i) The dc conductance was measured with a stand
ard potentiometer-type circuit by passing a constant
2 ].1'. Law, J. Chern. Phys. 30, 1568 (1959).
600
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current through the end contacts and me,~suring the
voltage drop between M and M'. The reproducibility
corresponded to 0.25 ,umho/O.
(ii) The sample lifetime was obtained from a low
frequency photoconductance measurement which was
calibrated against the decay time found by a fast light
pulse technique at various times during the experiment.
Variations in sample lifetime of 0.2,usec could be
readily detected.
(iii) The change in contact potential with light
(LlCP) L was found by means of a high-impedance
detector and wave analyzer connected to the electrode
D, which was held in a stationary position close to the
sample. Since the spacing between sample and plate
varied slightly from one run to the next, the sensitivity
of the circuit was always calibrated by introducing a
standard square-wave voltage of the same frequency
as the light to the sample and determining the resultant
voltage developed on the reference plate D. By carrying
out measurements with the sample biased alternatively
negative and positive, and with no applied voltage, both
the magnitude and sign of (LlCPh could be determined.
The sensitivity of the measuring circuit was such
that a (LlCPh equal to 3X 10-4 v could be detected.
We should like to know if this is sufficient to enable us
to make measurements when an accumulation region
is present at the surface. On using the notation of
Garrett and Brattain,3 the carrier concentration of our
samples was such that A= pO/ni= 1450. These authors
have shown that the limiting values of the ratio
(LlCP)do at extreme values of Yare equal to A and A-I
or 1.45X103 and 6.9XlO-4, in units of kT/e. The
larger value corresponds to an inversion layer, while
the smaller one refers to an accumulation layer. The
quantity is proportional to the number of carriers in
jected by the light. o=wtPiLlV2/Ilrpo2, where w is the
width and t the thickness of the sample, Ll V 2 is the
photovoltage obtained by the passage of current I
while illuminating a length II, Pi is the intrinsic resis
tivity, and po is the sample resistivity. We normally used
values of 0 near 2X 10-1 so that the smallest value of
(LlCPh/1i which we could detect was 3X 10-4/2 X 10-1 or
L5X10-s. In units of kT/e this is equal to 5.7XlO-2;
hence, since this is much larger than the accumulation
layer asymptote (6.9X 10-4), we will be unable to obtain
readings when an accumulation layer is present but only
when an inversion layer exists.
(iv) The contact potential difference (CPD) between
the reference plate and the sample was obtained by the
Kelvin method. With this technique, the reference
electrode is positioned near the sample and vibrated at
some fixed frequency (in the present work 400 cps). A
voltage applied to it is then varied until the sinusoidally
varying voltage as detected by an amplifier connected
to the sample is zero. The balancing voltage then gives
the difference in work function of the two surfaces. The
3 C. G. B. Garrett and W. Y. Brattain, Phys. Rev. 99, 376
(1955). FIG. 1. Schematic arrangement of the experimental tube.
sensitivity of our circuitry was ±O.Ol ev. The use of
this method during the adsorption of gas involves the
assumption that the work function of the reference
electrode is not changing. We have used an oxidized
molybdenum ribbon which has proved satisfactory
except for measurements carried out in oxygen at
pressures greater than 10-° mm Hg. At higher pressures,
as we will see later, some weakly adsorbed oxygen in
creases the molybdenum work function by several
tenths of an electron volt.
A possible problem, in using ion bombardment to
clean the silicon surface before a contact potential
measurement, is the danger of depositing a silicon film
on the reference electrode. One would then see essen
tially no change in CPD during gas adsorption as the
work function of the two surfaces would change by the
same amount. In the present tube, the reference elec
trode was retracted from the sample and protected by
the shield E during the bombardment process.
(v) Some attempts were made to obtain field effect
measurements, but the small capacitance « 1 ,u,uf)
between sample and plate gave much too small a
sensi tivi ty.
RESULTS AND DISCUSSION
A. Silicon-Oxygen System
The sticking probability of oxygen on silicon has
previously been determined4 so that it is possible to
calculate the fractional coverage f) as a function of the
product of the time and the pressure of oxygen to which
the surface was exposed.
The adsorption of a monolayer of oxygen (4X 10-5
mm of Hg X min) produced a very small increase in
4 J. Eisinger and J. T. Law, J. Chem. Phys. 30, 410 (1959).
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IP: 130.216.129.208 On: Mon, 24 Nov 2014 22:24:19602 ]. T. LAW
0'
0.4
<IJ
':i o
,. 0.3
z
~ I
<J
W -' W
~ 0.2
" II.
<I
0.1
o 1-1
•
4 4A FLASH I • ° BO ... BAROU
·0 I-
A •
~ •
~ • •
A
• 9=1
~
It> A 0
l':..t:.~ 0 ~ . 0
A ,.1 • 0
• • • •
-10 7 2 • - 4 e to e - 4 e 8 t 0 $ 4 6 a 1 0-4 2
...... OF Hg x ... INUTES
the surface conductance equal to 1 .umho/D. Because
small temperature fluctuations produced changes in the
bulk sample conductance equivalent to 0.25.umho/D,
we were unable to obtain sufficiently precise data to plot
surface conductance vs surface coverage.
"During the whole of the adsorption process no changes
were detected in the surface recombination velocity
and the value of (~CPh remained below the limit of
detection (i.e., a noninverted surface).
In spite of the small space charge effects which could
be induced by oxygen adsorption, the contact potential
difference between the sample and the reference elec
trode changed by several tenths of a volt. In Fig. 2 we
have plotted the change in contact potential difference
against the exposure product, pXf. Assuming that the
work function of the reference electrode is unchanged,
the value of ~ (CPD) gives us the change in the work
function of the silicon. One can see that the four runs
carried ~ut fall into two groups, the first of which (I)
shows an increase in work function until the monolayer
point is reached followed by a smaller increase, while
the second (II) shows an equal increase to (J= 1 which,
however, is then followed by a substantial decrease.
The difference between (I) and (II) can be attributed
to the treatment of the system before the runs. The
data of curve (II) were taken after the sample had been
bombarded and annealed, while those of curve (I) were
obtained after the sample had been cleaned by flashing
to 1450oK. Previous worko,6 has shown that both' of
these treatments will produce a clean surface, and in
fact the data agree quite well up to exposures of about
4XIo-° mm min (or (J=1). Beyond this point, we
believe that the data depend on the state of the reference
molybdenum plate. In an independent study,1 it has
been found that oxygen exposures greater than 10-4 mm
5 F. G. Allen, J. Eisinger, H. D. Hagstrum, and J. T. Law, J.
App!. Phys. 30, 1563 (1959).
6 H. E. Farnsworth, R. E. Schlier, T. H. George, and R. M.
Burger, J, App!. Phys. 29, 1150 (1958).
7 F. G. Allen and G. W. Gobeli (private communication). I -,,'-, -~.
A ~ .
• •
• n
0
.0 A
~-1-+-
0
• • • FIG. 2. The changes in
contact potential difference
(114)) produced during the
admission of oxygen as a
function of PXt. The data
of curve (I) were taken
after a cleaning flash and
those of curve (II) after
bombarding and annealing.
min will increase the work function of an oxidized
molybdenum ribbon by several tenths of a volt. If this
oxygen is not removed when the silicon (from which it
is protected by the slider E) is flashed, but is removed
during the bombardment process, the data of Fig. 2
can be explained as follows. Curve (I) represents the
true effect of oxygen on silicon when using a reference
plate already saturated with oxygen up to exposure
times of 1 X 10-2 mm min while curve (II) falls following
the completion of the monolayer on the silicon because
subsequent oxygen adsorption on the molybdenum
causes its work function to increase.
The increase of 0.1 ev in the work function beyond
the monolayer region was reproducible and is believed
to be a real effect. Its dependence on coverage could
not be established because of a lack of data for the rate
of adsorption in the multilayer range. The data of Fig. 2
below (J= 1 have been combined with the adsorption
data4 to give a curve of ~IP (the change in work func
tion) against (J (Fig. 3). All the data obtained at each
coverage value fall within the length of the bars shown.
~IP is a linear function of (J up to values of (J greater than
0.9 and is equal to 0.35 ev at (J= 1. Over this range one
can calculate an average value of the effective dipole
moment.u of an adsorbed oxygen atom from the equa
tion ~IP= 41r.uN a, where N a is the number of atoms
adsorbed per cm2• In this way we find that .u=0.12
de bye units (1 debye unit= 10-18 esu). Dillon and
Farnsworth8 reported values of ~IP for exposures of
5 X 10-° mm min which ranged from 0.20 to 0.44 ev
depending on the crystal face and the pretreatment of
the sample.
We also carried out measurements at 4000K and
again found that a monolayer of oxygen ,caused an
increase in work function of 0.35 ev while no changes in
either lifetime or (~CPh could be detected.
The rather large changes in work function at both
8 J. A. Dillon and H. E. Farnsworth, J. App!. Phys. 29, 1195
(1958).
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3000 and 4000K were apparently not accompanied by
an appreciable change in the space charge, as far as we
can tell from the conductance measurements. The
measured changes in work function, of course, would
include changes in both the space charge and in the
external dipole layer. Previous work on etched surfaces
has shown that at least 20% of the change in contact
potential arises from changes in the space charge region.
Two possible reasons for the present observations are
(a) a high density of surface states which is much larger
than what is normally found on etched surfaces, or (b)
the clean surface condition corresponds to an extreme
bending of the energy bands so that changes in the
surface potential could cause the mobility to decrease
as quickly as the hole concentration increased, hence
producing little or no change in conductivity. From the
(~CPh data, the position of the bands at the surface
would have to correspond to an accumulation layer, or
in other words a p-type surface. When we come to
discuss the hydrogen adsorption data we shall see that
the clean surface is indeed strongly p type but, even so,
a high density of surface states (very much greater than
1010 cm-2) is required to account for a change in surface
conductance of only 1 J.Lmho/D while the work function
increased by 0.35 ev.
B. Silicon-Hydrogen System
1. Contact Potential and Photoelectric Threshold
Before describing the present data, it is worth while
to say something about the quantities measured by
contact potential and photoelectric techniques and
0.40
0.35
0.30
II)
!:i 00.25 >
Z
~ t-hl 0.20
...J w
~
.J 0.15
~
<J
0.10
0.05
o /
V / /
/
V
V /
V /
I
V
o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
8
FIG. 3. The change in the work function of silicon produced by
oxygen adsorption as a function of the coverage (0). I
CLEAN
SURFACE
II
H+
OUTSIDE
SURFACE
lIT
H+
INSIDE
SURFACE EC-----~ I
I
I
I I , , , , ,
I
Y'~ET
I ,
I I
I I
EF------------------ ~==~~--i=
-y
Ey ____ .... _"""""=:: ___ ..L
EC-----~
IPPET
I
I , , ,
E ----------------- -----.:t---l--F I --'-___ L--y
Ey ------=-=----.
Ec
EF
Ev I
I
I
I
I
I ,
I
I
I , , ,
Y'PET
I I ----------------- -----~---+-..I-__ :L
-y
---~~:-------__r_
FIG. 4. The energy structure of the semiconductor surface and
the changes produced by the adsorption of protons in (II) "out
side" a sites and (III) deep {j sites.
how these should change during the adsorption of
hydrogen atoms.
The contact potential difference measures the differ
ence in work function between the reference electrode
and the semiconductor so that, assuming a constant
reference electrode work function, the values obtained
during the adsorption of gas give the change in the
energy difference between the Fermi level at the surface
and the vacuum level. The work function is given by
<PSi= (Ec-E F)-Y+x+ VB,
where Ec is the energy of the bottom of the conduction
band in the interior, EF the Fermi energy, Y the surface
potential (the difference between the electrostatic
potential at the surface and in the interior), X the barrier
above the bottom of the conduction band which an
electron would have to surmount in the absence of an
ionic double layer, and VB is the potential across the
ionic double layer. During the adsorption of gas, either
or both Y and VB may change.
Similarly, the photoelectric threshold, which me as-
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] IP: 130.216.129.208 On: Mon, 24 Nov 2014 22:24:19604 .1. T. LA \\.
0.35
0.30
<J)
~ 0.25
~
z 0 0.20 a: I-
<.l w 0.15 ..J w
~ 0.10 .. w .. 0.05 9-
<J
Cl 0 Z <
.~ I' I.' -0.05
<l
-0.10
-0.15 o -V
~
~
0.2 /
/. V
to. ,Y -rr-
~ci~
0.4 /" Vf'
I
I/'I'PET (EISINGER)
/
'Y ~
°V;< )(
/x
~~st
f
0.6
(J 0.8 ~ x
I
1.0
FIG. 5. The change in work function and photoelectric threshold
of silicon as a function of the fraction of the surface covered with
hydrogen atoms.
ures the distance from the valence band edge at the
surface to the vacuum level, is given by
where Ev is the energy of the top of the valence band
in the interior. The photoelectric threshold is, therefore,
sensitive only to changes in the ionic double layer
potential VB. These quantities are shown in the energy
diagrams of Fig. 4.
Let us consider two possible types of sites for the
adsorption of hydrogen atoms, in both of which the
hydrogen, through a charge transfer, is present as
protons.
(a) The hydrogen is adsorbed in such a way that on
the average a surface passing through the center of the
hydrogen atom is nearer the vacuum-solid interface
than a similar surface passing through the surface
silicon atoms. In other words, the protons are adsorbed
on top of the silicon atoms in positions such as are
labeled a sites in Fig. 10. This then would give us a
dipole with its positive end outwards and a resultant
decrease in VB as shown in Fig. 4 (II). At the same time,
such a positive charge would increase the density of
electrons in the surface region and produce a decrease in
-Y. (The negative sign arises because Y is measured
positively downwards.) Hence, both the photoelectric
threshold and the work function would decrease with
the latter quantity changing most.
(b) If, however, the protons resulting from the charge
transfer were adsorbed in deep-lying sites, such as the
(:3 sites in Fig. 10, so that on the average they were
below the surface silicon atoms, the potential drop
across the ionic double layer VB, would increase even
though the value of -Y would still decrease. As shown
in Fig. 4(III), this would lead to an increase in photo
electric threshold, while the work function could either
increase or decrease depending on the relative values of VB and -Yo Whenever ~VB were greater than
l1( -Y), the work function would increase.
With this background information, we will now
consider the available data at 3000K for the changes in
cfJSi and cfJPET produced by hydrogen atoms. In Fig. 5
we show the observed changes in contact potential as
a function of surface coverage (IJ) together with a
similar curve for the photoelectric threshold changes
obtained by Eisinger.9 The bars on the latter curve
indicate the spread of the published data. All of our
data were obtained as a function of time at a given
hydrogen pressure and tungsten filament temperature.
They were then converted to a fractional coverage basis
by carrying out separate IJ vs time measurements using
a flash filament technique. These latter curves had
shapes identical to those previously reported2 in an
adsorption study of this system.
Of the three runs shown, one was taken immediately
after bombarding and annealing the sample and the
other two after subsequent cleaning by heating alone.
Unlike the problem associated with oxygen affecting the
work function of the reference electrode, the pretreat
ment of the system in this case had no effect.
For values of 1J<0.35-0.40, the work function is
lowered from its clean surface condition while the
photoelectric threshold is increased much more slowly
than at higher coverages. Unfortunately, Eisinger's
threshold data at low coverages are open to question as
the sample was still cooling from its cleaning flash and
the values obtained may be too high. We can, however,
put an upper limit on the increase in the potential across
the ionic double layer VB in going from IJ=O to 1J=0.35,
of 0.07 ev. At the same time, the work function dropped
by about 0.09 ev, so that if the resulting protons were
adsorbed in a sites, the surface potential Y would have
had to change by 0.16 ev in the direction of a less p-type
surface. We shall see in the next section that the surface
conductance is unchanged in going from IJ=O to 1J=0.3
so that such a change in Y is unlikely. If the value of Y
were indeed unchanged, one would require that the
threshold decrease by an amount equal to the change
in work function on this coverage range so that all of
the changes were occurring in the external dipole layer.
Such a decrease in the threshold could then only be
explained if the hydrogen were adsorbed in a sites out
side the surface. From the available data, we feel that
this is the more likely explanation, but more accurate
data in the low-coverage region is required to decide
between the two possibilities.
For values of IJ> 0.35 there is a steady increase in
both the threshold and the work function so that the
hydrogen adsorption in this region must be occurring
in deep sites of the (3 type. Before one could reliably
deduce changes in Y from the two curves shown in
Fig. 5, one would require accuracies of better than 0.01
ev in both sets of data. This is obviously not the case at
9 J. Eisinger, J. Chern. Phys. 30, 927 (1959).
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] IP: 130.216.129.208 On: Mon, 24 Nov 2014 22:24:19SURFACE ELECTRICAL CHANGES IN SILICON 605
present so that we will use the transport properties in
later sections to arrive at this information. We can,
however, put an upper limit on ~(-Y) of 0.11 ev. in
going from 0=0.35 to 0= 1, with the surface becommg
less p type.
From the equation ~¢=47rN ajJ., we can calculate the
average dipole moment j.L of hydrogen atoms in the two
types of sites. The initial. decrease in ~ork fu?ction
gives a j.L of 0.08 debye umts for the a sItes, wh~le the
subsequent increase in ~¢ gives 0.15 debye umts for
the i3 sites.
Some contact potential measurements made at 3500K
were essentially identical with the 3000K data except
that the changes were apparently somewhat larger. In
going from 0=0 to 0= 1, the work function increased
by 0.3 ev rather than the 0.2 ev found at 300oK.
The only previous report on the effect of hydrogen
on the work function of silicon was by Dillon and
Farnsworth8 who found that exposures of 2XlO--8 mm
min with an ion gauge in the system produced a decrease
of 0.05 ev. This change is similar to what we observed
up to coverages of 0= 0.35. To increase the .coverage
beyond this region requires much longer tlmes (or
higher partial pressures of hydrogen atoms) as the
sticking probability is decreasing quite rapidly.2 This
may be the reason that the subsequent increase which
we observed was not reported by the previous authors.8
2. Surface Conductance
In Fig. 6, the change in surface conductance (in
j.Lmho/D) is plotted as a function of coverage for a
number of separate runs on two different samples. At
coverages below 0=0.2, little or no change occurs but
above this value, the conductance decreases steadily
until, at 0= 1, a decrease in surface conductance of
24 j.Lmho/D is found. In this respect, the data are
similar to those obtained for the work function de
scribed above.
If we knew the position of the energy bands in the
clean condition, we could deduce changes in the space
charge from the conductance measurements, since the
surface conductivity is a unique function of the surface
potential for a given bulk resistivity. Kingston and
Neustadter10 have given solutions of Poisson's equation
which can be used to construct a curve of ~u vs Y by
using the bulk mobility values for the carriers. It is
possible that, at the extremes of Y, such a curve is in
error as a result of a scattering reduction of the mobility
as described by Schrieffer.l1 Silicon, however, has a
sufficiently long Debye length that this should only
introduce appreciable errors near the degenerate condi
tion. From the values of (~CPh described below, we
know that the surface is becoming less p type during
the adsorption of hydrogen. The sign of the change in
10 R. H. Kingston and S. F. Neustadter, J. App!. Phys. 26, 718
(1955).
11 J. R. Schrieffer, Phys. Rev. 97, 641 (1955). 26
24
22
20
18
~ 16 o
"-
~ 14
:I
::; I
3-
b
<]1
I 2
0
\I
sf--
4
2
0 o I 1 .f r
I rr
'I fi
f
~
t /f
o jt-y.
0.2 0.4 i
0.6
e 10..
aV-
7
0.8 1.0
FIG. 6. The change in surface conductivity (i.n JLmho/D)
as a function of the fraction of the surface covered WIth hydrogen
atoms.
surface conductance then tells us that the clean surface
condition was at some negative value of Y. If this were
the case a 24-j.Lmho/D conductance change is possible
only if Y at 0=0 were equal to or more negative than
-8.4. If the clean surface were exactly at Y = -8.4,
the value of Y would have to change by + 24 since a
conductance decrease of 24 j.Lmho/D would be possible
only if the adsorption of a monolayer of hydrogen
shifted the energy bands at the surface to the point of
the minimum in conductivity.
We can obtain an estimate of the change in Y from
the photoelectric threshold and work function changes
described above. In going from 0=0.2 to 0= 1.0, the
threshold increased by 0.33 ev while the work function
increased by 0.25 ev. Assuming that the hydrogen is
adsorbed as protons in deep sites the difference of 0.08
ev must be equal to the decrease in -Y or about 3.
Even allowing for possible errors in the two sets of data,
the value of (-Y) should be between +3 and +6
rather than the change of +24 estimated above. This
means that the clean surface must be more strongly
p type than would correspond to a Y value of -8.4.
For extreme values of -Y, the surface conductance
derived from Poisson's equation reduces to
G=ej.LpTp=ej.Lpn,£, exp[!i3(¢-1f8)],
where £, is the Debye length, ni is the number of intrinsic
carriers, j.Lp is the hole mobility, i3 is equal to e/kT, ¢ is
the Fermi level, and 1f .• is the surface potential (1f.,=
1fo+ Y, where 1f() is the electrostatic potential deep in
the interior).
On using data calculated from this equation, we can
determine the values of Y for the clean surface which
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TABLE 1. Variations in barrier height as a function of the frac
tion (0) of the Silicon surface covered by Hydrogen atoms. (All
data except (EF-EV) are given in units of kT/e.)
0 Y (calc) (-Y) (calc) VB (exptl) EF-Ev (calc)
0 -9.0 0 0 0.12
0.2 -8.95 -0.05 0.5 0.12
0.4 -8.3 -0.70 2.8 0.145
0.6 -7.5 -1.5 8.2 0.16
0.8 -6.5 -2.5 11.8 0.19
1.0 -6.0 -3.0 12.5 0.20
will lead to a conductance change of 24,umho/D for
a change in -Y of +3 and [these are the extreme
values of (-Y) obtained from the work function and
photothreshold data] +6. The values of Y obtained
are between -8.5 and -9.0.
For these values of Y in the clean condition, the
valence band edge at the surface must lie below the
Fermi level by 0.12 to 0.14 ev. These may be compared
with the value of 0.13 ev previously reported! for a
similar sample cleaned in the same way.
The principal criticism of this type of analysis lies in
the use of bulk mobilities for the carriers when we are
dealing with extreme values of Y. For our particular sam
ple, the surface becomes degenerate (i.e., the valence
band coincides with the Fermi level) when Y = -13.6 so
that at values of Y near -9.0 we are farfrom this condi
tion. Even if we assume a 50% reduction in mobility at
Y = -10, a change of -Y of 6 to produce a conductance
change of 24 ,umho/D would still require a clean surface
condition of Y= -9.5 to -10. Unfortunately, without
extremely accurate threshold and work function data
on the clean surface, or independent measurements of
the carrier mobility, we are unable to obtain better
estimates from the present data of the energy state of
the clean surface.
It appears that for the clean surface the most reason
able value of Y is -9 so that we can now use the con
ductivity data and Poisson's equation to calculate the
variation in Y with coverage. In Table I we list such
calculated values obtained using bulk mobilities,
together with the experimental data for the potential
0.10
0.06
> 0.06
'"
~
~ 0.04
0.02
o 0.4
I~ l/ 8=1.0 ./
... k ~
~ V
V V
o 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
l>VB IN eV
FIG. 7. The change in potential across the ionic double layer
plotted against the change in Y for the adsorption of hydrogen
from 0=0 to 0= 1. change in the ionic double layer VB (from the photo
electric threshold). We can see that, as in the case of
oxygen adsorption, the changes in the ionic double
layer potential are considerably larger than the simul
taneous changes in the space charge or internal poten
tial. Calculated values of (EF-Ev), the distance of the
valence band edge below the Fermi level at the surface,
are also given. If one plots the change in internal barrier
against the change in external barrier (from Table I),
the data can be described by a straight line (Fig. 7) so
that over the whole range ()= 0 to ()= 1.0, VB is approxi
mately five times larger than the corresponding change
in Y. This ratio is essentially the same as the one ob
served by Brattain and Bardeen!2 for the changes
produced by gas adsorption on etched germanium
surfaces.
The change in conductance was also studied at 350°
and 485°K. At the latter temperature, the sample was
in the intrinsic range and no conductance charge was
observed. At 3500K, however, a decrease in conductance
of 15,umho/D occurred in going from ()=o to ()= 1.
While this was not studied in detail as a function of (),
it appeared to be much like the 3000K data, and the
difference in magnitude between the two could be
largely a result of a changed mobility, which in bulk
silicon!3 decreases by about a factor of 2 in going from
300° to 3500K.
Another possible reason for the difference is that the
hydrogen adsorption could occur at different sites at the
higher temperature, but a detailed study of both the
adsorption and the surface conductance would' be
required to support this hypothesis.
The only previous work with which the present data
can be compared is that of Heiland and Handler14 who
studied the change in surface conductance of germanium
in the presence of atomic hydrogen. They found that
the conductivity increased; i.e., an initially p-type
surface became more strongly p type. This would imply
that hydrogen on germanium ionizes to give a negative
ion, unlike the present indications on silicon where posi
tive ions must be postulated to explain the data.
3. Surface Recombination
To obtain values of the surface recombination velo
city (s), we need to know both the filament lifetime T!
and the body lifetime Tb since
where VB, the surface decay constant, is related to s.
Since the value of Tb changed during the vacuum
treatment, it was determined as follows at the end of the
experiment. The sample was removed from the tube
and given a chemical treatment to produce a low surface
12 W. H. Brattain and J. Bardecn, Bell Svstcm Tech. J. 32, 1
(1953). .
13 F. J. Morin and J. P. Maita, Phys. Rev. 96, 28 (1954).
14 G. Heiland and P. Handler, J. App!. Phys. 30, 446 (1959).
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recombination velocity.ls Values of Tf were then meas
ured and quite accurate estimates of Tb obtained since s
could be reduced to about 100.
In all of our measurements s was quite high (>4X 103
cm secl), but as the data shown in Fig. 8 indicate, it
increased during the adsorption of hydrogen. The data
show considerable scatter which could be caused by
small changes in the body lifetime, at least in going
from one run to the next. The increase during adsorption
is small, but the direction of the change is in agreement
with our picture that the bands are moving nearer the
fiat-band condition, if we assume that no new recom
bination centers are being created by the adsorption
process. No data on clean silicon surfaces are available
for comparison, but Buck and McKim,16 working on
etched silicon surfaces, found increases in s from 2X 102
to 3 X 103 over the same range in surface potential for
a sample of comparable resistivity. Surface recombina
tion velocities have been measured on clean germanium
surfaces and smalll6 or zerol7 changes obtained by
adsorbing a monolayer of oxygen.
4. Change in Contact Potential with Light (I1CPh
As described in the experimental section, the quantity
in which we are interested is the ratio of the measured
(I1CPh to the number of injected carriers o. Because
of limitation on the sensitivity of our measuring system,
we were unable to detect signals arising from an
accumulation layer, so that all of the experimental
values obtained were negative quantities indicating an
inversion layer at the surface.
xt03
S.O
5.S
T 5.2
U
UJ
III
::::E
U
~ 4.8
If)
i
A 4.4
•
4.0 o A
~ .4.
• A •
~
A
r
0.2 0.4 •
I-•
0.6
(3 •
A •
•
0.8 • •
• • A
1.0
FIG. 8. The surface recombination velocity s (cm sec1) as
a function of the fraction (fJ) of the surface covered with hydrogen
atoms.
15 T. M. Buck and F. S. McKim, J. Electrochem. Soc. 105, 709
(1958).
16 J. T. Law and C. G. B. Garrett, J. App!. Phys. 27, 656 (1956).
17 H. H. Madden and H. E. Farnsworth, Phys. Rev. 112, 793
(1958). -1.6
-1.4
-1.2
-1.0
il~ -0.8
"·-0. 6
• •
-0.4 •
• • •
-0. 2 ... ....
•
0 o 0.2 0.4 • "
• • •
•
0.6
e •
• i
.-~-
• .-I---
• ••
•• • •
A
0.8 1.0
FIG. 9. The change in contact potential with light per unit of
carrier injection (ACP)L/Il as a function of the fraction of the
surface covered with hydrogen atoms.
In Fig. 9 we have plotted (I1CPh/o (in units of
kT / e) as a function of the amount of hydrogen adsorbed.
The first thing to note is that these curves prove that
the adsorption of hydrogen atoms makes the surface
region progressively less p type. This is the only direc
tion of band movement which would produce an
increasingly large negative signal. On the other hand,
the observed decrease in surface conductivity could
arise from a surface which moved toward the fiat-band
condition from either extreme of surface potential.
Let us consider for a moment whether it is possible
to predict the value of the surface potential at which
(I1CPh/o goes through zero. Garrett and Brattain3
have given expressions for (I1CP)L/o as a function of Y
for various densities of surface states. If the surface
acceptor and donor trap densities, N a and N b, are very
large,
(I1CPh/o ex: (1-e2Y1)/ (1 +-2e2Y'),
where Y'=Y+!ln(AWa/N b) and (I1CPh/o=O at
Y= -! In(AWa/Nb).
If Na=N b, the zero point is at Y = -! InA2 which for
our sample is at Y=-7.25. A better estimate from
our data is that (I1CP)L/o=O at Y= -9.25. This would
require that the ratio N a/ N b were equal to 50 rather
than unity. About all we can say is that the ratio ob
tained is not unlikely. With no traps, on the other hand,
the zero value would have to occur at Y = O. This is
obviously not the case.
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t
>\!l a:
UJ z
UJ
I~-_" I PLANE THROUGH
CENTERS OF SURFACE I SILICON ATOMS
DISTANCE X _
FIG. 10. The potential energy profile for the adsorption of
hydrogen atoms on silicon. The two types of sites, a and {3, are
indicated.
The conclusions of this section are then that while
the surface photoeffect is of great value in determining
whether the surface is p or n type, the results do not
shed much light on either the density or location of the
surface traps.
5. Comparison of Electrical and Adsorption Data
In a previous paper2 we reported the variation in the
sticking probability (5') as a function of coverage. The
sticking probability represents the fraction of the
particles incident on the surface which are adsorbed, so
that the higher the sticking probability, the higher the
binding energy between the atom and the substrate.
For the 0<0.15, 5' is constant and equal to unity; it
then decreases by two orders of magnitude in going to
0=0.4, after which there is another constant region
until the effect of monolayer completion sets in. If we
compare this with Figs. 5 and 6, we see a striking
similarity in that most of the change in the electrical
properties is occurring during adsorption in the coverage
range 0=0.3-0.9, where 5' is only changing by a factor
of 5 (out of the factor of 104 in going from 0=0 to 1.0).
We shall postulate that there are two types of sites
for adsorption on the clean surface. In Fig. 10 is shown
a potential energy diagram of the surface region with
the two types of sites labeled a and {3, where the a sites
are "above" the average surface, while the {3 sites are
"below" it. Therefore, if adsorption into both a and {3
sites were accompanied by an electron transfer in such
a way as to produce protons, the a adsorption would
decrease the work function while the {3 adsorption
would cause it to increase. In passing, we should note
that the silicon work function is sufficiently high
(4.8-4.9 ev) that the formation of H-ions is unlikely.
It is also true that the changes in (ACP)L and con-ductivity observed in the present study are explicable
only if we are dealing with the adsorption of positively
charged particles in the 0 range 0.3-1.0 where the work
function is increasing.
If we accept the energy picture shown in Fig. 10, the
initial stage of the adsorption process will fill the a sites
with a high sticking probability (since there are no
activation energy barriers to overcome) and cause a
decrease in the work function. The experimental data
indicate that there need to be about 2.5X 1014 such sites
per cm2 or 0=0.2-0.3. However, since this adsorption
must involve a transfer of charge (the work function
changes), the reason why the charge in the space charge
region does not change must be that the charge transfer
occurs into surface states.
The decrease in work function amounted to 0.1 ev so
that, if we treat the external barrier region as a parallel
plate condenser with a charge separation of 3 A, this
corresponds to a charge in this region of 1O-IX3X10-6
or 3X 10-7 coul cm-2• If one charge goes into each
surface state, we require about 3X 10-7/1.6X 10-19 or
2X 1012 acceptor states cm-2, a number close to what
has been previously postulated for clean silicon surfaces.
If such a number is chosen, the states would need to be
close to the valence band edge for the occupancy to
approach unity. The physical origin of these surface
states, and, therefore, also of the high sticking-proba
bility adsorption sites is by no means obvious. When
a surface is formed, it is likely that some bond formation
takes place between neighboring silicon atoms, and
Farnsworth,18 using a low-energy electron diffraction
technique, has observed spacings between surface atoms
considerably different from what is found in the bulk.
If this pairing process occurs in a random way, then
some surface atoms will be left with no unpaired
neighbors and may be the source of the observed sites.
Our highest estimate for such unpaired surface atoms,
however, appears to be equivalent to 0=0.15 which is
of the right order, but somewhat lower than the cover
age at which we observe a change in the surface
conductance.
For coverages above 0=0.3, the sticking probability
drops by a factor of 100, and we assume that this
marked decrease in the rate of adsorption corresponds
to adsorption in the {3 sites, which involves an activation
energy as shown. If these sites are below the average
surface of the silicon, ionization of the hydrogen to
produce a proton will lead to an increase in the silicon
work function. This model of deep and shallow sites is
identical with one proposed by Gundry and Tompkinsl9
to explain the adsorption of hydrogen on nickel, and
they also pointed out that a decrease in work function
followed by an increase at high coverages could be
explained by it.
18 R. E. Schlier and H. E. Farnsworth, J. Chern. Phys. 30, 917
(1959).
19 P. M. Gundry and F. C. Tompkins, Trans. Faraday Soc. 52,
1609 (1956).
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SUMMARY
The present data show fairly conclusively that a clean
silicon surface is more strongly p type than the bulk,
with the valence band edge some 0.12 ev below the
Fermi level for a 150 ohm cm p-type sample. Adsorption
of oxygen appears to make the surface even more p type,
while atomic hydrogen makes it less p type.
Before detailed calculations of surface state distribu
tions can be meaningful, one preferably needs field effect data so that accurate comparisons of 2:,88 and
2:,sc can be made over a wide range of surface potential.
ACKNOWLEDGMENTS
We wish to acknowledge the expert glass blowing and
tube assembly of G. J. Gass and T. F. Chase, Jr., which
made the measurements possible, and the able assistance
of E. Ploom.
JOURNAL OF APPLIED PHYSICS VOLUME 32. NUMBER 4 APRIL. 1961
Response of a Thermocouple Circuit to N onsteady Currents
THOMAS T. ARAI AND JOHN R. MADIGAN
Roy C. Ingersoll Research Center, Borg-Warner Corporation, Des Plaines, Illinois
(Received August 26, 1960)
The response of a thermocouple circuit functioning as a Peltier cooler to time-varying currents was
determined by assuming that the current density could be represented by the sum of a dc and a time-varying
component. The time-varying component took the form of either an impulse applied at time to>O, a square
pulse lasting from to to t1, a step increase in the current at time to, or a sinusoidal ripple superposed on the
de current. The increased current results in an initial thermal cold spike at the cold junction but the time
average temperature difference between the junctions is reduced unless the dc current is well below the
optimum value. The possibility of using such thermal spikes in a very long wavelength infrared communica
tions system or in synchronous detection is discussed. In the case of a sinusoidal ripple the temperature
difference between the junctions may either follow the fluctuations in current or may not, depending on the
time constant of the couple and the frequency of the ac signa!. In the latter case the only effect is a reduction
in the temperature difference between junctions by the additional Joule heating due to the ac component.
I. INTRODUCTION
THE transient response of a thermocouple circuit
functioning as a Peltier cooler has previously been
calculated for steady currents.1-3 It is possible to show
that one cannot achieve greater temperature differences
between the hot and cold junctions by initially applying
currents greater than the optimum steady state current.
In fact, for arbitrarily large currents, the maximum
temperature difference between the junctions ap
proaches 2/1r of the value corresponding to the maxi
mum temperature difference for the optimum dc current.
However, it is experimentally observed that one can
momentarily achieve greater temperature differences by
superposing some time-varying current upon the opti
mum dc current. It is the purpose of this paper to
investigate quantitatively the effect of such time-vary
ing currents on the transient response of the couple
within the framework of certain simplifying assump
tions. A general expression is derived for the effect of the
1 A. D. Reich and J. R. Madigan, J. App!. Phys. 32, 294 (1961).
2 N. Alfonso and A. G. Milnes, "Transient response and ripple
effects in thermoelectric cooling cells," paper presented at the
AlEE Winter General Meeting, New York, January 31-February
5, 1960.
3 L. S. Stilbans and N. A. Fedorovich, Zhur. Tekh. Fiz. 28,
489 (1958) [English translation: Soviet Phys.-Tech. Phys. 3,
460 (1958)]. time-varying component on the transient response and
detailed results are obtained for several explicit forms
of the time-dependent current.
II. NONSTEADY CURRENTS
The one-dimensional heat equation applicable to a
thermocouple circuit with no refrigeration load is
ae[x,t,j (t)] a2e[x,t,j(t)]
Cv k +pp-Pjo(x-a/2) (1)
at ax2
with
e[O,t,j (t)] = e[a,t,j (t)] = ° (2)
and
e[x,O,j(O)] = 0,
where P is the Peltier voltage at the cold junction, p is
the resistivity, jet) is the current density, and k and
Cv are, respectively, the thermal conductivity and heat
capacity per unit volume of the material in the thermo
couple arms. It has been assumed that the thermocouple
arms are equivalent as far as the magnitudes of p, k,
and Cv are concerned and that these quantities and Pare
not temperature dependent. Since P=dST[a/2,t,j(t)],
where dS is the difference in Seebeck coefficients be
tween the arms of the couple and T[a/2,t,j(t)] is the
temperature of the cold junction at time t, the Peltier
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1.1703744.pdf | Exact Wave Functions in Superconductivity
Daniel Mattis and Elliott Lieb
Citation: Journal of Mathematical Physics 2, 602 (1961); doi: 10.1063/1.1703744
View online: http://dx.doi.org/10.1063/1.1703744
View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/2/4?ver=pdfcov
Published by the AIP Publishing
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IP: 131.111.164.128 On: Wed, 24 Dec 2014 09:08:22JOURNAL OF MATHEMATICAL PHYSICS VOLUME 2, NUMBER 4 JULY-AUGUST, 1961
Exact Wave Functions in Superconductivity
DANIEL MATTIS AND ELLIOTT LIEB
IBM Research Center, Yorktown Heights, New York
(Received February 9, 1961)
The ground-state wave function and some of the excited states of the BCS reduced Hamiltonian are
found. In the limit of large volume, the boundary and continuity conditions on the exact wave function
lead directly to the equations which Bardeen, Cooper, and Schrieffer found by a variational technique.
It is also shown in what sense the BCS trial wave function may be considered asymptotically exact in this
limit. Finite-volume corrections are included in an appendix, and explicit calculations are carried out for
a one-step model of the kinetic energy which has possible applications to the problem of the finite nucleus.
I. INTRODUCTION
WE wish to find the ground-state wave function
and some of the elementary excited states of
H=L E(k,s)Ck,8*Ck,8-VL L Ckt*C-H*Ck,~Ck't. (1)
k,. k k';;"k
The operators C and C* are the usual Fermi operators
and anti-commute. The sums are restricted to an
immediate neighborhood of the Fermi surface, which
includes 4n distinct states of momentum (k) and spin
(s= t or .), and which are popUlated by 2n electrons.
In other words, our eigenfunctions must be simul
taneously eigenfunctions of the number operator 'T/
(2)
k,.
with eigenvalue 2n.
Our Hamiltonian is the famous "reduced Hamilton
ian" of the BCS theory; and for an introduction to the
present work, we refer the reader to Sec. II of the
BCS paper.! In their notation, n=N(O)hw, where
N(O)=density of states at the Fermi surface and
hw=typical phonon energy. As has been stated, we
wish to investigate the nature of the exact solutions
to this problem, and we shall see that they are very
similar to what BCS found by a variational calculation.
For the purposes of finding the ground state, it is
convenient to think in terms of a pseudo-Hamiltonian
tJ which has the same ground state as (1). First, by
time-reversal symmetry, we may assume that E(k,t)
= E(-k,.). Second, it is clear that, in the ground state,
all electrons must be paired, as in Ckt*C_u*, because
unpaired electrons do not benefit from the attractive
interaction. Following BCS, then, we define
(3)
and consequently, the ground state of the pseudo
Hamiltonian
H=2L Ekbk*bk-VL L bk*bk, (4)
k k';;"k
coincides with the ground state of H. [We have set
Ek= E(k,t).] Indeed, every eigenstate of H is a state of
1 J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,
1175 (1957). H, but the converse is not true. The bk operators have
mixed commutation properties and may not be re
garded as Bosons for which the diagonalization of (4)
would be trivial. In fact, they are a set of Pauli
operators.I
A complete set of states for our problem consists of
all possible configurations of n pairs, of which a typical
member is
(5)
where {k} i is a set of n different k's chosen from the
2n permissible values. There are (2n)!/ (n!)2"-'22nj (1I'n)!
different rP/s. For comparison, the totality of con
figurations (allowing an arbitrary occupation number)
is 22n. For every rPi there is a corresponding amplitude,
which we may write as /ir=f[S(k 1)· • ·S(k2n)], where
S(kj) = 1 or 0 according to whether kj is in the set
{k} i or not. It is to be understood that f is not defined
for all possible values of its arguments (of which there
are 22n) but only for those values such that ~)j S(kj)=n.
The general eigenfunction of H is therefore
if;= L firPi. (6)
config.
The problem now consists of finding the ground-state
amplitudes 1> and the corresponding energy. For some
insight into the general problem, we first turn to the
strong-coupling limit which is well understood.
II. STRONG-COUPLING LIMIT2
We set Ek=O, and the Hamiltonian is simply
Hs.c.=-VL L bk*bk,.
k k';;"k (7)
As this is purely attractive, we may safely assume that
the ground-state wave function possesses all the
symmetry of the Hamiltonian. The outstanding sym
metry property is invariance under the interchange of
any two momenta k and k'. Therefore, one may pre-
2 The strong-coupling limit is generally well understood. An
exhaustive treatment of this limit, including a perturbation
theoretic approach to weak coupling, is given by Wada and
Fukuda, Progr. Theoret. Phys. (Kyoto) 22, 775 (1959).
602
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IP: 131.111.164.128 On: Wed, 24 Dec 2014 09:08:22EXACT WAVE FUNCTIONS IN SUPERCONDUCTIVITY 603
sume that
f( ... S(k)· . ·S(k') ... ) = f(· .. S(k')· . ·S(k)· .. ), (8)
i.e., that f is a symmetric function of its arguments.
Now we make use of the property that S(k)=O or 1,
which is expressible as
S(k)=S2(k). (9)
Consequently,3 the most general function which obeys
Eq. (8) can be written as
f(S(k 1)·· 'S(k2n»= fer. S(k»= fen). (10)
k
But as n is a constant, f must be constant and hence
all amplitudes are equal in the strong-coupling ground
state. We can check this directly:
(-vI: I: bk*bk,)·f(n) I: II bk*jO)
k k';"k i kS(kJ.
=E •. c.f(n) I: II bk*jO), (11)
i
with f(n) = n!(2n!)-!"-'2-n(n·n)1 for normalization. This
is the Schrodinger equation, and each complexion is
connected to n2 other complexions. Therefore,
E •. c.=-vn2, (12)
a well-known result. It may be useful to recall that n
and V-I are both proportional to the volume (for fixed
density), so that E is an extensive property of the
system. Eq. (12) is in perfect agreement with the
BCS result taken in the strong-coupling limit, but is
in slight disagreement with the calculation of Wada
and Fukuda,2 who include a diagonal term -v I:bk*b k
in their interaction. There is no particular significance
in their discrepancy.
TIl. ONE-STEP MODEL
The number of sign changes (or nodes) in the
amplitudes f is a good quantum number, and by the
adiabatic theorem, its value persists as Ek is changed
from a constant value to some arbitrary function. We
make use of this to solve for the ground state of a
model which is not quite so trivial as the strong
coupling limit, and which may be of interest in the
nuclear problem where energy levels are discrete. We
shall assume that Ek is a step function-zero over half
of the states and equal to a positive constant (E) over
the remaining states.
The ground-state amplitudes must be nodeless
functions which are symmetric under the interchange
of any two pairs within the same half-space. Let the
occupation numbers over each half-space be,
no= I: Sk and n.= I: Sk. (13)
k such that k such that
Ek=O Ek=E
3 This theorem was kindly pointed out to us by Dr. D. Jepsen
and Dr. T. D. Schultz of this laboratory. We can eliminate no by the relation
n= no+n.= const, (14)
and therefore the ground-state amplitudes are a
function of n. alone, and are denoted f(n.). The
equations for the amplitudes are simply
[2en.-E-2vn.(n-n.)]f(n.)
= v(n-n.)2f(n.+ l)+vnN(n.-l), (15)
where n. assumes integer values from zero to a maxi
mum of n. These equations are easily soluble when n
is a small integer. For example, if n= 1, there are only
two amplitudes, f(O) and f(l), and the eigenvalue
equation is the usual determinantal condition
I-E
Det -v -v 1 =0
2E-E ' (16)
which has the solutions
(17)
The lower of these E_ is the ground-state energy and
belongs to the nodeless solution f(l)/f(O) >0, as
expected.
For large n, the determinantal equation is impractical,
and we now use a method for isolating the ground-state
energy from all the other solutions in the limit of large
volume, n -t 00. Corrections in the form of an expansion
in n-1 are discussed in the Appendix, and may be of
value already for n;::: 3, when the determinantal method
is cumbersome.
Because the amplitudes can be chosen real and
positive in the ground state, we write
f(n.) = constenS(x), (18)
where X= n./ n, and S is a real function. Next, we
divide both sides of Eq. (15) by nf(n.) and find
2EX-W -2Xx(1-x)
=X[(1-x)2p(x+1/n)+x2/p(x)], (19)
where
p(x)=exp{n[S(x)-S(x-1/n)]}, (20)
W=E/n, and X=vn. (21)
The variable x goes from 0 to 1 in steps of 1/ n. One
can now proceed to the limit n -t 00, but first one
notes that
lim exp{n[S(x+1/n)-S(x)]}
n ..... '"
= lim exp{ +n[S(x)-S(x-1/n)]}
n ..... '"
=exp[a/axS(x)], (22)
provided Sex) is a sufficiently smooth function.
Therefore, to order l/n if Sex) is sufficiently smooth,
p(x)= p[x+ (l/n)], and Eq. (19) turns into an algebraic
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IP: 131.111.164.128 On: Wed, 24 Dec 2014 09:08:22604 D. MATTIS AND E. LEIB
equation
2EX-W -2Xx(1-x)=X{ (1-x)2p(x)+x2/ p(x)}, (23)
which is subject to the requirement that p(x) be real,
positive, and continuous. The conjecture that Sex)
approaches a continuous limit function as n --+ 00,
which implies that p(x)'""'p(x+1/n) and satisfies Eq.
(23), which in turn implies that Sex) has a limit
function is certainly self-consistent. But it need not
be true. Equation (19) is a nonlinear difference equation,
and in order to get from the point x=o to the point
x= 1/4 say, we must iterate it n/4 times. The assertion
that p(x+1/n) may be replaced by p(x) will result in
an error of order l/n. But since it takes n/4 steps to
get to x= 1/4, we may accumulate an error or order 1,
in which case p(1/4) will not satisfy Eq. (23). Once
p(x) ceases to satisfy the quadratic equation, we see
from Eq. (19) that p(x) will oscillate wildly. In the
Appendix we prove that the errors do not in fact
accumulate in the regions (O,m) and (n,l) where m
and n are the least and greatest points, respectively, at
which the discriminant of Eq. (23) vanishes. For the
ground state, the discriminant vanishes at only one
point and, hence, in this case, our smoothness assump
tion is justified everywhere except in a small neighbor
hood about the vanishing point. There are three critical
points: at x= 0 and 1, and at the turning point where
the discriminant vanishes.
The "boundary conditions" are as follows: at x=O,
p(O)=-W/X, (24)
which follows from Eq. (19) at x=O. Obviously, W
will have to be negative or zero. At x= 1,
p(1)=X/(2E- W), (25)
which follows from Eq. (19) at x= 1. At intermediate
points, the quadratic equation possesses two solutions
p(x) 2EX-W-2Xx(1-x)
2X(1-x)2
±_1_[(2EX- W-2XX(1-X»)2_x2]i.
1-x 2X(1-x) (26)
The boundary conditions impose the positive root
near x=O and the negative root near x= 1. Therefore,
at one intermediate point, the discriminant must
vanish so that the transition from positive to negative
root may be continuous. The reality condition is
translated into the requirement that the discriminant
have a minimum at this "turning point" where it
vanishes. Thus, simultaneously, we require
[(2Eh.-W-2Xh,(1-h,»)2 ] D= _h,2 =0,
2X(1-h.)
h.
. p(h,)=-, (27)
1-h. where x= h. is the turning point (by analogy with the
BeS notation) and
aDI -0 ax h.-. (28)
It does not follow, however, that
is discontinuous at the point x=h" although it always
remains finite. Equations (27) and (28) possess a
solution provided X~ E/2,
1( E) 1-E/2A
h'=2 1-2X ' p(h.) 1+E/2X' (29)
and
W= -X[1-E/(2X)]2. (30)
Recalling that X=vn and E=nW, we find for the
ground energy in the one-step model:
( E)2 E Eo .•. = -(vn)(n) 1--- , for (vn)~-.
2 (vn) 2 (31)
For (vn)=A<!E, the turning point sticks at h,=O,
and one finds that only the negative solution is required
for reality and continuity, provided W =0. Therefore,
Eo.s.=O for (vn)::::;;tE. (32)
Had we used the BeS trial function, the results would
have been identical. As we shall see in Secs. IV and V,
this is no coincidence, even though the BeS trial
function is not an eigenfunction and does not conserve
particles. It may also be easily verified that these
results agree with the strong-coupling theory if we
set E=O, even as to the constancy of the amplitudes
fen,) in that limit. For the excited states, we turn back
to the Hamiltonian in its original form given in Eq. (1).
The low-lying excited states are relatively easy to
find in the one-step model. We break up a pair, putting
one electron in an Ek= 0 state, and the other in an
Ek= E state. There are (n-1) remaining pairs for
which (n-1) Ek= 0 states are accessible, and an equal
number of Ek= E states. The energy of the "singles" is
E singles=O+E= E, (33)
and the lowest possible energy for the remaining pairs
is [substituting (n-1) for n in our previous result]
E(n-l) = -(v) (n-1){ 1 E ]2
2v(n-1) , (34)
provided v(n-1)~ !E, and zero otherwise. Thus, the
excitation energy ~ associated with such excited states
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is (calculated to leading order in the volume) from which it follows that in the limit n -'> co,
f!.=E singles+E(n-1)-E(n)=2(vn), if (vn);:::!E, (35) p",.=l/P.,.', E~E'. (43)
and
f!.=E, if (vn):::;;k (36)
It is interesting to note from Eq. (35) that unless E
exceeds a critical value, this energy of excitation is
independent of E, hence, is the same as for the strong
coupling limit. This shows an amazing rigidity in the
ground-state wave function.
IV. SOLUTION FOR ARBITRARY FUNCTION £k
Proceeding with a knowledge of the one-step model,
we can now derive the BCS equations for an arbitrary
function Ek. We do this by approximating Ek as closely
as we please by a staircase function. If we call the
number of states in the step about some discrete E, N"
then as n -'> ~, N, -'> ~. Thus, no matter how "fine"
the staircase, each step will always have an infinite
number of states associated with it. The limit to a
continuous function E(k) is taken after the limit n ---+ ~,
but always the number of steps on the staircase is
regarded as large. We shall assume that E varies from
a minimum value EF-hw to a maximum of EF+hw,
where EF is the unperturbed Fermi level and ftw is the
energy of the typical phonons responsible for the
attractive interaction v. We define the population of
the portion of phase space belonging to Ek in a given
complexion by
(37)
k such that
As before, n, can vary by integer steps from zero to a
maximum value N ,. If we denote a sum over distinct
energy shells, (i,e., a sum over the steps in the staircase)
by the usual summation symbol with superscript E, we
recall that Once again we have assumed that p.", approaches a
continuous limit function. If we extend this definition
to include the special case e'= E, p",= (p.,,)-l= 1, then
in our limit, Eq. (40) simplifies to
2L:'N.EX.-E
=!v L:' N, L:" N"
X {(l-x,)x"p",+ (l-x,.)x,/ P"..)' (44)
Each po"~, is required to be real and continuous in the
ground state, with respect to variations in any of the
independent variables x., or of the parameters E and E'.
For example, we must find
lim P.",= 1, and p." .. p.",.'= P.,", (45)
E'=E
but these conditions will be trivially satisfied by our
solution. To investigate the continuity with respect to
the independent variables, we isolate an arbitrary
term on the right-hand side of Eq. (44), and combine
all the other terms with the left-hand side. Thus,
a={3p",+ ("1/ p".,), (46)
where
(l-x"')x" }
X {(l-X")X"'p, .. "",+ ,
PE/,E"
~ (E e')
( /I '") , E ,E
~(ef,E), (46a)
(3=!vN,N.,(1-x,)x", (46b)
and
'Y=!vN,N,.(l-x.,)x,. (46c)
and (38) The "boundary conditions" are
p".,=a/(3 when "1=0, (47)
(48) L:'N.=2n.
The algebraic equations for the amplitudes are
[2 L:' en,-E-v L:' n,(N,-n.)Jf(·· "n."" "n,,·"·)
= V L:' L""'" (N,-n,)n" (39)
Xf(· . " (n.+ 1)· " . (n,,-l)· . "). (40)
We let
f(· . ·n.· . ·n .. · .. )= expnS(· . 'x,' . 'x,,' .. ), (41) P",'='Y/a when (3=0,
whereas the general solution is
(49)
Continuity might require that at some point p"" have
a cusp. That is, the discriminant must vanish at some
point, and
where x,=n,/N" and again divide both sides of the a=2({3'Y)!. (50)
equation by the amplitude f( "'n," ·n,,···). One The reality condition requires that
defines
f(··· (n,+l)'" (n,-l»)
f(· "n," ·n, .. ·) (42) (51)
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where t is representative of any variable in the problem.
This is already quite similar to the one-step problem,
and suggestive of the BeS equations, but the derivation
is not yet complete. Anyhow, for each pair (e,e'), there
exists a value of x. and x.' (which we shall denote h.
and h.,) for which
_~_ (~)t _ (l-h,,)h,)l p",'-- -
2f3 f3 (1-h.)h" (52)
by Eqs. (49) and (SO). At this point, Eq. (44) reads
E=2 L' N.eh.
-v L' N. L" N.{h.h.,(l-h,)(l-h.,)]t. (53)
We also investigate Eq. (44) in the neighborhood of
this point. Let n.=N.·h.+on, n.,=N,,·h.-fm, and
all other occupations remain fixed. For infinitesimal
on, one finds a differential equation, which after some
simplification reduces to
(l-hflf)h"} 1 d X{ (l-h")h'" --p'''.""lx,=h.
trE",E'" N~dxf:
-vN,(1-2h,)=const. (54)
In general, we don't know the value of djdx,(p.",.",),
not even at the point in question. However, it is
finite, and by Eq. (52), its coefficient vanishes.
2e-v( 1-2h, )
[h,(l-h.)]!
XL'" N"{h,,,(l-h, .. )]!=const. (55)
Following BeS, this is solved by defining the gap
parameter EO
EO=V L'" N.,,[h.,,(1-h.,,)], (56)
from which it follows that
(57)
where E= e-1 const. To determine this constant, we
refer back to Eqs. (38) and (39) which, upon being
combined, yield the condition
(58)
It is easy to see that this constant is the chemical
potential for a pair 2/.1, which is conventionally deter
mined by the condition that the total number of
particles be fixed, as here. If N. is approximately a constant function of E, then Eq. (58) can be written as
(59)
and it is seen that jJ is independent of eo and is equal
to its unperturbed value which we denote by EF•
Otherwise, one defines
rjJ(E)=N.jN2~,
and Eq. (58) becomes
iEF+IlW E
dErjJ(E) O.
EF-Ilw (E2+Eo2)t (60)
(61)
This is an implicit equation for the chemical potential
and, in general, jJ can be a function of Eo.
The ground-state energy is simply obtained by
substituting the values of h. determined by Eqs. (57)
and (58) into Eq. (53), as in reference 1.
This concludes our derivation of the equations of
superconductivity based on an analysis of the properties
of the exact eigenfunction of the reduced Hamiltonian
(1). In the following section, we conclude our verifica
tion of the BeS theory by showing that the point
{x,} = {h.} is a stationary point, in the sense that as
n ~ co, the contribution of the various configurations
to the wave function becomes essentially a delta
function centered about this point, and that, therefore,
the BeS trial function (or any other trial function
which is correct in the neighborhood of this point)
becomes asymptotically exact in this limit, and not
just the variational energy.
V. THE STATIONARY POINT
In the limit of infinite volume, only certain con
figurations contribute significantly to the wave-function
normalization integral, and also in the calculation of
matrix elements to the low-lying excited states. We
have seen that the BeS equations are exact in the
neighborhood of a certain point in occupation-number
space. We shall now show that this is also the stationary
point, and that the BeS wave function correctly
weights the relative amplitudes of different con
figurations in the neighborhood of this point, provided
care is exercised in conserving particles.
We investigate the one-step model,4 for which the
wave-function normalization requires
n [ nl ]2 1= L f(n,),
n.=O n,l(n-n,) 1 (62)
The first factor is the number of ways we can have the
occupation number n., i.e., the number of distinct
configurations belonging to the same value of n,. In
4 The generalization to the model of Sec. IV would be repetitious
and will be omitted.
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the limit n ----t 00, both this factor and .f(n.) are very
rapidly varying functions of n., and most of the
contribution comes from a neighborhood of the point
where the summand has a maximum. (The sum could
be replaced by an integral at this point and evaluated
by the method of steepest descents.)
Let the stationary point be at n., and let us factor
from the sum the value of the summand at this point.
( n!j(n.) )2( (n-n.)2.f(n.+1) n.2 1= 1+ +---n,l(n-n.)! (n.+1)2 .f(n.) (n-n.+1)
.f(n.-1) (n-n.)2(n-n.-l)2.f(n.+2)
X +------------------.f(n.) (n.+ 1)2(n.+2)2 .f(n.)
n.2(n,-1)2 f2(n.-2) ) + + ....
(n-n.+l)2(n-n.+2)2 p(n.) (63)
In our limit,
( n!j(n.) )2( (1-x.)2 (1) 1= 1+ P(x.)-O -
ii.l(n-ii.)i X.2 n
xl 1 (1) (1-X.)4 + 0 -+ _ p4(Xi)
(1-X.)2P(X.) n X.4
(1) X.4 1 -0 -+._--
n (l-x.)4 p4(X.) o(~)+ ... ). (64)
To order 1/ n, all the terms in the neighborhood of
x.=n./n must contribute equally, therefore,
p(x.)=x./ (1-x.). (65)
However, comparing this with Eq. (27), we see that
x.=h., (66)
and indeed the stationary point is the same as the
turning point at which the discriminant of Sec. III
vanished. As this is the only point of interest in the
calculation of the normalization integral (and of low
lying matrix elements), we must verify that the trial
function has the right amplitudes at and near this
point.
The BCS function is
1/1= II ([1-h(Ek)]t+[h(Ek) ]tbk+)! 0), (67)
k
and is evidently normalized. For the one-step model
. (Ek=O or E), h. is the same as in our Eq. (29), and
ho= 1-h •. Decomposing the function (67) into con
figurations of distinct no and n., we find that the trial
amplitudes do correctly depend only on these parame
ters, but that
no+n.;:rfn, (68)
so that the trial function does not conserve pairs, as has already been noted. For any fixed value of no+n.,
the ratio of the trial amplitude for the configuration
(no+n.) to the trial amplitude corresponding to
(no+q, n.-q) is
BCS ratio of amplitudes= (h./[l-h,])q, (69)
and is correct for any finite positive or negative integer
q (in the limit n ----t 00). Moreover, the average value of
no+n. in the trial function is n; therefore, such quanti
ties as the energy, which are insensitive to the exact
number of particles, can be accurately computed with
the trial function, as we have already discovered in
the preceding sections. This ratio is incorrect for very
large values of such that q/n;:rfO, except in strong
coupling, where the ratio is correctly given as unity
for all q. This suggests that the trial function (or the
equivalent Bogoliubov transformation) be handled
with some care; but because it is correct at the station
ary point, this function does asymptotically, and on
the average, approach the exact eigenfunction of the
problem as n ----t 00. Many investigators have already
shown that the variational ground-state energy of the
reduced Hamiltonian is exact in an asymptotic sense, 5-7
but as the variational theorem does not imply an
equivalent accuracy in the wave function, the present
analysis has not been in any sense redundant.
APPENDIX
This section is rather mathematical and concerns the
intrinsic error in approximating the nonlinear difference
equation for the p functions by a quadratic equation
such as (23) or (46). Once we establish that the error
is of order n-1, we can calculate this error to leading
order to see the effect of finite-volume corrections on
the theory.
The error analysis proceeds in several steps. We
shall show that:
(a) p(x) approaches a limit function as n ----t 00 and
that this limit function obeys the correct boundary
conditions provided the discriminant vanishes at least
at one point in the interval (0,1).
(b) The lowest energy is such that the discriminant
vanishes only at one point, the "critical point."
(c) The limit function which p(x) approaches is the
solution to the quadratic equation, except in the
neighborhood of the critical point.
Let the primitive equation be (for simplicity, we
depart slightly from the notation in the text)
a(y)p(y)p(y+ 1/n) -2lJ(y )p(y)+c(y )=0,
O~y~1, (A.I)
where this equation holds for all y=integer/n in the
interval; and let g(y) be the solution to the quadratic
6 P. W. Anderson, Phys. Rev. 112, 1900 (1958).
6 J. Bardeen and G. Rickayzen, Phys. Rev. 118,936 (1960).
7 N. N. Bogoliubov, D. N. Zubarev, and Yu. A. Tserkovnikov,
Soviet Phys.-JETP 12, 88 (1960).
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equation
a(y)g2(y)- 2b(y)g(y)+c(y)= o. (A.2)
The coefficients have the properties, c(O)=a(l)=O,
b(y)~O. First, we show that if Yo and Yl are, respec
tively, the least and the greatest points at which the
discriminant D(y) vanishes,
(A.3)
then p(y) approaches a continuous limit function as
n -'> 00, in the regions (O,Yo) and (Yl, 1). The proof for
the first region is as follows: let
g(y+1/n)=g(y)+(1/n)n(y), (A.4)
and
p(y)=g(y)+ (ljn)S(y). (A.S)
If we choose the correct solution to (A.2) in this
region, namely,
b(y)+[D(y)]i
g(y) a(y) , (A.6)
it can be directly verified that S(y) is of order unity in
the immediate neighborhood of the point y=O. We
must now show that this function remains finite on
the interval (O,Yo). The function n(y) can be obviously
calculated and is of order unity if we exclude a neighbor
hood of the point Yo. It is also of order unity in that
neighborhood if
~DI =0 (as in the ground state).
oy Y=YO
Now, we calculate p(y+ljn) by two different methods.
Using Eqs. (A.4) and (A.S),
p(y+ l/n)= g(y)+ (l/n)[n(y)+S(y+ l/n)], (A.7)
and using the primitive equation
( 1) 2b(y) c(y) P y+- .
n a(y) a (y)p(y) (A.8)
Eliminating p(y) by Eq. (A.S), we also assume that
S(y) is of order unity, and, therefore,
( 1) 2b(y) c(y)
P y+-
n a(y) a(y)g(y)[l + S(y)/n·g(y)]
1 c(y)S(y) ( 1 ) =g(y)+ +0 - .
na(y)g2(y) n2
Comparing Eqs. (A.7) and (A.9), we find
S(y+ lin) = M (y)S(y)-n' (y),
where (A.9)
(A. 10)
c(y)
O<M(y) <1 for y<yo, (A.lla)
a(y)g2(y) and
n'(y) = n (y) + order l/n. (A.llb)
This difference equation is far simpler than the original
equation (A.1). Now, we want to show that S(y+1In)
is finite. An upper limit to S is S,
S(y+ 1jn)=M (y)S(y)+w, (A.12)
where w=Maxln'(y)l, and is known to be finite. The
solution to this equation is
S(y+ljn)
=w(l+M(y)+M(y)M(y-l/n)
+M(y)M(y-ljn)M(y-2In)+···), (A. 13)
and if M(y) is the maximum value of M in (O,y),
S(y+l/n) <w/l-M(y), (A.14)
and is always finite for y<yo.
A similar proof goes through for the other interval,
except that one chooses the other root of the quadratic
equation to make p and g agree at y= 1.
Now, if we use the fact that bey) decreases mono
tonically with the energy eigenvalue, then we see that,
if the energy is too low, the discriminant can never
vanish in (0,1); and both boundary conditions cannot
be obeyed by a continuous function [which we have
shown p(y) to be]. The lowest value of the energy for
which D(y)=O in the interval is such that YO=Yl,
i.e., the discriminant vanishes only at one point. Then
we have shown that as n -'> 00.
b(y)+[D(y)]!
p(y)=g(y) y<yo, (A.1S)
a(y)
and
b(y)-[D(y)]t
p(y)= g(y) y>YO. (A.16)
a(y)
Our analysis does not include the immediate neighbor
hood of YO. If one wished, he could investigate this
critical region (which would involve an analysis similar
to that of the WKB approximation at a turning point),
and would undoubtedly find that a limit function does
not exist here. But as this region can be chosen as
small as we please, there is no real point to such an
analysis. Nevertheless, we should satisfy ourselves that
nothing untoward happens in this region, namely,
that our assumption is justified that the lowest energy
is that which gives one critical point. As we have
mentioned, below this energy there is no solution
(to order l/n) and, hence, our assumption yields a
lower bound; but it agrees asymptotically with the
BCS variational solution, which is an upper bound.
Hence, it is correct asymptotically, and it must indeed
be possible to continue our solution for p(x) through
the critical region.
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Finally, we should like to calculate the lowest-order
correction to the energy. We recall
and
Define =exp(OS _~ 02S + ... ), (A.17)
oy 2n oy2
expaS/ay=g(y)::g(y), (A. 18)
and to order n-2,
(1 02S) (1 0 ) exp --=exp --lng(y) ,
2n oy2 2noy (A.19)
where g(y) is given in Eqs. (A.1S) and (A.16).
With these substitutions, the primitive equation becomes
( a eXP[:n :y Ing(y) ])g2(y)_ 2bg(y)
+(cexpL~:y lng(y)]) =0, (A.20)
and if we note that both a and c are proportioned to
the interaction v, we see that the interactions off the
energy shell have been increased from a strength v to
an effective strength
v=v exp[~ ~ Ing(Y)]=V[l+~~ Ing(y)], (A.21)
2nay 2nay
which is greater than v because, in the important
region near Yo,
(d/dy) lng(y»O, y""yo. (A.22)
Consequently, the ground-state energy divided by the
number of particles actually must increase as the
volume is decreased (always at fixed density). For
n»l, this correction is quite negligible, and it always
vanishes in the strong-coupling limit (in which g(y) = 1,
%y[lng(y)J=O). In the weak-coupling limit, or for
the one-step model, this correction has the effect of
slightly increasing the critical temperature for very
small volume crystals.
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 2. NUMBER 4 JULY-AUGUST, 1961
Some Cluster Size and Percolation Problems
MICHAEL E. FISHER AND JOHN W. ESSAM
Wheatstone Physics Laboratory, King's College, London, England
(Received December 15, 1960)
The problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites
is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly discussed. The
cluster size and percolation problems are then solved exactly for Bethe lattices (infinite homogeneous Cayley
trees) and for a wide class of pseudolattices derived by replacing the bonds and/or sites of a Bethe lattice
by arbitrary finite sUbgraphs. Explicit expressions are given for the critical probability (density), for the
mean cluster size, and for the density of infinite clusters. The nature of the critical anomalies is shown to
be the same for all lattices discussed; in particular, the density of infinite clusters vanishes as R(p) ~C(p-Pc)
(P~ pc).
I. INTRODUCTION
RECENTLY Dombl has drawn attention to the
problem of determining the distribution of cluster
sizes for particles distributed in a medium in accordance
with a statistical law. In the simplest case, the particles
occupy at random the sites of a lattice (or, more
1 C. Domb, Conference on "Fluctuation phenomena and sto
chastic processes" at Birkbeck College, London, March 1959;
Nature 184, 509 (1959). generally, the vertices of a linear graph). Each site can
accommodate one (and only one) particle and is
occupied with a constant probability p. A group of
particles which can be linked together by nearest
neighbor bonds from one occupied lattice site to an
adjacent occupied site are said to form a cluster. The
main theoretical task is to evaluate the mean cluster
size and higher moments of the distribution as functions
of the density (or concentration) of the particles, this
being measured by the probability p.
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1.1753835.pdf | BANDFILLING MODEL FOR GaAs INJECTION LUMINESCENCE
D. F. Nelson, M. Gershenzon, A. Ashkin, L. A. D'Asaro, and J. C. Sarace
Citation: Applied Physics Letters 2, 182 (1963); doi: 10.1063/1.1753835
View online: http://dx.doi.org/10.1063/1.1753835
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BandFilling Current in Heavily Doped GaAs Diodes
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128.252.67.66 On: Wed, 24 Dec 2014 04:51:52Volume 2, Number 9 APPLIED PHYSICS LETTERS 1 May 1963
BAND-FIl;LING MODEL FOR GaAs INJECTION LUMINESCENCE
D. F. Nelson, M. Gershenzon, A. Ashkin,
L. A. D'Asaro, and J. C. Sarace
Bell Telephone Laboratories
Murray Hill, New Jersey
(Received 25 March 1963)
Reports of laser action 1,2 in 'GaAs diodes have
heightened the interest in the emission mechanism
of the fluorescent line used. We report measure
ments on the fluorescence in such diodes at for
ward bias and present a band-filling model which
accounts for the results.
The diodes were made by diffusion of Zn from a
2% Zn solution in Ga at 750°C for 16 h into a
floating-zone purified GaAs crystal uniformly doped
with 3.2 x 1018 Te atoms cm-3. The diodes were
etched to form mesas of area 9.0 x 10-4 cm2 and
length-to-w idth ratio 3.3. Capacitance measure
ments-.indicated linearly c$raded junctions having a
width at zero bias of 900 A.
Figure 1 shows the line shape at 200K for a range
of dc diode currents. The line shifts to higher
energy with higher diode current. 3 • The low energy
tail increases little if any in intensity at a given
energy as the line shifts. The intensity on the
high energy side of the line has an exponential
energy constant of "'2.5 kT. At the highest current
levels run continuously, the peak of the line be
comes more symmetric due to stimulated emission
and then shows strong line narrowing due to os
cillation in particular cavity modes. Spectra .taken
at higher currents, under pulse operation, show
sevecfal oscillating cavi~ modes separated by
3.4 A having widths of < 2 A.
In Fig. 2(a) we plotted the position of the emission
pe ak at 200K in electron volts vs the diode current.
In the middle region, between 2 x 10-4 A and
INDEXING CA TEGORIES
A. laser
A. GaAs diodes
B. emission mechanism
EIT
182 4 x 10-1 A, the peak posltlOn varies with current
1 as 1 = !IC exp (hv 18.7 meV). Below 2 x 10-4 A
the peak position fhifts more rapidly with current.
At a current of 4 x 10-1 A the laser threshold is
reached. Above threshold the rate of shift of the
peak with current is greatly reduced. ~ince the
diode lost contact with the liquid hydrogen when
run continuously at currents above .46 A, the points
.in Fig. 2 above this current were obtained using
I-p.sec pulses.
Also shown in Fig. 2(a) are the voltage-current
characteristics of the diode at 20oK. 'If the voltage
drop due to a 2-D series resistance in the diode is
removed, an exponential relation identical to that
for the shift of the peak of the luminescence results
for 1 above 2 x 10-4 A. Below 2 x 10-4 A an excess
current appears in the characteristic presumably
due to some nonsaturated recombination mechanism.
From Fig. 2(a) we see that for currents at least up
to 6 x 10-2 A the emission peak in volts matches
very closely the voltage applied across the junction.
5.---------r---------,---------,---------~
2 DIODE D-5
I05r---------T---------+---------~~--*_--~
5
2
IO,~~L-L-~~~~~~~~J-~~~~-L-L-L~
1.3 1.35 1.4 1.45 1.5
PHOTON ENERGY IN ev
Fig. 1. Fluorescent intensity vs photon energy for
different dc diode currents at 20oK. Dotted portions of
curves are resolution limited.
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At 77'1<. the data are identical to those at 20'1<. ex
cept for two effects. First, all voltages and photon
energies are shifted to lower values by 8.3 meV,
the band gap shift,4 and, second, the laser threshold
increases by a factor of 6. Note that the low energy
spectral envelope remains the same and that the
high energy tail.is still rv 2 k T.
Figure 2(b) shows the integrated fluorescent
intensity for different polarizations and directions
of view ing vs the diode current. For 2 x 10-4 < 1 <
4 x 10-1 A the curve corresponds to constant quan
tum efficiency. For 1 < 2 x 10-4 A the quantum
efficiency decreases coincidently w.ith the appear
ance of the excess current of Fig. 2(a). This indi
cates that the excess current undergoes nonradiative
recombination. At the laser threshold the emission
from the edge of the junction polarized parallel to
the junction, P ,.increases sharply with current due
to stimulated Yemission accompanied by marked
directional effects in its emission. For the diode
shown.in Fig. 2(b) the emission polarized perpendic
ular to the junction plane, P x' continues to .increase
linearly above threshold, which, when compared
with the emission out the back of the diode, indi
cates amplification. In other diodes stw:lied the
P x emission rises above linearity above threshold.
The fact that light of both polarizations has a
sublinear behavior above the laser threshold when
viewed out the back of the diode indicates that
laser action tends to prevent further upper state
population increases. We attribute the differences
in intensities of these polarization components to
scattering of light traveling initially in the plane
of the junction. The directions of polarization in
relation to the geometry of the sample are con
sistent with this hypothesis. The scattering co
efficient can be estimated from this effect to be
~ .01 cm -1 from the geometry of the diode (light
emitting region taken as 10-4 cm thick) and using
the polarization ratios either above or below
threshold.
A model5 -7 which can account for all the ob
servations reported here is the filling of an impurity
band by injected minority carriers. The density
of states in the "forbidden" gap in diodes such
as used .is sufficiently large for such a model
("-'1018 cm-3 eV-1 at 50 meV below the conduction
band edge8). The radiative recombination lifetime
is expected to be close to the minimum (intrinsic)
radiative recombination lifetime (3 x 10-10 sec)
since hv "-' E gap (ref 9). This lifetime is long •. 50
1.45
II!
~ o >
~
>1.40
'.35
I
~
~ '0-2
> : ...
~10" 3
" ;
>-
~ 10-4
Z w ...
~
... z .0-w
U II!
W
<r: 5
& (a) DIODE 0-4 / sY~ <>0
~ rncraf'" I
/ I
I ,If
Jbl Py ~
-
ED:;! Px
!- I~
!-/ ~ BACK
I-pz
/ {'>
!-
I-
/~
I I' L,z ~NCTION y ~. PLANE
EDGE ,.~- I BACK o 310-
IL 7:/ IIIEWINC t III EWING
DIRECTION DIRECTION
Py, Px p' p' y, Z
JUNCTION AREA 9X'0-4 CM2
- --- '03 ,02 ,0 , ,0'
I IN AMPS
Fig. 2 (a) Circles plot the emission peak position in
electron volts vs log I, squares the V-I characteristic of
the diode, triangles the V.I characteristic corrected for
a 2-D series resistance in the diode. All data for20oK.
(b) Fluorescent intens ity vs current for different di
rections of viewing ond lineor polarizations for diode at
20oK.
compared with equilibration times ("-' 10-11 sec)
thus allow.ing thermal equilibrium within the filled
band. "In such a model the emission peak would
occur at or just below the energy eV and would
move with V as observed. Lifetime shortening
from stimulated emission above the laser threshold
Vol ould oppose further band filling as observed. The
high energy tail of the emission Hne would fall
off with an exponential energy constant of "-'kT
due to the Fermi distribution while ""2 kT was
observed. The low energy tail would reflect the
density of occupied states (as modified by any
change in radiative lifetime as a function of
183
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energy) vs energy and would be expected to sat
urate in its emission strength as well as remain
the same at 20 and 77~, effects also observed.
Thus, the data are most easily interpreted in terms
of an exponentially varying density of impurity
states.
Since the voltage-current characteristic has the
same slope in the middle current region at both
20 and 77'1(, the inj ection current cannot be a
diffusion current.IO Transport through states in
the impurity band which is being filled is a likely
injection mechanism. Such transport, extensively
studied in Esaki junctions, is due to successive
tunnelings between impurity states.
The band-filling model is not inconsistent with
p-side injection reported in similar laser diodesII
nor with the terminal state of the fluorescent tran
sition being a Zn acceptor center. 12
We wish to thank E. O. Kane tor many very useful
di scussions and R. J. Archer, A. G. Chynoweth,
184 C. G. B. Garrett, D. A. Kleinman, J. M. Whelan,
and P. A. Wolff for a number of helpful suggestions.
IR. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys,
R. O. Carlson, Phys. Rev. Letters 9,366 (1962).
2M. I. Nathan, W. P. Dumke, G. Burns, F. H. Dill, Jr.,
G. Lasher, Appl. Phys. Letters 1,62 (1962).
3J. I. Pankove, Phys. Rev. Letters 9, 283 (1962).
4M. D. Sturg~, Phys. Rev. 127,768 (1962).
SJ. I. Pankove, Phys. Rev. Letters 4, 20 (1960).
6G. Lucovsky, Bull. Am. Phys. Soc. 8,110 (1962).
7M. Gershenzon, D. F. Nelson, A. Ashkin, L. A.
0' Asaro, J. C. Sarace, Bull. Am. Phys. Soc. 8, 202
(1962).
BE. O. Kane, Phys. Rev. (to be published).
9E. O. Kane, private communication.
lOThis effect has been independently seen by R. C. C.
Leite, S. P. S. Porto, J. M. Whelan, A. Yariv (to be
published).
IIA. E. Michel, E. J. Walker, M. I. Nathan, IBM}. Res.
Develop. 7,70 (1963).
12M. I. Nathan, G. Burns, Appl. Phys. Letters 1, 89
(1962).
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1.1729431.pdf | Magnetic Moment Distributions in Dilute Nickel Alloys
G. G. E. Low and M. F. Collins
Citation: Journal of Applied Physics 34, 1195 (1963); doi: 10.1063/1.1729431
View online: http://dx.doi.org/10.1063/1.1729431
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] IP: 131.111.164.128 On: Tue, 18 Nov 2014 05:13:44JOURNAL OF APPLIED PHYSICS VOL. 34, NO.4 (PART 2) APRIL 1963
Magnetic Moment Distributions in Dilute Nickel Alloys
G. G. E. Low AND M. F. COLLINS
Solid State Physics Division, Atomic Energy Research Establishment, Harwell, England
A neutron scattering technique has been developed which enables the spatial distribution of the magnetic
moment disturbance around an impurity atom in a ferromagnet to be examined over a distance of several
Angstroms. Measurements show that when vanadium or chromium is added dilutely to nickel the resulting
magnetic disturbance occurs not only at the impurity atom sites themselves, but also on the neighboring
nickel sites. In fact, most of the change in saturation magnetization observed on alloying is due to a de
crease in the moments on nickel atoms in the vicinity of solute atoms rather than to changes in moment at
the impurity atom positions. On the other hand, if iron or manganese is added dilutely to nickel the moment
disturbance appears to be confined to the solute atoms. These results are discussed in terms of a theory
suggested by Friedel.
INTRODUCTION
THE electronic structure of the metals of the first
transition period has long been a subject of interest
and one which has called forth a considerable amount
of discussion. A striking feature of the magnetic prop
erties of the elements of this series and of many of the
binary alloys formed from them is the systematic de
pendence of their saturation magnetization on electron
concentration. The relevant experimental results in
this connection are summarised in the well-known
Slater-Pauling diagram, which is shown in Fig. 1.
Not all 3d-3d alloys follow the Slater-Pauling curve,
however; for example, alloys of nickel with small addi
tions of V or Cr as impurities show sharp deviations
from the curve as can be seen from the data in the figure.
Such alloys have always presented a problem for,
whereas the general features of the main curve are ex
plicable in terms of a simple band filling process, the
alloys which deviate from the Slater-Pauling curve
require the introduction of some further mechanism.
A possible explanation is that the impurities exist in the
matrix in association with large negative moments.
However, it has been suggestedl that a more likely
process is that the presence of the impurities affects the
moments on the neighboring matrix atoms, and hence
that a large part of the diminution in magnetization
is due to a reduction in the moments on nearby nickel
sites. In the present experiments the actual spatial
distribution of magnetic moment around impurity
atoms in nickel has been determined by a neutron dif
fraction technique. Alloys containing small additions
of V and Cr have been examined and compared with.
two other alloys, namely NiFe and NiMn, which follow
the Slater-Pauling curve. The neutron scattering from
the former pair indicates a considerable spread in mag
netic moment disturbance around an impurity atom
site, whereas, the latter alloys show no evidence of
this behavior.
The work has been carried out by measuring the
elastic magnetic disorder scattering from the alloys
concerned. A new experimental technique has been
developed which embodies the use of long wavelength
1 J. Friedel, Nuovo Cimento Supp!. 7, 287 (1958). neutrons (,,-,5 A) and a time-of-flight technique to
eliminate the effects of certain competing scattering
processes which limit the applicability of more con
ventional measurements.2,3 The scattering pattern ob
served depends on the spatial extent of the magnetic
defects under examination, and, as pointed out by
Marshall,4 it is possible to demonstrate quite clearly
whether the magnetic disturbance is confined to the
impurity atom site or whether it is spread out to include
neighboring matrix atoms.
PRINCIPLE OF THE METHOD
The differential cross section for the elastic magnetic
scattering of unpolarized neutrons in a ferromagnet is
given by
du =( 'Ye2
)2sin2a 1 j drp(r) exp (ix·r) 12, (1)
dQ 2mc2 v.
where the integral is over the volume of the specimen
V •. The first bracket has a numerical value of 0.073
barn and x represents the usual scattering vector of
the neutrons [K= (4?r sin 8) /'11., where 8 is half the
scattering angle]. a is the angle between x and the
direction of magnetization of the sample. It is assumed
that the orbital contribution to the magnetic moment
density per) is quenched. If orbital moment is present
Eq. (1) is valid only for K=O although it still provides
a fair approximation at other values of K.
The neutrons scattered by the above cross section
fall into two categories: those which undergo coherent
processes form Bragg peaks, the study of which allows
a spin density map to be drawn showing the mean
density distribution inside the unit cells. The neutrons
which suffer diffuse scattering, on the other hand, give
information concerning fluctuations from the average
spin distribution, and it is with these that the present
work is concerned. If the magnetic defects in the sample
are randomly arranged, the above expression immedi
ately separates into two parts corresponding to the two
2 C. G. Shull and M. K. Wilkinson, Phys. Rev. 97, 304 (1955).
3 R. D. Lowde and D. A. Wheeler, J. Phys. Soc. Japan 17,
Supp!. B-II, 342 (1962).
4 W. Marshall, Conference on Neutron Diffraction, Gatlinburg,
April 1960.
1195
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] IP: 131.111.164.128 On: Tue, 18 Nov 2014 05:13:441196 G. G. E. LOW AND M. F. COLLINS
3
]
c .2 C 2
~ .,
c
'" o
E
c o E
.2 o V)
Cr Mn Cu
FIG. 1. Slater-Pauling diagram showing the saturation mag
netization against concentration for 3d-metal alloys.
types of scattering described. Thus,
dO' =( 'Ye2
)2sin2a{(iF[ L exp(i1l:0m) [2
dn 2mc2 m
+ [ L exp(i1l:·m) (lrn-i) [2J, (2)
where the m denote sites on the basis lattice and
Irn= 1 drp(r) exp[i1l:o (r-m)J,
Vm
the integration being over the volume of the unit cell
atm.
The second term given by Eq. (2) corresponds, of
course, to the disorder scattering. It may be trans
formed into a convenient form by introducing the
function p' (r-s) to represent the deviation in magnetic
moment density at r, arising from the presence of an
impurity atom at the lattice point s. The reference
value against which the deviation is measured is the
moment density appropriate to the unperturbed matrix.
Hence, assuming that effects in a dilute alloy may be
superposed, it is found after substitution and some
manipulation that
dudefect ('Ye2)2 --= --sin2(a)Nc(1-c) dn 2mc2
where c is the fractional concentration of defect sites in
the basis lattice. If the magnetic disturbances arising
from alloying are confined to the solute atom sites,
the last term in this expression reduces to a simple
atomic form factor. On the other hand, if the dis
turbances are more widespread and extend on to
neighboring matrix atoms, corresponding to larger
values of r, the rate of fall-off with increasing K is
much greater and the scattering is peaked in the
forward direction.
In fact the complete spatial distribution of the mag
netic moment disturbance may be obtained by carrying
out a Fourier inversion of scattered neutron intensity. However, this would require the use of a single-crystal
specimen and the collection of a great deal of data,
and for a first experiment in connection with the
problem outlined in the previous section, a simpler
investigation involving the use of a polycrystal is
adequate. In this case the cross section in Eq. (3) has
to be averaged over all directions of 11: relative to r. The
resulting expression may be expanded for small K in
terms of the second moment (r2) of the spin density
deviation. Thus,
dudefect=( 'Ye2)2 sin2(a)Nc(1-C)[ r drp'(r)]2
dn 2mc2 lv,
X[1-iK2(r2)+O(K4)]. (4)
In the forward direction (K = 0) the scattering is di
rectly dependent on the rate of change of saturation
magnetization with solute concentration since
1 drp'(r) =dii/dc, v,
where ii is expressed in Bohr magnetons per solute
atom.
In order to investigate the behavior of the scattering
for larger K, the deviation of the integrated magnetic
moment at a lattice point m from the value appro
priate to the unperturbed matrix will be denoted by
M' (m), where m= 0 corresponds to the impurity
responsible for the magnetic disturbance. If it is as
sumed that an atomic form factor f(K) is applicable to
all sites, it is then possible to write
1 jdrp'(r) exp(i1l:°r) 12
rv[j(K) J2[ L exp(i1l:0m)M'(m) [2
= [f(K) J21 L[M'(m) J2+oscillatory terms}.
For a polycrystal of high coordination number the
oscillatory terms largely cancel out and no trace of
oscillatory behaviour has been found in the experi
mental results. It has been assumed, therefore, that
any high scattering angle tails observed in the experi
ments may be interpreted simply in terms of
[j(K) J2L[M'(m) J2,
wheref(K) is a 3d form factor.
A final topic to be mentioned is the question of
alloy concentration. Clearly from an experimental
point of view, the enhanced cross section to be ob
tained makes higher concentrations desirable. On the
other hand, one wishes to examine a system in which
the impurity atoms are more or less isolated, so that
the assumption of superposition of magnetic distur
bance effects may hold. In practice a compromise
concentration of approximately It% has been used.
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IP: 131.111.164.128 On: Tue, 18 Nov 2014 05:13:44MAG NET Ie MOM E NT DIS T RIB UTI 0 N SIN N Ie K E L ALL 0 Y S 1197
For dilutions much greater than this the measurements
could not be made in many cases without great diffi
culty. In passing it is interesting to note that for ap
preciably higher concentrations (say, greater than
10%) the environment of all atoms, both solvent and
solute, departs very little from the average environ
ment appropriate to the constituent concerned. Thus,
in this concentration limit it is reasonable to assume,
not only that the solute atoms carry a certain fixed
moment, but that a single moment value may be as
sociated with all the solvent atoms, even though in the
case of dilute systems the impurity atoms lead to dis
turbances on neighboring matrix sites.
EXPERIMENTAL TECHNIQUE
From an experimental point of view the main diffi
culty in connection with the present work arises from
the small magnitude of the cross section corresponding
to magnetic disorder scattering from a dilute alloy.
It follows that great care must be taken to eliminate
the effects of other forms of diffuse scattering from the
measurements. In most cases the largest source of
diffuse background is the nuclear incoherent scattering,
which may be two orders of magnitude greater than the
scattering of interest. However, the magnetic component
can be successfully separated from the nuclear scatter
ing by taking the difference between two counts, one
with a=90° and one with a=O°, so that the magnetic
cross section is, in effect, switched on and off. Thus,
during a measurement the alloy sample forms an inte
gral part of the low reluctance magnetic circuit of a
simple lightweight electromagnet which is rotated
between the two settings of a by automatic control
circuits. In each position counts are recorded for an
interval corresponding to the observation on a beam
monitor of a preset number of incident neutrons. The
counting cycle is repeated, perhaps fifty times during a
measurement, and as a check on the apparatus the
variance among the fifty differences obtained is com
pared with the variance expected on the basis of count
ing statistics.
Although not nearly so large as the nuclear incoherent
scattering, a more insidious form of diffuse background
is that which arises in some part from magnetic inter
actions. Such scattering has a dependence on the direc
tion of sample magnetization and thus contributes to
the difference counts described above. In particular
such effects can arise from magnetic inelastic scattering
and also, in polycrystalline samples, from multiple
Bragg scattering in which at least one of the reflections
is of magnetic origin. In the present work the latter
type of scattering is obviated by carrying out the
measurements with neutrons whose wavelengths lie
beyond the Bragg cutoff. (This also effectively eli
minates the single transmission effect.) The inelastic
scattering is excluded by incorporating a form of
neutron time-of-flight analysis into the apparatus. Reactor
core
FIG. 2. Schematic diagram of the apparatus for measuring
magnetic impurity scattering with long wavelength neutrons.
Since long wavelength neutrons which are inelastically
scattered suffer, in general, considerable energy in
creases, a fairly crude velocity selection suffices in this
connection. In addition to eliminating the effects of
inelastic scattering, the time-of-flight arrangement also
provides a means for defining the wavelength resolution
of the apparatus. A check on the efficacy of the meas
ures described above for eliminating the effects of un
desired magnetic diffuse scattering is provided by
making observations on a pure element and confirming
the absence of a dependence in the scattering on direc
tion of magnetization.
A diagram of the apparatus, which is mounted at the
reactor Pluto, is shown in Fig. 2. The incident neutron
beam passes through a filter consisting of polycrystalline
beryllium and large single crystals of bismuth cooled in
liquid nitrogen. Those neutrons whose wavelengths
exceed the Bragg cutoff in Be (3.95 A) are transmitted
and pass on through a simple chopper spinning with
axis parallel to the beam. Neutrons scattered in the
specimen are recorded in a bank of BFa counters after
traversing a flight path of approximately 1 m. The
counter assembly is gated in synchronism with the
chopper, which contains eight neutron ports and
rotates at about 6000 rpm. A gating delay is arranged
to correspond to the time of flight of about 5-A neu
trons. The gatewidth, etc., define a wavelength resolu
tion of "-'25% (full width at half-height).
Because of the use of long wavelength neutrons in the
present measurements, angular definitions may be
relaxed considerably and yet a reasonable resolution
in (sinO) /t.. still maintained. Thus, the collimation of
the incident neutron beam is 2° (full width at half
height) and that of the scattered beam may be varied
from 2° upwards. This relaxation of angular resolution
results in a greatly enhanced counting rate and largely
offsets the losses occasioned by working in the long
wavelength tail of the Maxwell spectrum from the
reactor. The counting rate is also increased by the
use of a rather large incident beam area of 1 in. sq.
Alloy specimens, which take the form of disks 6 cm
in diameter and 6 mm thick, are placed so as to lie in
the plane containing K and symmetrically about the
beam.
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IP: 131.111.164.128 On: Tue, 18 Nov 2014 05:13:441198 G. G. E. LOW AND M. F. COLLINS
0
I
E 04 3
0 ~ a (a)F' In NI
II • . ~'V
?:
i: (b)Mn in Ni
" 0·4 a.
E ...1i. a a r
M • " I
"I .., 2·0
~ (c)er in Ni a
!i
" " 1·0 t-2l1{.
" E
0 ~ 1%'-...."
11 .......... 8 1% ...I!L .......
c; 2·0 Q .. u
" cJl 1·0
'" '" 0
U F-2-"" (d)V In Ni
He '-.... II
" '-... a
II ~ B ..1l.. II o 0·5 1·0 1·5
Scattering Vector K = 4 n Sin), 9 (.8.-~
FIG. 3. Magnetic impurity scattering from dilute nickel alloys.
For Fe and Mn impurities the scattering follows an atomic 3d
form factor (solid line) showing that the magnetic disturbance is
confined to impurity atom sites. The scattering from the Cr and
V alloys indicates that for these cases the magnetic disturbance
extends on to neighboring nickel atoms. The dashed curves repre
sent scattering calculated from models in which the disturbance
is restricted to the impurity and its twelve nearest neighbors.
RESULTS
Two samples of the NiFe alloy were prepared, one
by powder metallurgy followed by rolling and heat
treatment and a second by melting in an argon-arc
furnace and annealing to produce homogeneity. The
scattering data obtained with these specimens are
shown in Fig. 3(a). There is no indication of a rapid
variation of scattered intensity with K and, in fact, the
atomic form factor appropriate to 3d electrons, as
represented by the solid curve, fits the data very well.
The ordinate of this curve at K= 0 is adjusted so as to
correspond to the experimentally observed value of
dp,/de for this alloy (2.2/o1B per atom) so that no dis
posable parameters remain. Thus, for NiFe it appears
that the magnetic moment disturbance arising from
the Fe is confined to the solute atom sites.
Neutron diffraction measurements on concentrated
NiFe alloys5 show that the iron and nickel atom sites
have, respectively, moments of 2.6/o1B and O.6,uB inde
pendently of environment. This is in agreement with
our own conclusion that the presence of iron atoms
in a nickel matrix does not influence the moment on
nearby nickel atoms, and that the moment at the iron
atom sites, obtained from a best fit of a 3d form factor,
is 2.8±O.2/o1B'
Results obtained with a NiMn sample, prepared by
argon-arc melting as described above, are shown in
Fig. 3 (b). For this alloy also there is no indication of a
rapid fall-off of scattered intensity with K and the
5 M. F. Collins, R. V. Jones, and R. D. Lowde, J. Phys. Soc.
Japan 17, Suppl. B-III, 19 (1962). magnetic disturbance appears to be localized on the
solute atoms as is the case for NiFe.
Figures 3(c) and (d) show the results obtained with
lWo NiCr and two NiV samples. These were prepared
in each case, one by vacuum meltinglfollowed by
annealing at lOOOac, and one by argon-arc melting as
described above. For both alloys the scattering data
show a rapid fall-off with increasing K, indicating a large
spread in magnetic moment disturbance around a
solute atom. From the intensity at high scattering
angles a measure may be obtained of the amount by
which the moment localized on the impurity atoms
themselves differs from the O.6MB corresponding to an
undisturbed matrix atom. In the case of NiCr this
difference appears to be roughly zero with a maximum
error of ±1.1,uB. For NiV a difference of approximately
1.8±O.4/o1B appears to be appropriate assuming that
the moment on V is negative. Thus, in both cases, the
major part of the diminution observed in the saturation
magnetization of these alloys results from a loss of
moment on nickel atoms in the vicinity of impurities.
The dashed curves shown in the figures correspond to
calculations based on models in which it is assumed
that the whole of the disturbance in the nickel matrix
is confined to the nearest neighbors of solute atoms.
The moments assumed for the impurities themselves
are O.6/o1B and -1.2/o1B for NiCr and NiV, respectively.
Each nearest neighbor is assumed to have a moment
of O.23/o1B and O.32/o1B. respectively, in accordance with
the experimentally observed values of djl/ de.6 In the
case of NiV at least, it appears that the disturbance is
more widespread than nearest neighbors. Inspection
of the variation of intensity at small K indicates that the
second moment of the distribution of magnetic dis
turbance for this alloy has a value given by «r2» 1 =
3.2±O.2 A (see Eq. 4) whilst the distance between
nearest neighboring atoms is 2.5 A.
It should be pointed out that there is an unsatis
factory feature of the results for one of the NiCr
samples and for NiMn in that the scattering in the
forward direction is not in agreement with calculated
values based on a chemical analysis of impurity content
and published values of djl/ de appropriate to this
content. However, this does not invalidate the relative
values plotted, and deductions based on the shape of
the curves of scattered intensity remain sound. The
discrepancies mentioned may well arise from concen
tration gradients in the samples (although precautions
were taken in this connection) as only a few grams taken
from an edge were analyzed in each case. This possi
bility is being investigated. No appreciable gas con
tent was revealed in these samples either by metallo
graphic examination or chemical analysis.
It is clear from our results that the nickel moment in
NiCr alloys varies widely according to environment.
6 R. M. Bozorth, Ferromagnetism (D. Van Nostrand Company
Inc., Princeton, New Jersey, 1951).
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] IP: 131.111.164.128 On: Tue, 18 Nov 2014 05:13:44MAG NET Ie MOM EN T DIS T RIB UTI 0 N SIN N Ie K E L ALL 0 Y S 1199
Neutron diffraction results have recently been pub
lished7 for two nickel alloys containing 6.0% and 8.3%
chromium, in which cross sections have been inter
preted in terms of uniquely defined moments for the
nickel and chromium lattice sites. Although the
fluctuations of magnetic moment arising from varying
environments diminishes with increasing concentration
for the alloys examined, these fluctuations should still
be rather important.
DISCUSSION
The results described in the last section show a
striking difference between those alloys which lie on the
Slater-Pauling curve (NiFe and NiMn) and those
which deviate from it (NiCr and NiV). The scattering
from the first pair of these alloys may be understood
on the basis that the magnetic moment disturbance is
localized on the solute atoms themselves, but it is ap
parent that, in the case of the latter pair, the magnetic
disturbance spreads out to include neighboring solvent
atoms. It is this spread (corresponding to a decrease of
moment on neighboring matrix atoms) which results
in the rapid diminution of the saturation magnetization
with solute concentration.
FriedeJ1 has suggested a qualitative theory which
provides an explanation of the differing characteristics
of the two classes of alloy described above. Briefly he
suggests that the presence of an impurity atom in a
matrix such as nickel causes a bound state to be de
tached from each of the 3d half-bands. These states
are moved to higher energies with an energy separation
dependent on the strength of the perturbation. The
nickel matrix is assumed to have one full half-band and
one partially filled half-band. In the case of those alloys
which show a deviation from the Slater-Pauling curve,
Friedel considers that the impurity states belonging
to the full half-band are perturbed to such an extent
that they pass up through the Fermi level and so empty
their electrons into the partially filled half-band. If
each state is fivefold degenerate this results in a mag
netic moment decrease of lO,uB per added impurity
7 V. I. Gomankov, D. F. Litvin, A. A. Loshmanov, and B. G.
Lyashchenko, Phys. Metals Metallog. (USSR) 14,26 (1962). atom. In addition to this effect there is always, of
course, a process involving the screening charge around
a solute atom. Thus, if z is the difference in atomic
number between the solvent and solute atoms, one
would expect to have z less electrons at an impurity
atom site than on an unperturbed solvent atom. As
these electrons will nearly all have been removed from
the highly dense states in the unfilled half-band, an
increase in magnetic moment from this process of Z,uR is
expected. Thus, in the case of alloys which deviate
from the Slater-Pauling curve, a net change of
(Z-10),uB per solute atom ensues on the present model
and in fact this prediction is in fair agreement with
experiment. On the other hand for alloys which follow
the curve, it is considered that the impurity states from
the full half-band remain below the Fermi surface (as
the perturbations are smaller) and thus only the effects
of the screening charge have to be taken into account.
That the two mechanisms described above could lead
to differing spatial distributions for the magnetic
moment disturbance around a solute atom may be
argued as follows. On the one hand, the high density
of states available in the unfilled 3d half-band allows
the screening out of the difference in nuclear charge z
to be effectively accomplished at the solute atom sites
themselves. On the other hand, however, the impurity
states have a relatively large spatial distribution since
they are made up from a limited number of 3d wave
functions. The charge arising from the ~emptying of
these states has this distribution and is thus more wide
spread than that which results from the screening.
ACKNOWLEDGMENTS
We wish to thank Dr. W. Marshall for pointing out
that neutron scattering methods might detect extended
magnetic moment disturbances of the type described
in the paper. To Dr. W. M. Lomer and R. D. Lowde
we are grateful for a number of discussions. Also we
have to thank L. J. Bunce, N. S. Clark, M. S. Clarke,
R. F. Dyer, T. A. Hodges, and I. C. Walker for valued
experimental assistance. Finally thanks are due to the
International Nickel Company (Mond) Limited,
Henry Wiggin and Company Limited, and the B.S.A.
Research Centre for the preparation of alloy samples.
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IP: 131.111.164.128 On: Tue, 18 Nov 2014 05:13:44 |
1.1735130.pdf | Effects of Carrier Injection on the Recombination Velocity in Semiconductor
Surfaces
George C. Dousmanis
Citation: Journal of Applied Physics 30, 180 (1959); doi: 10.1063/1.1735130
View online: http://dx.doi.org/10.1063/1.1735130
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Published by the AIP Publishing
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IP: 193.0.65.67 On: Thu, 27 Nov 2014 16:42:27JOURNAL OF APPLIED PHYSICS VOLUME 30, NUMBER 2 FEBRUARY. 1959
Effects of Carrier Injection on the Recombination Velocity in
Semiconductor Surfaces
GEORGE C. DOUSMANIS
RCA Laboratories, Radio Corporation of America, Princeton, New Jersey
(Received July 8, 1958)
The effects of carrier injection on the surface recombination velocity are discussed on the basis of the
Shockley-Read model. The relevant parameters are the surface potential <1> .. bulk resistivity, the injection
density on, and the properties of the surface states. At low injection the behavior of the surface recombina
tion velocity s can be described by excursions over the usual s 'lJS </>. curves and, depending on the values of
the various parameters, s can either increase or decrease with injection. At high injection s approaches a con
stant value that depends only on the cross sections and densities of the recombination states, and can be
larger (as is predicted for the case of Ge) or smaller than the plateau value of the near-equilibrium curve.
The predictions of the theory are illustrated with curves of s vs the fractional excess carrier density fin/no
where, in the case of Ge, use is made of experimentally determined surface parameters. The curves can be
used in applied work as a guide in controlling and possibly utilizing the effects of injection on s. The effects
of appreciable injection on s could be used for studying the surface recombination centers.
INTRODUCTION
THE behavior of the recombination velocity (s)
with injection is an interesting aspect of semi
conductor surface physics. In addition, under certain
conditions, this behavior can appreciably influence the
operation of semiconductor devices because of the
effect on the minority carrier lifetime during a cycle
of device operation. That changes of s with injection
can be large is strongly suggested by the large variations
in s with changes in the surface potential rp. induced by
electric fieldsl-5 and illumination.3-.
The currently accepted model for the semiconductor
surface is shown in Fig. 1. Surface recombination
arises from recombination states (the "fast" surface
states) that are presumed to be located mainly at the
semiconductor-oxide interface. 6 From the HalF and
Shockley-Read8 model, the recombination velocity
in the absence of injection, is given bY,2,9,lo
s
(qrp, 1 Cp) (Et-E; 1 Cp) cosh ---In- +cosh ----In-
kT 2 Cn kT 2 C ... (1)
1 Henisch, Reynolds, and Tipple, Physica 20, 1033 (1954).
t Many, Margoniski, Hamik, and Alexander, Phys. Rev. 101,
1433 (1955); Hamik et ai., Phys. Rev. 101, 1434 (1955); A. Many
and D. Gerlich, Phys. Rev. 107, 404 (1957).
3 W. H. Brattain and C. G. B. Garrett, Bell System Tech. J. 35,
1019 (1956).
4 J. E. Thomas, Jr., and R. H. Rediker, Phys. Rev. 101, 984
(1956).
6 G. C. Dousmanis, Bull. Am. Phys. Soc. Ser. II, 2, 65 (1957);
Phys. Rev. 112, 369 (1958); G. C. Dousmanisand R. C. Duncan,
Jr., J. App\. Phys. 29, 1627 (1958).
6 Reference 5 indicates contributions to the surface recombina
tion velocity from recombination states distributed in the surface
space charge region. This is suggested by a variation of the width
of the S liS <1>, curves with resistivity.
7 R. N. Hall, Phys. Rev. 83, 228 (1951); 87, 387 (1952).
8 W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952).
9 D. T. Stevenson and R. J. Keyes, Physica 20, 1041 (1954).
10 G. C. B. Garrett and W. H. Brattain, Bell System Tech. J.
35, 1041 (1956). The quantities C p and C n are the capture probabilities
(each equal to the cross sectionXthermal velocity) for
holes and electrons at surface states of density Nil cm2
and at energy Et• E; denotes the middle of the for
bidden band and 4J. and rpb are the surface and bulk
potentials, respectively (see Fig. 1). Two curves
defined by Eq. (1) are plotted in Fig. 2. Curve A arises
from recombination states located at +7kT or -7kT
units from Ei, whereas in curve B the states are located
at + 2kT or -2kT from Ei• Curves as narrow as B, but
without any plateau region at 4J.=O, could arise from a
continuous distribution of levels throughout the
forbidden band.5 For simplicity in curves A and B, Cp
is assumed to be equal to Cn. If Cp is different from Cn
the curves are not centered about rp.=O, but about
rp.= (kTI2q) In(Cp/C n).
Injection terms had been included earlier in the
expression of Garrett and Brattainll for the flow of
carrier-pairs to the surface. The present treatment
presents, first, a physical illustration of changes in s
that arise from changes in rps induced by low-level
injection. These changes are conveniently represented
by excursions of an "operating point" along the curves
in Fig. 2. The complete formula describing the effects
of injection on s will then be derived. The treatment
includes, in addition to effects due to changes in rp.,
modifications of the usual s 'liS rp8 curve itself as a result
of moderate and high-level injection.
The effects of low-level injection can be easily
understood as follows: Consider material for which the
point a on Curve A in Fig. 2 represents rp.=rpb.12 This
II Because of obvious typographical errors formula (6) in
reference (10) is incorrect. The terms X-I exp( -v) and X exp"
should add to, rather than multiply, the other terms in the
denominator. With these corrections (6) becomes equivalent to
our formula (2).
12 The bulk potential cf>b is the distance of the Fermi level from
mid-gap and is determined by the impurity concentration. For
the relation between p and cf>. see W. Shockley, Electrons and Holes
in Semiconductors (D. Van Nostrand Company, Inc., Princeton,
New Jersey, 1950), p. 246.
180
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IP: 193.0.65.67 On: Thu, 27 Nov 2014 16:42:27RECOMBINATION VELOCITY 181
ExCESS £!..ECTRONS
AT SURFACE
C-8ANO
v-BAND
OXIOE LAYER
FIG. 1. Energy bands at semiconductor surface for p-type
material with n-type surface.
corresponds to completely flat bands up to the surface.
In a surface with a given cf>., injection always tends to
flatten the bands,3.6.13 i.e., cf>. moves from its equilibrium
value towards the point cf>.=cf>b (point a in this ex
ample). This holds even when an appreciable part of
the injected carriers are trapped in fast surface states.
Thus, if the original surface treatment and the surface
environment is such that cf>. (0) lies in the portions ab or
cd, injection would tend to increase s. If, on the other
hand, cf>. (0) is in curve portion ac, injection would tend
to decrease s. For intrinsic material the point a coincides
with Smax(cf>.=O) and low level injection would always
tend either to increase s or not affect it at all depend
ing on whether the initial cf>. is on the sides or the
plateau portion of the curve.
In the case where C p is different from C n the effects of
low-level injection can be deduced by the same pro
cedure, but using a curve symmetric about cf>.= (kT /
2q) In(Cp/C n) as noted above. In this case injection
always tends to keep constant or increase s in material
of such resistivity that cf>b= (kT/2q) In(Cp/C n).
EFFECT OF LARGE INJECTION
The simple aforementioned considerations apply,
provided the injection level is low enough so that (1)
holds, i.e., the system moves along the equilibrium
curves. As in the case of (1) in the foregoing, the general
formula for s has to be derived from Eg. (4.4) of
Shockley and Read by retaining, in the expression for
the recombination rate U(=son), the second power
terms in the excess electron density (on). Thus, one
obtains, for any injection one =op).
u /~n=s=s(o)(cf>.')[l+~l
no+po
[ Cp+Cn on 1-1
X 1+ S(O)(cf>.')--.
CpCnN I no+po (2)
13 E. O. Johnson, Bull. Am. Phys. Soc., Ser. II, 2, 66 (1957);
Phys. Rev. 111, 153 (1958). This reference includes a discussion
of distortions in the photovoltage curves that may arise from
charge changes in fast states. 100
90
eo
70
60
1/1 ~o
... >
~ 40
a:
30
20
10
FIG. 2. Curves of relative surface recombination vs surface
potential for the noninjection case. In Curve A (E;-E,)!kT= ±7
(case of Ge); in B, (E,-E i)kT=±2.
S(O) (cf>.') denotes expression (1) with cf>.' representing
the modified value of cf>. as a result of the injection.
cf>s' is to be considered as a parameter that approxi
mately represents the surface potential at low injection
levels. At higher injection levels s approaches a constant
that is entirely independent of cf>., as will be discussed
below. The quantities no and po represent the electron
and hole densities deep in the bulk in the absence of
injection.
Expression (2) can be rewritten with on/ (no+ po)
expressed in terms of the minority carrier injection
ratio on/no,
s=S(O) (cf>.')[I+ 1 .on]
1+exp(-2qcf>b/kT) no
[IS (0) (cf>.') On]-1
X 1+ "-
l+exp( -2qcf>b/kT) SI no (3)
In (2) and (3) it is implied that the minority carriers
are the electrons. If the material is n type, no is to be
replaced by po, on by its equalop, and exp( -2qcf>b/kT)
by exp2qcf>b/kT. SI is the constant
(4)
Equation (3) shows that s approaches this constant
asymptotically as (on)/ (no) becomes large. It is interest
ing to compare this value of s with the maximum
plateau value of (1). From (1), setting cf>.= (kT/
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IP: 193.0.65.67 On: Thu, 27 Nov 2014 16:42:27182 GEORGE C. DOlTSMANIS
... v
:. 2
or
~
.... ~ I .. SAMPLE #3
n-TYPE Gt 7ncm
INVERSION L A.VER
°p~-~~A-~~--~-I~OI~----~'-----~~----~IO'
-..... INJECTION RAfiO '*
:FlG. 3. Measured values of surface photovoltage and comparison
with theory (measurements of E. O. Johnson).!'
2q In(Cp/C), and from (-l) we obtain
1+COSh(Et-Ei_~ In Cp)
(Cpe,,)! kT 2 Cn (5)
5mnx(O) C p+C n cosh (2qtPb/ kT)
In the case of Cp"",C n and / (Et-Ei)/kT/»1,
!QCPb/kT/»l (5) implies that
10000 r---------
Et-EI·~7kT
1000
100 Nt-CTp • Ii;,'
CTn• up
1
'11 .. 10 CAVSEC
IQI
Sm(ll~ 1400(M IS[C
(F"OR n-TYP( Ge p-o.5 OHM-eM) S--.. 5000 eMISEC
'OR ~ VERY LARGE
"0 .----t--
q4>ttkT',:~+,
q.jl~l/kT ,:~~
s~~.lO' 1400 eMISEC
FIG. 4. Curves of the surface recombination velocity s vs on/no
for equilibrium surface potential qq,.(O)/kT of ±8, and q,b/kT=O
(intrinsic bulk), and ±S (in n-type Ge p=O.5 ohm-em.) The
surface recombination states are located at ±7kT from mid-gap
(curve A, Fig. 2). or
depending on whether the recombination state energy
levels are located, respectively, farther from or nearer
to mid-gap than the Fermi level. In n-type germanium
experimental studies" of the curves of s ,'S tP, in the
resistivity range of 0.2 to 12 ohm-em indicate that in all
cases (Et-Ei»qtPb, hence 51 is larger than 5max'I)) •
One also arrives at the same conclusion from the curves
of s ~'S tP, in silicon.5
In etched germanium surfaces, determinations of
Cp, Cn, and Xt from various experiments2.5.14.15 give,
from (4) in the foregoing values of 51 in the range of 103
to 104 em/sec. Equation (5) suggests, assuming the
properties of the fast states to be independent of
resistivity, that the departures of 51 from 5nH\x (0) are
at a minimum when
(6)
or simply, for C p,,-,C n, when the state levels and the
Fermi level are at equal energies from mid-gap.
The effect of injection on the shape of the s ~'S tPs
curves can be seen by plotting s vs tPs using (3) in the
foregoing and a particular value of lin/no. For instance,
the s values for intrinsic material and on/no= 10 in Ge
JD 000 ,------"""[t---==-E
I-. '-'tC::!.C""kTO:--
V-'-10-7 C-M-'-S-EC-----,
NtCTp '10 S-5OOO CM/SEC
CTn N CTp FOR ~ VERY LA_RG_E __ +-_
1000 ~
Q4>b +5 q~;' -2
100 kT'-~ kT '+~
(FOR n· TYPE G e
p -0.5 OHM-CM)
INTRINSIC MATERIAL
q()/kT' 2
5':1: 18 eMISEc
FIG. 5. Curves of svslm/noforqt/>.(O)/kT=±2 and qt/>b/kT=O, ±5;
(E,-E;)/kT=±7 (curve A, Fig. 2).
14 C. G. B. Garrett, Phys. Rev. 107, 478 (1957).
16 S. Wang and G. Wallis, Phys. Rev. lOS, 1459 (1957).
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IP: 193.0.65.67 On: Thu, 27 Nov 2014 16:42:27RECOMBINATION VELOCITY 183
are about 6 times larger than those of the equilibrium
curves.
Plotting of curves of s vs injection level from (3),
requires a knowledge of the equilibrium s tIS CPs curves
(i.e., Cp, Cn, Ee, and Nt) and, in addition, the change in
CPs produced by a given injection level. For the s vs cps
curves in Ge one can use curve A (Fig. 2) which is
fairly representative of several types of measurement2,3.5
for both p-and n-type material. The width of the curve
is to some extent resistivity-dependent,5 with (Et-Ei)
in the range of 6 to 9kT for material in the resistivity
range of 12 to 0.3 ohm-cm. For simplicity curve A with
Et-Ei=±7kT will be used here for all resistivities.
The height of the curves, of course, depends strongly on
resistivity and this is taken into account according to
(1) aforementioned. To obtain the change in CPa with
injection we use the experimental surface photovoltage
curves.!3 A set of such curves, relating Acp. with on/no,
is shown in Fig. 3. In germanium the photovoltage
effects agreed fairly well with calculated curves.6•13 The
calculated curves of Acp. vs on/no (or op/po) are based
on space charge considerations alone. Caution is to be
exercised in using the curves in cases, particularly with
other materials, where appreciable effects on CPR are
expected as a result of charge changes in fast surface
states.l• The direct influence of these changes on s is
'00000 r-----------------,
'0000
tOOO ~ kT ·j:z
Nt O'"p' ,0'
Un • O"p
V. 107 eM/SEc.
S_ SOOO CM/SEC
FOR 8 n VERY LARGE no
FIG. 6. Curves of s vs on/no for q,,(O)/kT=±8 and qq,b/kT=O,
±S; (Et-Ei)/kT=±2 (curve B, Fig. 2). Note added in proof.
The =r signs following qq,b/kT on the lowest left-hand side of this
figure should be changed to ±. '0000
'000 s~Jx.ln 000 eM ISEC
INTf'INSIC MATERIAL.
~'.±z
('!.:r2300 eM/SEC
Et-Ei .+ Z
kT -
Nt O'"p '10'
Un" tTp
V-IO"CM/SEC
S -SOOO CM/SEC
FOR ~ VERY LARGE
'00 OL.O-' -o..l..,----'--,"--o----1.'o". --"0.'--' ... 0.---"0.
FIG. 7. Curves of s vson/no for qq,,(O)/kT= ±2 and qq,b/kT=O, ±5;
(E,-Ei)/kT=±2 (curve B, Fig. 2).
taken into account in the basic formula from which (2)
is derived.
The procedure of plotting curves of s vs on/no as given
by (3) is as follows: Starting with a given CPR (0) one
finds, from the curves of ACP. vs on/no, the modified
potential c/>.' for specific values of on/no. sin (3) is then
evaluated as a function of on/no using the values of
5(O)(cp.') obtained from Fig. 2, curve A. The relative
values of s in this curve are changed to values in
em/sec by use of (1) and the values for the surface
parameters given in the following.
Figure 4 shows curves of s vs on/no for initial qCP.{O)
=±8kT and qCPb=O, ±5kT. (In n-type Ge, CPb=O, 5
correspond to a material that has intrinsic resistivity,
and 0.5 ohm-cm, respectively.) The curves cover seven
orders of magnitude of injection level on/no (10-2 to
100). To illustrate the differences arising from initial
surface treatment the same cases are plotted in Fig. 5
but with qCP8(O)=±2kT. In Figs. 4 and 5 /Et-E i/
~ /qCPb/ in all cases so that 51>Smax(O).
Figure 6 and 7 show curves that demonstrate the
behavior of s in a hypothetical case where (Et-Ei)
=±2kT (curve B, Fig. 2). The curveswithqCPb/kT=±5
show the behavior of s when the fast surface states are
closer to mid-gap than the Fermi level. In Figs. 4-7,
51 is taken for all cases as 5000 cm/sec, corresponding
to cross sections IT p~lTn, ]1{ tIT p= 10-3 and u= 107
em/sec.
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IP: 193.0.65.67 On: Thu, 27 Nov 2014 16:42:27184 GEORGE C. DOUSMANIS
DISCUSSION
The curves of Figs. 4 and 5 should represent the
behavior of s in Ge fairly accurately, since the surface
parameters introduced in the theory have been deter
mined experimentally.2,3,5,14,15
One notes in the curves of Figs. 4 and 5 that large
changes are introduced by injection starting in some
cases with as Iowan injection level (on/no) as 0.05.
Significant changes in s are indicated in some cases for
injection ratios of 10 to 103, usually encountered in
transistor devices.
The choices of qtPb/kT are such that these theoretical
curves should give the behavior of material in Ge that
is close to intrinsic as well as p-and n-type material
0.5 ohm-cm in resistivity. Interpolation between these
would give the theoretical curves for the resistivities
that are most commonly used. Likewise, the values of
qtP. (0) / kT cover the values one usually encounters in
germanium surfaces. The wide variety of behavior in
Figs. 4-7, coupled with the indication that the effects
on s are large, is suggestive of several modes of opera
tion that may be desired in present or future devices.
Aside from the cross sections and densities of the
recombination states, the main variables are bulk
resistivity, equilibrium surface potential, and surface
state energy. Since the state energies have not as yet
been definitely correlated to chemistry and atomic
structure at the surface, a desired behavior of s is to be
obtained by the choice of bulk resistivity and surface
treatment. A considerable amount of further flexibility
will of course be available when surface work is extended
towards a better understanding of the origin and control
of the fast state properties.
It is assumed throughout this paper that op=on.
The present scheme can be extended to the cases when
op is different from on by introducing effects due to differences in electron and hole lifetimes16 resulting
from heavy trap densities, etcP
It is interesting to note that the general formula (2)
does not involve any additional surface parameters
than those of the near-equilibrium formula (1). Thus,
measurement of the effects of large injection on s
provides another means of determining the properties
of the surface states. The very magnitude of the
injection effects of s suggests that this technique may
prove quite sensitive for studies of the surface states.
The scheme given here for determining the effects of
injection on s is quite general,18 For a new material one
needs to determine the equilibrium curve of s vs tP. and
the relation between tP. and injection level. The behavior
of s under all conditions is then given by introducing
this information in Eq. (2). A wide variety of behavior
is indicated and the curves given here represent only
a few cases. The theory could be used as a guide in
applied work in controlling or utilizing the effects of
injection on surface recombination and hence surface
lifetime.
ACKNOWLEDGMENTS
The author is obliged to E. O. Johnson for several
helpful discussions, and R. C. Duncan, Jr., for aiding
with the calculations; and for their reading and criti
cizing the manuscript.
16 There are several references on this subject. See for instance,
A. Rose, in Progress in Semiconductors (Heywood and Company,
Ltd., London, 1957), Vo!' 2.
17 For the involvement of surface recombination (surface
lifetine) and injection by illumination in the photoelectro
magnetic effect see Kurnick, Strauss, and Fitter, Phys. Rev. 94,
1791 (1954); L. Pincherele, Photoconductivity Conference (John
Wiley and Sons, Inc., New York, 1956), p. 307.
18 The curves in Figs. 4-7 can of course be applied to any
semiconductor surface where conditions are consistent with the
approximations involved in this treatment. As noted in text, the
surface parameters chosen for the curves of Figs. 4 and 5 are such
that, in addition to covering other possible cases, they should
approximately represent the expected behavior of a Ge surface.
At this time no quantitative data exist over any appreciable range
of injection for comparison with the present theory.
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1.1725107.pdf | Hyperfine Structure in the Microwave Spectrum of HDO, HDS, CH2O, and CHDO :
BeamMaser Spectroscopy on AsymmetricTop Molecules
P. Thaddeus, L. C. Krisher, and J. H. N. Loubser
Citation: The Journal of Chemical Physics 40, 257 (1964); doi: 10.1063/1.1725107
View online: http://dx.doi.org/10.1063/1.1725107
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/40/2?ver=pdfcov
Published by the AIP Publishing
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54THE JOURNAL OF CHEMICAL PHYSICS VOLUME 40, NUMBER 2 15 JANUARY 1964
Hyperfine Structure in the Microwave Spectrum of HDO, HDS, CH20, and CHDO:
Beam-Maser Spectroscopy on Asymmetric-Top Molecules*
P. THADDEUS
Columbia Radiation Laboratory, Columbia University, New York, New York, 10027 and
Goddard Institutefor Space Studies, NASA, New York, New York 10027
L. C. KRISHER
Columbia Radiation Laboratory and Department of Chemistry, Columbia University, New York, New York 10027
AND
J. H. N. LOUBsERt
Columbia Radiation Laboratory, Columbia University, New York, New York 10027
(Received 15 August 1963)
Hyperfine structure in the 220~221 rotational transition of HDO at 10 278.2 Mc/sec and HDS at 11 283.8
Mc/sec, and in the 211~2'2 rotational transition of CH20 at 14488.6 Mc/sec and CHDO at 16 038.1 Mc/sec,
has been investigated with a high-resolution beam maser microwave spectrometer. Linewidths of S kc/sec
have been obtained. The hyperfine Hamiltonian for an arbitrary number of nuclei with quadrupole, spin-ro
tation, and spin-spin interactions is discussed, and the matrix elements of the Hamiltonian and the intensities
of hyperfine transitions calculated in terms of the tabulated 6j coefficients. Quadrupole and spin-rotation
constants and bond lengths which have been determined are, for the 220 state of HDO: (eq.rQ)D=79.3±0.3
kc/sec, CH= -43.47±0.11 kc/sec, CD= -2.33±0.02 kc/sec; for the 221 state of HDO: (eq.rQ)D=79.6±0.3
kc/sec, CH= -43.63±0.13 kc/sec, CD= -2.20±0.02 kc/sec; for the 220 state of HDS: (eqJQ)D=42.9±0.4
kc/sec, CH= -2S.03±0.13 kc/sec, CD= -0.47±0.02 kc/sec; for the 221 state of HDS: (eq.rQ)D=43.3±0.4
kc/sec, CH = -2S.4S±0.13 kc/sec, CD = -0.22±0.02 kc/sec; for CH20: CH (211) -CH (2,2) = 2.26±0.13
kc/sec, CH(2u) =0.6S±0.SO kc/sec, (1/YHH3)-i=1.898±0.017 1; for CHDO: (eVt.Q)D=170.0±2.0
kc/sec (where Vu is the second derivative of the electrostatic potential along the CD bond), CU(211)
CH (2,2) = 2.42±0.SO kc/sec, CD (211) -CD (2,2) =0.25±0.10 kc/sec, CH (211) = 0.2±1.0 kc/sec, CD (211) =
0.13±0.20 kc/sec, and (1/YHD3)-i= 1.88±0.10 1. The VEt calculated from the deuteron quadrupole coupling
constants are, for HDO: l.S6X 10'6 statvolt/cm 2; for HDS: O. 76X 10'5 statvolt/cm 2; and for CHDO: 0.83X
10'6 statvolt/cm2•
I. INTRODUCTION
THE great majority of stable molecules have 12; elec~
tronic ground states for which the total magnetic
field produced by the electrons is small, and the effect
of nuclear electric quadrupole moments rather than
magnetic dipole moments is predominant in the hyper~
fine structure (hfs) of the rotation (or inversion)
spectrum. In microwave spectroscopy, quadrupole hfs
is usually well resolved, and has been extensively
studied.
The quadrupole moment of the deuteron is several
orders of magnitude smaller than that characteristic of
most nuclei, however, and produces hfs which is near
the limit of resolution of most microwave spectrometers.
Deuteron quadrupole hfs has been observed by micro~
wave spectroscopy for only a few molecules, and the
coupling constant has been measured for only several
cases to an accuracy of better than 10%.1 At the same
time, because of the precision to which the electronic
wavefunction for molecular hydrogen can be calculated,
* Work supported in part by the Joint Services (the U.S. Army,
the Office of Naval Research, and the Air Force Office of Scien
tific Research) and in part by the National Science Foundation. t Present address: Physics Department, University of the
Orange Free State, Bloemfontein, South Africa.
1 R. Bersohn, J. Chern. Phys. 32, 85 (1960). the deuteron quadrupole moment is known much more
accurately than the quadrupole moments of other
nuclei,2 and a measurement of its coupling constant in
other molecules therefore allows an accurate determina
tion of the gradient of the molecular electric field.
There are also other interactions which lie near or be~
yond the limit of resolution imposed by Doppler broad~
ening, which gives at room temperature for light mo~
lecules a linewidth of about SO kc/sec in the centimeter
region. A 12; ground state has zero electronic angular
momentum only in the idealized case that the nuclear
frame is fixed in space, or "clamped." Under rotation,
higher electronic states with nonzero angular momen~
tum are slightly excited, producing a small molecular
magnetic field and a magnetic hyperfine interaction
proportional to I· J. These "spin-rotation" inter~
actions were first observed in molecular hydrogen, and
subsequently many alkali halides, with molecular beam
magnetic resonance techniques.8 A number have also
been observed by high~resolution microwave absorption
spectroscopy.4,6 The energies of the I· J interactions
2 J. P. Aufiray, Phys. Rev. Letters 6, 120 (1961).
8 N. F. Ramsey, Molecular Beams (Oxford University Press,
Oxford, England, 19S6), Chap. 8.
4 R. L. White, Rev. Mod. Phys. 27,276 (19SS).
6 C. H. Townes and A. L. Schawlow, Microwave Spectroscopy
(McGraw-Hill Book Company, Inc., New York, 19S5), Chap. 8.
257
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54258 THADDEUS, KRISHER, AND LOUBSER
are typically of the order of a few tens of kilocycles per
second, but may be as large as 100 kc/sec or greater for
molecules with large rotational constants.
The direct magnetic dipole-dipole, or "spin-spin,"
interactions between nuclei are still smaller, with typi
cal energies of a few kilocycles per second. This inter
action depends only on the dipole moments and masses
of the nuclei and the geometry of the nuclear frame of
the molecule. These are usually more accurately found
by other methods, and this interaction is not as
interesting, therefore, from the point of view of molecu
lar structure as the hyperfine terms previously dis
cussed. In certain rare cases, however, such as CH20
studied here, a very precise measurement of a spin-spin
interaction may allow calculation of an internuclear
distance to an accuracy as great as or exceeding that
already known.
Two general methods have been used to investigate
these various interactions in 12: molecules. Since Dop
pler broadening decreases in direct proportion to the
frequency of a transition, while there is no systematic
decrease of the hyperfine interactions, absorption spec
troscopy at low microwave and radio frequencies offers
the possibility of observing hfs that cannot be resolved
in the centimeter region. There is a systematic decrease
of the intensity of absorption lines, however, that is
roughly proportional to the cube of the frequency for
rotational transitions, which severely limits the appli
cability of this method. Treacy and Beers6 have used
this approach to observe two rotational transitions in
HDO which lie as low as 825 and 487 Mc/sec.
The second approach consists in selectively observing
a limited class of molecules whose velocity spread along
the direction of signal propagation is a small fraction
of the average thermal velocity. This is most commonly
done with a molecular beam, which, if well collimated,
permits in practice a reduction in Doppler broadening
by a factor of 10 to 100. This technique has, of course,
been employed for some time in high-resolution optical
spectroscopy. With a microwave spectrometer operating
on this principle, Strandberg and Dreicer7 have ob
served a linewidth of only 12 kc/sec for the ammonia
3, 3 inversion line. Using the same transition, Newell
and DickeS have devised an ingenious method, em
ploying spatial as well as temporal Stark modulation,
for observing only those molecules moving at a given
velocity with respect to the wavefronts of the micro
waves. A linewidth of about 10 kc/sec was observed.
In practice, both of these schemes suffer from low
sensitivity. In the centimeter region they have only
been applied to the intense ammonia inversion spec-
6 E. B. Treacy and Y. Beers, J. Chern. Phys. 36,1473 (1962).
7 M. W. P. Strandberg and H. Dreicer, Phys. Rev. 94, 1393
(1954) .
8 G. Newell and R. H. Dicke, Phys. Rev. 83,1064 (1951). trum, and even in this case the sensitivity was not high
enough to allow investigation of hfs.9
The first spectrometer of comparable resolution to
overcome this limitation on sensitivity was the ammo
nia beam maser of Gordon, Zeiger, and Townes,I° which
also used a molecular beam to reduce Doppler broaden
ing. In addition, however, an electrostatic state selector
was employed to increase the population difference
between the states of the transition. It was the unique
feature of the maser that instead of increasing the popu
lation of the lower state of the transition with respect
to the upper state, it was found possible to remove
effectively the lower-state molecules from the beam.
The hfs of the transition was therefore observed in
emission instead of absorption. A linewidth of 7 kc/sec
and a signal-to-noise ratio of about 1000 with super
heterodyne detection and oscilloscope display were ob
tained for the ammonia 3, 3 line. This sensitivity
allowed new features of the inversion spectrum (due
mainly to magnetic interactions of the protons) to be
investigated.ll
In this paper we report the results obtained from a
beam-maser study of the light asymmetric-top mole
cules HDO, HDS, CH20, and CHDO. These molecules
are nearly symmetric tops, and their rotational spec
trum possesses closely spaced pairs of levels due to the
lifting of the ±K symmetric-top degeneracy (Fig. 1).
These K-type doublets resemble in some respects the
inversion doublets of ammonia. In particular, if transi
tions between the two components of a doublet are
allowed, the two states will repel each other under the
Stark effect to typically very high fields, due to the
isolation of the doublets from other rotational states.
Electrostatic state selection is therefore feasible. More
over, as in the case of ammonia, the great majority of
rotational intervals for these molecules lie in the milli
meter or submillimeter region, and the lower rotational
states are well populated.
The spectrometer used in this investigation and the
techniques of frequency measurement have been de
scribed elsewhere.12 Therefore, only those experimental
considerations which are peculiar to a given molecule,
such as the preparation of the sample, are discussed in
detail.
II. THEORY OF HFS
In this section we summarize the theory of hfs for 12:
molecules, in particular for asymmetric rotors. The
9 Recently in the millimeter region a molecular beam spectrom
eter has been used in studies of the rotational spectrum of the
alkali halides with good signal strength and linewidths of about
100 kc/sec. See J. R. Rusk and W. Gordy Phys. Rev. 127, 817
(1962). '
10 J. P. Gordon, H. J. Zeiger, and C. H. Townes, Phys. Rev.
99, 1264 (1955).
11 J. P. Gordon, Phys. Rev. 99, 1253 (1955).
12 P. Thaddeus and L. C. Krisher, Rev. Sci. Instr. 32, 1083
(1961) .
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54M I C ROW A V ESP E C T RUM ° F H DO, H D S, C H20, AND C H D ° 259
interaction terms diagonal in the rotational states and
the relation between the hyperfine coupling constants
and the more basic nuclear and molecular properties are
brought together and presented under a systematic
notation. Finally, the problem of writing down the
matrix elements for an arbitrary number of nuclei in a
given representation in terms of the tabulated 6j co
efficients is considered, and the calculation of intensities
is briefly discussed.
Foley13 originally considered the coupling of two
similar nuclei with electric quadrupole moments in a
diatomic molecule. His work was extended to dis
similar nuclei and to symmetric- and asymmetric-top
molecules by Bardeen and Townes,14 Myers and
Gwinn,15 and Robinson and CornwelU6 The case of
three nuclei with quadrupole moments has been treated
by Bersohn,17 who indicated how his results could be
generalized to an arbitrary number of nuclei.
In an attempt to understand certain previously un
resolved features of the ammonia inversion spectrum,
Gunther-Mohr, Townes, and Van Vleck18 undertook a
systematic investigation of all hyperfine effects which
could be observed under the highest microwave resolu
tion then available. Their treatment was comprehensive
enough to explain the results which Gordonll soon
afterward obtained with the first beam maser. These new
features were magnetic in origin, due either to the
direct dipole-dipole interactions between various nuclei,
or to the I· J interactions between the nuclei and the
molecular magnetic field. A review of the theory of the
I·J interactions and of the microwave work up to 1955
is given by White4 and in the monograph by Townes
and Schawlow. 5 Ramsey3 also gives an extensive dis
cussion of these interactions, with particular attention
to linear and diatomic molecules.
The work of Gunther-Mohr, Townes, and Van Vleck
on NH3 has been extended to the deuterated ammonias
by Hadley,19 while hfs in planar molecules with two
off-axis spins and one axial quadrupolar nucleus has
been treated by Okaya.20 The problem of the coupling
of an arbitrary number of nuclei having, in general,
both a magnetic dipole and an electric quadrupole mo
ment has been considered in some detail by Posener.21
a. Hamiltonian
The hfs of the molecules studied in this work is
interpreted on the basis of the Hamiltonian given by
13 H. M. Foley, Phys. Rev. 71, 747 (1947).
14 J. Bardeen and C. H. Townes, Phys. Rev. 73, 627 (1948).
IIi R. J. Myers and W. D. Gwinn, J. Chern. Phys. 20, 1420
(1952). I
16 G. W. Robinson and C. D. Cornwell, J. Chern. Phys. 21, 1436
(1953) .
17 R. Bersohn, J. Chern. Phys. 18,1124 (1950).
18 G. R. Gunther-Mohr, C. H. Townes, and J. H. Van Vleck,
Phys. Rev. 94, 1191 (1954).
19 G. F. Hadley, J. Chern. Phys. 26,1482 (1957).
20 A. Okaya, J. Phys. Soc. (Japan) 11,249 (1956).
21 D. W. Posener, Australian J. Phys. 11, 1 (1958). 3t2
220
221
3000 313
>-u 303
"': 0: ....
w " z::li
w~
2000 211
212 --312
220--..L313 321
202 .:>=== --322 221~303 220
221 __ 321
1000 110 322 --211 __ 220
III --212 221
--202 =312
312 313
101 __ 110 ~211 =313
212 --303 211 ... ~303 --III
III~IIO 110~212 --101 --202 III
--101 101-=:::202
0 000 --000 --000 --000
HOO HOS CH20 CHOO
FIG. 1. The lower rotational levels of HDO, HDS, CH20, and
CHDO, calculated in the rigid rotor approximation.
Gunther-Mohr, Townes, and Van Vleck18:
JC= LAg(Jg- Lg)2
+1" eQK V 67h(2h-l) :
X a [IKIK+ (IKIK)tr]-Idh+l) I}
+ eJ.!NLgKriK-3[riK x (Vi-'YKVK)]- IK
C i.K (la)
(lb)
(lc)
(ld)
X [(h, IK) -3rLK-2(h' rLK) (lK' rLK)]. (Ie)
Ig and Lg are, respectively, in units of h, the com
ponents of the total angular momentum excluding
nuclear spins, and the electronic orbital angular mo
mentum, along the principal axis of inertia of the
molecule. The Ag are the rotational constants given,
in terms of the principal moments of inertia, by Ag=
h2/2Ig. The index i refers to the electrons, with charges
-e and positions and velocities ri and Vi with respect
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54260 THADDEUS, KRISHER, AND LOU-BSER
to the molecular center of mass, assumed to be the
center of mass of the nuclear frame. Indices K and L
refer to the nuclei, with masses MK, charges eZK, spins
h, and electric quadrupole moments QK. 'YK is the
Thomas precession factor22 (1-ZKMp/gr.MK). The
nuclear g factors are defined so that the magnetic
dipole moment operator is YK=gr.ILNI K, where ILN is the
nuc!ear magneton. Mp is the proton mass, c the velocity
of lIght, r;j= ri-rj, and r;j= 1 ri-rj I. V is the nega
tive of the electric field gradient tensor at the Kth
nucleus:
(IKIK) tr stands for the dyadic transpose of the operator
IKIK.
Term (1a) is the energy of rigid rotation of the
nuclear frame and (1b) is the electric quadrupole
interaction of the nuclei with the molecular electric
field. Terms (1c) and (1d) are the (spin-orbit) ener
gies of interaction between the nuclear dipole moments
and the currents due to electronic and nuclear motion . ' respectIvely. Term (1e) is the magnetic dipole-dipole
(spin-spin) interaction of the nuclei. Not included in
Eq. (1) is the classical dipole-dipole interaction be
tween electrons and nuclei, or the Fermi contact term
proportional to S .. IK• For 12; molecules, rotation of
the nuclear frame slightly excites higher electronic
orbital states, but not higher spin states (to the same
order in perturbation theory, in any case), and these
terms do not therefore contribute to the hfs.18
The terms in the Hamiltonian of Eq. (1) are written
in order of descending magnitude for most 12; mole
cules. The first term, the energy of rotation of the
nuclear frame, is of the order of 105 times the largest
hyperfine term for all the molecules considered here.
Terms off-diagonal in the rotational states are there
fore neglected,23 and the perturbation problem reduces
to the averaging of the Hamiltonian over a given asym
metric rotor wavefunction. The result of this averaging,
together with the diagonal terms of the I·J interaction
considered below, we will call the hyperfine Hamil
tonian. For a given rotational state this will be a func
tion of J, the various nuclear spins IK, and the hyperfine
coupling constants.
The result of averaging the quadrupole term (lb)
over an asymmetric rotor wavefunction was originally
studied by Bragg,24 and is reviewed by Townes and
22 N. F. Ramsey, Phys. Rev. 90, 232 (1953). The sign for the
term ZxMp/gxMx given in Refs. 18 and 21 is in error. 2' ~econd-order hyperfine effects, produced by the hyperfine
matnx elements connecting different rotational states may be
expected to modify the hyperfine levels by an energy of ' the order
of the (hyperfine energy)2 /rotational energy or of the order of 1
cps for the molecules considered here. '
24 J. K. Bragg, Phys. Rev. 74, 533 (1948). Schawlow2li and Posener.21 For the Kth nucleus
(eqJQ)K
JCquadrupole 2h(2h-l) J(2J-l)
X[3(I KoJ)2+HI KoJ) -h(h+1) J(J+l)]. (2)
The qJ, in general, vary from rotational state to state,
and may be expressed in terms of the diagonal elements
Vgg=a2v /ag2 of the electric field gradient tensor, when
this tensor is written in the principal axis system of the
molecule. For an asymmetric rotor25•26
qJ=2L(Jg2)V gg/(J+l) (2J+3). (3)
9
The (Jrl) are the average values of the square of the
components of J along the principal inertial axes of
the molecule
(Ja2)=t[J(J+1) +E(K) -(K+l)aE(K)/aK], (4a)
(N)=aE(K)/aK, (4b)
(Jc2)=t[J(J + 1) -E(K) + (K-l)aE(K) /aK], (4c)
whe~e a, b, and c refer, respectively, to the least, inter
medIate, and greatest principal axes of inertia. E(K) is
the energy parameter of an asymmetric rotor with
asymmetry parameter K= (2B-A-C)/(A-C), and is
tabulated by Townes and Schawlow.27
The three Vgg are not independent, but satisfy
Laplace's equation: LgVgg=O. It is often a good
approximation to assume that the field gradient tensor
is cylindrically symmetric about a molecular bond, in
which case the tensor is determined by the single
derivative V H along the bond direction ~.
The result of averaging the spin-spin term (le) over
an asymmetric rotor wavefunction is given by Posener.21
It is also presented by Ramsey3 for diatomic molecules,
and by Gunther-Mohr et al.18 for ammonia.
This interaction may be put in a form in the molecule
fixed frame of reference which allows the well-known
results given above for the quadrupole interaction to
be applied, and at the same time suggests a consistent
notation. If we define the two symmetric dyadics
S=HYKYL+YLYK) , (5)
R=rKL-5(-3rLKrLK+rLK21), (6)
then the spin-spin interaction for the two nuclei labeled
K and L can be written as a contraction of these
dyadics, in analogy to the quadrupole interaction
JC.P in-Bp in = S: R.
26 See Ref. 5, Chap. 6.
26 J. K. Bragg and S. Golden, Phys. Rev. 75, 735 (1949).
27 See Ref. 5, p. 527. (7)
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54M I C ROW A V ESP E C T RUM 0 F H DO, H D S, C H20, AND C H D 0 261
R, like V, may be averaged over an asymmetric rotor
wavefunction to give
(R)= J(2~-1) t![JJ+(JJ)tr]-J(J+l)l). (8)
dJ is calculated from the dyadic R when written in the
principal axis frame, in the same way that qJ is calcu
lated from V.
dJ=2~(JD2)RD,/(J+l) (2J+3). (9)
If Eq. (8) is now substituted into Eq. (7) I the tensor
contraction can be written in terms of scalar-product
operators, and the spin-spin interaction becomes
3Cspin-spin= [gLgKf.LN2dJ/ J(2J -1)]
X {![ (IL' J) (IK' J) + (IK' J) (h' J)]-(h· IK) J2).
(10)
The rotation of the nuclear frame produces a small
magnetic field which contributes to hfs in first order.
If we make the substitution VK= (,) x rK, and
Wg= 2AgJg/li, then the term (ld) for the Kth nucleus
can be written in the reference frame fixed in the
molecule as
(11)
where N is a tensor whose components in the principal
axis system of the molecule are
Ng,,= -(2egKf.LNAg/hc) ~ZdLK--
~K
X [rLK' (rL-'YKrK) Ilgg,-(rLK)g(rL-'YKrK),,]. (12)
We may again use the results quoted for the quad
rupole interaction to average N over an asymmetric
rotor wavefunction. We find that
(N)=2~(Ji)Ngg/ J(2J-l) (J+l) (2J+3) ,
X {![JJ+(JJ)tr]-J(J+l)l), (13)
and Eq. (11) becomes, using the commutation rules
for the components of angular momentum
3Cnuo. I·J= [~(Ji)NDg/ J(J+l) ]IK,J. (14)
g
From Eq. (12) it can be seen that the diagonal ele
ments Ngg, and therefore the I·J constant due to nu
clear rotation, are always negative.
For l~ molecules, Term (Ie) in the Hamiltonian,
the interaction of the nuclei with the orbital motion of
the electrons, makes no contribution to the hfs in first
order. The cross terms in (la), however, proportional
to JgLg, connect excited electronic states to the l~ ground state, and (lc) makes a second-order contri
bution to the hfs.28
The perturbation calculation is carried out in detail
by White4 and by Townes and Schawlow.6 Written in
the molecular frame, the interaction has the same form
as Eq. (11). For the Kth nucleus
3Celectronio I.J = IK, E, J. (15)
E is a tensor fixed in the molecular frame with com
ponents
E"g=2(e/c)gKf.LNA g
,,(0 I Lg I n)(n I Pg' 10)+(0 I Pg' I n)(n I Lg I 0)
X~ . '.n Wn-Wo
(16)
The summation over n is over all excited electronic
states, and V stands for riK-3riK x (Vi-'YKVK).
The averaging proceeds as in the previous case. If
we include the contribution due to rotation of the
nuclear frame from Eq. (14), we find for the total I, J
interaction for the Kth nucleus
3Cr.J= [~(J,2)(Ngg+Egg)/ J(J+l) ]IK,J. (17)
Several points should be noted. While the components
of N can be accurately calculated on the basis of Eq.
(12) and the molecular geometry, the tensor E can only
be roughly estimated due to our ignorance of excited
state electronic wavefunctions. Electrons which are
spherically symmetric about a given nucleus, however,
do not contribute to the I, J constant for that nucleus,
since for these electrons (0 I LD In) is zero. It can be
shown alsos that those electrons which are spherically
distributed about other nuclei contribute in the same
way as the nuclear charges, but with opposite sign, to
N. For molecules with many-electron atoms, the ob
served I, J constant is therefore the difference between
two larger numbers. To determine the sum of Eq. (16)
over the excited states of the valence electrons from the
experimental constants, the components of N should
be calculated with an effective nuclear charge equal to
Z minus the number of closed-shell electrons.
The second-order perturbation treatment5 of (Ie)
reveals a second hyperfine term having the same de
pendence on I and J as the quadrupole interaction,
and usually referred to as the pseudoquadrupole effect.
This interaction is typically of the order of a few cycles
per second and is not further considered here. Higher
order terms in the perturbation expansion producing a
pseudomagnetic dipole interaction between nuclei and
decoupling of the electronic spins were considered by
28 This must not be confused with the "second-order" hyperfine
interactions between different rotational states mentioned above.
For the I·J interactions arising from the term (lc) in the Hamil·
tonian, such effects can appear only to third order in the pertur
bation expansion.
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54262 THADDEUS, KRISHER, AND LOUBSER
Gunther-Mohr et al.ls for ammonia, and found to be
equally small.
We are now in a position to write down the hyperfine
Hamiltonian for all the molecules studied. Hfs for
HDO, HDS, and CHDO is due to a proton and a
deuteron, and therefore the Hamiltonian is
:JC= (eqJQ)D
2ID(2ID-1) J(2J -1)
X[3(ID· J)2+HIo· J) -Io2J2J
+CH(IM"J)
+gDgHJ.l.N2dJ {;![(I . J) (I . J) J(2J-l) 2 D H
+ (1M" J) (lD· J) J-(1M" ID)J2)
+CD(lD·J). (18)
For CH20, hfs is due only to the two protons. In the
coupling scheme
11+12=1,
I+J=F, the Hamiltonian is:
g 2" 2d :JC= C I. J + H r-N J
H J(2J-l)
X a[(ll·J) (12·J)+(l2·J) (ll·J)J- (ll·12)J2). (19)
(FI' •• ·FL-/ I h·J I Fl·· ·FL-l) B. Matrix Elements
In Eqs. (2), (10), and (17) the hyperfine interactions
are given as products of the diagonal operators J2 and
IK2, and the scalar products h· J and h· IK. To calcu
late the hfs due to an arbitrary number of nuclei, it is
most convenient to select a representation in which off
diagonal matrix elements of the Hamiltonian are as
small as possible. For N distinguishable nuclei, num
bered in order of decreasing coupling to J, the appro
priate representation is
I II·· . IN, J, Fl·· ·FN-l, F), (20)
defined by the coupling scheme
J+Il=Fl,
Fl+12=F2,
(21)
The matrix elements of h·J and IL·IK diagonal in
the total angular momentum F, and all the inter
mediate angular momenta F;, can be calculated from
the vector model. The quickest and most elegant way
of calculating the off-diagonal elements is to use the
Wigner 6j coefficients29,3o which are now tabulated3l for
all integral and half-integral values of the coupling
spins up to 8. In the above representation we find that
= (-I)r {J(J+l) (2J+l)[(2F l'+I) (2Fl+1)]-· • [(2FL-l'+1) (2FL-l+l) Jh(h+1) (2h+l»)i
X{Fo' FI' II} ••• {FL-2' FL_l' IL_l}{FL IL FL-l'} (22)
Fl Fo 1 FL-l FL_2 1 1 FL-l 1£ '
where
L-l
r= (L-l)+ L(Fi-l'+Ii+F i) + (FL-l+h+h) ,
i=l
and, for K < L,
(FK'·· ·FL-l' I IL·IK I FK•· ·FL-l)
where = (-1)8 {h(h+1) (2h+l)[(2F'K+l) (2h+l)]-· . [(2F'L-l+l) (2FL-l+l)Jh(h+l) (2h+l»)i
X{IK FK' FK-l}{FK' FK+I' IK+l} .•. {FL-2' FL-I' IL-l}{FL 1£ FL-l'} (23)
FK h 1 FK+l FK 1 FL-l FL-2 1 1 FL-l h '
L-l
+ L (Fi-l'+Ii+F i) + (FL-l+h+F L). The { ) are the 6j coefficients, tabulated in Ref. 3l.
Where it simplifies the notation we have let J=Fo=Fo'
and F=FN. h·J and h·IK are diagonal in the Fi not
lying in the explicit ranges Fl·· ·FL-I and FK·• .FL-1,
respectively. The product of 6j symbols immediately
on either side of the suspension points in Eq. (22) i=K+l
29 A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1960), 2nd ed.
30 B. R. Judd, Operator Techniques in Atomic Spectroscopy (McGraw-Hill Book Company, Inc., New York, 1963).
31 M. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3j and 6j Symbols (The Technology Press, Massachusetts Institute
of Technology, Cambridge, Massachusetts, 1959).
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54M I C ROW A V ESP E C T RUM 0 F H DO, H D S, C H20, AND C H D 0 263
reduces to one symbol when L = 2, and vanishes when
L=l, leaving for the entire chain only the symbol on
the far right. Likewise in Eq. (23), when L=K+1 the
intermediate product vanishes, leaving for the entire
chain the product of the symbol on the far left and
that on the far right.
For the case of the spin-spin interaction, which
couples three spins, it is most convenient in practice
first to calculate the scalar product operators and then
(F/·· ·FL-l'\ :JCquadrupole \ Fl·· • FL-I) = (-1)1 (eqJQh perform the matrix products of Eq. (10). The quad
rupole interaction couples only two spins, however,
and while the matrix elements may be calculated in
this way, using Eq. (2), it is even simplier to revert to
the tensor form of the operator
:JCquadrnpole=iV:Q,
which can easily be shown to become for the Lth
nucleus in the above representation32
X[(2J+1) (2J+2) (21+3) [(2F '+1) (2F +l)J ... [(2F '+1) (2F +1)J(2h+1) (21£+2) (2h+3)]!
8J(2J -1) I 1 L-l L-l 81£(21£_1)
X{FOI FI' II} ••• {FL_2' FL-I' IL-l}{FL 1£ FL-I'} (24)
Fl Fo 2 FL-I FL_2 2 2 FL-I 1£ '
where
L-I
/= L(Fi-I'+I,+Fi)+(FL-l+h+F L).
;"1
In an asymmetric rotor there may exist one or more
pairs of equivalent nuclei-identical isotopes with the
same molecular environment and hyperfine coupling
constants. The protons in H20, CH20, and NH2D are
an example of such a pair, which always defines a two
fold axis of symmetry for the molecule. If there are
more than two equivalent nuclei the molecule will have
a higher-fold symmetry axis, and will be a symmetric
or spherical rotor (for example, NHa or CH4).
The Hamiltonian for equivalent nuclei is most con
veniently written in a representation where the total
angular momentum of the pair
(25)
is well defined. The various Is for equivalent pairs may
then be coupled together, and to the 1£ of the other
nuclei in the molecule, by the coupling scheme (21).
The requirement that the total wavefunction be
either symmetric or antisymmetric on exchange of
equivalent nuclei, however, restricts the values of Is
which can occur in a given rotational state. The sym
metry of the rotational wavefunction Y;JK_l.K on ex
change is a simple function of K-I and K if inversion
of the molecule is not considered.aa For example, in the
case of planar CH20 where the symmetry axis is the
least principal axis of inertia, the rotational state is
symmetric under exchange of the protons when K_I is
even, and anti symmetric when K_I is odd. The sym
metry of the spin state depends only on whether Is is
odd or even. Since the state for which Is has its greatest
value, 2Is(l), is always symmetric, in the case of
integral-spin nuclei even Is states will be symmetric
32 See Ref. 29, pp. 111, 115.
33 See Ref. 5, Chap. 4.
34 See Ref. 3, Chap. 3. and odd Is states antisymmetric. The converse will be
true for half-integral-spin equivalent nuclei. In either
case, only every other value of Is is allowed.
It often happens that when a given hyperfine inter
action is summed over an equivalent pair, terms off
diagonal in Is vanish, and the result is formally equiva
lent to the coupling of the single angular momentum
Is to the rest of the molecule. In particular, it is clear
that
and it is also easily shown that the spin-spin inter
actions of Is(1) and Is(2) with the Lth nucleus sum to
:JC = gSgLIJ.N2 ( dJ ) SL
88 J(2J-1)
X (![(Is· J) (h· J) + (h· J) (Is· J) J-(Is· IL) J2}.
(27)
The quadrupole interactions of an equivalent pair,
and their mutual spin-spin interaction, however, con
nect states which differ in Is by 2. In a representation
where Is is well defined, the elements off-diagonal in Is
cannot therefore be simply written down in terms of
the matrix elements given above, although the correct
expressions may be derived without difficulty as chains
of 6j symbols in terms of the general matrix elements
of tensor operators.29 The elements diagonal in Is,
however, may be written in terms of the above ex
pressions.34 Moreover, in the common case of two
equivalent spin! nuclei such as the protons in CH20
or NH2D, no terms off-diagonal in Is can exist, and all
hyperfine interactions vanish when Is=O. When Is=l
the mutual spin-spin interaction gives the quadrupole-
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54264 THADDEUS, KRISHER, AND LOUBSER
IY
I
(x.-.o4OO!Sl) I
y·.o9310A
.0
HOO
like term
:JC _ gs2/-1N2 (dJ ) SS
mutual B.-21s(21s-1) J(2J -1) FIG. 2. Geometry of
the HDO molecule.
X[3(IsoJ)2+!(I soJ)-Is2J2J, (28)
We have obtained the important result that the
hyperfine Hamiltonian, even when equivalent nuclei
are present, can often be written in terms of simple
products of the operators whose matrix elements are
given explicitly by Eqs. (22) and (23).
Co Hyperfine Intensities
Due to the complicated dependence of the state
selection process on both the parameters of the mole
cule and apparatus, there is no simple correspondence
between the intensities of transitions observed in
absorption and those observed in emission with a beam
maser. Details of the calculation of absolute intensities
are discussed in Ref. 12.
Although the state selection of various hyperfine
levels is often quite preferential, it is found in practice
that, for hfs of a few hundred kilocycles per second or
less, the passage of a molecule out of the high electric
field region of the state selector is highly nonadiabatic,
and relative intensities of hyperfine transitions subse
quently observed as the molecules pass through the
resonant cavity of the maser may be calculated, as in
N
u= 2::(I.+F i-1'+Fi)+N,
;'-1
and, as before, we let J=Fo=Fo' and F=FN for nota
tional compactness.
In the event that various Fi are not good quantum
numbers, we must transform C into the representation
where the Hamiltonian is diagonal. In matrix notation
C'=AiCAr1, FIG. 3. Observed hfs of the 220-+221 transition of HDO. (a) is
the strong central line, the sum of the six unresolved AF= AFt = 0
transitions; (b) shows the high-frequency hyperfine satellites seen
against the background of the cavity response. The weak "for
bidden" 3! ...... 11 transition can be seen to the extreme left, at the
foot of the central line. The low-frequency satellites were ob
served to be slightly weaker than the high-frequency ones (see
text) .
absorption spectroscopy, on the assumption that all
hyperfine states are essentially equally populated.
HDO has the largest coupling constants of the mole
cules which we have studied, and the low-frequency
hyperfine satellites are observed to be slightly weaker
than the high-frequency ones, due to preferential state
selection of the higher hyperfine levels of the 220 state.
A similar but more pronounced effect has been observed
for the quadrupole satellites of the NH3 inversion
line.u
In fitting the theoretical to the observed spectrum
we have therefore, as is usually done, taken the hyper
fine intensities proportional to the square of the elec
tric dipole moment matrix elements, summed over the
degenerate magnetic states of the transition.35 That is,
in the representation (20),
Sa'F.a'F= 2:: I (a', F', m/ I/-IE I a, F, mF) 12, (29)
mF",mF
where /-IE is the component of the dipole moment along
the electric field, and a stands for all the intermediate
Fi• It is one of the well-known results of atomic spec
troscopy that the summation over mp and m/ gives36
(30)
where, in arbitrary units,
FI'
{F' X 0
F1 Fo (31)
where Ai and AI are the unitary matrices which
diagonalize the hyperfine Hamiltonians of the initial
and final rotational states, respectively, of the transi
tion.
36 E. U. Condon and G. H. Shortley, The Theory of Atomic
Spectra (Cambridge University Press, Cambridge, ]England,
1959) po 98.
36 See Ref. 29, p. 76.
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54MICROWAVE SPECTRUM OF HDO, HDS, CH20, AND CHDO 265
FIG. 4. Calculated hfs of
the HDO 220 ..... 221 transi
tion. 100
75
50
25
In theory the line shape of transitions observed with
a beam maser is also a complicated function of :the
parameters of the apparatus, since it depends on the
velocity distribution of the molecules emerging from
the state selector, and may be modified by collision of
the more divergent molecules with the cavity walls.
In practice, however, we have found the true lineshape
to be well approximated by a Gaussian, and we have
used this lineshape, adjusting the full half-width of
about 5 kc/sec slightly from molecule to molecule, to
calculate the theoretical spectra presented in this work.
III. EXPERIMENTAL RESULTS
a. HD037
The water molecule has been extensively studied in
the infrared, and its geometry is well known. The
T ABLE I. Molecular constants of HDO, calculated in the rigid
rotor approximation, neglecting electrons, from the geometry of
Fig. 2. The rotation constants and asymmetry parameter are:
A = 694.45 kMc/sec, B = 273.80 kMc/sec, C= 196.37 kMc/sec,
and K= -0.6891.
State
3.9794
4.0000 (N)
1.2607
1.0000 0.7603
1.0000 -2.470
-2.446
37 Preliminary results have appeared in Nuovo Cimento 13,
1060 (1959). 100 I
I
I
t3~-4
L3!-3~
TABLE II. Hyperfine intervals of the HDO 220 ..... 221 transition,
measured relative to the strong central line which is the sum of
the IlF=IlF l =0 transitions. The uncertainties quoted are proba
ble errors.
Transition Frequency
F1F ..... F1'F' (kc/sec)
1! ..... 2!
+1! ..... 2! 167.07±0.40
3f ..... 2!
+3f ..... 3! 109.91±0.30
3! ..... 2f
+ 1 1 ..... 1 ! 77 .91±0.30
3f ..... l! 18.80±0.30
F1F ..... F1F 0.OO±0.20
1! ..... 3! -20.01±0.30
2! ..... 3!
+1! ..... 11 -77.41±O.30
21-+31
+3!-+31 -109.04±0.30
2! ..... 1!
+2! ..... H -16S.86±O.30
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54266 THADDEUS, KRISHER, AND LOUBSER
TABLE III. Hyperfine coupling constants for HDO.
State (eqJQ)0 (kc/sec) CH (kc/sec) Co (kc/sec)
220 79.3±O.3 -43.47±O.11 -2.33±O.02
221 79.6±O.3 -43.63±O.13 -2.20±0.02
least and intermediate principal axes of inertia of RDO,
calculated neglecting the electrons, are shown in Fig. 2.
The molecular geometry was taken from the high
resolution investigation of the deuterated waters from
1.25 to 4.1 J.L by Benedict, Gailar, and Plyler.38
Rfs in the microwave spectrum of RDO has been
observed by Posener,39.40 and by Treacy and Beers.6
Posener studied in particular the (220, 221) K-type
doublet which we selected as being the easiest to in
vestigate with a beam maser, and found the central
line of the hfs to lie at 10278.2455±0.001O Mc/sec.
As with the other transitions studied, there was there
fore no search problem.
The rotational energy levels of RDO, calculated
from the rotation constants of Table I, are shown in
Fig. 1. The Stark effect of the (220, 221) doublet is very
favorable to state selection, since the adjacent rota
tional states lie several hundred kilomegacycles away.
Other closely spaced pairs of levels exist in RDO,
moreover, which are suitable for beam-maser study.
The lowest lying K-type doublet (110, In) is split by
about 80 kMc, while the (330, 331) doublet studied by
Treacy and Beers, which lies above the (220, 221)
doublet and is not shown in Fig. 1, is split by 825
Mc/sec. If a TM010 cavity of the type used in the present
experiments were used for the 330----7331 transition, it
would be large (about 28 cm in diameter), but by that
(X--.02BIA )
Y •. 079aA
HOS b
FIG. 5. Geometry of the HDS molecule. a
38 W. S. Benedict, N. Gailar, and E. K. Plyler, J. Chern. Phys.
24, 1139 (1956).
39 D. W. Posener, Australian J. Phys. 10, 276 (1957).
40 D. W. Posener, Australian J. Phys. 13, 168 (1960). . TABLE IV. Molecular constants of HDS calculated in the
ngid rotor approximation, neglecting electro'ns from the geom
etry of Fig. 5. The rotation constants and asy~metry parameter
are: A = 290.24 kMc/sec, B = 145.25 kMc/sec, C= 96.81 kMc/sec
and K= -0.4991. '
State
3.941
4.000 (N)
1.446
1.000 0.613
1.000 -0.745
-0.661
token co~ld be made long enough to allow exceptionally
narrow hnes 1 kc/sec or less in width.
The other water molecules R20 and D20 have a
comparable series of K-type doublets. The doublet
transitions, however, are of the GQ branch, and require
a component of the dipole moment along the a (least)
principal axis of inertia. For R20 and D20 this axis is
perpendicular to the dipole moment, and the matrix
element for the transition vanishes. Unfortunately, no
other transitions exist which are nearly as favorable for
beam-maser study. In the case of RDO, however, the
symmetry of the molecule about the dipole axis is
destroyed, and the a and b inertial axes are rotated by
210 (Fig. 2). In absorption spectroscopy, the 220----7221
transition is a rather strong microwave line with an
absorption coefficient of 3X 10-5 cm-1.
The RDO sample was prepared by mixing equal
parts of ordinary and heavy water, the exchange of
hydrogen proceeding very rapidly to yield 50% RDO.
Since only about 1 mm Rg of vapor pressure is needed
behind the effuser, it is possible to cool the vapor in a
salt-ice bath. In practice, however, this was found to
give only a small improvement in intensity.
The observed hfs of the 220----7221 transition is shown
in Fig. 3. The hyperfine intervals were measured as
described in Ref. 12, and are listed in Table II.
Since the hfs is observed to be very symmetrical,
~n~ th~ most mtense l1F=l1Fl=O transitions overlap,
It IS eVIdent that the hyperfine coupling constants are
very nearly the same in the 220 and the 221 states. This
equality of the coupling constants is expected when the
K-type doublet is close to a symmetric-top level for
which K> l,l8 On the assumption of the equality of the
constants in the two rotational states, and with the
FIG. 6. The observed high
frequency hyperfine compo
nents and the central line of
the 220->22/ transition of HDS.
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54MICROWAVE SPECTRUM OF HDO, HDS, CH20, AND CHDO 267
r
100
25
,
~ -100: -75
, '
L2~_3~ : " ,I
" ~31_3~ ,2 2 o
FREQUENCY. 25 :::50 :i 75
III II
II II
::LI~-+2i :L3~+2i Ii 100 I
1: I
:: I
~ 2~_1~ , 2 2
I
~2i-'~ : 2~-1~
I
I
L2~_3~ Kc /SEC
It-tiJi !
, I
L3i -+2j t3i-+3~ II L3~""'2i
:L11 .... 2~ ! 2 Z
LI1 .... 2~
i' 2 2
FIG. 7. Calculated hfs of the HDS 220--+22, transition.
TABLE V. Hyperfine intervals of the HDS 220--+22, transition,
measured relative to the strong central line which is the sum of the
D.F=D.F,=O transitions. The uncertainties quoted are probable
errors.
Transition
F,F--+F,'F'
1!--+2!
+1!--+2!
3!--+2!
+3!--+3~
3~--+2!
+1!--+1!
2!--+3~
+1!--+1!
2J--+3!
+3~--+3!
2J--+l!
+2!--+1! Frequency
(kc/sec)
91.9±0.4
62.3±0.4
4S.8±0.4
lS.6±0.4
O.O±O.4
-17.4±0.4
-4S.6±O.S
-62.4±O.S
-89.S±O.S spin-spin constant dJ of Eq. (9) calculated in ad
vance from the known molecular geometry (Table I),
a close fit of the calculated to observed spectrum was
obtained on the basis of the Hamiltonian of Eq. (18).
The theoretical spectrum, calculated using a Gaussian
linewidth with full half-width of 5 kc/sec, is shown in
Fig. 4.
In the I F1, F) representation, the terms off-diagonal
in F 1, due mainly to the I· J interaction of the proton,
are appreciable with respect to the diagonal energies
due to the deuteron quadrupole coupling. The magnetic
interaction therefore plays an important role in even a
qualitative understanding of the spectrum, and the
intensities calculated in the I F1, F) representation are
only approximately correct. The transitions 1, 3/2t--7
3, 5/2 in particular are forbidden in the I Fl, F) repre
sentation, but are reasonably strong for HDO. One
may be seen quite clearly as a close satellite of the
central line in Fig. 3.
As long as the constants to be varied, (eqJQ)n, CR,
and CD were kept the same in either rotational state,
it proved feasible to perform the fitting calculations on
a desk calculator. A somewhat closer least-squares
fitting, varying all six constants independently, was
TABLE VI. Hyperfine coupling constants for HDS.
State (eqJQ)D(kc/sec)
42.9±0.4
43.3±0.4 CH(kc/sec) CD (kc/sec)
-2S.03±O.13 -0.47±O.02
-2S.4S±O.13 -O.22±O.02
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54268 THADDEUS, KRISHER, AND LOUBSER
H b
1.20A o
c • x=.598A
CH.O
FIG. 8. Geometry of the CH20 molecule.
subsequently performed using an IBM 7090 computer.
The best fit constants found in this way are listed in
Table III.
h. HDS41
The microwave spectrum of hydrogen sulfide has
been studied by Burrus and Gordy42 and by Hillger
and Strandberg.43 No hfs has been reported prior to
the present work.
Dousmanis44 has·"calculated"":the geometry of H2S
from the measured frequencies of Hillger and Strand
berg; his results are shown in Fig. 5. The rotational
levels of HDS are shown in Fig. 1. The 220-+221 transi
tion lies at 11 283.83 Me/sec, about 1000 Me/sec
higher in frequency than in HDO.
It is seen that the (220-+221) doublet is surrounded by
the (313,303) oblate symmetric top K-type doublet.
Such nearby states can sometimes sufficiently modify
the Stark effect of a given level to destroy the possi
bility of state selection. In the present case, however,
the dipole-moment matrix elements that connect the
220 and 313 states and the 221 and 303 states are very
small, and the cQ branch 311r~303 transition is strictly
TABLE VII. Rotation constants and asymmetry parameters for
CH20 and CHDO, calculated in the rigid rotor approximation,
neglecting electrons, from the geometry of Fig. 8 and Fig. 11.
The (J.) for the rotational levels of interest are independent of K.
(J i), (Jb2), (Jc2) are 1, 4, 1 and 1, 1, 4 for the 211 and 212 states,
respectively.
ABC
(kMc/sec) (kMc/sec) (kMc/sec) K
283.75
198.69 39.52
35.59 34.69
30.18 -0.9612
-0.9358
41 Preliminary results have appeared in Bull. Am. Phys. Soc.
5, 74 (1960).
42 C. A. Burrus and W. Gordy, Phys. Rev. 92,274 (1953).
43 R. E. Hillger and M. W. P. Strandberg, Phys. Rev. 83,
575 (1951).
4. See G. R. Bird and C. H. Townes, Phys. Rev. 94,1203 (1954). TABLE VIII. Hyperfine separations of the 211->212 CH20
transition. Uncertainties quoted are probable errors.
Transition Frequency
F->F' (kc/sec)
2->2 1O.12±0.20
3->3 0.OO±0.30
1->2 -8.5±1.0
1->1 -20.73±0.30
forbidden, since it requires a component of the dipole
moment perpendicular to the plane of the molecule.
On the basis of the rotational constants listed in Table
IV the 313 state lies higher than the 220 state by about
57 kMc/sec, while the 303 and 221 states practically
coincide. The exact location of the levels, however, is
uncertain by about 20 kMclsec due to centrifugal dis
tortion and the uncertainties in the rotation constants.
When first observed the HDS lines were about four
times weaker, compared to the HDO lines, than ex
pected, and we attributed this to inhibition of state
selection. However, a carefully prepared sample, made
by reacting deuterated sulfuric acid with iron sulfide,
finally gave the strong signals shown in Fig. 6, and it
appears that in fact the state-selection process is quite
efficient.
Hydrogen sulfide has a vapor pressure of several
hundred millimeters of Hg at dry-ice temperatures,
and it was found that the source could be cooled with
a dry-ice-acetone mixture to produce roughly an im
provement of 2 in signal strength. As with HDO, the
short hydrogen exchange time permits a sample con
taining at best only 50% HDS.
The measured hyperfine intervals are listed in Table
V. The fitting of the calculated to the observed spec
trum was done in the same way as for HDO: the spin
spin constants dJ were calculated in advance, and the
three constants (eqJQ) D, CH, and CD, considered equal
in the 220 and 221 states, were varied to give a best fit
using the Hamiltonian of Eq. (18). Subsequently, these
constants were varied independently in the two rota
tional states with an IBM 7090 computer to give a
slightly better fit. The theoretical spectrum is shown
in Fig. 7. The line envelope was calculated assuming a
Gaussian line shape with full half-width of 4 ke/sec.
The~best-fit hyperfine constants are listed in Table VI.
50 kc
i .1 A FIG. 9. Observed hfs of the 211->212
transition of CH20. A single AF= 1 tran
sition can be seen just to the left of the
strongest line.
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54M I C ROW A V ESP E C T RUM 0 F H DO, H D S, C H20, AND C H D 0 269
TABLE IX. Hyperfine coupling constants for CH20 and CHDO.
(eV~Q)D eH (211) -CH (212) CH(211) CD (211) -CD (212) CD (211) (1/r3)-1
Molecule (kc sec) (kc/sec) (kc/sec) (kc/sec) (kc/sec) (1)
CH20 2.26±0.13 0.65±0.50 1.898±0.017
CHDO 170.0±2.0 2.42±0.50 0.2±1.0 0.25±0.10 0.13±0.20 1.88±0.1O
C. CH2045
Bragg and Sharbaugh,46 Lawrance and Strandber?,47
and Hirakawa, Miyahara, and Shimoda48 have st.udied
the microwave spectrum of the common specIes of
formaldehyde. Magnetic hfs, the only kind occuring,
was first observed by Okaya,49 who resolved two of the
three intense l:!.F=O hyperfine components of the
414~4t3 K-type doublet transition at 48285 Mc/sec.
More recently Takuma, Shimizu, and Shimoda50 have
investigated the 312~3!3 K-type doublet transition near
28975 Mc/sec with a beam maser, and have resolved
all three l:!.F=O components. They were able to deter
mine the difference CH(312)-CH(313), and the proton
proton distance. They have a~so succeeded in usi?g a
beam maser at radio frequenCies to observe rotatIOnal
transitions and hfs at 4.57 and 18.275 Mc/sec.51-53
40
~30
in z
<oJ ..
;!;
<oJ >20 ;:
" ..J
<oJ
0:
10
~20
I ."'1 20
FIG. 10. Calculated hfs of the CH20 211-+212 transition.
45 Preliminary results have appeared in J. Chern. Phys. 31, 1677
(1959) .
4e J. K. Bragg and A. H. Sharbaugh, Phys. Rev. 75,1774 (1949).
47R. B. Lawrance and M. W. P. Strandberg, Phys. Rev. 83,
363 (1951). . .
48 H. Hirakawa, A. Miyahara, and K. Shimoda, J. Phys. Soc.
(Japan) 11,334 (1956).
49 A. Okaya J. Phys. Soc. (Japan) 11,258 (1956).
60 H. Taku~a, T. Shimi2u, and K. Shimoda, J. Phys. Soc.
(Japan) 14, 1595 (1959). • •
61 K. Shimoda, H. Takuma, and T. ShimIZU, J. Phys. Soc.
(Japan) 15, 2036 (1960). .. .
52 <:ee also the article by K. Shimoda 10 the Proceedmgs of the
Inter"'national School of Physics" Enrico Fermi," Topics on Radio
frequency Spectroscopy (Academic Press Inc., New York, 1962).
i>3 H. Takuma, J. Phys. Soc. (Japan) 16, 309 (1961). We have studied the K-type doublet transition
211~212 which lies at 14488.65 Mc/sec. A linewidth of
5 kc/sec was obtained, and a sensitivity great enough
to allow detection of one of the four possible l:!.F= 1
transitions. This allowed a calculation of the I· J con
stant for either rotational state, and the proton-proton
distance.
The molecular geometry is shown in Fig. 8, and the
molecular constants are listed in Table VII. The bond
lengths and the HCH angle were taken with slight
modification from the paper of Lawrance and Strand
berg.47 CH20 is a very nearly prolate symmetric top,
with an asymmetry parameter K= -0.9612, so that the
prolate K-type doublets are in general split by a much
smaller frequency than for HDO and HDS. The
(2u, 212) doublet, for example, in HDO is split by
several hundred kilomegacycles.
From the rotational energy levels of CH20 (Fig. 1),
it can be seen that the (2n, 212) doublet is well located
from the point of view of state selection. Although the
density of rotational states is greater than for HDO,
with consequent smaller fractional population of a given
state, the large component of the dipole moment along
the a (least) principal axis of inertia makes formalde
hyde very favorable for beam-maser study.
We first succeeded in observing the 211~2I2 transition
with a formaline solution as the source of vapor, the
water which came off being removed with a dry-ice
trap, since formaldehyde has a vapor pressure of more
than 10 mm Hg at -80°C. It was subsequently found
that heating the polymer para-formaldehyde to about
130°C gave a copious flow of the monomer and much
y b
I
I
I
I
I
I
I
I
CHDO o .=.636 A
y=-.0305A 5°15'
FIG. 11. Geometry of the CHDO molecule.
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54270 THADDEUS, KRISHER, AND LOUBSER
TABLE X. Hyperfine separations of the 211->212 CHDO transition.
Uncertainties quoted are probable errors.
Transition Frequency
F1F->F/F' (kc/sec)
1!->1! 30.93±0.30
1!->1! 20.00±0.30
31->31
+3!->3! 0.00±0.30
2!->2! -49.03±OAO
2!->2! -57.30±0.40
stronger lines. Cooling of the effuser, as with HDS,
was found to give some increase in signal intensity.
The observed hfs is shown in Fig. 9. Since K-l = 1
for the (211, 212) doublet, the hyperfine coupling con
stants vary greatly between the two rotational states,18
and the I1F=O transitions are well separated. The hfs,
however, is due to magnetic interactions alone, and the
entire structure is only some 35 kc/sec wide, or about
half the Doppler width in conventional absorption
spectroscopy. A single I1F= 1 transition can be clearly
made out next to the most intense central line.
Frequency intervals were measured to within 1
kc/sec using the Radiation Laboratory's frequency
standard as described in Ref. 12; they are listed in
Table VIII. A reasonably close fit to the observed
spectrum was first obtained using the Hamiltonian
of Eq. (19), with the spin-spin constant dJ for the 211
and 212 states being calculated from the molecular
geometry of Fig. 8. A closer fit was then obtained by
slightly varying the proton-proton distance. The best
fit gave (1/,3)-1/3= 1.898±0.017 X. This contrasts with
the value found by Shimoda et al.50 of 1.82±0.04 X,
but is in good agreement with the value of 1.92 X
obtained from the geometry of Fig. 8.
From the theoretical spectrum of Fig. 10 it can be
seen that all but one of the I1F= 1 lines lie so close to
the I1F= 0 transitions that they cannot be resolved.
The single one discerned in Fig. 9, however, has allowed
a determination of the absolute values of CH(2u) and
CH (212), only the difference being determined by the
intervals between the main I1F=O lines. The various
hyperfine constants are listed in Table IX.
d. CHD045
The 2u-*212 transition for CHDO lies about 1500
Mc/sec higher in frequency than that for CH20. The
exact frequency has been found to be 16038.08 Mc/sec
by Hirakawa, Oka, and Shimoda.54
64 H. Hirakawa, T. Oka, and K. Shimoda, J. Phys. Soc. (Japan)
11, 1207 (1956). The rotation of the inertial axes shown in Fig. 11
was calculated, neglecting the electrons, on the assump
tion that the bond lengths and the HCH angle are the
same as those for CH20. The intensity of the 2u-+212
transition was as strong as expected, compared to the
same transition in CH20, indicating that nearby levels
did not hinder state selection. In particular, the 303
level, shown in Fig. 1 immediately above the 211 level,
is not expected to give any difficulty, since the oR
branch transition 211-+303 is strictly forbidden.
As for CH20, the CHDO vapor was produced by
heating the polymer paraformaldehyde. The sample,
prepared by the Volk Company, was specified to be
90% CHDO, due to the stability of this molecule
against H-D exchange. Dry-ice cooling of the vapor
gave some increase in the signal strength.
The observed hfs is shown in the oscilloscope trace
of Fig. 12. It may be qualitatively interpreted in the
following way: the deuteron quadrupole interaction
splits both the upper and lower rotational state into
a triplet. The coupling constant (eqJQ)n is quite dif
ferent in either case since K-l = 1,18 so that the three
most intense I1Fl = 0 transitions are widely spaced
they correspond in Figs. 12 and 13 to the central line,
and the two strong doublets on either side. The doublet
structure in turn is due to the magnetic coupling of
the proton-the central transition is also split, but not
sufficiently so to be resolved. The remaining structure
is due to the hyperfine transitions for which I1F or
I1F1r£0.
The measured hyperfine intervals are given in Table
X. Using the Hamiltonian of Eq. (18) there are seen
to be in principle a total of eight hyperfine constants
to be varied to fit the observed spectrum. Considerable
simplicity resulted, however, in initially using the
proton-proton distance calculated from CH20, and in
calculating the (eq~)n from the quadrupole coupling
constant along the bond, assuming that the electric
field was cylindrically symmetric about the C-D bond
direction. The I1Fl=I1F=O transitions were then fit to
within 1 kc/sec in terms of only three constants:
(eV~Q)D, CH(2u)-CH(212), and CD(2u)-CD(2d.
CH(2u), CD (211), and (1/,s)-1/3 were then varied to fit
as well as possible the shape and frequency of the
manifold of the other hyperfine components. All cal
culations were performed on a desk computer. The
best-fit hyperfine constants are listed in Table IX. The
theoretical spectrum of Fig. 13, calculated using a
Gaussian line shape with full half-width of 5 kc/sec,
reproduces all aspects of the observed hfs. Relative
50 kc
FIG. 12. Observed hfs of the 211->212
transition of CHDO.
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35
30
25
>-20
f-
(f)
Z w f-
Z
w 15
?
f
<!
...J ..;
J::
10
5
20
FREQUENCY
(~c ISEC)
;ON t
,;;-N t
-~ N ,;;-
FIG. 13. Calculated hfs of the CHDO 211-+212 transition.
intensities were calculated in the 1 Fl, F) frame, since
off-diagonal terms in the Hamiltonian are relatively
small.
IV. MOLECULAR CONSTANTS
a. Electric Field Gradients
The hfs of HDO, HDS, and CHDO is well interpreted
on the assumption that the electric field at the deuteron
is nearly cylindrically symmetric about the bond. The
coupling constants along the bond, (eV ~&)n, derived
from the (eq.l2)n of Tables III, VI, and IX, are listed
in Table XI. The V H are calculated taking the deuteron
quadrupole moment to be2 Q= 2.82X 10-27 cm2•
Since (eqJQ)n has been found for two rotational
states in each molecule, we can also, in principle, calcu-
TABLE XI. Quadrupole coupling constants along the OD, SD,
and CD bonds.
Molecule (eVuQ)p
(kc/sec) VEE (statvolt/cm2) ."
HDO 318.6±2.4 1. 56X 1016 0.06±0.16
HDS 154.7±1.6 0.76X1016 -0.12±0.13
CHDO 170.0±2.0 0.83XlO '6 I." 1<0.15 late the asymmetry parameter for the field gradient
tensor,
(32)
where r and X are directions perpendicular to the bond
direction~, and perpendicular and parallel, respectively,
to the plane of the molecule. The fJ found for HDO
and HDS are also given in Table XI. For CHDO the
complexity of the hfs has allowed only a rough upper
limit to be set on the value of this parameter.
The quadrupole coupling constant along the bond
of 318.6±2.4 kc/sec, which we have found for HDO, is
in good agreement with the value of 31S±7 kc/sec
T ABLE XII. Comparison of measured and calculated values of
en for HDO.
Cn (calc.) -Cn(meas.)
State (kc/sec)
5" (53) 0.84
542(52) 0.83
330 (33) 1.36
331 (32) 1.31
220 (22) -1.85
221 (2,) -2.47
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54272 THADDEUS, KRISHER, AND LOUBSER
found by Posener39 from his study of the same rota
tional transition with a high-resolution absorption
spectrometer. It is somewhat higher, however, than the
values of 310.3±3.0 kc/sec and 314.3±1.5 kc/sec found
from the radio frequency transitions by Treacy and
Beers.6
The deuteron quadrupole coupling constant, depend
ing only on the ground-state distribution of charge, is
one of the molecular properties which, in principle, can
be most directly calculated from the theory of molecular
structure. Only for HDO, however, of the molecules
studied here, has an attempt been made to evaluate
the field gradient at the deuteron in terms of the elec
tronic wavefunction of the entire molecule, and the
agreement with experiment is not good. Using the
self-consistent molecular orbitals of Ellison and Shu1l55
for all 10 electrons of the water molecule, Bersohn1 has
calculated the coupling constant along the bond to be
about twice the value we have found.
b. Spin-Rotation Constants
In the rigid rotor approximation the I· J constant
of a given nucleus for all rotational states of an asym
metric rotor is a function of three molecular constants
the diagonal elements of the symmetric tensor MglI=
Ngg+Egg of Eq. (17). For HDO, the Url) differ
enough between the 220 and 221 states (Table III) to
allow two independent relations for the Mgg of the
proton to be established from the hfs. The work of
Treacy and Beers6 determines four further equations
for the Mgg, only two of which are independent, how
ever, since the Ul) vary only slightly between the
3ao and 331 states, or the 542 and 541 states.
Their least-squares fitting of the three Mgg to the
experimental data has revealed a clear discrepancy
between theory and experiment.6 No substantial im
provement results if the least-squares fitting is repeated
with our more recent value of CH for the 220 and 221
states (Table III), and with the more precise deter
mination of the water geometry of Benedict, Gailar,
and Plyler37 used to calculate the Ui). We find that
Maa= -43.78 kc/sec, Mbb= -41.15 kc/sec, and
Mcc= -60.30 kc/sec. A revised version of Treacy and
Beers' Table IV, calculated on the basis of these con
stants and the Ul) of our Table I, is given in Table
XII. By way of comparison, if the Mgg are calculated
on the basis of the three independent equations fur
nished by the four lowest-lying rotational levels studied,
the 220, 221, 3ao, and 381 states, we find that Maa= -55.8
kc/sec, M bb= -18.9 kc/sec, and Mcc= -19.8 kc/sec.
Since all our calculations are based on the rigid
rotor approximation, it is tempting to consider that
centrifugal distortion, which is notoriously large for
light asymmetric rotors, may be the cause of this dis
crepancy. The effect of centrifugal distortion on the
6Ii F. O. Ellison and H. Shull, J. Chem. Phys. 23, 2358 (1955). rotational energies of HDO may be as large as a few
percent for J in the range from 5 to 10. Since the rota
tion constants are proportional to the inverse square of
the molecular dimensions, however, while the nuclear
contribution to the I· J constant-which predominates
over the electronic contribution for HDO-is propor
tional to the inverse cube, we should also expect a
centrifugal effect of a few percent for the Mgg of the
542 and 541 states.
The actual discrepancy between theory and experi
ment is seen to be considerably larger than this; if it
is confirmed b'y further experiment it may prove
necessary to abandon the rigid rotor approximation
altogether, and consider in detail the effect of molecular
vibration on the molecular magnetic field. Of particular
interest in this respect would be an investigation of
the hfs of the 110-+111 K-type doublet transition, which
lies near 80 kMc/sec and is well suited for beam-maser
study. Centrifugal distortion will be slight for these
levels, and, since the doublet lies near a symmetric-top
state for which I K I = 1, the hfs will be very asym
metric, yielding two independent relations for the Mgg.
The same transition in HDS, which has been ob
served to lie at 51.073 kMc/sec,43 is of interest for
similar reasons. There are, in addition, a number of
higher-lying K-type doublets for which the doublet
transitions are scattered throughout the microwave
region43 that make this molecule particularly interesting
from the point of view of the I· J interactions.
Takuma53 has evaluated the Moo for CH20 from the
various observations of hfs,45.50 which yielded five inde
pendent relations for the three constants. All of the
experimental results were well interpreted on the basis
of the rigid rotor approximation. He found that Maa=
30.2±2.7 kc/sec, Mbb= -3.0±3.4 kc/sec, and Mcc=
-13.2±3.4 kc/sec.
V. CONCLUSIONS
The present work has shown the value of a molecular
beam in increasing resolution, and state selection in
increasing sensitivity, in the investigation of molecular
hfs. The properties found from experiment-electric
field gradients, and magnetic interaction constants
have not so far been calculated for many polyatomic
molecules. These calculations, however, and particu
larly those of field gradients, are coming within reach
of modern computing techniques and our knowledge
of molecular structure.
It is important to emphasize that the present results
were obtained without recourse to the most sensitive
detection techniques now available. The use of stabi
lized microwave oscillators, and narrow-band phase
sensitive detection, should allow an improvement in
sensitivity by a factor of from 10 to 100, and the study
of similar hfs in many microwave transitions. The
extension of the present techniques to the rotational
and rotation-inversion transitions of other light asym
metric rotors, to the rotational transitions of linear
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131.104.242.103 On: Sat, 29 Nov 2014 22:36:54M I C ROW A V ESP E C T RUM 0 F H DO, H D S, C H20, AND C H D 0 273
and symmetric-top molecules, and, perhaps, to mole
cules with hindered rotation, can be expected.
ACKNOWLEDGMENTS
We should like to acknowledge the support of Pro
fessor C. H. Townes, who gave aid and guidance at all
times. The staff of the Radiation Laboratory offered
invaluable assistance during the course of this work;
we would like to thank C. Dechert, T. Bracken, and
1. Beller in particular. H. Lecar assisted in the taking
THE JOURNAL OF CHEMICAL PHYSICS of data, while several of the experi.m.ental techniques
described were inspired by A. Javan. A correspondence
with D. W. Posener concerning HDO was of great
interest, as were a variety of conversations with A.
Okaya, B. P. Dailey, R. Bersohn, and M. Karplus.
Professor R. Jastrow kindly made available the com
puting facilities of the Institute for Space Studies, and
Ohseun Koh and Daniel Fife contributed greatly by
writing the machine program used in the final fitting
of the HDO and HDS spectra. T. Psaropolos prepared
many of the illustrations.
VOLUME 40, NUMBER 2 15 JANUARY 1964
Intercombination Spectra of Chromium Acetylacetonate Crystals at Low Temperatures*
PETER X. AromNDAREZ AND LESLIE S. FORSTER
Department of Chemistry, University of Arizona, Tucson, Arizona 85721
(Received 20 May 1963)
The spectra of chromium acetylacetonate and several derivatives have been recorded at 77° and 4°K.
The structure in the 12000-15000 cm-1 region has been assigned to vibronic components of 2E_4A2 and
the 2Tl state located. The effect of ligand halogenation on the trigonal field splitting is small.
INTRODUCTION
THREE spin-forbidden transitions can occur within
the t28 configuration of Cr m, 2E+-4A 2, 2 T1(;_AA2,
and 2T2+-4A2• In the ruby spectrum, the lowest energy
state at 14400 cm-l has been identified as 2E, split by
the trigonal field and spin-orbit interaction'> into two
Kramers doublets, 2A and E.I The 2T2 stat~ in ruby
occurs near 21 000 cm-l but the location of 2TI is in
dispute.2,3
The 2E-4A2 transition in chromium acetylacetonate
[Cr(aca)aJ has been detected in absorption4 and emis
sion6 at 12900 cm-l and the effect of substitution on the
ligand upon the transition energy assessed.6
The spectra of Cr(aca)3 and several derivatives have
now been examined at 4° and 77°K with the hope of
identifying the vibrations associated with the 2E+-4A2
transition and of locating the 2T1level.
EXPERIMENTAL
The spectra were recorded on Kodak I-N plates with
a grating spectrograph. The emission spectra were ob
tained in the first-order while the second-order (40
* Supported by a grant from the National Science Foundation.
IS. Sugano and Y. Tanabe, J. Phys. Soc. Japan 13, 880 (1958).
2 J. Margerie, Compt. Rend. 255, 1598 (1962).
8 R. A. Ford, Spectrochim. Acta, 16, 582 (1960).
4 T. S. Piper and R. L. Carlin, J. Chern. Phys. 36, 3330 (1962).
6 L. S. Forster and K. De Armond, J. Chern. Phys. 34, 2193
(1961) .
6 K. De Armond and L. S. Forster, Spectrochim. Acta 19, 1393
(1963). A/mm) was used for the absorption spectra. Wave
length calibration was made with a helium Geissler
tube. The positions of the narrow lines are reliable to
",,4 cm-1 but separations of closely spaced lines can be
determined to within 2 cm-l• At least two separate
plates of each spectrum were obtained and several
tracings of each plate were made with a Hilger re
cording microdensitometer in order to differentiate
plate noise from intrinsic spectral features. The crystals
were mounted in a glass Dewar in direct contact with
the coolant and the light incident on the spectrograph
slit was split into two oppositely polarized beams with a
Wollaston prism.
The absorption spectra of the broad spin-allowed
bands were obtained with a Cary Model 11 spectro
photometer.
Infrared spectra in the 325 to 4000 cm-l region were
determined with Beckman IR-4 (CsBr optics) and
Perkin-Elmer Infracord (NaCl optics) instruments
using the KBr pellet technique.
RESULTS
Chromium Acetylacetonate
At 77°K the absorption spectrum of Cr(aca)3 con
sists of a number of broad bands and the components
of 2E+-4A2 at 12 950 cm-l are not resolved.4 When the
temperature is reduced to 4 OK the absorption spectrum
is quite well resolved and a number of lines of 1 cm-1
width can be detected (Fig. 1). The two lines at 12895
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1.1729838.pdf | Influence of the Silicon Content on the Crystallography of Slip in Iron—Silicon
Alloy Single Crystals
S. Libovický and B. Šesták
Citation: Journal of Applied Physics 34, 2919 (1963); doi: 10.1063/1.1729838
View online: http://dx.doi.org/10.1063/1.1729838
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/9?ver=pdfcov
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J. Appl. Phys. 38, 4549 (1967); 10.1063/1.1709176
Magnetomechanical Damping in Iron—Silicon Alloys
J. Appl. Phys. 36, 2235 (1965); 10.1063/1.1714457
Temperature Dependence of Anisotropy and Saturation Magnetization in Iron and Iron-Silicon Alloys
J. Appl. Phys. 31, S150 (1960); 10.1063/1.1984640
Anisotropy Constants of Iron and IronSilicon Alloys at Room Temperature and Below
J. Appl. Phys. 30, S317 (1959); 10.1063/1.2185952
Complicated Domain Patterns on IronSilicon Single Crystals
J. Appl. Phys. 23, 1339 (1952); 10.1063/1.1702072
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For certain voltage and temperature ranges,2 the tunneling cur
rent density J(T) at temperature T(OK) is
J (T) =J (O)+aT". (1)
It can be shown3 that, as a first-order approximation,
a=[(87rmtk2)/h3]exp{ -(47r/h)J[2m",(x)]!dX} (2)
where 111 is mass of electron, t is charge of electron, It is Planck's
constant, k is Boltzmann's constant, and ",(x) is the potential
barrier measured from the Fermi level. The ratio of the incre
mental current density [defined as t:..J(T)=J(T)-J(O)] to
J(O) is, roughly,
-y = t:..J (T)/ J (0) = 32r11ls2k2T"/ (3h2<p), (3)
where s is insulating film thickness and <p is the average value of
",(x).
The tunnel current leT), of the BeO structure, was measured
as the temperature was varied from approximately 100° to 4OO0K
and the applied voltage was kept at 1 V. Two measurements
(7.6 p.A at 148°K and 9.49 p.A at 345°K), were used in (1) to cal
culate for 1(0). The resultant value of 1(0) was 7.17 p.A, which
was then subtracted from all measured values of leT) to obtain
the incremental currents t:..l(T)=l(T)-l(O) that are plotted on
a log-log scale in Fig. 1. The plot is a straight line of slope 2.04,
thus agreeing with the P dependence.
Measured by the technique of Simmons and Unterkofler,4 the
oxide thickness s was estimated to be 33 A. With the reasonable
assumption that <p= 1 eV, the value of -y calculated from (3) is
about 0.25 at 300°K. The data in Fig. 1 give -y=0.237.
Stratton obtained the TO dependence by expanding (6-y )!csc (6-y)!
for small -y. The present experimental result indicates that the P
dependence is still valid for values of -y somewhat higher than
those implied by Stratton's equation.'
1 J. G. Simmons. G. J. Unterkoller, and W. W. Allen, Ap]>!. Phys. Letters
2, 78 (1963).
, R. Stratton, J. Phys. Chern. Solids 23, 1177 (1962).
3 C. K. Chow. "Temperature Dependence of Tunnel Current Through
Thin Insulating Films," Burroughs Corporation, Burroughs Laboratories.
Internal Technical Report, TR62-57. December 1962 (unpublished). 'J. S. Simmons and G. J. UntNkoller, App!. Phys. Letters (to be
published).
Effect of Magnetic Field Reversal on the
Determination of Certain Thermo
magnetic Coefficients
JOHN A. STAMPER
Texas Instruments, Incorporated. Dallas, Texas
(Received 9 May 1963)
IN the measurement of the Nernst, Righi-Leduc, and magneto
Seebeck coefficients, it is customary to take data with the
magnetic field in each of two opposite directions. For the adiabatic
conditions generally assumed, it is then possible to eliminate cer
tain errors. It should be emphasized that the effect of magnetic
field reversal depends on experimental conditions and that there
are important cases where the coefficients can be evaluated even
under nonadiabatic conditions.
In these cases it is possible to separate the symmetric and anti
symmetric contributions to the temperature gradient and measur
able electric field. Errors due to superposition of effects and mis
alignment voltages can then be avoided. Adiabatic conditions (no
heat loss from the sides of the sample) can be closely approximated
in the laboratory and allow the separation of symmetric and anti
symmetric contributions. The constancy of heat current density
w on reversal of the magnetic field is a more general condition
which permits the separation. This is discussed below.
The components of w normal to the sides of the sample are de
termined by the temperatures of the sample surface and the sample surroundings. Reversal of the magnetic field B causes negligible
change in these temperatures when SB«1 where S is the Righi
Leduc coefficient. Thus, at the sample surface, w is the same for
both directions of the magnetic field. The following analysis shows
when this must be true throughout the volume of the sample.
Consider a sample in the form of a rectangular parallelepiped.
Let the magnetic field be applied in the Z direction and a tempera
ture gradient VT be applied in the X direction. It is assumed that
electric current density is zero. If V2T = 0 for both directions (in
dicated by + and -) of the magnetic field (allowing time for
steady-state conditions to be reestablished) then V2w = 0 for both
directions so that if w+=w-at the sample boundaries then
w+=w-throughout the volume of the sample.
An expression for V2T can be obtained from the equation
V·w=O. This relation comes from the theory of steady-state pro
cesses' and can be written
V2T= [1-K'(B)/ K(B)](l2T /(lz', (1)
where K'(B) and K(B) are thermal conductivities parallel to and
normal to the magnetic field, respectively. Thus V2T is zero if
either factor in the right-hand side of (1) is zero. Other than in
the adiabatic case ((IT /(lz=O), the condition (l2T /az2=0 is not
likely to be met experimentally. A positive heat flux into both
xy-faces or out of both xy-faces implies a2/Taz2~o. However, the
relation w+""'w-is valid in materials for which K'(B) =K(B) even
under nonadiabatic conditions.
Equations suitable for the evaluation of the adiabatic coefficients
can be derived from the relation w+-w-and the equations 2 for
heat current density and measurable electric field.
1 H. B. Callen. Phys. Rev. 73, 1349 (1958).
2 J. B. Jan. Solid Stale Physics (Academic Press Inc., New York. 1953).
Yol. 5, p. 8.
Influence of the Silicon Content on the Crystal
lography of Slip in Iron-Silicon Alloy
Single Crystals
S. LIBOVICKY AND B. SESTAK
I11slilute of Physics. Cuchoslovak Academy of Sciences.
Prague. Cuchoslovakia
(Received 29 April 1963)
IN earlier papers we found that the occurence of slip on single
crystals of Fe-3% Si alloy depended on the deformation rate
along the crystallographic or non crystallographic planes.,-a At
room temperature, during bending at deformation rates in the
surface of up to about 2X10-' sec-', the slip planes approach the
maximum resolved shear stress planes. At velocities above 10 sec'
slip occurs along the {110} planes. The transition from one type
of slip to another was studied at a lower temperature.4 At 78°K,
at velocities of about 10-7 sec-', the slip remains generally 11011-
crystallographic but there is a clear tendency of parts of the slip
planes to become {11O} planes, particularly on the tensile side of
the bent samples.
Up to now it has been universally accepted that, in an alloy of
iron with more than 4% Si, slip occurs only along the {110}
planes .. We have now found that when the silicon contents are
higher the slip planes differ markedly on the compression and
FIG. 1. Orientation
of samples.
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tensile sides of the same sample while on the compression side
the slip remains noncrystallographic up to higher deformation
rates than with an alloy with 3% Si.
The samples with the orientation shown in Fig. 1, cut from
single crystals grown by the Bridgman method, were chosen for
the study. They were deformed by three-point and four-point
hending at different deformation rates in an Instron tensile ma
chine and by the impact of a falling weight as in our earlier
papers.2,3 The samples were deformed at room temperature and
the slip bands were observed on the compression and tensile sides
of the samples. Figures 2 and 3 show photographs of part of the
surface on the compression and tensile sides of the same sample of
an alloy with 7.5% Si after deformation at two different deforma
tion rates. It is clearly seen that slip occurred quite differently on
the compression and tensile sides. On the compression side, slip
occurred along the maximum resolved shear stress planes in a
range of deformation rates of 6X 10-7 sec' to 5X 10-"2 sec'. With
the orientations used the planes with maximum resolved shear
stress are identical with the (H2) planes but it should be em
phasized that this is not crystallographic slip along these planes
since the character of the slip lines does not correspond to that of
crystallographic slip. The slip lines are slightly wavy according to
local inhomogeneities and deviate from the correct direction under
the influence of neighboring bands and as a result of inhomogen
eous stress. This is best seen in samples in which the middle knife
edge during three-point bending pressed the edge of the sample; as
a result of the inhomogeneous external stress field, the direction of
the slip bands changed like a fan [similarly as in Figs. 13(a), (b)
of Ref. 1]. Only at deformation rates of 4X 102 seC' do we ob-
Ca)
Cb)
),FIG. 2. Slip bands on compression (a) and tensile (b) side of same three
point-bent sample of Fe-7.5 % Si alloy with orientation shown in Fig. 1.
Deformation rate 1 XIO-6 sec-I, Direction of tensile and compression stress
are indicated. Oblique illumination. Magnification Xt50. Ca)
Cb)
FIG. 3. Same as Fig. 2. Deformation rate 4 X 102 sec'.
serve on the compression side a transition to slip along the {llO}
planes [Fig. 3 (a)]. In similar samples of the same orientation but
with 3% Si the slip under the same deformation conditions was
only crystallographic on both sides [cf, Fig. 13(c) in Ref. 1]. On
the tensile side of the samples with 7.5% Si slip always occurred
exclusively along the {llO} planes in a range of deformation rates
from 6XlO-7 sec' to 4X102 seC'.
A similar difference in the slip geometry was also observed on
samples containing 5.5% silicon. On the tensile and compression
sides of these samples at a deformation rate of 5 X 10-6 seC' the
slip is non crystallographic along the maximum resolved shear
stress plane [the same character as in Fig. 2 (a)]. When the defor
mation rate is increased to 4X 10-2 seC' sections appear on the
tensile side of the sample apart from non crystallographic slip
where the slip occurs exactly along the {llO} planes (Fig. 4).
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] IP: 130.102.42.98 On: Sat, 22 Nov 2014 08:07:57COMMUNICATIONS 2921
FIG. 4. Slip bands on tensile side of sample of Fe-5.5 % Si alloy with
orientation shown in Fig. 1. Deformation rate 4 X10~2 sec-I. Direction of
tensile stress b indicated. Oblique illumination. Magnification X150.
while on the compression side it remains quite non crystallographic.
When the deformation rate is raised still further to 4X102 sec1
slip occurs only along the {110} planes on both sides.
It follows from the above results that the range of deformation
rates, in which there is a difference between the slip on the tensile
and compression sides of samples, expands with increasing silicon
content.
The difference in the character of slip on the tensile and compres
sion sides of the same sample support the conception1,3 that slip
along the {110} planes is caused Ly the extension of dislocations
on these planes since it is plausible that the energy of suitable
stacking faults in samples of our orientation may decrease on the
tensile side as a result of deformation and increase on the compres
sion side, Several modes of extension have been proposed.6" It is
not yet possible to decide which of them plays a role in the plastic
deformation of the crystals studied hy us. Since it has not yet been
possible to observe extended dislocations in bcc metals after plas
tic deformation, it can be deduced that the extension is small or
exists only in the stress field. Due to the insignificant extension it is
Letter to consider the anisotropy of the dislocation core char
acterized by extension.
In the case of small energy of the stacking fault on the (110)
planes, the dislocations in this plane dissociate into their partials
and can then move only along these planes. It is not yet clear
whether the dislocations move also atomically along the non
crystallographic planes during noncrystallographic slip or whether
they alternately move along small sections of nonparallel {110}
planes. In the first case one would have to assume a pronounced
influence of the deformation rate and the influence of temperature
011 the energy of the stacking fault. In the second case the temper
ature and deformation rate would influence the alternation of the
sections of the {110(planes,
1 B. Se8tiik and S. Libovicky, Proceedings of Symposium on the Relation be·
tween the Structure and the Mechanical Properties of Metals (National
Physical Laboratory. Teddington, England. 1963).
2 B. Sestak and S. LibovickY. Czech. J. Phys. BI2, 131 (1962).
3 B. Sestak and S, Libovicky, Czech, J, Phys, B13, 266 (1963).
'B. Sestak and S. Libovicky, Acta Met. (to be published).
5 C. S. Barrett, G. Ansel, and R. F. Mehl, Trans. Am. Soc. Metals 25, 702
(1937).
r, J. Friedel, see discussion in Ref. 1.
7.1. B. Cohen, R. Hjnton, K. Lay, and S. Sass, Acta Met. 10,894 (1962). Observation of Continuous-Wave
Optical Harmonics
s. L. MCCALL AND 1,. \'1. DAVIS
Wt~stan Development Laboratories, Philco Corpora/ion,
Palo Alto, California
(Received 5 June 1963)
INVESTIGATORS previously have used the intense light from
a pulsed solid-state laser to generate optical harmonics in vari
ous substances, Here we report use of the light beam from a high
intensity gas-discharge laser to observe the production of continu
ous-wave (cw) second-harmonic light in potassium dihydrogen
phosphate (KDP).
For comparing experimental results with certain aspects of the
theory of nonlinear optical phenomena, the cw light from a gas
laser has some important advantages over the pulsed light from a
solid-state laser. For example, (1) harmonic conversion efficiency
could be measured more accurately with gas laser cw excitation
than with pulsed light; (2) Franken and Wardl suggest that ex
tremely monochromatic light, as is provided by the gas laser,
would permit further study of the possibility that phonon inter
actions shift or broaden the frequency of harmonic radiation.
For the experiment we used a helium-neon gas laser ,,·ith
",-,20-mWoutput (recently constructed by Spectra-Physics, Inc.).
The laser cavity was 3.5 mm in diameter and approximately 3 m
in length, Confocal mirrors, with focal lengths of 3 m, were used
as end reflectors. Oscillation was restricted to TEMoo modes,
Laser light of wavelength 6328 A was passed through a red-pass
filter and focused into a KDP crystal oriented at the index match
angle.2,3 The emergent light was passed through a NiS04 solution
filter to remove the 6328 A component, and then was detected.
Polaroid color film photographs gave a blue image having the
striking intensity pattern reported by Maker et aI.' The second
harmonic light was also detected by a photomultiplier. When the
KDP was rotated, the intensity of harmonic light was highly sen
sitive to the angle between the crystal z axis and the incident beam
direction, with the half-power rotation angle being less than S°.
On attenuating the intensity of the incident light with neutral
density filters, oscilloscope traces were obtained as shown in Fig,
1. The data agree well with the expectation that the second-
5msec -! -"
J ~ V"t'\, "-..;.,;',. 0rv'~
-- --.
.L
50mV
..... ./' T
FIG,!' Oscilloscope traces of lP28 photomultiplier output. Vertical scale
is 0.050 V /div; horizontal, 0,005 sec/div. The 120·cps ripple is due to modu
lation of the laser light and also to pickup in the detector circuit. Top to
bottom: Laser light unattenuated, attenuated by 0.3 (10-0 •• transmission)
and 0,5 neutral density filters, and completely attenuated. Maximum second
harmonic generated is approximately 8 X 10-1' W for 20 mW of excitation
light, corresponding to 5 X 10' red photons required to produce one ultra
yiolet photon.
harmonic production efficiency is proportional to the intensity of
the incident light.
The experiment was made possible through use of the ~3-m
laser developed by W. E. Bell and A. L. Bloom of Spectra-Physics,
Inc,
1 p, A. Franken and J. F, Ward, Rev, Mod. Phys, 35, 23 (1963). 'J, A. Giordmaine, Phys. Rev. Letters 8, 19 (1962).
3 P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys.
Rev, Letters 8, 21 (1962).
• See Ref. 3, Fig, 3,
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1.1728474.pdf | Temperature Dependent Luminescence of CaWO4 and CdWO4
G. B. Beard, W. H. Kelly, and M. L. Mallory
Citation: Journal of Applied Physics 33, 144 (1962); doi: 10.1063/1.1728474
View online: http://dx.doi.org/10.1063/1.1728474
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/33/1?ver=pdfcov
Published by the AIP Publishing
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] IP: 128.114.34.22 On: Mon, 24 Nov 2014 21:35:58144 B. T. BERNSTEIN
RESULTS
The individual contributions to the elastic shear
constants C and C' of beryllium are given in Table I.
DISCUSSION
The overlap-hole contributions to C and C' for the
nearly-free-electron and free electron cases were evalu
ated independently in a manner similar to that used by
Leigh and Reitz and Smith. It was not possible to fit
the measured elastic constants on the basis of this
model unless a reduction factor of 0.59 was applied to
the electrostatic contribution. In conclusion, it is felt that the valence electron con
tribution to the elastic shear constants of beryllium is
very sensitive to the shape of the energy bands and
hence the Fermi contributi:m is strongly dependent
upon the energy band model chosen, the free-electron,
and nearly-free-electron calculations greatly over
emphasizing the Fermi term as compared to the calcu
lations based on the work of Herring and Hill. On the
basis of the model of Herring and Hill the elastic
shear constants of beryllium, like the alkali metals, are
predominantly determined by the electrostatic energy
of the ion-cores.
JOURNAL OF APPLIED PHYSICS VOLUME 33. NUMBER 1 JANUARY. 1962
Temperature Dependent Luminescence of CaW0 4 and CdW0 4*
G. B. BEARD
Department of Physics, Wayne State University, Detroit, Michigan
AND
W. H. KELLY AND M. L. MALLORY
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan
(Received June 12, 1961)
The relative efficiencies and decay times of alpha particle induced scintillations of CaWO, and CdWO,
were investigated as a function of temperature in the range 10° to 375°K. Their behavior at intermediate
and high temperatures is in agreement with that expected from the Mott-Seitz-Kroger configurational
coordinate model. Values of EQ, thermal quenching energy, of 0.34 and 0.31 ev were found for CaWO, and
CdWO" respectively. As the temperature was decreased below 600K, an increase in the decay times and a
decrease in the relative efficiencies were found. This behavior can be explained qualitatively by assuming
the existence of a trapping level.
INTRODUCTION
BOTH CdW04 and Ca W04 crystals have been known
to be good scintilla tors for some time.! Because of
their high densities and high atomic numbers, the
crystals have relatively high photoefficiencies. However,
they have the disadvantage of a long scintillation decay
time which makes them poor for high counting rate
experiments. It is only recently that their scintillation
properties have been studied in some detail in connec
tion with their use for certain specific investigations. 2.3
These investigations included a search for the natural
alpha activities of tungsten in Ca W04 and CdW04 and
the relative energy response of Ca W04 to various
sources of excitation.
We have undertaken a study of the effect of tempera
ture on the scintillation efficiencies and decay times of
Ca W04 and CdW0 4 crystals primarily to ascertain
* This research was supported in part by the United States Air
Force through the Air Force Office of Scientific Research and
Development Command.
1 J. A. Birks, Scintillation Counters (McGraw-Hill Book
Company, Inc., New York, 1953), Chap. 5.
2 G. B. Beard and W. H. Kelly, Nuclear Phy~. 16, 591 (1960).
8 W. R. Dixon and J. H. Aitken, Nuclear lnstr. and Methods 9,
219 (1960). whether the scintillation response could be improved
by operating in a particularly favorable temperature
range. Kroger4 has previously investigated the relative
luminescent efficiencies of Ca W04 and CdW0 4 as a
function of temperature using excitation from ultra
violet light (;>..= 2537 A) in the temperature region from
80° to 480°K. In the experiment reported here alpha
particles from P02!O were used as the source of excitation
in the temperature region from 10° to 375°K. Measure
ments were also made over a limited temperature range
using CS137 gamma rays.
EXPERIMENTAL ARRANGEMENT
Various crystals of CaW04 and CdW0 4 with dimen
sions of about lOX5X3 mm were used. The crystals
were polished such that one side was flat. This side was
placed in direct contact with a light pipe. On the side
opposite the flat side, a drop of P0210N03 solution was
placed and evaporated to dryness. A counting rate of
roughly 6000 counts/min was used with all crystals.
4 F. A. Kroger, Some Aspects of the Luminescence Solids
(Elsevier Publishing Company, Inc., New York 1948) Chaps' 3,6. ' , .
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] IP: 128.114.34.22 On: Mon, 24 Nov 2014 21:35:58LUMINESCENCE OF CaWO. AND CdWO. 145
The crystal and the remaining exposed end of the light
pipe were covered with a thin layer of MgO, aluminum
foil, and black electrical tape. Using this arrangement,
little difficulty with light trapping is experienced.5 A
thermocouple and carbon resistance thermometer were
mounted in contact with the aluminum foil. The light
pipe and crystal were placed in the probe as indicated
in Fig. 1.
The temperature of the crystal was measured with a
Constantan-copper thermocouple in the region from
30° to 37soK. Below 300K it was measured with
an Allen-Bradley S6 ohm, 1/10-w carbon resistance
thermometer.
The measurements were divided into five overlapping
temperature regions, corresponding to the appropriate
temperature baths. In order to obtain temperatures
above those of the cooling baths, a variable current was
passed through a Manganin heating wire wrapped
around the probe. The temperature was held to within
a variation of 3°K during the 10-min counting runs in
the liquid helium region, while for similar runs in the
other temperature regions the deviations were less than
10K. For each point, 30 min were allowed for tempera
ture equilibrium to be reached throughout the crystal
before the data were taken.
To detect the scintillation, the light pipe was mounted
on an RCA 6342 photomultiplier. The anode RC time
constant was -'" 10-3 sec and the preamplifier consisted
of a conventional White cathode follower. As long as
the scintillation decay time is small compared to the
ancde time constant, the decay time can readily be
determined from the rise time of the output pulse and
the amplitude of the output pulse will be proportional
to the scintillation efficiency. The photomultiplier
temperature was monitored and maintained constant
to within 2°e. For various runs, light pipes of Lucite or
quartz were used. The failure to detect any differences
in the relative scintillation efficiencies and decay times
using the two different light pipes makes it reasonable
to assume that any wavelength shifts in the detected
luminescence radiation were relatively small. This
conclusion is in accord with the results of Kroger! who
found little change in the emission spectrum in going as
low as 800K using ultraviolet excitation.
As a further precaution, an RCA 6903 photomulti
plier (quartz window) was also used for runs in the
region of 10° to 900K. No noticeable change was
observed in the scintillations.
A Tektronix 531 oscilloscope was used to observe the
output of the preamplifier. The trace of the scope was
photographed using exposure times sufficiently long to
determine a reliable average of the amplitude and decay
time. As a check on and to supplement these measure
ments, a model A-61 amplifier (modified for long rise
time pulses) and a 256 channel pulse-height analyzer
were also used to determine the average pulse height at
5 R. H. Gillette, Rev. Sci. Instr. 21, 294 (1950). GermQn SllVlr
Tubino ---..
Heotina
Coil--+-
Thermocouple t To Photomult; .. ;.,
Bran
Tunqstate Crystal
Carbon Resislance
lhermometer
FIG. 1. Cross-sectional diagram of probe.
each temperature. The amplifier-analyzer response was
measured using a variable rise-time pulser and a
correction factor depending on the rise time of the pulse
was determined. This factor was used to correct the
pulse-height data. Measurements were made on the
crystal at room temperature before and after each
experimental run and were found to agree.
RESULTS AND THEIR INTERPRETATION
The analyzer data were compared with the photo
graphic data and the results agreed to within 10% for
the relative scintillation efficiencies. Figures 2 and 3 are
the results obtained for CaW0 4 and CdW0 4 with the
P0210 alphas. The efficiency data represent an average
of the results as determined by the two methods. The
reciprocals of decay time and relative efficiency are
plotted against the reciprocal of the temperature to
facilitate the comparison with theory. The efficiency
at 273°K is arbitrarily chosen as 100%. The uncer
tainties in the relative efficiencies and decay times are
estimated to be ± 10%.
Within the limits of the experimental accuracy, data
obtained using CS137 gamma rays as a source of excita
tion in a limited region above and below room tempera
ture agree with the results using P0210 alphas in the same
temperature range. The results are also in agreement
with those of Gillette.5 The absolute scintillation
efficiency is greater for gamma-ray excitation. 2,3 Al
though artificial crystals of Ca W04 were used in this
work, previous experience has shown that one obtains
similar scintillations using a natural Ca W04 (scheelite)
crystal at room temperature.
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IP: 128.114.34.22 On: Mon, 24 Nov 2014 21:35:58146 BEARD, KELLY, AND MALLORY
7<deg K)-1
10rr-r°~JTO~-,-,-,O~.~0~5~~~~0
.!!!! 5 .,
Clnd 2
~c.ec-·)
7" 1.0
0.5
0.2
..!l!
11
Clftd 5.0
..!!!:!c.ec -')
7" 2.0 Ca WO 4 -Low remp
/' //' High remp
r'
1.0 h--"--r-=T1o ....... ...,.... ....... -
0.5 _,~,~~_~T1-"-"--~~~~_,_, ___ 1
0.010 1 0.005 0
-(des K)-1
T
FIG. 2. Reciprocals of efficiency (f/o/f/) and decay time (10-5)/ (r)
sec! plotted as a function of the reciprocal of the absolute tem
perature for CaWO •. Upper diagram: entire temperature range.
Lower diagram: high-temperature range expanded by a factor
of 10.
Relatively little theoretical work has been done on
the problem of luminescence in pure (unactivated)
crystals. The most recent work on temperature quench
ing dealing with crystals of the type used here appears
to have been done by Kroger4 and Botden.6 The model
..!!2. /'
T1 // High remp
and 5.0 ,,/
10-5(sec _') I
'7" 2.0
1.0 ~ ____ ~ _____ ~_...-
0.010 1 0.005 0
,(deg K)-1
FIG. 3. Reciprocals of efficiency (f/o/f/) and qecay time (10-6)/ (T)
sec-! plotted as a function of the reciprocal of the absolute tem
perature for CdWO •. Upper diagram: entire temperature range.
Lower diagram: high-temperature range expanded by a factor
of 10.
6 P. J. Botden, Philips Research Repts. 6, 425 (1951). TABLE 1. Results obtained from the CaWO. and CdWO.
scintillation efficiencies and decay times.
EQ from
Scintillation
efficiency Decay time
data ev data ev TL, sec S, sec-1
CaWO, (O.34±O.03) (O.34±O.03) (6.7±O,7) XlO-6 (1.6±O.8) XIOIO
CdWO, (O.30±O.03) (O.32±O.03) (7.8±O.8) XIO-' (O,8±O.8) XlOlO
advanced to explain their work on temperature quench
ing using ultraviolet excitation also is in agreement with
the results reported here for CaW0 4 and CdW04 in the
temperature range above about 60oK. For this tempera
ture region, Kroger uses the picture of configuration
coordinates as applied to luminescence by Seitz,1 with
the modification as proposed by Gurney and Mott. 8
Figure 4 is a configurational coordinate diagram showing
the ground state and only one excited state of a lumi
nescent center. The ordinate of the curves is the total
energy of the system, including both ionic and electronic
terms. The abscissa is a configuration coordinate which
specifies the configuration of the ions around the center.
The equilibrium position of the ground state in Fig. 4
is at A. If the center is excited, it is raised to the excited
state at B. A new equilibrium is obtained at C, with the
energy difference between Band C given up as phonon
emission. The center then decays from C to D by photon
emission and again the energy difference between D and
A is given up as phonon emission. The decay from C to
D is assumed to be temperature independent. Gurney
and Mott proposed that an alternate return to the
ground state could occur by a nonradiative transition
at E if the excited state of C is given sufficient thermal
energy EQ• Thus, the photon is not emitted and thermal
quenching results. This leads to the following equations9
for the luminescence efficiency and decay time.
17= [l+S/ PL exp( -EQ/kT)J-I (1)
l/T-l/7£=S exp( -EQ/kT), (2)
where 17 is the efficiency for luminescence, S is a con
stant, PL is the probability of luminescence with no
thermal quenching and equal to 1/ T L, EQ is the energy
difference between states C and E of the excited state,
and T is the measured decay time. It is seen from the
data presented in Table I that the value of EQ deter
mined from the decay-time data and scintillation
efficiency data agree very well. The value obtained for
EQ=O.34 ev for CaW04 agrees with that obtained by
Botden6 using ultraviolet light (A=2537 A) as the
source of excitation. It is interesting to note that the
EQ values obtained for Ca W04 and CdW04 are approxi-
7 F. Seitz, Trans. Faraday Soc. 35, 79 (1939).
8 R. W. Gurney and N. F. Mott, Trans. Faraday Soc. 35, 69
(1939).
9 C. C. Klick and J. H. Schulman in Solid State Physics, edited
by F. Seitz and D. Turnbull (Academic Press, Inc., New York,
1957), Vol. 5, p. 97.
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] IP: 128.114.34.22 On: Mon, 24 Nov 2014 21:35:58LUMINESCENCE OF CaW0 4 AND CdW04 147
mately equal. Table I contains also the values TL and S
obtained for the two tungstate crystals.
The decay time of Ca W04 at room temperature,
T = 5 p.sec, is found to agree with the value given by
Dixon and Aitken.3 The decay time of CdW04 at room
temperature is 7.1 p.sec.
The model used here to picture the luminescence
behavior above "-'60oK does not describe the small dip
noted in the luminescence curve for Ca W04 at
1/T=0.004 (deg K)-I. This dip does not have a corre
sponding variation in the decay time data and does not
appear in the CdW0 4 data.
In the discussion above it is assumed that both the
radiative transition probability and the proportion of
the absorbed exciting radiation actually absorbed in
fluorescent centers are constant and independent of
temperature. According to this picture the luminescence
decay times and efficiencies should remain constant
below the temperature-quenching region. This is in
obvious disagreement with the experimental results.
However, by a relatively simple modification one can
obtain qualitative agreement between the model and
experiment for both the high-and low-temperature
regions.
The proposed modification is as follows: Assume there
exists a metastable level F lying an energy tlE below C
and that this level is preferentially excited from B.
(See Fig. 4.) The center may be thermally excited
to C from F with a probability proportional to
exp(-tlE/kT), where it may decay from C to D by
photon emission. Also the center may be de-excited
from F to D by an unobserved transition whose rate
mayor may not be a function of temperature. It should
also be assumed that the probability of the de-excitation
by the unobserved transition is small compared to that
of the thermal excitation to C at temperatures above
6OoK. From this picture one would expect the decay
time of the transition from C to D to increase and the
scintillation efficiency to decrease (approximately as
exponentials) as the temperature is decreased below
tlE/k. This is the trend that is observed. If one assumes
that the de-excitation of the trapping level by the
nonradiative transition is independent of the tempera
ture, then one obtains relations for the luminescence
efficiency and decay times that are somewhat similar
to Eqs. (1) and (2). Further, if one assumes that the Configuration Coordinate, r
FIG. 4. Configuration coordinate diagram.
state C is de-excited only by a luminescent transition
in this temperature region, then the luminescence
efficiency is given by the relation
7)= [1+P T/Sl exp(tlE/kT)J-\ (3)
where PT is the probability per unit time for the excita
tion of the trapping level via the unobserved transition,
and 51 is a constant.
Since the decay leading to luminescence goes by a
cascade of levels, the luminescence decay is not a simple
exponential in time and hence the decay time data are
not easily compared with the formulas. Applying Eq.
(3) to the data, one obtains tlE =0.0023 ev for Ca W04
and tlE=0.0026 ev for CdW0 4 with PT/51 =0.010 and
0.016 for the two crystals, respectively. Assuming the
decay time from the trapping level is temperature
independent, an order of magnitude for the decay time
from the trapping level by the nonradiative transition
can be estimated to ~SO p.sec.
ACKNOWLEDGMENTS
The authors wish to express their sincere appreciation
to Dr. F. J. Blatt for fruitful suggestions, and discussions
on the interpretation of the data. The assistance of
Dr. M. M. Garber and Dr. H. A. Forstat with the low
temperature measurements is gratefully acknowledged.
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1.1984588.pdf | The State of d Electrons in Transition Metals
Conyers Herring
Citation: Journal of Applied Physics 31, S3 (1960); doi: 10.1063/1.1984588
View online: http://dx.doi.org/10.1063/1.1984588
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128.138.73.68 On: Sat, 20 Dec 2014 13:28:45JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL 31. NO.5 MAY. 1960
Magnetism, General and Theory
The State of d Electrons in Transition Metals
CONYERS HERRING
Bell Telephone Laboratories, Murray Hill, New Jersey
This paper gives a brief critique, in elementary language, of the principal types of theoretical pictures
which have been advanced concerning the electronic states of transition metals, especially those of the iron
group. It also calls attention to the possibility that some of the properties of these metals can be correlated
by the use of concepts which have an exact, not just approximate, meaning for a many-electron system.
The Fermi surface is probably a concept of thiS type. Major conclusions are that in the iron group metals
the 3d electrons ought not to differ radically from those in the free atoms either in number or in spatial
distribution, and that in most, though perhaps not all, of these metals the 3d electrons, magnetic or non
magnetic, have an itinerant behavior.
INTRODUCTION
IN the last quarter century, a wide variety of pictures
of the electronic structures of transition metals have
been proposed and elaborated. Some of these are still
lIourishing; some have died. The diversity of viewpoints
is much wider for these materials than for, say, mono
valent metals or nonmetals. The reason for this diversity
is that, of the two main approaches to the electron
theory of solids-the free-electron idea and the idea of
electrons bound to particular atoms-neither seems ade
quate to describe a metal made from atoms with in
tomplete d shells. Each school of thought has therefore
been driven to use its own combination of crude ap
proximations, and usually the concepts of one school
have no clear-cut meaning in the framework of another
schooL
In this paper, I shall begin by enumerating briefly
the principal types of theories and indicating which
approaches appear to me to be wrong or unproductive.
The latter part of this first section will be devoted to a
tescription of some conclusions which have emerged
from one of these approximate approaches (the self
&Insistent field method), and to adding some words of
taUtion about its limitations. Finally, I shall try to show
that at least a partial picture of the state of electrons in
~ition metals can be constructed using concepts
which are not approximate, but which have an exact
:ning for a many-electron metal. The main object of
S paper, in fact, will be a plea for more experimental
and theoretical work aimed in this direction.
For simplicity, the discussion will be limited through
DIlt to pure elements of the iron group.
1. SURVEY OF PAST AND PRESENT THEORIES
Blending of Basic Ideas
th: In Surveying the various theories one is struck with -:fact that there are just a few basic ideas which enter
~ all the theories, and that the divergences of viewtt are due merely to their employing these ideas in
~ erent relative proportions. It is like mixing a few
Ors in various proportions to get a variety of
35 different cocktails. Table I lists these main ideas, liquors,
at the top, and the main categories of theories, cocktails,
in the first column.
(i) The first of the ideas is that of electronic energy
bands made up of states of an electron in which it moves
freely through the crystal. Such a state of motion re
sembles that of an electron in free space in being de
scribed by a traveling wave; however, this wave in a
crystal, called a Bloch wave, differs from that of a free
particle in being not of pure sinusoidal form, but rather
a sine wave periodically modulated by the interaction
of the electron with the fields of the atoms.
TABLE I. Approximate compositions of the major types of
theories of transition-metal electrons. The symbols X, x, denote
respectively a major and a subordinate use of the idea of a given
column; the subscripts a, b, c, refer to different aspects of ideas
(ii) or (iii), as enumerated in the text
(iii) (iv)
Liquors ...... (1) (ii) Coupled Valence
Cocktails 1 Bands Correlation atoms bonds
Itinerant X Xa 0 0
Minimum polarity x Xb Xa•c 0
s-d models X Xa Xa,b,c 0
Valence X Xb Xa,b,c X
(ii) The second idea, related to this, has to do with
the modification of the motion of itinerant electrons by
their mutual electrostatic repulsion. Two approaches to
this modified motion may be distinguished: (a) The
application of a "correlation" correction to the theory
of noninteracting electrons in Bloch states.l (b) The
conception of the state of a metal as a resonant super
position of states corresponding to various distributions
of neutral atoms and positive and negative ions.2•3
1 E. \vigner, Phys Rev. 46, 1002 (1934); Trans Faraday Soc.
34,678 (1938); D. Pines, in Solid State Physics edited by F. Seitz
and D. Turnbull (Academic Press, Inc, New York, 1955), Vol. 1,
p 367; J G. Fletcher and D C. Larson, Phys. Rev 111, 455
(1958).
2 J. C. Slater, Phys. Rev. 35, 509 (1930).
3 S. Schubin and S. Wonsowsky, Proc Roy. Soc. (London)
A145, 159 (1934), Physik. Z. Sowjetunion 7, 292 (1935), 10, 348
(1936).
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128.138.73.68 On: Sat, 20 Dec 2014 13:28:454S CONYERS HERRING
(iii) The third idea is the conception of electrons
bound in a specific configuration on a single atom. This
leads to three further conceptions which are sometimes
important in solid-state theories: (a) Intra-atomic cou
pling of electrons (Hund's rule, etc.). (b) Crystal-field
splitting (i.e., the development of a difference in energy
between states which in the free atom would have the
same energy, differing only in orientation). (c) The
ferro-or antiferromagnetic coupling of the total spin of
one atom to the total spin of a neighbor.
(iv) The last idea is that of directed valence bonds,
in which an electron on one atom joins with an electron
on a neighboring atom, these two becoming paired into
a singlet state and losing any coupling they may have
had to other electrons of their own atoms.
The theoretical schools have been grouped into four
main types, although within each type there may still
be quite a diversity (no two men mix a given cocktail
in quite the same way):
The first category, itinerant theories, is based mainly
on the idea of electrons moving freely through the
crystal (hence the capital X in the first column) with
rough corrections for the mutual repulsions of the elec
trons sometimes grafted on as an afterthought (hence
the small x in the second column). Theories of this type,
giving interpretations of magnetic and other properties
of transition metals, have been variously developed by
Slater,4.5 by Stoner6 and Wohlfarth,1 by Friedel,8 and
by many others.9 The various a priori calculations of the
electronic energy band structures which underlie such
theories have been summarized in a review by Calla
way1°; more recent work by Sternll and Wood12 may
also be cited.
The next group of theories, for which I use Van Vleck's
name "minimum-polarity," has as its central feature the
idea of fluctuating ionic states, the transitions between
these being crudely allowed for using the band-theory
concepts of the first column. While the basic idea goes
back to some old papers of Slater2 and of Schubin and
Vonsovski,3 in the thirties, the most detailed application
to iron-group metals is that of Hurwitz and Van Vleck,13
reported at the 1952 magnetism conference.
The third group, s-d models, assumes that the 4s
electrons (the ones that have large orbits in the isolated
4 J. C. Slater, Phys. Rev 49, 537, 931 (1936).
5 J. C. Slater, Revs. Modern Phys. 25, 199 (1959).
6 E. C. Stoner, Phys Soc. Repts. Progress Phys 11,43 (1948); J. phys. radium 12, 372 (1951)
T E. P. Wohlfarth, Revs. Modern Phys 25, 211 (1953).
8 J. Friedel, J. phys radium 16, 829 (1955); 19,573 (1958).
9 A B. Lidiard, Proc Phys. Soc. (London) A65, 885 (1952);
G S. Krinchik, Izvest. Akad. Nauk S.S.S.R. Ser. FIZ. 21, 869
(1957); Bader, Ganzhorn, and Dehlinger, Z. PhYSIk 137, 190
(1954).
10 J. Callaway, in Solid State Physics, edited by F. Seitz and
D. Turnbull (Academic Press, Inc., New York, 1958) Vol 7, p. 99.
11 F. Stern, Thesis, Princeton University (1955), Phys. Rev.
(in press).
12 J. H. Wood, Qttarterly Progress Report, Solid-State and Jl1 oleC1t
lar Theory Group (Massachusetts Institute of Technology, October
15, 1959), p. 4
13 J. H. Van Vleck, Revs. Modern Phys 25, 220 (1953) atom) become free electrons in the metal, but thattbe
3d electrons .( those in. incompletely fil~ed shells of m
compact orbIts) remam bound to the mdividual at ore
To this group belong, among others, the theOri;~
Vonsovski,14 Zener,15 Mott and Stevens,r6 and Lo
and MarshallP mer
The fourth group, valence-bond theories, is associatfl
mainly with the name of Pauling.Is It borrows id
from all columns, but relies mainly on the concePt e:
valence bonds formed from hybrid mixtures of s, P and
d states.
Divergent Opinions
As illustrations of the divergencies between the view
points of the different schools, one may list some of tbll
answers which different theorists have given to fOUr
important questions pertaining to metals of the ire
group:
(1) How many electrons per atom are in states with,.
angular momentum and charge distnbution characteristit,
oj 3d electrons? Most of the theories have assumed this
number to be the same as in the free atom or (mote
often) about one more, the number being nonintegra.lin
the itinerant theories, integral in some of the s-d theories
Paulingl8 has suggested, however, that 6 electrons per
atom are in conduction states formed from hybricme-d
orbitals differing widely from atomic d states; thisleavf!S
correspondingly fewer electrons with the compact 3t.t
distribution.
(2) Are these electrons in general, and ferromagnetill
3d electrons in particular, bound or itinerant? In t1Ie
itinerant theories they are all itinerant. In some of the
s-d theories they are all bound. According to the mini
mum-polarity model,13 there should be some itinerant
behavior of magnetic electrons for cases like Ni, whem
the average number of 3d electrons per atom is non
integral, but very little when it is integral. According to
Griffith19 and to :NIott and Stevens,16 there are two
classes of 3d electrons, one of which (symmetry til' is
itinerant, the other (symmetry eo) bound if of integral
occupation. The latter carry most of the magnetic
moment in Fe, but not in Ni. Pauling'sl8 view seellllli
similar to this.
(3) What is the origin of the exchange. forces whicj
align the spms in the ferromagnetic metals? According til
most of the itinerant models,4 it is the intra-atomic ex
change responsible for Hund's rule in the free atom!
electrons flitting from atom to atom find that their
14 S Vonsovsky, J. Phys. (U S SR.) 10, 468 (1946); S. V,
Vonovski and E. A. Turov, J. Exptl. Theoret. Phys. (U.S.S.R,}
24, 419 (1953). For a brief review see S V Vonsovski, Izvest.
Akad. Nauk. S S.S.R. Ser. Fiz. 21, 854 (1957) [translation: B~
Acad. Sci. U.S.S.R. (Columbla Technical Translations, Wlute
Plains, New York), 21, 854 (1957)J
15 C Zener, Phys. Rev 81,440 (1951),83,299 (1951); seealS!)
C. Zener and R R Heikes, Revs Modern Phys. 25, 191 (195~)}.
1fi N. F Mott and K. W. H Stevens, Phil. Mag. 2, 1364 (1951.
17 W. M. Lomer and W. Marshall, Phil. Mag. 3, 185 (195g~'_A
18 L. Pauling, Phys. Rev. 54, 899 (1938), Proc. Nat!. j\~
SCl 39, 551 (1953).
19 J. S. Griffith, J. Inorg. & Nuclear Chern. 3, 15 (1956).
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128.138.73.68 On: Sat, 20 Dec 2014 13:28:45d ELECTRONS IN TRANSITION" METALS 5S
interaction energy with other electrons momentarily on
the same atom is lower if their spins are parallel. In
'VllIl Vleck's discussion of the minimum-polarity mQdeJ,13
be suggests that although it is only rarely that an elec
tron belonging on one magnetic atom gets onto another
-magnetic atom, these occasional events may still suffice
to give the necessary exchange coupling. In the s-d
theories, the coupling of the free conduction electrons
to the bound d electrons affects both the energy and the
llloment. in some theories,14,16 these effects are taken to
be small, and the principal coupling is taken to be the
inter-atomic effect (iii) (c) above arising from the slight
overlap of d orbitals on neighboring atoms. Zener's
theory,!" on the other hand, pictures the inter-atomic
roupling as always antiferromagnetic, and the coupling
of the conduction electrons to the d electrons as being
so strongly in favor of parallel spin alignment as to
overbalance the former in the ferromagnetic metals. An
interesting speculation, diametrically opposed to this,
has been made by Vonsovski and Vlasov,2o by Mott
and Stevens,16 and by Anderson.21 This is that it is
energetically more favorable for a conduction electron
10 align itself antiparallel to the 3d electrons, because
in the antiparallel orientation it can lower its energy by
hybridizing with unfilled 3d orbitals; i.e., it doesn't have
to work so hard to keep away from the 3d electrons to
the extent required by the exclusion principle. If this
effect is large it can increase the effective ferromagnetic
coupling of the 3d cores with each other, just as in
Zener's theory, and at the same time will lower the
saturation moment.
{4) What is the origin of the large binding energies of
the iron-group metals? Although it has been arguedi5
~at the d shells in these metals have a purely repulsive
lIlteraction, the prevailing view is that in some way they
~tribute a large part of the binding energy. The
ltinerant and valence theories (though using rather
4ifferent concepts) both picture this as arising from the
ilybridization of s-p conduction states with d states, to :m bands of lower energy, or more effective valence
nds. A further suggestion has been made by Friedel
lnll by ~1ott,22 namely, that because of the small energy
:aratlOn of the 3d and conduction states in the metal,
~van der Waals interaction of the 3d shells can become
qUite large.
In all these matters, the views of the different schools
are hard to compare and evalua te because each school
::ks with an admittedly approximate picture ("good"
1\. Its adherents, "crude" to its rivals) and because the
t"cture sh .. . I· . ht I ' arp III Its simp est verSIOn, becomes Illcreas-~ when one tries to graft improvements onto it.
157~7~·1~~)sOVSki and K. B Vlasov, J. Exptl. Theoret. Phys. :j-:: Anderson (unpublished). \tott. PhlldeM1, Proc Phys. Soc. (London) B45, 769 (1952) N" F
, " ago 44, 187 (1953). ' Critique of the Not-Fully-ltinerant Theories
Now let me indicate some preferences. The valence
theories, as they have been formulated so far, are not
of the type that theoretical physicists like to use in that
there is almost no mathematical superstructure 'and the
adjustable parameters are almost simple transli~erations
of the experimental data. Thus, in spite of a qualitative
a?peal of some of their ideas, these theories have pro
Vided no framework for correlating the more complex
phenomena. Van Vleck's13 version of the minimum
polarity theory goes further in the direction of calcu
lating things from first principles, but still does not
provi~e ~ set of basic concepts adequate for a fully
quantitative theory. As for the s-d theories most of
them in the past have made assumptions ~hich can
now be shown to be incorrect. For example much of
this work has ignored the fact that at least s~me of the
3d electrons can move from atom to atom, a fact I shall
try to demonstrate later on. Other work has assumed an
antiferromag~etic coupling of the spins of neighboring
a.tom~, eve~ In som: ferromagnetic metals, an assump
tIOn Illconslstent With recent neutron diffraction data.
However, these theories have made a notable contribu
tion to the t~eory of ferromagnetism, in that they have
ca~led attentIOn to the fact that it is possible for the
SpIns. of .bound. electrons to be aligned via the spin
polanzatlOn which a magnetic ion can induce in the sea
of conduction electrons in which it is bathed. This
"indirect exchange" coupling is now believed to be the
dominant coupling mechanism in the rare earth metals
though of less importance in the iron group. '
The two most recent theories of the s-d group, those
of Mott and Stevens16 and of Lomer and Marshall,n
have ad?pte~ the idea, suggested long ago by Pauling18
an.d r:vlved III recent controversies23 over x-ray deter
mmatlOns of electron density, that in some transition
metals the number of electrons which have the compact
orbits characteristic of 3d electrons in a free atom is
much less than the number in the free atom. But there
are other types of experimental evidence against this
view, and it will be argued presently that it is mosi
unreasonable on purely theoretical grounds. So this
feature of these theories must be rejected. The most
carefully thought-out of these theories, that of Mott
and Stevens,16 has another feature which seems dubious
to me, but which is hard to reject definitively. This is
the assumption that the anisotropy of the electrostatic
field around an atom in a metal splits the 3d states into
~ .high-energy and a low-energy group, the former being
ItInerant, the latter bound. My own opinion is that it
is very doubtful that the crystal-field anisotropy is large
enough to make a clear separation of this sort.
23 R. J. Weiss and J. J. DeMarco, Revs. Modern Phys. 30,59
(1958); Phys. Rev. Letters 2, 148 (1959); B. W. Batterman
Phys. Rev. Letters 2, 47 (1959); KOffiura, Tomiie, and Nathans,
Phys. Rev. Letters 3, 268 (1959)
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128.138.73.68 On: Sat, 20 Dec 2014 13:28:456S CONYERS HERRING
Yardstick for Itinerant Theories
Only the itinerant theories are left to discuss. These
are the theories which have gone farthest in the direction
of setting up a quantitative description of the states of
the electrons. However, none of these theories takes
adequate account of the correlations in the positions of
the electrons due to their electrostatic repulsion. This
being the case, it is natural to gauge the adequacy of
all these theories by comparing them with the best
theory one can construct neglecting these correlations.
The "best" such theory (by the criterion of having the
lowest energy, and probably by most other criteria as
well) is that obtained by solving the self-consistent field
equations for electrons in a metal. This means that one
determines the motion of each electron in the field
produced by the nuclei and the average distribution of
all the other electrons. Several efforts at calculations of
this sort for iron group metals have been made, the most
ambitious being the current one of Wood.12 A study of
such calculations makes clear several points:
The first point has to do with the concept of a d band,
states in which an electron hops from atom to atom,
being in a 3d level on each atom as it does so. Some of the
simpler itinerant theories picture such a band of states
as existing side by side with a normal s-p or conduction
band, overlapping it in its energy range, but otherwise
independent. They also picture holes in this d band as
acting just like ordinary free particles, except for having
a heavier effective mass. There are two things wrong
with this picture. One, which has a crucial bearing on the
occurrence of ferromagnetism, 24 is that there are different
"NORMAL/I
d ELECTRON -
WOULD GIVE CHANGE UNIFORM CHARGE
OVER CELL
IN CHARGE DENSITY AND CHANGE
+ +
++ + +
• + + + + + + + + + + IN ELECTRONIC
POTENTIAL ENERGY r--------'-r
FIG. 1. C~ange of charge density and potential energy produced
by transferrIng an electron from a compact d state to a state with
a uniform distribution over an atomic cell.
24 Slater, Statz, and Koster, Phys. Rev. 91, 1323 (1953). IJ}
(!) a: w
cD o
>-a: -5
~
~ a: w z
W
.J « -10
;:::
Z
Ul fa 0. vt (MAJORITY SPIN)
FOR Fe ATOM 3d6 4S2-
(WOOD AND PRATT)
~~ -, "-....... -... _-----
10 2.0 FIG. 2. Top,:'
curve: sum of,
Coulomb, exchi
and . centrifugal
~en.tJals for a~,
Jonty-spin del
in atomic iron'
the self-c '
field calcula
Wood and
Top, other
changes in
tential due tovari
hypothetical"
tributions. Batt
charge per unit'
dius for maj'
and minority-s
electrons.
kinds of 3d states, differing in the orientation of
electron's orbit around the nucleus. Thus, for a give;
wavelength and direction of motion of an electron w~
formed from such states, there will be not just ,fnil
quantum state, as for a free particle, but several.'rF
certain values of the wave number, these states' '
have the same energy; in this case, one speaks
"degenerate" band. The other defect of the simp
theory is that it turns out not to be possible to se .
a band formed from the 3d atomic states from a
formed from states with larger orbits-the 4s and"
the actual states of electrons in a metal are usuall
little of both.
Charge Distribution
Another conclusion which has come out of self'
sistent field studies, is that in a metal such as iron nei
the number of 3d electrons nor their spatial clistribu.
differs very radically from that in the free atom'
recent x-ray studies23 have made this a much t'
about question, it may be worthwhile to elabor
little on the reason, which is basically one of pure'
trostatics. Figure 1 shows, in the upper left, the ra
compact charge distribution of a d sh~ll electron in .
spherically averaged. Suppose we were to remove
electron from this compact state and distribute
charge uniformly over the much larger atomic celi,
Sl'lOWri at the upper right. This would alter the ch·
distribution in the atom by the amount shown at:.
lower left. The corresponding alteration in the pote ']
energy of an electron would therefore be, as a func
of radius, as shown in the curve at the lower rigM;,:
other words, each d electron that we place in a spr
out state changes the potential in such a way as to
the compact state more stable.
Now, there are two influences which might'·
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eivably act to spread out the 3d electrons in the metal.
due which by itself would surely not suffice to produce
a Ill~jor redistribution, is t?e altered b~undary condi
tions which the wave functIOn must satIsfy. The other
is the electrostatic effect of compressing t,he valence
electrons of the atom into the smaller atomic volume
cf the metal. The upper part of Fig. 2 shows, in the full
curve, the sum of the electrostatic potential and the
effective potential of the centrifugal force arising from
the angular momentum of a d electron, for a majority
spin d electron in atomic iron. The dot -dash curve shows
the amount by which this potential would be changed if
the renormalization of the valence distribu tion amounted
to simply adding two electrons with a uniform distri
bution out to the radius r" of the cell. The dashed curve
is the one drawn schematically in Fig. 1, namely, the
!Change in potential resulting from shift of a single d
electron to a uniform distribution. Notice that such a
shift would more than compensate the effect of valence
electron renormalization. The shifting of a number of d
electrons to. a uniform distribution clearly would be
energetically unfavorable. To give an idea of the sensi
tivity of the 3d charge distribution to such changes in
the potential, the dotted curve shows the difference of
the effective potentials for majority and minority spin
eJectrons, the corresponding charge distributions being
drawn in at the bottom.
The conclusion from these simple arguments, which
is given quantitative form by the recent band calcula
~ions of Wood,12 is that the 3d electrons should be a little
more spread out in the metal than in the free atom, but
that neither their number nor their distribution will be
radically changed. Correlation effects, which tend to
make the d shell more compact, only reinforce this
.conclusion.
To justify this last statement I must take up another
major topic, namely, the shortcomings of the self-con
sistent field approach and the direction in which one
should correct it to take account of the fact that the
:electrons do not move as independently as this approach
assumes, but correlate their motions to stay out of each
ot~er's way. Being able to do this, the several electrons
whIch form the 3d shell of an atom will be less reluctant
to occupy this central region of the atom than they
W?uld be if they moved independently. Therefore, they
\VJ.II hUddle a little closer to the nucleus than self-con-
TABLE II. Principal conclusions of Sec. 1. ~ ================================
.~rge d~stribution of d-like electrons:
. S ~ly sl~ghtly different from that of free atom . . .
Iig~tly expanded, but not as much so as sell-conSIstent jjeld
II .. ca culations predict.
glllerant picture is adopted, d band is:
, l\' egenerate
C ot f,!Uy separable from s-p band .
. 're!atlOn correction to itinerant picture:
, enous, but fluctuations in the number of d electrons on an
:., altolll are encouraged by compensating fluctuations of s-p : e ectrons.
~~================================= FREE ELECTRON GAS
OCCUPATION OCCUPATION
PROBABILITY PROBABiliTY mill
Px-
WITHOUT
INTERACTIONS Px
WITH
INTERACTIONS
FIG. 3. Occupation of momentum states for a Fermi gas,
"'ith and without interactions between the particles.
sistent field theory predicts, and as the size of the 3d
distribution is known to be fairly sensitive to the form
of the potential energy function, we may expect this
contraction to be appreciable.
Itinerancy vs Correlation
A more fundamental aspect of the correlation correc
tion has to do with an objection sometimes raised by
opponents of itinerant theories, who argue that since it
costs quite a bit of energy to remove an electron from
one isolated atom and put it on another isolated atom,
the self-consistent field solution cannot be very near the
truth, because it allows the wrong number of 3d elec
trons to be on an atom for much of the time. This
objection has some validity, but I think not as much as
its extreme proponents claim, because the electrostatic
effects of having too many or too few 3d electrons on a
given atom in a metal can be largely compensated by
redistribution of the charge of the conduction electrons,
which latter are surely quite mobile.
Table II summarizes the conclusions of this section.
2. FERMI SURFACE OF AN ASSEMBLY OF
INTERACTING ELECTRONS
The remainder of this paper will be devoted to the
thesis that many properties of pure transition metals
can be understood and correlated using concepts which
have an exact meaning for a system of interacting elec
trons, and to a plea for more theoretical and experi
mental work oriented in this direction. Although there
are several such concepts, I shall discuss only the most
important one, that of the Fermi surface.
Definition and Properties of the Fermi Surface
Consider first the lowest energy state of a gas of
completely free electrons having no interactions with
each other. Since the exclusion principle says that no
more than one electron of each spin can occupy each
momentum state, the distribution in momentum will
have the familiar form shown at the Jeft of Fig. 3, with
all states below a certain limiting momentum occupied,
all slates above it empty. The occupation probability
thus drops from one to zero as we cross the boundary of
a sphere in momentum space, and this boundary is
called the Fermi surface.
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If the electrons repel each other, as real electrons do,
it will no longer be possible for each electron to continue
permanently in a state of a particular momentum; it
will suffer accelerations, and may sometimes go faster
than it ever would in the noninteracting case. Therefore,
the occupation probability of the momentum states will
no longer be one below the Fermi surface, but rather less
than one, and it will no longer be zero in the region
outside. One might suppose that the Fermi surface
would be washed out completely, but according to
current many-body theories25 this is not the case. In
stead, as shown on the right, there apparently remains,
in the exact true state of affairs, a discontinuity in the
momentum distribution at the Fermi surface.
A closely related property is that even in the presence
of interactions there still seem to exist excited states of
the many-electron system which can, in a certain ab
struse but nevertheless exact sense, be described as
having holes in certain states just below the Fermi sur
face and excited electrons in states just above it.25,26
This analogy of states of the coupled many-electron
system to states of a noninteracting assembly only holds
for states very near the Fermi surface, not for all states.
The reason for this is discussed in the appendix: if we
try to construct a state of the interacting-electron sys
tem having an electron or hole far from the Fermi sur
face, electron-electron collisions will shift the electron
or hole to other momentum states so rapidly that the
initial state can by no means be described as stationary;
however, if the electron or hole is placed very close to
the Fermi surface, these collisions become very rare,
and the initial state departs very little from stationarity.
These and similar arguments, put forward in em
bryonic form many years ago by Skinner27 and more
recently stressed by Mott,28 are believed by many to
apply as well to electrons in the crystal field of a metal
as to the free-electron gas. This view is beginning to
receive sophisticated theoretical attention from several
angles.29 If it is correct, then in the space of momentum
or wave-number vectors for an electron there exists a
surface across which the occupation probabilities of the
possible one-electron states are discontinuous, and in the
25 V. M. Galitski and A. B, Migdal, J. Exptl. Theoret. Phys.
U.S.S,R. 34, 159 (1958) (translation: Soviet Phys.-JETP 7, 96
(1958». Most of the current theories of the many-Fermion prob
lem, e.g., those of footnote 26, imply the existence of this sort of
discontinuity of occupation numbers, though they do not point
it out explicitly. However, none of the treatments yet has full
mathematical rigor.
26 L. D. Landau, J. Exptl. Theoret, Phys. 30, 1058 (1956);
M. Gell-Mann, Phys. Rev. 106, 368 (1957); N. M. Hugenholtz,
Physica 23, 481, 533 (1957); N. M. Hugenholtz and L. Van Hove,
Physica 24. 363 (1958),
27 H. W. B. Skinner, Trans. Roy. Soc. (London) 239, 95 (1940).
28 N. F. Mott, Nature 178, 1205 (1956).
29 P. Nozieres and D. Pines, Phys. Rev. 109, 1062 (1958);
V. P. Silin, J. Exptl. Theoret. Phys. (U.S.S.R.) 33, 495 (1957);
W. Kohn and J. M. Luttinger, Phys. Rev. (in press). It is note
worthy, also, that for dilute electrons or holes in a semiconductor
the many-body analysis can be carried through with much greater
rigor than for the metal or the Fermi gas: W. Kohn, Phys. Rev.
105,509 (1957); A. Klein, Phys. Rev. 115, 1136 (1959). neighborhood of which there can exist quasi-particl J
electron-like or hole-like, which can carry a current ;
accelerated, etc. The surface so defined may be cJle4
the Fermi surface; such a surface exists in metals ati(J
does not exist in insulators.z
In addition to these speculations, whi~h if true a~
exactly true, one c~n make other spec~latlOns about tliij
extent of the admIttedly only approxImate correspont
ence between the true states of interacting metallk!
~lectro~s and t?e states of the itin,erant theories wbic\\,
19nore mteractIOns. For example, JD the latter theoriii§
the volume of momentum space enclosed by the Fertlii
surface is exactly proportional to the number of elee;
trons per unit volume. On the basis of what we kno"
about the free-electron gas, it is plausible to speculate
that in the many-electron theory of a metal this relatioll
is still either exact or very nearly correct. [Note added i~
prooJ.-J. M. Luttinger (personal communication) has
recently shown that the relation in question is exact fot
any model of interacting electrons for which perturba,;
tion series converge.] Another plausible speculation i~
that there exists a set of one-electron states of the
traveling-wave type for which the occupation probS
ability is comparable with unity inside the Fermi surl
face, small outside it. In other words, it is reasonable to
hope that the discontinuity in occupation on crossing
the Fermi surface is a major fraction of unity in real
metals. This hope is supported by the smallness of the
departure from unity for the high-density free-electron
gas, by the modest magnitude of correlation energies
for an electron gas of typical metallic density, by the
comparative ideality of the momentum distribution
found for alkali metals in positron-annihilation experi~
ments, etc.; however, there is little evidence bearing oli
its reasonableness for transition metals.
Fenni Surfaces of Transition Metals
If the shape and location of the Fermi surface can be
determined (and I shall argue presently that there isa
reasonable chance this can be done), then a precise
meaning can be attached to certain statements which
the itinerant and sod theorists often argue about:
Figure 4 shows some typical examples of possible situa",
tions in a ferromagnetic metal. The top row, showing
Fermi surfaces for up-spin and down-spin electro~
which are both of rather simple form and not very
different from each other, corresponds to the picture of
all the 3d electrons being bound (or in filled bands),
only a single band of weakly polarized conduction elec:
trons being itinerant. The second row, with both Fermt
surfaces of complex form but only slightly different;,
corresponds to the conception of itinerant 3d electront
which carry very little of the magnetization, so that the
magnetization must be attributed mainly to bound
electrons. The third row, with a complex Fermi surfac~
for the minority spin direction and a simple one for the
majority spin direction, corresponds to the picture of
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128.138.73.68 On: Sat, 20 Dec 2014 13:28:45d ELECTRONS IN TRANSITION METALS 95
POSSIBLE FERMI SUFACES
'1'ORM:t ,--------------, i 0 [INTERPRETATION:
1 : d ELECTRONS
: l BOUND ! ! , , L _______________ J
;--G:::::CT:
'g@g'SOMEdELECTRONS I \ ITJNERANT, : 000 'MAGNETIC ELECTRONS
, 'MOSTLY BOUND
L~_J
r--\S.:U--] g' ~oo G: MAGNETIC ELECTRONS
1 " : tT)NERANT, , 0 , dt SAND FULL
L __ ~ __ j
. Fw. 4. Some possible forms of the Fermi surface
in a ferromagnetic metal.
3d and conduction bands, with the up spin
full i this is a picture with the magnetic elec
itinerant.
now to the question: how can one determine
rar.l.CLen:SUI:s of the Fermi surface experimentally?
most sources of information are indirect, it may
to start by discussing an experiment
in principle could give the desired information
, although in practice its accuracy does not
for this. When a positron is shot into a metal,
comes to rest and subsequently combines with
. with emission of two gamma-ray quanta,
go off in nearly opposite directions in order to
momentum. The small but usually finite mo
of the combining electron is transmitted to the
rays, which then go off at an angle slightly
from 180°. Measurement of the angular distri-
{)I gamma rays thus amounts to a measurement
'EIIUOno.entum distribution of the electrons of the
Since this momentum distribution is discon
across the Fermi surface, the location and shape
surface could in principle be determined.
it turns out that the directly measured yield
the practical type of slit geometry only have
of slope at the Fermi surface, and small
in the data can easily wash them out, as
averaging over crystallographic directions in a
sample.
are other techniques of mapping out Fermi
which have been used successfully in other
though they will be difficult to apply success
iron group. These are methods based on the
complete discussion of the theory of positron
mCludmg the effects of Coulomb interactions and
mo:meIltUln of the positron, see R. B. Ferrell, Revs.
308 (1956). The latter effects, though they have
consequences, do not round off the discontinuity
momentum.
and F. L. Hereford, Revs. Modern Phys. 28, 299 use of high magnetic fields and low temperatures, with
material of sufficient purity to make the mean free path
of the metallic electrons rather larger than the size of
their cyclotron orbits in the magnetic field. Under
these conditions, cyclotron resonance,32 anomalous skin
effect,33 Hall effect,34,35 magnetoresistance,35 etc., can be
made to yield information about the Fermi surface.
Although such detailed information is much to be
desired, we do not have it now for transition metals. We
do have, however, some information about certain
properties which can be expressed as averages over the
Fermi surface, and it should be possible to obtain still
more information of this kind in the near future. A very
well-known and very useful property of this sort is the
contribution which the electrons make to the specific
heat at low temperatures. This measures the number of
quantum states per unit energy for the quasi-particles
electrons and holes near the Fermi surface. This in turn
depends on the area of the Fermi surface and on the rate
of variation of energy normal to it. The high values
observed36 for the electronic specific heats of all the iron
group metals, except chromium, almost certainly mean
that Fermi surfaces of the simple form shown in the top
row of Fig. 4 do not occur for these metals. This con
clusion is confirmed by less direct information from
other sources. Of these, I shall mention only one, the
electrical resistivity, whose high value in the iron group
metals is, according to Mott,37 to be interpreted as due
to the availability of a large number of 3d-like states at
the Fermi surface into which the conduction electrons
or quasi-electrons-can be scattered. Here again, chro
mium seems exceptional in showing little or none of this
extra scattering. In the ferromagnetic and antiferro
magnetic metals, there is, of course, an additional
mechanism of scattering, namely scattering by the ex
change fields of the thermally disordered spins.as This
is responsible for the rapid rise of resistivity as the Curie
point is approached from below. But at room tempera
ture the spin ordering in iron, cobalt, and nickel is so
complete that this is a very minor effect. Collisions of
high-mobility conduction electrons with itinerant 3d-like
electrons can also introduce extra resistivity,39 but these
too are probably unimportant near room temperature.
If we grant from all this that at least some of the 3d
32 See for example the detailed application to bismuth by Galt,
Yager, Merritt, Cetlin, and Brailsford, Phys. Rev. 114, 1396
(1959).
3S See, for example, the application to copper by A. B. Pippard,
Trans. Roy. Soc. (London) 250, 325 (1957).
a4 J. A. Swanson, Phys. Rev. 99, 1799 (1955).
35 See, for example, E. S. Borovik, Izvest. Akad. Nauk S.S.S.R.
Ser Fiz. 19, 429 (1955) [translation: Bull. Acad. Sci. U.S.S.R.
(Columbia Technical Translations, White Plains, New York) 19,
383 (1955)].
au J. G.-Daunt, in Progl·ess if: Low-Temperature Physics, edited
by C. J. Gorter (Interscience Publishers, Inc., New York, 1955),
Vol. 1, p. 202.
31 N. F. Mott, Proc. Phys. Soc. (London) 47, 571 (1935), Proc.
Roy. Soc. (London) A153, 699 (1936); 156,368 (1936).
38 T. Kasuya, Progr. Theoret. Phys. (Kyoto) 16, 58 (1956);
P. G. de Gennes and J. Friedel, J. Phys. Chern. Solids 4, 71 (1958).
3l! W. G. Baber, Proc. Roy. Soc. (London) A158, 383 (1937).
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electrons must be pictured as itinerant, we may still ask
whether the bulk of the magnetic electrons of the ferro
magnetic metals are itinerant. In other words, do the
Fermi surfaces of up and down spin electrons differ by
enough to account for the magnetic moment? There are
again many possible sources of information, few of them
conclusive at present. I shall only mention some argu
ments which have been unduly neglected. One is that
if the 3d electrons are bound to specific atoms and yet
have an average magnetic moment per atom different
from that of a free atom with a partially filled 3d shell,
then at low temperatures a superlattice of spins should
form. If this does not show up in neutron diffraction,
then one should picture the magnetic electrons as
itinerant. Another related argument is that the presence
of bound 3d electrons, of a number insufficient to exactly
fill bne of the subshells (eg or t2y) into which the 3d levels
might be split by an environment of cubic symmetry in
a crystal, must entail a departure of an atomic cell from
full cubic symmetry, and either a superlattice or a non
cubic nature for the crystal as a whole.40 No indications
of such structures are known, except in manganese.
Finally, at least some of the possible models with bound
3d electrons would lead to an unquenched orbital mag
netic moment. in a cubic environment, hence to a g
factor departing seriously from the spin-only value 2,
contrary to observation.4o Arguments of this type indi
cate that the magnetic electrons are itinerant in cobalt
and nickel, but leave open the question whether they
are in iron.
Concluding Remarks
Before closing, I would like to call attention to a
group of experiments which, though only indirectly re
lated to the Fermi surface, hold great promise for giving
information about the states of magnetic electrons, at
least in some materials. These are experiments which
measure the magnetic fields H(O), which the ferro
magnetic electrons produce at the position of an atomic
nucleus. This field, due to orbital moments, polarization
of conduction electrons, and polarization of inner-shell
electrons by the 3d electrons, can be measured by its
orienfing effect on the nucleus, via nuclear resonance,4!
gamma-ray correlations,42 or nuclear specific heat.43 Of
the various contributions to H(O), one can reasonably
hope to estimate the orbital and inner-shel1 terms by a
combination of theory arid hyperfine-structure data
from nonmetals. The empirical value of the remaining
contribution, due to spin density of the magnetic and
conduction electrons at the position of the nucleus,
should give considerable insight into the extent to which
sand d states get mixed in the metal, and the extent to
which the Fermi surfaces of the two spins differ.
40 P. W. Anderson (personal communication).
41 A. C. Gossard and A. M. Portis, Phys. Rev. Letters 3, 164
(1959); A. M. Portis and A. C. Gossard, J. App!. Phys. 31, 205
(1960).
42 N. Kurti, J. phys. radium 20, 141 (1959).
43 V. Arp,N. Kurti,and R. Petersen, Bull. Am. Phys. Soc. Ser. II,
2, 388 (1957). The conclusions and admonitions of this section '
be summarized as follows: First, at least some of th~
electrons are to be pictured as itinerant in all or nearlj
all of t~ese metal~ .. Second,. the magnetic ~lectrons ~
~o be pIctu~ed as Itmer~nt m cobalt. an~ lllckel; ironia
m doubt. FI~ally, there.Is great promI~e m experimental
and theoretical work aImed at mappmg out the Fernjj
surfaces of these metals and the associated inertit4
properties of the quasi-particles. It would be very de~
sirable for the fundamental theorists to place the exi~
ence of a Fermi surface on as solid a basis of logic a§
possible, and to find out the quantitative significance 01
the volume of the Fermi surface for metals with fii~
from-free electrons. Experimen~alists might fin~ it verJ!
profitable to concentrate especIally on work wIth veti.
pure materials at low temperatures and high magne~
fields. '~f
ACKNOWLEDGMENTS .:r~
, ,,~
I am much indebted to Dr. W. Kohn and Dr. J.!I
Luttinger for some very illuminating discussiollS' OJ
many-electron theory, and to Dr. P. W. Anderson f6p;.~
number of suggestions regarding electrons in transitien metals.>
---'1;
APPENDIX: ASYMPTOTIC STATIONARITY OF:"'~~
QUASI-PARTICLE STATES .:1~~ "',
For noninteracting electrons the low-lying excit~
states consist, of course, of states with one or a numbe1l
of electrons removed from states just inside the Ferni
surface and placed in states just outside it. If one hltr~
duces a small electron-electron interaction into ti
theory, the Hamiltonian will acquire matrix elemeJiij
connecting these states with other, mostly higher, stat~
The matrix elements from one of these low-lying stltt~
a to states of considerably higher energy will lower •
energy Eu of the state in question, and change its W~~
function, but in a continuous and orderly manner, w4icj
will not introduce any confusion regarding which :~~
perturbed state it originated from. The matrix eleme~
connecting two states a, a' of nearly the same energy, ,.
the other hand, will cause the perturbed statio
states to be such complicated linear combinations of
unperturbed ones as to make any one-to-one co
spondence of the former with the latter rather mean
less. This is Just another way of saying that the st ._/
will be scattered into states a' with approximate~
servation of the unperturbed energy. The quantita~'
effect of this part of the perturbation can be meas.
by the lifetime Ta of the state a with respect to this ~
of scattering, .or by the associat:d na~ur~l width iiI. '.
Now the denSIty of states per umt eXCItatIOn energy'~
above the ground state goes to zero as Ea ~ 0, so in til
limit h/ru ~ 0 also. In fact, since an electron. of a gi~1
excitation energy f can only create holes within this~
energy interval f below the Fermi energy EJ, and ~
only a fra~tion of ord.er 4 Ef of these hole state~ calli
created WIthout puttmg a scattered electron eithet;,m
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,the Fermi surface or at an energy> (' outside it, it
C to see that h/T" must be of order ('2 as €<0.44
ore, for all sufficiently low-lying excited states
Ea. This means that one can define "asymptoti-
stationary states" a with the properties: they are
e-to-one correspondence with the unperturbed
ary states (electron-hole distributions) ; they are
from the unperturbed states by continuous
rbation, using only the high-frequency matrix ele-
~. ts of the interaction; though not true stationary
of the perturbed system, their level width s get«
excitation energies as the latter become small.
e picture given by the rather intuitive reasoning
e preceding paragraph has been verified in more
L. Ginzburg and V. P. Sinn, J. ExpO. Theoret. Phys. 29,
55); see also footnotes 27 and 39 above. careful studies of the general interacting-fermion or
free-electron gas.26 Although special difficulties have
plagued attempts to make calculations of comparable
rigor for electrons in the periodic field of a crystal, it is
very reasonable to suppose that adjoining the ground
state of a metal there exist asymptotically stationary
states compounded out of "elementary excitations"iden
tifiable as electrons or holes, each electron or hole having
a wave vector in the neighborhood of a certain surface
in the Brillouin zone. The surface thus defined may be
called the Fermi surface; it is clear that its size and
shape, and the normal derivative of the energy of an
elementary excitation near it, enter into electronic
specific heat and other properties in much the same way
as in the theory for noninteracting electrons.
APPLIED PHYSICS SUPPLEMENT TO VOL. 31, NO. S MAY, 1960
Microwave Resonance in Rare Earth Iron Garnets*
C. KITTEL
M iller Institute jor Basic Research in Science, University (lj Cali)ornia, Berkeley 4, Calijornia
This paper gives an elementary discussion of the theory of g values and line widths in ferromagnetic
resonance in certain rare earth garnets. The experimental facts are reviewed briefly.
three recent papers, de Gennes, Portis, and the
resent author have considered the theory of several
ts of microwave resonance in the rare earth iron
ts and also in yttrium iron garnet containing rare
ions as impurities. The papers are concerned suc
ely with g values,! line widths,2 and giant ani so
peaks.3 The purpose of the present note is to make
Ie the central results relating to g values and line
s in a brief and simple form.
exchange interactions in the rare earth iron gar
. may be characterized, according to the analysis of
henet,4 by two strong features: (1) a strong ex
e interaction among the ferric ions, as demon
by the approximate equality of the Curie tem-
ures of VIC and the several rare earth iron garnets;
uch weaker exchange interactions between the
and rare earth ions and also between the rare
ions themselves.
e relaxation time of the ferric ions alone is quite
i.this follows because the intrinsic line width in pure
IS very narrow, less than 1 oe. The relaxation times
e trivalent rare earth ions of Sm, Tb, Dy, Ho, Er,
d Vb are probably quite short (by virtue of their
.orted in part by the National Science Foundation.
Ittel, Phys. Rev. 115, 1587 (1959). 9t Gennes, C. Kittel, and A. M. Portis, Phys. Rev. 116,
t ~tel, Phys. Rev. Letters 3, 169 (1959); a more complete
p as been submitted to Phys. Rev.
. authenet, Ann. phys. 3, 424 (1958). strong orbital components) and above 100-200oK may
be the dominant aspect of the problem. Our belief in the
shortness of the relaxation times is based on the line
widths of rare earth ions in dilute paramagnetic salts
measured by Bleaney and coworkers at Oxford.
We may readily derive a relation for the g value in
the limit of infinitely rapid relaxation of the rare earth
ions. Denoting by MA the magnetization of the ferric
ions and by MB the magnetization of the rare earth
ions, we have, for the equation of motion of M A,
(1)
where AMB is the exchange field on A from B. If the re
laxation frequency of the B ions is taken to be infinitely
fast, then MB must at every instant point exactly in
the direction of the total effective field H+AMA acting
on the B iOllS. We must therefore have
MB= (H+AM. 4)/IH+AM A\
:::::(H+AMA)MB/AM A• (2)
On substituting (2) in (1), and noting that M.1XMA = 0,
we have
so that
(4)
The correction to (4) occasioned by a finite relaxation
frequency is considered in references 1 and 2. Generally
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1.1729723.pdf | GaP SurfaceBarrier Diodes
H. G. White and R. A. Logan
Citation: Journal of Applied Physics 34, 1990 (1963); doi: 10.1063/1.1729723
View online: http://dx.doi.org/10.1063/1.1729723
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] IP: 137.112.220.16 On: Mon, 01 Dec 2014 22:06:291990 IRVING STEIN
It is seen that if ko -> 00, then the ac noise spectrum
approaches the dc noise spectrum. This agrees with the
broadband calculation for ac noise. Thus, the ac noise
arising from the tape is truly a modulation noise, even
with the assumption that the tape is perfectly
homogeneous.
If the irregularities in the tape, such as surface
roughness, are taken into account, then as ko -> 00, we
do not expect the ac noise to approach the dc noise.
JOURNAL OF APPLIED PHYSICS Thus, the percentage modulation of the transduced ac
recorded signal, due to the roughness of the tape surface
acting as a variable head-tape spacing, increases as
ko -> 00. Therefore, the demodulated noise at a given
kl increases as ko -> 00 .12 Any irregularities in the tape
add significantly to the demodulated noise as ko -> 00 •
12 P. Smaller, Proc. Magnetic Recording Tech. Meeting, 4, 5 October
1956, Bul. 94 (Armour Research Foundation, Chicago, 1956).
VOLUME 34. NUMBER 7 JULY 1963
GaP Surface-Barrier Diodes
H. G. WHITE AND R. A. LOGAN
Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey
(Received 19 December 1962)
Surface-barrier diodes have been formed by depositing vaporized metals onto chemically cleaned GaP.
Detailed studies of diodes made with gold on p-type GaP show that current flow is by the Schottky emission
of hot holes from the gold over the barrier at the metal-semiconductor interface. The barrier properties
deduced from the current-voltage properties were checked by capacitance-voltage and photoresponse
studies. With the junction biased in avalanche breakdown, electrons were emitted through the gold film into
vacuum, and were studied as a function of junction bias, film thickness, and retarding potential. It was
found that the distribution of hot electrons in GaP has a Maxwellian temperature of 0.8 V, and that the
attenuation length for 4.7-to 9-V electrons in gold is 130±40 A.
INTRODUCTION
SURFACE-BARRIER diodes have been made from
n-and p-type gallium phosphide by vaporization of
metals onto them. The present work is a survey of the
properties of these diodes as deduced from their be
havior as surface-barrier junctions, as photocells, and
as sources of hot electrons; in the last case electron
emission from the metal film is observed when the
reverse-biased junction is operated in vacuum. To this
end, a description is presented of (1) the method of
diode preparation, (2) the current-voltage characteris
tics at various temperatures, and (3) the capacitance-
voltage behavior. These data show that, as in silicon
surface-barrier junctions,! current flow is by Schottky
emission over the barrier at the metal-semiconductor
interface. The barrier height and width were compared
with the built-in voltage and width deduced from the
capacity measurements.
The photoresponse of the diode was studied as a
function both of the wavelength of the incident mono
chromatic light and of the junction bias. The former
permits a determination of the threshold energy of
radiation for photoresponse due to injection from the
metal into the semiconductor, a quantity closely re
lated to the built-in voltage. Finally, electron emission
from the junctions into vacuum was studied as a func-
1 D. Kahng, IRE Solid-State Device Conference, Durham,
New Hampshire (1962). tion of junction bias, metal film thickness, and retarding
field. From these data one obtains an estimate of the
attenuation length of hot electrons (4.7-to 9-V energy)
in the metal film (gold).
DIODE PREPARATION
The gallium phosphide used was doped with mag
nesium and grown by the floating zone method in the
[111J crystal direction. The slices were cut roughly
perpendicular to the growth direction. The slices were
lapped with 303 t grit and etched in hot aqua regia
("" 700e) until the surface was shiny. This usually was
accomplished in 5-6 min. When etched in this way one
surface was smooth and the opposite side had a matt
finish. It was found that the diode properties were
identical when formed on either the matt side or the
smooth side. However, all data presented here were ob
tained using diodes made on the smooth side in order to
determine the junction area more readily. The treat
ment of the GaP after etching was varied to determine
if there was any correlation between this procedure and
the diode properties. Three procedures were used after
etching.
(1) To minimize the effect of any water absorbed on
the surface the crystal was transferred from the etch
bath into hot hydrochloric acid (",75°C) for two
minutes and then into ethyl alcohol. After a thorough
rinsing in flowing ethyl alcohol for one minute the crys-
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tal was transferred immediately into the waiting vacuum
station while still wet.
(2) To determine if the surface collected impurities
after the chemical treatment the crystal was transferred
from the etch bath into flowing deionized water for one
minute and then soaked in ethyl alcohol for a few
minutes. After removal from the ethyl alcohol it was
allowed to dry, then exposed to room atmosphere for
one hour, and finally placed in the vacuum system.
(3) To determine whether an oxide on the GaP sur
face affected the diode properties, a heat treatment2 was
used to remove the oxide from the GaP surface. In fol
lowing this process, the crystal was transferred directly
from the etch bath into flowing de-ionized water and then
placed on a heater strip in the vacuum station. After
evacuation to a pressure of ~ 10-3 mm Hg, hydrogen
was introduced to a partial pressure of ~ 200 jJ.. With
the hydrogen present the GaP was heated to 700°C for
five minutes and then allowed to cool down to room
temperature.
The reverse bias characteristic of the diodes obtained
by these three techniques were essentially indistinguish
able so the simplest treatment, number 2, was used for
all further diodes.
VAPORIZATION
Films of Au ranging from 50 to 1000 A were produced
by vaporization either over the entire surface or
through masks to give precise geometry control.
Bismuth and aluminum were vaporized to give films of
100-150 A thickness. Vaporization of all elements was
done as quickly as possible, taking approximately 30-45
sec for 100-A Au, to minimize trapping of residual gases
in the vacuum system.
When the film covered the entire surface, small area
diodes were delineated by conventional waxing and
etching techniques. All films of bismuth were made by
vaporizing through a mask. Although all metals pro
duced junctions with similar properties, the detailed
studies described below were made using gold.
OHMIC CONTACT
Ohmic contact to the GaP was made by alloying
gallium into the reverse side of the unit3 with an ul
trasonically vibrating soldering tool. The localized heat
ing generated by the ultrasonic vibrations caused alloy
ing with some crystal regrowth while the sample was
essentially at room temperature. Gold saturated solder,
(melting point 95°C) was then applied to the gallium
surface where it combined and formed a well wetted
solder contact. The resistance of these contacts was
estimated from an oscilloscope J-V characteristic to be
less than 10 Q on 5-Q· Cm material. While this technique
2 M. Gershenzon and R. M. Mikulyak, J. App!. Phys. 32, 1338
(1961).
3 This technique was suggested by R. M. Mikulyak. I/)
~ 10-5
w
a.
::; «
~ 10-6 IO-x
,",x
'"'x 0
oO-x
ooxo
f to xo
FORWARD
00 ~o BitS
:~. r ° I
0 x·eo-
a XeO
Do,.,
oQ(-
'" x
o .0 x
--o-~-
00 o x
0 ox
of------< x
ox
)(J 00 xo j ° 0
1 J 0 I fz
W
a:
~ 10-7 00-
xo 0 :xo REVERSE
xo » BIAS x ~
I x ! • :300 oK (RUN 1) 0)(0 'x 0 x 1960 K
xo x I 0 0 7SoK
° 3000 K (RUN 2)
x U 00
4 6 S 10 12 14
APPLIED BIAS IN VOLTS
FIG. 1. Semilogarithmic plots of the forward and reverse current
versus the applied bias at three different ambient temperatures.
gave an excellent Ohmic contact to low resistivity
GaP ( < 1 Q. em) and a higher impedance contact (up to
l-kQ impedance) to higher resistivity material, the con
tact was adequate for all of the studies described here
except analysis of the forward J -V characteristic. No
characteristics of the other properties measured could
be ascribed to any difficulties with the Ohmic contacts.
Contact to the metal film was achieved by pressing a
blunt point directly against the film.
DIODE CHARACTERISTICS
A typical semilogarithmic plot of the current-voltage
characteristic at three different temperatures is shown
in Fig. 1. The diode was made by evaporating 100 A of
gold onto ,.,.2-n· cm p-type GaP through a mask to give
an area of 6.2X 10-3 cm2• The two plots at 3000K were
made before and after the low-temperature measure
ments and indicate the degree of reproducibliity of the
data. It is evident that the junctions exhibit rectifica
tion with a soft breakdown voltage VB at about 14 V at
room t~mperature. Both the prebreakdown cur
rent at a given bias and VB decrease with decreasing
temperature.
The maximum field at VB is estimated to be ,.,.106
V / em using the junction width determined from the ex
trapolated capacitance data. Since grown junctions of
GaP have been shown to break down by avalanche
multiplication4 in fields of 5 X 105 to 1 X 106 V / cm it is
reasonable to assign this mechanism to the breakdown
which occurs in the surface barrier junctions. More
over, the decrease in VB with decreasing temperature is
characteristic of breakdown by avalanche multi-
4 R. A. Logan and A. G. Chynoweth, J. App!. Phys. 33, 1649
(1962). .
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P-TYPE Gap IU·
VR
\!--'------..L.i-- EF
~ -.
Au. FILM RETARDING
POTENTIAL
FIG. 2. The schematic energy diagram of a surface barrier junc
tion on p-type GaP under a reverse bias VJ, with a retarding po
tential at a bias V R.
plication. The temperature coefficient of Y B, f3 = 3.4
X 1O-4°C-I in the range 78°-300oK is only about one
half as large as that observed4 in grown junctions of GaP,
where Y B was much more clearly defined.
The prebreakdown currents of surface-barrier junc
tions in silicon and germanium have been foundl to
agree well with Schottky emission over the barrier at
the metal-semiconductor interface,6 given by
I =AT2e-iPo1kTe[(ifE/ €*P/kTJ (1)
where T is the temperature in degrees Kelvin, k is
Boltzmann's constant, q is the electronic charge, and E
is the field across the barrier of height <1>0. €* is the dielec
tric constant appropriate to the electron motion and
has been found to be unity in Si and Ge surface-barrier
junctionsl since the transit time is small compared to
the dielectric relaxation time. A is a constant, of magni
tude 120 A/cm2°K2 when evaluated using simple theory.
The potential distribution is represented schematically
in Fig. 2.
The field across the barrier may be approximated by
the maximum field given by
EM=2yl/W 1, (2)
where V is the total voltage drop across the barrier,
including the built-in voltage Vi, and WI is the width
constant (the barrier width at unity voltage drop across
it). From Eqs. (1) and (2), it follows that
I=A1'2e-iPolkT exp[(2if/€*W I)W1/kT]. (3)
At fixed T, log I (X Vi with slope varying as T-I and
the magnitude of the slope permits a determination
of W I, which may be compared with WI determined from
capacity measurements. The extrapolated current value
at V = 0 permits a calculation of <Po which as shown in
Fig. 2 may be compared to Vi+~p, where !;p is the
separation of the Fermi level from the valence band.
• H. K. Henisch, Rectifying Semiconductor Contacts (Oxford
University Press, New York, 1957), p. 173. Log I is plotted against Vl in Fig. 3 at 300° and
196°K. It is immediately evident that the linearity ob
served with the data obtained at 300° and 196°K is
reasonable in view of the leakage current apparent at
1< 10-8 A and the approach to breakdown at I> 10-s A.
The data at 78°K do not show the required decrease
with temperature required by Eq. (3), the observed
I presumably being largely influenced by surface cur
rent generation. The logarithmic slopes, observed in
Fig. 3, are 20 and 31 V-i at 300° and 196°K, respec
tively, and have the T-I dependence of the theory. The
value of WI deduced from the slopes is 1.9X1O-s cm
in satisfactory agreement with WI = 1.2X 10-6 deduced
from the junction capacity at a bias of one volt (includ
ing the built-in voltage Vi=0.75 V). Extrapolation of
the curves of Fig. 3 to V = 0 gives <Po= 0.68 and 0.58 V
at 300° and 196°K, respectively. In view of the large
extrapolation involved, these values, which would be
equal in the absence of resistivity change with tempera
ture, are in reasonable agreement with each other and
with Vi+!;p=0.88 V, where !;p=0.13 V was estimated
from the carrier concentration in the GaP.
It is evident by inspection of Eq. (1) or of the poten
tial energy diagram of Fig. 2 that for any junction, the
forward bias cannot exceed the built-in potential Vi.
The forward characteristics of Fig. 1 show current
densities of only '" 1 A/ cm2 at biases of 2.5 V and
indicate series resistance effects, which are common
phenomena in GaP junctions.2,4 The series-resistance
effects observed here were shown to be associated with
the Ohmic contact described above. Thus when Ohmic
contacts were formed by allOYIng tin into n-type samples
or by diffusing zinc into p-type samples, the forward
characteristics were in accord with diode theory. In
particular, the logarithmic slope of the forward char-
en w
0::
W
~ 10-6
«
~
f-
~ 10-7
0:: a:
::> u '/ I
/ '/ /1
300·Y x/
ifs·K
x" / ,VV
1.5 1.6 1.7 1.8 v/4 IN (VOLTS)1.I'" 1.9 2.0
FIG. 3. Semilogarithmic plot of the reverse current versus V Ti,
where V T is the total reverse voltage, using the data of Fig. 1.
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10
0&
o z « « 7
'" ..,
~6 a
N )...5 .II~-.
i ol
I • , • J
/ ,
~.
7 I
j V il A/ v: '/ ,x B.)
o -2 / I
-1 f ~ V" IJ~ i'--
~ ~
o I 2 3
BIAS IN \oLTS l/
;
\04 8 u
o z
7 :
'" ..,
8 ~
;:) u
51.
Ii:
4 ~
> ...
3 u
f «
2 ~
1
. ~IG: 4. Plots of (capacitance)-2 versus applied bias for typical
)UntctlOns on p-type GaP (A, B, and C) and on n-type GaP
(D).
acteristic was q/nkT with n ranging from 1.4 to 1.6 and
the built-in voltages estimated from the forward I-V
data were in good agreement with those deduced from
the reverse characteristics and from the capacity and
photoresponse data to be described in later sections.
The simple contact was used to avoid the use of heat
treatment which might produce undesirable changes in
the electrical properties of the base material. The series
resistance effects apparent in the forward characteristic
(:$1000 Q) would contribute negligibly to the higher
impedance reverse characteristics in the current ranges
discussed above (10-8 A <I < 10-5 A) but would ac
count for the soft breakdown at higher current levels
(I> 10-4 A).
CAPACITANCE MEASUREMENTS
The capacity of the units as a function of applied
bias was measured at 50 kc. Excellent linear plots of
C-2 versus bias were obtained for all junctions. Typical
plots are shown in Fig. 4. In obtaining the capacity data
it was noted that there was no drifitng in value of the
capacity as has been ascribed to slow surface states.6
The built-in voltages Vi indicated by the capacitance
data were 0.75 and 1.9 V for p-and n-type GaP, re
spectively. The built-in voltages were generally in the
range 0.6 to 0.8 V and 1.3 to 1.9 V for p-and n-type
GaP, respectively. There was no distinct correlation
between Vi and the doping level. In any given unit,
6 S. R. Morrison, Semiconductor Surface Physics (University
of Pennsylvania Press, Philadelphia, 1957), p. 169. the value of Vi deduced from capacitance agreed well
with that deduced from the photo response, as de
scribed in the next section. Moreover, since ~1I"'~p"'0.1
eV, the sum of the built-in voltages for p-and n-type
GaP was within experimental error equal to the sum of
the Schottky barrier heights or to the band gap, as ex
pected theoretically.7 The variation of Vi observed here
is presumably due to surface states. Since the reverse
bias variation of the capacity, as shown in Fig. 4 does
not depart from the expected quadratic dependence,
the surface state occupancy is independent of bias.
This suggests that the surface states lie closer to the
band edge than the barrier height and remain empty of
mobile charge throughout the experiments. The doping
level in homogeneous samples of the GaP estimated
from the capacitance data agreed to within 15% with
that estimated from Hall measurements at 300°-400°C.
At these temperatures the Hall coefficient approaches a
constant value indicative of essentially complete ioniza
tion of the impurities. The carrier concentrations de
duced from Hall measurements at room temperature
were generally 50% less than the impurity concentra
tion deduced from the capacity and were interpreted as
due to carrier freezeout. Further discrepancies of up to
a factor of two were observed in some samples and. were
ascribed to doping inhomogeneities since the Hall effect
indicates average carrier concentrations, whereas the
capacity determines local impurity concentrations in
the vicinity of the junction.
PHOTORESPONSE
Th.'! short-circuit photocurrent generated at the junc
tion was measured both as a function of the bias and of
the wavelength of the incident monochromatic light.
At fixed bias, typical photoresponse curves for junctions
made by depositing gold on both n-and p-type GaP are
shown in Fig. S, where the relative photocurrent I p
per unit incident photon is plotted against the frequency
I' in eV of the incident light. The relative photocurrents
for the two junctions have been adjusted to match at
energies greater than about 2.3 eV, where the current
is generated by intrinsic absorption in the GaP. It is
evident that the threshold voltage for photoresponse,
(injection of holes over the Schottky barrier from the
gold into the semiconductor) is about 0.7 and 2.1 eV in
the p-and n-type materials, respectively. Since the
photoresponse8 varies as (1'-1'0)2, where 1'0 is the thresh
old frequency, 1'0 may be more precisely located from a
plot of I pi vs I' as shown in Fig. 6 for a p-type structure
biased at VJ=O and 5 V. The value hl'o=O.71±O.03
eV is in excellent agreement with a previous measure
ment9, of O.715±0.035 eV. Similar measurements on
n-type structures are limited by the small range of 1',
7 See Ref. 5, p. 183.
8 A. L. Hughes and L. A. Dubridge, Photoelectric Phenomena
(McGraw-Hill Book Company, Inc., New York, 1932), p. 241.
9 C. R. Crowell, W. G. Spitzer, and H. G. White, App!. Phys.
Letters 1, 3 (1962).
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>u a: ... z ...
... z
~
U
~
... z :)
a:
~
... z
~IO-
a: :) u g
f I N TYPE
~
~-
X~
~l
.J~m' r
J I r
I 2
PHOTON ENERGY IN ev I t
l r rt
)
't'
3
FIG. 5. Semilogarithmic plot of the photocurrent per unit
incident energy in arbitrary units versus the photon energy for
surface barrier junctions on n-and p-type GaP.
where photo response is observed, and one can only
estimate a barrier of about 2 eV at the interface for
n-type samples.
From the potential diagram of the diode it is evident
that hllo= Vi+~p=<I>o, where ~p is several kT for the
junctions studied. It is evident that to within experi
mental precision, the energy barrier hvo=0.72±0.03 eV
and the built-in voltage Vi=0.7S±0.1 eV are in good
agreement with each other and with <I>o=0.68 eV de
duced above on a similar junction. In n-type samples,
the corresponding values hvo",,2 eV and Vi= 1.9±0.1
Vs are also in good agreement, in view of the limit range
of frequency used to estimate Vo. The structure in the
photoresponse curves in the ranges 1.3 to 2.3 eV and
near 2.3 in p-and n-type samples, respectively, may be
indicative of deep states in the GaP.1O
As shown in Fig. 6, Vo was unchanged using photore
sponse data at VJ= 0 and 5 V. The actual photoresponse,
however, increased markedly with bias as shown in
Fig. 7 when the short circuit photocurrent generated at
the junction is plotted as a function of bias at the indi-
10 M. Gershenzon and R. M. Mikulyak, Solid State Electron. 5,
313 (1962). cated values of v, for a junction similar to that of Fig. 1.
The increase in photocurrent with v at any given bias is
only qualitatively in agreement with the response curve
of Fig. 5 since no corrections have been applied to
account for the variation of the intensity of the light
source with v. The bias dependence of the photo current
I p parallels exactly that of the dark current I. Thus log
I p vs Vi is linear with logarithmic slope equal to that of
log I vs Vi, and the logarithmic slope varies as T-1 as
required by Eq. (3). The increase in photocurrent with
bias is thus ascribed to the lowering of the Schottky
barrier by the applied electric field.
The increase in photocurrent with bias observed using
a particular wafer of GaP could be enhanced if the
etched surface was treated prior to deposition of the gold.
Such treatmentll to produce inversion layers on silicon
would consist of boiling the sample in an aqueous solu
tion of NaOH or scratching the surface with a sharp
point. The increase is ascribed to a modification of the
Schottky barrier by surface states.
ELECTRON EMISSION
To study electron emission from the surface-barrier
junctions, the samples were placed in a vacuum system
at a pressure of ",.10-6 mm Hg. Emitted electrons were
collected at the first dynode of a Dumont SP 240 photo
multiplier tube, made of Be-eu approximately 1 in. in
diameter and positioned i in. above the junction. At
current levels above 10-13 A, the emitted electron cur
rent Ie was directly measured with an electrometer and
a brass collecting plate of dimensions equal to that of
the first dynode of the tube. In operation the first
dynode was biased positively at 1100 V and the bias of
each of the 12 stages in the tube increased by 245 V per
stage, giving an over-all gain of about 5X 106• The photo
multiplier tube output was amplified approximately
lOO-fold by the use of a cathode follower and a Tektronix
1.1 r-~--------'''''o-
1.0
II) 0.9 ...
~ as
>--'
~ 0.7
a: t: 0.6
dI
~ U5
!: 0.4
~.
-U; 0.3
......
H -0.2
0.1 / / I x
o 't. -= 0
x VJ ~ 5VOLTS
O_~-U~_J-~ __ ~~~
0.6 0.7 O.S 0.9 1.0 1.1 1.2 1.3
'PHOTON ENERGY IN ev FIG. 6. The square
root of the photocurrent
per unit incident energy
plotted against the pho
ton energy for a surface
barrier junction on
p-type GaP biased at
VJ=O and VJ=5 V,
respectively.
11 T. M. Buck and F. S. McKim, J. Electrochem. Soc. 105, 709
(1958).
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a~--~--~----------~--~-,
7
6
U)
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z
VI «4
iii
w
fIJ
0:
~ 3
w
0:
2
°OUU~~2--~3--4~~5~~6--~7~e~~9~'O
PHOTO CURRENT IN ARBITRARY UNITS
FIG. 7. Linear plots of the photocurrent versus junction bias
using incident monochromatic light with h" in eV indicated for
each curve.
type 121 preamplifier. The signal then entered a pulse
forming network consisting of a 47oo-#L#LF condenser
shunted to ground with a 1000-Q resistor, was ampli
fied with a Hammer model N301 amplifier, and was ob
served with a Berkeley counter and digital recorder.
Provision also was made to bias the junction independ
ently and measure the junction current h. Figure 8
shows typical plots of I. versus V J for two junctions
formed by evaporating 100 A of gold onto about 1-Q·cm
p-type GaP. The high-current electrometer readings
(> 10-11 A cm-2) were joined smoothly to the currents
given by the phototube output counts, by multiplying
the latter by 2. This factor is ascribed to the electron
losses in traversing the various stages of the phototube.
It is evident that the emitted current varies approxi
mately exponentially with junction bias. There was no
evidence of a correlation of Ie with junction current so
that one must assume that a significant fraction of the
junction current was surface generation current. In the
bias range studied, the junction was in breakdown (see
Fig. 1) with the junction current varying from about 0.1
to 10 rnA over the bias range studied.
Microscopic examination of the junctions after the
emission studies gave no evidence of damage to the
structures such as local heating or pinhole formation.
Gross heating effects were also ruled out by replacing
the steady dc bias to the junction with a 100-cps voltage
pulse which was varied in length from 1 to 6 msec. The
emitted currents observed, when corrected for the duty
cycle, were in good agreement with the dc values.
While in some junctions, a uniform white light was visible over the entire junction area at high reverse
current flow, it was more common to observe light emis
sion at the junction periphery under these conditions.
After waxing the edge of a junction with apiezon wax it
was observed that Ie, when corrected for the reduced
area due to waxing, was equal to values observed on the
prewaxed junctions. It was, therefore, verified that edge
effects did not contribute unduly to the emitted current.
The emission current was found to decrease upon in
creasing the thickness of the gold film used to form the
junction. This effect is demonstrated in Fig. 9 where a
plot is made of the loagrithm of the emission current
versus the thickness of the gold film at a constant bias,
VJ= 9 V, using junctions made on nearly identical sub
strate GaP. For film thicknesses less than 300 A, a
large scatter in emission currents was observed and is
taken as evidence of relatively large thickness variations
in the thin layers. From the slope of the curve in Fig. 9,
the attenuation length L of electrons in gold is found to
be 130±40 A. To compare this result with other meas
urements of L, it is necessary to estimate the energy of
the emitted electrons. To this end, one requires a
mechanism of electron flow.
It is seen that the characteristics of Ie can be ex
plained quantitatively with the following assumptions:
(1) The barrier junction is in avalanche multiplication
when emitted electrons are observed. This assumption
is supported by estimates of the junction field (,....., 106
VI cm) and the temperature dependence of VB. (2) The
holes which are emitted over the Schottky barrier are
multiplied in the high junction field and the energetic
electrons so produced are those which may overcome
1
x
x
x -
10 12 I x I x
I )
I • I x I -I
I -I
I
I x I
x -I
• I
I
I
I -I
I
I
I
COUNTER : ELECTROMETER .. I ..
i
14 16 18
BIAS IN VOLTS 20
FIG. 8. Semilogarithmic plot of emission current
versus junction bias. --
22
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IP: 137.112.220.16 On: Mon, 01 Dec 2014 22:06:291996 H. G. WHITE AND R. A. LOGAN
10' o
Z o u w
'" a: w n.
'" I-Z
:::> 8
~
I-10' Z w a: a:
:::> u
:z
Q
'" '" ~ w
10 \
~
\ t
1\
±
f
1 I
),-130±40A
I
r\
\
\
\
:\ ~ \
o 200 400 600 800 1000
FIL.M THICKNESS IN ANGSTROMS
FIG. 9. Semilogarithmic plot of emission current versus the thick
ness of the gold film used to form the surface barrier junction.
the work function of gold at the surface. (3) The energy
distribution of the electrons at the edge of the space
charge region is that calculated by Wolffl2:
N(~) ...... exp( -~/kTe),
where N(~) is the density of electrons of energy ~ above
the conduction band, k is Boltzmann's constant, and
Te is the steady-state electron temperature in the
Maxwellian distribution, defined as
kTe= (e2FJA2)/3~r,
where E is the electric field, A is the mean free path of the
electrons between collisions involving phonon emission
of energy ~r, the transverse optical or Raman phonon
(~r=O.052 V in GaP).
The above distribution pertains to energies below the
threshold. For energies above the threshold,13 the dis
tribution may be approximated closely with the same
exponential form, with the electron temperature now
given by
kT.* = e2 FJlAj3~rZ,
where I-I = li-I+ X -1, li being the mean free path for
ionization and Z given by
Z = !+[l+ (e2.b.'2lAj3m2)],
where r= lilA. For li»X, Te= T.* and the electron tem
perature is, therefore, approximately the same through
out the energy range. Wolff's theory is known4 to de
scribe avalanche mUltiplication in Jrn junctions of
12 P. A. Wolff, Phys. Rev. 95, 1415 (1954).
13 D. J. Bartelink, Technical Report No. 1654-2 Solid-State
Electronics Laboratory, Stanford University, Stanford, California. GaP where it was found that A=32 A. Hence if one
assumes that l;»32 A, the electron temperature may be
estimated to be 0.65 V in the Maxwellian distribution,
with E"" 106 VI cm. The two remaining emission studies
to be described, namely variation of the work function
at the surface and the effects of inserting a retarding
field in the path of the emitted electrons are seen to be
consistent with this model.
The energy distribution of Ie was determined by in
serting a grid between the gold emitting surface and the
first dynode of the photomultiplier tube. The grids were
equal in area to that of the dynode and were made by
winding 5-mil-diam gold wire on a 0.015-in.-thick frame.
The grids were thus two parallel arrays of wire, 0.015
in. apart. The transparency of the grid to the electro
static field was experimentally determined by using
two different spacings for the gold wires, 0.005 and 0.010
in. The identical retarding field resulting from both grids
confirmed that the grids formed equipotential surfaces
in the paths of the emitted electrons. The planar con
figuration of the electron source, grid, and dynode col
lector causes the retarding field to affect only the com
ponents of electron energy normal to the gold surface.
However, if the electron energy distribution is radially
symmetrical, then the distribution in the normal com
ponent would be proportional to that in the total elec
tron flow.
At various fixed values of VJ, the variation of Ie was
measured as a function of the grid bias V G, relative to
o z 8 w
I/)
a: w a..
~103 z
:::>
8
1!!:
I-Z w a: a:
:::> u
z
~102
I/)
I/)
:I! w
10 P ..
........
............
-........ ~ '~ 1"-. Xl -15.8 VOL.TS
""-'.\
-.:"'o! ~
~ •
'\ \
VJ -13.0 VOL.~~ ~ , '1
.i -.l
<\ \
1 •
\ 1\ , \ 1 -4 -2 o 2 4 8 8 10
RETARDING POTENTIAL IN VOL.TS
FIG. 10. Semilogarithmic plot of emission current
versus the retarding;.potential. 12
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] IP: 137.112.220.16 On: Mon, 01 Dec 2014 22:06:29GaP SURFACE-BARRIER DIODES 1997
the gold emitting surface. Two typical results are
shown in Fig. 10, where for a junction biased at V J= 13.0
and 15.8 V, log Ie is plotted against V G. It is evident
that the displacement in V G at low emission currents is
approximately that in VJ, in accord with the above
model. The slope of the curves of Fig. 10 represent the
energy distribution of the emitted electrons. In particu
lar the distribution in the high energy tail is the ex
pected exponential with temperature kTe=0.8±0.1 V,
in fair agreement with kT.=0.65 V estimated above.
It is also noted that the exponential rise of the emission
current with junction bias (Fig. 8) also has a logarithmic
slope of (0.8 V)-l which may be related to the electron
temperature. However the change in field with junction
bias should cause the temperature to increase somewhat
with bias. This effect is difficult to estimate in view of
the noted series-resistance effects which contribute to
the bias dependence of currents in breakdown.
By lowering the work function of the gold electrode
with an evaporated film of barium the emission current
was increased by a factor of 10 to 15. The barium source
was contained in an iron cartridge 0.025 mm in diameter
and 12 mm in length, slitted along one side. The barium
was evaporated by Joule heating of the cartridge and
monitored by observation of the I-V characteristic of
the diode. When slight deterioration of the characteris
tic was observed, the vaporization was terminated and
the emission current was again measured. The enhanced
emission current slowly returned to values characteris
tic of its initial condition, and this procedure could be
cycled repeatedly, producing the same effect on the
emitted current.
Consistent with the above model, the barium lowers
the barrier height (It'Ba=2.1 V) thus allowing a greater
fraction of the available electron distribution to escape.
At bias VJ> It' the number of hot electrons over the
work function barrier It' available to contribute to Ie, is
Ie'" /",'" exp( -e/kTe)de.
Hence the factor by which leis increased by lowering It'
from It'Au=4.7 to It'Ba=2.1 may be estimated. Using
the value kTe=0.8 V obtained in the retarding potential
studies, one would expect Ie to increase by a factor of
26, in reasonable agreement with the observed in
creases of 10 to 15 described above in view of the un-certainty in the work functions and the precision of
kTe•
Finally the observed attentuation length of hot elec
trons in gold was observed to be 130±40 A. The elec
tron energies are estimated to be in the range It' Au to
V J, i.e., 4.7 to 9 V. This value is very close to that ob
served by Meadl4 who found L"-' 100 A for electrons in
the range 5 to 10 V above the Fermi level, but is greater
than the value of ",30 A extrapolated from Quinn's
theory. IS
DISCUSSION
The present studies demonstrate that current flow in
the surface-barrier junctions formed by evaporating
gold onto p-type GaP is by Schottky emission over the
barrier formed at the metal-semiconductor interface.
The properties of the barrier, the height and width, de
duced from the current-voltage characteristics were
found to be in good agreement with those deduced from
capacitance and photoresponse studies. For ease in ex
amining a large number of samples and to avoid the un
explored effects that might arise from heat-treatment
used to alloy o'r diffuse at high temperatures, a deficient
Ohmic contact was used. This introduced a series re
sistance which gave rise to discrepancies with theory of
the forward characteristic and contributed to the
softness of the reverse bias breakdown. There was no
evidence that the Ohmic contacts influenced other prop
erties of the junctions measured at high impedance
levels.
The emission of hot electrons from the reverse biased
junction gives two important results: the electron tem
perature in the Maxwellian distribution of the hot elec
trons in avalanche multiplication in GaP is 0.8 V and
the attenuation length in gold of electrons in the energy
range 4.7 to 9 V is 130±40 A.
ACKNOWLEDGMENTS
The crystals used in these studies were grown by C. J.
Frosch and L. Derick. The Hall effect measurements
were made by M. Gershenzon and W. Feldmann. We
wish to thank C. A. Lee for assistance in instrumenta
tion of the electron-emission studies and D. J. Bartelink,
A. G. Chynoweth, and D. Kahng for many valuable
discussions about this work.
14 C. A. Mead, Phys. Rev. Letters 8, 56 (1962); Erattum, Phys.
Rev. Letters 9, 46 (1962).
16 J. J. Quinn, Phys. Rev. 126, 1453 (1962).
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1.1702674.pdf | Experiments on MercurySilicon Surface Barriers
D. K. Donald
Citation: Journal of Applied Physics 34, 1758 (1963); doi: 10.1063/1.1702674
View online: http://dx.doi.org/10.1063/1.1702674
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/6?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 19:28:41JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 6 JUNE 1963
Experiments on Mercury-Silicon Surface Barriers
D. K. DONALD
Scientific Laboratory, Ford Motor Company, Dearborn, Michigan
(Received 8 November 1962)
Barrier heights of Si-Hg surface barriers were examined experimentally as a function of the resistivity
of the silicon. The saturation reverse current and the resistance at zero bias for the diodes were measured.
The simplest Schottky metal-semiconductor theory reasonably explains the resistivity-dependent behavior
of "n" -type Si. However, for "p" -type Si a surface inversion layer is predicted if the published work functions
are superposed on the simplest model. Here the resistivity dependence for the thicker barriers is less strong
and the measurements more variable. The results are summarized in terms of the flat band condition
(UB= us) in units of kT /q from midband: Calculated 3±3.5. The experiments are consistent with 3±5,
using no surface states to improve the fit. A Shockley surface-charge distribution ±n •• $1012/cm2 is sufficient
to suit the experimental data to uB=3.
Sample treatments were ion bombardment with Hg ions, followed by submersion in Hg. Field enhanced
desorption in liquid Hg was also used. Cleaved samples produced what seemed to be patchy surfaces with
small regions degenerate "p" type.
INTRODUCTION
THE diodic behavior of two condensed phases in
contact should be closely related to the work
functions of the materials. Experiments on silicon and
selenium have indicated that this is the case1•2 for metal
semiconductor contacts, but have involved uncertain
ties of as much as tenths of volts in barrier height (V D)
and orders of magnitude in saturation currents (jo)
for diodes. Here experiments on the Hg-Si interface
were undertaken in the hope that the barrier heights
could be interpreted with some accuracy from the
vacuum work functions of the metal (cf>m) and semi
conductor (cf>.). The saturation reverse currents of
diodes were examined as a function of the doping in the
Si. This systematically shifted the barrier height at the
interface. Intervening surface states or a double layer
at the interface would then modify this systematic
change of barrier height with doping. The saturation
reverse current in these diodes is exponentially related
to the barrier height and it is important to note that a
decade change in jo implies only 0.06 eV (2.3kT/q)
difference in barrier height. A very strong inter
dependence.
The Hg-Si interface is examined here for two reasons.
The use of two bulk materials eliminates the difficulties
in defining the work function of a thin film grown on a
substrate of different ion size and perhaps crystal struc
ture. Further, the ion sizes and metallurgical data3 indi
cate very little interaction of mercury and silicon both
at room temperature and at elevated temperatures.
Also the work functions of the materials are such that
measurements should be moderately simple. The work
function for liquid mercury (cf>m) is 4.53±0.04 eV4 and
1 Eberhard Spenke, Electronic Semiconductors (McGraw-Hill
Book Company, Inc., New York, 1958), p. 364; R. B. Allen and
H. E. Farnsworth, J. App!. Phys. 27, 525 (1956); E. C. Wurst
and E. H. Borneman, J. App!. Phys. 28, 235 (1957); bibliography
in Bardeen, reference 10.
2 R. J. Archer and M. M. Atalla, Ann. N. Y. Acad. Sci. 101,
697 (1963).
3 M. Hanson, The Constitution of Binary Alloys (McGraw-Hill
Book Company, Inc., New York, 1958), 2nd ed. the electron affinity of silicon (x) is 4.05±O.08 eV.' If
there are no surface states the flat band condition for
contact of Hg and Si occurs for "n"-type Si doped with
the Fermi level (ip), 0.48 eV from the conduction band
or 0.07±0.09 eV ("'-'3±3.5 kT/q) from the center of the
forbidden gap. The impedance level of diodes from this
combination is not excessive.
Three methods of cleaning the Si were attempted.
Bombardment by Hg ions in vacuum was found to give
reasonably reproducible interfaces. Interfaces formed
in air were treated with direct current to "form" them.
This effect, attributed to mass transport from the inter
face, gave similar experimental results to those from ion
bombarded samples. Finally, cleaving under Hg was
attempted. The interfaces formed by cleaving ap
parently had appreciable patch effects which rendered
the experiments useless for this work.
EXPECTATION
Spenke6 has reviewed the behavior of an abrupt
(Schottky) single-carrier metal semiconductor contact
and noted the regions of distinct behavior. Where the
carriers accumulate at the barrier an Ohmic contact is
formed. Region (1) in Fig. 2 gives Ohmic contacts for
surface barriers where the flat band condition is at
Us = 3. If the barrier is exhausted of carriers as in
Figs. 1 (a) and 1 ( c), and if the barrier is thick compared
to a collision length the diode equation:
j = jo(eUelkT -1) = (kT / eRo) (eUelkT -1) (1)
applies where the saturation reverse current jo is
directly related by derivative to the resistance at zero
bias (Ro) by Ro=kT/ejo. The saturation reverse current
for the above case is6:
jo= niefJ.euBEB (UB,US) = nBefJ.EB, (2)
where the carrier density (nB) at the barrier and the
4 Wayne B. Hales, Phys. Rev. 32,950 (1928).
6 F. G. Allen and G. W. Gobeli, Phys. Rev. 127, 150 (1962).
6 See reference 1, Chap. IV, Sec. 4.
1758
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to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 19:28:41EX PER I MEN T SON MER CUR Y -S I L I CON SUR F ACE BAR R I E R S 17 S9
electric field (EB) at the surface of the semiconductor
'are controlling parameters. The barrier field (EB) is
obtained from the solution of Poisson's equation as
shown by Kingston and others.7 For moderately doped
n-type Si, Eq. (1) holds and the potential energy dis
tribution is that of Fig. 1 (a). Region (2) of Fig. 2
sketches jo(u.) for this abrupt junction.
Highly extrinsic material results in a thin barrier
where carriers are not "thermal" under an applied bias
as collisions become infrequent. Here the barrier field
EB loses meaning and the thermal velocity of the
carriers controls jo, and6
(3)
is noted in region (3) of Fig. 2. For degenerate and near
degenerate materials tunneling is controlling as has
been noted by Chynoweth et at.8 and sketched in
region (4) of Fig. 2.
For "p"-type Si on Hg, Fig. 1 (b) shows that the
majority carrier changes sign near the metal-semi
conductor interface. The sources of back current will
be minority-carrier generation9 in the diffusion length
adjacent to the interface, pair production in the space
charge region, and hole injection from the metal. For
material with reasonable lifetimes near the interface,
hole injection from the metal will predominate. The
saturation reverse current can be thus written directly
by analogy with the single-carrier case. In the barrier
with inversion the important carrier is the minority
carrier (in this case the hole) in the channel so
(4)
is identical in form with Eq. (1) and contains PB, the
minority-carrier density at the surface of the semi
conductor. The thin barrier case and the onset of
tunneling also occur and are sketched for p-type mate
rial (us<O) as regions (3) and (4) along with the p-type
thick barrier region (2).
Charge tied up in surface states is a complicating
factor for surface or interface experiments as Bardeen10
has noted. We consider below a charge Q'8 independent
of potential at the metal-semiconductor interface.
Noting that the accumulated charge and the electric
fIeld perturbation in the presence of a spatial "pulse"
of charge density are integrals and that the difference
in potential across a pulse of surface states (Q8') is a
double integral, we can say that with fewer than
1013/ cm2 surface states the effect on the potential energy
and therefore on UB is small (;50.03 eV). The effect of
surface charge on surface field is a perturbation
7 Robert H. Kingston and Sigfried F. Neustadter, J. Appl. Phys.
26, 718 (1955); c. E. Young, J. AppJ. Phys. 32, 329 (1961).
8 A. G. Chynoweth, W. L. Feldmann, C. A. Lee, R. A. Logan,
G. L. Pearson, and P. Aigrain, Phys. Rev. 118, 425 (1960).
• William Shockley, Electrons and Holes in Semiconductors
(D. Van Nostrand, Inc., Princeton, New Jersey, 1950), Sec. XII-So
10 John Bardeen, Phys. Rev. 71, 717 (1947). (a) (b)
(e) (d)
FIG. 1. Band bending at Hg-Si contacts. (a) Simple exhaustion
barrier. (b) Barrier with inversion. (c) Simple exhaustion barrier
with reverse bias (U). (d) Barrier with inversion and with reverse
bias (U).
(!1E=Q8./E) on any existing field from other causes at
the surface.
For example, a concentration of surface states of
6XI011/cm2 produces a perturbing field of "-'105 V /cm
in silicon and a differential double layer of potential of
'::;4X1Q-3 eV if the surface charge is distributed uni
formly over 4 A and is independent of energy. Cleaved
Si in vacuum, on the other hand, has surface states in
excess of SX1013/cm2 for some dopings5 which produce
(amps/emf)
'10 (4) Si Hg
10'
r 102
io
I 10'3
10-4
10-11
(kT/q) ;20 (p) -10 0 +10 (n)+2O
FIG. 2. Predicted saturation reverse current vs bulk doping is
sketched for the Hg-Si system. Regions of Ohmic accumulation
barriers (1), thick exhaustion barriers (2), thin exhaustion
barriers (3), and tunneling barriers (4) are noted using uB=3
and vT~1.1XI07 em/sec for both carriers.
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to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 19:28:411760 D. K. DONALD
band bending of as much as 0.7 eV. Thus large densities
of surface states on cleaved Si in vacuum control the
work function whereas a surface or interface with fewer
than 1012 surface states/ cm2 will be perturbed only in
internal electric field and carrier-concentration gradient.
EXPERIMENTS
Samples of Si were cleaned in groups in closed evacu
ated vessels with 500-to 1000-V Hg ions at current
densities of 10 pAl cm2 and up to 10 C of ions. At the
same time the Hg was cleaned in the same way. Samples
were dropped into the pool of Hg with the Hg plasma
on or immediately after turning it off. Samples were
measured in situ directly after contact with the Hg.
The gettering action of the plasma quickly pumped out
residual air when the trapped oil pump was sealed off
and spectroscopic examination of the plasma showed
the air lines disappear. However, one very weak line
in the green, not of the Hg spectrum, was frequently
present and was probably a hydrocarbon line from the
breakup of organics in the system. A disturbed layer
on silicon has been found by others after ion bombard
ment.H However, this layer's contribution to work
functions has not been found large.12 In the contacts
we used a few patches of very thin film on the Hg
formed on contact with the Si and presumably all or
part of the disturbed layer is removed as these patches.
(amps/cm 2)
10
101
i Ii?
10
I 103
104
105
106 041-
Si Hg
II
o +10 (n)+20 -us-
FIG. 3. Saturation reverse currents and directions for Si-Hg
diodes compared with predictions of the models in Fig. 2. Data
are presented as the extrema and geometric mean.
11 R. E. Schlier and H. E. Farnsworth, in Semiconductor Surface
Physics, edited by R. H. Kingston (University of Pennsylvania
Press, Philadelphia, 1957), p. 3.
12 F. G. Allen, J. Phys. Chem. Solids 8, 119 (1958; Rochester). The experimental values of jo and Ro were measured
and in agreement for 2-15 runs on samples of "n"-and
"p"-type Si and are shown in Fig. 3 along with the pre
diction outlined above. The spread in jo was higher for
"p"-than "n"-type material and high in both cases but
in terms of the spread implied in the barrier potential
was not excessive. The samples did not change behavior
in tens of hours sealed in vacuum nor in hours if air
was admitted to the vessel and the samples remained
immersed in the Hg. When the samples were removed
from the Hg in air the barrier heights immediately
changed.
The resistivities of the samples ranged from 330-
0.0008 Q cm "p" and 220-0.01 Q cm "n" and most of
the samples were nominally oriented (111). No Ohmic
contacts were found for clean conditions, and no indi
cations of capacitance from insulating barrier films or
from slow states were seen for clean conditions. Indica
tions of surface states as a multivaluedness for jo were
. seen for dirty conditions. However, the voltage vs
current behavior found for dirty contacts13 could be
avoided. Ohmic contacts to the Si were by sand blasting
and In solder. Nickel was the only metal in contact with
the Hg and its solubility in Hg3 is low. Silicon was used
as a counter electrode in one experiment with no dif
ference in behavior; therefore, we assume the Ni
counter electrode did not influence the experiments.
It is known that current passing through a metal or
insulator can result in ion migration. This mass trans
port process can presumably also desorb a film from an
interface. A number of samples were treated to examine
this process for the Hg-Si system. Samples were etched
in HF+NH0 3, rinsed in water, and submerged in Hg
either directly or after a minute in boiling waterl4 to
produce a stable oxide. There were effects for either
polarity of applied voltage that rapidly changed the
interfaces compared to changes occurring in quiescent
Hg. Application of a positive voltage to the Si produced
migration to the same sets of characteristics seen for
ion bombarded surfaces. In most cases charges of the
order of a coulomb produced the effect. The extreme
dopings (Fig. 3) were Si films doped in evaporation
measured in a low-resistance bridge. These results were
not reconfirmed by ion bombardment.
The high resistivity "n"-type samples, however, re
quired care in that the diode went Ohmic if the mass
transport were pushed too rapidly, if samples stood in
air or if the samples were not boiled in water. We infer
that the high resistivity n-type samples were close
enough to the flat band condition that they became
exceptionally sensitive to surface states.
Some samples were cleaved under Hg with vacuum,
He, and air ambients, and the properties of the resultant
diodes examined. Samples of "p"-type material from
13 George G. Harman and Theodore Higier, J. AppJ. Phys. 33,
2198, 2206 (1962).
14 John W. Beck, J. AppJ. Phys. 33, 2391 (1962).
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to ] IP: 136.165.238.131 On: Fri, 19 Dec 2014 19:28:41EXPERIMENTS ON MERCURY-SILICON SURFACE BARRIERS 1761
2-50 n cm were examined and in only two out of twenty
cases were non-Ohmic contacts formed. Probing with a
mercury drop established that the Ohmic contacts oc
curred as patches at local regions of very high densities
of cleavage steps. Further, samples were cleaved in air
and then immersed in Hg. Again Ohmic contacts were
formed. Samples of "n"-type material showed patches
of very high diffusion potential which were gradually
eroded. The interpretation of these apparently patchy
surfaces on cleaving was beyond the scope of this work
and was not continued since these experiments could
not ignore regions of high densities of cleavage steps.
There are indications15 that cleavage steps may be dif
ferent from smoothly cleaved planes.
DISCUSSION
The above experiments indicate that a stable inter
face can be achieved between mercury and silicon which
is reproducible, is nominally passive at the Hg-Si-air
interface at the edges of samples, and does not change
greatly in times of the order of hours unless fully ex
posed to air. Diodes to both "n" and "p" types of Si
were obtained as predicted by the simple model. The
experimental effects of an intervening film of insulator
could be seen in the capacitance it introduced or in
terms of the strong effect on the breakdown character-
(kTlq)
8
IIIi~I t 6
4
u: 2 I 0
-2 II -4
(kT/ql -20 -10 0 -us-
IkT/qlr-------.-------,
6
u: Or---~---+-±_--_+--~ I -2
-4 Eel-Eel + 100
iO calc.
-us-0.18' 1
FIG. 4. Experimental results expressed as an artificial barrier
potential UB* as in Eq. (5). (a) No free parameters: EB from cal
culation. (b) Using a barrier field reduced by two decades from
EB calculated.
16 F. G. Allen (private communication) has noted wide fluctua
tions in C.P.D. as his polycrystalline Mo probe was moved over
areas of cleavage steps. istic of the diode. Careful passivation of edges of struc
tures was not specifically required.
The predictions for jo as a function of doping are
sketched in Fig. 2, following the predictions of the thick
barrier [Eq. (2)J, thin barrier [Eq. (3)J, and tunneling
models. Smoothing has been used since the onset of
tunneling is not abrupt.s The experimental values of jo
for both n-and p-type material are noted in Fig. 3, and
are lower than the predictions for nondegenerate mate
rial. We note that the saturation reverse current io of a
sample is determined by the active area of the junction
.Ii and two other terms; a pre-exponential term which
reflects the electric field or mean velocity at the interface,
and an exponential term we associate with the barrier
potential by suitable normalization. We write thus:
If the experimental current is equal to prediction then
UB* equals UB from calculations. For simplicity EB is
En(3,u.) in Fig. 4, where (un*-3) is a measure of dis
parity between experiment and prediction. The dis
parity in UB* is not large so the density of surface states
on the Si is probably fewer than 1014/cm2. From the
data of Figs. 3 and 4 at least two interpretations are
possible.
The disparity between experiment and prediction is
only of the order of 5 kT/q and by assuming uB",8 for
"p"-type material and that UB"'O for "n"-type material
the fit is improved. This is more complicated than a
simple patchy interface since patches favor the lowest
(amPI/emil
10
IOT~~~~-u~~~~~~~
(kT/ql -20 (pl-/O o +10 (n)+20 -us-
FIG 5. Saturation reverse currents and directions for Si-Hg
diodes compared with predictions of the models in Fig. 2 using a
barrier field reduced two decades from Ea calculated.
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barrier, whereas here the fit uses a barrier higher than
prediction for both "n"-type and "p"-type Si.
For instance, partially compensating the normal
Shockley surface states for silicon 7 centered at u. = -9.7
would be theoretically reasonable and improves the fit
of expectation and experiment. Alternatively, surface
states can be used in several ways to bring the experi
ment into better agreement. A density n88 of 6X 1011/cm2
charged surface states produces a field at the interface
of E88"-'Q88/ E'"'-' 105 V /cm. This field would be in excess
of or comparable to the calculated barrier field EB(3,u.)
in the intervaF -16~u8::; 17 and therefore strongly in
fluence experimental jo in that interval. For example,
the surface-state distribution could be such that the
experimental barrier field is a fixed fraction of the calcu
lated barrier field. The results of such an approximation
JOURNAL OF APPLIED PHYSICS are noted in Fig. 4(b) and Fig. S. Extrema in surface
state density of about 1012/cm2 would be sufficient to
bring experiment and prediction together.
Experiments performed on the Hg-Si system as a
specific metal-semiconductor contact used the super
position of vacuum work functions with considerable
success. Surface states of nominal density are ap
parently present at the interface but further work
should be done before an attempt is made to specify the
surface-state distribution.
ACKNOWLEDGMENTS
The author gratefully acknowledges the work of
W. C. Vassell on cleaved Si and a continuing interaction
with Dr. R. C. Jaklevic and Dr. J. J. Lambe.
VOLUME 34, NUMBER 6 JUNE 1963
Improved Techniques for Studying the Growth of CdS Crystals
P. D. FOCHS AND B. LUNN
Research Laboratory, Associated Electrical Industries (Woolwich) Ltd" West Road,
Templefields, Harlow, England
(Received 20 December 1962)
Techniques have been developed for studying the nucleation and growth of CdS crystals on silica fibers
and quartz crystals. Results show that at least one type of growth, namely a relatively fast growing (1120)
plate habit, can be nucleated on crystalline quartz, although the actual mechanism is still in doubt. This
growth habit has not so far been observed on silica fibers.
I. INTRODUCTION
IN an attempt to control the growth frQm the vapor
phase of the large thin plate-type (1120) CdS crys
tals reported in a previous communication,! it soon
became clear that some factor other than the usual
experimental variables of temperature, gas flow, etc.,
was very important. Whereas most other types or
habits of CdS crystals were observed to grow by the
subsequent broadening of one or more whiskers, these
particular plates appeared to the unaided eye to grow
as platelets from the very beginning. This suggested
that there were at least two distinct nucleation pro
cesses in the growth of CdS crystals.
The nucleation and growth of other crystals, such as
ice and cadmium, have received much more attention
and it is probable that many of these findings will also
apply to CdS. For example, the habit of ice crystals has
been shown to be dependent on temperature and super
saturation2,3 and the seeding of ice crystals with com-
1 P. D. Fochs, J. Appl. Phys. 31, 1733 (1960).
2 J. Hallett and B. J. Mason, Proc. Roy. Soc. (London) A247,
440 (1958).
~ T, Kobayashi, Phil. Mag. 6, 1363 (1961). pounds such as AgI and PbI2 has been extensively
studied.4
In order to study the nucleation and growth of CdS
crystals in greater detail, techniques using silica fibers
and quartz crystals have been developed. These tech
niques form the main subject matter of this paper.
II. EXPERIMENTAL
The apparatus, shown in Fig. 1, is used for both fiber
and quartz runs, the only difference being in the manner
of suspending the two materials.
The furnace, which can be rotated from the horizontal
to vertical position, is heated by two tubular Kanthal
Al elements; the temperatures of the elements are con
trolled independently by thermocouples embedded in
the windings and connected to stepless saturable reactor
temperature controllers. The furnace is maintained at
the experimental temperatures continuously to ensure a
constant temperature distribution along its length.
The silica U-tube is fitted with a demountable optic
ally flat end window to facilitate visual and photo-
4 G. W. Bryant, J. Hallett, and B. J. Mason, J. Phys. Chern.
Solids 12, 189 (1960).
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1.1713666.pdf | Analytical Formulation of Incremental Electrical Conductivity in
Semiconductors arising from an Accumulation SpaceCharge Layer
VinJang Lee and Donald R. Mason
Citation: Journal of Applied Physics 35, 1557 (1964); doi: 10.1063/1.1713666
View online: http://dx.doi.org/10.1063/1.1713666
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/5?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 164.107.254.56 On: Mon, 08 Dec 2014 18:23:18JOURNAL OF APPLIED PHYSICS VOLUME 35, NUMBER 5 MAY 1964
Analytical Formulation of Incremental Electrical Conductivity in Semiconductors
arising from an Accumulation Space-Charge Layer*
VIN-JANG LEE AND DONALD R. MASON
Department oj Chemical and Metallurgical Engineering, The University oj Michigan, Ann Arbor, Michigan
(Received 3 September 1963)
Analytic expressions are derived which relate the incremental electrical conductivity in an accumulation
layer on a semiconductor to the concentration of surface ions. The theory is checked both by comparing the
predicted results with published graphs which were obtained by numerical integrations, and by evaluating
three separate sets of experimental data on different semiconducting materials. The data of Weller and
Voltz for the effect of oxygen adsorbed on Cr20, do not fit the assumptions of the theory, since their particles
are too small. The data of Smith for oxygen adsorbed on CuO, and the data of Molinari et al. for hydrogen
on ZnO indicate that the observed trends are all in the proper direction and of the proper magnitude to
support this work, but they are not of sufficient precision to support these derivations conclusively. Although
a definitive quantitative experimental check remains to be done, the reasonableness of the derivation has
heen estahlished.
INTRODUCTION
IN this paper, analytic expressions are derived which
relate the incremental electrical conductivity in a
semiconductor to the surface potential and the con
centration of surface charge creating an accumulation
layer.
This problem has been considered and solved by
many authors,1-4 using numerical integrations on digital
computers. However, only Sandomirskii5 has presented
an approximate analytic solution to this problem. By
restricting his analysis to a one-carrier semiconductor,
his results are not applicable to intrinsic materials. By
considering both holes and electrons in this work, addi
tional new relationships are obtained which satisfactorily
explain several previously inexplicable experimental
results. 6, 7
MODEL
The physical model assumed in this derivation is not
restrictive, but is only representative. Assume that a
homogeneous, relatively thin slab of nondegenerate
semiconductor material is oriented as shown in Fig. 1
with electrical contacts being made uniformly on the
x-z planes at the ends of the bar which are perpendicular
to the y axis. The extent of the bar in the y direction is
not important, and the electric field is applied in the y
direction. Furthermore the height W of the semicon-
* Contribution No. 17 from the Semiconductor Materials
Research Laboratory, The College of Engineering; The University
of Michigan, Ann Arbor, Michigan. This work has been supported
by Texas Instruments, Inc., Dallas, Texas.
1 C. G. B. Garrett and W. H. Brattain, Phys. Rev. 99, 376
(1955).
2 R. H. Kingston and S. F. Neustadter, J. Appl. Phys. 26, 718
(1955).
3 R. F. Greene, J. Phys. Chern. Solids 14, 291 (1960).
4 V. O. Mowery, J. App!. Phys. 29, 1753 (1958).
• V. B. Sandomirskii, Bulletin. Acad. Sci. USSR (English trans!.)
21, 211 (1957).
6 A. W. Smith, Actes du Deuxieme Congres International de
Catalyse, Paris, 1960 (Editions Technip, Paris, 1961), Pt. A, pp.
1711-1731.
7 A. Cimino, E. Molinari, F. Cramarossa, and G. Ghersini, J. Catalysis 1, 275 (1962). ductor slab is assumed to be much larger than the half
width L, which in turn is large in comparison with the
thickness of the space-charge region o. That is, W»L»o.
Therefore, the surface charge on the x-y planes at z=O
and z= W can be neglected. Also, the electron mobility
Jl.n and hole mobility Jl.P are assumed to be constant
throughout the space-charge region and equal to the
corresponding carrier mobilities in the bulk. Although
Shrieffer8 and Zemel9 have shown that this is not strictly
valid, it can be regarded as a zero-order approximation.
We shall now proceed to the formulation of the incre
mental conductivity dO's associated with one face of a
p-type semiconductor slab as a function of surface po
tential and surface charge concentration arising from
ionized acceptors [A -Jon that face, expressed as
charged centers/cm2• At a distance x beneath the sur
face, the differential of the incremental surface conduc
tivity is given by
d(dO',) = q[dn(x)Jl.n+ dp (x)Jl.pJdx. (1)
The carrier concentrations are related to the diffusion
potential u(x)= Y(x)/kT, where Boltzmann statistics
---
----y
L ............. X
FIG. 1. Slab of semiconductor of height Wand width 2L, with
electrical contacts on x-z planes and electric field f, in the y
direction.
8 J. R. Schrieffer, Phys. Rev. 97, 641 (1955).
9 J. N. Zemel, Ann. N. Y. Acad. Sci. 101,830 (1963).
1557
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35 0 .....
0
~ 30 E
:t
b;" 25
<l ,: 20 t-:;;
>= <.> 15 :> c z
0
<.> 10
w
<.>
~ 5 0: :>
IJl
-' 0 j'! z w
~ -5 w
0:
<.>
~ J I Y I
u' -u' ___ l.L' I--S
I---~- I---
J
\1~~RANC:H N-BRANCH
A02! J
~&E"'.'" \ I INITIAL
~H20 !} r--
I )) HF SOAK~/
H20 Ozt---
\ I
BOILING
1 TRErMEfrr
8 4 0 -4 -8 -12 -16 -20 -24
SURFACE POTENTIAL, U~, kT UNITS
FIG. 2. Incremental surface conductivity as function of surface
potential on 140-fl-cm p-type silicon (adapted from Buck and
McKim, Ref. to).
are used in the relationships
An(x)=nB exp[u(x)]-nB,
Ap(x) = PB exp[ -u(X)]-PB, (2)
u(x) being defined as positive when the space-charge
region becomes more n type. Since surface acceptors
create a more p-type space-charge region, the accom
panying diffusion potential is negative. Therefore, in
order to deal with positive values of the diffusion, sur
face, and bulk potentials in the equations, primed values
are defined such that u'(x) = -u(x); u.'= -u.;
UB'= -UB.
In the interior of the semiconductor, wherein u(x) = 0,
then
(3)
The incremental surface conductivity at a point x
then can be defined as
d(Aus)= q{JLnnB[exp( -u')-l]
+JLppB[exp(u')-l]}dx. (4)
By defining integral incremental carrier concentra
tions, Garrett and Brattain,! Kingston and Neustadter,2
Greene,3 and Mowery4 have also defined Eq. (4), and
the latter three authors have integrated the equation by
numerical methods.
Mathematical Formation
The general problem can be delineated more clearly
by referring to Fig. 2 adapted from Buck and McKim10
for p-type silicon. As surface acceptors are added, an
accumulation layer is formed and the incremental sur
face conductivity increases. As surface donors are added,
a depletion layer is formed producing a decrease in Au.
to some minimum value corresponding approximately to
10 T. M. Buck and F. S. McKim, J. Electrochem. Soc. 105, 709
(1958). the formation of an intrinsic surface. (The minimum is
shifted by differences in the electron and hole mobilities.)
As an inversion layer is formed the conductivity starts
to increase, but Aus does not become positive until the
gain in conductivity from the inverted region of the
surface layer compensates for the loss of conductivity
arising from the depleted region of the surface layer.
This compensation would not be expected to occur until
the inverted surface potential is greater than twice the
bulk diffusion potential. When the surface is sufficiently
inverted, then Aus becomes positive and appears to be
similar to an accumulation layer. To a good approxima
tion, a highly inverted layer can be approximated as an
accumulation layer.
For accumulation layers the diffusion potential can
be considered to increase smoothly from zero in the in
terior to some value us' on the surface of the semicon
ductor. For a highly inverted layer, the conductivity
can be assumed to be dominated by the inverted region
beyond the mirrored bulk diffusion potential UB, but the
limits of integration for the diffusion potential are not
so apparent. In the remaining derivations, only the
accumulation is considered.
By defining
q(JLnnB+JLppB) = 2UM cosh8, (5)
q(JLppB-JL nnB)=2uM sinhO, (6)
UM= qni CJLnJLp)!, (7)
exp8= (JLpPB/JLnnB)!, (8)
then Eq. (4) can be written as
d(Au.) = 2uM[cosh(8+u')-cosh8]dx. (9)
The total increment of current is the integral of the
incremental current density over the half-width of the
slab. The variable of integration in Eq. (9) can be
changed from x to u'. Lee and Masonll showed that for
accumulation layers and for highly inverted layers,
du'/dx= -4(L mPB)1 sinh(u'/2), (10)
where Lm = 21rq2/ EkT.
Since the origin has been changed in this work from
the surface of the semiconductor to a point inside the
semiconductor, then it is necessary to change the nega
tive sign in Eq. (10) to a positive sign. Equation (10) is
an exact expression for an intrinsic semiconductor, and
is a first-order approximation for acceptors on a p-type
semiconductor (or for donors on an n-type semiconduc
tor). Substitution of Eq. (10) into Eq. (9) then gives
2UM jU8' [cosh(8+u')-coshOJdu'
Aus'''' , (11)
4(LmPB)t 0 sinh (u'/2)
where the appropriate boundary conditions have been
substituted.
II V.-J. Lee and D. R. Mason, J. Appl. Phys. 34, 2660 (1963).
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The integral in Eq. (11) can be simplified with hyper
bolic trigonometric identities and integrated.
8UM /,,,,'/2
Aus= sinh (6+u'/2)d(6+u'/2)
4(LmPB)! 0
2UM --{cosh(6+u.'/2)-cosh(O)}. (12)
(LmPB)!
Note that when u.'=O, then A<T,=O, which is required
from the boundary conditions. Equation (12) can also
be written as
du8=---
(LmPB)!
X{2 coshe sinh2(u//4)+sinhe sinh(u//2)}. (13)
These equations relate the incremental surface con
ductivity of the accumulation layer on the surface of the
semiconductor to the surface potential and the bulk
properties.
Relationships to Surface Ion Concentrations
This equation now can be extended to define the in
cremental electrical conductivity in terms of the surface
charge concentration. Lee and Masonll have also shown
that, in general, the surface potential can be related to
the surface ion concentration by the relationship
sinh (us'/2)= (Lm/PB)![A-]/2. (14)
By using Eqs. (5)-(7) and (14) and additional hyper
bolic transformations, it follows that
(Lm/PB)!(/J.pPB-/J.nnB) } + [A-]. (15)
2 (/J.ppB+/J.nnB)
Using the model defined in Fig. 1 and Eq. (3), the
fractional change in total conductivity is
Luo <To 10
Simpler relationships can be obtained for high surface
coverage and low surface coverage. High Surface Coverage (u.';:::8)
For high surface coverage, when us';::: 8, then
sinh(us'/4)"'cosh(u.'/4) within 4%, and Eq. (13) can
be written
2UM
d<Ts= {(coshe+sinhe) sinh(u.'/2)}. (17)
(LmPB)!
Substitution of Eqs. (5)-(7) gives
Au/"q/J.p[A- ]. (18)
The incremental surface conductivity then is propor
tional to the hole (majority carrier) mobility and the
surface ion concentration.
The fractional change in total conductivity is
For semiconductors, wherein the electron contribution
to the total conductivity is negligible, then Eq. (19)
becomes
(do/uO)= (M/lo)= [A-]/LpB. (20)
this last expression has also been derived by
Sandomirskii.5
Low Surface Coverage (u/5:.1)
For low surface coverage, when u.' 5:.1, then
sinh(u.'/2)"'(u.'/2) within 4%. Similarly,
(sinhu.' /4)2= (u.' /4)2= u.'2/16= (Lm/ pB)[A-]2/16.
Equation (13) gives the incremental surface con
ductivity as
q {(/J.pPB+/J.nnB)(Lm/PB)! du8=- [A-]2
2PB 4
+ (p.pPB-/J.nnB)[A-]t· (21)
The fraction change in total conductivity is
+(~pPB-/J.nnB)[A_J}. (22)
\p.pPB+/J.nnB
This expression and Eq. (16) are significantly different
from that derived by Sandomirskii5 in that his expression
contains no quadratic dependency of the excess con
ductivity on surface ion concentration. This quadratic
dependency would be most apparent on intrinsic or
lightly doped semiconductors.
The above formulations are for an intrinsic or p-type
semiconductor with a negative surface charge. The re-
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o
"o 100 _
E 10 :t
b
<l
-1101 10· 10' 10'0 10" 10'2 10"
SURFACE ION CONCENTRATION, [A-] OR [D+],IONS/em2
FIG. 3. Computed relationships between .::lU8 and surface
ion concentrations on germanium at 300oK.
suIts for an intrinsic or n-type semiconductor with posi
tive surface charge are similar and can be easily written
down by analogy.
From the foregoing analysis it is apparent that two
types of tests can be made. First, the results obtained
from the analytical mathematical solutions can be com
pared with results obtained by numerical integration
methods. Second, the derived equations can be checked
against experimental data.
Numerical Evaluation
By using numerical integration techniques, Mowery!
has presented graphs relating incremental surface con
ductivity t:.rTs (which he calls t:.G) to the electrical con
ductivity and surface potential for germanium and sili
con. An analytic relationship which gives the same
results for accumulation layers is represented by Eq.
(13) above. Mowery has also presented a graphical cor
relation between surface charge and surface potential
for germanium and silicon. These same results are ex
pressed in analytic form for accumulation layers on any
semiconductor by Eq. (14) above. By inserting the
appropriate numbers used by Mowery into the equations
derived above, the results given on his graphical correla
tions have been obtained within the ability to read the
published graphs.
The p branch of the relationship between t:.rT 8 and us'
shown in Fig. 2 can be computed from Eq. (12), and the
agreement appears to be within ±20% from the pub
lished curve. A more precise comparison with a curve
derived for 15000 12·cm p-type silicon showed an aver
age variation of about ±25%. These comparisons then
constitute a satisfactory check on the mathematical
operation.
Inversely, a measurement of t:.rTs now can be used to
ascertain the surface ion concentration. In Fig. 3, Eq.
(15) is plotted showing t:.rT8 as a function of ionized sur
face acceptor concentration [A-] on intrinsic and 1 12· cm p-type germanium. Similar curves are also shown
for ionized surface donor concentration [D+] on intrinsic
and 1 12·cm n-type germanium. The minimum in the
curve for ionized acceptors on intrinsic germanium
arises from the coefficient of the linear term, which is
negative because the hole mobility is lower than the
electron mobility. However, when donor ions are placed
on germanium, there is no minimum in the ArTs VS [D+ ]
curve, as shown on Fig. 3.
Experimental Evaluation
Three sets of experimental data are available6,7.12
which may be used to ascertain the validity of the
theories presented above. Although none of these works
gives a good quantitative check of the theories in all
aspects, the observed trends are semiquantitatively cor
rect. Each set of data is discussed separately.
Smith6 has published data showing changes in con
ductivity in thin films of CuO as a function of oxygen
adsorbed on the surface. However, it is difficult to ascer
tain exactly what the author has done experimentally,
and his theoretical section contains errors. A careful
reading of the manuscript seems to support the con
clusions that his fullv covered surface (not achieved)
would contain about' 2X 1014 oxygen atoms/cm2, that
his reference conductance (go) for film 3 is 44 t-tmho for
the conditions reported in Smith's Fig. 4, and that the
film thickness of film 3 is 0.1 t-t. We have further as
sumed that the reported measurements of conductance
as a function of surface coverage were made at 127°C
(not critical). With these assumptions Smith's data are
plotted in Fig. 4 showing t:.rT / rTo as a function of ad
sorbed oxygen concentration [0].
Smith observes that some oxygen is adsorbed im
mediately which has no influence on the film conduc-
~
rIOOOIr----,-----.----.--,--r----.-
<l
~ >
§ 800
:::>
~
~ 600
;;: t;
~
~ 400
1! z w
~ w a: u
~
fa
N 200 o
o 0
~ 0 2 4 6 8 10x10"
~ ADSORBED OXYGEN ON CuO, [0] (ATOMS/Cm2)
FIG. 4. Normalized incremental electrical conductivity of a
CuO film as a function of surface oxygen concentration (after
Smith).
12 S. W. Weller and S. E. Voltz, Advances in Catalysis (Aca
demic Press Inc., New York, 1957), Vol. 9, pp. 215-223.
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to ] IP: 164.107.254.56 On: Mon, 08 Dec 2014 18:23:18INC REM E N TAL E LEe T RIC ALe 0 ~ Due T I V I T Y I ~ S E M leo N Due TOR S 1561
tance. These atoms presumably form a nonionic surface
dipole layer, or fill covalent surface states, but do not
ionize or form a space-charge region. Above 1.8X 1013
oxygen atoms/cm2 but below 6X 1013 oxygen atoms/cm2,
the fractional conductivity increases linearly with ad
sorbed oxygen concentration as expected from Eq. (20).
By making an approximate computation using an as
yet unpublished adsorption theory,13 and assuming a
shallow surface acceptor level, then it appears that most
of the oxygen atoms on the surface are singly ionized.
With this assumption, the slope of the line in Fig. 4 in
dicates that pB=5X1017 holes/cm 3. Smith indicates
that the conversion from conductance to conductivity
in his system of units requires a factor of 4X 103, There
fore, uo=0.176 mho/cm, from which it follows that the
hole mobility JLp'::::::!.2 cm2/V· sec. This is a reasonable
value, since it is a factor of 10 less than that for CU20.14
By assuming that the dielectric constant of CuO
is 10, then Lm=3XI0-6 cm, the screening length,
Ls= HLmPB)!=4X 10-7 cm so that the film thickness is
equal to about 25 L8•
When the incremental conductivity has increased 3.5
times, the net surface ion concentration contributing to
the space charge is 2X1013 oxygen atoms/cm2, and the
surface potential u.' = 7.8. It appears, therefore, that
these data confirm the assumptions made in this theory
for moderate surface coverages. If the monolayer cover
age is greater than 2X 1Q14 atoms/cm2, the magnitudes
of these conclusions will not be grossly affected.
Cimino et aU have measured changes in the electrical
conductivity of compressed beds of ZnO powder as a
function of surface treatment, hydrogen gas pressure,
time, and temperature. Again, important experimental
details are omitted from their paper, but it appears that
the relationship between these data and our theory is
that
AUs,l-Au.,o= AXhd/4S, (23)
where Au 8,1 is the final surface conductance, mho/
square; Au.,o is the reference surface conductance,
mho/square; AX is the reported conductance change,
mho; h is the distance between platinum contacts, cm;
d is the average particle dimension, cm; and S is the
area of platinum contacts, cm2• We shall assume then
that hiS is approximately unity, that AUs,ois zero or
small relative to AU.,I, and that the average particle
dimension is given by d=6f/pA, where f=roughness
factor, p= density, and A = surface area, cm2/g. For a
surface roughness factor of 2, then d"-'4X 10-4 cm. These
authors also find an initial large adsorption after surface
cleaning which occurs so rapidly that its influence on
surface conductivity cannot be followed. This initial
uptake is a function of temperature and gas pressure.
The changes discussed here are those occurring after
13 V.-J. Lee, Ph.D. thesis, University of Michigan,Ann Arbor,
1962.
14 Pekar, quoted in A. F. loffe, Physics of Semiconductors
(Academic Press Inc., New York, 1960), pp. 178-179. ~
bO
..3 2000 Hydrogen Pressure
>-!::
2: f-a 1600 :J 0 Z 0 a o 650mm Hg[HJ =7.5 x 1013 Atoms'cm2
A 232mm [HJ=6xIO'3
o 83mm [HJ=4.,OI3
o 53mm [Ho] =32,10'3
...J «
<.) 1200 a: f-a w
...J
W
...J 800 « f-z W ::;; o
W
Q: 400 a
~
0
W
N o :J 0 « ::;; 0 Q: 2 4 6 8 10,10'3
0 Z NET H ATOMS ON ZnO, [Hl-[Ho],(ATOMS/cm2)
FIG. 5. Normalized incremental electrical conductivity of com
pressed ZnD powder as a function oi surface oxygen concentration
(after Cimino et at. Ref. 7).
the reference conditions have been established as a re
sult of initial gas uptake.
For ZnO heat treated in vacuum, the reference con
ductance was measured as a function of temperature. At
57°C, Xp::1O-s mho. After subtracting out the initial
amount of hydrogen adsorbed at various pressures,
their data at 57°C showing AX/Xo vs net adsorbed hy
drogen concentration are given in Fig. 5. From the high
coverage theory give in Eq. (20) and adapted to n-type
material, it follows that
AX/Xo=Aa/uo= {[H]/LnB}{[H+]/[H]}. (24)
Since L""-'d/2, then
nB/{[H+ ]j[H]} = 2[HJXo/dAX= 2.3XI016. (25)
Since a reasonable carrier concentration for ZnO is
about 1017 carriers/cm 3, it appears that most of the
hydrogen atoms are ionized. When these data are con
sidered in conjunction with adapted forms of Eqs. (18)
and (24), then
!1xhd
AU8=-=QMn[H]{[H+]/[H]}
S
or
Axd (26)
JLn 5.4XI0-3 cm2/V·sec. (27)
q[HJ{[H+]/[H]}
By also computing JLn from the bulk conductivity rela
tionship for an n-type semiconductor,
xo=uoS/h= (S/h)(nBqJLn) = 10-5 mho. (28)
Using the assumptions and conclusions above it is
found that JLn'::::::!.2. 7X 10-3 cm2/V· sec which agrees
within a factor of 2 of the value found from the incre
mental conductivity theory above. This can be com-
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~
bO
.g 30 ,:
; 25
o z
820
...J
j g 15
....
~ 10 z ....
~ ....
II:
U
~
o
~
~
II:
~ 5 10 15 20 25 30
NET Oz ADSORBEO ON Cr203,q (p. rnoles/gm)
FIG. 6. Normalized incremental electrical conductivity increase
of sintered Cr20a as function of amount of adsorbed oxygen (after
Weller and Voltz, Ref. 12).
pared with a value of about 100 cm2/V· sec for single
crystals of ZnO, so that this indicates a large decrease
in mobility in the compressed particles form of the
material.
These data indicate that Lm=4Xl0-6 cm. At the
beginning of the linear region (SX1013 ions/cm2), Eq.
(14) indicates that u8= 13, which supports the assump
tion of conditions of high surface coverage.
Weller and Voltz12 have published data showing
changes in electrical conductivity of sintered Cr203 as
a function of the concentration of oxygen adsorbed on
the surface. Their data in Fig. 6 show 1:1(1/(10 as a func
tion of net O2 adsorbed in micromoles/ g. In another
publication,15 they reported that the surface area of this
material was equal to 35 m2/g. Using the measured
density of 5.1 (by water immersion) and assuming a
smooth surface, they computed the average particle
diameter to be 335 A. This particle size would increase
linearly as a function of surface roughness, and since a
roughness factor of from 5 to 10 is reasonable, the par
ticle size probably is in the range of 0.1 to 0.3 JJ,. The
data from Fig. 7 show that
(1:1(1/(10) = 4.4X 10-15[OJ ([0-J/[OJ) = [O-J/ Lp B, (29)
where a shape factor in the numerator determined from
15 S. E. Voltz and S. Weller, J. Am. Chern. Soc. 75, 5231 (1953). the particle geometry and the surface-to-volume ratio
has been assumed as unity. This is not unreasonable in
view of other assumptions used in the computation. A
comparison of this result with Eq. (20) shows that PB
is equal to about 1019 holes/cm3•
Weller and Voltz also report an original "conduc
tivity" (10, in the absence of adsorbed oxygen, as
1.4X 10-4 Q. cm. This presumably is for the particulate
solid, and should be somewhat greater for a homo
geneous solid. However, using this value of (10 with the
hole concentration obtained above, it follows that the
hole mobility up~10-3 cm2/V·sec at SOOae. This then
represents a minimum value, and an actual value 10 or
100 times greater is not unreasonable. Independent
mobility measurements do not seem to have been made
on this material. However, Chapman, Griffith, and
Marsh16 reported Hall and conductivity measurements
on 70% Cr20a-30% AbOa which is an n-type semicon
ductor. In this material the free electron concentration
was 3.SX 1013 electrons/cm 3 and the Hall mobility was
2 cm2/V· sec at 442°e. Therefore, it appears that the
hole mobility obtained in this work may be somewhat
low.
Chapman et al. have also shown that from 400° to
500°C, the effective energy gap of Cr203 annealed in
oxygen is 1.22 eV. This apparently represents a deep
acceptor level, since gap values of 2.50 and 2.86 eV
were obtained on materials annealed in hydrogen and
vacuum, respectively.
By assuming that the concentration of acceptor levels
is 3X102°/cm3 (from 7SJJ, moles excess 02/g, and 1.5
excess 0 atoms create one Cr vacancy, which creates
one acceptor level), that holes are created by ionizing
these acceptors, and that the concentration of states
in the valence band is equal to 4.83X 1015 Tt"'1020, then
it follows that the Fermi level is at Ea/2, and PB~1Q16.
holes/cm 3. The agreement with the previously ascer
tained value of 1019 holes/cm 3 is poor. For 1:1(1/(10=0.1,
then the surface potential computed from PB= 1019 and
Lm= 10-6 gives u.'~3.8, which is marginally low.
It is apparent then that these data do not completely
check the theory, although qualitative trends are fol
lowed. The discrepancies may be ascribed to the
small particle size of this material, since it is not
large when compared with the computed screening
length [L8=!(LmPB)!= SX 10-6 cm].
16 P. R. Chapman, R. H. Griffith, and J. D. F. Marsh, Proc.
Royal Soc. (London) A224, 419 (1954).
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1.1729262.pdf | Thermoelectric Figure of Merit in Silver Selenide Powders
A. S. Epstein
Citation: Journal of Applied Physics 34, 3587 (1963); doi: 10.1063/1.1729262
View online: http://dx.doi.org/10.1063/1.1729262
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] IP: 128.59.222.12 On: Sun, 30 Nov 2014 23:05:15JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 12 DECEMBER 1963
Thermoelectric Figure of Merit in Silver Selenide Powders
A. S. EpSTEIN
Central Research Department, Jl;[ onsanto Chemical Company, St. Louis, Missouri
(Received 5 July 1963)
The thermoelectric figure of merit of silver selenide powders prepared by a low-temperature process is
examined. By careful attention to the process variables including the compaction, sintering temperature,
and cooling rate through the {32 phase it is possible to obtain an n-type thermoelement having a figure of
merit of 3X 10-3 per degree at room temperature.
I. INTRODUCTION
SIL VER selenide as a thermoelectric material has
been reported by a number of investigators.I-a The
combination of a low-temperature solid-state phase
transition and relative ease in altering the composition
of silver selenide, as well as the opportunity of preparing
the material in powder form at a low temperature,
makes the material desirable for studying some of the
fabrication parameters and properties of a powder as
they effect the thermoelectric figure of merit. It is of
special interest since previous investigations have been
concerned mainly with silver selenide prepared directly
from the melt and there is little comparative informa
tion with silver selenide prepared directly as a powder
at low temperatures. In this paper, we are primarily
concerned with the effect of a variation in the silveri
selenium ratio, the effect of compaction, the sintering
temperature, and the effect of cooling through phase
transitions4 on the thermoelectric quantity-figure of
merit.
II. MATERIAL PREPARATION
Silver selenide was prepared by a low-temperature
wet process developed by Kulifay5 and Stearns.6 The
process involves a simultaneous reduction of mixed
solutions of the elements in question (silver, selenium)
with a resultant formation and precipitation of the
binary which can then be separated. The resultant
silver selenide powder is blue-black in color and finely
divided. Average particle size is 0.2 iJ.. The powders
prepared were pelletized and treated by procedures de
scribed below for the individual experiments to be re
ported. The compacted pellets were cylindrical-! in.
in diameter with length ranging from 0.2 to 0.4 cm.
1 J. B. Conn and R. C. Taylor, J. Electrochem. Soc. 107, 977
(1960).
2 R. Simon, R. C. Bourke, and E. H. Lougher, "Preparation
and Thermoelectric Properties of {3-Ag 2Se," Special Report
Battelle Memorial Institute, July 1962, Advan. Energy Con
version (to be published).
3 P. Junod, Helv. Phys. Acta 32, 567, 614 (1959); G. Busch,
B. Hilti, and E. Steigmeier, Z. Naturforsch 1611, 627 (1961);
G. Busch and E. Steigmeier, Helv. Phys. Acta 34, 1 (1961).
4 A. Baer, G. Busch, C. Frohlich, and E. Z. Steigmeier, Z.
Natorfursch. 1711, 886 (1962).
• S. M. Kulifay, J. Am. Chern. Soc. 83, 4916 (1961).
6 R. I. Stearns, "Chalcogenide Synthesis Using High Pressure
Hydrogen" Inorg. Chern. (to be published). III. MEASUREMENTS
The electrical and thermal measurements were those
necessary to calculate the figure of merit Z through the
relation Z=S2/Kp where S is the Seebeck coefficient in
units of iJ.v/ deg, p is the electrical resistivity (in Q-cm),
K is the thermal conductivity in W/cm deg.7 Measure
ments'of these parameters were carried out individually,
at room temperature using the apparatus described
below.
A. Seebeck Coefficient
The Seebeck coefficient was measured with the appa
ratus shown in Fig. 1. A range of samples varying in
length from 0.1 to 2.5 cm could be accommodated. Cali
brated Chromel-Alumel wire, Band S gauges No. 36
or No. 40, was used to measure the temperature dif
ference across the sample while a potential difference
between the hot and cold chromel wires served to give
a measure of the potential across the sample caused by
the temperature difference. The thermocouples are im
bedded in tantalum cylinders above and below the
sample. Heaters wrapped around the tantalum cylin
ders act as a source of heat and can be used to provide
fixed temperature differences of desired amount. At
room temperature, a 5° gradient has been used. The
upper tantalum block is nominally thermally insulated
from the external support whereas the lower block may
or may not be, depending on the experimental problem.
If it is not insulated, then the lower tantalum cylinder
rests on a copper sink. Pressure, on the column con
sisting of the tantalum cylinders and sample, is pro
vided by a spring-like arrangement shown at the top of
the figure. The degree of pressure can be adjusted as
can the height of the sample. The thermocouples are
carefully insulated and heat shields and bell jar, as well
as vacuo, can be accommodated. Heaters are supplied
by direct current. The potential difference and tempera
tures are read with an Land N type K-3 potentiometer.
B. Thermal Conductivity
Thermal conductivity has been measured by a com
parative method in apparatus of the type shown in
7 A. F. Ioffe, Semiconductor Thermoelements and Thermoelectric
Cooling, P. I. (Infosearch Ltd., London, 1957).
3587
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HIG~ VACUUM
I
AIR INLET
FIG. 1. Apparatus for measuring Seebeck coefficient.
Ref. 8. The unknown is placed between materials whose
thermal conductivities are known. Two types of stand
ards have been used: Armco iron and quartz. The
Armco iron and quartz were carefully prepared and the
calibrations of Armstrong and Dauphinee9 and LuckslO
for Armco iron and Devyamkova, et al.H for quartz
were followed. Calibrated Chromel-Alumel thermo
couples, Band S gauge No. 36, were used for tempera
ture detection. Temperature and temperature differ
ences were measured with the aid of an Land N type
K-3 potentiometer and carefully insulated switch boxes.
C. Electrical Resistivity
Measurements of de electrical resistivity were made
by a four-in-line probe and an Land N type K-3 po
tentiometer.12 These results were checked by ac re
sistivity measurements using a setup similar to that de
scribed by UreP
8 A. S. Epstein and B. Wildi, Symposium on Electrical Conduc
tivity in Organic Solids, edited by H. Kallmann and M. Silver
(Interscience Publishers, Inc., New York, 1961), Chap. 24, p. 337.
9 L. D. Armstrong and T. M. Dauphinee, Can. J. Res. 25A,
357 (1947).
10 c. F. Lucks, Battelle Memorial Institute (private com
munication).
11 E. D. Devyamkova, A. V. Pemrov, 1. A. Smirnov, and
B. Ya. Moizhes, Fiz. Tverd. Tela. 2, 738 (1960) [English trans!.:
Soviet Phys.-Solid State 2, 681 (1960)].
12 Handbook oj Semiconductor Electronics, edited by L. P. Hunter
(McGraw-Hill Book Company, Inc., New York, 1956), Sec. 20.2.
13 Thermoelectricity: Science and Engineering, edited by R. R.
Heikes and R. W. Ure, Jr. (Interscience Publishers, Inc., New
York, 1961), p. 326. IV. EXPERIMENTAL RESULTS
A. Silver/Selenium Ratio
The ratio of silver to selenium used in the preparation
of the silver selenide powders was varied from 2.00 to
2.30. This was accomplished by the following procedure:
Initially, predetermined amounts of silver and
selenium were carefully weighed and introduced in the
form of solutions in the low-temperature wet processo,6
to give silver selenide. The resultant product was
analyzed by x-ray diffraction for stoichiometric silver
selenide, free silver, and free selenium. The initial
amounts of added silver and selenium were then varied
until only stoichiometric silver selenide was found in
100% yield with no free silver or selenium detectable
by x-ray diffraction procedures. The amounts of silver
and selenium added were then correlated with the
x-ray diffraction findings. Following this, various ratios
of silver to selenium were selected with the aid of x-ray
diffraction, and different silver selenide powders made.
The powders were compacted at 70000 Ib/in.2 into
!-in.-diam pellets, sintered at 200°C for 15 min (cooling
rate ,,-,3 deg/min) and the thermal and electrical prop
erties necessary to determine the figure of merit were
measured at room temperature. The fabrication pro
cedure to prepare the thermoelectric elements was kept
as identical as possible.
The results of this experiment are shown in Fig. 2
where the room temperature figure of merit is plotted
against the silver to selenium ratio as determined above.
Each point associated with the curve represents the
average of a number of samples, the number varying
from as many as ten to as few as two. Wherever pos
sible, however, ten samples have been used as a repre
sentative sampling unit for averaging.
It is noted from Fig. 2 that the highest figure of merit
is found for a silver/selenium ratio of 2.00 and for a
silver/selenium ratio "-' 2.28. The latter case represents
a condition of excess silver added to the composition.
Some difficulty resulted in using the low-temperature
190 2.00 2.10 2.20 2.30 2!10
SILVER/SELENIUM RATIO
FIG. 2. The effect of silver/selenium ratio on the room tempera
ture figure of merit. All the samples have been sintered and pre
pared as described in the text.
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2.6
o
~2.4
f-a:
"'2.2 :::;
"o
1.8'--ri<-Ir;,..--I.--~",*"----';',....-,,----,--,-~ 05 .10 .15 .20 .25 .30
GRAMS/ ADDITION 20
FIG. 3. Variation of figure of merit with compaction using
a small addition technique (see text).
process to obtain a silver/selenium composition con
siderably less than 2.
B. Compaction
The compaction study involved methods of loading
the die (!-in.-diam die) with silver selenide powder and
then applying the final pressure of 70000 Ib/in.2 to
form the pellet. The total amount of material used was
kept constant (2 g) and the sintering process was kept
as identical as possible (200°C for 15 min and cooling
rate ",,3 deg/min). The silver/selenium ratio was 2.28.
The loading or "small addition" method involved
taking a certain fraction of the total amount of mate
rial, placing it in the die, and with the aid of the plunger
applying a hand pressure of about 20-30 lb. The opera
tion was repeated until the total amount of material
allocated for the particular sample was reached. Follow
ing this step, a final pressure of 70 000 Ib/in.2 was
applied and the compacted sample was then subjected
to the routine sintering (200°C for 15 min) and testing
for figure of merit. The results are shown in Fig. 3.
Since the total amount of material used for each
sample was fixed at two grams, the abscissa in Fig. 3 is
actually a measure of the fraction of grams used in each
addition to load the die with a hand pressure of 20 lb.
100 150 200 250 300 50
SINTERING TEMPERATURE ('C)
FIG. 4. The effect of sintering temperature on figure of merit. Smaller and smaller additions mean that less and less
material was used in each addition and consequently
more additions are required. For example, if 2 g were
used in one addition (2 g/addition) then only one addi
tion was necessary, but if 0.05 g/addition were used,
then 40 additions would be required.
From Fig. 3 we note that the figure of merit appears
to increase with decrease of amount of material per
addition. This may possibly be ascribed to (1) insurance
of better particle contact between grains, and (2) the
role of excess silver.
C. Sintering Temperature
The effect of sintering temperature for a fixed time
(15 min) on figure of merit using samples prepared from
silver/selenium ratio> 2 is shown in Fig. 4. It is noted
that for a final compaction pressure of 70000 lb/sq in.,
the optimum figure of merit occurs at a temperature
between 150° and 200°C. Further work revealed that in
order to insure good sintering and to fully assure that
/\----
2 4 6 10 12 14 16 I 20-60
TIME TO coa... IN , PHASE IN MINUTES
FIG. 5. Figure of merit vs cooling rate. The cooling rate is given
in the abscissa as the time to cool from 11 r to 93 °c.
the sintering occurred above the fN phase transition
region a temperature of 200°C was decided on. Attempts
to vary the sinter time with fixed sinter temperature
produced no change in the figure of merit. Sinter times
from 15 min to 1 h were tried. In these experiments the
cooling rate was maintained at ",,3 deg/min. No sig
nificant changes with heating rate in these temperature
ranges were noted and a heating rate of 3 deg/min was
adopted.
D. Cooling Rate
In the sintering operation it was found that variations
in the cooling rate in the range of temperatures from the
supercooled a-/32 phase transition point (117°C) to
93°C (/32-/31 phase transition point4) had a direct in
fluence on the figure of merit of the silver selenide
samples.
A study was conducted to determine the nature of
the variation in figure of merit of the silver selenide
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] IP: 128.59.222.12 On: Sun, 30 Nov 2014 23:05:153590 A. S. EPSTEIN
with the rate of cooling in the above region. Silver
selenide powders14 were compacted using 0.05 g/addition
and a pressure of 70000 Ib/in.2• The disks were heated
at a rate of ",3 deg/min and sintered at 200°C for
15 min. Sample temperatures were monitored using an
Land N recorder which recorded the voltages of the
thermocouples inserted in the sample. The samples
were cooled at various rates. It was noted that the most
important region to control in the cooling curve in
order to directly influence the figure of merit was where
(3 phase grain growth occurred, i.e., the temperature
14 The silver/selenium ratio was > 2.0 and between 2.0 and
2.28. Different batches of silver selenide may have slightly dif
feren t ratios. range mentioned above. The results of our investigation
are shown in Fig. 5. The abscissa in Fig. 5 is actually
plotted in terms of the time to cool from the super
cooled phase transition temperature (117°C) to 93°C.
This is the cooling in the (32 phase and is the region of
(32 phase grain growth. It is noted from Fig. 5 that a
maximum in figure of merit appears to occur around
7 min. Extremely slow or fast cooling rates give figures
of merit at room temperature of "'2XlO~3 per degree.
ACKNOWLEDGMENT
I would like to acknowledge the assistance of J. F.
Caldwell on many phases of the work.
]OUR:-\AL OF APPLIED PHYSICS VOLUME 34. NUMBER 12 DECEMBER 1963
Radiation Effects in GaAs*
L. w. AUKERMAN,t P. W. DAVIS, R. D. GRAFT,t AND T. S. SHILLIDAY
Battelle Memo-rial Institute, CO'lumbus 1, OhiO'
(Received 7 January 1963; in final form 5 July 1963)
Comparison of the annealing properties of radiation-induced conductivity changes in GaAs indicates that
about 10% of the damage created by reactor irradiations anneals in a manner quite similar to but not iden
tical with that created by 1-MeV electrons. The remaining neutron damage requires much higher annealing
temperatures and is presumed to result from complicated damage structures characteristic of highly energetic
knock-on atoms (e.g., disordered regions). Heavy neutron irradiation of either p-or n-type GaAs results in
very high resistivities which appear to be influenced by the presence of slow surface states. Energy levels
resulting from neutron irradiation are estimated to lie at approximately 0.1 and 0.5 eV below the conduction
band and at 0.6 eV above the valence bane\. Moderate irradiation of GaAs by fast neutrons gives rise to a
continuous optical absorption spectrum for wavelengths beyond the fundamental absorption edge, with the
absorption increasing as the inverse square of the wavelength. Similar behavior occurs in CdTe and CdS
after neutron irradiation. Although this effect is not well understood, it is suspected of being associated with
defect structures characteristic of fast neutron bombardment, since heavy bombardment with electrons does
not produce the same behavior.
INTRODUCTION
THE effects of energetic neutrons and of various
types of charged particles on the electrical and
mechanical properties of solids have been studied in
tensively for about 15 years. Of all the semiconducting
materials, germanium and silicon have received by far
the greatest attention. Other semiconductors, including
compounds, have been studied on a more modest scale.
The technological importance of compound semicon
ducting materials has recently increased to the point
where a knowledge of radiation effects in these materials
is of value to the design engineer. This paper is con
cerned with the effects of electron and neutron irradi
ation on the electrical and optical properties of the
compound semiconductor GaAs.
* This work was supported by the Aeronautical Research
Laboratory, U. S. Air Force.
t Present address: Electronic Research Laboratories, Aerospace
Corporation, EI Segundo, California. t Present address: North American Aviation Inc., Columbus,
Ohio, Most of the electron irradiations are discussed in a
previous publication.l The major conclusions of that
paper can be summarized as follows: Irradiation with
i-MeV electrons at room temperature decreases the
carrier density of both n-and p-type samples. The
annealing of lightly irradiated n-type samples can be
described phenomenologically in terms of the sum of two
first-order reactions with activation energies 1.10 and
1.55 eV, respectively. The rate constant for the higher
energy process was Fermi-level dependent, the anneal
ing occurring faster for specimens of greater carrier
density. It WfLS pointed out that this could be under
stood if it is assumed that the motion of the defect
involved required the occupation of an electronic state
which has a low probability of occupation. Since the
annealing was consistent from specimen to specimen, in
spite of rather large variations in impurity content and
dislocation density, it was suggested that intrinsic
defects were involved.
1 L. W. Aukerman and R. D. Graft! Phys. Rev. 127, 1576 (1962).
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1.1733456.pdf | Investigation of Bulk Currents in MetalFree Phthalocyanine Crystals
George H. Heilmeier and George Warfield
Citation: The Journal of Chemical Physics 38, 163 (1963); doi: 10.1063/1.1733456
View online: http://dx.doi.org/10.1063/1.1733456
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129.24.51.181 On: Sat, 29 Nov 2014 15:51:50THE JOURNAL OF CHEMICAL PHYSICS VOLUME 38, NUMBER 1 1 JANUARY 1963
.Investigation of Bulk Currents in Metal-Free Phthalocyanine Crystals*
GEORGE H. HEILMEIERt AND GEORGE WARFIELD
Department of Electrical Engineering, Princeton University, Princeton, New Jersey
(Received 24 August 1962)
The bulk current in many single crystals of metal-free phthalo
cyanine has been found to exhibit Ohmic behavior up to fields of
1()4 V /cm and square-law dependence on voltage for higher fields.
Photocurrents in these crystals were Ohmic over the entire range.
If one interprets these results as space-charge-limited currents
(SCLC), trap densities of 1012 to 1Ol4/cm3 are found from the
temperature behavior of the I-V characteristic. From the transi
tion between Ohmic and square-law regions, the concentration of
EVEN though a study of the dc bulk currents in a
single crystal can yield much information about
the electronic traps in a solid, there are very few
reports in the literature of such studies on molecular
crystals. No doubt this stems from the very low con
ductivities of the molecular crystals and the apparent
difficulty of making good, relatively permanent Ohmic
contacts to these materials. In the absence of Ohmic
contacts, dc measurements are plagued by polarization
effects.
Kleitman, and Fielding and Gutman1 have reported
on some measurements of the current-voltage charac
teristic in crystals of metal-free phthalocyanine. They
found an Ohmic characteristic over the range of fields
covered in their experiment-up to 103 V f cm. It is the
purpose of this paper to report on more extensive
studies of the dc bulk currents in single crystals of
metal-free phthalocyanine.
Metal-free phthalocyanine is a planar molecule
which crystallizes in a base-centered monoclinic lattice
in its most stable polymorph. This molecular crystal
is of interest because it exhibits semiconductor proper
ties. The crystals used in these experiments were grown
by sublimation 1 and were found to contain the typical
metallic impurities listed below in parts per million:
Cu, 100-1000; Na, 1-10; Ca, 1-10; Fe, 6-60; Si, 0.1-1;
Mg, 3-30; AI, < 1. No analysis of the organic impuri
ties is available at this time, although every effort
was made to reduce them by chemical treatment of
the starting material. Typical samples had dimensions
of length, 3 cm; width, 1 mm; and thickness, 0.2 mm.
The measuring apparatus consisted of a Cary vibrat
ing reed electometer and a 2400-V battery power supply
connected in series with the center contact of the
guard-ringed sample. Silver paste was found to make
Ohmic contact for electrons to this material for fields
from 15 to 104 V fcm. The sample was prepared on a free carriers was calculated to be approximately 1()6 to 107/cm3
in fairly good agreement with that found from Hall measurements.
Samples were measured which had both dark and photo currents
which varied as V at low fields, but as Vl.5-1.7 at higher fields.
These observations are interpreted qualitatively in terms of a
model in which a layer of higher resistivity than the bulk extends
from the contact into the bulk.
notched quartz substrate with silver paste guard-ring
electrodes along the C' axis. This is shown in Figs. 1
and 2. The quartz sample holder was mounted on
degreased Teflon supports in a copper box with low
leakage high-voltage connectors to minimize the
effects of leakage currents and pickup. Due to the
extremely high impedance of the crystals in the dark,
it was necessary to wait at least one hour between
changes in voltage to ensure that the true equilibrium
current was attained. Several samples were measured
by this technique.
A typical current-voltage characteristic in air at
room temperature is shown in Fig. 3. It is character
ized by a linear region which extends to a field of
approximately 104 V fcm followed by a region in which
the current rises as the square of the applied voltage.
It was not possible to raise the field to values higher
than 9X 104 V fcm because a discharge occurred be
tween the outside guard rings. The small size of the
crystals made it impossible to keep the rings a satis
factory distance from the edge of the sample to avoid
this handicap. An unsuccessful attempt was made to
select crystals of varying thickness, but crystals suit
able for electroding all seemed to be of the same
thickness. The mean conductivity of the crystals in
the Ohmic region was found to be 1.9X 10-13 (O-cm)-1
at room temperature.
The measurement of the I-V characteristic of sam
ple number 39-3 under illumination by white light for
various light levels is shown in Fig. 4. The photocur
rent is linear over the range of measurement in con
trast to the dark current which shows a departure
from Ohmic to square-law behavior at fields of 104
V fcm. This is interpreted as an indication that the
mobility is not field dependent up to fields of 3.6X 104
V fcm in this crystal. If the non-Ohmic behavior of
the dark current were due to a field-dependent mobil
ity, the photoexcited carriers would also behave in
this manner, because they experience the same field.
* Work supported by RCA Laboratories. The possibility that the field dependence of mobility t Present address: RCA Laboratories, Princeton, New Jersey. ld b . d d
1 P. E. Fielding and F. J. Gutman, J. Chern. Phys. 26, 411 cou e con~entratlOn epen ent, and thus account
(1957). D. Kleitman, U. S. Tech. Servo Rept. PSl11419-1953. for the OhmIC photocurrents seems remote in view
163
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129.24.51.181 On: Sat, 29 Nov 2014 15:51:50164 G. H. HEILMEIER AND G. WARFIELD
GUARD RING ELECTRODE
FIG. 1. Electrode arrangement for measurements normal to the
ab crystal plane (C' axis). The heavily shaded areas are silver
paste electrodes and terminals.
of the low density of free carriers (approximately
106/cm3) measured by the Hall effect.2
Several samples yielded an Ohmic dependence of
current on voltage up to approximately 300 V, and
then the current increased as the 1.5-1. 7 power of the
voltage. The photo currents in these crystals exhibited
a similar 1.5-1. 7 power dependence on voltage above
300 V. This behavior could be indicative of a field
dependent mobility as discussed previously. It is to
be noted that these crystals were from a different run
of the crystal growing furnace than those yielding
Ohmic photocurrents, although, to the best of our
knowledge, they were prepared by the same tech
niques. We return to these observations later.
The Ohmic region of the current-voltage character-
FIG. 2. Photograph of a typical crystal with silver paste center
electrode and surrounding guard-ring electrode (50X).
2 G. H. Heilmeier, G. Warfield, and S. E. Harrison, Phys. Rev.
Letters 8, 309 (1962). istic followed by a square-law dependence of the cur
rent on voltage at higher fields is characteristic of
space-charge-limited currents (SCLC) in solids. The
theory of SCLC has been examined in detail for vari
ous trap distributions in insulators with field-inde
pendent mobilities by Rose3 and Lampert.4 The case
with field-dependent mobility was later treated by
Lampert.D
The band theory models of an insulator carry with
them implicitly the suggestion that free carriers in
jected into either the conduction band or valence band
t-
--/
: x/x
/ /' /
/SLOPE"'Z
x / x
IOI3r-:-_l'----:-t'~I'--:-+I_-+I--'I_tI-L-'.l......L..j IL
100 200 300 400 600 1000
v (VOLTS)
FIG. 3. Typical current-voltage characteristic for metal-free
phthalocyanine single crystal at room temperature.
could move freely through the solid. The magnitude
of the current 10 that could be passed through a
"perfect" insulator would be limited only by the
space charge of the carriers themselves similar to the
SCLC in a vacuum diode. The relation governing this
beha vior in a solid is
(1)
where K is the relative dielectric constant of the solid,
Jl is the mobility of the carriers in cm2/V-sec, V is the
applied voltage in volts, d is the separation between
electrodes in em, and A is the area of the sample in
cm2•
If the Fermi level, which rises when charge injection
occurs, is farther from the bottom of the conduction
band than Et, the trap depth of traps with density
Nt, and moves in a region of the forbidden gap where
the trap density is much less than Nt, the ratio of free
electrons 1t to trapped electrons nt is constant and
3 A. Rose,'Phys. Rev. 97, 1538 (1955).
4 M. A. Lampert, Phys. Rev. 103, 1648 (1956).
5 M. A. Lampert, J. App!. Phys. 29, 1082 (1958).
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129.24.51.181 On: Sat, 29 Nov 2014 15:51:50BULK CURRENTS IN PHTHALOCYANINE CRYSTALS 165
independent of the applied voltage. In this range the
current It is given by
11=810, (2)
8=nlnt= (NcINt)exp( -Et/kT), (3)
where N is the effective density of states in the con
duction band. This relation is extremely useful in the
experimental determin~tion of the trap ?epth and
density. It is seen that If one plots the loganthm of the
ratio of the experimentally deten:nined SCLC to .the
theoretical trap-free SCLC at a gIVen voltage agamst
the reciprocal of the absolute temper~ture, ~he slope
of the curve is equal to -0.43 Etlk, whIle the mtercept
(at l/T=O) yields the ratio No/Nt. Since Nc is usually
known, the trap density can be dete:mined.
The initial current for low fields IS governed by the
intrinsic free carriers in the material and it will be
Ohmic in the presence of Ohmic c?n.tacts. The tran:i
tion to currents governed by the mJected charge wlll
occur at that voltage VI for which the intrinsic current
would equal the SCLC. Thus
noep,( V tid) = (10-13K8p, V t2) / tf3,
V t= 1013noed2(K8)-1, (4)
where no is the concentration of free carriers before
injection. Thus the voltage at which deviation fr?m
Ohmic behavior is observed is a measure of the denSIty
SAMPLE 39-3 ~l
/ x
,.10 15 x / ;l,x
PHOTONS/, x /'
"SEC / x x/
5 x/
FIG. 4. Current-voltage characteristic of crystal 39-3 with
light intensity as a parameter. of the normal volume-generated carriers. The measure
ment of the number of intrinsic carriers using this
transition voltage is perhaps redundant in cases where
SCLC are known to exist, but provides consistency in
the present case where the interpretation is not un
equivocal. It is seen that, at a fixed temperature, this
transition voltage increases as the normal volume
generated carrier concentration (or conductivity) in
creases. Results of this nature are clearly indicated in
Fig. 4 where the critical voltage is varied by shining
light on the phthalocyanine crystal. Indeed, in these
measurements the transition voltage occurred beyond
the range of the measurements.
What has been described would certainly be the
case if the Ohmic and SCLC were in physically sepa
rate paths; however, this is not the case, and the
actual process is evidently different due to the fact
that the two types of currents have different potential
distributions. It seems likely that the process intro
ducing the larger density of free carriers would deter
mine the potential distribution. Hence higher volume
generated carrier densities mean that higher voltages
are necessary before the injected carrier density pre
dominates, although this is not a sharp change in the
current-voltage dependence.
As the voltage is increased, the square-law region
terminates in a steeply rising current which rises to
the trap-free curve as the traps are filled. Under this
interpretation, the trap density Nt can be determined
from the voltage Vtll at which the traps become filled
and the current rises sharply4:
Nt=10-13(KVt/I)/(ed2). (5)
Using this value for Nt, we can determine the trap
depth from Eq. (3). Thus the voltage at the trap
filled limit can provide an independent check of the
trap density and depth.
Caution must be taken in determining whether a
sharp rise in current with voltage is indeed due to the
fIlling of traps. The observation of such phenomena is
common in semiconductor measurements, and any
one of several effects can be used to explain such
results. The effects include field emission from elec
trodes, from traps or from the valence band; poor
contact; barriers, heating effects; and collision ioniza
tion of trapped or valence electrons.
The trap-free SCLC can be observed in the presence
of traps, if transient measurements are made. In this
case, the current which is due to space charge that is
injected into the conduction band is observed before
any of it can be captured by the localized centers.
This trap-free current eventually decays to a steady
state value determined by the density and depth of
the trapping centers.
Figures 5 and 6 illustrate the temperature depend
ence of the current-voltage characteristics of samples
21-5 and 39-3, respectively. The curves are charac
terized by an Ohmic region out to fields of approxi-
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129.24.51.181 On: Sat, 29 Nov 2014 15:51:50166 G. H. HEILMEIER AND G. WARFIELD
169.--___________ -.
SAMPLE 21-5
I I
IOI0f-_______ ----.f--/ __ -:-------l I T=IIS'C
x
/ ! x
/ X/X
~ 10 If I---------,"------,f------I
~ !
I
x/T:59'C
x/
./ I
-13
10 f-~~~~~~-L1_~~~~
10 20 40 60 100 200 400 600 1000
V (VOLTS)
FIG. 5. Current-voltage characteristic of crystal 21-5 with
temperature as a parameter.
mately 104 V /cm at low temperature, and then an
approximate square-law dependence of current on
voltage for higher fields. At higher temperatures, the
transition from Ohmic behavior at lower fields is
contrary to the prediction of the simple theory pro
vided the Fermi energy is a slowly varying function
of temperature over the range of measurement. While
one can postulate several mechanisms which might be
responsible for this behavior, it is evident that further
experiments are necessary to clarify this point.
The theoretical trap-free SCLC 10 for a sample with
known constants can be calculated from Eq. (1). For
sample 21-5 at 400 V (d=2X1O-2 cm, A =3X1ij-4
cm2, K=3, ,u=0.1 cm2/V-sec), Io=1.9X1O-7 A. The
experimentally observed current II for this sample at
400 V and room temperature is 10-12 A. Hence 8=
It/1o=5.4X 10-6.
From Fig. 5, it can be seen that at room tempera
ture the transition from Ohmic to square-law behavior
occurs at approximately 250 V. Thus from Eq. (4),
the normal concentration of free carriers is found to
be no=8.8X106/cm3• This is in approximate agreement
with that found from Hall-effect measurements.2 This
information would be redundant if SCLC were known
definitely to be responsible for the observed behavior.
In the present case, however, it serves as a check on
internal consistency. The density and depth of traps in sample 21-5 are
found from logarithmic plot of Eq. (3) where it is
assumed that N c and N I are not strong functions of
temperature. This is shown in Fig. 7. The intercept
at 1/T=0 yields a ratio of Nc/Nt=4X106• If we
assume that the effective density of states in the con
duction band is of the order of the density of mole
cules (approximately 1021/cm3), the trap density is
found to be Nt=2.SX1014/cm3• The trap depth is
found from the slope of this line, and is approximately
0.8 e V below the bottom of the conduction band.
It is of interest to compute the traps-fill ed-limit
voltage for this sample by using in Eq. (5) the trap
density found from the information in Fig. 7. The
result is Vtjl=SX104 V. This voltage could not be
applied to the sample because of arcing between the
guard rings. Hence the sharp rise in current associated
with the filling of traps could not be observed in this
crystal.
An analysis of the data for sample 39-3, which are
shown in Figs. 6 and 7, indicate a trap depth of 0.8 eV
below the conduction band, and a trap density of 1014/
cm3• The traps-fill ed-limit voltage for this trap density
was computed to be approximately 7X 104, a voltage
higher than could be safely applied to the sample.
The observation of a steeply rising current charac
teristic that could quite possibly be due to the traps
filled limit was observed in samples 21-2, and 21-3.
H /
1",0 SAMPLE 39-3 / I <.>- x
I" /
8;1 r5;' T~ 1650 C .; ~------+--'-'-'----'---i
t#'IQf
ti]1Q.1i..
~/~ '" /
I xl
/ l
IOI3'---j-J'-+'-I-W+--+--4-'4-W4--+--.LLi~
I :2 4 6 10 20 4060 100 200 600 1000
V{VOLTS)
FIG. 6. Current-voltage characteristic of crystal 39-3 with
temperature as a parameter.
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129.24.51.181 On: Sat, 29 Nov 2014 15:51:50BULK CURRENTS IN PHTHALOCYANINE CRYSTALS 167
These data are shown in Fig. 8. The sharp rise occurs
at a voltage of 550 V for 21-2 and would predict a
trap density of Nt=2X1012/cm3 and a trap depth of
0.81 eV. Sample 21-3 did not yield exact square-law
behavior (slope approximately 1.7). This could have
been due to a true field-dependent mobility or experi
mental error, although all crystals were measured in
the same careful manner described previously. The
trap density of sample 21-3 as given by the voltage at
the sharp rise in current is Nt= 1.6XI012/cm3• More
extensive measurements on these samples could not
be made due to the failure of the contacts. However,
the trap depth is not contrary to that found from
thermal data on the other crystals.
o
~I~IO
" CD
-2
10 SAMPLE 21-5 (500V)
2 3
I/T(10-3K-1) 4
FIG. 7. Ratio of experimentally observed SCLC, 1 I, to theoreti
cal trap-free SCLC, 10, as a function of liT for two crystals with
the applied voltages indicated.
The data presented on the bulk currents in metal
free phthalocyanine crystals seem to indicate the
presence of SCLC. A crucial test of this would be the
magnitude of the current which flows in response to
voltage pulses of various heights. These measurements
did not yield meaningful results on these crystals be
cause of the large displacement currents due to the
geometry of the sample and the externally low lifetime
of the injected carriers in this material. The transit
time in this material is of the order of 10-6 sec. Thus
a pulse with a rise time of at least 10-7 sec would be
necessary for the observation of the trap-free current.
Pulses with this rise time would produce displacement
currents, effectively in parallel with the sample, which
are much greater than the expected SCLC. A change
in sample geometry or electrode area is not possible IOflr----------_~~-
~ -12 :s 10 ...
-13 10 L-__ ~~L-~~~ __ L-L-LL~
100 200 300 400 1000
VIVOLTS)
FIG. 8. Current-voltage characteristics for two crystals with
indications that the trap-filled limit has been reached. Both
curves are for room temperature.
with the present crystals, and schemes to cancel this
component of current have not given convincing re
sults to date. Another test for space-charge-limited
currents, which could not be performed due to the
physical nature of the crystals, is the thickness depend
ence of the SCLC. Theoretically, the current density
should be inversely proportional to the cube of the
electrode separation.
The very low trap densities obtained by considering
the bulk currents to be space-charge limited are sur
prising considering the high metallic impurity content
FIG. 9. Model of a crystal ex
hibiting non-Ohmic dark and
photocurrents at high fields, with
an equivalent circuit to describe
the physical behavior of the
model. GUARD
RING
REGION OF LOWER
CON DUCT IVITY
THAN BULK
BULK
OHMIC I=f(v2)
RESISTANCE
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129.24.51.181 On: Sat, 29 Nov 2014 15:51:50168 G. H. HEILMEIER AND G. WARFIELD
of these crystals (approximately 0.1%). It would be
of interest to check these trap densities by other ex
periments, e.g., by thermally stimulated conductivity
measurements or by photoconductive decay measure
ments. However, similar low trap densities have been
reported in iodine by Many6 and in anthracene by
Mark and Helfrich.7 Both find trap densities of the
order of 10l2/cm3 in these molecular crystals.
These low trap densities in the rather impure molec
ular solids contrast sharply with the larger trap densi
ties in the much purer inorganic semiconductors and
insulators. For example, the lowest substantiated8 trap
density in CdS is of the order of 1013/cm3, and this
was observed in very specially prepared crystals.
Thus it would appear that in these molecular solids
the metallic impurities are neither the origin for elec
tronic traps nor the source of the carriers. It is quite
possible, as is the known case for Cu, that all the
metallic impurities are essentially "buried" in the
center of the large molecules and have difficulty in
giving up electrons to or taking electrons from the
conduction band of the solid.
The observation in some crystals of both dark and
photocurrents which varied as the 1.5 to 1.7 power of
the applied voltage remains a perplexing problem which
we discuss only qualitatively. Lampert5 has discussed
space-charge-limited currents in crystals with field
dependent mobilities. The mobility becomes field de
pendent when the carriers, on the average, gain from
the field in one mean free path an energy comparable
with the mean thermal energy. Under these conditions,
the drift velocity no longer varies linearly with applied
field, but in general varies less than linearly with the
field. Hence the currents are sub-Ohmic.
It is difficult to see how this model can be applied to
phthalocyanine since the width of the conduction band
is only of the order of 0.5 kT wide at room temperature.
If the carriers are restricted to this one band they
cannot, on the average in one mean free path, gain
energies from the field comparable to the mean thermal
energy. More to the point, however, to account for the
observed variation of current with voltage, the mobil
ity would have to increase with increasing field. Such
an increase might result from the tunneling of carriers,
6 A. Many, S. Z. Weisz, and M. Simhony, Phys. Rev. 126, 1989
(1962).
7 P. Mark and W. Helfrich, J. App!. Phys. 33, 205 (1962).
8 R. H. Bube and L. A. Barton, RCA Rev. 20, 564 (1959). under the influence of the field, to a higher conduction
band where the effective mass of the carriers is de
creased. However, there is no evidence from optical
absorption data for the existence of such a higher band.
A possible explanation of the observed behavior in
volves a region of higher resistivity than that of the
bulk, extending from the surface into the crystal. This
region could possibly be the result of a metallic phthalo
cyanine or other surface contamination. It acts as a
series resistance, and causes a deviation from the ideal
square-law behavior of the SCLC over more than an
octave of voltage. To explain the photocurrent, we
must require this layer to absorb strongly thus leaving
a region below it virtually unaffected by the incident
radiation. The crystal is then effectively divided into
two regions consisting of a photo conductor and an
insulator in the dark. The insulator is capable of sup
porting SCLC flow when the injected carrier density
in this region is greater than the intrinsic carrier den
sity. The situation can be described by the equivalent
circuit of Fig. 9.
To summarize, the bulk current in many metal-free
phthalocyanine single crystals has been found to ex
hibit Ohmic behavior up to fields of 104 V / cm and
square-law dependence on voltage beyond this value.
Photocurrents in these crystals were Ohmic over the
entire range. If one interprets these results as SCLC,
trap densities of 10l2-10l4/cm3 are found from the
temperature behavior of the I-V characteristic. The
transition between Ohmic and square-law regions was
used to calculate the number of free carriers, and this
value showed good agreement with that found by Hall
measurements. Other samples were measured which
yielded both photo and dark currents which were
Ohmic at low fields and exhibited 1.5-1.7 power de
pendence on voltage at higher fields. These data were
interpreted qualitatively in terms of a model which
consisted of a higher resistivity layer which extended
into the bulk, and behaved like a photoconductor in
series with the bulk material.
ACKNOWLEDGMENTS
The authors wish to thank G. Gottlieb and J.
Corboy for supplying the crystals, Mrs. E. Moonan
for the tedious job of electroding the crystals, and
S. E. Harrison and A. Rose for many halpful discus
sions and for their critical reading of the manuscript.
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1.1726972.pdf | Electronic Structure of Diatomic Molecules. III. A. Hartree—Fock
Wavefunctions and Energy Quantities for N2(X1Σg+) and N2+(X2Σg+,
A2Πu, B2Σu+) Molecular Ions
Paul E. Cade, K. D. Sales, and Arnold C. Wahl
Citation: J. Chem. Phys. 44, 1973 (1966); doi: 10.1063/1.1726972
View online: http://dx.doi.org/10.1063/1.1726972
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v44/i5
Published by the American Institute of Physics.
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Journal Homepage: http://jcp.aip.org/
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Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME H. NUMBER 5 1 MARCH 1966
Electronic Structure of Diatomic Molecules. III. A. Hartree-Fock Wavefunctions
and Energy Quantities for N2(X I~g+) and N2+(X 2~g+, A ~u, B 2~ .. +)
Molecular Ions*
PAUL E. CADE AND K. D. SALEst
Laboratory of Molecular Structure and Spectra, Department of Physics, University of Chicago, Chicago, Illinois
AND
ARNOLD C. WAHL
Chemistry Division, Argonne National Laboratory, Argonne, Illinois, and
Laboratory of Molecular Structure and Spectra, Department of Physics, University of Chicago, Chicago, Illinois
(Received 3 August 1965)
The problem of the convergence of a sequence of Hartree-Fock-Roothaan wavefunctions and energy
values to the true Hartree-Fock results is examined for N2(X 12:.+). This critical study is based on a hier
archy of Hartree--Fock-Roothaan wavefunctions which differ in the size and composition of the expansion
basis set in terms of STO symmetry orbitals. The concluding basis set gives a total Hartree--Fock energy
of -108.9956 hartree and R.(HF) =2.0132 bohr for N2(X 12:.+).
Results are also presented from direct calculations for three states of the N2+ molecular ion (X 22:.+,
A 2IIu, B 21:u+) which are also thought to be very close approximations to the true Hartree--Fock values.
The results give EHF=-108.4079, -108.4320, and -108.2702 hartree and R.(HF) =2.0385, 2.134, and
1.934 bohr for the X 21:.+, A 2IIu, and B 21:u+ states of N2+, respectively. Extensive calculations for various
R values establish that the X 21:.+ and A 2IIu states are reversed in order relative to experiment, a short
coming ascribed to the Hartree--Fock approximation.
I. INTRODUCTION
THIS is the third in a planned series of papers
whose objective is to obtain analytical Hartree
Fock wavefunctions for the ground state and certain
excited states of diatomic molecules, and from these
wavefunctions to calculate many expectation values
of the electronic coordinates, certain molecular proper
ties, transition probabilities, and the electronic charge
and momentum density for each molecular orbital as
well as for the entire molecule. All of these calculations
are made for several internuclear separations. The
longer range objective of this series is to provide a
solid and extensive platform from which to begin a
critical re-examination of the theory of the electronic
structure of small molecules. This extensive platform
will consist of calculations for homologous and iso
electronic series of diatomic molecules to approximately
the same level of accuracy.
The first paper in this series ' dealt with the Hartree
Fock-Roothaan equations for diatomic molecules, es
pecially for homonuclear diatou{ic molecules, and dis
cusses at length the organization of the computer
program to calculate efficiently the supermatrix ele
ments and perform the SCF procedure. Results are
also presented by Wah)! for F2(X I~q+) as a prototype
molecule. The second paper of this series2 deals pri-
* Research reported in this publication was supported by
Advanced Research Projects Agency through the U.S. Army
Research Office (Durham), under Contract DA-ll-022-0RD-
3119, and by a grant from the National Science Foundation,
NSF GP 28. t Present address: Department of Chemistry, Queen Mary
College, London, England.
1 A. C. Wahl, J. CheIlL Phys. 41,2600 (1964).
2 W. Huo, J. Chem. Phys. 43,624 (1965). marily with extensive calculations on CO(X I~+) and
BF(X I~+) and also discusses the partner hetero
nuclear diatomic SCF computer program with empha
sis on any differences from the description given
by Wahl.1 Further members of this series will in
clude results for Li2(X I~g+), Be2(I~q+), B2(X 3~g-)
C2(a I~g+, A' 3~g-), and 02(X 3~q-, a l.6g, b I~g+) to
complete the study of the first-row homonuclear di
atomic molecules; results for N a2(X I~g+), CI2(X I~g+),
and eventually all ground configuration states for all
second-row homonuclear diatomic molecules; results
for the first-and second-row hydride molecules, AH;
results for first-row oxides, AO, and fluorides, AF, a
number of other important heteronuclear diatomic
molecules [including BN(a'~+), NO(X2TI, A2~+),
CN(X 2~+, A 2TI), LiCI(X I~+)], and a number of
other small heteronuclear diatomic molecules some of
which have not been experimentally identified.3 As a
matter of course each presentation will usually include
calculations for several positive and/or negative ions
of the parent system and potential curves for all
ground configuration states as well.
The objectives of the present paper are:
(i) To report the Hartree-Fock-Roothaan wave
functions for N2(X I~g+) and N2+(X 2~g+, A 2TI",
B 2~" +), molecules and molecule ions for a range of
internuclear distances. A number of expectation val
ues of electronic coordinates, certain molecular prop
erties, and charge-density contours of the various mo-
3 The scope of this study will eventually include certain excited
states of a given diatomic system and thus the specific state (s)
involved will have to be clearly designated. It is therefore conven
ient to employ the spectroscopic designation for the state in
question where it is available and applicable.
1973
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1974 CADE, SALES, AND WAHL
lecular orbitals are given in subsequent members of
this series.
(ii) To discuss the study of the convergence of the
expansion method4 toward limiting behavior with re
spect to the total energy and certain molecular proper
ties. This limit is the Hartree-Fock result. This study
is essentially a problem in numencal analysis, but since
the results in this series are intended to represent the
Hartree-Fock results as closely as practically possible,
it is a rather important item to be considered.
The first major attempt to calculate the electronic
density of N2 (and also F2), after the advent of quan
tum mechanics, was the effort by Hund in 1932 using
the Thomas-Fermi approximation. The valence bond,
or Heitler-London-Slater-Pauling, method could sug
gest a reasonable interpretation for the nitrogen mole
cule, but since at least three "bonding pairs" were
involved, ab initio valence bond calculations for the
full six (or 10) valence-shell electrons are very diffi
cult. Kopineck6 did, however, make certain valence
bond calculations for both the six-and lO-electron
cases following the methods of Hellmann. The most
extended valence bond (or atomic orbital) method cal
culations seem to be the unpublished results of T. Itoh,6
which yielded a dissociation energy of 5.03 eV. Huber
and Thorsen7 have also made certain valence bond
calculations for the X l1;g+-A 31;,.+ excitation energy
for N2•
Molecular orbital calculations have proved much
more manageable and there has been a succession of
LCAO-MO-SCF calculations for N2• Scherrs made
calculations in which the molecular orbitals were ap
proximated by Is, 2s, 2pu, and 2fJ1r± STO orbitals on
each nucleus, using the Slater orbital exponents with
no optimization of these nonlinear parameters. Ransil,9
and Fraga and RansillO determined LCAO-MO-SCF
wavefunctions for N2 at R.(exptl), and for a range of
internuclear distances, again using the minimal valence
shell Slater-type expansion functions, but the orbital
exponents were optimized at R.( exptl). Fraga and
Ransilll also made limited configuration interaction
calculations based on the minimal basis set just men
tioned. Clementp2 made a small improvement by add
ing a 3d7r STO to the minimal set and optimizing the
4 The notation given in this section follows that employed in
C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951); 32, 179
(1960); C. C. J. Roothaan and P. S. Bagus, in Met/wds in Com
putational Physics, edited by B. Adler, S. Fernbach, and M.
Rotenberg (Academic Press Inc., New York, 1963), p. 47ff.
These three papers are referred to as T1, T2, and T3, respectively.
6 H. Kopineck, Z. Naturforsch. 7a, 22, 314 (1952).
6 R. S. Mulliken, in lectures on "Problems Concerning the
Electronic Structure of Diatomic Molecules," Autumn quarter,
1961, University of Chicago, Chicago, Illinois.
7 L. M. Huber and W. Thorsen, J. Chern. Phys. 41, 1829
(1964).
8 C. W. Scherr, J. Chern. Phys. 23,569 (1955).
lB. J. Ransil,Rev. Mod. Phys. 32, 245 (1960).
10 B. J. Ransil and S. Fraga, J. Chern. Phys. 35, 669 (1961).
11 S. Fraga and B. J. Ransil, J. Chern. Phys. 36, 1127 (1962).
12 E. Clementi, Gazz. Chim. Ital. 91, 722 (1961). orbital exponent of the new function. Richardson's
"double-f" expansion set,13 in which the molecular
orbitals were approximated by one Is, two 2s, two
2pu, and two 2fJ1r STO functions on each nucleus, was
the first extended basis set LCAO-MO-SCF calcula
tions for N2• He was able to perform only crude opti
mization of the nonlinear variational parameters, how
ever. Richardson also made similar calculations for
certain N2+ states. Only recently, Nesbet14 has pub
lished a much more extended LCAO-MO-SCF calcula
tion for N2• This last calculation and parallel ones for
several other molecules are near the depth of study
intended in this series.
In the present endeavor, no critical comparison of
these various LCAO-MQ-SCF results for N2 with the
results here given is attempted. This is chiefly because
the present calculations contain a hierachy of different
results, certain of which are comparable to the earlier
ones, and these seem more suitable for the comparisons
discussed.
II. HARTREE-FOCK AND
HARTREE-FOCK-ROOTHAAN EQUATIONS
In the Hartree-Fock approximation the electronic
wavefunction is written
IPIl= [(2N) !]t(4)la) 11 (4)J/3) 2 ••• (4)Na)2N-l(4>Nf3)21IJ,
(11.1)
for the system containing 2N electrons with the closed
shell configuration specified by the set of N ortho
normal space orbitals 4>;, and having the state symme
try symbol 0(11;+ or 11:g+).4 A single Slater determinant
suffices for a closed-shell configuration, for a closed
shell configuration plus one extra electron in an open
shell or a closed-shell configuration with one hole to
form an open shell, and for those open-shell configura
tions which arise if all open-shell 4>. have the same spin
function (that is, states for M 8= ± S). If there are
several electrons in a single open shell, or electrons
distributed in open shells of different symmetry, the
wavefunctions for the open-shell configuration of defi
nite state symmetry 0 can be written in the form,
IPIl= ECKIP(Il)K, (11.2)
K
where the IP(Il)K are single Slater determinants of the
form in Eq. (11.1) ordered by the superscript K. This
index K identifies the possible various choices from
the degenerate members of molecular spin orbitals
available to the electrons in the incomplete shells,15
18 J. W. Richardson, J. Chern. Phys. 35,1829 (1961).
14 R. K. Nesbet, J. Chern. Phys. 40, 3619 (1964).
16 Numerous examples of wavefunctions of the form of (11.2)
occur in the literature and basic texts. A convenient collection
of such forms for D",; or C",y symmetry molecules is given by
S. Fraga and B. J. Ransil in "Formulae for the Evaluation of
Electronic Energies in the LCAO-M0-SCF Approximation,"
Tech. Rept. 1961, p. 236ff, Laboratory of Molecular Structure
and Spectra, University of Chicago, Chicago, Illinois.
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I r. A 1975
The CK in Eq. (II.2) are determined entirely by sym
metry requirements, that is, by combining the <I>(O)K
to form <1>0 of a definite state symmetry, n. It is quite
proper to refer to both expressions (ILl) and (II.2)
as "single configurations" since in both cases a unique
set of space orbitals, tPi, is implied. In open-shell cases
this set of space orbitals, tPi, may give rise to several
states (n) in which only the set of coefficients CK will
differ (for example, the X 3~o-, a l~g, and b l~g+ states
of O2).
If the molecular orbitals, tPi, are expanded in terms
of certain known functions,4
tPiXa= LXpXaCiXp,
p (II.3)
the problem of calculating the molecular wavefunction
is reduced to finding the linear expansion coefficients,
C'AP' and an adequate expansion in terms of the expan
sion functions, XpXa' Following Roothaan and Bagus1•4
in T3 the energy expression for a molecule can be
written
Eo=HtDT+tDtT<PDT-tDotQDo+ L(Z"Z~/ Ra~),
a>p
(II.4)
where <P and Q are now supermatrices with elements,1·4
and
The supermatrix elements are thus constructed from
electronic integrals involving the expansion functions
XpX", and the explicit form of the matrices and super
matrices is given by Wahl.l The Hartree-Fock-Rooth
aan equations are defined by4
FCC=ESCj. and
FOC=ESC (II.6)
This series of papers deals with the solution of Eqs. (11.6)
for diatomic molecules and specifically for the lowest state
of any symmetry type arising from closed-shell configura
tions or configurations which have several open shells of
different symmetry (for example fT, 7I"n, fT7I"n, and so
forth) .
In choosing to solve the Eqs. (11.6), one has thus
bypassed the direct numerical solution of the Hartree
Fock coupled integrodifferential equations for diatomic
molecules and substituted the relatively well developed
machinery of matrix methods and calculation of elec
tronic integrals over analytic functions. In addition
one has also assumed the task of exploring the con
vergence of expansions given by Eq. (II.3) such that
true Hartree-Fock wavefunctions are obtained.
The expansion method has been referred to in the
literature as the "Roothaan method," "Roothaan
scheme," "extended basis set SCF," and by various other names. We would like to propose that one refer
instead to the expansion method in terms of the Har
tree-Fock-Roothaan equations as given by Eq. (II.6).
This would thus imply an expansion form as systema
tized by Roothaan4 in T1, indicate the adoption of
the open-shell formalism given by Roothaan and Bagus4
in T3, and, of course, imply the iterative solution to
threshold invariance in the CrAP coefficients. This sug
gestion is primarily motivated by an attempt to clarify
the term "Self-Consistent-Field." The solution of the
Hartree-Fock equations yields the Hartree-Fock wave
function which is identical with the Self-Consistent
Field wavefunction, that is, only if the tPi are exactly
determined is the true self-consistency of the inde
pendent particle model really achieved. In contrast,
one may have a hierachy of the Hartree-Fock-Roothaan
equa tions and H artree-F ock-Roothaan wa vefunctions
where the members of the hierachy arise from different
size or kinds of expansions of the form of Eq. (11.6).
The hierachy of Hartree-Fock-Roothaan wavefunctions
thus gradually approaches the Hartree-Fock-wavefunc
lion in a limiting manner. One must remember that in
speaking of LCAO-MO-SCF wavefunctions, or, as we
propose, of Hartree-Fock-Roothaan (HFR) wavefunc
tions, that the self-consistency merely means a certain
degree of invariance in the Cxp coefficients and does
not mean the true self-consistency of the independent
particle model, except as the expansion converges to
the true Hartree-Fock orbitals tPi.
A remaining practical point may be noted before
concluding this section. The basic limitation on how
large an expansion basis set may be employed depends
on the total number of unique <PXpQ.l'r8 and QXpq.l'rs
supermatrix elements generated. The present versions
of the homonuclearl and heteronuclear2 diatomic SCF
programs perform the contraction of the supermatrices
such that the entire <P or Q supermatrix must occupy
the rapid access memory of the computer (the 32K
core for an IBM 7094). Thus approximately 20000
total supermatrix elements are permitted. In terms of
the total number of expansion functions XpX" this re
quires that
Ltnx(nx+1) ~Nmax, (II.7)
x
where nx is the number of expansion functions of sym
metry A (fTg, fTu, 71"", 7I"g, etc., for homonuclear diatomic
molecules and fT, 71", etc., for heteronuclear diatomic
molecules). The present limit for N max is 144 for homo
nuclear diatomic molecules and 172 for heteronuclear
diatomic molecules.
III. DETERMINATION OF THE BEST
HARTREE-FOCK-ROOTHAAN WA VEFUNCTION
The most time-consuming and tedious task leading
to the results presented here was the exhaustive study
to determine the best basis set expansion, the optimal
nonlinear parameters (orbital exponents) of the ex-
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1976 CADE, SALES, AND WAHL
pansion functions, the behavior towards convergence,
and the effects of these various characteristics of the
calculation on certain expectation values and molecular
properties of the nitrogen molecule, to be discussed
subsequently. This study on nitrogen and that for
Li2,16 were particularly exhaustive in order to provide
a useful guide for similar studies on other first-row
homonuclear diatomic molecules. This section outlines
the methods employed and concludes by presenting
the final choice of the Hartree-Fock-Roothaan wave
function for nitrogen. Evidence is presented in support
of the belief that this final result is a very close ap
proximation to the true Hartree-Fock wavefunction.
A. Basis Expansion Functions and Molecular
Parameters
The ground electronic configuration of nitrogen in
terms of molecular orbitals as established by molecular
spectroscopy is written
N2(X l};g+) loi1u u22u g22uu2bru43oi,
where the molecular orbitals are in the order of de
creasing orbital energy, the outer three being obtained
from the analysis of the relative positions of the several
Rydberg series ionization limits of N2(X l};g+). The
use of the simply numbered symbols lUg, 2u g, 3u g, 1u u,
br", ... , is more suitable when dealing with extended
basis set expansion for the molecular orbitals. The
familiar symbols, ug1s, ug2s, ug2p, uu1s, 7ru2p, ••• ,
which denote the parentage of the molecular orbitals
with respect to the separated atoms limit, is employed
here instead to denote the individual STO symmetry
basis functions, as is explained below. Another useful
notation suggested by Mulliken which consists in writ
ing only the outer-shell molecular orbitals as ZU, yu,
xu, W7r, V7r, ••• , is most useful when dealing with
homologous series of molecules involving different num
bers of closed inner shells of the separated atoms and
for isoelectronic series of homonuclear and heteronuclear
diatomic molecules.
The general expansion form of the various molecular
orbitals was given in Eq. (11.3). The expansion func
tions XpAa employed in this series of papers for homo
nuclear diatomic molecules are given byl
XpAa= 2--t[XnA.ZA.mA,,(r a) + UAXnA.IA.mA" (rb) J, (111.1)
or explicitly
XpAa= (2.1AP) nAp+i[2 (2nAP) IJ-1
X {ranAp-l exp( -.IApra) YZA.mA,,(8a, cp)
+uArbnAp-1 exp( -.IAprb) Y lA.mA" (Ih, cp) }. (111.2)
Detailed definition of the spherical harmonics Yz ... (O, cp)
and the coordinate systems employed are given by
Wahl, Cade, and Roothaan.17
16 P. E. Carle, K. D. Sales, and A. C. Wahl, "Electronic Structure
of Diatomic Molecules. IV. Li2(X 1~.+) and Lh+(2~.+, 2IIu)
Ions", J. Chern. Phys. (to be published).
17 A. C. Wahl, P. E. Cade, and C. C. J. Roothaan, J. Chern.
Phys. 41,2578 (1964). A simple shorthand procedure is employed in dis
cussing the expansion basis sets. This is most easily
conveyed by a few examples: uo1s, uo2p, and 11,3d are
defined
U gls= 2--t[X18(ra) +XIB(rb) J,
11 g2p=2--t[X2p~(ra) +X2p~(rb) J,
U g3d= 2--t[x3d~(ra) +X3d~(rb) J, (1II.3)
and analogous symbols for corresponding U u symmetry
basis functions except with minus signs. The 7ru, 7ry,
and higher type STO symmetry orbitals are similarly
abbreviated by 7ru2p, 7rg2p, 7ru3d, 7rg3d, where, for ex
ample,
_j2--t[X2P .. (ra) +X2pr(rb) J]
7ru2p= ,
2--t[x2p;;(ra) +X2pi' (rb) J (111.4)
that is, with inclusion of the two degenerate members
of subspecies a. The distinction between basis func
tions having the same symbol but different orbital
exponents is made by primes. The basis set composition
refers to the specific makeup (that is the set of nAp, lAP,
.lAP of the expansion functions for each symmetry).
The restriction of these calculations to the employ
ment of symmetry-adapted expansion functions XpAa
and hence symmetry-adapted molecular orbitals CPiAa
is not necessary and there have been doubts expressed
by Lowdin18 that such a choice really represents an
absolute minimum even to Hartree-Fock approxima
tion. The employment of symmetry-adapted molecular
orbitals does, however, provide considerable simplifica
tion in dealing with the large number of supermatrix
elements, as is discussed by Wahl.1 Even if it is useful
to relax the restriction that the molecular orbitals,
CPiAa, be symmetry adapted, it is important to realize
that in so doing the expansion basis set size would
have to be considerably reduced for molecules as small
even as nitrogen, especially if any optimization of or
bital exponents is desired, not to mention other diffi
culties.
It should be noted that basis set compositions for
Ug-and uu-type molecular orbitals CP,Aa are completely
independent. Thus, the corresponding I1g and 11" basis
functions (if indeed they are corresponding sets) do not
have to have the same orbital exponents as was the
case in the calculations by Ransil and Fraga. The
usefulness of this extra degree of functional freedom
was first pointed out by Huzinaga,I9 and Phillipson
and Mulliken,20 and is discussed in Part B of this
section.
B. General Principles in Synthesizing the Expansion
Basis Set; Illustration for N2(X l};g+)
In these calculations on nitrogen and in general in
seeking solutions to the Hartree-Fock-Roothaan equa-
lS P.-O. Liiwdin, Rev. Mod. Phys. 35, 496 (1963).
uS. Huzinaga, Proc. Theoret. Phys. (Kyoto) 19,125 (1957).
20 P. E. Phillipson and R. S. Mulliken, J. Chern. Phys. 28,
1248 (1958).
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE L E C T RON I CST R U C T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1977
tions, the problem of synthesizing the expansion basis
set may be conveniently considered in the following
terms21•22:
First, how is the basis set composition decided? That
is, how many STO symmetry basis functions are needed
to adequately represent each molecular orbital symme
try type and what kind of STO symmetry orbitals
with respect to the np, Ip, and r p values? Associated
with this question is the problem of deciding what
role the related atomic Hartree-Fock-Roothaan wave
functions should play in contributing to representing
at least the inner-shell molecular orbitals.
Second, what sequence, combination, and extent of
optimization of the orbital exponents, r p, of the STO
symmetry basis functions chosen to form the basis set
composition is necessary and useful?
Third, how close is the best Hartree-Fock-Roothaan
wavefunction to the true Hartree-Fock wavefunction
and how is this to be measured?
Fourth, what role can the various molecular proper
ties and expectation values playas criteria of the
convergence towards Hartree-Fock results? The con
vergence of the total energy may mask serious deficien
cies in the wavefunction and the problem is to discover
how to measure this and correct for it. This aspect will
be more fully considered for N2 and N2+ molecular ions
in Paper III.B.
These four items stress the purely numerical and
"experimental" nature of the problem under consider
ation. This is in contrast to many previous calculations
using small basis sets chosen by means of physical or
chemical intuition and computational feasibility. There
is good reason to retain these ideas if possible, but
when the basis set is large, physical or chemical in
tuition is of less use in constructing the basis set.
The problem, especially for the outer molecular orbit
als, becomes more properly one of a purely mathematical
nature, namely to represent the molecular orbitals with
as few terms as possible.
Previous calculations of the kind considered in this
series have also probed the problem of choosing the
basis set composition. These calculations include the
results of Richardson13 for N2, Lefebvre-Brion, Moser,
and Nesbet23 for CO, Clementi24 and Nesbet26 for HF,
Manneback26 for Li2, Kahalas and Nesbet27 for LiH,
21 The basic problem considered in this section has also been
dealt with extensively by Bagus, Gilbert, Roothaan, and Cohen,
for the first-row atoms. While the general procedures to obtain
very accurate approximations to the Hartree-Fock wavefunction
are basically similar in atoms and diatomic molecules, practical
difficulties prevent the more exhaustive methods used for atoms
from being used for diatomic molecules.
22 P. S. Bagus, T. L. Gilbert, C. C. J. Roothaan, and H. D.
Cohen, "Analytic Self-Consistent Field Functions for First-Row
Atoms," Phys. Rev. (to be published). Referred to as BGRC
subsequently.
23 H. Lefebvre, C. Moser, and R. K. Nesbet, J. Chern. Phys.
35,1702 (1961).
24 E. Clementi, J. Chern. Phys. 36, 33 (1962).
26R. K. Nesbet, J. Chern. Phys. 36,1518 (1962).
26 C. Manneback, Physica 29,769 (1963).
27 S. L. Kahalas and R. K. Nesbet, J. Chern. Phys. 39, 529
(1963) • -108.6
E l \ Nl'S)· Nl'S) lHartree- Fock) -IOB.7
-108.8 • L 9
\ L \ \
\ I
~"' .. x. •• .x ·LOI.IOI
-lOB.
-109. 0
EXPERIMENT .
ABCDEFGHI JKLMNOPQRS
Proor.ss in Bali, Set Synthesis -
FIG. 1. Synthesis of expansion basis set-improvement in total
energy for N2(X I~Q+)' R=2.068 bohr; E in hartrees .• -. is
Basis Set 1 and X - - - - X Basis Set 2.
and more recently the results of McLean28 for LiF,
Yoshimine29 for BeO, Nesbet14 for the 14-electron sys
tems, N2, CO, and BF, and finally the HCI results of
Nesbet.30 In general these efforts, all for extended basis
sets, seem to suffer from two defects. First, practical
considerations have forced the authors to restrict, often
severly, the size of the expansion basis set, and the
extent of optimization of orbital exponents. The second
defect is associated with the fact that these are, with
one exception, studies of individual molecules and do
not permit the study of whole homologous and/or
isoelectronic series. The present investigations gener
ously relax these constraints.
Two different and relatively independent schemes
were employed in synthesizing the expansion basis set
for N2(X l~g+) at R.= 2.068 bohr. The first scheme
consisted of building up the N2 wavefunction completely
from scratch. The second scheme considered the N2
molecule to be formed by two distorted N(4S) atoms
separated by Re, and the Hartree-Fock-Roothaan
wavefunctions for the N(4S) atom playa key role. In
the second scheme the BGRC calculations for atoms
are employed.
A summary31 of the progress of these two schemes
to build up the expansion basis set, expressed in terms
of certain energy quantities, is given in Figs. 1, 2, 3,
and 4, and in Tables I and II. In Table I, the total
energy, kinetic and potential energy, virial, and orbital
28 A. D. McLean, J. Chern. Phys. 39, 2653 (1963).
29 M. Yoshimine, J. Chern. Phys. 40,2970 (1964).
30 R. K. Nesbet, J. Chern. Phys. 41, 100 (1964).
al More details of these results are given in Tech. Rept, 1965,
pp. 130ft Laboratory of Molecular Structure and Spectra, Uni
versity of Chicago, Chicago, Illinois.
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsTABLE I. Synthesis of basis sets 1 and 2-energy quantities-N 2(X l~a+), R=2.068 bohr. -\0
~ 00
Basis seta Symmetry STO, XPA«
Basis set compositionb EC T V -V/T Eta,. -=1.,. E2.. E2.. E3v. t)7. E3v.
Set lA lTals, O'a2s, O'a2Pi O'uls, O'u2s, O'u2Pi -108.6336 108.7306 -217.3642 1.9991 -15.6471 -15.6442 -1.4211 -0.7137 -0.5555 -0.5454 1.2262
(BMMO g=u) 1ru2Pi (g=u)
Set 1B O'a1s, lTa2s, 1T.2Pi O'uls, lTu2s, lTu2p; --108.6459 108.7541 -217.4000 1.9990 -15.6436 -15.6405 -1.4262 -0.7185 -0.5499 -0.5435 1.3608
(BMMOg~u) 1ru2p; (g¢u)
Set 1C
(4X4X1) Set IB+O'.ls' and IT.Js' -108.6908 108.7386 -217.4294 1.9996 -15.6573 -15.6539 -1.4307 -0.7150 -0.5434 -0.5397 1.4020
Set 1D
(5X5Xl) Set 1C+0'.2s' and lTu2s' -108.7713 108.5154 -217.2867 2.0024 -15.7089 -15.7056 -1.4718 -0.7477 -0.5932 -0.5691 0.9002
Set IE Set 1D+1ru2p' -108.8691 108.7975 -217.6666 2.0007 -15.7116 -15.7082 -1.5069 -0.7702 -0.6193 -0.6172 0.8848
(5X5X2)
Set IF Set lE+O'q2p' -108.8896 108.8933 -217.7829 2.0000 -15.7140 -15.7104 -1.5257 -0.7770 -0.6291 -0.6272 0.8807 (')
(6X5X2) >
t:1
Set 1G Set IF+lTu2p' -108.8914 108.9084 -217.7999 1.9998 -15.7157 -15.7122 -1.5293 -0.7780 -0.6309 -0.6298 0.8417 l'1
(6X6X2) ~
Ul
Set 1H Set IG+1r.2p" -108.8926 108.9286 -217.8212 1.9997 -15.7108 -15.7072 -1.5249 -0.7750 -0.6275 -0.6277 0.8446 >
(6X6X3) t'"'
l'1
Set 1I Set 1H +1T.2p" -108.8992 108.9698 -217.8691 1.9994 -15.7070 -15.7034 -1.5235 -0.7739 -0.6293 -0.6257 0.8468 Ul
(7X6X3) ~
Set IJ Set 1I+O'a2s" -108.9022 108.8488 -217.7510 2.0005 -15.7115 -15.7078 -1.5246 -0.7738 -0.6294 --0.6256 0.8474 >
Z (8X6X3) t:1
Set 1K Set 11+0'.3$ --108.9022 108.8522 -217.7544 2.0004 -15.7113 -15.7076 -1.5246 -0.7738 -0.6294 -0.6255 0.8474
::i!l (8X6X3) >
Set 1L Set 1K+IT.3d -108.9298 108.8556 -217.7854 2.0007 -15.6988 -15.6952 -1.5119 -0.7742 -0.6369 -0.6144 0.8596 ::tt
(9X6X3) t'"'
Set 1M Set 1L+1ru3d -108.9819 108.8291 -217.8110 2.0014 -15.6725 --15.6689 --1.4644 -0.7692 -0.6262 -0.6080 0.8757
(9X6X4)
Set IN Set 1M+0'a4f -108.9832 108.8361 -217.8192 2.0014 -15.6716 -15.6679 -1.4640 -0.7690 -0.6258 -0.6078 0.8760
(10X6X4)
Set 10 Set IN +1ru4f -108.9855 108.8387 -217.8242 2.0013 -15.6694 -15.6657 -1.4612 -0.7680 -0.6237 -0.6077 0.8775
(lOX6X5)
Set IP Set 1O+1ru3d' -108.9868 108.8208 -217.8076 2.0015 -15.6753 -15.6716 -1.4662 -0.7718 -0.6281 -0.6110 0.8740
(lOX6X6)
Set 1Q Set IP+lTu3d -108.9869 108.8208 -217.8076 2.0015 -15.6752 -15.6716 -1.4660 -0.7719 -0.6280 -0.6109 0.8710
(10X7X6)
Set 1R Set lQ+ITg3d' -108.9875 108.8249 -217.8124 2.0015 -15.6749 -15.6713 -1.4665 -0.7721 -0.6283 -0.6112 0.8707
(l1X7X6)
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Basis seta Symmetry STO, Xp).a
Basis set composition b Eo T V -V/T tIer" Et,u E2.rq E2~" Ea., firu Ea.u
Set IS Set lR+u.,3d" and u.,3s -108.9888 108.8083 -217.7971 2.0017 -15.6747 -15.6710 -1.4657 -0.7720 -0.6275 -0.6104 0.8437
(12X8X6)
Set 2A u,ls, u.ls', u,2s, u.2s', u,3s, u.2P, -108.8967 108.8987 -217.7954 2.0000 -15.7169 -15.7133 -1.5381 -0.7823 -0.6374 -0.6311 0.8887
(8X7X3) u.2P', uq2p"; u"ls, u .. ls', u,.2s,
u .. 2s', u,,3s, uu2p, uu2p'j 7r,.2p,
"'u2p', 7r .. 2p"
Set 2B Set 2A+u,3d, u.,3d', u.,3d", 0',4/; -108.9897 108.8067 -217.7964 2.0017 -15.6791 -15.6755 -1.4723 -0.7761 -0.6339 -0.6126 0.8748
(12X8X6) uu3dj 7r .. 3d, ",,,3d', ",,,4/.
Set 2C Set 2B +optimization of all r's -108.9926 108.7954 -217.7880
(12X8X6)
Set 2D Set 2C+optimization of all r's -108.9928 108.7911 -217.7839
(12X8X6)
n The designations of the various basis sets given here are those employed in the text. The symbols under
the basis set designation, for exampJe (6X6X2) under Basis Set lG, refer to the number of symmetry basis
functions in the 0' •• 0'", and 11'" symmetry, respectively.
b This column gives only the "1' and 11' of the symmetry STO basis {unctions for each stage of the buildup
"'"".".
e~''>j
I ".S e"" -:4
I:l;jg:-> 0
*l:;en " 0
~
'~a .:-: 0
W~;. !It 0 i
• • •
,....'"1~ r ill
':-Z1j;' C'I b
>;<~a 0 i : I ~,.
I ~.e I" II' X +1:' S. CI
::.--teo
r.~~ r ::I:
!-
rp ""II' I Co
enoi!!·
::? &; '" III :lit
t:tT~ 1, ,.. f C
~~ .' II' J z
..,~ 0
S'gO "CI
(I
~5· ::11
a'* lit .. a
~e 2.0018 -15.6826 -15.6790 -1.4746 -0.7788 -0.6357 -0.6161 0.7975
2.0019 -15.6820 -15.6783 -1.4736 -0.7780 -0.6350 -0.6154 0.8077
of Basis Set 1. The bas .. sel composition refers in addition to the orbital exponents of the basis functions which
may change from case to case. The full wavefunction including the orbital e:<pOnents and the vectors for each
member of the buildup is available from the authors upon request.
o All energy quantities are in hartrees.
s-i g
~ .. ~::;;
::?X~·P ...
"" I'" "" . . .,
;
-:.....xg~ .. S!: I '1' I
0 ~ !) 0 ~ •
.. '<: CD :." 9
! '.f!.~ II n 0 0 UI
~ ...... ~ 0
O~~. n
I~(fl 0
O:j;s' ~ '" ,,-" i :.e ...
.~!O • II)
[> ~~. 5' I . 0 CD % §;I:' I: t> 00 0" ..
~ g-~. I Co
~ ~w U»,.;
en'" ~ 1 Oftlt""'t' ,.. s:sd l c s-;§. • z
,. ... 1»
en q-::t. 0
~~g "CI
............
~ ::; (I
0...0 :III
~1Cl-go ::;: '" o..ee. tci
t-'
tci
(")
~
:;d
o
Z
.....
(")
en
~
:;d
c:1
(")
~
c::
:;d
tci
o
"'1
t;
.....
;..
~ o
:s:
.....
(")
:s: o
t-'
tci
(")
c:1
t-'
tci
en
.....
..... .....
;..
.'C -l 'C
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsTABLE II. Synthesis of basis sets 1 and 2-energy differences-N.(X l~/), R=2.068 bohr. .....
~ 0
Symmetry STO, XpM
A(-V/T) Basis set Basis set composition AEa AT AV AELr. AELr. A .... A .... A .... .6.Eb'. A ....
Set lA .,..ls, .,..2s, .,..2p;
(BMMOg=u) .,..ls, .,..2s, .,..2p; 1ru2p; (g=u)
Set 1B .,..ls, .,..2s, .,..2p; -0.0123 +0.0235 -0.0358 -0.0001 +0.0035 +0.0037 -0.0051 -0.0048 +0.0056 +0.0019 +0.1346
(BMMOg~u) .,..ls, .,..2s, .,..2p; 1r.2p; (g~u)
Set lC Set IB+.,..ls' and .,..ls' -0.0449 -0.0155 -0.0294 +0.0006 -0.0137 -0.0134 -0.0045 +0.0035 +0.0065 +0.0038 +0.0412
(4X4Xl)
Set ID Set 1 C +.,. .2s' and .,. .2s' -0.0805 -0.2232 +0.1427 +0.0028 -0.0516 -0.0517 -0.0411 -0.0327 -0.0498 -0.0294 -0.5018
(5X5Xl)
Set IE Set ID+1r.2P' -0.0978 +0.2821 -0.3799 -0.0017 -0.0027 -0.0026 -0.0351 -0.0225 -0.0261 -0.0481 -0.0154
(5X5X2) (")
> Set IF Set lE+.,..2p' -0.0205 +0.0958 -0.1163 -0.0007 -0.0024 -0.0022 -0.0188 -0.0068 -0.0098 -0.0100 -0.0041 tj
(6X5X2) t>i
Set IG Set IF+.,..2p' -0.0018 +0.0151 -0.0170 -0.0002 -0.0017 -0.0018 -0.0036 -0.0010 -0.0018 -0.0026 -0.0390
(6X6X2) Ul
>
Set IH Set IG+1ru2p" -0.0012 +0.0202 -0.0213 -0.0001 +0.0049 +0.0050 +0.0044 +0.0030 +0.0034 +0.0021 t"" +0.0029 t>i (6X6X3) Ul
Set 11 Set IH +.,..2p" -0.0066 +0.0412 -0.0479 -0.0003 +0.0038 +0.0038 +0.0014 +0.0011 -0.0018 +0.0020 +0.0022 > (7X6X3) z
Set I} Set 1I+.,..2s" -0.0030 -0.1210 +0.1181 +0.0011 -0.0045 -0.0044 -0.0011 +0.0001 -0.0001 +0.0001 +0.0006 tj
(8X6X3)
~
Set lK Set 1I+.,..3s 0.0000 +0.0034 -0.0034 -0.0001 +0.0002 +0.0002 0.0000 0.0000 0.0000 +0.0001 0.0000 >
(8X6X3) ::r::
t""
Set lL Set lK+.,.q3d -0.0276 +0.0034 -0.0310 +0.0003 +0.0125 +0.0124 +0.0127 -0.0004 -0.0075 +0.0111 +0.0122
(9X6X3)
Set 1M Set lL+1r u3d -0.0521 -0.0265 -0.0256 +0.0007 +0.0263 +0.0263 +0.0475 +0.0050 +0.0107 +0.0064 +0.0161
(9X6X4)
Set IN
(10X6X4) Set IM+.,..4f -0.0013 +0.0070 -0.0082 0.0000 +0.0009 +0.0010 +0.0004 +0.0002 +0.0004 +0.0002 +0.0003
Set 10
(10X6X5) Set IN +1r.4f -0.0023 +0.0026 -0.0050 -0.0001 +0.0022 +0.0022 +0.0028 +0.0010 +0.0021 +0.0001 +0.0015
Set IP Set 1O+1r u3d' -0.0013 -0.0180 +0.0166 +0.0002 -0.0059 -0.0059 -0.0050 -0.0038 -0.0044 -0.0033 -0.0035
(10X6X6)
Set lQ Set IP+.,..3d -0.0001 +0.0001 0.0000 0.0000 +0.0001 0.0000 +0.0002 -0.0001 +0.0001 +0.0001 -0.0030
(10X7X6)
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1981
i o
I
o
I
o
I
o
I
o
I
o +
o +
o
I
~ o +
§
o
I o ....
~
o
I
§
o +
o +
o +
o +
o +
o +
o +
"" '" o
o +
:g o
o
I
"" §
o
I ~
<5
o
I
'" 00 -o
o +
'" §
o +
o +
00 8 o +
00 ....
8 o +
00 ....
8
o +
....
§
o +
o §
o
I
~
8
o
I
§
o
I ~ .... o
o
I
'" §
o
I
00 §
o
I
'" §
o
I
'" §
o
I
o +
§
o +
"" --o
o
I
0-§
o
I ~
<5
o +
o +
o +
o +
o §
o +
o +
§
o +
o +
o +
o
I
o
I
Ul
"" -;
'0
~
.~
.~
S
''= 0-o + u
'" ....
'" r.n y. -217.2r-----------------,
-217.5
-217.6
-217.7
-217.8
-217.9
A 8 C D E F G H 1 ~ K L M N 0 P Q R S
'""""" In ..... Set s,ntMtII -
FIG. 4. Synthesis of expansion basis set-variation in total
potential energy for Nz(X 1~.+), R==2.068 bohr; V in hartrees. (e-e Basis Set 1, X----X Basis Set 2.)
energies are documented for the first and second
schemes. The variation of these quantities as the basis
set synthesis progresses for both schemes is illustrated
in Figs. 1 (total energy), 2 (orbital energies), 3 (kinetic
energy), and 4 (potential energy). In these figures the
lines connecting the points do not correspond to any
curve fitting, but merely serve to connect the points
in sequence.
The construction of Basis Set 1 (see Table I)
started with Ransil's best minimal-molecular-orbitals
(BMMO) set,9 called Set 1A here, with the rp's of
symmetry ITa equal to those of ITu (g=u constraint).
The next step was relaxation of the g=u constraint
which was obtained from Set 1A by double (i.e., simul
taneous) optimizations of the rp's for each of the ITuts
and ITu1s; ITa2s and IT,,2s; and ITu2p and ITu2p pairs.
Then, in addition, certain other double optimizations
were carried out of rp's only in ITa or ITu symmetry and
several single optimizations which finally gave what
may be termed the BMMO (g,eu) set (Set lB). The
general procedure was then to add new STO symmetry
basis functions one or two at a time and optimize
certain combinations of orbital exponents. Thus for each
entry in Table I the orbital exponent of the new Xp)."
as well as certain others already present were opti
mized. The over-all basic logic was simply to add
Xp).a STO symmetry basis functions of lowest permitted
lp until no further improvement was evident and then
add Xp).a with the next highest lp value. For example,
in ITa symmetry, ITans and ITunp, and ITuns' and ITanp', etc.,
types were practically exhausted before starting to add
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1982 CADE, SALES, AND WAHL
11 gnd STO symmetry basis functions. Sets 1 C through
1G (Table I), represented by the first, and steepest
drop, of the solid curve in Fig. 1, correspond to the
gradual improvement to the "double zeta" approxima
tion (Set 1G). The first plateau of the solid curve in
Fig. 1, corresponding to Set 1H through Set lK (Table
I), represents the addition of new STO symmetry basis
functions, such as 7ru2p", 11 g2p", and 0" g3s, of already
existing kinds, differing only in different tp's or np
values, but introducing no new lp values. The second
and smaller drop of the solid curve in Fig. 1, was
obtained by the introduction of a new lp value, namely
introducing I1g3d (Set lL) and 7ru3d (Set 1M) STO
symmetry basis functions. The last stage (IN through
IS) gives rise to the second plateau of the solid curve
in Fig. 1 and here the next new lp value, involving
lp= 3, shows no significant change. This led to the
final result of Scheme 1, Basis Set IS, which consists
of 12 STO symmetry basis functions representing 0" g
molecular orbitals, eight representing l1u molecular or
bitals, and six representing the single 17r" molecular
orbital, or in short this is a 12X8X6 set. The cor
responding wavefunction32 is given in Table III and
the total energy for Set IS is -109.9888 hartree.
Before sketching the second scheme of synthesizing
the expansion basis set the striking appearance of the
solid curve in Fig. 1 should be noted. This curve
displays the progressive improvement in the total en
ergy (see Table II for energy differences), and can
be taken as evidence of the convergence towards the
true Hartree-Fock results. Particularly encouraging is
the fact that this curve seems to have leveled off and
is only affected in the third of fourth decimal place
by adding more STO symmetry basis functions of
types already present or even new types of basis func
tions from the remaining possibilities (STO symmetry
basis functions with np::;6 and lp::;3 permitted). The
leveling off follows two substantial drops in the energy
improvement curve associated with doubling the mini
mal basis set and the introduction of basis functions
with lp=2.
The synthesis of the basis set in the second scheme
was based on the use of the BGRC Hartree-Fock
Roothaan wavefunctions for the N(4S) atom.22 The
basic plan was to employ STO symmetry basis func
tions constructed from the atomic basis sets and then
to add XpAa, especially those with lp= 2, which seem
necessary from the results of Scheme 1. The orbital
exponents were then all singly optimized twice. The
dashed line in Fig. 1 shows the energy values for
Basis Sets 2A, 2B, 2C, and 2D and indicates an energy
limit in the close neighborhood of the limit of the
curve from the gradual buildup scheme.
The preceding two computational schemes suggested
the following answers to the questions posed at the
beginning of this section. For calculations on diatomic
molecules it seems imperative to start with atomic
32 The intermediate wavefunctions are available from the
authors upon request. Hartree-Fock-Roothaan wavefunctions whose basls
sets are large enough to adequately represent the
atomic orbitals, but small enough to permit the addi
tion of new XpAa and if possible leave room for further
exploration.33 In addition to the XpAa arising from the
atomic results, it is essential to have at least one, and
preferably two, XpAa STO symmetry basis functions
with lp= 2. Several XpAa, differing only in tp, for exam
ple, O"g2p, O"g2p', and O"g2p", are necessary either from
atom parentage or possibly for addition functions (for
example, 7ru3d and 7ru3d' in Basis Set 1S and 2D). It
is also clear that the "double-zeta" approximation13
leaves much to be desired since this result would do no
better than level off on the first and upper plateau of
the solid curve of Fig. 1.
The optimization of orbital exponents is very im
portant for small basis sets but can never alone absorb
the deficiency due to a lack of expansion functions. The
number and secondly, the kind of STO symmetry basis
functions are the most important considerations and
the energy improvement to be gained by exponent
optimization decreases sharply with skill in picking a
starting basis set composition. After the set of nAp and
lAp for the XpAa representing each symmetry type is
decided and a set of tAp is chosen from either atomic
results, interpolation, extrapolation, or elsewhere, opti
mization of certain orbital exponents is desirable. It
seems essential to do these optimizations with either
the whole set present (which is preferable) or to back
optimize the tp's very extensively as the basis set is
built up. The calculations of WahP and HUO,2 as here
also, indicate that optimization of certain t p's makes
no significant improvement (most notably the tp's of
the XpXa's important for inner shells). In the calculation
of the potential curve points of N2(X 12:g+) and for
the N2+(X 21;g+, A 2IIu, and B 22:u+) molecule-ions this
economy was employed. The major argument against
extensive double and/or triple optimizations of orbital
exponents is the tremendous time investment necessary
for molecules.34 The general problem of the simultaneous
optimization of all orbital exponents is thus not defini
tively solved and this represents a defect in the present
efforts. It is our belief, however, that for large basis
sets, carefully chosen series of single optimizations are
as effective as such a general solution would be.
The question of measuring how closely our best re
sult (Basis Set 2D) compares with the true Hartree-
33 This practical constraint arises from the limitation of the
total number of expansion basis functions (see Sec. II).
34 The computing time for A2 systems, in general, depends
mainly on the basis set size and the size of certain numerical
integration grids employed, and secondarily on the number of
electrons. The computation of the supermatrices, supervectors,
and the solution of the Hartree-Fock-Roothaan equations at a
single R value, commonly called a single SCF run, takes ~2 min
for Set lA and ~25 min for Set 2D for N2 on an IBM 7094.
To optimize a single orbital exponent then takes ,.....,10 min for
Set lA and ",100 min for Set 2D. For a basis set having XpAa
with large !p larger numerical integration grids are needed so
that a minimal orbital set for CI. takes ~15 min and a large set
for Cb comparable to Set 2D takes ",90 min. The expense of
optimization of orbital exponents is thus evident.
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. II I. A 1983
Fock results is not easy to answer conclusively. This
comparison refers to a point-by-point comparison of
the final Hartree-Fock-Roothaan molecular orbitals
with the numerical Hartree-Fock molecular orbitals
and a comparison of certain expectation values and
molecular properties. In recent studies by BGRC this
question is considered at length for first row atoms and
the very favorable comparison of the Hartree-Fock
Roothaan orbitals with the numerical Hartree-Fock
orbitals is obtained. Lacking even one numerical mo
lecular Hartree-Fock wavefunction our discussion must
be based almost entirely on observing the convergence
of the calculation. If all the functional parameters
were exercised to eXlhaustion with no further improve
ment, it would be very strong evidence of convergence.
In the results for Li2(X l~g+) this was comfortably
achieved,16 that is, by virtue of having only two 0' g
molecular orbitals and one 0'" molecular orbital, rela
tively huge basis sets could be employed to go far
beyond what is really needed. For N2 the situation is
somewhat less favorable since the problem and present
computer program do not permit comfortable margins
for such exploration. The problem is thus to determine
from a series of "improving" approximations how close
the last result is from the unknown quantity sought.
The quantity sought may be considered as any Hartree
Fock computed value, such as the total energy, orbital
energies, expectation values, or molecular property,
but especially the molecular orbitals and the molecular
orbital charge densities. In Table II are presented
I1E= EN+1-EN, I1T, 11 V, and the set of I1Ei for the
basis set synthesis in both Scheme 1 and 2. The differ
ences in Table II arise from the results given in Table 1.
The quantities considered here other than the total
energy are not bound to a predictable course toward
the Hartree-Fock result and their differences show
this. Most disturbing is the erratic behavior of the
kinetic and potential energy values displayed in the
solid lines of Figs. 3 and 4. Thus while the total energy
in the solid curve of Fig. 1 is relatively well behaved
and always descending, the differences in Table II
show that the lowering in total energy may be due
chiefly to a decrease in the kinetic energy (always
positive) or an increase in the potential energy (always
negative), or finally a combined change of a subtle
nature. The major feature of the I1T and AV differ
ences is that they also gradually decrease in size,
and though less convincing than I1E values, indi
cate that the kinetic and potential energy are also
converging, but more slowly, to the true Hartree-Fock
values. Not much else can safely be stated. The dashed
curves in Figs. 3 and 4 for the synthesis of Basis Set 2D
show a more reasonable behavior which suggests that
the erratic behavior of I1T and 11 V in Scheme 1 arises
from the crude approximation of the molecular orbitals
at the intermediate stages and the relative sensitivity
of (T) and (V) to these changes. The various E/s
shown in Fig. 2 and I1E/S in Table II indicate that the
E/S are not greatly sensitive to improving the wave-function, but also show an unsystematic behavior (see
I1Ei values) at the intermediate points of Scheme 1.
These I1Ei also exhibit decreasing values for all Ei col
umns as larger basis sets are obtained.
The general conclusions drawn here closely parallel
those discussed by Wah]! on F2 and Hu02 on CO and
BF. In assessing the relative approach to the true
Hartree-Fock results for the N2 calculations compared
to the calculations on CO and BF, one must remember
that relatively larger basis sets are employed for F2
and N2 compared to CO and BF. Therefore, the 12 O'g,
8 0'0., and 6 "11"0. basis set for N2 corresponds, in terms
of the heteronuclear problem, to at least 12 O'-type
functions and six "II"-type STO's on each nitrogen nu
cleus, while the largest set employed by Hu02 has
eight O'-type STO's on 0, eight u-type STO's on C,
four "II"-type STO's on 0, and four "II"-type STO's on C
for CO. Certain of the conclusions presented here are
also well known from the works of Nesbet (who first
draws attention to the necessity of introducing d-type
orbitals), McLean (who stresses the importance of
optimizing orbital exponents with the entire basis set
present), and others. The separation of the O'g and 0'"
basis function sets is most important in our opinion
only in permitting different numbers of XPAa for the
two symmetries due to perhaps having more 0' g-type
molecular orbitals than 0' .. , and as has been discovered,36
that fewer 0' .. expansion functions are necessary even
when an equal number of O'g and 0'" molecular orbitals
is present. Similar basis sets with identical SP values
could be confidently used for inner shells and if the
basis set was large enough completely identical O'g and
0'" basis functions would probably be equivalent to the
present results.
It may be useful to pinpoint certain deficiencies of
the preceding results. These are
(1) More basis functions should be added to clinch
our view that convergence is achieved.
(2) Simultaneous optimization of all orbital expo
nents would be desirable.
(3) Symmetry basis functions with lp>3 should be
added to confirm our belief that, at least for first-row
diatomic molecules, XPAa beyond O'g4j, O' .. 4f, "II",,4f, and
"II" o4f are unnecessary.
(4) It is not now feasible to attempt to weed out
the one or two functions, if indeed there are any func
tions, not really needed, as has been done by BGRC
for the first-row atoms. This defect is most objection
able if these basis sets are to be used as the starting
point for further calculations, such as those involved
in obtaining the electric polarizability or magnetic sus
ceptibility of diatomic molecules.34
C. Final Hartree-Fock-Roothaan Wavefunctions for
N2(Xl~g+) and N2+(X2~g+, A 2II " , B2~,,+) Ions
The concluding wavefunctions from the synthesis of
the expansion basis set using Scheme 1 (Set 1S) and
36 J. B. Greenshields (private communication).
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1984 C-A DE, SAL E S, AND WAH L
TABLE III. HFR wavefunction IS for N2(1"..21".J2oi2"."23"..21,,. • .', X 12:.+), R=2.068 bohr.
E= -108.9888 hartree, T= 108.8083 hartree, V = -217.7971 hartree, V /T= -2.00166
£I~g= -15.67467, E2~.= -1.46565, £3~.= -0.62749, £1",,= -15.67101, E2.,,= -0.77203, E!~,,= -0.61037
Ci~p Ci~p Ci~
'Xtw. C1 •• ,P C2<.,P C ... ,P 'X_ CI~",p C2<u,p 'XP~ CI~,P
"..1$ (r=6.34808) 0.91893 -0.26088 0.07517 ".,,1$ (6.42643) 0.95623 -0.21563 ".,,2p (1. 63543) 0.67240
"..ls' (10.44794) 0.08480 0.00624 -0.00172 "."ls' (11. 79782) 0.04946 0.00128 ".,,2p' (3.29947) 0.20651
"..2s (1.15656) -0.00156 0.01473 -0.30107 ".,,2s (1.35580) -0.01279 0.27631 ".,,2p" (7.66609) 0.00797
"..2s' (2.19475) 0.00657 0.72687 -0.28580 "'u2s' (2.12224) 0.01276 0.79805 ".,,3d (0.51707) 0.05309
uo3$ (8.34237) 0.00149 -0.02834 0.00347 "..3s (9.60739) -0.00250 -0.02039 ".,,3d' (2.33801) 0.06928
"..2p (1. 34282) -0.00178 0.00835 0.23567 "..2p (1.44387) -0.00380 -0.24628 ".,,4f (3.45922) 0.01041
".g2p' (2.36027) 0.00228 0.31756 0.58490 ".,,2p' (3.04456) 0.00294 -0.16958
u.2p" (6.31659) 0.00089 0.01389 0.03064 ".,,3d (4.36861) 0.00037 -0.00240
uo3d (1. 34301) -0.00110 -0.01061 -0.03648
".o3d' (2.62151) 0.00144 0.05116 0.05095
"..3d" (5.36005) -0.00023 -0.00134 -0.00199
ug4f (3.31101) 0.00041 0.00876 0.00443
Scheme 2 (Set 2D) are given in Tables III and IV,
respectively. Basis Set 2D is the final result at R.( exptl)
and is taken as the Hartree-Fock wavefunction for
N2(X l~g+).
The wavefunction for N2(X l~g+), Set 2D, was now
used for the starting basis set composition to obtain
the Hartree-Fock-Roothaan wavefunctions for the fol
lowing singly ionized states of N2:
N2+( 10"ilO",,220120",,2h- u43u g),
N~/( 10" g210" ,,220" g220" ,,2h-,,330" g2), ments.36 It should be emphasized that these calculations
on the N2+(X 2~g+, A 2IT", B 2~,,+) ions are direct cal
culations for the open-shell system resulting from the
removal of one electron from a 30"g, 17r", or 20"" molecu
lar orbital, respectively, and are not taken from the
N2(X l~g+) results. Accordingly, full rearrangement of
the molecular orbitals (and therefore full rearrangement
within the Hartree-Fock approximation) is achieved
for the N2+ ions resulting from the ionization processes
N2(" .20",,2h-,,430"i, X l~g+)
N2+( 10" ilO",,220" i20" ,,17r,,430" g2),
The order of the molecular orbitals and the state specifi
cations are in accordance with experimental assign- 1 N2+( •.• 20",,17r,,430" i,
-? N2+(··· 20",,21?r,,330"g2,
N2+(·· ·20",,21?r,,430"g,
TABLE IV. HFR wavefunction 2D for N2(1"..11". .. 12"..22"."23"..21,,.,,', X 12:.+), R=2.068 bohr.
E= -108.9928 hartree, T=108.7911 hartree, V= -217.7839 hartree, V/T= -2.00185
£1 •• = -15.68195, £2..= -1.47360, £3 •• = -0.63495, £1 ... = -15.67833, E2 ... = -0.77796, Elr .. = -0.61544
CiAp Ci~p
')(Ptll1 C1 •• ,P C2f1II'P C".'P 'Xtwu Clll1u,p C2<u,p Xpru
"..1$ (r= 5. 68298) 0.92319 -0.27931 0.07484 ".,,1s (5.95534) 0.93406 -0.24370 ".,,2p (1.38436)
"..ls' (10.34240) 0.15204 -0.00615 0.00262 "."ls' (10.65879) 0.11483 0.00000 ". .. 2p' (2.53288)
u.2$ (1. 45349) 0.00090 0.14106 -0.45859 ".,,2$ (1. 57044) -0.01156 0.36437 ". .. 2p" (5.69176)
"..2s' (2.43875) -O. 00003 0.59948 -0.17662 ".,,2s' (2.48965) 0.00479 0.54702 "..3d (2.05707)
"..3s (7.04041) -0.08501 -0.02333 -0.00678 ".,,3s (7.29169) -0.05343 -0.03054 ".,,3d' (2.70650)
"..2p (1. 28261) 0.00041 0.11602 0.42914 ". .. 2p (1.48549) -0.00679 -0.41355 ".,,4/ (3.06896)
"..2p' (2.56988) 0.00104 0.25907 . 0.48453 ".,,2p' (3.49990) 0.00294 -0.10945
"..2p" (6.21698) 0.00109 0.01092 0.02478 ". u3d (1. 69003) -0.00121 -0.03553
"..3d (1. 34142) 0.00017 0.03626 0.04696
"..3d' (2.91681) 0.00098 0.03984 0.03065
".o3d" (5.52063) -0.00018 -0.00275 0.00158
"..4/ (2.59449) 0.00032 0.01334 0.01086 Ci~p ----
C1"'u,P
0.46921
0.39869
0.03141
0.05938
0.01738
0.01233
36 R. S. Mulliken, in
1957). p. 169. The Threshold of Space, edited by B. Armstrong and A. Dalgarno (Pergamon Press, Inc., New York,
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1985
TABLE V. HFR wavefunction for N2+(11T.211T,,'2ITi21T,,131T.I7r u4, X 22:.+), R=2.113 bohr.
E= -108.4037 hartree, T= 108.1612 hartree, V = -216.5649 hartree, V /T= -2.00224
<1 •• = -16.18273, <2<.= -1.88073, <3 •• = -1.12345, <1 •• = -16.18002, <2 •• = -1.15688, <1 ... = -1.02369
C~i" Ci~" Ci~"
X]HI, Cu."" C"".,,, Ca..,,, X]HIU C11r",p C""",,, XP.,-u C1'Jl'u,,,
1T.ls (r=5.68298) 0.92380 -0.28356 0.08485 IT,,ls (5.95534) 0.93479 -0.25955 7ru2p (1.53784) 0.47956
1T.1s' (10.34240) 0.15194 -0.00622 0.00249 IT,,ls' (10.65879) 0.11476 0.00040 7r,,2p' (2.54788) 0.38728
1T.2s (1. 67662) 0.00156 0.20359 -0.43210 1T,,2s (0.95819) -0.00098 -0.01418 7ru2p" (5.61486) 0.03526
1T.2s' (2.50020) -0.00082 0.58070 -0.14364 1T,,2s' (2.20484) 0.00109 0.88861 7ru3d (2.24311) 0.07876
1T.3s (7.04041) -0.08537 -0.02826 -0.00497 lTu3s (7.29169) -0.05302 -0.02107 7ru3d' (3.68750 -0.00183
a.2p (1. 47252) 0.00014 0.09449 0.37297 lTu2p (1. 49449) -0.00080 -0.38798 7ru4J (3.04069) 0.01516
1T.2p' (2.56008) 0.00064 0.22626 0.51555 lTu2p' (3.52453) 0.00112 -0.10141
1T.2p" (6.51387) 0.00053 0.00962 0.02345 1T.3d (1. 58465) 0.00007 -0.03958
1T.3d (1. 57033) 0.00021 0.02886 0.02899
1T.3d' (2.87912) 0.00034 0.03387 0.02357
1Tu3d" (5.52063) 0.00010 -0.00235 -0.00070
IT.4J (2.79841) 0.00020 0.01271 0.01570
Thus we have obtained Hartree-Fock-Roothaan wave
functions for these three states of N2+ which are to be
employed in subsequent research on the reorganization
of the electronic charge distribution of nitrogen upon
ionization, and for calculating the transition moments
for the first negative and Meinel systems of N2+.
The calculations to obtain Hartree-Fock-Roothaan
wavefunctions for N2+(X 22;0+' A 2II", B 22;,,+) molecu
lar ions all followed the same general procedure which
is now outlined.31 The starting basis set comp0i;iition
for each was Basis Set 2D (Table IV) for N2(X 12;0+)
and after a single SCF run a sequence of single re
optimizations of certain orbital exponents was carried
out for R=R.(exptl), that is, R=2.113 bohr for
N2+(X22;0+), R=2.222 bohr for N2+(A 2II,,), and
R= 2.0315 bohr for N2+(B 22;,,+). Thus the reorganiza
tion of the charge distribution, and hence modification
of the molecular orbitals, relative to N2(X 12;0+) is
effected through both the linear expansion coefficients
and the orbital exponents. The sequence of single re
optimizations of t ,,'s was really three separate se
quences, one involving U 0 basis functions, one involv
ing UII basis functions, and one involving 1I'u basis func
tions. The relative order of these three sequences of
single reoptimizations was related with the N2+ state
involved. Thus for the X 22;g + state which results from
a loss of a 3u 0 electron from N2(X 12;0+) the U 0 sequence
came first, then the u" sequence, and finally the 11'"
sequence. This ordering was based on the anticipated
order of importance as viewed in terms of the relative
number of matrix elements which would involve the
tPiAa that lost the electron. The t p's of XpAa which in
volve mainly core molecular orbitals lUg and 10' .. were
not usually reoptimized.
The starting basis set composition gave an energy
of -108.2533 hartree for N2+(B 22;,,+). Reoptimization
of tp's for all eight u" symmetry basis functions lowered
the energy to -108.2549 hartree and reoptimization
of t ,,'s for nine of the 12 U /I symmetry basis functions gave an additional lowering of 0.0011 hartree to
-108.2560 hartree. Finally the tp's for all six 11' ..
symmetry basis functions were reoptimized to give the
final energy of -108.2596 hartree at R= R.( exptl)
for N2+(B 22;,,+). The concluding wavefunction for
N2+(B 22;,,+) is given in Table VII. The final results
for N2+(X 22;0+' A 2IIJ') are given in Tables V and VI
and are the results of a similar procedure.s1
Before noting the changes in the Hartree-Fock
Roothaan wavefunctions in going from N2(X 12;/1+)
to N2+(X 22;0+' A 2II". B 22;,.+), the major features
of the sequences of reoptimizations of the t,,'s for the
N2+ ions are now summarized. The three sequences of
reoptimizations, for Uo, U'" and 11'" symmetry, give
identical total improvements of .1E=0.0063 hartree
for each the X 22;0+' A 2II", B 22;,,+ states. Thus, the
reorganization effected by reoptimization of the t"
values was apparently insensitive to the state involved,
although the details differ somewhat. The second
major feature was that for each state the total gain
from reoptimizing the orbital exponents came from
the uo2s, u02p, u,,2s', and 1I',,2p STO symmetry basis
functions only. These basis functions are the most
important contributors to the 2uo, 3ug, 2u .. , and 111' ..
molecular orbitals and they were usually (except for
u02p) the first STO symmetry basis function of that
symmetry optimized. Thus, there was little connection
between the symmetry of the molecular orbital which
lost the electron and the symmetry of the XpAa basis
functions which make the larger improvements upon
reoptimization of the orbital exponents as anticipated.
In as much as t" reoptimizations are able to reflect
the reorginization of the tPiAa, spatial readjustment is
relatively symmetry independent, but depends sharply
on spatial overlap between the outer molecular orbitals.
It may have been desirable to completely parallel
the N2(X 12;/1+) calculations for each ion, but the
computation time for each ion would have increased
very considerably. By starting with the N2(X 12;0+)
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1986 CADE, SALES, AND WAHL
TABLE VI. HFR wavefunction for N2+(1<Til<T"t2<T.t2<T,,13<T.2h· .. 3, A tll,,), R=2.222 bohr.
E= -108.4270 hartree, T=108.1820 hartree, V= -216.6089 hartree, V/T=-2.00226
El.a= -16.18050, E ••• = -1.86435, '3 •• = -1.03491, EI •• = -16.17820, '2.,,= -1.20616, El .. " = -1.02888
C2<g,p Ca.a,p C2< .. ,P
<T als (r = 5.68298) 0.92406 -0.27186 0.09871 <T"ls (5.95534) 0.93494 -0.26169 1r .. 2p (1. 54195) 0.50577
<T.ls' (10.34240) 0.15190 -0.00599 0.00238 <T"ls' (10.65879) 0.11463 0.00030 1r .. 2p' (2.55318) 0.36203
<T a2s (1. 62293) 0.00115 0.18359 -0.44381 <Tu2s (2.21503) -0.00296 0.87726 1r,,2p" (5.60093) 0.03573
<T.U (2.48547) -0.00073 0.56677 -0.18261 <T,,2s' (3.13473) 0.00366 0.00836 1r,,3d (2.15014) 0.09281
<T.3s (7.04041) -0.08552 -0.02640 -0.00558 <T,,3s (7.29169) -0.05416 -0.02388 1r,,3d' (4.00627) -0.00193
<T.2p (1. 42818) -0.00014 0.13471 0.39371 <T,,2p (1. 50366) -0.00211 -0.38275 1r,,4/ (2.80565) 0.01410
<T.2p' (2.55362) 0.00123 0.23594 0.48071 <T"2p' (3.54115) 0.00254 -0.10310
<Ta2p" (6.21698) 0.00164 0.01123 0.02620 <T .3d (1. 65063) 0.00002 -0.04804
<T.3d (1. 52969) -0.00010 0.04759 0.05017
<T.3d' (2.87017) 0.00100 0.03538 0.03206
<T.3d" (5.52063) -0.00027 -0.00224 -0.00208
<T.4/ (2.37367) 0.00019 0.01767 0.01161
wavefunction, the major features are already present,
and presumably the loss of a 30'g, bru, or 20'" electron
from Nt is a less drastic change than that resulting
from the formation of N2+ from an N(4S) atom and
an N+(3P) ion. However, it was decided to make an
alternative calculation for the X 2~g+ state using
results from N+(3 P) calculations. This was moti
vated by the observation that the X 2~g+ and A 2I1"
states are reversed relative to experiment and the
desire to remove any doubts that this result is indeed
a Hartree-Fock result and not a shortcoming of the
Hartree-Fock-Roothaan results themselves. A Har
tree-Fock-Roothaan wavefunction for the N+(3 P)
was thus obtained of a quality comparable to the
results of BGRC.22 A 12X8X6 basis set was then
chosen for Nt+(X 2~g+) which took the tp values for
Xp}.a with Ip=O, 1 as the average of the tp's for N(4S) and N+(8 P) results. The Xp}.a with Ip= 2, 3 were taken
from Table V, the previous calculation for N2+(X 2~g+).
Three sequences of single optimizations were carried
out in the manner described before. The starting energy
was -108.4009 hartree and after the three sequences
of single optimizations the energy was -108.4031
hartree. This investigation indicates that the pro
cedure of using the Nt Set 2D as the starting set for
the Nt+ ions is probably satisfactory when gauged
only in terms of the total energy.
Therefore with slight trepidation, we contend that
these results for Nt+ are very near the Hartree-Fock
values. This is important since the energy results for
Nt+(X2~g+) and N2+(A 2IT,,) are reversed relative to
experiment at their respective R.( expd) values.
If this reversal persists as E(R) is obtained for these
ions, and if one is convinced that the error between
TABLE VII. HFR wavefunction for N2+(1<T.21<T,,22<T.22<T,,3<T.211r,,4, B 22;,,+), R=2.0315 bohr.
E= -108.2596 hartree, T= 107.8350 hartree, V = -216.0946 hartree, V /T= -2.00394
EI •• = -16.16271, <2<.= -1.89302, '3 •• = -1.00569, '1.,,= -16.15837, E2<u= -1.25973, El ... = -1.03593
C;Xp C'XP C,AP ----
X".. CI •• ,p C2tru,p Ca..,P X_ C1fTU'P C,.",p Xp:ru CI ..... p
<Tals (r=5.68298) 0.92217 -0.28050 0.09191 <T"ls (5.90826) 0.93825 -0.25604 1ru2p (1.53707) 0.47174
<T.ls' (10.34240) 0.15250 -0.00611 0.00323 <Tuls' (10.69717) 0.11717 -0.00078 1ru2p' (2.55041) 0.38718
tT .2s (1. 61093) 0.00087 0.15461 -0.49521 tTu2s (1. 65689) -0.00279 -0.40083 1r,,2p" (5.61323) 0.03537
<T.U (2.49340) 0.00018 0.56562 -0.15239 <Tu2s' (2.10568) 0.00229 1.13692 1r,,3d (2.10647) 0.06439
<Tu3s (7.02110) -0.08461 -0.02788 -0.00612 tTu3s (7.31519) -0.05994 -0.02072 1r,,3d' (2.70650) 0.01331
<T.2p (1.39078) 0.00070 0.12146 0.38610 <Tu2p (1. 51874) -0.00141 -0.47613 1ru4/ (3.06104) 0.01352
tT.2p' (2.58075) 0.00088 0.28087 0.48916 <Tu2P' (3.48920) 0.00045 -0.11921
<T.2p" (6.25817) 0.00012 0.01305 0.02453 <Tu3d (1.37167) -0.00014 -0.03593
<T.3d (1.41676) 0.00021 0.03355 0.04293
<T.3d' (2.93434) 0.00106 0.04265 0.02525
11.3<1" (4.46937) -0.00019 -0.00512 -0.00121
<T.4/ (2.59449) 0.00039 0.01354 0.01213
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsELECTRONIC STRUCTURE OF DIATOMIC MOLECULES. III. A 1987
TABLE VIII. Energy quantities for certain states of N,H, N,3+, and N24+ (R=2.0132 bohr).·
Ion Ou ter shells E(hartree) ftG', flG'u <2." <2 •• <2." Ehr" -(V)/(T)
N.I+(32:.-) 2"u23".21".u' -107.3871 -16.7170 -16.7129 -2.5000 -1.6373 -1.5207 -1.6495 1.9894
N22+(I.<l.) 2",;3".21".u· -107.3344 -16.7201 -16.7160 -2.5036 -1.6386 -1.5223 -1.5977 1.9890
N2'+ (12:.+) 2"u'3".o1".u· -107.2985 -16.7400 -16.7368 -2.4089 -1.5570 -1.0668b -1. 5242 1.9966
N23+(2IIu) 2"u23".'1"'u1 -105.6941 -17.3286 -17.3247 -3.0792 -2.1202 -2.0114 -2.0114 1.9796
N4+(I2:.+) 2"u'3"i1".uo -103.6322 -17.9820 -17.9783 -3.6956 -2.6293 -2.5256 1.9684
• These calculations are all direct calculations for the ion and state in question. No claim is made that these are Hartree-Fock results or that the states can·
idered are bound states. All results employ Set 2D of Table IV.
b Virtual orbitals.
these results and the true Hartree-Fock values is too
small to explain the discrepancy, then this reversal
must be due to a differential shortcoming of the Har
tree-Fock approximation for these two states, A 2ll"
and X 22;g+, of N2+. This indeed is the conclusion of
this research, as is discussed fully in a later section.
In the course of the calculations to obtain the final
wavefunctions of N2+(X 22;0+' A 2ll", B 22;,,+) several
recurrent features were noted which are clearly as
sociated with the readjustment of the molecular or
bitals. As expected, the major change, expressed in
C iAp and t p values, was that the rPiAa, and the total
wavefunction in general, contracted. Thus one notices
that relative to the N2 wavefunction, the CiAp of the
N2+ ions in 20'0' 3ug, 20'", and 111·" molecular orbitals
are shifted to favor the XpAa with larger t p values
for the important vector components and upon optimi
zation of the tp's for each state of N2+, the tp values
usually increased also, shrinking the orbitals. This
discussion in terms of CiAp and tp values is obviously
awkward and in this series of papers we seek to examine
these questions more directly. In a subsequent con
tribution, Wahl and Cade37 consider the reorganization
of the electronic charge distribution when N2 is ionized
to form the three N2+(X 22;0+' A 2II", B 22;,,+) mo
lecular ions. This study extensively employs charge
density contours directly.
Calculations have also been made for several states
of N22+, and N2H, and N2H molecular ions but it is
most likely that the highly ionized systems are domi
nated by the Coulombic repulsion. These results are,
however, unoptimized Hartree-Fock-Roothaan results
using Set 2D and no attempt is made to modify and
improve the basis set composition. In Table VIII
the energy values for these calculations are presented
with no claim that these results are very close to the
Hartree- Fock values.
D. Calculation of Potential Curves for N2(X 12;g+) and
N2+(X 22:g+, A 2II". B 22;,,+) Molecular Ions
It is well known that the regular Hartree-Fock
wavefunctions for molecules go over into usually a
mixture of ground-and/or excited-state wavefunctions
37 A. C. Wahl and P. E. Cade, The Reorganization of the
Electronic Charge Distribution in the (Nitrogen Molecule
Nitro~en Molecular Ion) System," J. Chern. Phys. (to be pub
lished). for the separated constituent parts (e.g., atoms or
ions for diatomic molecules) as the internuclear dis
tance(s) become very large. The exceptions are cases
in which the separated constituent parts are them
selves closed-shell systems or one separated part is a
bare nucleus. This behavior is well illustrated for
HeH+(12;+) and NeH+(12:+) by the calculations of
Peyerimhoff.38 Thus with relatively few exceptions,
potential curves for diatomic molecules are expected
to be rather poorly represented by the usual Hartree
Fock results when viewed over the whole range of
internuclear separations. Especially, however, the
calculated potential curve, EHF (R), deteriorates rapidly
at intermediate to large R values as EHF(R) rises
very steeply and often exceeds even the dissociation
limit at intermediate R values (e.g., two or three
times Re). For R values less than Re(exptl) and for
perhaps a restricted range of R values on both sides of
Re (exptl), EHF( R) might be exec ted to be more
successful in representing at least the shape of the
true potential curve (that is, a potential curve con·
structed from a Rydberg-Klein-Rees analysis39 of
experimental results for the molecule and state in
question), although EHF (R) calculated is elevated
substantially by virtue of the intrinsic shortcomings
of the Hartree-Fock approximation.
If consideration is limited to R values in a narrow
range around Re( exptl) , the quality of the representa
tion of the shape of the true potential curve is meas
ured by
A(R) =ERKR(R)-EHF(R)
= J ['IF-'lFHF]H['IF-'lFHF]dV
where 'IF is the exact wavefunction and the true po
tential curve is taken as that obtained from an RKR
38 S. Peyerimhoff, J. Chern. Phys. 43, 998 (1965).
39 See J. T. Vanderslice, E. A. Mason, W. G. Maisch, and E.
Lippincott, J. Mol. Spectry. 3, 17 (1959); 5, 83 (1960); and F.
R. Gilmore, J. Quant. Spectry. Radiative Transfer 5,369 (1965).
Rydberg-Klein-Rees abbreviated RKR henceforth, although
Gilmore's results do not include Rees' quadratic procedure.
F. R. Gilmore kindly furnished detailed unpublished numerical
data which was employed by the authors.
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1988 CADE, SALES, AND WAHL
TABLE IX. HFR wavefunction for N2(1",.'1", .. '20-.'2", .. !3cr.'111'u', X 12:.+), R=1.85 bohr.
E= -108.9635 hartree, T= 109.7690 hartree, V = -218.7325 hartree, V /T= -1.99266
01 •• = -15.63888, Et..= -1.55836, Et..= -0.64458, El ... = -15.63096, E2.v= -0.73869, El .... = -0.67089
C'I.J> C'Ap C'Ap
X1W. Ct..,P C",.,p C30.,P X1W .. Cl ... ,p C"""p Xp .... Cln,p
",.ls (r= 5. 94056) 0.91434 -0.29387 0.06704 ",,,Is (5.93353) 0.93359 -0.23852 1I' .. 2p (1.41982) 0.43150
",.ls' (10.34240) 0.12958 0.00030 0.00166 ", .. Is' (10.65879) 0.11715 0.00002 1I' .. 2p' (2.53835) 0.41612
",.2s (1.41866) -0.00036 0.09577 -0.47425 ", .. 2s (1. 54143) -0.01416 0.28143 1I' .. 2p" (5.69176) 0.03159
"'.2s' (2. 44464) 0.00411 0.61729 -0.17010 ", .. 2s' (2.47023) 0.00476 0.53707 11' .. 3d (1. 97411) 0.03959
",.3s (7.13249) -0.04846 -0.03655 -0.00149 ",,,3s (7.25156) -0.05577 -0.02971 1I' .. 3d' (2.68119) 0.04179
",.2p (1.31201) -0.00051 0.08691 0.38339 ",,,2p (1.48871) -0.00764 -0.48016 11',,41 (3.27677) 0.01276
",.2p' (2.60843) 0.00229 0.31261 0.49517 ",,,2p' (3.57859) 0.00307 -0.10993
tT.2p" (6.42640) 0.00088 0.01146 0.02187 ", .. 3d (1.68711) -0.00113 -0.03842
",.3d (1. 32508) -0.00032 0.02099 0.02244
",.3d' (3.01776) 0.00162 0.04471 0.03548
",.3d" (6.01250) -0.00018 -0.00212 -0.00156
",.41 (3.41264) 0.00062 0.00890 0.00740
analysis, i.e., obtained by employing the turning points,
Rmin and Rmax, the height of each known vibrational
state, the dissociation energy, Do, and the energy of
the separated atoms. If t:.(R) is constant, or slowly
varying, over the range of R values considered, then
the Hartree-Fock potential curve, EHF(R), is a good
approximation to the shape of the RKR potential
curve, &KR(R). This clearly depends on the differ
ential shortcomings of the Hartree-Fock results as a
function of R, or more conventionally stated, it de
pends on the variation of the "correlation" energy
with R. It is true that for each R value within the
narrow range around Re, EHF(R) is correct to second
order, but this knowledge offers no security that
t:.(R) is slowly varying or constant, it merely means
that second-and higher-order corrections are more
important for some R values than for others, pre
sumably in a systematic manner. The authors know of
no general predictions as to the quantitative behavior
of t:.(R) over a restricted range of R values. For
H2(X 12:g+) and He22+(12:g+) , Kolos and Roothaan40
have given curves for the variation of the correlation
energy over a range of R values around Re.
These few preliminary remarks are intended to
support the value of the calculations now presented
for potential curves for N2(X 12:g+) and N2+(X 22:g+,
A 2II .. , B 22:,,+) molecular ions. The Re(HF) value
may be slightly displaced and the EHF(R) curve may
be slightly arcuated or flattened relative to ERKR(R),
but as mentioned earlier, EHJ!CR) may be an accurate
representation of the shape of the true potential curve
over a narrow range of R values. Thus one objective
was to obtain a quantitative measure of the accuracy of
the shape of EHFCR) over a small range of R values
around ReC exptl). The evaluation of the quality of
40 W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys. 32,
231 (1960), see Fig. 6. the shape of the calculated potential energy curve
might also be considered by solving the vibrational
rotational problem of the nuclei using EHF(R). A
second objective is to give expectation values and
molecular properties for specific vibrational states
(for example, v::; 5) and therefore quantities more
readily comparable to experimental results. Thus a
careful consideration of the shape of EHFCR) between
Rmin(V) and Rmax(v) will give some idea of the quality
of the molecular property calculated for the vibra
tional state v. The final objective was to obtain spec
troscopic constants by several independent means
and to consider the relative merits of the different
methods.
The preceding remarks are applicable for the Har
tree-Fock potential curve, so that before considering
the results obtained argument is necessary to explain
how the calculated potential curve was obtained
and to support our belief that this curve is a very
close approximation to EHF(R). A related matter
studied is the practical problem of discovering what
extent of optimization of orbital exponents as a func
tion of internuclear distance is necessary. Tables IX
through XIII give the final wavefunctions and energy
quantities for N2(X 12:g+) for R= 1.85, 1.95, 2.05,
2.15, and 2.45 bohr. For these R values, and for R = 1.65
and 2.90 bohr, which are not presented, considerable
reoptimization of orbital exponents was carried out.
Other results for N2(X 12:g+) were obtained using
interpolated t p values and a number of parallel calcu
lations (but without optimizing tp values) were made
for N2+(X 22:g+, A 2II .. , B 22:g+) molecular ions. In
the calculations by Nesbet14 on nitrogen, five R values
were chosen which are roots of an appropriately
scaled fifth-order Chebyshev polynomial.41 This permits
41 A. F. Tidman, Theory of Approximation of Functions of a
Real Variable (The MacMillan Company, New York, 1963),
Chap. II.
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1989
TABLE X. HFR wavefunction for N!(1er,21er .. 22er,'2er .. '3er,'l,.. .. ', X 12:,+), R= 1.95 bohr.
E= -108.9914 hartree, T=I09.2666 hartree, V= -218.2580 hartree, V/T= -1.99748
Et,,= -15.65864, E2a.= -1.51944, f3,,= -0.64009, El,.,,= -15.65324, E2au= -0.75699, El",,= -0.64381
CQ.p Ci~p CQ.p
XP" C1",p C!,.,P Ca..,P XP''' Clau,p C~,p XJ"tu Cl .... ,P
er.ls (r=5.68298) 0.92264 -0.28799 0.07170 er .. 1s (5.95534) 0.93372 -0.23944 7r .. 2p (1.40557) 0.45036
er,ls' (10.34240) 0.15208 -0.00630 0.00302 er .. ls' (10.65879) 0.11493 0.00012 7r .. 2p' (2.53288) 0.40619
er,2s (1. 41988) 0.00080 0.11221 -0.45954 er .. 2s (1. 57044) -0.01325 0.36698 7r .. 2p" (5.69176) 0.03177
a.2s' (2.43875) 0.00035 0.61377 -0.18152 a.2s' (2.48965) 0.00495 0:52425 7r .. 3d (2.05707) 0.05238
er,3s (7.04041) -0.08473 -0.02472 -0.00460 er,,3s (7.29169) -0.05315 -0.02888 7r .. 3d' (2.70650) 0.02669
er.2p (1.30312) 0.00042 0.10205 0.40187 er .. 2p (1.49395) -0.00747 -0.43100 7r .. 4f (3.15158) 0.01267
er,2p' (2.58558) 0.00138 0.28682 0.48928 er .. 2p' (3.49990) 0.00308 -0.11382
er.2p" (6.21698) 0.00105 0.01172 0.02407 er .. 3d (1. 69003) -0.00123 -0.03346
er.3d (1.34142) 0.00015 0.02816 0.03107
er.Jd' (2.95065) 0.00122 0.04322 0.03434
er.Jd" (5.52063) -0.00016 -0.00254 -0.00176
erN (3.02076) 0.00045 0.01087 0.00858
TABLE XI. HFR wavefunction for N2(lerl1er,,'2er,22er .. 23er.sl,..u', X 12:.+), R=2.05 bohr.
E= -108.9944 hartree, T= 108.8567 hartree, V = -217.8511 hartree, V /T= -2.00127
Et,,= -15.67829, E2a.= -1.48034, Ea •• = -0.63550, EI ... = -15.67446, E2au= -0.77472, El",,= -0.61940
Ci~ C;~ C;~
XP" Cla"p C""p C"".,P X_ Clag,p C"",p Xp .... Cl .... ,P
er,ls (r=5.68298) 0.92312 -0.28047 0.07462 eru1s (5.95534) 0.93400 -0.24310 7ru2p (1.39040) 0.46819
er.ls' (10.34240) 0.15205 -0.00625 0.00277 er .. ls' (10.65879) 0.11485 0.00012 7r .. 2p' (2.53288) 0.39794
er,2s (1.42218) 0.00085 0.13329 -0.44444 er .. 2s (1. 57044) -0.01201 0.33824 7r .. 2p" (5.69176) 0.03166
1T,2s' (2.43875) 0.00006 0.60508 -0.18936 er .. 2s' (2.44572) 0.00495 0.56306 7r,,3d (2.05707) 0.04664
r.3s (7.04041) -0.08498 -0.02420 -0.00480 er .. 3s (7.29169) -0.05335 -0.02721 7r .. 3d' (2.47469) 0.02968
:r.2p (1. 29712) 0.00042 0.11600 0.41592 er .. 2p (1.48549) -0.00690 -0.41678 7r .. 4f (3.06896) 0.01231
er.2p' (2.56733) 0.00108 0.26133 0.48640 er .. 2p' (3.49990) 0.00296 -0.11016
er,2p" (6.21698) 0.00108 0.01149 0.02476 er .. 3d (1.69003) -0.00121 -0.03467
<7 ~d (1.34142) 0.00017 0.03561 0.03763
r.Jd' (2.91681) 0.00103 0.04156 0.03355
er.Jd" (4.95183) -0.00018 -0.00357 -0.00236
erN (2.59449) 0.00034 0.01344 0.01011
TABLE XII. HFR wavefunction for N2(ler,21er .. 22er,22er .. I3cr,II1r,,', X 12:,+), R=2.15 bohr.
E= -108.9799 hartree, T= 108.5254 hartree, V= -217.5053 hartree, V /T= -2.00419
Et,,= -15.69694, E2a,= -1.44205, E2a,= -0.63057, El ... = -15.69411, E2a .. = -0.79183, Et .... = -0.59736
C;~ C;~p C;~
XP', C,-"p C""p C"""p Xp.,. Clag,p C~,p XP.u C1 .... ,P
er,ls (r=5.93769) 0.91506 -0.27096 0.07613 er .. ls (5.93368) 0.93446 -0.24717 7r .. 2p (1.37528) 0.48352
er,ls' (10.34240) 0.13012 -0.00004 0.00062 er .. ls' (10.65879) 0.11689 -0.00026 7r .. 2p' (2.53388) 0.39163
er.2s (1.42470) -0.00063 0.14616 -0.42794 er .. 2s (1. 56045) -0.01043 0.33335 7r .. 2p" (5.69176) 0.03156
er.2s' (2.42328) 0.00425 0.60892 -0.19210 er .. 2s' (2.43781) 0.00474 0.58812 7r .. 3d (2.08982) 0.06310
er.Js (7.08574) -0.04949 -0.03311 -0.00298 er .. 3s (7.25407) -0.05648 -0.02702 7r .. 3d' (2.56540) 0.01117
er.2p (1. 29398) -0.00072 0.11748 0.42535 a .. 2p (1.48774) -0.00621 -0.40233 7r .. 4f (2.93505) 0.01228
er,2p' (2.55118) 0.00148 0.24127 0.48416 er .. 2p' (3.53093) 0.00285 -0.10346
er,2p" (6.12719) 0.00101 0.01085 0.02593 er .3d (1. 68538) -0.00115 -0.03610
er.Jd (1.35983) -0.00052 0.03900 0.04278
er.Jd' (2.94342) 0.00100 0.03636 0.03053
er,3d" (5.53943) -0.00024 -0.00295 -0.00220
er,4f (2.58964) 0.00015 0.01333 0.01015
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1990 CADE, SALES, AND WAHL
TABLE XIII. HFR wavefunction for N2(lullu,,22ui2ui3ug21'11'",,\ X 12:.+), R=2.45 bohr.
E= -108.8792 hartree, T= 107.8664 hartree, V = -216.7456 hartree, V /T= -2.00939
EJ,.= -15.74469, E-, •• = -1.33758, E3 •• = -0.61231, Et.,,= -15.74327, E2vu= -0.83927, Et",,= -0.54184
C,Ap
Ct.u, ..
u.ts (\"=5.68298) 0.92428 -0.26335 0.07791 u"ls (5.95534) 0.93493 -0.25651 'll'"u2p (1. 33036) 0.51817
u.ls' (10.34240) 0.15191 -0.00607 0.00175 uu1s' (10.65879) 0.11462 0.00010 'II'",,2p' (2.53355) 0.38388
u.2s (1. 43321) 0.00102 0.19474 -0.36773 u u2s (1. 62946) -0.00755 0.42890 'll'"u2p" (5.69176) 0.03135
ug2s' (2.40158) -0.00069 0.61766 -0.19982 u,,2s' (2.47750) 0.00457 0.56616 'II'",,3d (2.02067) 0.06791
ug3s (7.04041) -0.08571 -0.01902 -0.00644 u,,3s (7.29169) -0.05404 -0.02780 'II'",,3d' (2.92400) -0.00057
u.2p (1. 32165) 0.00036 0.12585 0.47105 u,,2p (1.48555) -0.00484 -0.32965 '11'",,4/ (2.69290) 0.01139
ug2p' (2.56043) 0.00042 0.17350 0.45587 u,,2p' (3.49774) 0.00265 -0.09231
Ug2p" (6.21698) 0.00125 0.00827 0.02573 u,,3d (1. 65747) -0.00106 -0.03239
ug3d (1.40283) 0.00020 0.04476 0.05141
ug3d' (2.90518) 0.00052 0.02455 0.02455
ug3d" (5.52063) -0.00020 -0.00221 -0.00207
uN (2.15310) 0.00016 0.01583 0.01344
excellent interpolated values of E(R) and is a com
mendable method for getting flexibility in the range of
E(R) values. We have, alternatively, obtained results
for the eight R values indicated above to permit
freedom in selection of the R values and also to obtain
tp(R) curves which afford full flexibility to calculate
the wave/unctions and subsequently expectation values
for any R value. The details of the calculations for
N2(X l~u+) are now briefly considered.
The development of EHF(R) and the wavefunctions
in Tables IX through XIII for N2(X l~g+) was the
result of a rather lengthy series of calculations.42
Two R values (R= 1.85 bohr and 2.15 bohr), one on
either side of R.(exptl) and substantially away from
the calculated minimum, were chosen as starting
points to obtain EHF(R). The calculations began with
Basis Set 2D (Table IV) and all tp values except
those for O'gls', O'u1s', and 11'u2p" basis functions,
which have relatively large tp values, were indi
vidually reoptimized for these two R values. For
R= 1.85 bohr, Basis Set 2D gave an energy of
-108.9629 hartree before any optimization, and after
the 23 single reoptimizations, this value was de
creased only slightly to -108.9635 hartree and for
R= 2.15 bohr, Basis Set 2D gave a starting energy of
-108.9798 hartree and after the same 23 single r p
optimizations the energy was -108.9799 hartree.
Using these results for R= 1.85 and 2.15 bohr, as well
as the result for R= Re( exptl) = 2.068 bohr, calcula
tions were made for R= 1.95 and 2.05 bohr starting
again with Set 2D (obtained for R=2.068 bohr), but
this time singly optimizing only those tp's which
produced any significant improvement. Thus for these
42 We refer to the calculated Hartree-Fock-Roothaan potential
curve as simply EHF(R). That is, we believe our best result
is sufficiently close to the true Hartree-Fock potential curve to
avoid introducing an EHFR(R) curve. two additional points, at most only eight r p's were
singly optimized. The final wavefunctions for R= 1.85,
1.95,2.05, and 2.15 bohr are given in Tables IX, X, XI,
and XII, respectively, and careful comparison of the
rp values and vector components, C,'}.p, with the results
in Table IV for Set 2D, indicates which tp's were opti
mized and how these quantities changed for the various
R values.
Finally, to obtain values for the potential curve
to the limits of Rmin and Rmax given from the RKR
analysis (v=21) and to obtain more points to employ
for interpolation purposes, calculations were also
made for R= 1.65, 2.45, and 2.90 bohr, again starting
with Set 2D. For each of these internuclear separa
tions, 17 t p's were reoptimized (neglecting those
XpAa which are significant only for the inner shells)
and the final wavefunction for R= 2.45 bohr is given
in Table XIII. The results now included reoptimized t l' values for the same basis set composition at eight
R values and curves for interpolation purposes were
drawn for all t l' which show a significant variation
with R [these were curves for tp(R) for the 0'02s,
O'g2s', O'g2p, O'g2p', O'g3d, O'g3d', 0'04/; O'u2s, O'u2p; 11'u2p,
11'u3d, and 11'u4/ STO symmetry basis functions]' The
XpAa functions which have r p's which change sig
nificantly as a function of R are either important in
the 20' g, 30'0' 20' u, or 111' u molecular orbitals (that is,
have large vector components) or involve high lp
values (11'= 2 or 3). The STO symmetry basis func
tions in this latter category contribute improvements
in the energy upon optimization of r p's for various R
values not expected from the size of their vector com
ponents. It was relatively easy to interpolate rp values
for these STO symmetry basis functions for inter
nuclear separations between R= 1.80 and 2.50 bohr to
three or four significant figures. These results permit
the calculation of EHF(R) at many points, as further
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investigations required, with the expectation that the
results (waveiunction, energies, and so forth) are
accurate and essentially equivalent to optimizing the
orbital exponents again for each R value needed.
The preceding procedure of starting with the Har
tree-Fock-Roothaan wavefunction for N2 with R=
Re(exptl) =2.068 bohr (Set 2D) and carrying out a
long sequence of single r P optimizations, with no
double optimizations or reconsideration of the basis
set composition, is believed to be quite satisfactory to
obtain the Hartree-Fock potential curve over a small
range of R values around Re. This belief depends
strongly on the quality of the Hartree-Fock-Roothaan
result at Ro( exptl) , that is, on the quality of ap
proximation to the true Hartree-Fock result as dis
cussed in Sec. IILB. This is supported by the rela
tively small improvements in the energy upon optimi
zation of the orbital exponents, and is presumably due
in large part to the small change in the Hartrec-Fock
field with R over this limited range of R values and
adequacy of the large expansion basis set used to
absorb these changes via the CAP vector components
which indicate only a small readjustment as R goes
from 1.85 to 2.45 bohr.
The conclusion suggested from these calculations is
that reoptimization of orbital exponents for different R
values is not very important for a large basis set com
position if the R values are near the R for which the
basis set was originally constructed. Little direct
evidence is available, but presumably for small ex
pansion basis sets, including minimal or double-r basis
sets, reoptimization of the r p for different R values
becomes more important. These calculations show that
as 1 R-R.( exptl) 1 becomes larger, reoptimization of
the nonlinear variations parameters becomes more
important, especially for XpAa with lp= 2 or 3. Figure 5
clearly illustrates this former point in which energy
improvement I1E is the magnitude of the difference
between EHF(R) calculated using Set 2D with re
optimized rp values and EHF(R) calculated using rp
values of Set 2D. The squares are for R= 1.85, 1.95,
2.05, 2.15, and 2.45 bohr, as discussed above, and the
solid circles are at R values using interpolated r p
values. The plot of I1E versus R thus clearly shows the
sequence of reoptimizations of rp performed here con
tributes an additional lowering of less than 0.001
hartree for R near Re( exptl). The fact that I1E is not
zero at R.( exptl) is no doubt indicative that another
sequence of single reoptimizations of rp at R.( exptl)
would lower the energy by "-'0.00005 hartree, which is
consistent with our earlier belief expressed in Sec.
III.B. The scale in Fig. 5 is such that this improve
ment appears large, but a clearer perspective of this
relatively small effect is more evident in Fig. 6.
The final results for the potential curve of N2(X 12;g+)
include the original result from Set 2D at R= 2.068
bohr, results for seven other R values for which rp
values were reoptimized, results using interpolated r p 0.0010 •
dE •
..,
• •
0.0005 • • •
41> • • o.
0.0000 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5
R
FIG. 5. Improvement of EHF(R) for N2(X 12:.+) by reoptimi
zation of orbital exponents for various R values; • indicates
interpolated IP values and D are results using reoptimized IP
values. I1E in hartrees; R in bohrs.
values for the 12 turning points,39 Rmin(V) and Rmax(v),
for v=O, 1, 2, 3, 4, 5, and finally a number of addi
tional points using interpolated orbital exponents which
are chosen to fill gaps in the curve. A summary of the
energy quantities for N2(X 12;g+) is given in Table
XIV for a selected group of R values.
The quantitative comparison of the calculated
potential curve EHF(R) with the curve obtained by
Gilmore39 for N2(X 12;g+) is considered next. The
EHF(R) curve, the solid line in Fig. 6, is constructed
from the calculations at the turning points, Rmin(V)
and Rmax(v) , given by Gilmore.39 Thus for Rmin(v=O,
1, 2, 3, 4, 5) = 1.994, 1.941, 1.905, 1.878 1.858, and
1.839 bohr, respectively, and Rmax( v= 0, 1, 2,3,4,5) =
2.166, 2.239, 2.292, 2.340, 2.383, and 2.423 bohr,
respectively, two sets of calculations are shown on
the solid curve of Fig. 6. The solid circles are results
using interpolated or reoptimized r p values and the
squares are results using the rp values directly from
Set 2D. Several other points not at Rmin(V) or Rmax(v)
are also shown, but this curve emphasizes how in
consequential reoptimization of the r p values was for
various R values. The dashed curve was derived from
Gilmore's results for the first six vibrational states,
use of Eexptl= -109.586 hartree for N2(X 12;g+) , and
vertically elevating the resulting "experimental" curve
so that the minimum was parallel with the minimum
of EHF(R). This maneuver is to facilitate examination
of the displacement of R. and assess the quality of the
shape of EHF(R) for various vibrational states. The
experimental curve was therefore uniformly elevated
by the amount 1 Eexptl-EHF(Re) 1= 1109.586-
108.99561 =0.590 hartree, in which both E(R) values
are taken at their respective minimum.
The shortcomings of EHF(R) when compared to the
ERKB. curve are that (i) the EHF(R) curve is gen
erally much too high; (ii) the EHF(R) curve is shifted
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-108.89 . ~IOae9
-108.90 -IOUO
-108.91 -108.91
-108.92 -108.92
-108.93 ":108.93
-108.94 'i i -108.94 I ,
E , , E , , ,
\ 0 0
-108.95 , I -108.95 , , , ,
\ , " ,
0 0 -108.96 \ ,
I -108.96
\ I \ I
\ I , , , I
-108.97 0 0
~108.97 \ ,
\ I ,
\ ,
\ , , I , 1
-108.98 0 0 -IOUB \ 1 \\, 1 ,I
I
I
I
-108.99 ,
.. 0 0 -108.99 '-'" , , , ,
-109.00 -109.00
1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50
R
FIG. 6. Comparison of the shape of the EHF(R) and the elevated EBKR(R) potential curves for N2(X ll;g+); • indicates calcu
lated results using interpolated I'p values and 0 indicates calculated results using reoptimized r" values. The 0 points derive from
the RKR analysis of Gilmore. E is in hartrees; R is in bohrs.
as a whole inward to smaller R values relative to
ERKR; and (iii) the EHF(R) curve is sharply arcuated
in contrast to ERKR(R) and this is especially evident
on the large R side of EHF(R). If the solid curve of
Fig. 6 is moved to the right such that the minimum coincides with that of the dashed curve, then one
would see that for V= 0 or 1 the shape is quite reason
able, but for v;:::2 the EHF(R) already rises much too
high on the large R side. Thus one might expect that
molecular properties computed for v= 0, 1 using
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1993
TABLE XIV. Summary of energy quantities as a function of internuclear distance.
N2(ltT.21tTu22tT.22tT .. '3tT.21,.. .. 4, X 12:/)
R Ea T V VIT E2., E2.. Ea.r, fl ••
1.65b -108.7887 111.1275 -219.9162 -1.97895 -1.6292 -0.7002 -0.6534 -0.7355
1.82 -108.9489 109.9386 -218.8874 -1.99100 -1.5699 -0.7331 -0.6459 -0.6798
1.85b -108.9635 109.7690 -218.7325 -1.99266 -1.5584 -0.7387 -0.6446 -0.6709
1.905 -108.9825 109.4792 -218.4617 -1.99546 -1.5372 -0.7489 -0.6422 -0.6558
1.95b -108.9914 109.2666 -218.2580 -1.99748 -1.5194 -0.7570 -0.6401 -0.6438
2.0132 -108.9956 108.9959 -217.9914 -2.00000 -1.4953 -0.7687 -0.6379 -0.6285
2.05b -108.9944 108.8567 -217.8511 -2.00127 -1.4803 -0.7747 -0.6355 -0.6194
2.068b -108.9928 108.7911 -217.7839 -2.00185 -1.4736 -0.7780 -0.6350 -0.6154
2.0741 -108.9922 108.7698 -217.7620 -2.00205 -1.4712 -0.7790 -0.6346 -0.6140
2.09 -108.9904 108.7152 -217.7055 -2.00253 -1.4651 -0.7818 -0.6338 -0.6106
2.15b -108.9799 108.5254 -217.5053 -2.00419 -1.4421 -0.7918 -0.6306 -0.5974
2.20 -108.9679 108.3824 -217.3502 -2.00540 -1.4235 -0.8001 -0.6280 -0.5872
2.292 -108.9397 108.1586 -217.0983 -2.00722 -1.3902 -0.8149 -0.6226 -0.5693
2.34 -108.9226 108.0587 -216.9813 -2.00799 -1.3736 -0.8224 -0.6196 -0.5606
2.45b -108.8792 107.8664 -216.7456 -2.00939 -1.3376 -0.8393 -0.6123 -0.5418
2.90b -108.6828 107.4409 -216.1236 -2.01156 -1.2217 -0.9009 -0.5759 -0.4824
a All energy quantities are in hartree units (27.2097 eV) and internuclear sepa
rations are in bohr units (0.529172 A). b The rp values for these R values were separately optimized. Interpolated r p values were used for the remaining R values.
EHF(R) would reflect properly the shape effects of
the true potential curve, but that for v= 2 or higher
the shape effects would be progressively less well
represented. There would still be defects, however,
since the nuclei would come too close together in
vibration and would vibrate in a potential well much
too shallow. A more significant gauge of the quality
of EHF(R), and the effects of (i), (ii), and (iii) above,
would be to solve the vibrational-rotational problem
using EHF(R) and compare the results with experiment.
It seems certain that the major features of EHF(R)
are only slightly altered by reoptimization of orbital
exponents for A2 molecules when a large expansion
basis set is employed and in light of the over-all poor
quality of the potential curves given by EHF (R) ,
reoptimization of !p values for N2+(X 22;.+, A 2II", B 22;,,+) molecular ions was ignored. The largest
energy gain in the reoptimizations for N2(X 12;.+)
was "'0.001 hartree and a check reoptimization (that
for the N2+, X 22;.+ state at R= 2.0132 bohr) gave an
energy gain of 0.00006 hartree relative to the results
obtained using !P's optimized at R=R.(exptl) =
2.113 bohr for N2+(X 22;.+). It is a reasonable estimate
that reoptimization of the orbital exponents would
produce no gains greater than ",0.002 hartree and
probably half of this size. Therefore EHF(R) curves
for N2(X 22;.+, A 2II", B 22;,,+) were calculated using
the wavefunctions given in Tables V, VI, and VII,
respectively, obtained at the R.( exptl) values. The
resulting points for the potential curves are found in
Tables XV, XVI, and XVII for these three states
of N2+.
TABLE XV. Summary of energy quantities as a function of internuclear distance.
N2+(ltT.lltT .. 22tT.22tT .. I3tT.h,f, X 22:.+)
R Ea T V VIT E2.. E2.. Ea.r, Ell'.
1.80 -108.3389 109.5230 -217.8618 -1.98919 -2.0226 -1.1092 -1.1531 -1.1223
1.85 -108.3674 109.2387 -217.6061 -1.99202 -2.0002 -1.1170 -1.1485 -1.1045
1.95 -108.3999 108.7539 -217.1539 -1.99674 -1.9543 -1.1322 -1.1393 -1.0713
2.00 -108.4064 108.5488 -216.9552 -1.99869 -1.9313 -1.1398 -1.1344 -1.0559
2.0132 -108.4073 108.4984 -216.9056 -1.99916 -1.9252 -1.1417 -1.1331 -1.0519
2.035 -108.4079 108.4184 -216.8263 -1.99990 -1.9153 -1.1450 -1.1310 -1.0454
2.0375 -108.4079 108.4095 -216.8174 -1.99998 -1.9141 -1.1453 -1.1307 -1.0447
2.0385 -108.4079 108.4060 -216.8139 -2.00002 -1.9137 -1.1455 -1.1306 -1.0444
2.040 -108.4079 108.4007 -216.8086 -2.00007 -1.9130 -1.1457 -1.1304 -1.0440
2.050 -108.4078 108.3657 -216.7735 -2.00039 -1.9084 -1.1472 -1.1294 -1.0411
2.113b -108.4037 108.1612 -216.5649 -2.00224 -1.8807 -1.1569 -1.1235 -1.0237
2.15 -108.3986 108.0582 -216.4568 -2.00315 -1.8635 -1.1618 -1.1188 -1.0134
2.30 -108.3643 107.7187 -216.0830 -2.00599 -1.7995 -1.1832 -1.1012 -0.9757
2.40 -108.3333 107.5569 -215.8902 -2.00722 -1. 7597 -1.1970 -1.0883 -0.9529
2.60 -108.2626 107.3457 -215.6083 -2.00854 -1.6878 -1.2232 -1.0598 -0.9118
• Altenergy quantities are in hartree units (27.2097 eV) and internuclear b The rp values for this R value were re-optimized. The calculations for the
separations are in bohr units (0.529172 A). other R values employed the same set of orbital exponents.
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TABLE XVI. Summary of energy quantities as a function of internuclear distance.
N2+(luilUu22012Uu23uillTu3, A 211.)
R Ea T V
1.80 -108.3088 109.8977 -218.2065
1.85 -108.3484 109.6076 -217.9560
1.95 -108.4010 109.1093 -217.5102
2.00 -108.4165 108.8962 -217.3127
2.0132 -108.4196 108.8436 -217.2632
2.05 -108.4262 108.7045 -217.1307
2.125 -108.4319 108.4524 -216.8843
2.134 -108.4320 108.4248 -216.8568
2.140 -108.4320 108.4067 -216.8386
2.145 -108.4319 108.3918 -216.8237
2.15 -108.4318 108.3770 -216.8088
2.222b -108.4270 108.1820 -216.6089
2.30 -108.4154 108.0020 -216.4174
2.40 -108.3936 107.8136 -216.2072
2.60 -108.3357 107.5462 -215.8819
• All energy quantities are in hartree units (27.2097 eV) and internuclear
separations are in bohr units (0.529172 A).
IV. DISCUSSION OF RESULTS
The principal energy results are summarized in
Tables XIV, XV, XVI, and XVII for N2(X l~u+) and
N2+(X 2~u+, A 2IIu, B 2~u +), respectively. The varia
tion of EHF(R), THF(R), VHF(R), and Ei(R) with
internuclear separation for these four molecular sys
tems are shown in Fig. 7(a), 8(a), (b), and 9(a),
(b), (c), (d) in that order. The results for EHF(R)
are claimed to approach the Hartree- Fock accuracy
within at least 0.005 hartree and less for the other
quantities.
The potential curves for N2(X l~g+) and N2+(X 2~u+,
A 2IIu, B 2~ .. +) are given in Fig. 7 (a) and (b). Figure
7 (a) is the result of the present calculations as already
described and Fig. 7 (b) is replotted from data of
Gilmore.39 The Rydberg-Klein curves of Gilmore and
Fig. 7 (b) have been juxtaposed so that the calculated V!T E2.t. E<.?O'" <36, <1 ...
-1.98554 -2.0767 -1.1570 -1.0838 -1.1779
-1.98851 -2.0517 -1.1633 -1.0780 -1.1573
-1.99351 -2.0006 -1.1755 -1.0664 -1.1187
-1.99559 -1.9750 -1.1814 -1.0607 -1.1006
-1.99610 -1.9682 -1.1830 -1.0592 -1.0960
-1.99744 -1.9494 -1.1872 -1.0549 -1.0833
-1.99981 -1. 9117 -1.1956 -1.0463 -1.0586
-2.00007 -1.9072 -1.1966 -1.0452 -1.0558
-2.00023 -1.9043 -1.1973 -1.0445 -1.0539
-2.00037 -1.9018 -1.1978 -1.0440 -1.0523
-2.00051 -1.8993 -1.1984 -1.0434 -1.0507
-2.00226 -1.8644 -1.2062 -1.0349 -1.0289
-2.00383 -1.8278 -1. 2143 -1.0256 -1.0065
-2.00538 -1. 7833 -1.2245 -1.0132 -0.9799
-2.00734 -1. 7031 -1.2435 -0.9874 -0.9324
b The r p values for this R value were reoptimized. The calculations for the
other R values employed the same set of orbital exponents.
and "experimental" minima of N2(X l~u+) are exactlv
parallel. But note, that although the ordinate scaie
is the same in Figs. 7 (a) and 7 (b), and the abscissa
scale and range is identical, the ordinate range of
Figs. 7 (a) and 7 (b) is different. The objective is
simply to measure the internal spacings and relation
ships among the four states as calculated and compare
these with the experimental results. The small num
bered horizontal struts in Fig. 7 (b) indicate the various
vibrational states.
The first general impression in comparing the calcu
lated EHF(R) curves with the ERKR(R) curves is
that the N2+ states are all nearly correctly spaced
above the N2(X l~u+) curve. However, all Ew(R)
curves are shifted inward relative to the corresponding
ERKR(R) curves, all EHF(R) curves rise much too
rapidly at large R, and some significant relative
shifting has occurred among the N2+ states. The most
TABLE XVII. Summary of energy quantities as a function of internuclear distance.
N2+(luD21u,,22uD22uu3uit,,· .. , B 22:.+)
R Ea T V V!T E2.t, E2.r. ea., Elr ...
1.75 -108.2159 109.3476 -217.5635 -1.98965 -2.0205 -1.2217 -1.0288 -1.1281
1.80 -108.2430 109.0145 -217.2575 -1.99292 -1.9990 -1.2291 -1.0246 -1.1100
1.85 -108.2600 108.7123 -216.9724 -1.99584 -1. 9767 -1.2362 -1.0205 -1.0926
1.90 -108.2685 108.4384 -216.7069 -1.99843 -1.9540 -1.2430 -1.0164 -1.0761
1.925 -108.2700 108.3112 -216.5813 -1.99962 -1.9424 -1.2463 -1.0143 -1.0681
1. 9325 -108.2702 108.2743 -216.5445 -1.99996 -1.9390 -1. 2473 -1.0137 -1.0657
1.934 -108.2702 108.2670 -216.5372 -2.00003 -1.9383 -1.2475 -1.0136 -1.0653
1.95 -108.2699 108.1902 -216.4601 -2.00074 -1.9309 -1.2496 -1.0123 -1.0603
2.00 -108.2652 107.9655 -216.2307 -2.00278 -1.9076 -1.2559 -1.0083 -1.0451
2.0132 -108.2631 107.9098 -216.1728 -2.00327 -1.9015 -1. 2575 -1.0072 -1.0413
2.0315b -108.2596 107.8350 -216.0945 -2.00394 -1.8930 -1.2597 -1.0057 -1.0359
2.05 -108.2554 107.7621 -216.0176 -2.00458 -1.8845 -1.2620 -1.0042 -1.0306
2.15 -108.2240 107.4122 -215.6362 -2.00756 -1.8387 -1.2735 -0.9960 -1.0034
2.30 -108.1565 107.0061 -215.1626 -2.01075 -1.7728 -1.2892 -0.9833 -0.9665
2.40 -108.1034 106.7994 -214.9028 -2.01221 -1. 7314 -1.2989 -0.9745 -0.9443
2.60 -107.9885 106.5032 -214.4917 -2.01395 -1.6558 -1.3163 -0.9557 -0.9046
a All energy quantities are in hartree units (27.2097 eV) and internuclear b Thc r p values for tbis R value were re-optimized. The calculations for the
separations are in bohr units (0.529172 A). other R values employed the same set of orbital exponents.
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsELECTRONIC STRUCTURE OF DIATOMIC MOLECULES. III. A
-101.10
-101.15 , ,.
".
l /
i' 82 r:
\ I
-101.20 \ //
\\ ./l
-101.25 '. , .. x·
-101.30
-10135
-108.40
-108.45
-108.50
E(hartrH'
-1011.55
-108.60
-108.65
-108.70
-108.75
-108.80
-108.85
-101.95
-109.00 '&_ .... ~ I'
i ,ll
~ I // '\ i ,i'x' E,-/
1.1 "i ,.' /
• : .-' ,rf
-.;. ....... x# ., ...
ri /. j'n-."<DIJoo"'.".... A2 TT, , , , , , , ,
1 : ,
! ,
1 ,
! : :
l , , :
!
1 , , , , , , , , ,
! , , , , , , , , , , , , : , , , ,
2.0 N; STATES
2.2 2.4 LI 2.0 2.2
(a) (b) . -108.75
-108.80
-108.85
-108.90
-108.95
-109.00
N; STATES -109.05
-109.10
E(harINe)
-109.15
-109.20
-109.25
-109.30
-109.35
-101.40
109.45
-109.50
-101.55
-101.10
2.4 2.. 1995
FIG. 7. Potential curves, E(R), for N2(X 12;q+) and Nz+(X 22;q+, A 2n", B 22; .. +): (a) calculated EBJ'(R) results; (b) ERKa(R) results
of Gilmore. Note ordinate scale the same but ranges are different.
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111.O -214.0
110.0
109.0
T(hartr .. ) V (hartr .. \
108.0
107.0
106.0
1;8 2.0 2.2 2.4 2.6 1.8 2.0 2.2 2.4 2.6 -219.0
R (I;C1hr) R (bohr)
(a) (b)
FIG. 8. Calculated kinetic energy, T(R), and potential energy, VCR), curves for N2(X 1~.+) and N2+(X2~.+, A 2IIu, B 21:u+): (a)
kinetic energy curve and (b) potential energy curve.
significant feature, however, as noted earlier, is the
reversal of the N2+(X 2~g+) and N2+(A 2IIu) states
over a substantial range of R values. This reversal of
levels involves energy differences of the order of
",,0.02 hartree, which is considerably too large to be
explained as due to differential shortcomings of our
approximation to the Hartree-Fock result for these
two states. We are confident that this reversal is a
characteristic feature of the Hartree-Fock approxi
mation and hence must find an explanation in the
differential shortcomings of the Hartree-Fock results,
or differential correlation energy, for these two states
of N2+.
It may be recalled that the concept of bonding,
antibonding, or nonbonding orbitals first arose in
considering the change in R.( exptl) which resulted
when an electron vacated the particular molecular orbital in question. In as much as our claims of having
obtained to good approximation the molecular orbitals
are valid, that is the orbitals satisfying the physical
self-consistency requirement of this model, we can now
examine how the model behaves in this regard. Thus,
although all Re(HF) values are shifted inward relative
to the corresponding R.(exptl) values, and in spite
of the fact that the N2+(X 2~g+) and N2+(A 2IIu)
states are reversed, there is strikingly close analogy
between the ARe shifts of theory and experiment
upon ionization of N2(X l~g+). Thus for the loss of
the strongly bonding electron in 111'.. symmetry,
Re(exptl) goes from 2.0741 (N2, X l~/I+) to 2.222
bohr (N2+, A 2IIu), an increase of 0.148 bohr or +6.13%,
while R.(HF) goes from 2.0132 (N2' X l~g+) to 2.134
bohr (N2+, A 2IIu), a parallel increase of 0.121 bohr
or +6.0%. In an exactly analogous fashion AR.( exptl)
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1997
-1.00
-120
-1.40
((H)
-1.10
-1.10
-2.00
-0.80
-0.80
-1.00
e(H)
-1.20
-1.40
-1.60 Rlliohrl
(e) 2.80
N.(X'I;)
2.80 ·1.00
.1.20
.0-0-<1"-000
-1.40
£(H)
-1.60
-1.80
-2.00
1.60
-tOO
-1.20
-1.40
e(H)
-1.60
-1.80
-2.00
L80 ""'OoOo·':""·-o __ ..... o_-o-... __ .o.oa (2..,
2DO
2.00 220 2.40
III......,
(el)
2.20 2.40
Rlliohrl
(b) 2.60 2.80
FIG. 9. Variation of orbital energies, E2,., <36., E2.,., and Ei .. ,. with internuclear separation: (a) N2(X 12:.+), (b) NI+(X 12).+), (c)
N2+(A, 20,,), and (d) N2+(B,22),.+).
for the loss of a weakly bonding 3u g electron is + 1.88%
compared to the calculated value of + 1.26% and for
the loss of an antibonding 2u" electron, ~Re(exptl) =
-2.05% and ~Re(HF) = -3.93%. Thus in every
case the model gives good agreement and predicts
in each case less bonding in terms of ~R. than the
experimental results give. To good approximation
then, the far reaching concept of bonding and anti
bonding orbitals predicted by the theory seerns rela
tively immune to the shortcomings of the Hartree
Fock approximation. In a companion paper to the
present effort37 the rearrangement of the electronic
charge distribution upon loss of either a 3ug, h",
or 20-.. electron is considered in terms of the electronic
charge-density contours for the various orbitals them
selves. This latter study could not be done by using
only virtual orbitals, but requires directly calculated wavefunctions for NJ+(X 22;g+, A 2ll", B 22;,.+) ions
as obtained here.
Let us now consider certain quantitative aspects of
the total energy values, EHF(R), ionization potentials,
dissociation energies, and the energy quantity ~ (R),
defined in Eq. (IlL8). Since relativistic effects are
completely neglected in these calculations, ~ (R) is
not the variation of the usually defined correlation
energy with R because ERKR(R) contains implicitly
all the physics of the problem as communicated by the
experimental data used in the RKR analysis. There
fore, we refer to ~(R) as a quantity which measures
the variation with R of the shortcomings of a given
approximate wavefunction, or in the present case, of the
Hartree-Fock wavefunction. The ~(R) = ERKR(R)
EHF(R) curves for N2(X 12;g+) [md N2+(X 22;g+,
A 2ll", B 22;,,+) states are given in Fig. 10. The numbers
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissions1998 CADE, SALES, AND WAHL
-0.50..-----------------.
-0.55
4(R)
R (bohr)
used to obtain these curves come from the calculated
results for EHF(R) at Rmin(V) and Rmax(v) and the
results of Gilmore for ERKR(R) at the corresponding R
values. These curves all show the characteristic de
generation as R increases and are not slowly varying
functions of R around R.( exptl) (the gradient varys
from about 0.11 hartree/bohr for N2+, X 21;g+ to about
0.23 hartree/bohr for a part of the N2+, B 21;,,+ curve).
It is also observed that the shortcomings of EHF(R)
are only slightly worse for N2+(X 21;g +) than for
N2(X 12;0+)' but these two t::.(R) curves are very
close together over a substantial range of R values.
This is also seen in the potential curves for N2(X 12;0+)
and N2+(X21;0+) states in Fig. 7(a). Thus, except for a
constant and small relative elevation of the EHF(R)
curve for N2+(X 22;0+)' these two curves behave very
similarly to their counterparts in Fig. 7 (b). The point
to be made is that breaking of the 30'02 pair in going to
the X 21;0+ state of N2+ is of but minor consequence as
far~as the correlation energy is concerned. Or another
way of putting it, these Hartree-Fock calculations on
N2(X 12;0+) and N2+(X22;0+) are closely comparable
in quality. In contrast to this, the t::.(R) curve for the
A 2II" state of N2+ indicates that, relatively speaking,
the Hartree-Fock result is much better for this state
than for N2, and the final t::.(R) curve shows that the
B 21;,,+ state of N2+ is, relatively, the worse treated
by Hartree-Fock approximation. Thus the calculated
EHF(R) curve for the A 2II" state of N2+ is relatively
lower than its experimental counterpart in Fig. 7 (a) ,
(b) because the Hartree-Fock result for N2+(A 2II,,) is
much better than corresponding results for N2+(X 21;0+)
and N2(Xl1;g+). It is probably no coincidence that this behavior is associated with the loss of a strongly
bonding 1'11'" electron in forming N2+(A 2ll,,) from
N2(X l1;g+). In terms of breaking electron pairs the
results of t::.(R) for the N2+(B21;,,+) state resulting
from breaking up the 20',,2 pair is also significant.
Clearly there is substantial information in considera
tions of this kind about correlation energy, but until
more results are obtained for other homonuclear
systems, the authors will not attempt to suggest any
general rules. The authors have no explanation of
the reversal of the X 21;0+ and A 2llu states of N,+
relative to experiment and know of no explanations in
the literature. Several attempts have been made by
Clementi43 and others to construct empirical correla
tion energy corrections to be added.on to the Hartree
Fock energies for molecules to give accurate dissocia
tion energies. Our view is that insufficient data is pre
sently available to construct such empirical recipes
and this is why we introduce the quantity t::.(R). It is,
we feel, necessary to get more results for more electron
configurations, for variable differences of nuclear
charges, and to obtain these results as a function of
internuclear separation.
In Sec. IILD the defects of calculated potential
curve for N2(X 11;0+) were ascribed to an elevation
factor, a shift factor, and a shape factor. We may now
ask what these divisions mean, if anything, in terms of
the t::.(R) quantity and whether any useful quantita
tive or interpretative advantage may be gained from
them? Let us first define the quantity
oR=Re(exptJ) -R.(HF), (IV.1)
that is, the difference between the calculated equi
librium value, R.(HF), and the experimental R.
value. For the N2(X 11;0+) and N2+(X 21;0+' A 2ll '"
B 21;,,+) states the shift in the position of the minimum
is oR= +0.061, +0.075, +0.088, and +0.098 bohr,
respectively. A constant elevation factor for each state
may also be defined as
8= ERKR[R.( exptJ)]-EHF[R.(HF)], (IV.2)
which in magnitude is just the height of the minimum
of EHF(R) above the minimum of ERKR(R). The
quantity t::.(R) can now be written
t::.(R) =ERKR(R) -EHF(R) =8+ERKRO(R) -EHFO(R)
= 8+ [ERKRO(R) -EHFO(R-oR)]-a(R),
(IV.3)
in which ERKRO(R) and EHFO(R) refer to energies
relative to their respective minima and are always
positive. By definition a(R) is
(IV.4)
4. E. Clementi, J. Chern. Phys. 38, 2780 (1963); 39, 487 (1963).
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I I I. A 1999
The constant 8 is thus the elevation factor, a(R) is the
shift factor, or the correction to be subtracted at each
R value to bring EHF(R) upon ERKR(R) such that the
minima are coincident, and ERKRO(R) -EHFO(R-oR)
is the shape factor44 measuring how the two curves
with coincident minima differ in details of their shape
with R.
The major theoretical shortcomings of EHF(R)
relative to ERKR(R) are that (i) EHF(R) does not go
to the correct atom states upon dissociation; (ii)
EHF(R) is a result which does not take account of the
instantaneous interactions between the electrons, the
usually defined correlation defect; and (iii) EHF(R)
is the result of a calculation in which relativistic
effects are neglected. Can these theoretical defects be
meaningfully associated with the elevation, shift, and
shape factors just defined? Now the constant 8 is
-0.590, -0.605, -0.539, and -0.627 hartree for
N2(X 12:g+) and N2+(X 22:g+, A 2IT", B 22:,,+), respec
tively, and it is reasonable to associate this R-inde
pendent elevation factor primarily with the neglect
of correlation and relativistic effects in the inner shells,
defects also present in the corresponding calculations
for the separated N(4S) atom and N+(3P) ion. A
part of 8 must, however, be associated with another
source since, as is discussed below, the rationalized
dissociation energies are still only about 50% of the
experimental value. This must be due to the inclusion
in 8 of a large part of the change in the correlation
energy of the molecule with R, which may be viewed
as an averaged background. Superimposed on this
average background (and not a part of 8) would be the
change in the correlation energy of the outer electrons
in going from the valence shells of the atoms to the
outer molecular orbitals. These conjectures are con
sistent with the relative sizes of the 8 values for these
four electronic systems of N2 and N2+, and, pre
sumably, the differences in 8 for these states are largely
associated with their differences in correlation energy
since one might expect that relativistic defects for the
molecules are approximately the same as for the sepa
rated atoms, except if R is very small.
The shape factor given by ERKRo (R) -EHFO (R -oR) ,
seems to be dominated by the fact that EHF(R) does
not approach the energy of the separated ground-state
atoms (or atom and ion) as R increases. The shift
factor, expressed either in terms of a(R) or oR, indi
cates that, in the Hartree-Fock approximation for the
N2 molecule and the N2+ molecular ions, the nuclei
are more shielded from one another by the electron
charge distribution between them than is actually
the case.45 Thus, as might be anticipated, the neglect
of the instantaneous interactions between the elec
trons will increase the electron density in regions where
44 A related quantity is defined by A. D. McLean, J. Chern.
Phys. 40,243 (1964).
45 Most aR values calculated have been positive for other un
published diatomic molecules and ions but some are negative. it is already high (and, of course, the instantaneous
interactions are more important). The shift factor is
thus largely ascribable to the R-dependent portion
of the correlation energy although it must also be
associated with a sort of "stovepipe" effect due to the
rapid rise of EHF(R) for large R.l4,44
In summary then, the arbitrary division of the
defects manifested in ~(R) and the association of the
elevation, shift, and shape parts with known theoretical
shortcomings seems at this point only heuristic. This
discussion was intended to attempt an assessment of
the shortcomings of EFtF(R) , staying as close to
experiment as possible, and to emphasize the im
portance of considering the various defects as a func
tion of internuclear separation. The usefulness of
~(R), 8, a(R), and oR will be critically examined for
entire homologous and/or isoelectronic series of mole
cules at a later date. These concepts may offer an
alternative or a supplement to the present few empirical
schemes to obtain dissociation energies for diatomic
molecules by estimating correlation energies, and
possess the advantage of being a quasidirect com
parison with experiment. The proposed analysis in
terms of this division of defects of an arbitrarily calcu
lated potential curve, from results beyond the Hartree
Fock approximation by configuration interaction or
perhaps by completely alternative methods, for ex
ample, might also serve as an objective standard for
interpretive comparison of merits. It must be re
membered, however, that such a division is completely
arbitrary. Some useful data is expected to emerge,
however, such as studies of oR versus I ZA -Z's I or
other parameters characteristic of the molecular
systems.
The calculated dissociation energy, D.( R), is the
"rationalized" value, which is defined for a diatomic
molecule, AB, by
where D.(R) includes the zero-point energy and is
given here as a function of R. The energy quantities,
EHF(AB)(R), EHF(A), and EHF(B) are, respectively, the
calculated Hartree-Fock energies of the diatomic
molecule, AB, and the infinitely separated parts,
A and B, all in the appropriate electron configuration
and state. De(R) as defined can be positive or negative,
the latter simply corresponding to no binding if true
for all R values. The rationalization is that, ideally,
one expects the molecular Hartree-Fock result to
approach the appropriate atomic Hartree-Fock results
as R becomes very large, so that De(R) at R.(HF)
would be a good approximation to the true dissociation
energy (D.) except for the difference between the
correlation (and relativistic) corrections for the AB
and A + B systems.
In calculating De(R) we have used EHF(N, 4S) =
-54.40093 hartree22 and EHF(N+, 3P) = -53.88799
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hartree and find that D.(R=2.0132 bohr) =0.1937
hartree (5.27 eV) for N2(X l~g+), D.(R=2.0385
bohr) =0.1190 hartree (3.24 eV) for N2(X 2~g+),
D.(R= 2.134 bohr) =0.1431 (3.89 eV) for N2+(A 2IIu) ,
and De(R= 1.934 bohr) = -0.01874 hartree (-0.509
eV), unbound, for N2+(B 2~u+). These D.(R) values
are quoted for R=R.(HF) and not for R=R.(exptl).
These results reflect the shortcomings already dis
cussed, but are far better than most previous approxi
mate results, being 53% and 37%, respectively, of the
experimental value [9.90 eV for N2(X l~g+) and 8.86
eV for N2+(X 2~g+)]. In Fig. 1, which shows the
improvement in the total energy as the expansion
basis set is built up in Scheme 1, it is evident that
N2(X l~g+) is bound relative to the Hartree-Fock
atom results (also indicated on Fig. 1) from Set lE
onwards. The D.(R) curve can be obtained from
polynomial curves for EHF(R) and EHF(N, 4S) and
EHF(N+, 3 P). There have been several recent pro
posals of alternative ways of calculating De, such as
that given by Stanton,46 and the method suggested by
Richardson and Pack.47
The ionization potential 1.(R) is defined by
-1.(R) = EHF(AB)(R) -EHF(AB+)(R) , (IV.6)
for the ionization of AB to form AB+. This definition
includes ionization potentials corresponding to excited
states of the molecular ion AB+, in which case the
appropriate EHF(AB+)(R) curves must be employed,
and double ionization potentials can be similarly
defined. The ionization potential, 1.(R), is the "ver
tical" ionization potential for AB if R=R.(HF),
except that 1.(R) includes the zero-point energy of
AB, and with this correction is the quantity allegedly
measured as electron-impact ionization potentials.
The related "adiabatic" ionization potential is defined
by
-1.(a)= EHF(AB) (R.) -EHF(AB+) (R.') , (IV.7)
that is, the difference between the minimal energy of
AB at R. and AB+ at R.' (R. and R.' are the Hartree
Fock minimal values). The experimental "adiabatic"
ionization potential measure the difference between
the energy of AB(v=O, J=f1.) and AB+(v=O, J=f1.),
so that Va) is slightly modified by the difference of the
zero-point energies of AB and AB+. It should be
recalled, however, that the energy values, EHF(AB+) (R),
are for direct calculations for the molecular ions in
question.
The "vertical" ionization potentials of N2(X l~g+)
to form N2+(X 2~g+, A 2IIu, B 2~u+) molecular ions,
using results for each obtained for R= R.(HF) =
2.0132 bohr, the minimum for the EHF(R) curve of
N2(X l~g+), are calculated to be 1.(R= 2.0132 bohr) =
16.01, 15.67, and 19.93 eV, respectively. The corre-
48 R. E. Stanton, J. Chern. Phys. 36, 1298 (1962).
47 J. W. Richardson and A. K. Pack, J. Chern. Phys. 41, 897
(1964). sponding experimental values of Frost and McDowell48
after adding the zero-point energy of N2(X l~g+) are
15.77,16.98, and 18.90 eV (although no state specifica
tion is obtained by the electron-impact measurements).
The directly calculated "vertical" ionization potentials
are thus 1.5% and 5.4% too high for formation of
N2+(X 2~g+) and N2+(B 20'u+) and 7.7% too low for
the formation N2+(A 2II,.). Figure 9(a) indicates the
variation of the orbital energies, E2<ru' E3Vg' and Ex..u,
of N2(X l~g+) over a small range of internuclear
separation and for R= 2.0132 bohr, the "vertical"
ionization potentials using Koopmans' theorem are
17.36 eV (+ 10.1 % in error) for the loss of an 30'g
electron, 17.10 eV (+0.71 % in error) for the loss of an
111'0. electron, and 20.92 (+ 10.7% in error) for the
loss of an 20'0. electron. These values indicate that, as
expected, the directly calculated values are generally
better and are especially good (1.5%) when the cor
relation effects in the neutral and ionized state are
very similar. It is also to be noted that the "vertical"
ionization potential to form N2+(A 2IIu) given by
Koopman's theorem is astonishingly, and probably
fortuitously, good. The full 1.(R) curves can be
obtained by use of the EHF(R) polynomial curves.
One final set of "vertical" ionization potentials
might also be mentioned which involves the loss of two
electrons by N2(Xl~g+) to form N22+(?). Certain
double-ionization processes involving nitrogen have
recently been reexamined by Dorman and Morrison,49
who find one ionization potential at 42.7 eV and an
other at 43.8 eV, but no clearcut association with
particular states of N22+ is available from these elec
tron-impact measurements. The N22+ calculations
given in Table VIII are crude compared to the N2
and N2+ results in that no reoptimization of orbital
exponents was attempted, and it must be remembered
that the Coulombic repulsion of the N+ ions can easily
dominate the potential curve of any N;+ species.5O
For the double "vertical" ionization potentials of
N2(X l~g+) we calculate that 1.(R= 2.0132 bohr) =
43.77 eV to form N22+(20'u230'g2111'u2, 8~g-), 45.20 eV to
form N22+(20'u230'g2111'o.2, l.1g), and 46.18 eV to form
N22+(20'u230' gOb'u\ l~g+), respectively. The
N22+(20'u230'g217ru2, l~g+)
state is probably also close by as are a number of
other states involving 20'u230'g111'u3, 20'u30'g111'u\ and
20' .. 30'g217ru3 electronic configurations. Explicit knowl
edge as to which of these states are bound, if any, is not
available from calculations performed. The 1.(R=
2.0132 bohr) given are probably too high by about
48 D. C. Frost and C. A. McDowell, Proc. Roy. Soc. (London)
A230, 227 (1955).
49 F. H. Dorman and J. D. Morrison, J. Chern. Phys. 39, 1906
(1963).
00 It is for this reason that highly ionized states of molecules,
such as N23+, N24+, etc., are not useful in understanding correlation
energy in molecules. This is in contrast to the situation for atoms
and their successive ions.
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0.2-0.5 eV due to relative crudeness of the N22+ wave
functions when compared to the N2(X l~u+) results.
We cannot make any convincing identification of the
states of Nt2+ involved in the measurements of Dorman
and Morrison.
The "adiabatic" ionization potentials of N2(X l~g+)
to form N2+(X 2~g+, A 2II", B 2~,,+) are well known
from spectroscopic observations of Rydberg series of
N2 and, in general, ionization potentials measured in
this way are the most accurate obtained. For ioniza
tion of N2(X l~o+) to the X 2~g+, A 2II", and B 2~,,+
states of N2+, the experimental adiabatic ionization
potentials are [.<8)= 15.585, 16.74, and 18.744 eV,
respectively, using the values quoted by McDowell51
and the relationship
I.(a)= lo(a)+!(w.-w.') -i(w.x.-w.x.')
+i(w.y.-w.y.') , (IV.8)
where the unprimed quantities are for N2(X l~g+)
and the primed values are for the various states of
N2+, and lo(a) is the experimental "adiabatic" ioniza
tion potential. The zero-point corrections which are
smaller than the uncertainty in 10(8) are neglected
[usually the last two terms in Eq. (IV.8)]. The adia
batic ionization potentials calculated here are I.(a) =
15.99 eV (X 2~g+), 15.34 eV (A 2II,,); and 19.74 eV
(B 2~,,+) and it should be noted that the calculated,
and not the experimental, R. values are taken as the
minimal point in the EHF(R) curves of N2 and N2+.
The adiabatic ionization potentials are again in very
good agreement with experiment asmight be expected
since the R.(HF) values are shifted only slightly in
going from N2 to N2+.
From these calculations on the ionization potentials
we see that ionization potentials can be calculated to
within 10% at worse (using Koopman's theorem), but
to within about 5% using directly calculated results.
Furthermore, in those few cases where the Hartree
Fock approximation is equally good for both neutral
and ionized system, ionization potentials can be pre
dicted to within 1% or 2%. Therefore it is likely that
Hartree-Fock calculations of the type described here,
guided by more experience, can help to identify the
state of the ion that a Rydberg series is convergent
upon. It should also be feasible to construct empirical
schemes based upon Hartree-Fock results and esti
mates of differential correlation energies of AB and
AB+ to give I.(R) or I.(a) just as schemes are now
being proposed to calculate D. by Clementi43 and
others.
It is traditional to calculate spectroscopic constants
once a potential curve, such as EHF( R), has been
calculated. This is usually done by performing the
61 C. A. McDowell, in Mass Spectrometry, edited by C. A.
McDowell (McGraw-Hill Book Company, Inc., New York,
1963), p. 544. analysis introduced by Dunham52 in which a poly
nomial fit is made to EHF(R). We have calculated
spectroscopic constants, but with certain misgivings.
As is evident from the potential curves in Fig. 6, the
sharp rise of EHF (R) for large R casts doubt on any
conclusions which might be drawn about quantities
calculated using EHF(R) at large R values. Stanton,46
Leies,53 Goodisman,54 and McLean,44,55 among others,
have recently expressed certain ideas about the quality
of potential curves, and especially the quality of spec
troscopic constants calculated from Hartree-Fock
results for various internuclear separations. In par
ticular, Stanton has conjectured, " .•• that Hartree
Fock potential energy surfaces and exact potential
energy surfaces are parallel over short distances."
McLean44 has questioned certain of Stanton's ideas
based on calculations for H2 and LiF and the results
of the present paper are also in disagreement with
Stanton's conjecture quoted above. The spectroscopic
constants presented in Table XVIII are not given with
the expectation that results of great accuracy are
possible, but rather to examine their deficiencies.
A number of different polynomial fits were made to
EHF(R) for N2(X l~g+) in which the number of points
was varied from 4 to 10, and the over-all symmetry of
distribution of points was also independently studied.
Certain of the second-order spectroscopic constants
were observed to vary wildly, especially when a small
left or right asymmetric distribution of points on
EHF(R) was employed, and generally the results were
worse when one point was very near the minimum of
the curve. For example, it was observed in this study
for N2(X l~g+) that 2681::S;w.::S;2773, 2.85::S;w.x.::S;
3975, -7.072::S;w.y.::S;5.588, 2.119::S; B.::S; 2.124, and
O.0091::S;a.::S;O.0187 for the various extremes of these
results. Clearly then, a reasonable choice and dis
tribution of points is obligatory and the use of a
symmetrical 10-point polynomial in R was adopted
for these states of Nt and N2+.
The resulting spectroscopic constants obtained from
the usual Dunham analysis are compared in Table
XVIII with the experimental results. It is concluded
that most spectroscopic constants are quite unreliable,
but that at least for B., a., and w. the calculated results
are always wrong in the same direction, i.e., too high
or too low. It may be possible for the theory to dis
count the assignment of a certain observed state if the
observed B. or w. values are outside the ranges
implied above, but positive predictions seem im
possible with only this accuracy for A2-type molecules.
These remarks, however, are made from the results on
N2 and N2+ molecular ions, and for other cases, for
example, the diatomic hydrides, AH, the situation may
be more favorable.
&2 J. L. Dunham, Phys. Rev. 41,713,721 (1932).
63 G. M. Leies, J. Chern. Phys. 39,1137 (1963).
64 J. Goodisman, J. Chern. Phys. 39, 2396 (1963).
II A. D. McLean, J. Chern. Phys. 40,2774 (1964).
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TABLE XVIII. Comparison of calculated and experimental spectroscopic constants ior
N2(X 12:.+) and N2+(X 22:.+, A 2II", B 22:,,+).
System Source B. a. "'. w.x .. "'.Y. R.(bohr) k.XlO-s
N2(X 12:.+) Experimental- 1.9987 0.01781 2358.07 14.188 -0.0124 2.0741 2.291
Calculatedb 2.121 0.01347 2729.6 8.378 -0.4745 2.0134 3.073
% Error +6.1% -24.4% +15.8% -41.0% -2.9% +34.1%
N2+ (X 22:.+) Experimental 1.932 0.020 2207.19 16.14 2.113 2.009
Calculated 2.065 0.01481 2570.5 9.809 -0.2462 2.0405 2.726
% Error +6.9% -25.9% +16.4% -39.2% -3.4% +35.7%
N2+(A 2II,,) Experimental 1. 740 0.018 1902.84 14.91 2.222 1.493
Calculated 1.887 0.01550 2312.5 6.082 0.9114 2.1344 2.206
% Error +8.4% -13.9% +21.5% -59.2% -4.0% +47.8%
Nz+(B 12:,,+) Experimental 2.073 0.020 2419.84 23.19 2.0315 2.415
Calculated 2.296 0.01280 3101.8 19.88 2.087 1.935 3.969
% Error +10.8% -35.9% +28.2% -14.3% -4.8% +64.3%
-The experimental values are taken from the review of Loftus.
b The calculated values are the results from a Dunham analysis using 10 points of the EHP(R) curves.
It has been suggested by Goodisman54 that spec-It is interesting to note that the order of the four states
troscopic constants might be more accurately calcu-is identical for both T(R) and VCR) and that no
lated employing the force on the nuclei as a function of curve crossing between the A 2Il ... and X 22;g + states
internuclear separation. This is explored in Paper of N2+ is observed in either T(R) or VCR). In fact,
IILB, where expectation values and molecular proper- except for the B22;,,+ state of N2+, the T(R) and
ties are considered. Leies,53 who states that "vibrational VCR) curves seem nearly completely parallel over
properties derived from the SCF function would be the entire range of R values considered. That T(R)
valid," which contradicts the results cited, suggests and VCR) for the A 2Il" and X 22;g+ states of N2+ are
the direct comparison of t:.G values from the numerical nearly parallel is not inconsistent with the result shown
solution of the nuclear vibration-rotational problem in Fig. 7, in which the EHF(R) curves for these two
using EHF(R) and this study is in progress for N2 states cross at approximately R= 1.92 bohr. It is
and N2+ molecular ions. These two alternative methods also noted that the loss of an electron from N2(X 12;0+)'
for the calculation of spectroscopic constants are regardless of the resulting N2+ state, lo'wers the kinetic
important since the Dunham analysis is, generally energy curve, T(R), much less than it raises the po
speaking, rather crude. In a Dunham analysis per-tential energy curve, V (R). These curves are plotted
formed using the turning points and energy values from results given in Tables XIV through XVII.
calculated as described in Sec. IILD for N2(X 12;0+)' A relatively narrow cross section through the
the spectroscopic constants were found to be B.= molecular orbital correlation diagram for N2 and N2+ is
2.00087, w.=2271.4 cm-t, a.=0.0406, and w.x.=71.65 shown in Figs. 9(a), (b), (c), (d). These plots of
cm-1• This included only the potential curve for V=O E2<r.(R), E2<r.(R), E2<r.(R), and El.-.(R) are linear except
through 5, which is why W.Xe was so bad, but it does for slight curvature over a narrow range of R values.
indicate the best quality of results for B. and w. that The Ei for the N2+ ions are, of course, substantially
can be obtained from a Dunham analysis. Embellish- lower than those for N2 although the internal spacings
ments of the Hartree-Fock potential curve to improve and gross characteristics are comparable. The only
the spectroscopic constants, such as suggested by feature of interest here is the crossing of the Eau. (R),
McLean,55 seem to be relatively ineffectual in view E2u.(R), and f1".(R) curves. The order of the orbital
of the shortcomings of EHF (R) . energies at large R is E2u. < E21T. < fau. < El.-. for N2(X 12;0+)
The variation ofthe kinetic and potential energy with and N2+(X22;g+, A2Ilu, B22;,,+) although their rela
internuclear separation for N2(X 12;0+) and N2+(X 22;g+, tive separations vary. The E1r. is seen to cut across both
A 2Il", B22;,,+) is shown in Figs. 8(a) and 8(b), E2u. and EalT. (except for the X 22;g+ and B22;,,+ states,
respectively. The potential energy curves include in which cases the second crossing seems to be for R
the nuclear repulsion term. The major, and well-known less than the smallest R shown). Similarly the E2<r"
feature is the decrease in magnitude of both the curve rises and apparently crosses both E3IT. and El.-._
kinetic and potential energy as R increases. The rate of . The order at small R thus is E2u.<El.-.<EalT.<E2<r ••
decrease is large when R is small, for example, less than&Therefore the actual Ei values seem to behave well in
R.( exptl), and gradually becomes smaller as R~ co •• correlating with the separated and united atom situ a-
Downloaded 23 Apr 2013 to 129.174.21.5. 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_permissionsE LEe T RON I CST Rue T U REO F D I A TOM I C MOL E C U L E S. I r I. A 2003
tions and the only new information is the precise
behavior of the Ei(R) over a narrow range of Rand
the intersections of these E;(R).
v. CONCLUSIONS
The extensive set of calculations which form the
basis of the presentation in this paper and the studies
by WahP for F2 and by Hu02 for CO and BF comprise
the only thoroughly documented attempts at a critical
study of the convergence of a hierachy of Hartree
Fock-Roothaan wavefunctions to the true Hartree
Fock wavefunction for molecules, except for the study
by Kolos and Roothaan40 for the hydrogen molecule.
On the basis of these calculations the following con
clusions are submitted:
(1) The solution of the Hartree-Fock-Roothaan
equations for a carefully selected, and sufficiently
large, expansion basis set in terms of STO functions
seems to approach the true Hartree-Fock solution as
evidenced in terms of certain energy quantities. This
conclusion is not very surprising in light of the re
markably successful Hartree-Fock calculations for
first-row atoms by BGRC22 and the reasonable ex
tension of these methods to another electronic system
which differs basically only in having a bicentric char
acter. Moreover, the accurate representation of the
molecular orbitals is realized by a practicable pro
cedure, feasible for all first-row homonuclear diatomic
molecules once sufficient information is available from
a few exhaustive pilot calculations.
(2) The starting basis sets for the molecular calcu
lations should incorporate Hartree-Fock-Roothaan
basis sets and wavefunctions for the atoms which are
simultaneously as accurate and as small as possible.
(3) It is imperative to introduce expansion basis
functions in Ug or Uu and 71'u or 71'g symmetry which
have lp= 2 and 3 (d-type and f-type STO functions)
as is also emphasized by Nesbet,14 This is evident
from the discussion given in Sec. III and other im
portant manifestations will be considered in Paper
III.B in regard to certain molecular properties for N2
and N2+ ions. The minimal basis set9•10 and the double
zeta basis setl4 are seen to have serious shortcomings
when viewed comparatively in the over-all perspective
of the basis set synthesis. The conclusion is that
there is no particular reason to obtain such inter
mediate results if the objective is to obtain the Hartree
Fock wavefunctions and the concomitant molecular
properties. These observations should suggest the
exercise of caution in considering the merits of results
for diatomic molecules or polyatomic molecules which
employ minimal basis sets. Ramifications of this
problem are also considered in Paper III.B in relation
to certain molecular properties.
(4) A certain degree of ambiguity should be recog
nized in employment of the term "molecular orbital."
We are referring here to the molecular orbital, that is the canonical set of functions cf>iAa which satisfy the inde
pendent particle model, and not functions which are
simply of proper symmetry and are "delocalized." The
molecular orbitals inferred in the study of molecular
electronic spectroscopy are clearly only approximately
the molecular orbitals as given by the independent
particle model. This is exhibited in the present study
in the reversal of the levels of the N2+(A 2llu) and
N2+(X 2};g+) states and the Ea.-. and El,.. for N2(X l};g+)
relative to experiment. It is claimed by certain authors
that major vestiges of the orbital concept are retained
in elaborate theories beyond the Hartree-Fock model
which include the instantaneous interactions of the
pairs of electrons of the system. This or similar ex
planations must reconcile the molecular orbital of
theory and experiment in certain situations.
(5) Although the calculated "rationalized" dis
sociation energies are still quite bad and the potential
curves are all too high relative to experimental results,
both not especially unexpected results, the calculated
internal spacings of ionized states are quite well pre
dicted. The prospects of careful studies of the varia
tion of the correlation energy with internuclear sepa
ration and upon selective ionizations are thus quite
good. For complete homologous series and certain
isoelectronic series such raw data for the possible
success of empirical correlation energy are thus close
at hand.
(6) Spectroscopic constants from Hartree-Fock
calculations obtained by the Dunham analysis of the
potential curve are quite unreliable.
(7) A very encouraging result of these calculations
is the relatively high accuracy obtained for the directly
calculated ionization potentials. The accuracy, which
should be about the same for other systems, indicates
that we can calculate ionization potentials to within
about 5%. We have also seen that in some cases the
Koopmans' theorem ionization potential may be
much more accurate.
ACKNOWLEDGMENTS
The authors have profited considerably from the
assistance of our colleagues in the execution of the
research described herein. We are grateful to Dr.
P. S. Bagus and Dr. T. L. Gilbert, of the Solid State
Division, Argonne National Laboratory, for free use of
their unpublished Hartree-Fock wavefunctions for the
first row atoms and for some useful advice. Frequent
conversations with Dr. J. B. Greenshields, Dr. Winifred
Huo, Dr. B. J. Ransil, and Dr. B. Joshi, among others,
have been most helpful in serving as useful criticisms
of this work in progress. Professor Clemens C. J.
Roothaan has generously provided seasoned and
welcome criticisms of the arguments and results given
here and it is a pleasure to acknowledge his sustained
encouragement and support. Finally, it is a privilege
to thank Professor Robert S. Mulliken for many useful
suggestions and criticisms.
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1.1703254.pdf | Investigation of TripletState Energy Transfer in Organic Single Crystals by
Magnetic Resonance Methods
Noboru Hirota and Clyde A. Hutchison Jr.
Citation: The Journal of Chemical Physics 42, 2869 (1965); doi: 10.1063/1.1703254
View online: http://dx.doi.org/10.1063/1.1703254
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Published by the AIP Publishing
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IP: 141.218.1.105 On: Sun, 21 Dec 2014 23:17:23CLASSICAL EQUATION OF STATE 2869
range both Be and B. decrease from 0 through negative
values to -00 never satisfying condition (B). Con
dition (A) must determine the singularity in P. There
is an explicit singularity in either B expression at the
volume of close packing (y= 1) but it is possible as
V~OO that at a density less than that of close packing
an exact expression for B(p) would have another singu
lar point determined by other singularities not on the
positive real p axis but in the complex plane. In any
case, our directly deduced B. series only gives us a van
der Waals-like loop in the equation of state and a
compressibility K that is never infinite. An infinite K
is characteristic of a critical point for real gases such
as occurs for transitions between liquid and gas. We
deduce here no critical point for the dense-gas-solid
transition and this is consistent with the completely
repulsive potential model involved, because in this
model T is only a scaling parameter since the virial
coefficients are all temperature independent and a
hard-cube or hard-sphere gas is in a sense like an ordi
nary gas with an attractive tail in its intermolecular
potential at infinite temperature.
THE JOURNAL OF CHEMICAL PHYSICS VI. CONCLUSIONS
We conclude that a van der Waals loop could be ob
tained for the hard-sphere problem also, if enough of
the lower virial coefficients (assuming some of these
definitely negative) were explicitly available and these
were combined with an asymptotic form similar to
that of Eq. (22).
The asymptotic volume dependence of the Bl for
llarge may prove a fruitful point of departure for future
work on the behavior of real gases and liquids, since
the behavior of these substances can be expected to be
dominated by the behavior of hard cores, either at suf
ficiently high temperature or at high enough J,liquid
densities, even for ionic melts.36
ACKNOWLEDGMENT
The author wishes to acknowledge the computational
assistance of Muhammad Agha in preparing the tables
and in checking the algebra on which they are based.
36 H. L. Frisch, Advan. Chern. Phys. 6, 272-285 (1964).
VOLUME 42, NUMBER 8 15 APRIL 1965
Investigation of Triplet-State Energy Transfer in Organic Single Crystals by Magnetic
Resonance Methods*
NOBORU HIROTA AND CLYDE A. HUTCHISON JR.
The Enrico Fermi Institute for Nuclear Studies and the Department of Chemistry, The University of Chicago, Chicago, Illinois
(Received 21 August 1964)
The transfer of triplet-state energy from phenanthrene-d 1o molecules to naphthalene molecules in a
single crystal of diphenyl has been investigated by electron magnetic resonance methods. The decay rates
of the magnetic resonance signals of triplet-state phenanthre.ne-~lo and n~phthalene, both when they ?c
curred as single solutes and also when they occurred together m diphenyl smgle crystals, have been studied
over a wide range of temperatures. A number of processes leading to a satisfactory kinetic model have .been
considered and numerical values of concentrations of postulated species and of rates have been determmed.
Predictions concerning delayed optical emission, made on the basis of the proposed kinetic model, have
been experimentally confirmed.
1. INTRODUCTION
IT has been shown in previous electron magnetic
resonance work by Brandon, Gerkin, and Hutchi
son1 that energy is transferred in single crystals of
diphenyl, containing low concentrations of phenan
threne and naphthalene molecules, from optically ex
cited phenanthrene to unexcited naphthalene with
creation of triplet states in the naphthalene molecules.
The observation of the characteristic triplet-state naph
thalene2 magnetic resonance spectrum in such crystals
when they were illuminated under conditions such that
only phenanthrene could absorb light proved that the
* This work was supported by the U. S. Atomic Energy Com
mission and the National Science Foundation.
1 R. W. Brandon, R. E. Gerkin, and C. A. Hutchison Jr., J.
Chern. Phys. 37, 447 (1962).
2 C. A. Hutchison Jr. and B. W. Mangum, J. Chern. Phys.
34,908 (1961). energy was transferred from phenanthrene to naphtha
lene. High-precision measurements of spin Hamilto
nian parameters1•3 for the triplet-state magnetic reso
nance spectrums of these two solutes in a variety of
hosts showed variations large with respect to the ex
perimental uncertainties. Nevertheless, the parameter
values for either of these guests in diphenyl were not
detectably affected by the presence or absence of the
other. In this way the absence of complex formation
or even of juxtaposition of different guest species in
the diphenyl host was also shown by magnetic methods.
The investigation of the anisotropy3 which arises from
both the electron spin-electron spin interactions and
the proton-electron interactions showed that both guest
molecules were well oriented in the diphenyl structure
and had their principal magnetic axes almost parallel
8 R. W. Brandon, R. E. Gerkin, and C. A. Hutchison Jr., J.
Chern. Phys. 41, 3717 (1964).
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IP: 141.218.1.105 On: Sun, 21 Dec 2014 23:17:232870 N. HIROTA AND C. A. HUTCHISON JR.
35000 em-' I 4 B2u -3.591 x 10
4 , 3.350 x 10
_, --3B3u~3.1554 ___________________________ _
30000 em B3u ------, 4
A,---2.8275 x 10
-, 25000 em
3'
20000 em-' B2o-2.1I0
15000 em-'
10000 em-'
5000 em-' 3B- 2•301
'Ag--- 0 , A,-O 'Ag---O
NAPHTHALENE
H H PHENANTHRENE-dIO OIPHENYL
H H H H HCOH
H H
H H otQ)o HODH
H H H H o 0 0 0
FIG. 1. Energies of low-lying singlet and triplet states of naphthalene, phenanthrene-d lO, and diphenyl. The sources of the numerical
values given in this energy-level diagram are as follows: Naphthalene: 3B2u, maximum jj for phosphorescence in diphenyl single crystal
host measured in present work; see also IB3u, 0, 0 band absorption spectrum in durene single-crystal host measured by McClure.' IB2u,
0,0 band of absorption spectrum in host measured by Sponer and Nordheim.5 Phenanthrene-d lo: 3BI, maximum jj for phosphorescence in
diphenyl single-crystal host measured in present work. IAI, 0, 0 hand of absorption spectrum in pure phenanthrene single crystals of
ordinary light phenanthrene measured by McClure.8 Diphenyl: 3B, maximum jj for phosphorescence in dibenzyl single-crystal host
measured in present work.
to the principal magnetic axes of the host molecules.
Thus the fact that both guests occurred substitutionally
in the diphenyl crystals was also shown by magnetic
methods.
In the present paper we report the investigation, by
magnetic methods, of the rates and mechanisms of
this energy transfer process over a range of tempera
tures. We also have investigated the decays of the
phenanthrene and naphthalene triplet-state magnetic
resonance signals at various temperatures when these
two molecules occurred separately as solutes in diphenyl
single crystals and these results are reported and dis
cussed here.
2. DIPHENYL PHENANTHRENE NAPHTHALENE
SINGLE-CRYSTAL SYSTEMS
In the studies reported here the host single crystal
of diphenyl contained either naphthalene or phenan
threne-d lO or both as guests.
The detailed information on the crystal structure of
diphenyl, available in the literature, has been sum
marized in a previous paper.3 We have investigated
the orientations of the principal axes of the fine struc
ture tensors of the phenanthrene and naphthalene mole
cules in their lowest triplet states. The angles between
these axes and the biphenyl crystallographic axes have been determined. The results have been presented pre
viously.3
The energies of the low-lying singlet and triplet states
are summarized in Fig. 1.4-6 A 1 cm thickness of the
12.8 g liter-! solution of naphthalene in iso-octane
described by Kasha7 provided a high-frequency cutoff
filter which permitted only the phenanthrene-d lO mole
cules to absorb light from an A-H6 high-pressure Hg
arc by singlet-singlet absorption from the ground state.
The broken line in Fig. 1 marks the 0.05 transmission
point of the filter. In fact, the insertion of this filter
completely extinguished the characteristic triplet-state
magnetic resonance spectrum of naphthalene2 in a
single crystal of diphenyl containing only naphthalene
as solute, but changed the intensity of the magnetic
resonance spectrum of phenanthrene3 by only the fac
tor ",,0.5 (the filter overlaps the absorption band of
phenanthrene to this extent) in a diphenyl crystal
containing only phenanthrene.! In a crystal of diphenyl
containing both phenanthrene and naphthalene, inser
tion of this filter extinguished neither spectrum thus
proving the energy transfer.!
4 D. S. McClure, J. Chern. Phys. 22, 1668 (1954).
5 H. Sponer and G. Nordheim, Discussions Faraday Soc. 9, 19
(1950).
6 D. S. McClure, J. Chern. Phys. 25, 481 (1956).
7 M. Kasha, J. Opt. Soc. Am. 38, 929 (1948).
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The paramagnetic resonance spectra of diphenyl sin
gle crystals containing both naphthalene and phenan
threne have been discussed in another paper.3
3. EXPERIMENTAL PROCEDURES: MAGNETIC
RESONANCE ABSORPTION
The magnetic measurements were made in a manner
similar to that described by Hutchison and Mangum 2
and by Brandon, Gerkin, and Hutchison.3
The crystals used for the measurements described
here were single crystals of diphenyl containing (a)
0.07±0.02 mole % naphthalene, or (b) 0.22±0.OS
mole % phenanthrene-d 1o or, (c) 0.07±0.02 mole %
naphthalene plus 0.45±0.05 mole % phenanthrene-d 1o•
The diphenyl and naphthalene starting materials were
obtained from Eastman Kodak Company and were
zone refined for at least 40 passes before being used
for crystal growing. The phenanthrene-d 1o was obtained
from Merck, Sharp and Dohme of Canada Ltd. and
was also zone refined for at least 40 passes. All the
crystals were grown from the melts by methods similar
to the one described previously.2 The concentrations
of the solutes were determined by ultraviolet absorp
tion spectrum analysis of the crystals used for the
magnetic resonance measurements.
The crystals were approximately rectangular paral
lelepipeds with edge lengths in the range from 1 by 3
by 5 to 2 by 5 by 7 mm. They were mounted in the
cavity so that H could be rotated in the cleavage plane
(the be plane, Fig. 1, Ref. 3). H was rotated until it
was as close as possible to being parallel to the short
in plane axis of one of the two phenanthrene or naph
thalene molecules per unit cell (the y axes of Figs. 1,
4, 6, Ref. 3). The low-field absorption lines2.3 were
used for the energy-transfer studies.
A Cu-constantan thermocouple was soldered to the
brass cavity, and its emf was measured during all runs.
Independent experiments, with one thermocouple sol
dered to the cavity and another inserted in a crystal
in the cavity, and with all conditions otherwise the same
as those during the actual magnetic measurements,
showed that the crystal temperatures were never more
than 0.5°K higher than the cavity thermocouple tem
peratures. In all experiments the light from the A-H6
high-pressure Hg arc source was filtered through a
1-cm path length of a solution of 250 g liter -1 NiS04•
6H20 in H20 for removal of heat. The temperature
was controlled by controlling the pressure of N2 gas
in the vacuum space of the inner Dewar of a double
Dewar cryostat, while keeping liquid N2 in the outer
Dewar.
The magnetic resonance measurements were all made
at a carrier frequency within SX107 cycle sect of
9.60X 109 cycle sec1• The magnetic field was modulated
at a frequency of ,-...;1.25 X 105 cycle sec1•
Magnetic resonance signal decay curves were ob
tained by setting I H I at a value such that the pen >I-
~ 101-1 -+---t~
UJ
~ 71----+--+-
5,1---t--i---t-
o
TIME. SEC 9
FIG. 2. Intensity of naphthalene magnetic resonance absorption
signals vs time after cessation of illumination. ---e- denotes naphthalene signal intensity in the three
component system, 0.07 ±0.02% naphthalene and 0.45±0.05%
phenanthrene-dlO in diphenyl. ---e----denotes naphthalene signal intensity in the two
component system, 0.07±0.02%.naphthalene in diph~ny!. The
various temperatures correspondmg to the curves are mdlcated
as follows: 1, 77.3°K; 2, 77.3°K; 3, 88.4°K; 4, 94.0oK; 5,
103.0oK; 6, 107.0oK; 7, 113.3°K; 8, 124'soK.
recorder was at the deflection maximum of the first
derivative curve. After a period of darkness, the crystal
was illuminated for a period long enough to reach
essentially maximum signal intensity but short enough
to avoid undue temperature rise, e.g., ,-...;25 sec in the
77°K experiments, ,-...;10 sec at 1000K and ,-...;5 sec at
150oK. A shutter was then closed and it in turn initi
ated a sequence of 1-sec time markers made by another
pen on the same recorder. The time constant of the
combined detection and recording system was ,-...;0.05
sec. The linearity of the system was verified by varying
the amplitude of the 1.25 X 105 cycle sec1 modulation
amplitude in a precisely known manner and observing
that the recorded signal amplitude was proportional
to the 1.25 X 100 cycle sec1 modulation amplitude
within the experimental error. From three to five decay
curves were recorded for each crystal at each tempera
ture. For each system the results presented in this
paper were all obtained from the same single crystal.
Other crystals from different growths were examined
and in the case of each system gave results in agreement
with the conclusions presented here.
The signal-to-noise ratios ranged from 200 to 20 for
the phenanthrene-d 1o measurements depending upon
the condition of the experiment. For naphthalene they
varied from 40 to 10.
4. EXPERIMENTAL RESULTS
4.1. Magnetic Resonance Absorption
The experimental results of the magnetic-resonance
measurements are presented in Figs. 2, 3, and 4, and
in Table I. In addition, the experiments of Brandon,
Gerkin, and Hutchison1 were repeated by A. Forman
at the boiling point of He, and the transfer was found
to occur at this temperature also. The points plotted
in Figs. 2 and 3 are the averages of the points read
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IP: 141.218.1.105 On: Sun, 21 Dec 2014 23:17:2300 -t
N S ., ... Ul
~ ...
1'1 .,
'"
~
~
~
S
'" ti
~ ..... .::
'" 1'1 0
~ 8
~
~ cg
D
8
H
cg
D
8
H 77
I 80
I 90
I TABLE I. Summary of magnetic resonance absorption results.
100
I 110
I Temp. (OK)
120
1 130
I 140
I 150
I
Exponential de-Decay not exponential. Decay described by Decay rate ap-Decay too rapid for determination of rate.
proximated by cay. 1'=10.0 sec. d[PTJ/dt= -kl[PT J-k2[PT J2 sum of first-and
with kl = -1/10.0 sec-l at all T's. Slope of intensity vs time at second-order
large times is approximately same at all T's. CPT J denotes terms .
concentration of triplet species.
Exponential decay, 1'=2.3 to 2.0 sec. Decay not exponential. First e-I
time falls from 2.0 to 1.0 sec.
Decay close to exponential. Slope of inten-Decay not exponential. First
sity vs time at large times changes with T. e-l time 3-4 sec less than in Decay too rapid for determination of rate.
e-l time changes from 0.4 sec less than that two-component system.
in two-component system to 3-4 sec less
than that in two-component system.
Decay close to Decay not exponential. Pronounced Decay not exponential. Pronounced decrease in first Decay almost ex-Decay not exponential. First e-l
exponential. e-l increase in first e-l time from "'-'2.4 sec e-l time from 5.2 sec to """'2.1 sec. ponential. First e-l time falls from 2.0 to 1.0 sec,
time approxi- to 5.2 sec. time same as for same as for two-component
mately 0.3 sec two-component system.
greater than in system.
two-component
system.
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from the three to five recorder curves run at each
temperature. The signal was, in each case, normalized
to 35.0 for the time at which the shutter was closed,
and the base line for no signal was obtained by inspec
tion of the recorded signal at large times. The brackets
in the figures show the maximum spread of the three
to five points. The curves in Figs. 2 and 3 were visually
drawn through the points.
The points in Fig. 4 were obtained in all cases,
except for the case of phenanthrene-dlo in the three
component diphenyl phenanthrene-dlo naphthalene sys
tem, by reading from Figs. 2 and 3, or from curves
run for additional samples and not presented in Figs.
1 and 2, the values of the time corresponding to
the ordinate value 12.9 on the appropriate curve
(12.9/35= e-l) and corresponding to the ordinate value
4.7 (4.7/35.0=e-2). The first of these times was taken
as the first e-l time plotted in Fig. 3 and the difference
between these times was taken as the second e-l time
plotted in Fig. 4. For the case of phenanthrene-dlo in
the three-component system, diphenyl phenanthrene-dlo
naphthalene, the points for the first e-l times were ob
tained as described just above, but the points for the
second e-l times were obtained by least-squares fits of
straight lines to the points of Fig. 3 in the range of
intensities from 35.0 e-l to 35.0 e-2• The parameters
of this straight line yielded the second e-l times plotted
in Fig. 4. Standard deviations for the first and second
e-l times plotted in Fig. 4 range from 0.2 to 0.5 sec.
The brackets on the points in Fig. 4 represent the
maximum spreads in the first and second e-l times
obtained from curves similar to those of Figs. 2 and 3
drawn:for each of the three to five recorder curves
obtained for each sample at each temperature.
4.2. Delayed Optical Emission
The delayed optical emission spectra reported in
this paper were obtained with a continuous A-H6 Hg
~ 10
~
I.&J 7 I-
~ 51---+~-:
o 4 6 8 10. 12
TIME, SEC
FIG. 3. Intensity of phenanthrene-d lo magnetic resonance ab
sorption signals vs time after cessation of illumination. --e-- denotes phenanthrene-d lo signal intensity in the
two-component system, 0.22±0.05% phenanthrene-d u in di
phenyl. ---e--~ denotes phenanthrene-d lO signal intensity in the three
component system, 0.07 ±0.02% naphthalene and 0.45±0.05%
phenanthrene-d lO in diphenyl. The number labeling each curve
is the absolute temperature in degrees K. 12 r-----r-----.----~----_.----_,
80 100 120 1.40 160
T. OK
FIG. 4. e-I times of magnetic resonance absorption signals after
cessation of illumination vs temperature.
- --A---denotes naphthalene e-I time in three-co~nent
system, 0.45±0.05% phenanthrene-dlo plus 0.07±0.02'lo naph
thalene in diphenyl.
----. --denotes naphthalene e-I time in three-component
system, 0.24±0.05% phenanthrene-dlo plus 0.05±0.02%
naphthalene in diphenyl.
---0 - - -denotes naphthalene e-I time in two-component
system, 0.07 ±0.02% naphthalene in diphenyl.
In all three of these curves the plotted e-1 time is the time re
quired for the magnetic resonance signal intensity to decrease by
factor e from its value at the cessation of illumination.
---e---, --e-- denote phenanthrene-d 1o e-1 times in two
component system, 0.22±O.05% phenanthrene-dlo in di
phenyl. The broken line indicates the first e-1 times, i.e., the
times for decay of signal e-1 X its initial value; the solid line
indicates the second e-1 times, i.e., the times for decay from
e-1 to e--2X the initial value. ----.--, --II-- denote phenanthrene-d1o e-1 times in
three-component system, O.24±O.05% phenanthrene-d 1o plus
O.05±O.02% naphthalene in diphenyl.
Solid and broken lines have the same significance as in the im
mediately preceding case.
---A---, --A-- denote phenanthrene-d 1o e-1 times in
three-component system O.45±O.05% phenanthrene-d 1Q plus
0.05±O.02% naphthalene in diphenyl.
Solid and broken lines have the same significance as described
in the preceding cases.
arc source, a conventional rotating disk phosphoro
scope, a Bausch & Lomb monochrometer, photomulti
plier, dc amplifier, and chart recorder. The samples
were placed in fused-silica tubes which were mounted
in an AI block holder in a quartz-window Dewar cryo
stat. The emission intensity vs wavelength was dis
played on the recorder at various temperatures as the
Al block slowly warmed from nOK to higher tempera
tures. The temperature was measured by means of a
thermocouple inside the fused-silica tube and in contact
with the single-crystal sample. The lifetimes of the
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300 400 500 600 ~m120K
~ F P w I
~
300 400 500 600
A.mJL A.mJL
~~ irT"9 ~l}~ ~~
300 400 500 600 300 400 500 600
'A,mp. 'AJIlp.
FIG. 5. Delayed emission of phenanthrene-d lo.
delayed fluorescence reported here were obtained by
closing a shutter and simultaneously photographically
recording an oscilloscopic display of the intensity of
the most intense part of the spectrum vs time using a
sweep rate of 0.5, 1.0, or 2.0 cm sec\ depending upon
the rate of the decay which was being observed.
The results of the delayed emission studies are pre
sented in Fig. 5 and in Table II. The spectra shown
in Fig. 5 were obtained using the phosphoroscope, and
the relative intensities were reliable to 2% and the
wavelengths were reliable to 50 cm-I• The P and F
denote the characteristic phenanthrene-d lO phosphores
cence and fluorescence spectra, respectively. The decay
e-I times given in Table II were obtained from the
photographic recordings of oscilloscope traces described
above.
5. DISCUSSION OF RESULTS
5.1. Magnetic Resonance Measurements
5.1.1. Rates of Transfer
We may reach two obvious conclusions concerning
the rates of the triplet-state energy transfer from phe
nanthrene-d lO to naphthalene by an examination of the
magnetic resonance experiment results.
(i) For T:::;85°K the transfer from phenanthrene-d io
to naphthalene was very fast relative to the naphtha
lene intersystem (triplet~singlet) conversion rate from
the lowest triplet state to the ground singlet state.
This is clearly the case as is seen from the fact that
the naphthalene triplet-state decay rate is negligibly
affected by the presence or absence of the phenan
threne in this temperature range. Thus the transfer
of energy from phenanthrene-d lO to the naphthalene
in this temperature range must have been essentially
completed by the time the measurements of the rates
of decay of the magnetic signals were begun, i.e.,
within 0.05 sec after interruption of the illumination.
Transfer for periods longer than this from the phenan
threne-d lO (whose mean life is ",,10 sec) with generation
of triplet-state naphthalene would have produced a decrease in the decay rates of the naphthalene mag
netic signals. Transfer of energy from phenanthrene-d io
to naphthalene by this very fast low-temperature proc
ess has been found to occur at phenanthrene-dlO con
centrations as low as 0.1 mole % in the presence of
naphthalene at concentrations as low as 0.05 mole %.
In our dilute crystals the average distances of separation
of the two species, phenanthrene and naphthalene, was
large. The energy separation between the lowest vibra
tional state of the lowest triplet state of the guest and
lowest triplet state of the host was also relatively large.
Therefore, for the observation at T:::;85°K (in which
essentially all the triplet phenanthrene molecules are
in their lowest vibrational states and at quite large dis
tances from nearest naphthalene molecules) the prob
ability of the occurrence of the observed energy transfer
by the mechanism proposed by Robinson and Frosch8
and by Sternlicht, Nieman, and Robinson9 is very re
mote.
(ii) An additional temperature-dependent triplet
state energy-transfer process sets in at T> 85°K and
the transfer rate becomes comparable in magnitude
with the naphthalene intersystem conversion rate as
the temperature is increased. This is clear from the
fact that the decay rate of the naphthalene triplet
state magnetic resonance signal was very appreciably
reduced in this temperature range by the addition of
phenanthrene-d lO to the crystal. This second process
in turn eventually became so rapid as temperature was
raised that it had all occurred before the start of the
recording of the naphthalene signal decay. Thus at
1300K the decay rate of the naphthalene signal is
again about the same in the presence or absence of
phenan threne-d lO•
TABLE II. Lifetime of the delayed fluorescence of phenanthrene-dlo;
""'().2 mole % phenanthrene-d lO in diphenyl.
Temp lie time Rough esti-
mation of (J'
(deg K) (sec) (sec) Remarks
101.0 4.5 0.3 Extremely close
nential decay. to expo-
112.0 3.0 0.2 Appreciable deviation from
exponential decay.
122.2 1.4 0.1 Still larger deviation from
exponential decay than at
lower temperatures.
129.0 0.9 0.1 Still larger deviation from
exponential decay than at
lower temperatures.
140.5 0.4 0.05 Still larger deviation from
exponential decay than at
lower temperatures.
8 G. W. Robinson and R. P. Frosch, J. Chem. Phys. 38, 1187
(1963).
9 H. Sternlicht, G. C. Nieman, and G. W. Robinson, J. Chem.
Phys. 38, 1326 (1963).
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5.1.2. Kinetic Model for the Magnetic Resonance Results
A kinetic model which displays the essential features
of the observations on the decays of the magnetic reso
nance signals and on the slow, temperature-dependent
transfer process for T> 85°K may now b~ described.
For the two-component phenanthrene-d lO-dlphenyl sys
tem we consider the following four processes:
(1)
(intersystem conversion from the lowest triplet state
to the ground singlet state) ;
(2)
(vibrational excitation from the lowest triplet state
and de-excitation into the lowest triplet state) ;
(3)
(energy transfer from vibrationally excited t.riplet
state to host molecule which is in its ground smglet
state before the transfer. This yields host in one of
the vibrational states of its lowest triplet band plus
guest in its ground singlet state) ;
(4)
(triplet-triplet annihilation producing excited singlet
state of guest and ground singlet state of host from
lowest triplet states of both) ;
where P and D represent phenanthrene-d lO and di
phenyl, respectively; the SUbscripts S! T, T* st~nd
for singlet state, triplet state, vibratlOnally excIted
triplet state, respectively (the particular singlet states
or triplet states in question being denoted by the state
ment in parentheses after the listed process) ; and the
k's are rate constants. We assume that the rates at
which the Processes (2) and (3) are proceeding are
very large compared with either the intersystem con
version rate, (1), or the triplet-triplet annih!lation, .(4);
i.e., we assume that over a very short penod of tIme,
relative to the intersystem conversion time, the rates
of creation and destruction of PT by (2)---? and (2)~
become equal and the rates of creation and destruction
of DT by (3)---? and by (3)~ become equal. A con
stant ratio [PT* ]/[PT], is thus established, where [ J's
denote concentrations; this ratio is assumed to have
the thermal equilibrium value, exp( -!1E/kT), where
!1E is the difference in energies of PT* and PT. A
steady-state value of [DT], namely
[DT J=ka[PT*][Ds]/k![PsJ
is also established. We now add the relatively slow
process of intersystem conversion (1) and triplet
triplet annihilation (4) which relatively slowly alter [PTJ, leading to the expression:
dePT J/dt= -kl[PTJ-k4[PT][DTJ
= -kl[PT]-k4(ka[PT ]2[DsJ/k![Ps])
Xexp( -!1E/kT) (I)
for the decay of [PT] with time.
For the three-component system containing naphtha
lene we consider, in addition to the above-mentioned
processes, one additional relatively slow process,
(5)
where N represents naphthalene. This process removes
DT'S which are replaced by the very fast process (3)---?,
being created from PT.'s as described previously. ~he
PT.'S are in turn very rapidly generated by the eXClta
tion (2)---? from PT'S. Thus we have a third contribution
to the decay of [PT] in addition to the intersystem
conversion and the triplet-triplet annihilation discussed
for the two-component system. We may hence write
dePT J/ dt= -kl[PT]-k4[PT ][DT]-k5[DT ][N s]
= {-kl-k.ka [Ds][Ns] exp( -!1E/kT)}[PT J
k3 CPs]
-{k4ka [Ds] exp( _ !1E/kT) }[PT J2. (II)
k3 CPs]
5.1.3. Decay of Phenanthrene-d lO Magnetic
Resonance Signals
The kinetic model which has just been described
gives a satisfactory account of the essential featur:s
of the decay rates of the phenanthrene-dlo magnetic
resonance signals as a function of temperature. The
application of Eqs. (I) and (II), given ~bove, to. the
consideration of the decay of the populatlOn of trIplet
states of phenanthrene-dlo in the two-component di
phenyl phenanthrene-dlo system and the three-compo
nent diphenyl phenanthrene-dlo naphthalene system
leads to the three following points of agreement with
the experimental observations.
(i) At low enough temperatures both Eq. (I) and
Eq. (II) lead to a first-order temperature-independent
decay of the population of triplet-state phenanthrene-dlo
molecules. This was what was observed in both the two
component and three-component systems as is shown
by the results which are presented in Figs. 2, 3, 4, and
Table I. The thermally activated energy-transfer proc
ess does not take place in either case and only the
intersystem conversion is responsible for the disappear
ance of triplet-state molecules.
(ii) At higher temperatures Eq. (I) leads to a decay
rate for the two-component diphenyl phenanthrene-d lO
system which is the sum of the same first-order tem
perature-independent decay of. triplet-st~te p~ena~
threne-d lO (intersystem converslOn) mentlOned m (1)
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plus a second-order temperature-dependent decay. In
spection of the solid curves of Fig. 3 for phenanthrene-d 1o
in the two-component system shows that this is just
what was observed for the decay of the magnetic reso
nance signals. As the temperature increases, these ex
perimental curves show an increasing curvature in the
short-time region in agreement with increasing contri
bution of the temperature-dependent second-order term
of (I). In the long-time region, however, where the
first-order process should again predominate, even in
this higher temperature range, all the curves become
straight lines of the same slope, independent of tem
perature, and with the same slope as the lowest
temperature curve which is a straight line over the
whole range of times.
(iii) At higher temperatures Eq. (II) leads to a
decay rate for the population of triplet states of phe
nanthrene-d lO which is a sum of a first-order decay and
a second-order decay but in this case, in contrast with
the two-component case of (ii), the first-order decay
is temperature dependent as well as the second-order
process. Inspection of the broken curves of Fig. 3 shows
that this is in agreement with observation. In the long
time region these curves become straight lines but the
slope is a function of the temperature.
(iv) Examination of Eq. (I) shows that the term
which is second order in CPT] should decrease in mag
nitude as CPs] is increased. Preliminary investigations,
not discussed in the experimental results section of this
paper, agreed with this prediction.
5.1.4. Evaluation of tlE
The tlE which occurs in Eq. (II) was evaluated by
consideration of the magnetic resonance results sum
marized in the broken curves of Fig. 3 plus additional
such curves for other temperatures not presented in
Fig. 3. All of the points on each individual such loga
rithmic curve at each temperature for times (after
cessation of illumination) greater than the e-l time
were least-squares fitted by straight lines of the form,
log intensity= -kt+e. In this region of time the term
in CPT J2 is almost negligible. The value of k obtained
in this way at each temperature was then decreased
by kl which was taken to be 10.0-1 sec1• Six such
resulting rate constants, k-kl' at six different tem
peratures over the range of the experiments were least
squares fitted by the relation,
log(k-k 1) = -(tlE/k) T-l+C.
In one set of experiments this gave 1411 cm-1 as the
best value for tlE/ he and in another set, the value
1410 em-I, with standard deviations 33 cm-1 in the
first case and 21 cm-1 in the second. Taking for kl the
value obtained from the second e-I times (times for
decay from 35.0e-I to 35.0e-2) of the solid curves of
Fig. 3 the resulting tlE/he's for the two sets of experi
ments were 1393 and 1392 em-I with standard devia
tions 41 and 22 em-I, respectively. tlE was also estimated from the measurements on
the two-component diphenyl phenanthrene-dIG system
using Eq. (I). kl was assumed to be 10.0-1 seci•
Measurements near nOK and at the very highest tem
peratures were discarded, leaving curves for six differ
ent temperatures in one set of experiments, five different
temperatures in a second set, and, six different tem
peratures in a third. The rate constants which fitted
these curves gave the value 1521 cm-1 for tlE/he in
the first set of experiments, 1517 cm-l in the second,
and 1591 in the third with standard deviations 47.
22, and 61 cm-l, respectively, using a least-squares
fitting procedure similar to that described above. The
best value of tlE/ he for the three sets of experiments
was thus 1543 cm-1 with a standard deviation, 50 em-I.
It is very clear that the values of tlE/ he required by
our kinetic model to account for the observed decay
rates of the phenanthrene-d 1o magnetic resonance signal,
namely 1.41OX103 em-I or 1.392X103 cm-l depending
on choice of value for kl in the three-component system,
and 1.543 X 103 cm-l in the two-component system, are
close to the experimental values of the wavenumber
difference between the lowest triplet states of the phe
nanthrene-d lo molecule and the host diphenyl molecule.
These experimental values and their sources are given
in Fig. 1 and tlE/he=1.60X103 em-I. The measure
ment of the triplet-state term value in the present
work was for diphenyl in dibenzyl yielding the value
2.301 X 1()4 em-I. This may be compared with the meas
urement by Lewis and Kashalo whose value was 2.280X
1()4 cm-1 for diphenyl in diethyl ether isopropane ethyl
alcohol mixture at 77°K and with the value 2.300X 104
cm-1 given by Ermolaevll for diphenyl in an ethyl
alcohol diethylether mixture at n°K. These three val
ues for the lowest triplet-state energy of diphenyl may
arise from solvent effects. Solvent effects on the ener
gies of triplet states of such molecules are known to
be as large as 200 cm-I. There are no experimental data
available on the diphenyl triplet exciton level in the
diphenyl single crystal. Therefore, any numbers at
which we have arrived here for the energy separation
between the triplet state of phenanthrene-dlo and the
host triplet exciton band are subject to considerable
uncertainty.
Nevertheless, these results give strong evidence
that the temperature-dependent energy-transfer proc
ess which is found to occur at the higher temperatures
requires the vibrational excitation of the triplet state
phenanthrene-d lO to a level near that of the host crys
tal's triplet exciton band.
We thus conclude that there are two quite distinct
energy-transfer processes in these crystals. There is a
process which occurs very rapidly during the populat
ing of the phenanthrene-dIG triplet states. This presum-
10 G. N. Lewis and M. Kasha, J. Am. Chern. Soc. 66, 2100
(1944).
11 V. L. Eramolaev, Usp. Fiz. Nauk. 80, 3 (1963); [English
trans!. Soviet Phys.-Usp. 6, 333 (1963)].
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ably takes place from some excited state, through
which the phenanthrene-d lQ molecules pass after the
absorption of light and during the internal intersystem
conversion process by which they get from an excited
singlet state to their lowest triplet state. This state
can apparently feed energy to the triplet band of the
host which in turn transfers it to the relatively deep
naphthalene traps. The second process transfers energy
from a vibrationally excited state of the lowest triplet
state of phenanthrene-dlO. The energy of this state is
essentially the same as that of the triplet exciton band
of the host diphenyl crystal and the energy is rapidly
transferred to the host from the thermal equilibrium
population of such states.
The kinetic model is thus seen to adequately account
for all of the essential features of the phenanthrene-dlo
experimental decay rates as a function of temperature
in both the two-component and the three-component
systems.
5.1.5. Decay of Naphthalene Magnetic
Resonance Signals
The same kinetic model also gives agreement with all
the essential features of the measured decay rates of
the naphthalene magnetic resonance signals described
in Figs. 2, 4, and Table I. In particular, the striking
minimum with respect to temperature in the triplet
state naphthalene decay rate, evidenced by the maxi
mum in Fig. 4, as the phenanthrene-d lO decay increases
with rising temperature, is clearly understood in terms
of the model. It is a result of the fact that, as this
temperature range is entered from lower temperatures,
the high-temperature thermally activated transfer proc
ess feeds energy to naphthalene at a rate comparable
with the loss rate by naphthalene intersystem conver
sion. As the temperature rises to still higher values this
higher-temperature process becomes sufficiently rapid,
as pointed out earlier, that it is essentially completed
by the time the measurements begin and the naphtha
lene decay rate again rises to its low-temperature value.
5.2. Delayed Emission
Of the five processes considered in connection with
the kinetic model, one of them, namely (4), leads to
interesting predictions concerning the optical properties
of the two component diphenyl phenanthrene-d 1Q sys
tem. Inasmuch as this process (4) generates singlet
state diphenyl molecules and singlet-state phenanthrene
molecules with excess energy of excitation there should
be a fluorescence of phenanthrene molecules with at
tendant emission of light that is associated with this
process. Moreover, because the sources of the diphenyl
triplet exciton which generates an excited singlet phe
nanthrene from a triplet phenanthrene are the long
lived triplet phenanthrene molecules themselves, this
"fluorescence" should appear a relatively long time
after the cessation of illumination. In addition, at tem-peratures too low for the thermally activated transfer
process to take place, only the triplet-singlet transition
(phosphorescence) should be apparent in the delayed
emission; but at the same higher temperatures at which
the processes, (2) and (3), have been invoked to ac
count for the behavior of the magnetic resonance sig
nals, the singlet-singlet emission (fluorescence) should
appear strongly in the delayed luminescence.
These predictions agree precisely with the observed
optical behavior in the experiments on the two-compo
nent diphenyl phenanthrene-d lO system which are sum
marized in Fig. 5 and Table II. At the lowest tempera
tures, 77°K, the phosphorescence of phenanthrene-dm
was the major constituent of the light emitted ,..,.,1 X 10-3
sec after cessation of illumination. As the temperature
was raised the fluorescence appeared, with the long
lifetime indicated in Table II. At still higher tempera
tures where the triplet-state lifetime of phenanthrene-dm
became short the fluorescence lifetime became short
and the phosphorescence intensity became small be
cause a large fraction of the triplet-state energy was
removed by the thermally activated transfer process,
generated singlet phenanthrene-dlO's, and appeared as
fluorescence.
A process similar to (4) of course occurs in the two
component diphenyl naphthalene system and similar
optical results have been obtained in this case. They
will be reported in the future.
In the three-component diphenyl phenanthrene-d lo
naphthalene system the delayed fluorescence of the
phenanthrene-d 1Q has also been observed but not that
of naphthalene.
These predictions concerning the delayed emissions
are not only qualitatively in agreement with observa
tion, as described above, but they are also in reason
able quantitative agreement. Equation (I), when inte
grated to give CPT] at time t yields
CPT] [PT]t-O exp( -kIt)
1+ (Kjk1)[l-exp( -kIt)]' (a)
where
K k4k3[Ds] exp( -AE/kT)
kt[Ps]
The intensity of the delayed light which is emitted at
the fluorescence frequency is proportional to K[PT]2.
In the region of lowest temperature at which the ther
mally activated process occurs, the condition Kjk1«1
is satisfied. Hence Eq. (a) becomes
((3)
and the fluorescence intensity IF is given by
f,a: K[PT ]=K[P T ]2t-o exp( -2klt). ('Y)
Thus we expect to find a decay rate for the "delayed
fluorescence" at the lowest temperatures at which it
was observed which is exponential and with a lifetime
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TABLE III. Numerical values.
Process
(1)
(2)
(3)
(4)
(5) Numerical value
[P7]~1017 molecules cm-a
kl[PT]~1016 molecules cm-a'seC!
[PT* ]/[PT ]~10-9
k2~101I sec-!
[PT.]~l()8 molecules cm-a
ki[PT, ]~1019 molecules cm-a'seCl
k2[PT] = k-[PT* ]~1019 molecules cm-a'sec!
k2~H)2
(k3[Ds][PT* ]lk-[Ps][D T ])~1
[DT] = 1011 molecules cm-3
k5~1Q--l2 cma molecule-I. seC!
just one-half of the observed phosphorescence lifetime
at the lower temperatures. This is very clearly in good
quantitative agreement with the experimental results
summarized in Table II. Table II shows that, as the
temperature increased, the fluorescence decay deviated
increasingly from exponential as expected on the basis
of these same considerations.
Similar triplet-triplet annihilation processes have
been discussed by Kepler, Caris, Avakian, and Abram
son.12 The theoretical considerations of such annihila
tions by ]ortner, Choi, Katz, and Rice1s have led to
an understanding of the basis of such processes and
also to numerical values in good agreement with the
experiments.
6. NUMERICAL VALUES
A set of possible numerical values for the various
rates and lifetimes which have been discussed in connec
tion with the proposed model, and for the concentra-
12 R. G. Kepler, J. C. Caris, P. Avakian, and E. Abramson,
Phys. Rev. Letters 10, 400 (1963).
13 J. Jortner, S. Choi, J. L. Katz, and S. A. Rice, Phys. Rev.
Letters 11, 323 (1963). Source
Estimates of signal intensity.
k1",1Q--! sec-l corresponding to 10-sec mean life.
Experimentally determined AE and steady-state Boltz-
mann distribution assumption.
Estimated.
From AE and value of [PT] given above.
From numerical values given above.
From steady-state Boltzmann distribution assumption.
From numerical values given above.
Assumed.
From uv analysis.
From estimated lO-cm-I triplet bandwidth for host
crystal with factor 10 decrease for vibrational overlap.
Steady-state assumption for [DT].
From numerical values given above.
From Eq .. (I) and numerical values given above plus
(d[PTJ/dt) =i[PT] sec-I from measured 8 sec mean
life at lOOoK.
From (a) measured 5 sec time for phenanthrene decay
from e-I to e-2 of initial value in three component sys
tem, (b) Eq. (II) neglecting [PT]2 term, (c) meas
ured [Ns] by uv analysis, (d) numerical values given
above.
tions of the various species which have been postulated,
have been estimated from the measurements reported
in this paper and from other available information re
lated to these various mechanisms. This set of values
is summarized in Table III. These values will be seen
to be consistent with all of the magnetic resonance
measurements and with experimental and theoretical
information from a variety of other sources. It is to
be particularly noted that the numerical values given
under (4) and (5) in Table III are in order-of-magni
tude agreement with the theoretical considerations of
]ortner, Choi, Katz, and Rice.1s
ACKNOWLEDGMENTS
We acknowledge the assistance of Dr. Arthur Forman
with the 4 OK experiments; the assistance of Professor
Donald S. McClure, Steven L. Murov, and Chen
Hanson Ting with the optical experiments; the helpful
discussions with Professor Stuart A. Rice and Professor
Joshua Jortner; and the construction by Clark E.
Davoust, Edward Bartal, and Warren Geiger of ap
paratus and equipment used in the experiments.
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1.1777040.pdf | Magnetotunneling in Lead Telluride
R. H. Rediker and A. R. Calawa
Citation: Journal of Applied Physics 32, 2189 (1961); doi: 10.1063/1.1777040
View online: http://dx.doi.org/10.1063/1.1777040
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/32/10?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 130.70.241.163 On: Sun, 21 Dec 2014 14:54:44VALENCE BANDS I~ LEAD TELLURIDE 2189
C. Magnetoresistance Anisotropy
We have begun a study of the magnetoresistance as
a function of carrier concentration at 296° and 77°K.
As yet we have seen little if any change in the anisotropy
with changing carrier concentration at either tempera
ture. At 77°K, this is consistent with the assumption
that all the occupied band edges lie at nearly the same
energy, as Stiles concluded from his results at 4.2°K.
At room temperature there may be no large effects be
cause there are not enough carriers in the second band
or because this band consists of another set of (111)
ellipsoids of similar anisotropy.
D. Hall Mobility at 77°K
Table I reveals a very strong dependence of the Hall
mobility on carrier concentration at 77°K. This large
effect cannot be a two-carrier effect since a large
mobility ratio would be required, and this would dis
agree violently with the Hall data of Fig. 1. A careful
analysis of the mobility-temperature data also shows
that ionized impurity scattering cannot be the cause of
more than a small part of the total decrease in the mo
bility at this temperature. We believe that the principal
cause of the mobility variation is the effect of statistics
on the lattice mobility; i.e., the average energy be
comes a function of carrier concentration, and this
affects the mobility through the energy dependence of
the scattering time. Such an effect has been observed at room temperature in n-type PbTe samples with very
high carrier concentrations.21
CONCLUSIONS
A low-temperature two-band model (111) ellipsoids
plus a sphere) for p-type PbTe in which the mobilities
of the two carriers are nearly equal and in which the
two band edges occur at nearly the same energy is the
simplest way of explaining the lack of any of the usual
two-band effects in the magnetic field dependence and
the carrier concentration dependence of the Hall data
at 77QK. Above about 1S00K, on the other hand, effects
appear in the temperature dependence of both the Hall
coefficient and the resistivity which are characteristic
of a two-band model with an energy difference between
the band edges of about 0.1 ev and with the lower
mobility in the lower energy band.
ACKNOWLEDGMENTS
I am greatly indebted to Bland B. Houston, Jf. and
Richard F. Bis, without whose crystal-growing efforts
this work could not have been carried out. I also wish
to record with thanks the many valuable discussions
with Frank Stern, the advance information given me
on their own work by Philip Stiles, Jack Dixon, and
H. R. Riedl, and the much-needed experimental as
sistance of J. R. Burke, Jr.
21 T. S. Stavitskaya and L. S. Stil'bans, Fiz. Tverdogo Tela 2,
2082 (1960) [translation, Soviet Phys.-Solid State 2, 1868
(1961)].
JOURNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 32, NO. 10 OCTOBER, 1961
Magnetotunneling in Lead Telluride
R. H. REDIKER AND A. R. CALAWA
Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington 73, Massachusetts
Rotation of a large magnetic field (""60 kgauss) in a plane perpendicular to the direction of junction current
in PbTe tunnel diodes produces a periodic behavior of this current. Diodes in which the junction current
flow is along the [toOJ, [110J, or [111J crystallographic axis have been investigated. The observed anisot
ropies are consistent with a crystal with cubic symmetry whose constant energy surfaces in k space are
ellipsoids of revolution oriented along the (111) crystalline axes. The magnetotunneling results are interpreted
in terms of the effective motion of each ellipsoidal valley as the magnetic field is increased, valleys oriented at
different angles to the electric field contributing with different weights to the tunneling current. Quantitative
comparison awaits theory. Heavier mass bands close in energy to these ellipsoids, although unimportant by
themselves in tunneling, may be necessary to explain the apparent position of the Fermi level with respect
to the band edges. The excess current has the same anisotropy as the tunneling current; however, the thermal
current does not show this anisotropy and therefore must be of different origin. Hump current, which was
ohserved in two diodes, disappeared in magnetic fields above 3 kgauss.
INTRODUCTION
ON the basis of the magneto tunneling effects in
InSb,! it was predicted that similar large reduc
tions in tunneling current with magnetic field would
occur in tunnel diod€s of other low-gap semiconductors.l
* Operated with support from the U. S. Army, Navy, and Air
Force. The marked decrease in tunneling current of PbTe
tunnel diodes with application of magnetic fields up
to 88000 gauss has been observed.2 In this paper the
effects of magnetic fields on the tunnel current will be
described for PbTe diodes fabricated so the current
flow is along either the [100J, [110J, or [111J crystal
lographic axes and for magnetic fields both parallel
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and perpendicular to the current flow. The theory of
tunneling for the case of isotropic effective mass has
been considered for both longitudinal l.3-5 and trans
verse3.4 magnetic fields. Several questionable approxi
mations, however, have been perforce necessary in the
solution for the transverse field.6 For the presently
assumed multivalley band structure of PbTe having
prolate energy ellipsoids along the (111) axes,7·s it is
impossible for the magnetic field to be longitudinal at
the same time to the differently directed current
components associated with each of the four ellipsoids.
Thus there is no true longitudinal case. While no
acceptable theory is presently available for quantitative
comparison, the magneto tunneling experiments do
yield information both about the tunneling process
and about the band structure of PbTe.
DIODE FABRICATION
The diodes were fabricated using Bridgman-grown
single-crystal Ag-doped PbTe which had approximately
3 X lOIS net acceptors per cm3 and a Hall mobility
of approximately 1000 cm2 "I secl at 77°K and 5000
cm2 "I secl at 4.2°K. The tunneling contact was
made by alloying indium spheres into one face of an
oriented wafer of this p-type material while at the same
time ohmic contact was made to the opposite face by
alloying a 0.OO5-in. thick thallium disk backed by a
gold-clad tantalum tab. In order to obtain appreciable
tunnel current and negative conductance at forward
biases it was found necessary to utilize an alloy cycle
of about 3-sec duration and a peak temperature of
approximately 320°C. Longer times and/or higher
temperatures resulted in little or no sensible tunnel
current at forward biases.
The [100J oriented diode whose magneto tunneling
is described below was fabricated elsewhere9 by alloying
to vapor grown p-type PbTe grown in an excess Te
vapor.
DIODE CHARACTERISTICS
Typical I-V characteristics obtained at zero and
higher magnetic fields are shown in Fig. 1. The largest
peak-to-valley ratio obtained at zero magnetic field was
1.5 to 1 at 4.2°K. In order to explain the zero magnetic
field I-V characteristic shown in Fig. 1 it appears
necessary to postulate that there are other heavier mass
1 A. R. Calawa, R. H. Rediker, B. Lax, and A. L. McWhorter,
Phys. Rev. Letters 5, 55 (1960).
2 R. H. Rediker and A. R. Calawa, presented at the Symposium
on Electron Tunneling in Solids, Philadelphia, Pennsylvania,
January 30-31, 1961.
3 R. R. Haering and E. N. Adams, J. Phys. Chern. Solids 19,
8 (1961).
4 P. N. Argyres and B. Lax, J. Phys. Chern. Solids (to be
published).
6 P. N. Argyres, Bull. Am. Phys. Soc. 6, 345 (1961).
6 P. N. Argyres (private communication).
7 C. D. Kuglin, M. R. Ellett, and K. F. Cuff, Phys. Rev.
Letters 6, 177 (1961).
8 R. S. Allgaier, Phys. Rev. 112,828 (1958).
9 At General Electric Research Laboratories, Schenectady,
New York. FIG. 1. Voltage-current characteristics at 4.2°K of a PbTe
tunnel diode in different transverse magnetic fields. The junction
current is parallel to the [110J crystallographic axis.
energy bands not too far in energy above the four
(111) ellipsoidal minima assumed for both n-and p-type
PbTe.7•s The net acceptor density in the base region is
3 X lOIS cm-3 and from results on other diodes we are
led to believe that the net donor density in the n region
is above 1019 cm-3• Using a density of states effective
mass of O.lmo for the (111) ellipsoidal valleys the Fermi
level would be 75 mv below the edge of the valence
band and 170 mv above the edge of the conduction
band. This degeneracy is inconsistent by a factor larger
than 2 with the I-V characteristic of Fig. 1. The
existence of other heavier-mass band edges would
reduce the degeneracy of the material, while the tunnel
ing current would still be produced mainly by the
much lighter carriers in the (111) ellipsoidal valleys.
Capacitance measurements on several diodes yielded
values for C / A of approximately 10 J..tf/ cm2• These
values are significantly larger than the value of 1.6
J..tf/cm2 calculated at zero bias using a net acceptor
density in the base region of 3 X lOIS cm-3 and assuming
a built-in junction potential of 0.2 v, a dielectric
constant of 2510 and an abrupt junction. A similar
discrepancy in capacitance values has been reported for
indium antimonide alloy diodes,ll The zero-bias
junction width calculated from the net acceptor density
is 135 A and the average junction field is 1.5 X 105 v
cm-I• The junction width and average junction field
(23 A and 9X 105 v cm-I) determined from the measured
capacitance value yield values for tunnel current density
which cannot be reconciled with the experimental tunnel
current, indicating that simple diode theory cannot be
used to determine junction width from capacitance
measuremen ts.
In order to investigate the effects of magneto-
, 10 T. C. Harman (private communication).
11 C. A. Lee and G. Kaminsky, J. Appl. Phys. 31, 1717 (1960).
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resistance in all PbTe tunnel diodes fabricated in our
laboratory, a "dummy" was fabricated in which the
tunneling contact was replaced by an ohmic contact.
This "dummy" had zero magnetic field resistance of
less than 0.5 ohms compared to over 20-ohm resistance
in the tunneling region at zero magnetic field for the
diode of Fig. 1. The application of 66.5 kgauss increased
the resistance of the dummy to less than 0.7 ohms for
longitudinal and less than 0.8 ohms for transverse
magnetic field. On the other hand in Fig. 1 the effective
diode resistance is increased to over 200 ohms by
application of 66.5 kgauss. For all the diodes described
below, except the [l00J diode, magnetoresistance can
be neglected compared to magnetotunneling effects.
The [100J diode, for which no dummy was made, did
show small magnetoresistance effects at high currents
as described below. At normal currents the results are
in agreement with those obtained on the [100J oriented
diodes fabricated by us, for which magnetoresistance
can be neglected.
EXPERIMENTAL PROCEDURE
The tunnel diode current was measured as a function
of voltage at constant values of magnetic field and as a
function of both the magnitude and direction of
magnetic field at constant values of diode voltage. All
data were taken on an X-Y recorder. Because the source
resistance and associated lead resistance could not
easily be made small compared to the resistance of the
diodes, in recording the current-magnetic field charac
teristics the voltage across the diode was maintained
constant by a closed loop servo-system. The error
between the diode voltage, as measured across voltage
probes, and a constant reference voltage was reduced
to less than 50 JJ.V by using the servo to vary the source
voltage. In order to rotate the diodes with respect to
the magnetic field (directed along the bore of the Bitter
magnet) the diodes were mounted in a right-angle gear
drive mechanism capable of rotation through 320°. The
angular displacement was determined from the voltage
across a linear potentiometer which was linked to the
gear assembly. The zero point angle was set visually
introducing a possible error of approximately 10°. The
angular error relative to the zero is within 5°. Each
diode and the entire gear mechanism were immersed
in liquid helium to insure a constant temperature.
All the data presented in this paper were taken with
the diodes at 4.2°K.
EXPERIMENTAL RESULTS
[100J Diode
As is evident from Fig. 1, large magneto tunneling
effects have been observed for the PbTe tunnel diodes.
Also, rotation of large magnetic fields in a plane
perpendicular to the direction of the tunneling current
produces a periodic behavior of this current. Figure 2
shows this behavior for a diode in which the junction 3 ~----------------------------,
2
60 r---~~----------------~~-------' 0000000000000000000000000000
+ + + + + 00 + + 40
20 [001] [OIl] [010] [Oil] [ooT] [oli] [010]
o LiI __ ~ __ ~I __ ~ __ ~I __ -L __ ~~I __ ~
o 90" 180" 270·
MAGNETIC FIELD DIRECTION IN (100) PLANE IN
DEGREES FROM [001]
FIG. 2. Diode current as a function of the direction of a 6O-kgauss
magnetic field in the (100) plane perpendicular to the direction
of junction current. The behavior of the current is shown at
different fixed diode voltages: (a) 10 mv forward in tunneling
region; (b) 10 mv reverse in tunneling region; (c) 80 mv reverse
in tunneling region; (d) 80 mv forward in valley region; (e) 175
mv forward in thermal current region. The zero magnetic field
currents are respectively: (a) 6.1 rna; (h) 8.3 rna; (c) 112 rna;
(d) 9.9 rna; (e) 78 rna.
current is parallel to the [l00J axis and a 60-kgauss
field is rotated in the (100) plane perpendicular to the
current. The 90° periodic anisotropy expected from the
presently accepted cubically symmetrical four valley
modeF is obtained for the tunneling current [Fig. 2(a),
(b), (c)J as well as for the excess current [Fig. 2(d)].
In all PbTe diodes investigated the periodic anisotropy
disappears as the forward bias is increased in the thermal
portion of the current characteristic [see Fig. 2(e)].
Thus there seems to be two components of current as
one goes from the excess current region into the thermal
region, the excess-type current which decreases in
magnitude as the bias is increased and the thermal
current which seems of different origin. The excess
current on the other hand does have the same anisot
ropy as the tunneling current.
In Fig. 2 at high currents [see Fig. 2(c) and (e)J a
180° periodic anisotropy is observed. This 180° periodic
anisotropy (which just means that the effect is in
dependent of the sign of the magnetic field) can be
explained in terms of magnetoresistance. While the
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current at the n-p junction is parallel to the [100J axis,
for this particular diode the bulk current is not since
the junction and the ohmic contact are not concentric;
rather both contacts are on the same face of the wafer.
As indicated above this diode is the only one to be
discussed made9 using vapor grown PbTe. In all other
diodes considered, the junction and ohmic contact
are concentric and only very small periodic anisotropy
is observed at high thermal currents.
The 90° anisotropy can be explained by investigating
the effects of the magnetic field on the four valleys.
Since the electric field is in the [100J direction, all
valleys contribute equally to the H = 0 tunneling
current. The minimum current (which is the maximum
effect) occurs when the magnetic field is parallel to the
[010 J and [001 J directions. In these directions the
effective cyclotron resonance masses m/ for all four
(111) oriented ellipsoids are equal. As the magnetic
field is rotated from these directions, two of these
effective masses increase :;md two decrease. At 60 kgauss,
since the heavier m/ valleys are significantly lower in
energy they contribute more to the tunneling current
than the smaller m/ valleys. Thus on the average one
would expect the smallest "average" effective cyclotron
mass"in the [OlOJ and [OOlJ directions and the largest
effects" in'" these directions.
The~inimum effect occurs when the magnetic field
is parallel to the [Ol1J and [Ol1J directions. In these
directions the effective cyclotron mass for two of the
ellipsoids is at its maximum for magnetic fields in the
(100) plane. Thus at 60 kgauss where the heavier m/
valleys contribute more to the tunneling, the minimum
effect occurs in these directions. At lower magnetic
fields where the light and the heavy m/ valleys con
tribute nearly equally to tunneling the anisotropy may
be small as will be described below.
0.4
~ 0.3
5 0. ,.
<t :::;
...J :; 0.2
". r
[ !
0. 30. 60. 90. 120. 150. 180. 210. 240. 270
MAGNETIC FIELD DIRECTIo.N, DEGREES FRo.M [110.]
FIG. 3. Diode current a~ a function of the direction of a 53.2-
kgauss magnetic field in the (111) plane perpendicular to the
direction of junction current. The diode is in the tunneling region
and is biased 20 mv forward. [111J Diode
Figure 3 shows the 60° periodic anisotropy in the
tunneling current as 53.2 kgauss is rotated in the (111)
plane perpendicular to the current. The variation in
amplitude of the current oscillation as the magnetic
field is rotated is believed due to slight misalignment
of the magnetic field with respect to the (111) plane.
For tunneling from an ellipsoidal band with principal
effective masses ml*, m2*, ma*, the number of electrons
leaking from the valence to conduction band is12
p2 (m1*m2*m3* )t ( n m/ exp
181rh2Eot m/3
where
3
(m/)-l= L [cos2'Y;/m;*J,
i=l
and 'Yi are the angles between the direction of the
electric field and the principal axes of the effective
mass tensor. For tunneling in the [111 J direction three
1.6 ,----,------------ ·---,-----~I
t.4 I
<J) w a: w
'" t.2
t.O
~ 0.8
::::;
...J
~
0.6
0.4
0.2 J
PbTe tt5S
to 40 50 60
KILOGAUSS
FIG. 4. Diode current as a function of magnetic field for the
diode of Fig. 3 and for the magnetic field directions corresponding
to the three maxima and three minima of Fig. 3. The diode is
biased 20 mv forward.
12 L. V. Keldysh, Sov. Phys.-JETP 6, 763 (1958).
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of the valleys have an effective force mass13 mJ*
=0.0137mo while the fourth valley has the large effective
force mass m/=0.06me.
Substituting in Eg. (1) these values for effective
mass, Eg=0.2 v andF/e= 1.5X105 v/cm (as determined
above) it is seen that less than 0.1% of the tunnel
ing current is carried by carriers from the heavy valley.
The heavy force mass valley also exhibits a large mass
to the magnetic field in the (111) plane. This cyclotron
resonance mass me * = 0.027 mo does not change as the
magnetic field is rotated in the (111) plane and the
current carried by this valley should show no anisotropy.
For the other three valleys m/ varies with the magnetic
field direction varying from a minimum of O.013mo to
a maximum of 0.027mo. At high magnetic field the
maximum effect (minimum current) occurs as expected
in the [121J direction where one m/ is a minimum and
the other two are equal. The minimum effect occurs in
the [110J direction as expected where one mc* is a
maximum and the other two are equal although smaller
than the two equal masses in the [121J direction.
While at high magnetic fields the anisotropy is
dictated by the heavy m/ valleys this is not true at low
magnetic fields where both light and heavy m/ valleys
must be considered since they all make sizable contri
butions to the tunneling current. In Fig. 4 the currents
for the magnetic field in the directions corresponding
(0 three maxima and three minima of Fig. 3 are shown
as a function of magnetic field. As seen from the figure
the anisotropy at low fields is small and is different
from that at high fields. Also at low fields where
contributions from the light m/ valleys are more
important, the magnetic field effects are as predicted
larger.
o
0.4
0.1 ---------------- AT
! \J:,
ll: :;
" ::;
4-'
:E
, [·to! ,i.Il), [+1 ~ , J
-45* o· 45'" 90" 135" 180tl 225-
ANGLE FROM [OOIJ OF TRANSVERSE MAGNETIC FIELD
FIG. 5. Diode current as a function of the direction of a 53.2-
kgauss magnetic field in the (110) phne perpendicular to the
direction of junction current. The top curve (and corresponding
left ordinate) are for a diode reverse bias of 30 mv and the bottom
curve (and corresponding right ordinate) are for a diode reverse
bias of 85 mv.
1B All effective mass values used are reduced values derived from
the values in reference 7 assuming identical conduction and
valence bands.
14 B. Lax, H. J. Zeiger, R. N. Dexter, and E. S. Rosenblum,
Phys. Rev. 93, 1418 (1954). .FIG. 6. An artist's view of the relationship of the ellipsoidal
band structure of PbTe to the (110) plane. The valleys 1 and 2
have a smaller mass as seen by the [110J directed electric field
than valleys 3 and 4.
[110J Diode
Figure 5 shows the behavior of the tunneling current
for a [110J diode as a 53.2-kgauss field is rotated in the
(110) plane perpendicular to the current. For tunneling
in the [110J direction two of the valleys have an effec
tive force mass mJl*=mf2*=0.0125mo while the other
two have a heavier effective force mass, mJ3*=mJ4*
=0.029mo. Thus the contribution, using Eq. (1) and
the associated assumptions, to the tunneling current
by the last two valleys is about 10% of the total current.
Figure 6 shows the relation of the four-valley ellipsoidal
band structure of PbTe to the (110) plane. In Fig. 7
four cyclotron resonance masses are plotted as a
function of the direction of the magnetic fields in the
(110) plane. While this figure is for the conduction
band in germanium and the absolute values for the
masses are incorrect for PbTe, the general shape of the
curves is correct.
FIG. 7. Effective mass of
electrons in germanium at 4°K
for magnetic field directions in
a (110) plane. (After Lax,
Zeiger, Dexter, and Rosen
blum14.) While the absolute
values do not apply, the general
shape of the curve is the same
as that for PbTe.
o
[~OO] 30 60
[IH]
8 (degrees) 90
[~IOI
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(/)
UJ a:
UJ
Q. 50
:::E 30 «
:J
...J
:E
20
10
o 10 40 ) 30
c
E 20
10
40 80 120 160
mv
60mv (DIVIDE
ORDI NATE 8Y 2)
! I
20 30 40 50 60
KI LOGAUSS -j
I
FIG. 8. The effect of magnetic field on hump current. The
arrows in the inset, which shows the H=O, I-V forward charac
teristic, indicate the fixed voltages at which the magnetic field
effects on the diode current are shown. The magnetic field is
parallel to the [111 ] direction of junction current.
Looking at Fig. 7 and following the rules developed
above, at 53.2 kgauss one expects the maximum effect
(minimum current) in the [OOlJ direction where all the
masses are equal and minimum effects in the direction
perpendicular to the [111J direction and in the [1ioJ
direction where the heaviest masses occur. The mini
mum effect, however, may not occur in the [lioJ
direction because the two heaviest m/ valleys (valleys
3 and 4) are the heavymf* valleys and only produce 10%
of the H = 0 tunnel current. If one neglects these two
valleys, one expects a maximum effect in the [lioJ
direction because the masses of the valleys 1 and 2
are again equal. What one sees in the [lioJ direction at
high magnetic fields and low reverse biases is a minimum
effect within a maximum effect (top curve Fig. 5). In
creasing the reverse bias as shown in the figure tends to
accentuate the maximum effect associated with valleys 1
and 2 as does reducing the magnetic field. Increasing
the magnetic field, reducing the bias or going into
forward bias accentuates the minimum effect associated
with valleys 3 and 4. The experimental results can be
explained by noting that the magnetic field raises the energy of valleys 1 and 2 with respect to valleys 3 and
4. Increasing the reverse bias increases the relative
tunneling current due to valleys 1 and 2 (which are
the easy tunneling valleys) just as increasing the reverse
bias in a germanium tunnel diode increases the relative
tunneling current due to the more probable transitions
to the k=O conduction minimum. This explanation is
consistent with the Fermi level not being too far
removed from the (111) valley band edges.
Effect of Magnetic Field on Hump Current
Two PbTe tunnel diodes showed a hump in the current
in the "thermal" region at about 165 mv (see inset Fig.
8). In Fig. 8 the effect of longitudinal magnetic field
on this hump current is illustrated. The current as a
function of magnetic field is plotted at fixed forward
biases both larger and smaller than 165 mv. Less than
3 kgauss is necessary to reduce the hump current so it no
longer can be distinguished. One explanation of hump
current is that it is due to tunneling through the
intermediary of a discrete energy level in the forbidden
gap. If this explanation is correct, our results indicate
that at least for PbTe this energy level is extremely
sensitive to magnetic field.
SUMMARY
Experimental results on magneto tunneling in PbTe
confirm in detail, although not quantitatively, the
multivalley band structure having prolate energy
ellipsoids oriented along the (111) crystalline axes.
Heavier mass bands close in energy to these ellipsoids
are necessary to explain the apparent position of the
Fermi level with respect to the band edges. Quanti
tative interpretation in terms of the band structure
awaits the development of magneto tunneling theory
applicable to PbTe. The experiments have brought
out differences between the excess current and the
"thermal" current and have shown a large magnetic
effect on hump current.
ACKNOWLEDGMENTS
The authors wish to thank T. M. Quist, C. R. Grant,
and J. M. McPhie for help in taking the data, and H. H.
Bessler for assistance in fabricating the diodes. We
should also like to thank J. H. Racette and R. N. Hall
of the General Electric Research Laboratory who
graciously supplied us with 2 PbTe tunnel diodes. We
are indebted to B. Lax, J. G. Mavroides, and P. N.
Argyres for many helpful discussions. The authors are
grateful to F. Smith and W. Mosher of the M.I.T.
National Magnet Laboratory for providing us time in
the Bitter magnet and helping run the facility for us.
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1.1729305.pdf | Drift and Diffusion of Activation in OxideCoated Cathodes
Koji Okumura and Eugene B. Hensley
Citation: Journal of Applied Physics 34, 519 (1963); doi: 10.1063/1.1729305
View online: http://dx.doi.org/10.1063/1.1729305
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/3?ver=pdfcov
Published by the AIP Publishing
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] IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47INVESTIGATION OF ELECTRON TRAJECTORIES 519
has been discussed extensively in betatron theory. These
fields are shown to have radial focusing. When n~ -1,
however, the particle orbit will not in general exhibit
radial stability. Stringent conditions must be imposed
on the angular aperture of the injected beam for all
~>O in order to have the maximum radius of the particle
orbits only slightly changed for particles injected with
a small radial component.
If ~>O the particle will move toward the center of
the field and be reflected off some finite inner radius.
For n S; -1, increasingly more of its kinetic energy is
transferred to radial motion until at some radius, given
by x at xmo.x, the motion is entirely radial. For n> -1,
the motion is never purely radial. Figure 7 shows the
qualitative behavior of the particle in a field for which
nS; -1, and should be compared with Fig. 6. V. CONCLUSIONS
Comparison of the experimental results with the
theoretical behavior of single-particle trajectories in this
geometry justifies such a treatment for qualitative
studies of this type of problem. It is possible to choose
the parameters of the field and beam such that the
particle describes many radial periods before returning
to the injection point. Therefore, the particle may be
confined by such a field for a sufficiently long time so
that it is possible to form a plasma in the region of
motion.
ACKNOWLEDGMENTS
The authors wish to acknowledge valuable discussions
with Dr. J. W. Flowers. The preparation of the photo
graphs by H. W. Schrader is appreciated.
JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 3 MARCH 1963
Drift and Diffusion of Activation in Oxide-Coated Cathodes*
KO]I OKUMURAt AND EUGENE B. HENSLEY
Department of Physics, University of Missouri, Columbia, Missouri
(Received 30 July 1962)
Two experiments pertaining to oxide-coated cathodes are described. One of these involves the drift of
activators in oxide-coated cathodes under the influence of an electric field and the other involves the dif
fusion of these activators in the absence of an electric field. The drift experiment revealed a temporal non
uniform increase in the activation of the conductivity. However, it failed to show any significant spatial
nonuniformity. Pulse measurements of the current-voltage characteristics showed the origin of the tem
poral nonuniformity to be associated with nonohmic properties of the coating which were related to the
Nergaard effect. The results of the diffusion experiment indicated a diffusion coefficient at 10000K for the
crystalline defects undergoing diffusion of (1.5±0.8)XlO-7 cm2/sec and an activation energy of the dif
fusion of 0.67±0.4 eV. Comparison of this diffusion coefficient with data obtained from the literature makes
possible the identification of the mobile defects involved in activation processes in oxide-coated cathodes
as being barium vacancies. On the bases of these results, a "mobile acceptor model" for oxide-coated cath
odes is presented which suggests that activation is achieved by the removal of the highly mobile acceptors
(barium vacancies) which compensate the relatively immobile donors (oxygen vacancies) and not by an
increase in the density of the donors as has usually been assumed. Nergaard's theory explaining the milli
second decay of thermionic emission from oxide-coated cathodes is modified in terms of the mobile acceptor
model, and the kinetic processes involved in the activation of oxide-coated cathodes are discussed.
I. INTRODUCTION
SINCE the early experiments by Beckerl-2 it has been
generally believed that the activation processes in
oxide-coated cathodes which lead to a low effective work
function are associated with a stoichiometric excess of
barium within the cathode. However, the form in which
this stoichiometric excess of barium exists and the de
tails of the kinetics involved in obtaining this state have
not been well understood. In the present investigations,
two experiments which shed light on these processes
* Supported in part by the U. S. Office of Naval Research. t Present address: Division of Pure Physics, National Research
Council, Ottawa, Canada.
1 J. A. Becker, Phys. Rev. 34, 1323 (1929).
2 G. Herrmann and S. Wagener, The Oxide Coated Cathode
(Chapman and Hall, Ltd., London, 1951), Vol. II, p. 156. are described. One of these involves the drift of the
crystalline defects responsible for a cathode's activation,
hereafter referred to as the activators, under the in
fluence of an electric field and the second involves the
diffusion of these activators in the absence of an elec
tric field. Most of the conclusions to be drawn from the
present investigation are based on the results of the
diffusion experiment, however, the results of the drift
experiment are consistent with these conclusions and
contribute to the understanding of the electrolytic
activation of oxide-coated cathodes.
The drift experiment was originally motivated by ob
servations of one of the authors in connection with
another experiment3 of a peculiar behavior in the in-
3 E. B. Hensley, J. App\. Phys. 27, 286 (1956).
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] IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47520 K. OKUMURA AND E. B. HENSLEY
PROBE
THEllMOCOUPLE
PIIOBES FIG. 1. Cross section of
elements in tube for drift
experiment.
crease of the conductivity of an oxide-coated cathode
during electrolytic activation. The variation of the con
ductivity of an oxide coating sandwiched between two
pure nickel planar electrodes was observed to increase
in two distinct steps. Since it was suspected that the
oxide layer activated first at one electrode and that this
activation then spread progressively to the second elec
trode, the present experiment was designed to include
three probes in the oxide layer to observe such an effect.
The results of the present experiment confirmed the
temporal nonuniform increase in conductivity but failed
to show any significant spatial nonuniformity. Pulse
measurements of the current-voltage characteristics re
vealed the origin of the above phenomena to be associ
ated with nonohmic properties of the oxide layer at
intermediate levels of activation.
After it became apparent that the drift experiment
could not yield direct information regarding the mo
bility of the activators in oxide-coated cathodes, an ex
periment to measure the diffusion of these activators
based on thermionic emission measurements was de
vised. A sprayed cathode was prepared on a pure plati
num ribbon 2X30 mm in size. The ribbon had a small
tab of active cathode nickel inserted at one point. When
heated, the profile of the thermionic emission distribu
tion could be observed by collecting the current through
a transverse slit in a movable anode. The emission in
the vicinity of the nickel tab was observed to be much
larger than in the other regions of the cathode. Pro
longed heating at selected temperatures caused this
region of high emission to spread laterally. This diffusion
process was analyzed and the·diffusion coefficient of the
mobile defect in the oxide coating responsible for the
change in activation was obtained as a function of the
temperature. By comparing this temperature-dependent
diffusion coefficient with other data obtained from the
literature the mobile defects were identified as barium
vacancies.
II. EXPERIMENTAL PROCEDURES
A. Drift Experiment
The arrangement of the sample in the drift experiment
is shown in Fig. 1. Two planar electrodes, each of which
had a tungsten heating coil inside the cup were pressed
together by tungsten springs projecting from the tube press. Sandwiched between these two electrodes was a
layer of (BaSr)O about 1 mm thick. Three fine platinum
probes were located in the coating parallel to the elec
trode faces and at approximately equal intervals through
the coating thickness. A nickel-molybdenum thermo
couple was welded to the back of each electrode.
The electrode cups and heaters were commercial
cathode parts. On the flat surfaces of these cups were
welded flat buttons of either pure electrolytic nickel or
pure platinum. These buttons were machined to flat
disks 8 mm in diameter. The coating material used was
Raytheon CSl-2 cathode coating mixture.4 The probe
wires were approximately 0.001 in. in diameter and were
bent into a flat hook shape to prevent their working
loose from the coating during the tube assembly process.
All tube parts were carefully cleaned and the finished
tubes were processed following standard procedures for
vacuum tube construction.6 The planar electrodes were
heated in high vacuum at temperatures over 1000°C
prior to coating. Each electrode was first coated by
spraying to a thickness of about 0.2 mm and then was
placed in a special jig which held a probe flush to the
coated surface. A second layer of the coating was then
applied so as to embed the probe. Two such electrodes
were then mounted on the tube press with the aid of a
special assembly jig and were pressed together with a
third probe between them.
An all glass vacuum system with a three stage oil
diffusion pump was used to evacuate the tubes. After a
tube was baked out at 450°C the carbonate coating was
converted to oxide by heating. Because of the large
thickness of the coating the conversion process took an
unusually long period of time. Difficulties in the assem
bly process and in the conversion process limited the
number of successful tubes.
During the electrolytic activation of the oxide coating
the potential difference between the two electrodes was
maintained constant at 10 volts. This was achieved by a
specially designed regulated power supply in which the
control voltage was obtained by comparing the voltage
on the positive electrode with a reference voltage. The
voltage was maintained constant within approximately
0.01 V throughout the activation process in which
the sample resistance changed about three orders of
magnitude. The current through the sample was ob
tained by measuring the voltage drop across a resistor
R., in series with the positive electrode, several values
of which were available for the different current ranges.
The amount of the Joulean heating in the sample and
the Peltier heating at the electrode-sample contacts due
to the conduction current varied by large amounts dur
ing the course of the experiment. This made it necessary
to provide automatic control of the temperatures of each
of the two electrodes. Each of the two temperature con-
4 A spray suspension of approximately equal molar (BaSr)C0 2•
6 Tube Laboratory Manual, 2nd edition, Research Laboratory
of Electronics, Massachusetts Institute of Technology, Cam
bridge (1956).
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IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47ACTIVATION IN OXIDE-COATED CATHODES 521
trol circuits was arranged to be activated for a lO-sec
period every 2.5 min. During this period the potential
from the thermocouple was compared with a preset
reference potential. The difference voltage was used to
control a servo motor which in turn drove a set of fine
control Variacs suppl ying heater power to the electrodes.
By proper adjustment of the servo amplifier gain, the
amount of correction for each period could be made ap
proximately equal to the error measured.
The dc conductivity of each of the four coating sec
tions set off by the three platinum probes was measured
at regular intervals. Connections for making these meas
urements were made by means of microswitches which
were actuated by projections on a slowly rotating drum.
Successive pairs of the five leads from the electrodes and
probes were connected to the input of a General Radio
type 715A dc amplifier, the output of which was con
nected to an Esterline Angus recorder. Thus, the poten
tial drop across each of the four sections of the coating
were recorded alternately during one revolution of the
drum. The voltage drop across the resistor R8 was also
recorded to obtain a measure of the current through the
sample. Two additional microswitches were used to
energize the relays controlling the previously described
temperature control circuits.
Voltage-current characteristics of the four sections of
the coating were made intermittently during the activa
tion process. It was important that these measurements
not interfere with the dc electrolytic activation process.
Accordingly, millisecond pulse techniques were adopted.
A multivibrator circuit fed square pulses of approxi
mately 2-or 3-msec duration to a Western Electric 275-B
mercury relay, causing a 40-J,lF electrolytic capacitor
C" to charge through a 25-Q resistor Rp from a 45-V
battery. At the end of the pulse, the capacitor was con
nected to discharge through the same resistor Rp. The
resulting voltage across Rp was two successive approxi
mately triangular pulses with a total duration of ap
proximately 5 msec.
The voltage pulses which appeared across resistor
Rp were applied across the experimental tube and the
series resistor R •. The voltage across R8 was proportional
to the current and was fed to the vertical amplifier of a
Dumont type 304A oscilloscope through a difference
amplifier. The potential between any pair of the probes
or electrodes was applied to the horizontal amplifier of
the oscilloscope through a second difference amplifier.
The pulse also triggered a relaxation oscillator which fed
a synchronized unblanking pulse to the z axis of the
oscilloscope. The resulting current-voltage curves were
recorded by a Polaroid-Land oscillograph record camera.
B. Diffusion Experiment
A cross-sectional view of the principal components of
the tube used for the diffusion experiment is shown in
Fig. 2. The cathode ribbon was made of 99.9% pure
platinum 70X 4 mm in area and either 0.003 or 0.004 in. TANTALUM
ANODE BOX PORCELAIN
CYLINDER
IBG 8r)0
COATING
FIG. 2. Cross section of principle elements in
tube for diffusion experiments.
thick. A 1-mm edge was folded down on each side of the
ribbon to give it rigidity at high temperatures. A 900
bend was made 20 mm from each end of the ribbon to
connect with the press leads of the tube. Notches were
cut in the folded sides to facilitate making this bend and
to provide extra heating to compensate for the conduc
tion losses to the press leads. A piece of very thin nickel
foil 2 mm in length was welded to the platinum surface
a few millimeters from the center of the ribbon. This
nickel contained about 0.5% aluminum and is known to
be a very effective activator material for oxide-coated
cathodes. 6 To prevent the decrease in cathode tempera
ture in the area of the nickel tab caused partially by the
additional thickness and also by the higher thermal
emissivity of the nickel surface, the width of the plati
num ribbon was reduced along the length of the nickel
activator. Two Pt, Pt-lO% Rh thermocouples of 0.002-
in. wire were welded to the ribbon at points 10 rom either
side of the center. The ribbon was directly heated by
passing an alternating current through it. The upper
surface of the ribbon was sprayed with Raytheon C51-2
coating mixture to a thickness of about 0.1 mm.
The anode consisted of a tantalum box 7 in. long and
about lOX 10 mm in cross section. The top of the anode
structure was supported by two pivots located at the
top of two parallel glass rods such that the bottom of
the anode structure could swing parallel with the
cathode ribbon. The bottom of the anode was made of
a platinum plate in which a slit of 0.1 to 0.3 mm wide
was cut at right angles to the length of the ribbon. Inside
the anode box and immediately behind the slit was a
platinum collector plate supported on a thin porcelain
tube. A fine wire inside this porcelain tube provided for
electrical connection from this collector to a press at
6 M. Benjamin, Phil. Mag. 20, 1 (1935).
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IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47522 K. OKUMURA AND E. B. HENSLEY
CI!
:II c(
""0_
~
...
Z
III 1.11
~ I
;:)
U
II: o
\:; 0.15
III
j "ICKEL TAB o
U ---AFTER ACTIVATION
I INITIAL DISTRIBUTION I
I 2 3 4 !5 6
DISTANCE IN MM
FIG. 3. Distribution of thermionic emission along ribbon
cathode of diffusion tube at the end of a diffusion experiment
(dashed line), and after activation in preparation of a new
diffusion experiment (solid line).
the top of the tube envelope. In order to outgas the lower
part of the anode structure by electron bombardment, a
pair of tungsten filaments were enclosed in a small box
located at one end of the platinum ribbon.
Standard procedures for vacuum tube construction
were followed in processing these tubes.5 Prior to spray
ing, the cathode ribbon was heated to approximately
lO00°C in high vacuum. The completed tube was sealed
to the vacuum system and baked at 450°C for one hour.
The anode structure was outgassed by rf induction
heating and electron bombardment. After the carbonate
was decomposed at a temperature of 11000K, the anode
was again outgassed. A getter in a side chamber was
then flashed and the tube was sealed off at a pressure of
approximately 1 X 10-7 Torr with the cathode hot.
In the measuring apparatus, the tube was mounted
so as to swing freely about a horizontal axis which co
incided with the axis of the pendulum anode inside the,
tube. By slowly rotating the tube about this axis while
the internal pendulum anode hung free, a profile of the
thermionic emission from the cathode could be obtained.
The standard scan rate used was 10.5 mm/min.
An anode voltage of 67.5 V was normally used. As the
cathode-anode spacing was approximately 3 mm, this
provided good collimation of the electron emission from
the cathode. A potential of 6 V was placed on the collec
tor with respect to the anode to minimize the loss of
secondary emission current from the collector surface.
The collector current was measured using a Keithley
model 210 electrometer, the output of which was fed
into an Esterline-Angus recorder.
The emission measurements were usually made at a
cathode temperature of 800oK. This produced a collec
tor current in a range from 10-10 to 10-6 A. This low
cathode temperature was preferred in order to minimize
any interference with the distribution of cathodeactiv
ity during the measurements. The time constant of the
circuit, however, prevented the use of cathode tempera
tures lower than 800oK. Immediately after sealing a tube off from the vacuum
system the emission distribution was usually not flat
but showed a small rise in the neighborhood of the
nickel activator. This was considered to be the result of
the repeated heat treatments of the cathode during the
tube processing which activated the cathode coating
area near the activator. However, the initial emission
distribution was often unstable and the above mentioned
emission rise over the activator region tended to de
crease when the cathode was heated at low tempera
tures, i.e., about 6OO0K, for several hours. At this stage,
the collector current at 8000K was usually in the range
between 10-10 and 10-9 A.
When a cathode was heated to a temperature above
11000K, the emission level over the activator region of
the coating rose sharply, while in the other regions the
emission remained almost unchanged. Following several
minutes at about 11500K, the sharp peak of emission
in the activator region was often more than two orders
of magnitude higher than in the unactivated regions.
However, this sharp peak was not stable and the height
of the peak gradually decreased when the cathode was
held at temperatures lower than the activation tempera
ture. Although the rate of emission decrease was very
slow, the peak often decreased by one or more orders of
magnitude canceling out most of the activation gained.
Because of the instability of the emission described
above, no diffusion experiments were successfully car
ried out after the first activation of the cathode. In
stead, it was often necessary to age the cathode at tem
peratures between 800° and lO00oK, the range in which
most of the diffusion experiments were made, for long
periods of time in order to stabilize the emission. A
second activation was made after aging and the emission
stability was again checked. With some tubes, the emis
sion stability was so bad that repeated activation and
aging could not bring the cathode into a stable condition.
When the cathode emission became sufficiently stable
the cathode was given a final activation for 10 to 20
min at a temperature of about 1150° to 1200°K. The
emission distribution was then measured at 8000K to
obtain the initial distribution for the diffusion experi
ment. Following this the cathode temperature was held
constant at the temperature at which the diffusion was
planned to take place. Emission distribution measure
ments were made intermittently with the cathode tem
perature at 8000K.
The diffusion temperatures possible were limited. It
was necessary that they be higher than 8000K so that
the emission measurements would not interfere with
the diffusion process. On the other hand, they should
not be as high as the activation temperature, since an
additional activation would then take place during the
diffusion process.
After the diffusion measurements at one temperature,
which usually took 100 to 200 h, the cathode was again
activated at about 12000K for from ten to twenty min
utes to establish a new initial condition for the second
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IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47ACTIVATION IN OXIDE-COATED CATHODES 523
diffusion measurement at a different temperature. The
additional activation raised the emission level over the
area of the nickel activator and at the same time lowered
the emission level outside the activator region. The exact
cause of this latter effect is not fully understood but
appears to be due to some deactivation process ~hich
fortunately in the successful tubes is only present at
the elevated cathode temperatures. One possibility
is an increase in gas evolution which leads to a poison
ing of the cathode in the regions away from the nickel
activator. In Fig. 3 the dashed line shows the emission
distribution at the end of a diffusion experiment. The
solid line represents the initial condition of the distribu
tion immediately following the activation process in
preparation for a second diffusion experiment.
Although the circumstances which lead to suitable
initial conditions for the diffusion measurements are
not fully understood, the process makes possible these
measurements at different temperatures with one tube.
This was of particular importance in the present in
vestigation since the absolute values of the constants
in different cathodes were not likely to be reproducible
because of different coating conditions such as the physi
cal dimensions of the coating particles and the density.
Usually after several repetitions of a diffusion experi
ment it became difficult to reactivate the cathode and
to establish a sharp emission rise in the nickel activator
region. In such exhausted conditions the cathodes
often looked greyish, particularly in the area over the
activator.
III. EXPERIMENTAL RESULTS
A. Drift Experiment
Although a total of sixteen tubes were constructed and
measured, several of these failed to yield significant re
sults due to mechanical failures of various types. On
the first ten of these tubes, dc measurements only were
made, after which the desirability of making the pulse
measurements became evident.
To make measurements of the activation processes
the samples were heated to a constant temperature,
usually about 1100oK, and a constant voltage, usually
10 V, was applied between the two electrodes. With al
most all of the tubes tested, the direct current through
the oxide increased from an initial value of between
10-4 and 10-3 A to a final value between 10-2 and 10-1
A. This corresponds to an increase in dc conductivity
from about 10-5 to 10-3 n-1cm-1• The final value of the
oxide conductivity after activation compared favorably
with that of oxide-coated cathodes activated by other
means. The rate of activation, however, was very slow
compared with the activation utilizing the reduction of
the oxide by base metal impurities. With most tubes it
required from a few hours to over 20 h to bring the cath
odes to their fully activated state.
The data obtained from tube No. 14, which were typical
of data obtained from the other tubes, are shown in Fig. ~~O~--~----~-----T-----r-----r~
a::
c[
a:: t-
iii TUBE NO. 14 a::
~
)-10-1
t-:;:
t
U
::> o
~IO"1/
u
FIG. 4. Relative conductivities of the four sections of the oxide
coating for drift tube No. 14 as a function of time. The number!
refer to the layers in sequence starting from the negative electrode
as shown in Fig. 1. An arbitrary scale is used since the thicknesses
of the layers are not exactly equal. The bottom curve shows the
variation of the total current. The temperature and total voltage
across sample were held constant at 11500K and 10.3 V,
respectively.
4. The conductivities are shown on an arbitrary scale
since the thicknesses of the oxide layers were not exactly
equal. The numbers refer to the layers in sequence start
ing from the negative electrode and proceeding to the
anode. Also shown is the total current through the oxide.
All tubes exhibited an initial rise in the conductivity
for all four sections which slowed down after a relatively
short period of time and was followed by a period in
which the conductivity underwent a very sharp rise
during which the major part of the final activation was
gained. Although the above pattern was followed by
most of the tubes measured, the details regarding the
time required for various phases in the activation proc
esses varied considerably from one tube to another.
In general, it was observed that the rates of increase
of the conductivity for the four sections of the oxide
were not entirely uniform. This was most pronounced
during the early stages of the activation process. At
first it was suspected that these nonuniformities were
evidence for the transport of localized activation as
was initially postulated. However, a more extensive
examination of the data from all of the tubes failed to
support the existence of this phenomena.
The millisecond pulse measurements of the current
. voltage characteristics of the coating sections were made
intermittently during the activation processes on all
tubes later than No. 10. Typical data for these measure
ments are shown in Fig. 5. Shown in this figure are direct
tracings of the oscillograph photographs. The abscissa
is the voltage axis with positive voltage to the right and
the ordinate represents current. The crossings of the
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TIME PULSE IS APPLIED IN HOURS
o 2 3 2 4 2 57 8 9 102 12 8 15 8 21.8
-----./ / ./ /' ./ ./ / ./ ./ / I
2 / / / ./ / / / / I {
3 c---; r-.
--&)4 ./ ./ / " I I /' ./ / ,,-
/ L
TOTAL / / ./ ./ ./ / / ./ ./ / ./
160 160 40 40 40 40 20 10 5 2.5 1.2
CURRENT MAGNIFICATION RATIO
FIG. 5. Oscillograph tracings of the current-voltage character
istics for tube No. 14. The times at which the curves were ob
tained are indicated by the numbers at the top. The sections as
numbered in Fig. 1 are indicated by the numbers on the left. The
abscissa is the voltage axis with positive voltage to the right, the
same direction as the dc voltage. The current scales are magnified
by the factors shown at the bottom of the figure. The current
scale for the total thickness has been reduced in each case by an
additional factor of 3/8.
abscissas and ordinates represent the apparent origins
of the characteristics. These differ from the true origins
by the amounts of the superimposed dc components.
Because of the increase in current during activation,
the ordinates of the curves were scaled down as the acti
vation increased. The scaling factors are indicated at
the bottom of each set of curves. The abscissas for
the total thickness curves were also scaled down by
a factor of 3/8 in order that the figures would be of
comparable size.
Starting from the preactivation characteristics, the
I-V characteristics of each section of coating evolved
through a reproducible sequence of variations until
they reached their final form. The I-V characteristics
of the cathode portion which was initially curved up
ward changed to curve downward in the early period
of the activation. This curvature was then maintained
throughout the latter part of the experiment. The curves
for sections 2 and 3, which were both essentially ohmic
during the initial period of activation, changed to curve
downward during the intermediate stages and then re
turned to the ohmic characteristics when the coating
reached its fully activated state. However, section 3 was
faster than section 2 in both of these transitions and its
characteristics were practically ohmic throughout the
latter period of the activation process. The characteris
tics of the anode portion of coating maintained its ini
tial form for about one-third of the total activation
period, although the curvature became somewhat more
complicated in the transition period after about 9 to
10 h of activation.
During the intermediate period of activation the I-V
characteristics of all four sections were curved down
ward as were also the curves for the total thickness of
the coating. Since the direct current producing the acti
vation corresponds to a positive current on these curves, it is evident that during activation the oxide sections
are all conducting in the "reverse direction" of the recti
fying characteristics represented by these curves. The
rectification characteristics of all the sections tend to
be more pronounced during the plateau period of the
activation. This is probably the principle reason for the
retardation in the build up of the conduction current
in this period. It can be observed in tube No. 14 that
the direct current actually decreased slightly during this
period.
With some of the tubes an attempt was made to ob
serve the effect on the pulse measurements of suddenly
reversing the dc conduction current. The results of such
a procedure were so rapid that it was difficult to obtain
quantitative data. In general, the principal effect was
that during intermediate levels of activation the curva
ture of the characteristics for section 2 and 3 tended to
become ohmic and then curve in the opposite direction.
These transitions would take place in a time span of only
a few seconds although the details of the curves tended
to be somewhat unstable for a longer period.
Several of the tubes were broken open at the conclu
sion of the activation process. It was observed that a
thin layer of the coating at the surface of the positive
nickel electrode was somewhat dark while the coating
at the cathode side remained white. These sections of
the oxide coating were subjected to a spectroscopic
analysis. In two tubes approximately 100 ppm of nickel
was observed in the oxide of the anode section whereas
only a trace of nickel could be detected in the oxide from
the cathode section. Also observed was a small amount
of magnesium in both the cathode and anode section.
Analysis of the nickel used for these electrodes revealed
approximately 100 ppm of magnesium. The presence
of nickel in the coating near the anode suggests that the
anode buttons were somewhat eroded during the elec
trolytic activation. Tube No. 16 was run using platinum
button electrodes in order to check whether or not this
contamination of the anode section had any noticeable
influence on the experiment. No particular differences
were observed between the data obtained for this tube
and the others in which nickel electrodes were used.
B. Diffusion Experiment
Although the principles of the diffusion experiment
are straightforward, it was necessary to expend several
tubes in order to learn the proper procedures for con
ditioning the cathodes to obtain reproducible results
and to learn the range of temperatures and diffusion
times that could be used. This limited the number of
tubes from which a significant sequence of data was
obtained to two.
Figure 6 shows a typical set of emission distribution
curves obtained with tube No.8. The origin of the
abscissa was taken at the center of the nickel activator.
Since the nickel was welded about 4 mm from the center
of the cathode ribbon in order to obtain the best tem-
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IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47ACTIVATION IN OXIDE-COATED CATHODES 525
2.0
D. --0 h
2 ---12.2h
'" ----46.9h
"'0 I. --------118.5 h.
~
....
Z
UJ 1.0 II:
II:
::>
(,)
II: 0
t; 0.5
UJ
..J ..J 0
(,)
0 -2 -I o I 234 6
DISTANCE IN MM
FIG. 6. Typical set of thermionic emission distribution curves ob
tained from the tube No.8 showing effects of diffusion.
perature uniformity on one side, the emission distribu
tions were measured on that side only. It will be noticed
that the emission peaks at the center of the nickel acti
vator remain at almost the same level despite the slowly
extending tail during the diffusion period, making the
area under the distribution curves increase. This sug
gests a slow additional activation due to the nickel
activator.
Data similar to that obtained from tube No.8 were
also obtained from tube No.9. However, the distribu
tion curves were very patchy with this tube making the
measurements rather inaccurate. The surface condition
of the coating of this tube was not good. However, it
was felt that the results obtained from this tube sub
stantially verify the results obtained from tube No.8.
In order to analyze the diffusion using the thermionic
emission data, it is necessary to assume a relationship
between the electron emission and the density of the
crystalline defect undergoing diffusion. A general re
view of such relationships has been given by Hensley. 7
Probably the most appropriate approximation to be
used in the present situation is the model in which the
Fermi energy is expressed as a function of a partially
filled donor, given by Eq. (26) in the above reference.
Substituting this value for the Fermi energy in the
Richardson-Dushman equation results in the expression
Na-N a 41rmek2J'2 (E.-Ed) Na-N a J= exp --- = C
Na h3 kT Na' (1)
where C is constant for any given temperature.
From Eq. (1) it can be seen that the thermionic emis
sion is approximately proportional to the number of
donors. However, if we consider changes in the emission
brought about by changes in either the donor or accep
tor densities, a much better approximation may be ob
tained. Since it is highly improbable that both the donor
and acceptor concentrations are altered by the diffu
sion, only two cases need be considered; either the
7 E. B. Hensley, J. App!. Phys. 32, 301 (1961). increase in J by an amount dJ is caused by an increase
in Nd by an amount dNd or it is caused by a decrease
in Na by an amount -dNa. For the first case we obtain
dJ= (C/Na)dNa,
and for the second case we obtain (2)
(3)
For the first case, changes in the thermionic emission
are directly proportional to the changes in the density
of donors. For the second case, changes in the thermionic
emission will be proportional to changes in the density
of the acceptors as long as these changes are not too
large.
The boundary conditions for the diffusion are those
of a long thin prism. Since the diffusion is observable
for a distance of only a few millimeters from the nickel
activator and the coating is 30 mm long, it is reasonable
to assume infinite boundary conditions. It is also as
sumed that the total number of mobile activator centers
(either excess donors or missing acceptors, depending
on which is moving) remain constant. The general solu
tion for this geometry is given as follows8:
N(x,t) = No/2(1rDt)i
X i~ f(x') exp[ -(x-x')2j4Dt]dx', (4)
where f(x') is the initial distribution of the activator
centers and No is their total number per unit length.
The shape of the initial distribution is undoubtedly
complex. However, since it is the result of diffusion dur
ing the activation process and the residuals of previous
diffusion experiments, it seems reasonable to assume that
for areas somewhat removed from the nickel activator
the initial distribution could be expressed as a series of
Gaussians. Thus
f(x') = f'(x')+ L: (a;/1r)i exp( -aix'2), (5)
i
where l' (x') is the non-Gaussian part of the initial dis
tribution in the vicinity of the nickel activator. Sub
stituting this expression into Eq. (4) we obtain the
solution for any time t:
No /-00 ( (X-X')2) N(x,t)= f'(x') exp - dx'
2 (1rDt)i -00 4Dt
No(a;)i ( X2) +L: exp .
i [1r(1+4aiDt)]1 4Dt+1/ai (6)
Figure 7 shows an example of diffusion data for tube
No.8 plotted on log J vs x2 coordinates. The 'portions
of these curves at values of x2 less than 5 are probably
dominated by the first term of Eq. (6). At larger values
8 H. S. Carslow and J. C. Jeager, Conduction of Heat in Solids
(Clarendon Press, Oxford, England, 1947), pp. 33-34.
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~ 10..--.,,--r----,--...,----,-----,
z TUBE NO.8
::;) TEMP. 900"K
>-0: c(
0: I
iii
0:
c(
~
I-
Z W
0:
~ ,-o .
~ -
Io
W
...J 5 0 o 25 FIG. 7. Diffusion
data for tube No.8
plotted on a loga
rithmic scale vs the
square of the dis
tance from the center
of the nickel tab.
of x2 the presence of what appears to be straight line
segments somewhat justifies the correctness of Eq. (6).
Initially it was attempted to analyze the data directly
using the second term in Eq. (6), however, it proved to
be impossible to obtain consistent results. Careful con
sideration showed that greatly improved results could
be obtained by replotting the data with the initial dis
tribution subtracted off. The data shown in Fig. 7 have
been replotted in this manner in Fig. 8. The basis for
this procedure is essentially twofold. First, as can be
seen from Eqs. (1), (2), and (3), changes in the thermi
onic emission are accurately proportional to changes in
the concentration of the diffusion defect, whereas the
actual value of the thermionic emission is only crudely
proportional to the actual number of defects. The second
and probably more important consideration is that the
initial distribution as shown by the thermionic emission
probably contains contributions from defects other than
those which undergo diffusion. Consequently, the initial
distribution of the defects undergoing diffusion is proba
bly considerably smaller than the apparent distribution
observed. In order to analyze the data, it is necessary
to assume that the actual initial distribution of the de
fects undergoing diffusion in the region being analyzed
is negligible.
The data as plotted in Fig. 8 are analyzed on the basis
of the second term in the Eq. (6). The slope of this line
at time tl should be
SI = -loge/ (4Dlt+ 1/ai), (7)
and at time t2 a similar expression is obtained. Solving
these two equations for the diffusion constant we obtain
In this manner the width of the initial distribution ai is
eliminated. The diffusion constant was then calculated
using all the pairs of the curves shown in Fig. 8 with the
results D= (6.98±O.98) X 10-8 cm2/sec.
The results for all of the diffusion experiments carried
out with tubes Nos. 8 and 9 are listed in Table I in the
order in which they were obtained. The first two runs
for tube No.8 were insufficiently stable to obtain an
estimate of the spread in the data. Also for tube No.9 TABLE I. Diffusion constants for activator centers.
Tube No.
8
8
8
8
8
9
9
9 Temperature
915°K
968°K
1015°K
900 oK
1000 oK
891°K
947"K
lOOooK DXI08
9±? cm2/sec
lO.9±? cm2/sec
32.0±6.6 cm2/sec
6.98±O.98 cm2/sec
20.3±4.0 cm2/sec
4.17±2.15 cm2/sec
6.05±1.69 cm2/sec
8.S4±4.86 cm2/sec
the condition of the coating was somewhat inferior to
tube No.8 and consequently, the data were more
dispersed.
From the variation of the diffusion coefficient with
temperature, the activation energy for the diffusion
process can be obtained. A plot of log D vs liT for the
data in Table I is shown in Fig. 9. Straight lines repre
senting the equation D=Do exp-ElkT are drawn
through the data for each of the two tubes. The activa
tion energies represented by the slopes of these lines are
shown in the figure.
The results of the diffusion experiment may be sum
marized by estimating the diffusion coefficient at lOOOoK
for the defect undergoing diffusion as being (1.S±O.8)
X 10-7 cm2/sec and the activation energy of the diffusion
as being O.67±0.4 eV. The accuracy of these results is
obviously rather poor. Most of this should be attributed
to the very porous structure of the oxide coating which
can result in considerable variability in the continuity
of the diffusion paths. Nevertheless the accuracy
achieved should be sufficient to distinguish between the
alternative diffusion mechanisms to be considered in
the next section for which the diffusion coefficients differ
by several orders of magnitude and the activation ener
gies differ by a factor of S.
IV. DISCUSSION AND CONCLUSIONS
A. Drift Experiment
The results of the drift experiment are not to be
considered as part of the major contributions of this
investigation. They do, however, illuminate some
10..----.---,.--.,....---.-----.
(I)
l-
i:
::;)
>-0:
c(
0: I-iii
. ~ ,
~
0 ..,
I ..,
0 TUBE NO.8
TEMP 900"K
109.8 ~
6.·4 h
38.8 h
o 17 h
5 10 20
DISTANCE IN MM SQUARED 25 FIG. 8. Data
shown in Fig. 7 re
plotted with the ini
tial distribution sub
tracted .
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IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47ACTIVATION IN OXIDE-COATED CATHODES 527
z o
en ::> ... ... 5
-8 o TUBE NO.8
• TUBE NO·9
IO~ ______ ~~ ______ ~ ______ ~
M ~
FIG. 9. Diffusion coefficients from Table I plotted as a function
of the reciprocal temperature. The activation energies obtained
from the slopes are shown in electron volts.
of the details regarding an experimental technique
which has frequently been used.9-11 The results are
also shown to be compatible with a model for oxide
coated cathodes to be presented shortly, as well as pro
viding evidence in support of a suggestion regarding
the processes involved in electrolytic activation.
The conductivities of the oxide layer were increased
two or three orders of magnitude during the electrolytic
activation which required from about 10 to more than
20 h. The major part of this increase was considered
to be due to the electrolysis since very pure nickel was
used for the electrodes.
The activation proceeded in two distinct steps in most
of the tubes confirming the previous observation made
by Hensley. The activation appeared to be spatially
homogeneous throughout the coating thickness. Al
though the relative rates of increase between the four
sections of the coating were rather irregular during the
first few hours, no significant indication of localized
ionic transport due to the electrolysis was detected.
The millisecond pulse measurements on the current
voltage characteristics of the coating revealed that
every section of the coating was nonohmic during most
of the activation period. The I-V -characteristics varied
almost continuously during the activation. Although
the details of the I-V characteristics for an individual
coating section were hardly reproducible the general
features of their variations as observed in several tubes
were acceptably similar.
At the beginning of the activation the two inter-
mediate sections of the coating were approximately
9 R. Loosjes and H. J. Vink, Philips Res. Repts. 4, 449 (1949).
10 J. R. Young, J. App!. Phys. 23, 1129 (1952).
11 R. C. Hughes and P. P. Coppola, Phys. Rev. 88, 364 (1952). ohmic while the cathode and anode sections showed rec
tification characteristics in opposite directions. At the
end of the activation when the coating was fully acti
vated the I-V characteristics were again rather similar
to those at the beginning except that the directions of
the rectifications at the cathode and the anode section
were reversed.
An explanation for the latter condition is rather easily
obtained on the basis of a semiconductor-metal contact
in which the work function of the semiconductor is
smaller than the work function of the metal.12 There
would thus be a barrier to the flow of electrons whether
these electrons were in the crystallites of the oxide or
in the pores. It is important to note that even though
the total I-V characteristics show no appreciable recti
fication, a pronounced rectification can still be present
at the two electrodes in such a manner as to offset each
other.
The rectification at the beginning of the activation is
somewhat more difficult to understand. It is tempting
to postulate that in this early stage of activation the
charge transfer between the nickel and the semicon
ductor is primarily by means of holes in the semicon
ductor. The rectification characteristics would then cor
respond to a p-type semiconductor-metal contact in
which the work function of the semiconductor is larger
than that of the metal. The difficulty with this hy
pothesis is that the work function of nickel is 4.6 eV and
although the work function of the semiconductor is
largest in this early stage of the activation it is doubtful
if it could be this large. One possibility is that the work
function of the nickel surface has been reduced as a
result of a thin-film chemical reaction between the nickel
and the barium oxide.
The two intermediate sections of the coating showed
the strongest nonlinearity of the I-V characteristics
during the middle period of the activation where the
conductivity reached the plateau. The total thickness
itself also showed similar I-V curves during this period.
As has already been pointed out, the direct current was
in the direction of hard flow for this rectification and
consequently represents the principle reason for the
retardation in the build up of the conduction current
in this period.
The above rectification characteristics of the two in
termediate sections of the cathode may be shown to be
related to the millisecond decay phenomena studied by
Sprou1P3 and Nergaard14 and sometimes referred to as
the "Nergaard effect." A revision of the explanation of
this phenomena as originally proposed by N ergaard
is discussed in a later section of the paper, but this
does not affect the present discussion. At the tempera
tures used in the drift experiment the conductivity of
the oxide layer was due to the electrons in the inter-
12 A. J. Dekker, Solid State Physics (Prentice-Hall, Inc., Engle
wood Cliffs, New Jersey, 1957), pp. 348-354.
13 R. L. Sproull, Phys. Rev. 67, 167 (1945).
14 L. S. Nergaard, RCA Rev. 13,464 (1952).
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IP: 129.22.67.107 On: Mon, 24 Nov 2014 01:41:47528 K. OKtJMtJRA AND E. B. HENSLEY
stices between the oxide grains.s Under the influence of
the electric field the side of the grains facing the anode
became deactivated by the Nergaard effect whereas the
sides of the grains facing the cathode became more ac
tive. Consequently, during the pulse when the electron
flow was from the cathode to the anode, the effective
work function of the surfaces contributing to this flow
was larger than when the current flow was in the oppo
site direction. In well-activated cathodes the Nergaard
effect tends to disappear. This is compatible with the
. disappearance of the rectification in the two inte.r
mediate sections in the fully activated cathode. Also m
support of this explanation is the reversal of the non
ohmic characteristics when the direct current is reversed.
B. Diffusion Experiment
In order to identify the mechanism responsible for
the diffusion in the present experiment the results ob
tained here are compared with data from other dif
fusion experiments to be found in the literature. For
this purpose we may summarize the results of the pres
ent experiment as showing that the mobile defect re
sponsible for the activation of an oxide-coated cathode
has a diffusion coefficient at lO00oK of approximately
(1.5±0.8) X 10-7 cm2/sec with an activation energy of
approximately 0.67±0.4 eV.
Color centers may be induced into single crystals of
barium oxide by heating the crystals in barium vapor.
The resulting blue coloration has an optical absorption
centered at 2.0 eV.15 Sproull, Bever, and Libowitz16
measured the diffusion of these color centers and ob
tained a diffusion coefficient at 10000K of 1.5X 10-12
cm2/sec, with an activation energy of 2.8 eV. They pre
sented arguments for believing these color centers to
be oxygen vacancies. Carson, Holcomb, and Ruchardt17
attempted to observe paramagnetic spin resonance
from these centers and concluded from the absence
of such a resonance that the centers contained two
electrons.
Although the experimental data in the present in
vestigation are not very precise it is clearly evident that
the diffusion mechanism in the present investigation is
entirely different from that of the oxygen vacancies.
Consequently, since these oxygen vacancies are the
most probable candidates for the principle donors in
barium oxide it would seem highly improbable that the
activation processes in oxide-coated cathodes involve
any significant change in the density of the donors.
Redington18 measured the self-diffusion of barium in
single crystals of barium oxide using the radioactive
isotope Bal40• At temperatures higher than 13S0oK he
observed the diffusion to be characterized by a large
15 W. C. Dash, Phys. Rev. 92, 68 (1953).
16 R. L. Sproull, R. S. Bever, and G. Libowitz, Phys. Rev. 92,
77 (1953).
17 J. W. Carson, D. F. Holcomb, and H. Ruchardt, J. Phys.
Chern. Solids 12, 66 (1959).
18 R. W. Redington, Phys. Rev. 87, 1066 (1952). activation energy which implies the formation of de
fects. Such processes are not involved in the tempera
ture range of the present experiments. At temperatures
lower than 13500K he found the diffusion depended on
the previous heat treatments of his crystals. In well
annealed crystals only one process was observed. For
this process, the diffusion coefficient at lO00oK was
7Xl0-12 cm2/sec, with an activation energy of 0.44 eV.
For crystals quenched from 14600K two processes were
observed. These had diffusion coefficients at lO00oK of
7X 10-12 and 9X 10-12, respectively, and the correspond
ing activation energies were 0.44 and 0.3 eV. The latter
of these two processes was found to be accelerated by
an electric field. The electric charge carried by this proc
ess was calculated as (1.7±0.3)e. Redington presented
arguments for believing that the neutral defects w;re
interstitial barium atoms, whereas the charge-carrymg
defects were barium vacancies. Redington also observed
a surface diffusion with a diffusion coefficient at l0000K
of 1.6XI0-7 having an activation energy of 0.16 eV.
Considering first the neutral barium atoms as the
defect responsible for the diffusion in the present ex
periment, this can be rejected because the diffusion co
efficient is about 4 or 5 orders of magnitude too small.
Also it should be recognized that these defects represent
a stoichiometric excess of barium and Sproull et al.14
has shown that a stoichiometric excess of barium exists
predominantly in the form of oxygen vacancies.
Although the diffusion coefficients for the barium
vacancies, as observed by Redington, are also four or
five orders of magnitude smaller than the diffusion co
efficients in the present experiment, it must be recog
nized that an experiment involving radioactive tracers
measures the self-diffusion of one of the principle con
stituents of the crystal lattice. A radioactive barium
ion will be advanced one lattice site each time a barium
vacancy moves past it. Consequently, the diffusion co
efficient for the radioactive barium ions will be related
to that of the barium vacancies by the ratio of the num
ber of barium vacancies to the total number of ion pairs
in the crystals.19 Although the densities of the barium
vacancies in Redington's single crystals are not known,
a concentration of one part in 1()4 or 105 is entirely con
sistent with estimates that have been made. Thus,
within the accuracy of the measurements, the magni
tude and temperature dependence of the diffusion of the
barium vacancies observed by Redington are the same
as for the diffusion observed in the present experiment.
This constitutes a strong reason for identifying the dif
fusion mechanism in the present experiment as being
barium vacancies.
As was mentioned above, Redington also observed a
surface diffusion with a diffusion coefficient of 1.6X 10-7
cm2/sec with an activation energy of 0.16 eV. Although
this activation energy is smaller than that observed in
the diffusion experiments described in this paper, it is
19 N. F. Mott and R. W. Gurney, Electronic Processes in Ionic
Crystals (Clarendon Press, Oxford, England, 1940), pp. 33-34.
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not sufficiently small to rule out this mechanism on this
basis alone. However, a series of experiments carried
out be Bever20 seems to indicate that surface diffusion
does not playa primary role in these considerations.
Bever measured the self-diffusion of radioactive bar
ium in a (BaSr)O cathode coating using practically the
same technique as employed by Redington. At tempera
tures below 12800K she obtained a diffusion coefficient
evaluated at 10000K of SX 10-12 cm2/sec with an activa
tion energy of 0.40, in good agreement with the results
obtained by Redington. However, Bever failed to ob
serve a diffusion mechanism which corresponds to the
surface diffusion of barium measured by Redington.
Thus while a direct extrapolation from Redington's
data would indicate that surface diffusion would domi
nate in the porous coating ofa normal oxide-coated cath
ode, the experimental evidence fails to show the exis
tence of this diffusion. The most probable interpretation
of this result is that a negligible number of mobile bar
ium atoms exist on the surfaces of the coating grains.
C. Mobile Acceptor Model
As a consequence of the above considerations, the
hypothesis is suggested that the principle acceptors in
barium oxide consist of barium vacancies which are
highly mobile and that the principle donors are oxygen
vacancies which are much less mobile.n The diffusion
coefficient for the acceptors at 1000oK, according to the
results of the present experiment, is of the order of from
10-6 to 10-7 cm2/sec while that of the donors at the same
temperature as estimated from the color diffusion ex
periments is about 10-12 cm2/sec. Consequently, the
various phenomena in oxide-coated cathodes which have
previously been interpretated as being caused by the
transport of donors should now be interpreted using the
diffusion and electrolytic drift of the acceptors. In the
present experiment, the variation of the emission dis
tributions along the cathode ribbon was caused by the
diffusion of acceptors from the regions adjacent to the
nickel activator into the central region where the accep
tor concentration was greatly reduced by the initial
activation process. In other words, an oxide-coated
cathode is activated not by introducing additional do
nors, but rather by the removal of acceptors which have
been compensating the existing donors in the cathode
coating.
20 R. W. Bever, J. Appl. Phys. 24, 1008 (1953).
21 An alternate hypothesis for explaining the properties of
oxide-coated cathodes has been suggested by R. H. Plumlee,
RCA Rev. 17, 231 (1956), namely that the principle donor is an
impurity group of the form (OH-·e). It is felt that this hypothesis
does not adequately account for the well-established need for the
chemical reduction of the oxide coating to maintain cathode
activation. Also his basic assumption that an excess barium con
tent in the oxide would result in a high vapor pressure of barium
over the cathode is questionable in that thermodynamic equi
librium probably can never be achieved at the normal operating
temperatures of the cathode. For example, the experiments by
C. Timmer, J. Appl. Phys. 28, 495 (1957) cited by Plumlee, could
not be extended to temperatures below 1400oK. It is planned that
these considerations wiII be expanded in future publications. D. Millisecond Decay Phenomena
When an electric field is applied to a poorly activated
oxide-coated cathode with the purpose of drawing
thermionic emission, it is observed that this emission
decays with a time constant of the order of a few milli
seconds, depending upon the temperature. This effect
was first observed by Blewett who considered the phe
nomena to be due to the depletion of barium atoms from
the coating surface resulting in an increase in the work
function of the cathode.22 By plotting the decay and re
covery constants observed in this phenomena as a func
tion of temperature, he obtained an activation energy
for the process of approximately 0.7 eV.
A more careful analysis of this process was carried
out by Sprou1l23 using essentially the same point of view
as Blewett. While his experimental results were pre
sented in much greater detail and were in qualitative
agreement with the results reported by Blewett, Sproull
did not attempt to obtain an activation energy from
his data.
Nergaard14 proposed a different theory for the above
emission decay phenomena. He considered that the
donors in the oxide coating are highly mobile and that
before the electric field is applied these donors are uni
formly distributed throughout the coating. If the cath
ode is poorly activated, the Fermi energy is relatively
low, such that the donors are partially ionized. When an
anode voltage is applied, the ionized donors, being posi
tively charged, drift in a direction away from the sur
face leaving a donor depletion layer at the surface of
the coating. This drift of the charged donors is counter
balanced by a diffusion of donors back towards the sur
face as a result of the newly created concentration gra
dient. For a fixed anode voltage, a steady-state situation
results in which the drift of the charged donors is just
counterbalanced by this diffusion. The net result is a
depletion of the donors from the surface proportional
to the anode voltage with a corresponding increase in
the effective work function at the surface. When the
electric field is removed, the cathode will regain its
original work function with a time constant controlled
by the diffusion process alone. For a well-activated
cathode, the percentage of the donors which are ionized
is quite small, accounting for the fact that the emission
decay is hardly detectable in well-activated cathodes.
This explanation of the millisecond decay phenomena
has enjoyed wide success as is indicated by the fact that
the phenomena is frequently referred to as the Nergaard
effect.
Employing Nergaard's theory, Frost24 obtained the
diffusion coefficients and the activation energy for what
he thought to be the diffusion of donors from emission
decay measurements. His observations were essentially
confined to the emission recovery time constants at
22 J. P. Blewett, Phys. Rev. 55, 713 (1939).
23 R. L. Sproull, Phys. Rev. 67, 166 (1945).
2' H. B. Frost, Ph.D. thesis, Massachusetts Institute of Tech
nology (1945).
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various temperatures since the transient solution for
the combined drift and diffusion process was difficult to
analyze. He. obtained a diffusion coefficient at lOOOoK
of 5.9XIQ-6 cm2/sec with an activation energy of 0.435
eV. It is to be noted that these diffusion constants are
quite close to the diffusion constants obtained in this
paper.
Additional confirmation of Nergaard's hypothesis
was obtained by Hensley3 using an experimental tube
in which two planar cathodes were facing each other
with a very small separation. The saturation current
density from either of the cathodes could be obtained
using short pulses of moderate amplitude voltage. It was
observed that the amplitude of the saturation current
density was decreased when a direct current was drawn
in the same direction as the pulse current and that the
amount of this reduction depended upon the magnitude
of the direct current. When this direct current was in
the opposite direction to the pulse current, the satura
tion current density was observed to be enhanced, as
would be expected from N ergaard's theory. These effects
were more pronounced when the cathodes were rela
tively inactive and when the cathode temperatures were
moderately high, the conditions for a large percentage
of ionized donors.
In view of the mobile acceptor model discussed above,
some minor modifications of Nergaard's theory are re
quired. Instead of the drift and diffusion of donors as
was originally suggested by Nergaard, the drift and dif
fusion of acceptors should be considered. When an elec
tric field is applied to draw thermionic emission from
the cathode the acceptors, being negatively charged,
are attracted to the surface. This increases the density
of acceptors and increases the compensation of the uni
formly distributed donors. This results in a decrease in
the work function of the surface just as before. Offsetting
this drift of the acceptors is a diffusion of the acceptors
brought about by the newly established concentration
gradient. In a well-activated cathode, according to the
present point of view, the density of acceptors is greatly
reduced and consequently, the millisecond decay phe
nomena will tend to disappear. Thus, all of the essential
features of Nergaard's theory are retained with the only
change being the substitution of mobile acceptors for
the previous mobile donors.
E. Activation of Oxide-Coated Cathodes
A fairly detailed description of the processes involved
in the activation of an ordinary oxide-coated cathode
is suggested by the mobile acceptor model presented
above. Cathodes are normally fabricated by depositing
a rather porous coating of (BaSr)CO a on a nickel base
containing small amounts of active reducing elements.
After the initial exhaust of the vacuum tube, this coat
ingis converted to the oxides by thermal reduction.
It is normal procedure to heat the cathode to rather
high temperatures for a short interval of time in this
early stage of the processing schedule. It is suggested that the density of the donors in the form of oxygen va
cancies is established either during the initial formation
of the oxide particles or at the time of the highest tem
perature by the production of Schottky vacancy pairs.
At this stage the cathode is relatively inactive with ap
proximately equal densities of both donors and accep
tors. During the activation schedule the active reducing
elements in the nickel react with the barium oxide coat
ing forming an interface compound and releasing free
barium. This free barium finds its way through the coat
ing either by gaseous Knudsen flow or by diffusion along
the surface of the crystallites. Since the acceptors are
highly mobile, they make frequent excursions to the
surface of the crystal where the presence of free barium
can result in their annihilation. This results in an in
crease in the activity of the oxide coating.
The activation process should be considered a dy
namic one in that gases are usually present in tubes
which will interact with the oxide coating and pro
duce new barium vacancies. In order to maintain a
cathode in a high state of activation, it is necessary to
produce free barium at the oxide-metal interface at a
sufficient rate to offset this poisoning. It is well-known
that in exceptionally clean tubes only a passive nickel
is required in order to maintain a cathode in a high state
of activation although the initial activation may be a
rather slow process. On the other hand, a tube contain
ing appreciable quantities of oxidizing gases normally
requires an active base metal containing a relatively
large percentage of reducing elements in order to pro
duce free barium at a sufficient rate to offset the poison
ing processes. This, of course, results in cathodes hav
ing a relatively short life.
Some suggestions regarding the processes involved in
the electrolytic activation of an oxide-coated cathode
when very pure base metals are used may be implied
from the drift experiment. It is suggested that a small
fraction of the electric current is carried by the nega
tively charged acceptors, the cation vacancies. As the
concentration of these acceptors increases on the anode
side of the individual oxide particles, the Fermi energy
is lowered in this region. This results in an increased
fraction of the acceptors being neutrally charged. It is
suggested that when such neutral acceptors reach the
surface of an oxide particle there will be a finite proba
bility that a neutral oxygen atom is released. This would
result in a net reduction in the total density of accep
tors present and would result in an increase in the ac
tivation of the cathode.
In the drift experiment reported in this paper the
early and late reversals of the rectification characteris
tics of the cathode and anode sections, respectively,
are suggestive evidence for cation vacancies being trans
ported toward the anode. The fact that oxygen is re
leased during electrolytic activation of an oxide-coated
cathode has been reported by Isensee.25 This is further
26 H. Isensee, Z. Physik. Chem. B35, 309 (1937); also see
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supported by the spectroscopic evidence for a greater
amount of nickel oxide in the cathode coating near the
anode electrode suggesting the presence of atomic oxy
gen. These effects are also observed to essentially cease
as the cathode becomes fully activated.
The discussion in this section represents a qualitative
description of the activation processes in an oxide
coated cathode. While we have not attempted to carry
out a quantitative analysis of these processes, it is felt
that such a study should be very informative. However,
one precaution should be taken. Although it is fre
quently attempted to apply thermodynamics directly
to the processes involving cathode activation, it is not
clear that all of these processes represent thermody-
G. Herrmann and S. Wagener, The Oxide-Coated Cathode (Chap
man and Hall Ltd., London, 1951), Vol. II, p. 159-253. namic equilibrium. For example, although the accep
tors in the oxide particles have a high mobility and con
sequently are probably capable of reaching thermody
namic equilibrium with the system in rather short in
tervals of time, it is not at all clear that this is also
true for the donors. At the normal operating tempera
tures for oxide-coated cathodes, the diffusion times for
oxygen vacancies are extremely long so that their ex
cursions to the surface, where they could be annihilated
are very infrequent. The possibility also exists that the
energy of an oxygen vacancy in the surface layer of
atoms is higher than in the bulk of the particle so that
a potential barrier would exist, repelling the oxygen
vacancies away from the surface. This would prevent
their recombination with anions on the surface originat
ing from the ambient gas. Further study of these con
siderations is currently in progress.
JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 3 MARCH 1963
Interaction of a Bunched Electron Beam with a Gyrotropic Plasma*
BASIL W. HAKKI
The National Science Council, Damascus, Syrian Arab Republic
(Received 16 July 1962)
The interaction of a prebunched beam with an anisotropic medium is considered for the generation of
coherent electromagnetic energy in the low and submilJimeter region. Generalized expressions are obtained
for the fields created by a prebunched beam of general shape in an anistropic medium. The permeability is
assumed to be a scalar constant whereas the permittivity is tensorial. Two special cases are considered:
one is that of a helical prebunched beam and the other is that of a rectilinear beam, both interacting with
gyrotropic gaseous plasmas. Ionization, scattering, and rf field effects on the beam are neglected.
In addition to the solutions of the inhomogeneous vector wave equation, a method is suggested whereby
the free mode solutions could be directly deduced.
I, INTRODUCTION A PLASMA in its general aspects would encompass
the phenomena taking place in a diversity of
media. The most common plasma is the gaseous type
consisting of ions and electrons that have somehow been
separated from each other but altogether constitute
an electrically neutral gas. In addition to the gaseous
plasma, it is possible to regard the charge carriers in a
semiconductor to form a plasma. Such a "semicon
ductor plasma," which is composed of the usual
cond1.\ction electrons and holes in the semiconductor,
has many interesting features. In the first place, the
charge carrier concentration that can be obtained is
several orders of magnitude greater than that in a
gaseous plasma; it follows that the plasma frequency
in a semiconductor can be much higher than that in a
gaseous plasma. In the second place, the effecti~e
* Most of the work reported here was done when the author
was at the University of Illinois, Urbana, Illinois, and supported
by the U. S. Atomic Energy Commission under contract No.
AT (11-1)-392. masses of the charge carriers in a semiconductor can
be one tenth or smaller than the free electron mass.
Hence, the effective cyclotron frequency in a semi
conductor medium can be ten times or greater than the
cyclotron frequency of a free electron, for the same
magnetic field. Therefore, the higher plasma and
cyclotron frequencies of a semiconductor plasma in
crease the upper frequency limit at which it can be
utilized by several orders of magnitude over that of a
gaseous plasma. This makes a semiconductor plasma
an extremely'attractive medium for investigation in the
low and submillimeter region. This is not to imply that
gaseous plasmas cannot be utilized in this region. The
intense magnetic fields now available or being con
templated, in the range between lOS and 5X 10· G,
have renewed interest in gaseous plasmas in the ultra
microwave frequency range. However, these intense
magnetic fields can be obtained only at a relatively
great cost, and are in addition too physically
cumbersome.
However, the cloud of charge carriers in a semi-
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1.1735314.pdf | Thermal Conductivity and Thermoelectric Power
T. Geballe
Citation: Journal of Applied Physics 30, 1317 (1959); doi: 10.1063/1.1735314
View online: http://dx.doi.org/10.1063/1.1735314
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
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Thermal Conductivity and Thermoelectric Power of GermaniumSilicon Alloys
J. Appl. Phys. 29, 1517 (1958); 10.1063/1.1722984
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IP: 129.49.170.188 On: Sat, 20 Dec 2014 22:51:22JOURNAL OF APPLIED PHYSICS VOLUME 3D, NUMBER 8 AUGUST, 1959
Discussion
(Nole.-The Program Committee of the "Radiation Effects in Semicon
ductors" meeting and the editorial staff of The Journal of Applied Physics
have attempted to complete the publication of the "Proceedings" in the
shortest possible interval of time. It was consequently impossible to dis
tribute the pages of recorded discussion to the respective speakers for
editing, manuscript revision. or additional comments. ]. H. Crawford. Jr..
and J. W. Cleland of the Oak Ridge National Laboratory must therefore
take full responsibility for the following, very abbreviated, summation of a
portion of the lively discussion that actually contributed a great deal to the
meeting. They. in turn, wish to acknowledge the assistance of their col
leagues in the Solid State Division, who helped greatly in editing the
discussion.)
Infrared Absorption and Photoconductivity
in Irradiated Silicon
H. Y. FAN AND A. K. RAMDAS
R. J. Collins: We have investigated the optical absorption
spectra of silicon Irradiated with 2-Mev electrons and have ob
tained some rather different results. The 1.8-micron absorption
band is not observed even though the Fermi level is 0.3 ev below
the conduction band and the 3.3-micron band has disappeared.
The 1.8-micron band did not appear until the Fermi level had
been depressed below 0.37 ev by the electron bombardment.
H. Brooks: During the course of annealing studies on the 1.8-
micron band was it possible to obtain a frequency factor, and if
so, did this correspond to a single step or a multistep process?
A. K. Ramdas: The frequency factor we determined was orders
of magnitude smaller than the value of 1012 to 1013 normally ex
pected for simple annealing.
H. Brooks: This would indicate that the mobile defect must
migrate several lattice spacings before annihilation.
R. J. Collins: Was any fine structure such as exhibited by the
3.3-micron band observed in the case of the 5.5-micron band?
A. K. Ramdas: No. This band was examined at liquid helium
temperature with the highest resolution available and no fine
structure was observed; nor was any observed for the 3.9-and
1.8-micron bands.
R. J. Collins: With regard to the low temperature structure of
the 3.3-micron band, did you always find the 2.9-micron line when
others that did not fit the expected spacing were observed, and
were the several lines always present in the same relative
concentration?
A. K. Ramdas: Of this we are not sure because the structural
lines of the 3.3-micron band appear on a very high background
and many of them are comparatively weak. We have not estab
lished whether the weak lines always have the same relative
strength with respect to the strong ones. It could be that two
different defects with two different ground states contribute to
absorption in this range. However, they would be expected to
have nearly the same excited states because in the large orbits
(excited states) the electrons would presumably have the same
energy levels. This point would have to be established by means
of an annealing experiment.
Mechanism and Defect Responsible for
Edge Emission in CdS
R. J. COLLINS
A. K, Ramdas: In the reflectivity curve for the infrared range
(Fig. 3 of Collins' paper) did you use polarized light?
R. J. Collins: No.
A. K. Ramdas: Inspection of this curve indicates a slight differ
ence between the experimental points and the calculated wave
length dependence of reflectivity. The slight dip in the region of
the maximum may indicate two maxima in the experimental curve. R. J. Collins: The crystal is hexagonal and therefore one would
expect two closely spaced maxima, but these should not differ in
position very much. In order to separate the two curves accurately
careful examination with polarized light is necessary. The error
introduced by neglecting this effect is small, but it might shift the
value obtained for the longitudinal phonon frequency from 305
cm-I to 295 cm-I.
J. H. Crawford, Jr.: In addition to the direct displacement of
sulfur atoms by the 300-kev electrons, there is another mechanism
for defect introduction by ionization which should be kept in mind,
namely the Varley mechanism. This operates by multiple ioniza
tion of an anion which is then forced into an interstitial site by the
positive crystal potential at the anion site. Even though CdS is
appreciably covalent such an ionization process may well produce
a defect. However, either the displacement process or the Varley
mechanism has the same end result in that both produce sulfur
vacancies and interstitials.
Diffusion-Controlled Reactions in Solids
H. REISS
G. Leibfried: I wish to make one remark about the Coulomb
forces in a diffusion equation. Your parameter A'. is essentially an
annihilation radius in the case of diffusion with a force. I have
recently tried to obtain this quantity for a vacancy and an inter
stitial pair. If 5 to 6 ev is taken as the pair energy and a 1/R6 po
tential is assumed, your formula gives the annihilation radius.
This gives 3 to 4 lattice distances, which is in accord with the
values which Waite has used for his analysis. It does not depend
very much on temperature, because the radius is proportional to
temperature to the one-sixth power. Furthermore, the radius is
very well defined if the potential decreases rapidly with distance,
but is not very well defined for the case of a l/r potential.
With respect to Waite's analysis, I have examined this in detail
and have the impression that it can only be applied when there is
initially a correlation only between interstitials and vacancies, but
otherwise the pairs themselves are distributed at random. Even
in this case a correlation is built up between the interstitials and
between the vacancies in the course of time. It may be possible to
neglect this correlation for low defect densities. On the other hand,
for high densities such as encountered in neutron irradiation, a
correlation, e.g. between interstitials, already exists in the begin
ning stage and seems to make Waite's assumptions invalid.
A. G. Tweet: I would like to comment on the use of the power of
tiT in the exponent of the equation relating to precipitation for
determining the shape of the precipitate particle. As has been
pointed out, (t/T)! dependence has been taken to indicate a
spherical or spheroidal precipitate, whereas platelet and linear
shaped particles are expected to exhibit a power different from !.
Recent work by Ham has shown that as the precipitation process
proceeds toward completion, the precipitation equation becomes
much more complex, and these criteria are no longer valid. This
complication should be remembered when attempting to deter
mine the shape of precipitate particles, and the criteria normally
used should be applied only during the early part of the process.
Thermal Conductivity and Thermoelectric Power
T. GEBALLE
J. H. Crawford, Jr.: You have mentioned the influence of the
electron-phonon interactions on the thermal conductivity at low
temperature in n-type Ge. Have there been any attempts to de
termine the extent of the effect of hole-phonon interactions?
T. Geballe: I don't think any information is available on hole
phonon interactions as yet. It is a worthwhile experiment and I
am planning to do it.
J. H. Crawford, Jr.: There is one experiment that may be of
some interest in this connection. Dr. A. F. Cohen irradiated some
fairly impure n-type Ge, measured the marked decrease in thermal
1317
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IP: 129.49.170.188 On: Sat, 20 Dec 2014 22:51:221318 DISCUSSION
conductivity, annealed and observed almost complete recovery
of thermal conductivity in the region of the maximum. However,
after standing for a couple of months, a remeasurement of thermal
conductivity showed that the thermal conductivity had increased
above its original value by an appreciable amount. It should be
noted that the irradiation was sufficient to produce p-type ma
terial after all of the activated Ge70 had decayed to gallium. One
possible explanation of the enhancement is that the hole-phonon
interactions have a much smaller effect on the conductivity.
Transport Properties
R. K. WILLARDSON
G. K. Wertheim: The energy level structure of electron irradi
ated Si is different for vacuum floating zone and quartz crucible
grown crystals. The results agree with those obtained by G. D.
Watkins using spin resonance techniques. Do you find a difference
in energy level structure for neutron irradiated Si of the two types?
R. K. Willardson: The energy level structure of neutron irradi
ated Si is the same for n-type material of either type as far as we
can ascertain.
Recombination
G. K. WERTHEIM
H. Y. Fan: It seems to me that the Hall effect and the recom
bination type of measurements are very sensitive, in some respects
much too sensitive. You are likely to see energy levels or defects
which are introduced to a very small extent but which are very
effective in pinning down the Fermi level when the resistivity is
high, or they are very effective for recombination when the capture
cross section is large. For instance, in n-type Si irradiated with
neutrons we see only two definite absorption bands, whereas all
previous measurements of Hall effects indicated you might have
a spread of levels. I think in such cases, if you want to spot the
major levels, some measurements which are a little less sensitive,
like optical absorption, perhaps should be made.
G. K. Wertheim: I think I differ with you fundamentally because
my feeling is that any level that you can see is of interest because
it contains some information about the nature of the bombard
ment damage. The mere fact that it is introduced in a small
density does not make it less interesting, and perhaps this is a
good argument for the use of lifetime measurements because, if
the cross section is large, it provides a rather sensitive tool to get
us something that we cannot see with optical means.
Radiation Effects on Recombination
in Germanium
O. L. CURTIS, JR.
H. Y. Fan: I would like to point out another factor in connec
tion with the recombination type of measurements, that is, the
surface effect. Photoconductivity does depend upon the carrier
lifetime, and some previous work at Purdue by StOckman showed
a distinct photoconductivity peak corresponding to some energy
level toward the middle of the energy gap such as shown here at
0.32 ev in the case of 14-Mev neutron radiation. However, some
subsequent measurements by Spear at Purdue showed that this
effect was purely a surface effect. We are all aware that the
trapping and the carrier recombination surface effects can be very
important. So here is another thing that we must bear in mind.
O. L. Curtis, Jr.: I believe that we do not have surface effects
in these samples. These samples are about 7 or 8 millimeters in the
smallest dimension; and, whereas you might well expect surface
effects in small samples, even with fairly short lifetimes, still with
post-irradiation lifetimes of the order of 20 microseconds or so it
seems hardly possible that the surface can be playing an important
role in our measurements. Now there is something to be borne in
mind. That is, if you use, say, a white light to excite the carriers, you might excite the carriers only at the surface, essentially; and
you might find predominantly surface effects, whereas you think
because of the size of your specimen you should be eliminating
them. For these measurements we used a germanium filter of the
order of a half-millimeter in front of our specimens so that the
carriers that are excited are excited fairly uniformly inside the
specimen.
J. J. Loferski: I would like to speak in defense of devices. There
seems to be the feeling abroad that if one attaches to a piece of
germanium anything other than a couple of ohmic contacts the
measurements that one makes on that device are to be regarded
at least with suspicion and perhaps to be ignored entirely. Now
this is not true. Careful measurement made on properly made
devices can, for instance, follow lifetime changes with an accuracy
of 1% or better; and that is pretty difficult to do if you are
measuring the lifetime directly. Usually plus or minus 10% is
pretty good for direct lifetime measurements.
Also, the great sensitivity that one gets on such pieces with
other than only ohmic contact makes it possible to follow recom
bination-center concentrations of the order of 1010 or even less per
cm3 in germanium.
P. Rappaport: We have tried to compare the results that one
gets when measuring lifetime on a slab of germanium with just
two ohmic contacts to those one gets from lifetime measurements
on junction diodes, which is perhaps the simplest type of device.
As Curtis suggested, we had difficulty with that experiment. We
have in the past, however, had satisfaction from such devices.
The difficulty is that, when using junction diodes to measure life
time changes when one is concerned with these changes as a
function of resistivity, there is another parameter that changes in
the junction. It is the collection efficiency for the excess carriers
that are induced in the semiconductor, and that is the thing that
we have not been able to pin down well enough to be able to com
pare the results with those obtained on bulk specimens.
Electron-Bombardment Induced Recombination
Centers in Germanium
J. J. LOFERSKI AND P. RAPPAPORT
O. L. Curtis, Jr.: Because of the possibility of multiple levels, it
seems apparent that in order to know anything about the proper
ties of recombination centers one must make lifetime measure
ments both as a function of temperature and carrier concentra
tion. The temperature dependence of p-type material shows that
such an analysis as you have made in the p-type region is mean
ingless. Our measurements on C060 gamma-irradiated, p-type
material, mentioned in the previous paper, indicated a very similar
dependence on carrier concentration to that you show for 1-Mev
electron irradiation; but our observations of the dependence of
lifetime on temperature reveal that recombination did not take
place at the 0.26-ev level, rather that the occupation of a level in
this region probably determined the number of upper levels
available for recombination. One cannot safely determine energy
level position solely on the basis of measurements as a function of
carrier concentration.
Magnetic Susceptibility and Electron
Spin Resonance
E. SoNDER
H. Brooks: If your susceptibility data are interpreted on the
basis of clustering, then perhaps it might mean that the clusters
are considerably larger than we have been accustomed to thinking
in the past, and that the flux necessary to produce overlap is con
siderably less than 1018 to 10'9.
G. Leibfried: The closed-shell repulsion in covalent materials is
much smaller than in metals; this would cause the damage due to
one fast neutron to be distributed over an area a factor of 5 to 10
larger.
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IP: 129.49.170.188 On: Sat, 20 Dec 2014 22:51:22 |
1.1735313.pdf | DiffusionControlled Reactions in Solids
H. Reiss
Citation: Journal of Applied Physics 30, 1317 (1959); doi: 10.1063/1.1735313
View online: http://dx.doi.org/10.1063/1.1735313
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
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J. Appl. Phys. 30, 1141 (1959); 10.1063/1.1735284
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IP: 142.157.212.201 On: Mon, 24 Nov 2014 22:19:14JOURNAL OF APPLIED PHYSICS VOLUME 3D, NUMBER 8 AUGUST, 1959
Discussion
(Nole.-The Program Committee of the "Radiation Effects in Semicon
ductors" meeting and the editorial staff of The Journal of Applied Physics
have attempted to complete the publication of the "Proceedings" in the
shortest possible interval of time. It was consequently impossible to dis
tribute the pages of recorded discussion to the respective speakers for
editing, manuscript revision. or additional comments. ]. H. Crawford. Jr..
and J. W. Cleland of the Oak Ridge National Laboratory must therefore
take full responsibility for the following, very abbreviated, summation of a
portion of the lively discussion that actually contributed a great deal to the
meeting. They. in turn, wish to acknowledge the assistance of their col
leagues in the Solid State Division, who helped greatly in editing the
discussion.)
Infrared Absorption and Photoconductivity
in Irradiated Silicon
H. Y. FAN AND A. K. RAMDAS
R. J. Collins: We have investigated the optical absorption
spectra of silicon Irradiated with 2-Mev electrons and have ob
tained some rather different results. The 1.8-micron absorption
band is not observed even though the Fermi level is 0.3 ev below
the conduction band and the 3.3-micron band has disappeared.
The 1.8-micron band did not appear until the Fermi level had
been depressed below 0.37 ev by the electron bombardment.
H. Brooks: During the course of annealing studies on the 1.8-
micron band was it possible to obtain a frequency factor, and if
so, did this correspond to a single step or a multistep process?
A. K. Ramdas: The frequency factor we determined was orders
of magnitude smaller than the value of 1012 to 1013 normally ex
pected for simple annealing.
H. Brooks: This would indicate that the mobile defect must
migrate several lattice spacings before annihilation.
R. J. Collins: Was any fine structure such as exhibited by the
3.3-micron band observed in the case of the 5.5-micron band?
A. K. Ramdas: No. This band was examined at liquid helium
temperature with the highest resolution available and no fine
structure was observed; nor was any observed for the 3.9-and
1.8-micron bands.
R. J. Collins: With regard to the low temperature structure of
the 3.3-micron band, did you always find the 2.9-micron line when
others that did not fit the expected spacing were observed, and
were the several lines always present in the same relative
concentration?
A. K. Ramdas: Of this we are not sure because the structural
lines of the 3.3-micron band appear on a very high background
and many of them are comparatively weak. We have not estab
lished whether the weak lines always have the same relative
strength with respect to the strong ones. It could be that two
different defects with two different ground states contribute to
absorption in this range. However, they would be expected to
have nearly the same excited states because in the large orbits
(excited states) the electrons would presumably have the same
energy levels. This point would have to be established by means
of an annealing experiment.
Mechanism and Defect Responsible for
Edge Emission in CdS
R. J. COLLINS
A. K, Ramdas: In the reflectivity curve for the infrared range
(Fig. 3 of Collins' paper) did you use polarized light?
R. J. Collins: No.
A. K. Ramdas: Inspection of this curve indicates a slight differ
ence between the experimental points and the calculated wave
length dependence of reflectivity. The slight dip in the region of
the maximum may indicate two maxima in the experimental curve. R. J. Collins: The crystal is hexagonal and therefore one would
expect two closely spaced maxima, but these should not differ in
position very much. In order to separate the two curves accurately
careful examination with polarized light is necessary. The error
introduced by neglecting this effect is small, but it might shift the
value obtained for the longitudinal phonon frequency from 305
cm-I to 295 cm-I.
J. H. Crawford, Jr.: In addition to the direct displacement of
sulfur atoms by the 300-kev electrons, there is another mechanism
for defect introduction by ionization which should be kept in mind,
namely the Varley mechanism. This operates by multiple ioniza
tion of an anion which is then forced into an interstitial site by the
positive crystal potential at the anion site. Even though CdS is
appreciably covalent such an ionization process may well produce
a defect. However, either the displacement process or the Varley
mechanism has the same end result in that both produce sulfur
vacancies and interstitials.
Diffusion-Controlled Reactions in Solids
H. REISS
G. Leibfried: I wish to make one remark about the Coulomb
forces in a diffusion equation. Your parameter A'. is essentially an
annihilation radius in the case of diffusion with a force. I have
recently tried to obtain this quantity for a vacancy and an inter
stitial pair. If 5 to 6 ev is taken as the pair energy and a 1/R6 po
tential is assumed, your formula gives the annihilation radius.
This gives 3 to 4 lattice distances, which is in accord with the
values which Waite has used for his analysis. It does not depend
very much on temperature, because the radius is proportional to
temperature to the one-sixth power. Furthermore, the radius is
very well defined if the potential decreases rapidly with distance,
but is not very well defined for the case of a l/r potential.
With respect to Waite's analysis, I have examined this in detail
and have the impression that it can only be applied when there is
initially a correlation only between interstitials and vacancies, but
otherwise the pairs themselves are distributed at random. Even
in this case a correlation is built up between the interstitials and
between the vacancies in the course of time. It may be possible to
neglect this correlation for low defect densities. On the other hand,
for high densities such as encountered in neutron irradiation, a
correlation, e.g. between interstitials, already exists in the begin
ning stage and seems to make Waite's assumptions invalid.
A. G. Tweet: I would like to comment on the use of the power of
tiT in the exponent of the equation relating to precipitation for
determining the shape of the precipitate particle. As has been
pointed out, (t/T)! dependence has been taken to indicate a
spherical or spheroidal precipitate, whereas platelet and linear
shaped particles are expected to exhibit a power different from !.
Recent work by Ham has shown that as the precipitation process
proceeds toward completion, the precipitation equation becomes
much more complex, and these criteria are no longer valid. This
complication should be remembered when attempting to deter
mine the shape of precipitate particles, and the criteria normally
used should be applied only during the early part of the process.
Thermal Conductivity and Thermoelectric Power
T. GEBALLE
J. H. Crawford, Jr.: You have mentioned the influence of the
electron-phonon interactions on the thermal conductivity at low
temperature in n-type Ge. Have there been any attempts to de
termine the extent of the effect of hole-phonon interactions?
T. Geballe: I don't think any information is available on hole
phonon interactions as yet. It is a worthwhile experiment and I
am planning to do it.
J. H. Crawford, Jr.: There is one experiment that may be of
some interest in this connection. Dr. A. F. Cohen irradiated some
fairly impure n-type Ge, measured the marked decrease in thermal
1317
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IP: 142.157.212.201 On: Mon, 24 Nov 2014 22:19:141318 DISCUSSION
conductivity, annealed and observed almost complete recovery
of thermal conductivity in the region of the maximum. However,
after standing for a couple of months, a remeasurement of thermal
conductivity showed that the thermal conductivity had increased
above its original value by an appreciable amount. It should be
noted that the irradiation was sufficient to produce p-type ma
terial after all of the activated Ge70 had decayed to gallium. One
possible explanation of the enhancement is that the hole-phonon
interactions have a much smaller effect on the conductivity.
Transport Properties
R. K. WILLARDSON
G. K. Wertheim: The energy level structure of electron irradi
ated Si is different for vacuum floating zone and quartz crucible
grown crystals. The results agree with those obtained by G. D.
Watkins using spin resonance techniques. Do you find a difference
in energy level structure for neutron irradiated Si of the two types?
R. K. Willardson: The energy level structure of neutron irradi
ated Si is the same for n-type material of either type as far as we
can ascertain.
Recombination
G. K. WERTHEIM
H. Y. Fan: It seems to me that the Hall effect and the recom
bination type of measurements are very sensitive, in some respects
much too sensitive. You are likely to see energy levels or defects
which are introduced to a very small extent but which are very
effective in pinning down the Fermi level when the resistivity is
high, or they are very effective for recombination when the capture
cross section is large. For instance, in n-type Si irradiated with
neutrons we see only two definite absorption bands, whereas all
previous measurements of Hall effects indicated you might have
a spread of levels. I think in such cases, if you want to spot the
major levels, some measurements which are a little less sensitive,
like optical absorption, perhaps should be made.
G. K. Wertheim: I think I differ with you fundamentally because
my feeling is that any level that you can see is of interest because
it contains some information about the nature of the bombard
ment damage. The mere fact that it is introduced in a small
density does not make it less interesting, and perhaps this is a
good argument for the use of lifetime measurements because, if
the cross section is large, it provides a rather sensitive tool to get
us something that we cannot see with optical means.
Radiation Effects on Recombination
in Germanium
O. L. CURTIS, JR.
H. Y. Fan: I would like to point out another factor in connec
tion with the recombination type of measurements, that is, the
surface effect. Photoconductivity does depend upon the carrier
lifetime, and some previous work at Purdue by StOckman showed
a distinct photoconductivity peak corresponding to some energy
level toward the middle of the energy gap such as shown here at
0.32 ev in the case of 14-Mev neutron radiation. However, some
subsequent measurements by Spear at Purdue showed that this
effect was purely a surface effect. We are all aware that the
trapping and the carrier recombination surface effects can be very
important. So here is another thing that we must bear in mind.
O. L. Curtis, Jr.: I believe that we do not have surface effects
in these samples. These samples are about 7 or 8 millimeters in the
smallest dimension; and, whereas you might well expect surface
effects in small samples, even with fairly short lifetimes, still with
post-irradiation lifetimes of the order of 20 microseconds or so it
seems hardly possible that the surface can be playing an important
role in our measurements. Now there is something to be borne in
mind. That is, if you use, say, a white light to excite the carriers, you might excite the carriers only at the surface, essentially; and
you might find predominantly surface effects, whereas you think
because of the size of your specimen you should be eliminating
them. For these measurements we used a germanium filter of the
order of a half-millimeter in front of our specimens so that the
carriers that are excited are excited fairly uniformly inside the
specimen.
J. J. Loferski: I would like to speak in defense of devices. There
seems to be the feeling abroad that if one attaches to a piece of
germanium anything other than a couple of ohmic contacts the
measurements that one makes on that device are to be regarded
at least with suspicion and perhaps to be ignored entirely. Now
this is not true. Careful measurement made on properly made
devices can, for instance, follow lifetime changes with an accuracy
of 1% or better; and that is pretty difficult to do if you are
measuring the lifetime directly. Usually plus or minus 10% is
pretty good for direct lifetime measurements.
Also, the great sensitivity that one gets on such pieces with
other than only ohmic contact makes it possible to follow recom
bination-center concentrations of the order of 1010 or even less per
cm3 in germanium.
P. Rappaport: We have tried to compare the results that one
gets when measuring lifetime on a slab of germanium with just
two ohmic contacts to those one gets from lifetime measurements
on junction diodes, which is perhaps the simplest type of device.
As Curtis suggested, we had difficulty with that experiment. We
have in the past, however, had satisfaction from such devices.
The difficulty is that, when using junction diodes to measure life
time changes when one is concerned with these changes as a
function of resistivity, there is another parameter that changes in
the junction. It is the collection efficiency for the excess carriers
that are induced in the semiconductor, and that is the thing that
we have not been able to pin down well enough to be able to com
pare the results with those obtained on bulk specimens.
Electron-Bombardment Induced Recombination
Centers in Germanium
J. J. LOFERSKI AND P. RAPPAPORT
O. L. Curtis, Jr.: Because of the possibility of multiple levels, it
seems apparent that in order to know anything about the proper
ties of recombination centers one must make lifetime measure
ments both as a function of temperature and carrier concentra
tion. The temperature dependence of p-type material shows that
such an analysis as you have made in the p-type region is mean
ingless. Our measurements on C060 gamma-irradiated, p-type
material, mentioned in the previous paper, indicated a very similar
dependence on carrier concentration to that you show for 1-Mev
electron irradiation; but our observations of the dependence of
lifetime on temperature reveal that recombination did not take
place at the 0.26-ev level, rather that the occupation of a level in
this region probably determined the number of upper levels
available for recombination. One cannot safely determine energy
level position solely on the basis of measurements as a function of
carrier concentration.
Magnetic Susceptibility and Electron
Spin Resonance
E. SoNDER
H. Brooks: If your susceptibility data are interpreted on the
basis of clustering, then perhaps it might mean that the clusters
are considerably larger than we have been accustomed to thinking
in the past, and that the flux necessary to produce overlap is con
siderably less than 1018 to 10'9.
G. Leibfried: The closed-shell repulsion in covalent materials is
much smaller than in metals; this would cause the damage due to
one fast neutron to be distributed over an area a factor of 5 to 10
larger.
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1.1702801.pdf | Impurity Conduction and Negative Resistance in Thin Oxide Films
T. W. Hickmott
Citation: Journal of Applied Physics 35, 2118 (1964); doi: 10.1063/1.1702801
View online: http://dx.doi.org/10.1063/1.1702801
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/35/7?ver=pdfcov
Published by the AIP Publishing
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] IP: 155.97.178.73 On: Fri, 28 Nov 2014 21:30:112118 11,\ C I '\ 0, T.\ K.\ 11,\ S 1-1 I, .\ '\ I) \\'.\ J) '\
ment with a vacuum ditTusioll pump at work. From the
experiments concerning the equilibrium of the photo
emissive yield and the resistance vs Cs pressure in the
closed sYstem, we can confirm that the photoemissive
yield a~d the resistance change reversibly with the
change in amount of Cs.
In the Cs-Sb photocathode, Cs ions are mobile. We
can change this material from p type to n Lype by con
trolling the Cs amount. vVe can form a p-n junction
in this photocathode film by flowing Cs ions through it.
The formation of the p-n junction has been confirmed
by the measurements of photovollaic effect and current
vs voltage.
From the reversible change in the photoemissive
yield with Cs amount, the mobility of Cs ions, and also
JOl'RN"AL OF APPLIED PIIVSICS the experimental facts in Sec. 3.3, we can explain the
decrease and the increase of the photoemissive yield
during operation by the deviation of the Cs concentra
tion from the optimum composition. Furthermore, we
have shown the conditions necessary for the stable
operation of Cs-Sb photocathode, and suggested a pre
ferred structure for these conditions.
ACKNOWLEDGMENTS
The authors would like to express their hearty thanks
to former Professor S. Hamada of Tohoku University
for his encouragement throughout the work. Thanks
are also due to Y. Watanabe for making all of the
glass bulbs,
VOLUME 35, N"UMBER 7 JULY 1964
Impurity Conduction and Negative Resistance in Thin Oxide Films
T. W. HrCKMOTT
General mectric Research Laboratory, Schenectady, New York
(Received 13 December 1963; in final form 12 March 1964)
The conductivity of AI-AJ,03-metal diodes that show low-frequency negative resistance in their current
voltage characteristics depends on impurities in the oxide and on the metal used as counterelectrode. For
heavily doped AhO", development of diode conductivity by application of voltages occurs at ~4 V, inde
pendent of oxide thickness. For oxide films that are not deliberately doped, the field in the insulator is more
important than voltage in developing conductivity. AI-AJ,O,,-metal diodes have been constructed with
Ag, Au, Cu, Co, Sn, In, Bi, Pb, AI, and Mg counterelectrodes. The current-voltage characteristics which
develop depend on the metal and on polarity of the diode voltage during development of conductivity.
With Ag as counterelectrode, most diodes were initially shorted; with Mg as counterelectrode, no diode
conductivity could be developed. Other metals fall in between and give peak currents in the current-voltage
characteristics in the sequence Au, Cu, Co, Pb, Sn, Bi, In, AI. There is no correlation between AI-AI 20a-
metal diode conductivity and metal radius or work function.
A CRITICAL step in the establishment of con
ductivity and negative resistance in insulating
films is the "forming" process, the development of
conductivity by the application of potentials to the
insulator.1-8 Procedures for making metal-insulator
metal sandwiches and the measuring circuit used have
been described previously.I.2 The oxide films which have
been studied, whether produced by anodization or by
evaporation, were initially characterized by high re
sistance. Leakage currents were typically between 10-9
and 10-12 A for 1 V applied to the diode. In some films,
the logarithm of current was proportional to voltage,
1 T. W. Hickmott, J. App!. Phys. 33, 2669 (1962).
21'. W. Hickmott, J. App!. Phys. 34,1569 (1963).
3 H. Kanter and W. A. Feibelman, J. App!. Phys. 33, 3580
(1962).
4 G. S. Kreynina, L. N. Selivanov, and T. I. Shumskaia, Radio
Eng. Elec. Phys. 5, 8, 219 (1960).
.; G. S. Kreynina, Radio Eng. Elec. Phys. 7, 166 (1962).
6 G. S. Kreynina, Radio Eng. Elec. Phys. 7, 1949 (1962).
7 S. R. Pollack, W. O. Freitag, and C. E. Morris, Electrochem.
Tech. 1, 96 (1963).
8 S. R. Pollack, J. AppJ. Phys. 34, 877 (1963). in some films to (voltage)!, Such current-voltage re
lationships may be expected for ionic motion, for field
assisted electron emission from traps in the insulator,9--11
for Schottky emission from the metal electrodes,12 or
for space-charge-limited currents in an insulator with
traps.13 All these processes probably contribute to the
leakage currents in oxide films thicker than 100 A.14 An
unequivocal determination of conduction mechanisms
in oxide films of very high resistivity is difficult because
of the small magnitude of the currents involved, the
difficulty in reproducing current-voltage characteristics
from sample to sample, and polarization effects which
result in a steady decrease in diode current with time at
a constant voltage.15
9 J. Frenkel, Phys. Rev. 54, 647 (1938).
10 J. Frenkel, Tech. Phys. USSR. 5, 685 (1938).
11 D. A. Vermilyea, Acta Met. 2, 346 (1954).
12 P. R. Emtage and W. Tantraporn, Phys. Rev. Letters 8 267
(1962). '
13 A. Rose, Phys. Rev. 97, 1538 (1954).
14 c. A. Mead, Phys. Rev. 128, 2088 (1962).
15 A. F. Ioffe, The Physics oj Crystals (McGraw-Hill Book
Company, Inc., New York, 1928).
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Forming of conductivity in insulating films depends
on the field in the oxide, on the purity of the oxide, on
the electrode metals, and on the environment. Initial
development of conductivity for all films has been done
in vacuum of 1 Torr or better.! While vacuum is not
essential to the development of high conductivity and
good negative resistance characteristics,1 the final
characteristics appear to have their optimum values
and develop most easily if such an environment is used.
The amorphous nature of the oxide films makes char
acterization of the structure and impurities difficult.
Anodization in molten KHSOcNH4HS04 eutectic
which has been used to produce many of the Ab03
films leaves about 5% sulfur in the oxide.! However,
such deliberate doping is not essential for development
of conductivity. It may be an unavoidable adjunct to
the formation of many thin insulating films. When
SiO is deposited by evaporation the film contains Si02
and Si as well as SiO.!6.!7 If diodes such as Al-SiO-Au,
Au-SiO-Au, Pb-SiO-Pb, or Sn-SiO-Sn are formed, the
structure of the oxide and the nature of the impurities
are different from the anodized films but negative
resistance can be developed in the same manner. Volt
ages required to develop conductivity in AI-Ab03-Au
sandwiches with oxide film thIcknesses between 200 and
300 A, but prepared in different ways which deliber
ately varied the impurity content, have ranged from
2.5 to 14 V. In general, the lower the voltage at which
FIG. 1. Development of conductivity in an AI-Ah03-Au diode
with 1-mm' area. Oxide thickness, 300 A. Au= +, AI= -.
16 D. B. York, ]. Electrochem. Soc. 110, 271 (1963).
17 G. W. Brady, J. Phys. Chern. 63, 1119 (1959). Vi
'"' :I:
Q
.... u
Z lOS -r---------r--.---"f'-- --.-T -~ r --~-r ---I
IR> 10~ OHMS FOR V < 4 VOLTS
o o
o
Im
;'!
~ 103
a::
....
'" o o
o o
o o
RESISTANCE
10 4 5 6 7 8 10-4
HIGHEST PREVIOUS DIODE VOLTAGE
FIG. 2. Change of diode resistance and maximum diode current
as the conductivity of the diode of Fig. 1 is developed.
conductivity first develops, the higher the maximum
current the diode will eventually exhibit.
A typical sequence for the development of conduc
tivity and negative resistance in an AI-Ab03-Au sand
wich of I-mm2 area, with a 300-A oxide film, and with
Au positive is shown in Fig. 1. Log current is plotted
because of large variations in diode current. In curve 1,
the potential was slowly raised until conductivity first
appeared at 4.1 V and then it was slowly decreased. The
diode current increased with decreasing voltage and
negative resistance appeared even at this stage in the
forming of conductivity. On successive runs, the max
imum voltage applied was gradually increased until the
full negative resistance characteristic was developed.
Not all of the curves in the sequence of developing
conductivity are reproduced in Fig. 1; the numbers of
the curves indicate the position of each trace in the
sequence. A striking feature is the rapid increase in
conductivity and maximum current for the narrow
voltage range between 4.4 and 4.6 V. This is shown
further in Fig. 2 in which diode resistance at small
voltages and 1m, the maximum diode current for in
creasing voltage, are plotted against the highest previ
ous voltage applied to the sandwich after initial devel
opment of diode conductivity. Forming of conductivity
in these oxide films by application of voltages produces
a permanent and irreversible change in the oxide. The
original high resistance is not recovered.
During development of conductivity, the maximum
currents for decreasing voltage tend to be higher than
for increasing voltage, as shown in Fig. 1. Once con
ductivity is fully developed, the reverse tends to be
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to ] IP: 155.97.178.73 On: Fri, 28 Nov 2014 21:30:112120 T. W. HICKMOTT
true; the peak current for decreasing voltage is less than
for increasing voltage, and hysteresis appears. Likewise,
if the previous maximum diode voltage is not exceeded
during development of conductivity, the currents for
decreasing voltage are less than for increasing voltage.
During de forming of conductivity, diode conductivity
is increased when the previous maximum applied
voltage is exceeded. The conductivity of a diode with a
fully developed characteristic can be reduced to a low
value by applying voltages greater than V m, the voltage
for maximum current, and then rapidly decreasing
voltage through the negative resistance range. A semi
permanent change in oxide conductivity is produced.
Low diode conductivity formed in this way is stable if
the diode voltage does not exceed 1.8 V. If voltage
is raised across a film of low conductivity and V m is not
exceeded, full conductivity will be restored between
1.8 V and V m, and this high conductivity will be main
tained indefinitely in the oxide. For all diodes which
have been studied, establishment of conductivity with
one polarity of the applied voltage results in the estab
lishment of conductivity and negative resistance for the
opposite polarity as well. The reproducibility of the
current-voltage characteristic of a given diode, once
conductivity has been fully developed, is about ± 10%.
For impure oxides such as bisulfate-anodized alumi
num films, the voltage at which diode currents exceed
about 10-5 A and conductivity first appears is nearly
constant and independent of thickness. Six sets of
Al-AIzOa-Au diodes of 10-mm2 area and with different
oxide thicknesses were made and conductivity was
developed. A variation of eight in oxide thickness,
determined by the voltage to which anodization was
carried, resulted in differences of about 0.4 V in Va, the
voltage at which conductivity first developed. All
developed conductivity between 3.7 and 4.1 V. With
SiO diodes, on the other hand, or with Alz03 diodes
which were not deliberately doped, the field in the
insulator seemed to be a more important factor in
developing conductivity; the thicker the oxide the
higher Va. THE INFLUENCE OF THE EVAPORATED
COUNTERELECTRODE
To investigate the influence of the evaporated
counterelectrode metal in determining the negative
resistance characteristic of AI-A1203-metal diodes, a
set of sandwiches of 1-mm2 area with oxide thickness
",,300 A was prepared. Relatively thin, impure oxide
films were used to minimize difficulty in developing
conductivity. Two different metals were used as coun
terelectrodes for the oxide film on each glass substrate.
Bismuth, tin, lead, indium, and copper of 99.999%
purity, silver of 99.96%, and magnesium of 99% purity
were evaporated from tantalum boats. Aluminum and
gold of 99.99% and 99.96% purity, respectively, were
evaporated from tungsten helices, and cobalt was
evaporated from an overwound tungsten helix on which
the metal had been electrodeposited. Pressures during
evaporation were generally about 10-5 Torr and no
stringent precautions were taken to deposit clean films.
Thick opaque films of all the metals were deposited.
Conductivity in some diodes was established with the
counterelectrode as anode; in other diodes the base
aluminum electrode was positive during development
of conductivity. Varying the metal electrode and the
polarity used for establishing conductivity principally
affected the magnitude of the peak current 1m and
shifted the voltage for maximum current V m. Results
for typical diodes with different counterelectrodes are
summarized in Table 1. Ag and Co were deposited on
the same substrate, as were Au and Sn, and each of the
succeeding pairs of metals. Va, in column 1, is the volt
age at which conductivity developed as the voltage was
raised with the given metal as anode. Values of V m and
I m in column 2 are those after conductivity was fully
developed. On reversing polarity, values of V m and 1m
in column 3 were obtained, and values in column 4 were
found when the original polarity was restored. Columns
5 to 8 give the same quantities for a diode in which the
aluminum base was initially anode. With counter
electrode positive, Va is nearly constant between 3.5
and 4 V; with the aluminum base as anode, a much
TABLE 1. Effect of metal electrode on negative resistance in Al-AI203-metal diodes.
Metal anode to develop conductivity Base aluminum anode to develop conductivity
Evaporated 2 3 4 5 6 7 8
metal + + + counter- Va Vm 1m Vm 1m Vm 1m Va Vm 1m Vm 1m Vm lm electrode (V) (V) (rnA) (V) (rnA) (V) (rnA) (V) (V) (rnA) (V) (rnA) (V) (rnA)
Ag 3.5 Film Shorts 3.5 2.8 6.4 2.8 21.5
Co 3.5-3.7 2.4 96 2.5 29 2.7 33 4.7 Poor characteristic
Au 4.0--4.2 2.7 138 3.2 6 2.8 106 4.5 3.6 3.0 2.7 130 3.2-3.6 12
Sn 4.1 2.3 58 2.8 47 2.8 41 4.2 3.8 5.7 2.9 20 3.1 20
Cu 3.0-3.5 Shorts 2.7 46 2.6 61 3.4 2.7 66 2.7 100 2.8 50
In 3.2-3.6 2.4 23 2.7 16 2.4 14 6.0 4.6 1.0 2.4 6.0 4.5 2.0
Al 7.0 4.3 2 3.9 9.0 3.9 8 9.0 4.5 0.6 3.9 0.4
Bi 3.6-4.0 2.7 42 >6 3.8 1.0 2.8 13 3.8 9 Pb 3.5 2.6 76 2.7 66.0 2.8 60 9.0 3.2 12 2.7 24 3.2 10
Mg Conductivity could not be developed up to 17 V
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<i:
~10-2
I-
ffi
~ a
....
'" o
C 10-3, AI-A1203-Au
AI'+,Au'-',ORIGINAL POLARITY
o)Au' -
blAu'+
c)Au' -
AI'-, Au' + ; ORIGINAL POLARITY
dl Au'+
01 Au'-
FIG. 3. Dependence of current-voltage characteristics of
AI-~129a-Au diodes on po~arity of voltage used to develop con
ductivity and on the polanty of the voltage in tracing out char
acteristics. Oxide thickness, 300 A.
wider range of values of Va was found. Once conduc
tivity was formed, a wide range of values of 1m was
found, with those with the evaporated counterelectrode
positive being consistently higher than with the alu
minum base positive. V m also tended to be significantly
higher with the aluminum base as anode.
Gold is the most satisfactory electrode metal of those
examined and has been used most extensively. Figure 3
shows fully developed current-voltage characteristics
for two AI-AbOrAu diodes in which conductivitv was
initially developed with opposite polarities. With -AI as
anode during development of conductivity, 1m was only
3 mAo When Au was anode, the peak currents were
large regardless of whether gold or aluminum was anode
during development of conductivitv. When aluminum
was made anode after gold had been anode during
development of conductivity, as in curve e, 1m was
decreased by a factor of 25 and V m was shifted from
2.8 V to 3.2 to 3.6 V. Subsequent restoration of original
polarity resulted in a curve so similar to curve d that it
is not included in Fig. 3. No shorts developed with
AI-AbO a-Au diodes; the peak current which was con
trolled with one diode in the set was 340 mA corre
sponding to a current density of 34 A/cm2•
AI-AlzOa-Ag diodes were difficult to work with
because of the development of shorts. With all other
metals, the diodes initially had very low conductivity.
Only 3 out of 13 AI-AbOa-Ag diodes were not shorted
initially. When an attempt to develop conductivity of one of the unshorted diodes was made with silver as
anode, negative resistance started to develop at 3.S V
as with other films but the diode quicklY shorted before
a full characteristic could be formed.- With silver as
cathode during forming, good characteristics were
developed. When polarity was reversed, a steep rise in
current occurred and the film shorted. In general, to
develop negative resistance in thick oxide films silver
is the best anode material4 but it will produce shorts in
thin oxide films.
The behavior of AI-AIzOa-Cu diodes was inter
mediate between those in which gold and silver were the
electrode metals. Stable current-voltage characteristics
with high current densities and high peak/valley ratio
were often found. However, intermittent shorts would
appear and disappear which were similar to those found
with ~ilver. diodes although more easily removed by
changmg dIOde voltages. At the opposite extreme from
the high currents found in diodes using the noble metals
as counterelectrodes are those in diodes using aluminum
or ~agnesium. It was not possible to develop negative
r~slstance or c?nductivity in any of the AI-Ah03-Mg
dlodes. Establtshment of negative resistance in Al
Alz03-AI diodes was difficult, values of 1m were low, and
V m was poorly defined, somewhat erratic, and shifted
to around 4 V. For all AI-Alz0 3-metal diodes when
aluminum was anode in developing conductivity, 1m
was lower and V m was higher than when it was cathode.
AI-A\z03-AI diodes represent the extreme of this
tendency. Diodes with counterelectrodes of Sn Pb Co . h d ' , , In, or Bl a properties that fell in between those of the
noble metals or aluminum. In general, negative resis
tance characteristics could be established without
difficulty if the counterelectrode were positive and with
somewhat more difficulty if the aluminum base were
anode. The differences in I m with differing polarity that
were observed with gold counterelectrodes were not as
pronounced with these metals.
Differences in tunneling characteristics of AI-Alz03-
metal .diodes have been attributed by Handy to differ
ences m atom size of the metal electrodel8; the greater
the atom size, the higher the tunneling resistance
provided the atom radius is less than 1.S A. For atomi~
radii greater than 1.S A tunneling resistance was nearly
:onstant. Simmonsl9,20 and Mead14 have emphasized the
lmportance of work function differences between the
two metals of the sandwich in determining tunneling
curves as well as in effecting conduction in thicker
oxides. Neither effect seems to account for the vari
ations of conductivity which are observed in the present
case. For example, Ag, Au, and Al have nearest inter
atomic spacings of 2:88, 2.87, and 2.86 A, respectively,
?ut the ra.nge of ~axlillu.m conductivities after develop
mg negatIve reslstance lS several hundred to one. For
18 R. M. Handy, Phys. Rev. 126 1968 (1962)
19 ]. G. Simmons, Phys, Rev. Letters 10 10 (i963)
20 ]. G, Simmons, ]. App!. Phys, 34, 2581 (1963) ..
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the same elements, the work functions are .t.31, .t.70,
and 4.20 V, respectivelyY There is no simple correlation
between work function difference of base metal and
counterelectrode and the maximum conductivity which
can be developed. Another possibility is that the tem
perature of evaporation of the metal determines the
development of conductivity.ls In Table I, Ag and Co
were deposited on the same substrate. Co requires more
than a 4000K higher temperature for comparable
evaporation rates22 but the conductivity which devel
oped was appreciably smaller than for Ag.
DISCUSSION
The dependence of the forming of conductivity of
metal-oxide-metal diodes on the doping of the insulator
suggests that conduction and negative resistance are
both manifestations of impurity conduction in wide
band gap insulators. The temperature dependence of
diode conductivity provides support for this view. It
has been reported1 that if the full current-voltage
characteristic of an AI-Ab03-Au diode is traced out as
diode temperature is lowered, the shape of the curve
remains independent of temperature while the max
imum current 1m decreases steadily until a temperature
is reached at which no negative resistance region is
21]. C. Riviere, Proc. Phys. Soc. (London) B70, 676 (1957).
22 R. E. Honig, RCA Rev. 23,567 (1962).
JOURNAL OF APPLIED PHYSICS found in the current-voltage characteristic. However,
if V m, the voltage for maximum current through the
diode, is not exceeded as temperature is lowered, con
ductivity and current-voltage characteristics of diodes
with fully developed conductivity are independent of
temperature, within about 10%, from room temperature
down to "OK. Injection of charge carriers from the
metal into the insulator is not thermally activated in a
diode with fully developed conductivity. Such inde
pendence of conductivity on temperature is character
istic of conduction in heavily doped impure insulators.
Thus, conductivity in metal-oxide-metal diodes may
be through an impurity band, or impurity states, within
the forbidden band gap of the insulator. The dependence
of forming of conductivity and of the final current
voltage characteristics on the metal electrode of
AI-Ab03-metal diodes may then depend on the match
ing of the Fermi level of the metal and the impurity
band of the insulator at the metal-oxide interface. If
the match is poor, forming of conductivity is difficult
and the final conductivity is small.
ACKNOWLEDGMENTS
Conversations with F. S. Ham, who also read the
manuscript, were of particular assistance during the
course of this work. In addition, discussions with R. J.
Charles and D. A. Vermilyea have been of great help.
VOLUME 35, NUMBER 7 JULY 1964
Effect of Mechanical Stress on p-n Junction Device Characteristics
]. ]. WORTMAN, ]. R. HAUSER, AND R. M. BURGER
Research Triangle Institute, Durham, North Carolina
(Received 24 January 1964)
A theoretical model is developed for the effect of mechanical stress on the electrical characteristics of Ge
and Si p-n junction devices. This model is based upon the stress-induced variations in energy band structure
and their effect on minority carrier densities. The changes in minority carrier densities are shown to depend
upon the type of stress applied, with anisotropic stresses causing larger changes than hydrostatic stresses.
From the calculated dependence of minority carrier densities upon stress, the equations are developed for
the current-voltage characteristics of diodes and transistors under stress.
Following the general analysis several examples are given in which stress is applied to a small section of a
junction in a diode or transistor. The results show that at stress levels greater than 1010 dyn/cm2, the device
currents can change by several orders of magnitude when stress levels are changed by a factor of 2. The
theory is compared with published experimental data and found to be in good agreement.
I. INTRODUCTION
IT has been recognized for some time that stress has
a significant effect upon the electrical characteristics
of p-n junction devices. Early work in this area treated
the effect of hydrostatic pressure upon diodes.! It was
found that the major effect of a hydrostatic pressure
is to change the band gap of a semiconductor. Recent
1 H. Hall, ]. Bardeen, and G. Pearson, Phys. Rev. 84, 129
(1951), experimental investigations have shown that anisotropic
stress has a larger effect upon p-n junction character
istics than does hydrostatic pressure and, in addition,
the effects are considerably different.2-6 For example,
2 W. Rindner, Bull. Am. Phys. Soc. 7, 65 (1962).
• W. Ri~dner and I. Braun, Bull. Am. Phys. Soc. 7, 331 (1962).
'W. Rmdner and I. Braun, Proceedings of the International
Conference on !he Physics .of Semiconductors, Exeter, England,
July 1962 (InstItute of PhYSICS and The Physical Society, London,
1962), p. 167.
5 W. Rindner, J. App\. Phys. 33, 2479 (1962).
6 W. Rindner and I. Braun, ]. App!. Phys. 34, 1958 (1963).
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1.1696189.pdf | GroundState Energy and Excitation Spectrum of Molecular Crystals
C. Mavroyannis
Citation: The Journal of Chemical Physics 42, 1772 (1965); doi: 10.1063/1.1696189
View online: http://dx.doi.org/10.1063/1.1696189
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:25THE JOURNAL OF CHEMICAL PHYSICS VOLUME 42, NUMBER 5 1 MARCH 1965
Ground-State Energy and Excitation Spectrum of Molecular Crystals
C. MAVROYANNIS*
Division of Pure Chemistry, National Research Council, Ottawa, Canada
(Received 15 July 1964)
The Green's function method has been used to investigate the properties of the Frenkel excitons in
molecular cryst~ls at zero te~peratur~. Excitation energies have been determined for crystals havin
o~e or more eXCIted molecules m the ~mt cell. Configuration effects resulting from the interaction betwee!
different molecular states hav~ been mcluded. G~neral formulas are developed for the ground-state ener
of a mole~ular cr~stal, first without ~nd then with mixing of different electron configurations. It is fou!~
that the mteractlOn between two different exciton states contributes appreciably to the gro d-t t
energy of the crystal. un s a e
1. INTRODUCTION
AMICROSCOPIC theory of the dispersion of elec
tromagnetic waves in molecular crystals was
developed by Agranovich. l If we neglect retardation
effects and assume the molecules in the crystal to be
rigidly attached to the lattice sites, we find that the
lowest elementary excitations are excitons and photons.
When the retarded interaction between the electrons
and transverse photons is taken into consideration a
new excitation appears, the so-called polaritons which
are, roughly speaking, a mixture of electrons and
transverse:photons.l
The different molecular states in the crystal, which
are stationary in isolated molecules, are mixed under
the influence of the Coulomb interaction between the
molecules and the resulting energy splitting is given
by the nondiagonal matrix elements of the crystal
Hamiltonian operator. The mixing of different molecu
lar states was first studied by Craig2 by means of
perturbation theory. It was shown that these second
order effects may be large in comparison with the
energy-level separation of the molecule. Agranovichs
solved this problem rigorously by employing the
Bogolyubov's canonical uv transformation4 to the non
diagonal part of the crystal Hamiltonian, which was
expressed in the second quantization representation.
The use of the canonical transformation has the
advantage of taking into account contributions to the
energy spectrum of the elementary excitations resulting
from the simultaneous existence of more than one
excited molecule in the unit cell and it is free from the
limitations of the perturbation theory.l.s
In the present study we are concerned with the
ground-state energy and excitation spectrum in molecu
lar crystals. We restrict ourselves to the tight-binding
approximation (Frenkel exciton) in which one takes
* National Research Council Post-doctorate Fellow 1963-1964
1 V. M. Agranovich, Soviet Phys.-JETP 10 307 (1960)'
[Zh. Eksperim. i Teor. Fiz. 37, 430 (1959) J. '
2 D. P. Craig, J. Chern. Soc. 1955, 2302.
3 V. M. Agranovich, Soviet Phys.-Solid State 3, 592 (1961)
[Fiz. Tverd. Tela 3, 811 (1961) J. into accou~t the. excited ~tates of the molecule only
at that lattice pomt at which the molecule is found in
the ground state. The crystal is supposed to be at zero
temI?erature and only the low-lying excited states are
conSidered. The retarded interaction between the
electrons and transverse photons is disregarded.
We use the double-time temperature-dependent
Green's function in a matrix form to calculate the
exci~ation energies, first without and then with mixing
of different electron configurations, for crystals having
o~e or more excited molecules in the unit cell. The
smgle-particle Green's functions and the excitation
e~ergies are expressed in terms of the effective poten
tials ~(k) of the pair interaction.5•6 The results
obtained for the excitation energies are identical to
those of Agranovich.1•3 From the Green's functions and
excitation energies which we calculate we derive the
distribution functions and the averag; energy of the
system. Both are reduced to the distribution functions
of the .ground state. and to the ground-state energy,
respectively, by takmg the limit at zero temperature.
Expressions are derived for the ground-state energy of
the crystal first without and then by taking into account
~he effect of the mixing of different exciton states. It
~s show~ that the second-order effects arising from the
mteractlOn between two different molecular states are
of the same order of magnitude as the second-order
perturbations which result from the interaction be
tween the electrons in the isolated molecules. Finally
in the Appendix, the formulas for the ground-state
energy are rederived by means of the canonical trans
formation method.
2. THEORY
The Hamiltonian, corresponding to a molecular
crystal in which all the molecules are fixed and where
the retarded interaction between the electrons and
transverse photons is neglected, is given in the second
6 s. :r. ~eliaev, .Soviet Phys.-JETP 7, 289 (1958) [Zh.
E~spenm.l Teor. Flz. 34, 417 (1958)J.
• N. N. Bogolyubov, J. Phys. USSR 11, 23 (1947). N. M. Hugenholtz and D. Pines, Phys. Rev. 116 489 (1959)
1772 ' .
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:25ENERGY OF MOLECULAR CRYSTALS 1773
quantization representation in the forml•8
Hl=Eo+ 'Lt:."B.//B."
.,;
+ 'L (OJ. I V,., l/iO)B.//B. u;
BF81ii,j
+! 'L (00 I V88lIJd;)(B."B. u+B.//B.u'+), (1)
,.741;',;
where
t:.,,=E/'-E.O
+ 'L[ (OJ. I VIIl I 0J.)-(00 I V"' I OO)J, (la)
Eo= 'LE.O+!'L(OO I V .. ,I 00), (lb) ,'''',
and
iff'1 V"llf''JIII)= f 1/J.*'1/J'1*"V .. 11/J/"I/J./llldT.dT' 1
+ exchange terms, (lc)
I/J/ being the wavefunction of the s molecule located at
the jth excited state and V"l> the Coulomb interaction
between the charges of the molecules s and SI. Eo is the
energy of the crystal in which all the molecules are in
the ground state and E/i_E.o is the energy of thejith
excitation of the isolated molecule. The index s stands
for s=n, a where n is the unit lattice vector and a
enumerates the molecules in the unit cell, a= 1,2, •. ',17.
B.,/ and B." are the creation and annihilation opera
tors of the j.th excited state, respectively. They are
Pauli operators, i.e., they satisfy the commutation rela
tions of the Fermi type for the same lattice sites and
of the Bose type for different lattice sites. At zero
temperature the operators B.,/ and B." are approxi
mately boson type.l In (1) the term corresponding to
the interaction of the excitons has been omitted. The
Hamiltonian (1) has been derived by Agranovitchl to
whom we refer for details.s
We introduce the exciton creation and annihilation
operators by the Fourier transformation
B •. /./= 1/Nt'LB",,/*(k) exp( -ik'fna), k.",
B •. /.p= I/Nt'LBa",p(k) exp(ik· fna), (2)
k,l'i
where N is the total number of unit cells in the crystal
and the sums over k are over the first Brillouin zone;
#1-. enumerates the exciton bands corresponding to the
ith molecular term when i=j, p is one of the degenerate
molecular states p and fna is the position vector of the
molecule a at the site n. For a nondegenerate ith
molecular term, !J.i at a fixed i runs through the 17 values
where u is the number of molecules in the unit cell,
while when the multiplicity of the degeneracy p is not equal to 1 then !J.i runs through the up values.s The
commutation relations for Ba,,/(k) and Ba",(k) (we
drop p for convenience) at zero temperature are of the
same type as for B.,,+ and B.,o i.e., of the boson type7
[Ba". (k), B",,/( q) J-= c5k.q,
[Ba",(k), Ba", (q) J-= [Ba,,/(k) , Ba,,/(q)J-=O. (3)
Using the fact that
c5k.q = (1/N) 'L exp[i(k-q). fna],
n
we find that th{Hamiltonian (1) becolllel!
HI = Eo+ 'L t:."Ba", +(k) Ba", (k)
k,i;"li,a
+ 'L rai.,Bp) (k) Ba,,/(k)B,B"1 (k)
k,i.i;I1,l'i,PI
+! 'L rai,,Bp)(k)[Ba",(k)B,B", ( -k)
k, i,jj /." ,Pi ,fJ
(4)
with the notation
r .. i.,Bp)(k) = raip.,Bi(J(I)(k)
= .E'L' (Ojip I Vna.m,B l/iqO)X expik'(fna-fm,B), (5)
q=1 m
= t'L' (00 I Vna.m,B lfip/iq)X expik· (fna-fm,B), (5a)
q-I m
where the prime in the summation indicates that the
term m= n has to be omitted. In the Hamiltonian (4),
terms with i=j correspond to the energy states of the
isolated molecules while terms with i¢j arise from the
interaction between different molecular states.
If we introduce the matrices
(rai,,Bp)(k)
L ",B"(k) = o:t, J r ",B"(2)(k) aI, j
8~G ~) (6)
then the equation of motion for the operator Aa.(k)
will be
di(dldt) Aai(k) =t:."Aa.(k)+ 'L Lai.,Bi(k)A,Bj(k). (7)
,B,f,"j
7 We use the system of units with 11= 1.
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:251774 C. MAVROYANNIS
We define the Green's matrix functionS as
=( «Bal',(k);B"p/(k)) «Bap,(k);Bap;(-k»»), (8)
«Bal'/( -k); Bal'/(k») < (Bap/( -k); Bap;( -k»)
where, for example, «Bap; (k); Bap/(k) » is the re
tarded single-particle double-time Green's function as
defined by Bogolyubov and Tyablikov.9 The time
arguments t and t' have been omitted from the operators
Bap.(k) and Bap/(k), for the sake of brevity. The
properties of the Green's functions have been discussed
at a great length by Zubarev10 and we refer to his
review paper for a detailed discussion.
If we define the Fourier components of the Green's
function as
1['" = -dE exp[ -iE(t-t') ] «Aa.(k); Aaj+(k) ) )(Eh
211" -CX)
(9)
and use the ~ function representation
1[00
~(t-t') = - exp[ -iE(t-t') JdE,
211" -0>
then the components satisfy the following equations
of motion:
[aE-A/, -L"i,"i(k) J( (A,,;(k); Aaj+(k) ) ) (E) = Ib,j
+ L: L,,;,~j(k) < (A~j(k) ; A,,/(k) ) )(E)
j,{J,I';;U1"i) with a similar expression for «AfJj(k); A,,/(k) ) )(E).
Here bij stands for the Kronecker delta.
We can write the equation for ( (Aa;(k); Aa/(k» )(E)
in the general form
[aE-A/, -2;(k) J{(A"i(k) ; Aa;+(k) »(E)
= Hij+S(k), (12)
where
2; (k) = 2;i(k) + (1-bij) ~i;(k) + .• ,
and
S(k) = (l-~ij) Sii(k) +., '.
The matrices 2;(k) and S(k) are functions of k and E.
~i(k) and ~ij(k) involve the elements of the matrix
2;(k) with i=j and i~j respectively. Moreover, the
elements of the matrix ~ (k) satisfy the symmetry
relation
~22(k, E) =2;u(k, -E), 2;12(k, E)=~21(k, E), (13)
and they are the so-called effective potentials of the
pair interaction.5,6
From (12) one finds the set of algebraic equations
+ L:Lai,{Ji(k) «A{Ji(k) j Aaj+(k) ) )(Eh
f)r'a (10) +G(O) (k) ~12(k)Gij(k) +G(O) (k) Su(k),
[aE-A/j- Laj,aj(k) ] «Aaj(k) i Aa/(k) ) )(E)
= 1+ L"j,ai(k) «Aa;(k) ; Aaj+(k) ) )(E)
+ L: Laj,{JI(k) < (A{JI(k) ; Aa/(k) ) )(E)
1,1'1;(fJr'a)
L: Laj,al(k) «Aal(k) ; Aaj+(k) ) )(Eh (11)
1,1'1; (l1"i) ,(I"';)
8 YU. A. Tserkovnikov, Soviet Phys.-Doklady 7, 322 (1962)
[Dokl. Akad. Nauk SSSR 145, 48 (1962)].
D N. N. Bogolyubov and S. V. Tyablikov, Soviet Phys.-Dok
lady 4, 589 (1959) [Dokl. Akad. Nauk SSSR 126, 53 (1959)].
10 D. N. Zubarev, Soviet Phys.-Usp. 3, 320 (1960) [Usp.
Fiz. Nauk 71, 71 (1960)]; see also, D. ter Haar, Kgl. Norske
Vindenskab. Selskabs Forh. Skrifter 34, 77 (1961); A. I. Alekseev,
~oviet Phys.-Usp. 4, 23 (1961) [Usp. Fiz. Nauk 73, 41 (1961)]. Gij(k) =G(O)(k)~21(k)Gij(k)
+G(O) (k) ~22(k)Gii(k) +G(O)(k) S21(k) , (14)
where Gii(k) and Gij(k) denote the Fourier transforms
of the single-particle Green's functions
Gii(k) = «Bal" (k) ; Bap/(k) ) )(Eh
Gii(k) = < (Bap,+( -k); Bap/(k» )(Eh
respectively, G(O) (k), G(O) (k) are the unperturbed
Green's functions
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:25ENERGY OF MOLECULAR CRYSTALS 1775
respectively; ie is the infinitesimal positive imaginary part of E with ~+O. Solving the system of Eqs. (14)
we have
G .. (k) = [E+~f,+:i:22(k) J[5ii+Su(k) J-:i:dk) S21(k)
'3 {E-![:i:u(k) -:i:22(k) JIL {~I;+![:i:u(k) +:i:22(k) JI2+:i:12(k) :i:21(k) +ie ' (15)
G .. (k) = _ :i:21(k) 5ii+[E-~1; -:i:u(k) JS21(k) + SuCk) :i:21(k)
'3 {E-![:i:u(k) -2;22(k) J12-{~I;+![:i:u(k) +:i:22(k) JI2+:i:12(k) :i:21(k) +ie . (16)
For i=j we obtain the Green's functions Gii(k) ==Gi(k) and Gii(k) ==Gi(k) in the form
E+~f,+:i:22i(k)
Gi(k) {E-![:i:ui(k) _ :i:22i(k) J}2-{~I;+![:i:ui(k) +:i:22i(k) J}2+ :i:12i(k) :i:2Ii(k) +ie ' (17)
G'Ck) = - :i:21i(k) (18)
• {E-![:i:ui(k) -:i:22i(k) J12-{~f,+![:i:ui(k) + :i:22i(k) J12+ :i:12i(k) :i:21i(k) +ie .
Thus, the Green's functions Gi(k) and Gi(k) have been
expressed in terms of the effective potentials :i:i(k) and
they are similar in form to those derived by Beliaev.5
This should be expected since the statistical Green's
functions differ from the usual field-theoretical Green's
functions only in the way in which the averages are
taken. When the temperature tends to zero the statis
tical Green's functions average over the ground state.IO
3. GREEN'S FUNCTIONS AND ENERGIES OF
EXCITATION
a. i=j
The energy of excitation is obtained from the poles
of the corresponding Green's function. Thus from (17)
we have
8", (k) = ![:i:ui(k) -:i:22i(k) J
+ [{ ~I;+![:i:ui(k) + :i:22i(k) JIL :i:12i(k) :i:21i(k) ]!,
(19)
where the matrix :i:i(k) can be calculated exactly from
(10) and (11) for i=j in successive orders depending
on the number of molecules simultaneously present in
the unit cell. For a crystal containing one molecule per
unit cell (q= 1)
(20) and the energy of excitation will be
8",.1(k) = {[~/;+ ru(l) (k) JL[ru(2) (k) J2}i, (21)
where the index i, indicating the ith molecular term,
has been omitted for the sake of convenience. Then
the corresponding Green's functions are derived from
(17) and (18):
Gi.1(k) = [E+~I;+ ru(l) (k) J/[E2-8", i(k) +ieJ,
Gi.l(k) = -ru(2) (k) /[EL8", .12(k) +ieJ, (22)
~ith Gi.1*(k, E) =Gi.1(k, -E) and Gi.1*(k, E) =
Gi.1(k, E). If the ith molecular term is nondegenerate
then J.Li refers to only one exciton band.
For a crystal containing two molecules per unit cell,
q= 1, 2, the matrix :i:i.12(k) takes the form
:i:i.l2(k) = L1.l (k)
+ Ll,2(k) [aE-~/; -L2•2(k) J-IL2,l(k). (23)
For convenience in deriving the corresponding excita
tion energies, we set r12(l) (k) = r12(2) (k) == r12(k). This
assumption is valid since the matrix elements of
r12(1) (k) and rl2(2) (k) differ only by an exchange of
terms which are negligibly small.3 With this assump
tion, the elements of the matrix :i:i.12(k) become equal
and the evaluation of the energies is greatly simplified.
Then from (17), (18), (19), and (23), we find
where 8", .12(1) (k) and 8", .12(2) (k) are the energies of excitation given by the two positive roots of the equation
(26)
with 8",.1(k) and 8",.2(k) obtained from (21). The above results are valid provided that 8",.u(k) ~8,,;.2(k).
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:251776 C. MAVROYANNIS
Finally for a crystal containing three molecules per unit cell (u= 1, 2,3)
2;i,l23(k) = Lu(k) + L,,2(k) {&E-~,,- Lu(k) -L2,3(k) [&E-~/;- L3,a(k) ]-l}-l
X {Lu(k) + L2,a(k) [&E-~,,- L3,a(k) J-IL3,1(k) 1+2+-t3, (27)
and the corresponding energies of excitation are derived from the roots of the equation
8j1,,12l(k) =8j1,}(k) + {41 rl2(k) 12 ~,,2[8j1',12l(k) -8j1,i(k) J+2r12(k) r23(k) ral(k) ~"a}
X {[8j1j,1232(k) -8j1,i(k) J[8j1',1232(k) -8j1,i(k) J-41 r23(k) 12 ~,,21-1+2+-t3, (28)
provided that 8j1;.l2a(k) differs from both 8j1',2(k) and 8j1"a(k). In (27) and (28) we have indicated by 2+-t3 that
there is a further term to be obtained by interchanging 2 and 3 in the second term.
b. i~j
Here we include in the calculation of the Green's functions and excitation energies the second-order effects
which result from the mixing of two different molecular terms i,=j. The energies of excitation are obtained from
the poles of the Green's function Gii(k) given by (15), i.e.,
(29)
where 2;(k) = 2;i(k) +2;ii(k). Using the assumption that ri,j(l)(k) = ri,/2) (k) =ri,j(k) in addition to r;,p)(k) =
r.)2)(k)=ri,i(k), (29) reduces to
8j1"jI/(k) = ~/;2+2~,,2;l1i(k) +2~,,2;l1ii(k) =8j1,2(k) +2~,,2;l1ij(k). (30)
2;l1ii(k) is determined from (10) and (11) jitisfound that
2;nii(k) =2 L: II rai,/lj(k) 12 ~t;/[8"',jI/(k) -8j1/2(k) J}. (31)
a,fj,#li;i~i
Then (30) becomes
8j11,,,j2(k) =8j1,2(k)+4 L: II rai,/lj(k) 12 ~J,At;/[8j1',jI/(k) -8j1j2(k)JI, (32a)
a,/l,jI;;;"';
that is, the energies of excitation are determined by the roots of the equation
8j1I,jllCk) =![8j1,2(k) +8j1/2(k) J±!{[8j1,2(k) -8j1/(k) J2+ 16 L: 1 rai,/li(k) 12 ~,,~It It. (32)
.. ,/l,;;;"'i
The Green's functions Gij(k) and Gij(k) are obtained from (15) and (16) after the substitution of the expres
sions for 2;(k) and Sij(k); the latter is found to be
SOCk) = Lli,lj(k) [&E-~Jj- Lli,lj(k) ]-1. (33)
Then Gij(k) and Gij(k) take the form
Gij(k) = rij(k) (E+~JJ (E+~JJ/{[P-8"',jll(1)2(k) J[P-8""jI/2)2(k) J+iel,
Gij(k) = -rij(k) (E-~JJ (E+~JJ/{[EL8"',jI/l)2(k) J[EL8"",,/2) (k)2J+iel, (34)
(35)
and Gi/(k, E) = Gij(k, -E), Gi/(k, E) =Gij(k, -E), r'i,lj(k)=rij(k), wE-ere 8", ,,,Y) (k) and 8""jI/2)(k) are
given by the roots of (32) with u = 1. Now the Green's functions G/l) (k) and G/l) (k), which include the second
order effects, are obtained from (11). One easily finds
GP) (k) =Gj(k) + { (E+~Jj) rji(k) /[E2-8"12(k) +ieJ} [Gii(k) +Gij(k) J,
G;<')(k) =Gj(k) + { ( -E+~JJ rji(k) /[P-8j1/(k) +ieJ} [Gij(k) +Gij(k) J. (36)
(37)
In a similar fashion, one can include third or higher
order effects, i.e., contributions to the energies of excita
tion resulting from the mixing of three or more different
molecular states. The expressions (21), (26), and (32)
for the energies of excitation are identical with those
derived by Agranovich,1.3 who used Bogolyubov's canonical transformation to diagonalize the Hamil
tonian (4). He also compared his results with those
obtained by means of the Heitler-London approxi
mation and by the use of perturbation theory in
calculating the effects which result from the mixing
of different exciton states; we refer to his papers for
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:25ENERGY OF MOLECULAR CRYSTALS 1777
details.1.3 From our calculations we can see, as should
be expected, that the canonical transformation and
the Green's function technique yield identical results
for the energies of excitation.
4. GROUND-STATE ENERGY
a. i=j
We calculate here the average energy for a crystal
having a single particle, (1= 1, per unit cell. First, the
effects resulting from the configuration interaction are
disregarded, that is, we consider only terms in the
Hamiltonian (4) with i=j. Then the average energy of
the system is
(H1)i=Eo+ L: .:lJ;(B+(k)B(k»
k,i.~i
+! L: r(l)(k) [(B+(k)B(k»+ (B(k)B+(k»J
k,i,~i
+! L: r(2)(k)[ (B( -k)B(k»+ (B+( -k)B+(k) )J.
k,i,~",
(38)
In (38) the subscripts i, J.l.i and 1 are omitted for con
venience. To calculate (H1)i we need to know the
distribution functions involved in (38) at t=t'. These
can be determined from the corresponding Green's
functions. The expressions for Gk(E) and Gk(E) given
by (22) can be written in the form
Gk(E) =_l_[BI'.(k)+.:lJ;+r(l)(k)
2BI" (k) E-BI" (k)
BI'.(k) -.:lJ;-r(l)(k)]
+ E+BI'.(k) ,
Gk(E) = -[r(2) (k)/2BI" (k) J
X {[E-BI'.(k) J-L[E+BI'.(k) J-l}.
The spectral density is given by the relation1o
(i/27r) [Gk(E+iE) -Gk(E-iE) J= h(E) (EfJE-1), (39)
(40)
( 41)
where (3= l/kBT, kB is Boltzmann's constant, and T,
the absolute temperature. Using the fact that
lim[E-BI'.(k)±iEJ-l= {P/[E-BI'.(k)J}
=Fi7ro[E-BI" (k) J, (42)
where P means that the principal value of the corre
sponding integral (or summation) must be taken,
Jk(E) takes the form
Jk(E) = [2BI'.(k) ]-1{[BI',(k) +.:lh+ r(l) (k) J
Xo[E-BI'.(k) J+[BI',(k) -.:l/.-r(I)(k) J
Xo[E+B",(k)J} (efJE-1)-I. (43)
Here the spectral function has the form of a delta
function and there is no damping. The correlation function (Bk+(t) Bk(t') ) is related to Jk(E) by
(Bk+(t)Bk(t'»= L:""'h(E) exp[ -iE(t-t') JdE, (44)
and from the correlation function at t= t' we obtain the
boson distribution function
(B+(k)B(k»= La> Jk(E)dE. (45)
Substituting (43) into (45) and performing the inte
gration, we find
(B+(k)B(k) )
=![{[.:lJ;+ r(l) (k) J/BI" (k) } coth!{3BI" (k) -1], (46)
and
(B(k)B+(k) )
=![{[.:l/.+r(I)(k)J/BI'.(k) I coth!{3BI'.(k)+1]. (47)
Similarly, we evaluate (B( -k)B(k» by using (40)
and the result is
(B( -k)B(k»= (B+( -k)B+(k»
= [ -r(2) (k) /2BI" (k) J coth!{3BI" (k) . (48)
Then from (38), (46), (47), and (48), we have
(H1)i=Eo+! L: [-.:lJ;+BI'.(k) coth!{3BI'.(k)J. (49)
k,i,}li
Equation (49) gives the average energy of the system.
In particular, we get from (49) the ground-state energy
of the crystal by taking the limit{3-too, coth!{3BI" (k)-t1
and then (49) reduces to
(H1)/O)=E o+! L: [-.:l/.+BI'.(k)J. (SO)
k,i.}Ji
An alternative way of deriving (SO) is the following.
The Green's functions «Bal',(k); Bal'/(k») and
«Bal'/( -k); Bal'/(k») at zero temperature go over
to the field-theoretical Green's functions and average
over the ground state.I° Thus for T= 0 and t= t', we have
(Hl)i(O)=Eo+~ L: .:lJ;jdEGk(E)
27r k,i,l'i C
1 .
+2 2: k~ir(l)(k) LdE[Gk(E)+Gk*(E)J
+ 2~ k~ir(2)(k) LdEGk(E) , (51)
and substitution of Gk(E), Gk(E), and Gk *(E) into
(51) yields
(H ).(O)=E +~ " .:l .jdEE+.:lJ;+r(l)(k)
1 • 0 2 ~ I. DO 2(k) +. 7r k,i,l'i C n--BI" ~E
(52)
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:251778 C. M A V ROY ANN) S
where the E integration is to be taken with a small
detour into the upper half-plane. Performing the inte
gration, we obtain
(HI)/O)=Eo+t 2: [-~/;+81',(k)J. (50)
k,i,J,1,i
If we set r(I)(k)=r(2)(k)=r(k) in the expression
(21) for 81'.(k) , then (50) reduces to
(HI)/O) = Eo+t 2: ~/;(-1+{1+[2r(k)/~fJ}i).
k,i.J.l.i
(53)
Where 2r(k) «~f" which corresponds to the case
when the interaction between the molecules is suffi
ciently weak or to the low-density limit in the crystal,
we may expand the square root in (53) in powers of
2r(k)/~/; with the result
<HI)/O)~Eo+t 2: [r(k) -t I r(k) 12/ ~f'
+t I r(k) 13/~f,Lt 1 r(k) 14/~f,3+ •.• ]. (54)
The square brackets on the right-hand side of (54)
involves terms which result from one-, two-, three-, and
generally many-body interactions, respectively, and
give finite contributions to the ground-state energy of
the crystal.
The lattice sum r(k) =r(1)(k) as given by (5) with
p= 1 has been the subject of many studies,u-13 It is usually divided into two partsI3; the first part involves
the dipole-dipole-type interactions and it converges
slowly with increasing separation between the mole
cules. Dipolar sums have been studied by Cohen and
Kefferl4 and others.1l•I5 The second part includes the
short-range interactions and converges rapidly with
increasing intermolecular distance. Both parts should
be considered for the computation of r(k) .12,13 An
estimate of the expression (54) has been madel6 for
the rare-gas solids assuming the atoms in the crystal
to interact via dipole-dipole forces. The ground-state
energy has been expressed in terms of the static molec
ular polarizability and the density of the crystal. The
main contribution to the ground-state energy comes
from the two-body interactions.16 Among the inter
actions of order higher than the second, the triple
interaction energy is most important and amounts
to 3 to 11 % of the cohesive energy for the rare-gas
solids. Thus an accurate computation of (50), including
short-and long-range contributions, is expected to give
(apart from anharmonicities) a good estimate of the
cohesive energy of molecular crystals.
b. i.,t-j
We calculate now the ground-state energy for the
crystal, including mixing of two different exciton
states. Averaging the Hamiltonian (4) for i.,t-j we have
1 i" 1 -- 42 L.. ruCk) dE[G;<I)(k, E)+GP)*(k, E)+G;<I)(k, E)+GP)*(k, E)J
1f' k",l'i C
Ii" 1 --+2" 2 L... rij(k) dE[Gij(k, E) +Gi/(k, E) +Gij(k, E) +Gi/(k, E) J,
"If' k,J.l.i • .uj;J~1, C (55)
where we have again set riP)(k)=riP)(k)==rij(k)
in addition to riP) (k) = riP) (k) == rii(k), and we
refer to a crystal having a single molecule per unit cell.
The Green's function Gij(k, E), Gij(k, E), GP)(k, E),
and GP) (k, E) are given by (34), (35), (36), and (37),
respectively. The next step is to substitute the expres
sions for the Green's functions into (55) and then
integrate over E. The path of integration C is a contour
consisting of the real axis from -00 to + 00, together
with a semicircle in the upper half-plane. Carrying out
the integration over E in (55) and after some tedious
algebra, we find the final formula for the ground-state
11 W. R. Heller and A. Marcus, Phys. Rev. 84, 809 (1951).
12 R. S. Knox, J. Phys. Chern. Solids 9, 238, 265 (1959).
13 A. A. Demidenko, Soviet Phys.-Solid State 3, 869 (1960)
[Fiz. Tverd. Tela 3, 1195 (1961) J. energy
(HI)i/O)=Eo+l 2: [-~/;-~fi
+8I'i,I'/!)(k) +8I'i.I'/2)(k) J, (56)
where the energies of excitation 81', ,IIY) (k) and
81'1 ,I'i (2) (k) are given by the positive roots of the
equation
81', .I'/(k) = t[81',2(k) +81'/(k) J
±t{[81',2(k) -8I'i2(k) J2+162: 1 riJ(k) 12~fA!ili. (57)
ir'i
14 M. H. Cohen and F. Keffer, Phys. Rev. 99, 1128 (1955).
15 B. R. A. Nijboerand F. W. De Wette, Physica24, 422 (1958).
16 C. Mavroyannis, thesis, Oxford University, Oxford, England,
1963.
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:25ENERGY OF MOLECULAR CRYSTALS 1779
To study the form of (56) we use the following
perturbation procedure. We introduce a small (non
dimensional) parameter fJ which we set equal to unity
in the final results, and we perform the substitution
rii(k)~r'i(k), rjj(k)~rjj(k), and rii(k)~
fJr ij(k). This substitution is valid for crystals with low
densityP The result is
[ 1 r··(k) 12
(Hl)i/O)~Eo+t~ rii(k)+rjj(k) -;~f;
J rjj(k) 12 2:E 1 rij(k) 12_ ••• J,
2~1i ir'i ~f;+~fj (58)
where summation is implied over repeated indices
(i, fJ.i andj, fJ.j respectively). From the point of view of
perturbation theory, the first two terms in the square
bracket of (58) are of the first order in the energy
while the last three terms, on the other hand, are second
order. Also, the last term with i,ej is of the same order
of magnitude as that of the third and fourth term. We
conclude that the effect of the mixing of two different
electron configurations is comparable with the second
order perturbations which result from the interactions
between the electrons in the isolated molecules. Thus
these effects should be taken into account in calculating
the ground-state energy of the crystal. It is demon
strated in the Appendix how the formulas (50) and
(56) for the ground-state energy can be derived from
the Hamiltonian (4) by means of the canonical trans
formation method. It is shown that the techniques
yield identical results.
In the present study by using the Green's function
technique we have shown how one may calculate
exactly the excitation spectrum and the ground-state
energy for a molecular crystal, with and without
taking into account the mixing of different electron
configurations. We have assumed that the molecules
are rigidly fixed at their equilibrium positions, i.e., the
crystal is at zero temperature. In a later publication,
the excitation spectrum and the interaction of excita
tion waves at finite temperatures will be discussed.
APPENDIX
We use here Bogolyubov's4 canonical uv transfor
mation to determine the ground-state energy. We
consider first the terms in the Hamiltonian (4) with
i=j and assume that the crystal has only one molecule
per unit cell.
a. i=j
We introduce the new quasiparticle operators
!Xk=UkBk+VkB_k+,
eLk + = VkBk+UkB-k +, (A1)
17 This perturbation procedure does not apply to molecular
crystals which have large oscillator strengths, i.e., where the
interaction between the molecules cannot be regarded as weak. where Uk and Vk are functions determined by the
relations
that is, !Xk+ and !Xk are the creation and destruction
operators, respectively, of quasiparticles (elementary
excitations) with wave vector k and obey Bose statis
tics. From (A2) we have
(A3)
with Ak being defined as
and 0", (k) is the energy of excitation given by (21).
We express the operators Bk and B_k+ in terms of
the !Xk'S as
Inserting (A4) into the Hamiltonian (4) and con
sidering only terms which contribute to the ground
state energy, we find
(H1)/O) = Eo+ :E ~fiVk2+! :E r(1) (k) (Uk2+Vk2)
k.i.~i k,i,,ui
-:E r(2) (k)UkVk. (AS)
k,i,,ui
Substitution of the expressions for Uk and Vk into (AS)
gives
(H1);<O)=Eo+! :E [-~,,+0"i(k)]. (A6)
k,i,J.'i
b. i,e j
In a similar way, one can calculate the ground-state
energy from the Hamiltonian (4) with i,ej, i.e., in
cluding the mixing of two different electron configura
tions. In this case for a given wave vector k, there are
two normal modes described by the two quasiparticle
operators !Xki and !Xkj defined as
where coefficients Uki, Vki, Ukj, Vkh etc., are determined
by the relations
(A8)
0,,; ."Y) (k) and 0", .,,/2) (k) are the energies of excitation
given by the two positive roots of (57). From (AS)
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130.18.123.11 On: Fri, 19 Dec 2014 21:53:251780 C. MAVROYANNIS
one finds
(A9)
where
Aki=[8p; ,/lY' (k)L8p;2(k)]2+4L: 1 rij(k) 12 A/;A/;
j,.<i
and
Uk?-Vki2+Uk/-Vk/= 1.
The coefficients Uk;', Vk/, Uk/, and Vk/ are obtained
from the expressions for Uk;, Vki, Ukj, and Vkj, respectively,
by replacing 8p;,pyl(k) by 8p;,p/2l(k). 8p;(k) is given
by (21).
THE JOURNAL OF CHEMICAL PHYSICS The next step is to express the operators Bki, Bkj,
etc., in tenus of the l¥ki and l¥k/S. Finally, the ground
state energy in terms of the coefficients Uki, Vki, etc.,
takes the form
(H1)i/Ol=E o+! L: A/; (Vk?+Vk/2)
+! L: A//(Vk/+ Vk/2)
k,j,!li
k,i,Pi
+! L: rjj(k) [(Ukj-Vkj)2+ (Uk/-Vk/)2]
k,j''''j
+! L: r;j(k) [(Uki-Vki) (Ukj-Vkj)
k .Ili ,J.Lii ir6i
+ (Uk/ -Vk;') (Uk/ -Vk/)]. (AlO)
Substituting the coefficients into (AIO) and after some
rearrangement, we find
{H1)i/O'=E o+! L: [-A/;-A//
k,i,ji/Ji.Jli
VOLUME 42, NUMBER 5 1 MARCH 1965
Magneto-Optical Rotation of Transition-Metal Complexes
SHENG-HSIEN LIN AND HENRY EYRING
Department of Chemistry, University of Utah, Salt Lake City, Utah
(Received 5 November 1964)
The Faraday rotations of certain tetrahedral and octahedral transition-metal complexes are examined
by using Kramers' equation from the viewpoint of the crystal-field theory.
1. INTRODUCTION
UNLIKE natural optical rotation, which is restricted
to a special class of molecules without a plane or
a center of symmetry, the magneto-optical rotation
is a quite general phenomenon. It can be observed in
all molecules. The angle of rotation per unit length
along the direction of propagation of light for mole
cules with fixed orientation, according to Kramers,l
can be expressed as
4Tr2112iN (j=--
chQ
L: { (n 1 X 1 n') (n' 1 Yln)-(n 1 Yin') (n' 1 X 1 n) I
X nnl 1I2-1I2(nn')
Xexp( -En/kT), (1)
1 H. A. Kramers, Koninkl. Ned. Akad. Wetenschap. Proc. 33,
959 (1930). where
Q= L: exp( -En/kT);
n
(n 1 X 1 n') and (n 1 Yin')
are the matrix elements of the x and y components of
electric moment in the presence of the magnetic field
corresponding to the transition from the lower state n
to the upper state n'. The corresponding equation for
molecules with random orientation has been given by
Serber.2 Serber has shown that at relatively high tem
peratures, the Faraday effect is independent of elec
tron spin, if the spin-orbit interaction is negligible and
the over-all multiplet width is small. As in natural
optical rotations,3 we may define the rotatory strength
2 R. Serber, Phys. Rev. 41, 489 (1932).
3 W. J. Kauzmann, J. Walter, and H. Eyring, Chern. Rev. 26,
339 (194D).
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1.1714519.pdf | GasSurface Interactions and FieldIon Microscopy of Nonrefractory Metals
E. W. Müller, S. Nakamura, O. Nishikawa, and S. B. McLane
Citation: Journal of Applied Physics 36, 2496 (1965); doi: 10.1063/1.1714519
View online: http://dx.doi.org/10.1063/1.1714519
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/36/8?ver=pdfcov
Published by the AIP Publishing
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] IP: 130.209.6.50 On: Thu, 18 Dec 2014 23:59:592496 J. RICHARD CUNNINGHAM, JR.
the potential use of garnets in the radio frequency
region, e.g., magnetic tape recorder heads and magnetic
thermal switches.
(3) In YIG the rotational mechanism accounts for
an average of 20% of the total permeability over the
temperature range 77°K to ""0.9 Te.
(4) It has proved possible using the temperature de
pendence of the initial permeability to derive K1(T) for
the Sc3+ substituted garnets. The derived K1(T) seem
to be in good agreement with the experimental K1(T)
measured on single crystals. A comparison of the results for the anisotropy con
stants of scandium-substituted garnets and the one-ion
model of anisotropy will be the subject of a future
paper.
ACKNOWLEDGMENTS
The author wishes to thank L. J. Schwee for the
experimental magnetization data, R. F. Stauder for
experimental anisotropy measurements, and W. E.
Ayers for his invaluable assistance in the materials
preparation.
JOURNAL OF APPLIED PHYSICS VOLUME 36. NUMBER 8 AUGUST 1965
Gas-Surface Interactions and Field-Ion Microscopy of Nonrefractory Metals*
E. W. MULLER, S. XAKAMURA, O. NISHIKAWA, AND S. B. McLANE
Department of Physics, Pennsylvania State University, University Park, Pennsylvania
(Received 23 December 1964)
The performance of the helium field-ion microscope depends critically upon accommodating He atoms
of 0.15-eV kinetic energy to the specimen tip. The small accommodation coefficient requires the field
trapped He atom to make several hundred contacts with the cold tip surface. The hopping He atoms diffuse
preferentially to tip regions where the high local field permits ionization before full accommodation is
reached. Improved accommodation is achieved with the provision of an intermediate collision partner in the
form of adsorbed neon or, preferably, hydrogen or deuterium. Now a high-resolution He ion image is ob
tained at 70% of the field used before. As the addition of hydrogen promotes field evaporation, its partial
pressure must be carefully controlled to achieve image stability of the nonrefractory metals. Low-field
evaporation by the hydrogen reaction permits easy conditioning of the tip surface of the nonrefractory
transition metals so that artifacts caused by yielding to He evaporation field stress are no longer a problem.
The field evaporation end form obtained with hydrogen added to He more closely approaches the desirable
spherical shape of the emitter than does field evaporation in vacuum or in a single imaging gas. As ex
amples, ion images of niobium, nickel, iron, and high carbon steel are shown.
I. INTRODUCTION
A MORE complete understanding of the field-ion
microscopel was achieved when it was realized
that field ionization actually takes place a few ang
stroms above the metal surface2 and occurs predomi
nantly after the helium atoms repeatedly rebound3 from
the emitter surface. Since the attainment of high resolu
tion in a point projection microscope requires that the
tangential velocity component of the imaging ions be
negligible, it is desirable to transfer most of the kinetic
energy taF02+~kTgaB (a=polarizability, F 0= field at the
tip surface) of the arriving gas molecules to the solid
surface before ionization. The breakthrough in field-ion
microscopy was . consequently achieved4 by cooling the
emitter in order to provide a sufficiently large accom
modation coefficient. As soon as the impinging molecule
transfers to the surface more energy than ~kTgaB' it
* Supported by the National Science Foundation and the U. S.
Office of Naval Research.
1 E. W. MUller, Z. Physik 131, 136 (1951).
2 M. Inghram and R. Gomer, J. Chern. Phys. 22, 1279 (1954).
3 E. W. MUller and K. Bahadur, Phys. Rev. 102, 621 (1956).
4 E. W. MUller, J. App!. Phys. 27, 474 (1956) i and 28, 1 (1957). remains trapped by its polarization in the inhomogene
ous field near the emitter tip, returning to the surface
in a number of hops of gradually decreasing height
until it is fully accommodated by being either adsorbed
or confined to low jumps of kinetic energy ~kTtip and
average height 3 kTro/8aF 02 (ro= tip radius). Field ioni
zation by tunneling of an electron from the gas molecule
into the metal can occur when on such a hop the mole
cule gets far enough away from the electronic surface
of the emitter to raise its ground level above the Fermi
level in the metal. Ionization will happen with greatest
probability above protruding surface atoms because the
local field at that location is at a maximum.
II. GAS-SURFACE THERMAL
ACCOMMODATION
Of all gases, helium is the most suitable for field-ion
microscopy because of the high electric field required
for its ionization, which assures high resolution. How
ever, the accommodation coefficient of helium on sur
faces of heavy metals is very small unless the solid
surface is cooled. Quantitative data on the accommoda-
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tion coefficient in the temperature range of interest
do not exist. The classical theory of Baule5 and re
cent treatments of the problem by Goodman 6 and by
Trilling7 calculate the accommodation coefficient (Eo
-E1)/ Eo of a gas of kinetic energies Eo and El before
and after the collision with the surface but neglect the
temperature of the solid and, of course, any possible
effects of the high field.
For large impact energies (!aFo2 for He at a typical
field of 450 MV jcm is 0.14 eV, or equivalent to HOOOK)
the theoretical accommodation coefficient approaches
a=2mjM, which gives for He-W (m=4, M=184)
a=0.0435, about one half of the fraction of the energy
4mMj(m+M)2 that can be transferred by central col
lision between free hard spheres. With the exception of
He, for which the measured accommodation coefficient
is nearly independent of temperature and can be as low
as 0.015, predictions about the temperature dependence
of a for the noble gases agree rather well with the experi
mental data obtained by Thomas and Schofield8 and
by Silvernai19 for the temperature range 77° to 303°K.
It has been pointed out by Brandonlo that below the
Debye temperature the surface might be unable to
absorb energy from a fraction of the collisions when all
possible vibrational modes are occupied. However, a
quantitative application of Frauenfelder's formula for
the Mossbauer effectll does not seem justified because
the Debye temperature as a bulk property of a crystal
cannot be adequate for the description of the behavior
of surface atoms, and the observed lack of temperature
dependence of the accommodation coefficient in the
wide range below the Debye temperature of tungsten
(1:1= 315°K) down to 78°K underlines this conclusion.
No direct experimental data are available in the 78° to
4 oK region of interest to field-ion microscopy. The
theories of Goodman and Trilling suggest a fast rise
of the accommodation coefficient to unity when at a
very low impact energy the gas molecule becomes
trapped in the adsorption potential well, but for helium
on tungsten this could be expected only near 4 oK. How
ever, it seems possible that under the prevailing condi
tions of field-ion microscopy, where surface atoms are
near field evaporation, the energy transfer in collisions
with protruding surface atoms on the edges of net
planes might approach the condition of free particle
collision. It has been shown by Nishikawa and Miiller12
that the fraction 4mM j (m+ M)2 of incident kinetic en
ergy is actually transferred when the gas molecule
promotes field evaporation. As a guide for our experi-
& B. Baule, Ann. Physik 44, 145 (1914).
6 F. O. Goodman, J. Phys. Chern. Solids 23, 1269 (1962).
7 L. Trilling, J. Mecan. 3, 215 (1964).
8 L. B. Thomas and E. B. Schofield, J. Chern. Phys. 23, 861
(1955).
9 W. L. Silvernail, Ph.D. thesis, University of Missouri (1954).
10 D. G. Brandon, Brit. J. Appl. Phys. 14,474 (1963).
11 H. Frauenfelder, The Mossbauer Effect (W. A. Benjamin, Inc.,
New York, 1962), p. 30.
12 O. Nishikawa and E. W. MUller, J. Appl. Phys. 35, 2806
(1964). mental approach we assume, therefore, that the accom
modation coefficient reaches the classical hard-sphere
collision value somewhere between 78° and 21°K.
The number n of consecutive collisions that would
accommodate the gas molecule from the original energy
Eo to the energy En, assuming a constant accommoda
tion coefficient a, is determined by the relation
(1)
Using the experimental a=0.015, it takes n= 260 colli
sions to reduce the kinetic energy of a helium atom of
HOOoK temperature equivalent, to 21°K. Since all of
the gas molecules arriving at the tip have the kinetic
energy !aFo2 plus the superimposed Maxwellian distri
bution according to the temperature Tgas from where
they originate [either the fluorescent screen at room
temperature or partially accommodated to the lower
temperature (25° to 1000K) of the accelerating elec
trode of the FIM], there are no slow molecules that
would accommodate easily. Thus, the accommodation
process under ordinary operating conditions is slow
although all molecules that reach the sphere of cap
ture4,13 determined by dipole attraction and the gas
temperature with a low enough initial gas kinetic ap
proach velocity Vo defined by !mvo2<a(!aFo2+!mv2) are
eventually trapped in the tip field and ionized. This is
a large fraction of all molecules entering the sphere of
capture when the average gas temperature is kept low
by the metal electrode in contact with the cold finger of
the conventional FIM design.
The large cross section of the emitter shank facilitates
the trapping of the gas molecules.14 Many molecules
will collide first with the shank and then, after partial
accommodation, diffuse towards the tip, following the
field gradient. Accommodation at the shank is con
siderably more efficient than at the tip cap since the
shank is covered with an adsorption layer of residual
gases, while the tip cap is usually field desorbed and
atomically clean.
Ionization occurs in a very shallow, disk-shaped ioni
zation zone about 4 A above each protruding surface
atom. Measurement of the energy distribution in field
ionization by Tsong and Miillerl5 reveals that under
best image conditions one half of the ionizations occur
wi thin a depth of 0.17 A and 95% wi thin a zone of
0.6-A depth. During their many hops of slowly dimin
ishing height the accommodating gas molecules traverse
this ionization zone with a fairly large average velocity.16
In order to achieve ionization, the field has to be rela
tively high to provide a sufficient tunneling probability.
Because of the remaining large average tangential ve
locity component, optimum resolution condition is not
13 M. J. Southon and D. G. Brandon, Phil. Mag. 8, 579 (1963).
14 E. W. MUller, 10th Field Emission Symposium, Berea, Ohio
(1963).
15 T. T. Tsong and E. W. MUller, ]. Chern. Phys. 41, 3279
(1964).
16 E. W. Miiller and R. D. Young, J. Appl. Phys. 32, 2425
(1961).
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80
70
(f) 60
(f)
'~ 50
~ 40
to
0:.: 30
<D ..l. 20
3 10
9 10 II 12 13 14 15 16 17 18
TIP VOLTAGE (kV) FIG. 1. Brightness of
111 region of a tungsten
tip, field evaporated in
He at 78°K as a function
of tip voltage .
reached at BIV. An ideal emitter would have a uniform
field over the whole imaged cap, and a reduction of the
applied voltage would result in better accommodation
of the molecules until their transit times through the
shallow ionization zone are long enough for tunneling.
In reality, due to the crystallographic variation of field
evaporation, the tip has regions of locally smaller radius
of curvature and, hence, higher fields. When the applied
voltage is lowered, the hopping gas molecules simply
diffuse to these high-field regions, from which they are
drained away as ions, while there is no ionization above
the region of larger local radius of curvature. This
concept explains two experimental facts of field· ion
microscopy:
(1) The net plane edges around close-packed, low
index planes, such as 011 on bcc crystals,17 111 and 001
on fcC,17 and 0001 on hcp crystals,18 are never well re
solved into single atoms but instead appear as fairly
smooth rings. They indicate regions of relatively large
local radius of curvature. If the applied voltage is high
enough, molecules will be ionized here before reaching
full accommodation and will image this region with
poor resolution; if the voltage is lowered, most of the
molecules will diffuse away to be ionized more efficiently
above the protruding tip areas.
(2) This diffusion process is found to be strongly
dependent upon the tip temperature. In Fig. 1 the local
brightness of the 111 region of a tungsten crystal
which was field evaporated to the end form at 78°K
has been plotted against the applied voltage.I9 Since
the intensity is proportional to the beam power, the
data can be corrected for local average current density
by dividing each datum point by the voltage. The inten
sity peak for the 111 plane shifts towards lower voltages
as the tip temperature is decreased. The effect of tem
perature on total current has been suppressed by nor
malizing the intensity at a voltage near field evapora
tion; this was necessary because of the uncertainty in
volved in determining the actual gas pressure in the
vicinity of the tip. The relative heights of the voltage
peaks increase considerably with decreasing tempera-
17 E. W. MUller in Advances in Electronics and Electron Physics,
edited by L. Marton (Academic Press Inc., New York, 1960), Vol.
13, pp. 83-179.
18 E. W. MUller, Proc. 111 European Regional Conf. Electron
Microscopy, Prague 1, 161 (1964).
19 B. J. Waclawski assisted in these experiments, which were
carried out in 1962. ture until about 21°K is reached. For the tip shown in
Fig. 1 this peak is 8 times higher than the emission of
the same area near evaporation field or at BIV, but
this ratio, which has been observed as high as 15: 1,
depends upon the area size chosen for the measurement,
the actual size of the 111 plane, and particularly upon
the ratio of local tip radius near the 011 region and the
111 region. As can be seen from the field-evaporation
equation F=e-3(A+I-~-kTlnt/to)2 (A=heat of va
porization, 1= ionization energy, ~= work function,
t= evaporation time, and to= the vibrational time of
surface atoms)p the dynamic end form will be such
that high ~ regions (around 011) assume a larger radius
of curvature and low ~ regions (around 111) a small
radius of curvature. Also, at a higher evaporation tem
perature, the ratio of local radii increases in agreement
with observation. In the ll1-brightness measurements
this is reflected by an increasing peak height. As an
example, a tip evaporated at 21°K gives a relative
111 peak of about 3 times the BIV level, as shown in
Fig. 2. The position of the peaks, for which additional
measurements have been made using melting nitrogen
and liquid oxygen as temperature baths, can be de
scribed by the empirical formula V max= Vo[1 + (T)!/26]
volts, (in Fig. 1, Vo= 9900 V). It appears that this effect
can be conveniently used as a means to monitor the
emitter temperature by simply measuring the local
screen brightness, which should be useful for in situ
annealing of lattice defects and similar ion microscopical
observa tions.
When the tip voltage is lowered from the over-all
best image voltage (BIV), the brightness of the 111
regions increases, while the regions of larger local radius
of curvature, particularly near 011, become correspond
ingly dimmer. It can be expected that the application
of a sudden voltage drop from BIV to 111 peak voltage
should result in a delayed increase in 111 brightness,
since the gas molecules trapped over the entire tip cap
need time to diffuse to the 111 regions. However, ex
periments with microsecond pulses and fast-responding
screen phosphors did not show any measurable delay.
Also, the relative intensity of the 011 vicinity compared
to the 111 region did not change when an ac voltage,
whose peak-to-peak value equaled the difference be
tween BIV and 111 peak voltage, was applied with a
frequency of up to 350 kc/sec. In the latter case a slow
60
50
~ 40
W
Z f-30
I
to
ir 20
<D
..!..
== 10
O~~~~~L-~~
5 6 7 8 9 10 II 12
TIP VOLTAGE (kV) FIG. 2. Brightness of 111 region
of a tungsten tip, field evaporated
in He at 21°K as a function of tip
voltage. For this specific tip the
field evaporation voltage FEV (1
layer/min) was 12 kV; the best
image voltage in He was 9.6 kV.
After addition of 1% H2 a second
brightness hump appears at the
low-field BIV.
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(a) (b)
FIG. 3. Ion image of the 111 region of a tungsten tip of 450-1 radius. (a) Pure.He, BlV at 13.2 kV after ~eld evaporation in
the presence of 1% H2, (b) pure He at 9.0 kV, and (c) He wIth 1% H2 added, at 9.0 k\.
screen response would not interfere with the observa
tion. These negative results indicate that the diffusion
time for the helium molecules is less than 10-7 sec at
21 oK; hence there is no temporary adsorption between
the hops but rather a continuous bouncing. Thermal
velocity at 21°K permits the helium atoms to travel
a distance of the order of the tip radius within 10-9 sec.
III. IMPROVED ACCOMMODATION WITH AN
INTERMEDIATE COLLISION PARTNER
During the investigation of the properties of neon
as an imaging gas12 it was observed that the addition
of a few percent of neon to the conventional helium
gas improved the image quality. Neon ions excite the
fluorescent screen some 6 times less efficiently than do
helium ions and their direct contribution to the image
is negligible. However, the best image voltage is found
to be a few percent lower, and more image details ap
pear in the interior of net planes. The 3.14-A square
array of atoms on the 001 plane of tungsten could be
seen for the first time. The effect of the small neon
admixture can be explained by an improved accom
modation of the helium atoms when they collide at the
surface with weakly adsorbed neon atoms, to which
they can transfer a maximum of 55% of their kinetic
energy. The neon atoms are not visible in the ion image
as they are presumably mobile at 21°K during the
bombardment with helium atoms of up to 0.14 eV. The
average degree of coverage in the adsorption layer of
neon seems to be small and cannot be derived from otber
adsorption data of neon because of the uncertainty of
the effect of the field. Further observations at a tem
perature below 21°K would be desirable.
It has been known for a long time20 that an adsorp
tion layer of gases increases the accommodation coeffi-
20 J. K. Roberts, Proc. Roy. Soc. (London) A129, 146 (1930). cient and Becker21 found experimentally that a hydro
gen-dovered tungsten surface accommodates helium
with a=0.07. We have now utilized this effect in field
ion microscopy by adding hydrogen and deuterium, of
which the latter should give the best match in collision
masses. With the addition of some 0.1% to 10% of one
of these gases a new sharp image appears at about 70%
of BlV for helium, comprising only the high-field regions
of the tip (Fig. 3). The accommodation of additional
helium atoms that would ordinarily not be trapped by
the field is clearly demonstrated by an increase of total
image brightness by 8% when 1 X 10-5 Torr hydrogen is
added to the helium gas. The hydrogen alone would have
increased the screen brightness in the form of a diffuse
background by 3%. Above 10-5 Torr hydrogen partial
pressure the further increase in screen brightness (at t~e
BlV) is entirely due to the diffuse background and IS
exactly linear with hydrogen partial pressure (Fig. 4).
The observed accommodation effect is somewhat in dis
agreement with a study made by Ehrlich and Hudda22
who believe that hydrogen does not remain adsorbed
on tungsten under the conditions of a helium ion image.
They assume desorption to occur by an electron shower
coming from the ionizing helium. Based on the experi
ments by Nishikawa and Milller12 on the specific varia
tion of the field evaporation end form by pure and mixed
image gases we are convinced that bombardment by gas
molecules having the full dipole attraction energy is
responsible for the observed desorption processes.
The sharp atomic images obtained by the addition
of hydrogen at 70% of BlV are entirely due to helium
ions, as was observed directly by magnetic deflection
of the image in a microscope fitted with a slot i of the
way between tip and screen.23 Only below 55% of
21 F. E. Becker, See footnote in Ref. 8.
22 G. Ehrlich and F. G. Hudda, Phil. Mag. 8, 1587 (1963).
23 T. C. Clements and E. W. MUller, ]. Chern. Phys. 37, 2684
(1962).
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150
125
::l100
'" z
~ i; 75
~
CD
-J 50
~ o
f-25
o 1oa.-L...J.......l-L....J.....JL....J..-'-'--'--L...l..
Q 0.5 1.0
Hi PARTIAL PRESSURE (TORR) FIG. 4. Total screen
brightness for a tungsten
tip of 700-A radius at
EIV = 17 k V as a function
of hydrogen partial pres
sure. For the He-H 2 mix
ture above 10-6 Torr H2,
the brightness increases lin
early with hydrogen partial
pressure with the same
slope as hydrogen alone.
helium BIV is the-image of an individual metal atom
made up of H2+ and H+ ions and the He+ ion is com
pletely absent. There is no voltage range in which hydro
gen ions and helium ions contribute simultaneously to
an image dot representing a surface atom. Interestingly,
under these conditions, i.e., in the presence of He, no
H3+ ions were seen in any voltage range, while in a pure
hydrogen ion image H3+ is quite abundant23 near hydro
gen BIV.
The best condition for the low-field helium ion image
due to the presence of hydrogen is characterized by an
additional hump in a brightness vs voltage plot for the
111 region, as shown in Fig. 2 for a tungsten tip in
1XH)-3 Torr He with 1% H2. While without hydrogen
the ionization probability vanishes rapidly as the volt
age goes down (Fig. 1), the small amount of hydrogen
once more increases the helium ionization at a field as
low as 320 MV fcm. The resolution in this image is also
better than in the pure helium image at the 111 bright
ness peak (Fig. 3). We conclude that only now is there
complete accommodation of the helium atoms to the
tip temperature.
The low-field helium ion image and the corresponding
111 brightness peak appear in a wide range of hydrogen
partial pressure down to below 10-6 Torr, indicating
10-4 o
o o
Nb TIP
1~9~-L..~ __ ~~ __ ~-L-
8 10 12 14 16 18 20
FIELD EVAPORATION VOLTAGE (kV) FIG. 5. Field evaporation
voltage of a niobium tip
of 6OO-A radius as a func
tion of hydrogen partial
pressure. Evaporation rate
is four 011 layers/min. that an adsorption film on the emitter rather than con
tinuous supply from the gas phase is essential. This
adsorption film reaches its saturation value above 10-5
Torr hydrogen pressure, as indicated by the beginning
of a linear increase of image brightness in Fig. 4. At
lower partial pressure the impinging He atoms limit
the amount of adsorbed hydrogen.
The hydrogen is introduced by a previously hydro
genated and electrically heated zirconium foil having
20 cm2 free surface. Adjusting the temperature of the
foil between 350°C and 800°C maintains reversibly
a hydrogen partial pressure between 10-6 and 10-3
Torr, which is monitored using a Consolidated Electro
dynamics Corporation mass spectrometer. An additional
advantage of the hot zirconium foil is that it acts as
a very active getter. Assisting the cryogenic pumping of
the liquid-hydrogen cold finger, the residual gases (water
vapor, nitrogen, carbon monoxide) were kept in the
10-9 Torr range although the conventional, unbaked,
field-ion microscope design with greased joints was used.
It may be noted that in Fig. 1 the 111 brightness plot
shows an indication of a low-field hump at liquid
hydrogen temperature, which is not present at liquid
helium temperature. This is probably due to cryogenic
pumping of some residual hydrogen in the imaging gas.
The low-field hump and the corresponding helium ion
image due to hydrogen adsorption are just barely notice
able at 78°K tip temperature. However, the resolution
in the very weak low-field helium image is not as good
as at 21°K, when a tip radius near 1000 A is used.
Temperature reduction obtained by solid nitrogen
(about 600K) is not yet effective in improving the image.
IV. HYDROGEN-PROMOTED FIELD
EVAPORATION
The essential features of the low-field helium image
as described above for a tungsten emitter can also be
observed with other metals. However, it is more difficult
to obtain quantitative data because of complications
due to the field-induced reaction17 between hydrogen
and the metal, becoming apparent as enhanced field
evaporation. For all metals except tungsten, rhenium,
tantalum, molybdenum, iridium, and platinum, field
evaporation in vacuum or in pure helium sets in at or
below BIV for helium, making it difficult to record stable
images without the use of efficient image intensifica
tion.24 When hydrogen is introduced into the FIM while
a barely stable, normal helium ion image of tantalum
or niobium is displayed on the screen, the field evapora
tion rate increases to a removal of one to ten atom
layers per second, and the voltage must be reduced in
order to again obtain a stable image.25 Figure 5 gives
the field evaporation voltage for a niobium tip of ap-
24 S. B. McLane, E. W. Muller, and O. Nishikawa, Rev. Sci.
Instr. 35, 1297 (1964).
25 E. W. Muller and S. Nakamura, shown in a 16 mm film at
the 11th Field Emission Symposium at Cambridge, England
(1964).
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(a) (b) (e)
FIG. 6. (a) Helium ion image of 111 region of a niobium tip at 22.7 kV with lattice rlefects at the 111 region caused by yielding to
field stress and many displaced, low-coordination surface atoms (extra bright spots). (b) Niobium tip field-evaporated in a He-l% D2
mixture. Image taken at 21.3 kVafter replacing gas by pure He. Average tip radius 800 A. (c) Same niobium tip after 5 atomic layers
of field evaporation imaged at 16.2 kV in He-1% D2, with a much more perfect 111 region. Surface vacancies are produced by field
reaction involving adsorbed residual gas molecules and deuterium.
proximately lOoo-A radius at an evaporation rate of 4
atomic layers/min. When a niobium tip is field evapo
rated in vacuum or in the presence of He as an imaging
gas [Fig. 6(a)], many lattice defects appear in the 111
region. Also, there are a large number of bright spots,
indicating that surface atoms have been rearranged to
low-coordination sites.26 After reducing the evaporation
field by adding 1% Dz and imaging again in pure He,
the surface particularly in the 111 region appears with
a better crystallographic perfection, Fig. 6(b). The low
field He ion image obtained with the addition of 1%
Dz to the image gas gives a very regular surface at 75%
of the field strength required for pure He, Fig. 6(c).
This greater perfection is due to the conditioning of the
tip surface at a lower field, and, hence, reduced me
chanical field stress. It has been pointed out repeat
edly26-28 that for the applicability of field-ion micros
copy to the less refractory or nonrefractory metals, it
is decisive whether field evaporation or yield to the field
stress (j= F2/87r occurs first. Numerically, this stress
amounts to 1 ton/mm2 at a typical evaporation field
of 475 MV /cm. Of all the metals studied so far, only
W, Ta, Ir, Pt, Rh, and Au have enough strength at
their respective evaporation fields to develop a perfect
field evaporation end form without yielding to the field
stress. Re, Mo, Nb, V, Pd, Fe, Ni, and Co yield to form
dislocation networks or slip bands, although some for
tunately only in restricted crystallographic areas.26 By
shaping the tip at a reduced field stress using hydrogen
or deuterium-promoted field evaporation it should be
possible to retain the original lattice perfection of the
26 E. W. MUller, Surface Sci. 2, 484 (19M).
27 E. W. MUlier, 10th Field Emission Symposium at Berea,
Ohio (1963).
28 E. W. MUller, Bull. Am. Phys. Soc. 9, 104 (1964). specimen, and the improved accommodation due to the
adsorbed light gas atoms should permit the use of the
advantageous helium for imaging at a reduced field,
provided that the hydrogen field reaction is slow enough
at low-field BIV.
It appears that hydrogen promotion of field evapora
tion occurs with all metals of interest to field-ion micros
copy, but the rates seem to vary considerably for differ
ent metals. Field evaporation of tungsten in the presence
of helium proceeds at a slightly lower field than in high
vacuum,29 and the end form in helium is more evenly
curved than in vacuum12 although the 111 region is still
more sharply curved than the rest of the tip. The 111-
brightness plots in Fig. 2 represent the helium-field
evaporation end form. If further field evaporation is
performed with hydrogen added, the 111 region begins
to evaporate at a field about 5% lower so that the end
form in a helium-hydrogen mixture approaches a shape
60
50 8%H
'" ~ 40 z t-G 30
~ 20
I
:::;. 10
O~~~L-~~ __ L-~~L-~~ __ L-
o 2 4 6 8 10 12 14 16 18 20
TIP VOLTAGE (kV)
FIG. 7. Brightness as a function of tip voltage of 111 region
of a tungsten tip, field evaporated in 1 % and 8% H2-He mixture
to remove the strong 111 protrusions.
29 R. D. Young, 7th Field Emission Symposium, McMinnville,
Oregon (1960).
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(a)
having a fairly uniform radius of curvature. As a result
the high-field helium image at BIV has less contrast in
the various crystallographic regions [Fig. 3(a)] and
there is no longer a preferred diffusion of hopping He
molecules to the 111 region when the voltage is lowered.
As shown in Fig. 7, the previously large 111-brightness
peak has entirely disappeared, but there is still the
small hump in brightness indicating the low-fleld helium
BIV. In this diagram the recording of 111 brightness has
been extended to zero voltage, and the two peaks at 6
and 4.5 kV represent the hydrogen image of the 111
region. In the voltage range below 8 kV the image con
sists only of both hydrogen ions, the upper peak being
due to the predominance of H+ and the low one to H2+'
(a) (b) FIG. 8. (a) Nickel tip,
imaged with He just below
evaporation field at 12.7
kV. The field stress caused
the 102-113 regions to
develop a large dislocation
density. (b) Nickel tip
after rapid additional field
evaporation in a He-l0%
H2 mixture at 9.9 kV and
imaged in pure He at
13.05 k V after further field
evaporation in He.
Hydrogen and deuterium also enhance field evapora
tion of Ta and Mo, beginning at the protruding 001
and 111 planes of the helium evaporation end form.
The reduction of the evaporation field is 15% for Ta
and 8% for Mo. These metals also give highly re
solved low-field helium ion images with the improved
accommodation.
(b) V. HELIUM ION IMAGES OF
NONREFRACTORY METALS
The use of the refractory metals in the foregoing ob
servations was convenient to study various effects of
hydrogen addition in clearly separated voltage ranges.
(0)
FlG.9. (a) Iron tip field-evaporated in He just above imaging voltage of 16.5 kV. The 111 region is not imaged because of pitting by
easy field evaporation of the stress-induced, highly defect structure. (b) The same iron:crystal additionally field-evaporated in He with
addition of 10-6 Torr H2• Then further evaporation was done in pure He eliminating locally attacked areas and taking pictures with the
hydrogen removed by gettering. The 111 net plane rings are perfectly developed. Image voltage is 17.1 kV. (c) The same iron crystal
imaged with neon at 16.4 k V.
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For the lower melting metals, such as platinum, nickel,
and iron, the surface stability under field stress and
hydrogen promotion of field evaporation is marginal.
Platinum and nickel show a slow reaction with hydrogen
at helium BlV, but in contrast to the refractory metals
the field evaporation rate increases when the tip voltage
is reduced to about 70% of pure helium BlV. At this
field the rapidly evaporating surface is quite disordered,
particularly around 135 on Pt and 012 on Ni. An almost
perfect surface can be recovered by some subsequent
field evaporation in pure helium and imaged at helium
BIV [Fig. 8(a) and (b)]. The effect of the hydrogen
treatment is then merely to produce a more evenly
curved tip so that the field stress at the formerly highly
protruding 102-113 region is reduced to below the yield
strength.
The field evaporation of iron is found to be very sen
sitive to the presence of small amounts of hydrogen.
Without hydrogen the 111 plane cannot be observed
[Fig. 9(a)J as the field stress always produces a random
structure in this region, which field-evaporates below
the ionization field of helium but can be imaged with
neon.12 Traces of hydrogen, in the 10-6 Torr range,
suffice to attack iron at the evaporation field, partiCll
larly in the 112-114 region, so deeply that details can
not be seen even with neon. However, if hydrogen is
removed by either replacing the gas mixture by a new
filling of pure He or Ne or by using a titanium getter,
a subsequent field evaporation will bring out a perfect
111 plane [Fig. 9(b) and (c)]. A strange observation
is that prolonged field evaporation in these imaging
gases retains the perfect condition of the 111 region of
iron, even if the high voltage is temporarily turned off or
the tip heated to 80oK. However, warming up the entire
cold finger by replacing the liquid-hydrogen coolant by
liquid nitrogen finally removes the beneficial effect of
the original hydrogen treatment. It appears then that
after the initial conditioning with hydrogen-promoted
field evaporation, an extremely low residual pressure of
hydrogen suffices to effect the field evaporation end form
of iron.
The attainment of reasonably good imaging condi
tions for iron opens the way for field-ion microscopy of
steel. Figure 10 shows a sample of high-carbon steel
wire which had been annealed to 850°C and then slowly
quenched (3 sec). The surface was conditioned with
hydrogen, but the final imaging gas was pure helium.
The developed net planes correspond to a bcc crystal
structure, and the two grain boundaries running diago
nally across the picture seem to indicate a lens-shaped
30 E. W. MUller, Proceedings of the 4th International Congress on
Electron Microscopy, 1958 (Springer-Verlag, Berlin, (1960), Vol.
1, p. 820. FIG. 10. Helium ion image of a high.carbon steel tip, field-evapo
rated with 10-6 Torr H2 added and then imaged after further field
evaporation in pure He at 17.9 kV.
martensite plate. The many bright spots randomly
strewn over the entire surface might represent inter
stitial carbon atoms.
VI. CONCLUSIONS
The gas-surface interactions at the emitter tip of a
field ion microscope are quite complicated, and a quan
titative understanding is difficult because of the many
uncertain factors of thermal accommodation, particu
larly in the presence of an adsorption layer. However,
for practical purposes, considerable progress has been
made with the use of hydrogen or deuterium additions
to helium and neon as imaging gases. The possibility
of a low-field helium ion image and the utilization of
hydrogen-promoted field evaporation extend the appli
cation of field-ion microscopy from the refractory metals
to the technically important common transition metals,
including steel.
ACKNOWLEDGMENTS
The authors are pleased to acknowledge the tech
nical assistance of Douglas F. Barofsky and Gerald L.
Fowler.
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1.1696953.pdf | Study of Collision Narrowing by Comparison of MolecularBeam and GasPhase
Nuclear Resonance Spectra
L. M. Crapo and G. W. Flynn
Citation: The Journal of Chemical Physics 43, 1443 (1965); doi: 10.1063/1.1696953
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:03THE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 5 1 SEPTEMBER 1965
Study of Collision Narrowing by Comparison of Molecular-Beam and Gas-Phase
N uc1ear Resonance Spectra *
L. M. CRAPO
Department of Physics, Lyman Laboratory, Harvard University, Cambridge, Massachusetts
AND
G. W. FLYNNt
Department of Chemistry, Mallinckrodt Laboratory, Harvard University, Cambridge, Massachusetts
(Received 12 April 1965)
The second moments of molecular-beam nuclear resonance curves are compared through a simple motion
narrowing theory with linewidths from gas-phase NMR data for CF4, SiF4, SFs, CH3F, CFaH, and CH2F2 at
300oK. From this comparison the average number of collisions necessary for these molecules to change their
angular momentum is estimated to be on the order of three.
I. INTRODUCTION
IN an NMR study of SF6 near the critical point,
SchwartzI was unable to explain the observed values
of ljTI and 1jT2 for the fluorine nuclear spins by a
dipole-dipole relaxation mechanism. Subsequent mo
lecular-beam measurements on SF6 by Baker and Ram
sey2,3 indicated that the spin-rotation interaction was
the predominant broadening mechanism, and thereby
Purce1l4 was able to show that Schwartz's measure
ments could be explained through a simple motion
narrowing theory using the second moment of the
molecular-beam curve. More accurate molecular-beam
fluorine nuclear resonance measurements on CF4, SiF4,
SF6, CFaH, CHaF, and CH2F25 have shown that the
spin-rotation interaction is the predominant intra
molecular magnetic interaction for all of these mole
cules. Anderson6 has pointed out that it should be
possible to relate the second moments from the molec
ular-beam curves to the corresponding gas-phase NMR
linewidths for these cases.
There has been considerable current interest in the
spin-rotation interaction as a mechanism for nuclear
spin relaxation.7-27 This interaction is the coupling
* Work supported by the National Science Foundation and the
Office of Naval Research (Harvard).
t Present address: Department of Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts.
1 J. Schwartz, Ph.D. thesis, Harvard University, 1957 (unpub
lished).
2 M. R. Baker, H. M. Nelson, J. A. Leavitt, and N. F. Ramsey,
Phys. Rev. 121,807 (1961).
3 N. F. Ramsey, Am. Scientist 49,509 (1961).
4 E. M. Purcell (private communication).
5 L. M. Crapo, C. H. Anderson, 1. Ozier, and N. F. Ramsey (to
be published).
6 C. H. Anderson (private communication).
7 M. Bloom, Physica 23,237,378 (1957).
8 H. S. Gutowsky, 1. J. Lawrenson, and K. Shimomura, Phys.
Rev. Letters 6,349 (1961).
9 C. S. Johnson, Jr., J. S. Waugh, and J. N. Pinkerton, J. Chem.
Phys. 35, 1128 (1961).
10 C. S. Johnson, Jr., and J. S. Waugh, J. Chern. Phys. 35, 2020
(1961).
11 M. Lipsicas and M. Bloom, Can. J. Phys. 39, 881 (1961).
12 J. G. Powles and D. K. Green, Phys. Letters 3, 134 (1962).
13 C. S. Johnson, Jr., and J. S. Waugh, J. Chem. Phys. 36,2266
(1962) . between a nuclear magnetic moment and the magnetic
field generated by rotational motion of the molecule
in which the nuclear spin is located. Collisions between
molecules modulate the rotational magnetic field ran
domly so that the intramolecular field seen by a nu
clear spin is no longer constant in a given molecule
but rather oscillates randomly in time and consequently
contains frequency components of appreciable inten
sity up to the frequency of collision. Frequency com
ponents in the rotational magnetic-field power spec
trum which are at a nuclear spin Zeeman transition
frequency can induce transitions between the nuclear
Zeeman energy levels providing a nuclear spin relaxa
tion mechanism.
A molecular-beam nuclear resonance curve represents
the probability of transition versus frequency for the
case of no collisions between molecules. Thus the sec
ond moment of this curve is just proportional to the
mean-square intramolecular magnetic field (H?) which
is defined by the equation
( 1)
where 'Y is the nuclear spin gyromagnetic ratio and
(w2) is the second moment of the nuclear resonance
14 M. Bloom and H. S. Sandhu, Can. J. Phys. 40, 289 (1962).
16 G. W. Flynn and J. D. Baldeschwieler, J. Chem. Phys. 37,
2907 (1962).
16 C. D. Cornwell, E. O. Stejskal, and L. G. Alexakos (private
communication) .
17 K. Krynicki and J. G. Powles, Phys. Letters 4, 260 (1963).
18 R. J. c. Brown, H. S. Gutowsky, and K. Shimomura, J. Chem.
Phys. 38,76 (1963).
,. G. W. Flynn and J. D. Baldeschwieler, J. Chem. Phys. 38,
226 (1963).
20 P. S. Hubbard, Phys. Rev. 131, 1155 (1963).
21 M. Lipsicas and A. Hartland, Phys. Rev. 131, 1187 (1963).
22 J. H. Rugheimer and P. S. Hubbard, J. Chern. Phys. 39, 552
(1963).
23 J. S. Blicharski, Acta Phys. Polon. 24, 817 (1963).
24 W. R. Hackleman and P. S. Hubbard, J. Chem. Phys. 39,
2688 (1963).
26 M. Bloom and 1. Oppenheim, Can. J. Phys. 41,1580 (1963).
26 K. Krynicki and J. G. Powles, Proc. Phys. Soc. (London) 83,
983 (1964).
27 U. Haeberlen, R. Hausser, and F. Noack, Phys. Letters 12,
306 (1964).
1443
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:031444 L. M. CRAPO AND G. W. FLYNN
curve in the absence of collisions. Order-of-magnitude
motion-narrowing theories28,29 suggest that the nuclear
resonance linewidth in the presence of collisions is given
approximately by
(2)
where 0", is the gas-phase nuclear resonance linewidth
in units of w= 2'11"v, and w. is the average effective
modulation rate of the interaction responsible for the
line broadening.
In the present work a theory is developed for spheri
cal molecules which gives a relationship identical in
form to Eq. (2) except for a numerical factor. This
relationship is then combined with experimental meas
urements of 0", and (w2) to give estimates for We, which
in turn can be related to rotational relaxation rates and
rough-sphere collision processes. The effective modula
tion rate w. is given by
(3)
where N mJ is the number of collisions necessary on the
average to change mJ by an appreciable amount and
Zc is the average kinetic collision frequency. The term
"appreciable amount" is taken here to mean changes in
mJ which are effective in averaging out the local field
Hi. The first-order spin-rotation interaction is propor
tional to mJ and therefore fluctuations in mJ give
fluctuations in the local magnetic field that the spins
see. For gases at moderate pressures Zc can be calcu
lated by kinetic theory and thus it is possible to deter
mine the rotational collision numbers N mJ through use
of Eqs. (2) and (3).
In essence the spin-rotation interaction is used here
as a probe to study collision processes. By comparing
its magnitude in a collision-free environment to its
magnitude in gas phase, conclusions can be drawn
about the effect of collisions in averaging out the inter
action by rotational modulation and about details of
the collision mechanisms.
II. THEORY
To discuss the motion-narrowing phenomenon, con
sider a system of nuclear spins plus lattice to be de
scribed by the Hamiltonian
X=Xo+Xp+F,
where Xo is the Zeeman interaction of the spin system
with the constant external field H, F is the Hamil
tonian for the lattice and contains only lattice coordi
nates, and Xp is the Hamiltonian for the interaction
between the spins and the lattice. F commutes with
Xo but not with Xp, thereby introducing a time de
pendence into Xp' The noncommuting property of F
28 N. Bioembergen, E. M. Purcell, and R. V. Pound, Phys. Rev.
73.679 (1948).
29 C. P. Slichter, Principles of Magnetic Resonance (Harper and
Row, New York, 1963), p. 154. and Xp is responsible for the motion-narrowing phe
nomenon since it causes a time-dependent perturbation
of the spin system described by Xo.
In general the nuclear magnetic resonance absorption
intensity can be expressed as
I(w) = L: G(t) exp( -iwt)dt. (4)
G(t) is the correlation function given by3°,31
G(t) = (exp[ix(t) J) exp(iwot), (5)
where
x(t) = lt: (a I Xp(t') I a)-({31 Xp(t') I (3)}dt'
o
= ltw(t')dt" (6)
o
In Eq. (5) the brackets indicate an average over the
distribution of random x(t) values; a and {3 denote
nuclear spin states with the restriction Ea-EfJ= nwo
(wo is the central resonance frequency), also
Xp(t) = exp( -iFt)Xp exp(iFt).
Equation (6) has been derived assuming that off
diagonal elements such as (a I Xp(t) I a') can be ne
glected. This assumption is valid only when the fre
quencies contained in Xp are very slow compared to
Waa' = (Ea-Ea,) Iii. In the cases of interest in this work
Xp contains frequencies equal to and higher than W"a'.
However, it can be shown that the contribution of
these higher frequencies to the linewidth is just equal
to the zero frequency contribution when Xp is the spin
rotation interaction in spherhical molecules.20 There
fore, Eq. (5) can be used to calculate a linewidth and
this width is just half of the total width.
The problem now is to calculate G(t) in Eq. (4),
which necessitates certain assumptions about the ran
dom modulation characteristics in order to make the
equations mathematically tractable. By assuming a
Gaussian probability distribution for x(t) of the form
Anderson3o and Abragam31 both show that
G(t) =exp(iwot) exp{-(w2) ~'(t-T)g"'(T)dT}' (7)
where (w2) is the second moment of the collision-free
resonance curve (which we assume is the molecular
beam curve) and g", (T) is a dimensionless correlation
function satisfying the restriction gw(O) = 1. This func
tion is appreciable only for T ;$Tc where Tc is the cor
relation time for the collision process. In the extreme
30 P. W. Anderson, J. Phys. Soc. Japan 9. 316 (1954).
31 A. Abragam, The Principles of Nuclear Magnetism (Oxford
University Press, London, 1961), Chap. 10.
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:03COLLISION NARROWING 1445
narrowing limit, i.e., ((2)Tc2«1, we have
[(t-T) g",( T) d~t f"" g",( T) dT~tTe
o 0
if the integral over g",(T) is taken as approximately
equal to the area of the rectangle of height 1 and
width Te. Thus for extreme narrowing
G(t) =exp(iwot) exp( -«(2) I t I Tc)
and
lew) = i: exp[ -«(2) I t I Te] exp[ -i(w-wo)t]dt
(W-WO)2+ ({W2)Tc)2'
This is just a Lorentz curve with full width at half
height 5",0 [0 indicates that only the zero frequency
contribution is considered and thereby the conditions
of Eq. (6) are satisfied] given by
5",0= 2 {w2 )Te.
By changing from angular frequency units w to fre
quency units JI and assuming the correlation time to
be Te=NmJ/Zc (where Zc is the kinetic collision fre
quency and N mJ is the number of collisions necessary
to change mJ and thereby change the local magnetic
field seen by the nuclear spins) the following is ob
tained:
The total width is twice 5.0 as is shown from the
Hubbard20 and Blicharski23 theories of spin-rotation
interaction relaxation; hence
(8)
This expression is derived assuming a Gaussian distri
bution for X(t) ; however, a more realistic distribution
for gas-phase collision modulation is the Markoffian
distribution. It can be shown from Anderson's motion
narrowing theory30 that in this case the expression for
5. is identical to that of Eq. (8).
When Xp is the spin-rotation interaction in a spheri
cal molecule containing N identical spins, then
(9)
;=1
where Ii and C (i) are the nuclear spin and spin-rotation
tensor of Nucleus i, respectively, and J is the rotational
angular momentum of the molecule. Hubbard20 shows
that the transverse relaxation time T2 for the spin
rotation relaxation mechanism in this system can be
derived from the correlation function
CiP(T) = CA,2( -1) Z+k ([Ji-Z] t+T[li-k]t)
+¥(~C)'""t._ "t,G : ;'x:' : :',)
X ([lr D01,(2) (ni) ]t+r[I/"Dok,(2) (ni)] t) (10) in which
and
~C=CIi-C.L.
CII and C.L are the diagonal components of C(i) in a
coordinate system whose z axis passes through Spin i,
so that Crx= Cyy= C.L and Czz= CII, liz is the lth com
ponent of the first-order spherical tensor formed from
the laboratory components of J for the molecule con
taining Spin i, DOz'(2) (ni) is the second-order rotation
matrix as a function of the Euler angles ni= (ai{3il'i)
between the laboratory coordinate system and the co
ordinate system (fixed in the molecule) whose z axis
passes through Spin i. The brackets ( ) indicate an
ensemble average over all molecules and
C : :',)
is a 3-j symbol.
In order to evaluate the correlation function given
by Eq. (10), the following assumptions are made:
([Ji-l]t+r[h-k]t)= ([Ji-Z li-k]t) exp( -T/rmJ,
(lla)
([IF'D o1,(2) (ni) ],+r[lik"Dok,(2) (ni)]t)
= ([N" lr]t) ([DOI,(2) (ni)DOk,(2) (ni)]t)
exp( -T/T'mJ) ' (llb)
where TmJ is the characteristic time for rotational ran
domization which is interpreted as the average time
between collisions which change mJ by an appreciable
amount; T'mJ is a characteristic time depending upon
the complex correlation between orientation and angu
lar momentum. Thus CiP is given by
C;P(r) =i (J(J + 1) )( -1) k5_I.d CA,2 exp( -r/rmJ)
+-H~C)2 exp( -r/r'mJ)}, (12)
since
and
The transverse relaxation time is now determined from
(15)
where
ll(w) = H~""[C;ik(r) exp(iwr)
+Ciikl(r) exp( -iwr) ]dr} (-1)kLI.k (16)
and Wo is the resonance frequency of Spin i. The evalu-
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:031446 L. M. CRAPO AND G. W. FLYNN
TABLE I. Experimental gas-phase fluorine NMR linewidths 0,
and molecular-beam fluorine second moments (p2) for spherical
molecules at 300°K.
0,(5 atm) 0,(10 atm) (p2)
Molecule (cps) (cps) (kc/sec)2
CF4• 48±1O 20±3 22 OOO±2 000
SiF4 9.00±2 4.6±0.8 4550±300
SF6 53±10 34±5 26 000±3 000
• Unpublished data from Cornwell, Stejskal, and Alexakos" gives 0,(5 atm)~
35.4 cps and 0,(10 atm)~18.7 cps with undesignated error limits. Their lO-atm
data are in good agreement with the data above. Since the width should change
linearly with pressure, the true 5-atm width is most likely near 40 cps.
ation of ll(W) from Eqs. (12) and (16) gives
ll(WO) =t(l(l+1) {CAV\+(T:~mY
( 17)
In the extreme narrowing limit, i.e., WOTmJ«1, WOT'mJ«1,
and ll(W)~ll(O)
1/T2'""2l1(0) .
By changing from units of W to frequency units v, a
factor of 471"2 is introduced so that
This expression is identical to that derived by Blichar
ski23 for dilute gases if TmJ=T'mJ and is used for a com
parison of NMR and molecular-beam data.
For spherical molecules in the classical limit
(1(1+1) )=3(kT/2b), (19)
where b=n,2j2A and A is the moment of inertia. The
second moment (/)2) of the molecular-beam resonance
curve is given by5
(20) ratus constructed by Baker et al.,32 and are described
in considerable detail elsewhere. 5 It is only necessary
to mention here that the molecular-beam nuclear mag
netic resonance curves were all single symmetric enve
lopes roughly Gaussian in shape. The fluorine resonance
curves were centered at about 7.36 Mc/sec and were
approximately 350 kc/sec wide, whereas the hydrogen
curves were centered at 7.82 Mc/sec and had widths
near 40 kc/sec.
Gas-phase fluorine NMR absorption experiments at
low rf power were carried out at pressures of 5 and
10 atm on a 40-Mc/sec Varian V-4300 NMR spectrom
eter. All samples were studied in standard 5-mm-o.d.
NMR tubing. The reported pressures, which were de
termined from ideal gas considerations, are estimated
to be accurate to about 10%. The SFs, CF4, and CFsH
were Matheson gas samples while the CH2F2 was the
same as that used by Flynn and Baldeschwieler.15 The
CHsF was prepared according to the method of Edgell
and Parts.33 The Matheson gases contain nearly 1.5%
air or 0.3% O2 which can contribute to the nuclear
relaxation. A calculation similar to one by Abragam34
shows that 1 % O2 in a lO-atm sample contributes
about 0.2 cps to the linewidth. Since all samples were
condensed at liquid-N2 temperature and pumped on
for several minutes during preparation, it is highly
likely that very much less than 1 % O2 is contained in
the samples. Each gas-phase NMR curve for SFs, CF4,
and SiF4 was a single Lorentz envelope whose full
width at half-height O. was measured by sideband
techniques using a Hewlett-Packard 201CR audio oscil
lator and 521 C frequency counter. Table I gives the
experimental values of (/)2) from molecular-beam meas
urements and results of the NMR linewidth measure
ments for CF4, SiF4, and SF6 at 300oK. The linewidths
of the 19F NMR curves for the nonspherical molecules,
which show the expected proton splittings, are given
in Table IV.
IV. RESULTS
The kinetic collision frequencies Z of spherical mole
cules can be calculated from the expression
for CH4, CF4, SiF4, and SFs. Writing TmJ= NmJ/Zc,
T'mJ=N'mJ/Zc, and combining Eqs. (10) and (12) gives where (23)
1/T2= 871"2(V2){NmJ+~(LlC/CAv)2N'mJ 1(1/ Zc). (21)
But 1/T2=7ro" where 0, is the full width at half-height
of the gas-phase NMR resonance curve and thus in
the limit (LlC/C Av)2N'mJ«N mn we have
0,= 87r (v2)N mJ/ Zc,
which is in agreement with Eq. (8).
III. EXPERIMENTAL (22)
The molecular-beam experiments on SFs, CF4, SiF4,
CF3H, CH3F, and CH2F2 were carried out on the appa-is the collision frequency for ideal rigid spheres from
Hirschfelder, Curtiss, and Bird35 and Y is a correction
factor derived from the Chapman-Enskog theory of
gases.36 In the expression for the collision frequency,
32 M. R. Baker, H. M. Nelson, J. A. Leavitt, and N. F. Ramsey,
Phys. Rev. 121, 807 (1961).
33 F. Edgell and L. Parts, J. Am. Chern. Soc. 77, 4899 (1955).
34 Reference 31, p. 352.
35 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular
Theory of Gases and Liquids (John Wiley & Sons, Inc., New York,
1954), Chap. 1.
36 See Ref. 35, p. 635.
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:03COLLISION NARROWING 1447
TABLE II. Calculated values of the kinetic collision frequency Zc at 5 and 10 atm and 3000K f?r spherical molec'-!les; the Chapman
Enskog correction factor Y, the ideal-gas collision frequency Zo, and the molecular diameter (T are also mcluded.
Zc(5 atm) Zc(lO atm)
Molecule (sec-1 X 10-10) (secl X 10-1°) Y(5 atm)
CF4 3.223 6.553 1.016
SiF4 3.569 7.291 1.022
SF6 3.045 6.225 1.022
P is the pressure, u the molecular diameter, m t~e
molecular mass, k the Boltzmann constant, and T IS
the temperature in Kelvin degrees. The correction fac
tor Y is given by
Y = 1 +0.6250(b o/V) +0.2869(b o/V)2
+0.115(b o/V)3+ ' .. ,
where bo equals (jhrNu3 and is the seco~d virial coeffi
cient for rigid spheres of diameter u, V is the molar
volume, and N is Avagadro's number. Table II gives
the calculated values of Z for CF4, SiF4, and SFs at 5
and 10 atm and 300°K.
Using the experimental values of 0, and (v2) from
Table I and the calculated values of Zc from Table II,
NmJ has been computed for CF4, SiF4, and SFs at 5 and
10 atm from Eq. (8). The calculated Nm/s are included
in Table III and we notice that they are all very nearly
the same. Theories for rotational relaxation, which are
discussed below, predict that NmJ should have about
the same value for CF4, SiF4, and SFs.
If Eq. (8) is correct, the results for N mJ should be
good to approximately 20% since the pressures of the
samples in the NMR tubes were known only to 10%
accuracy and the values of (v2) are accurate only to
10% because of the difficulties in evaluating the second
moments; the wings of the molecular-beam resonance
curves introduce considerable uncertainty into second
moment determinations.
Gas-phase NMR widths 0, and molecular-beam sec
ond moments (v2) were also measured for the fluorine
and hydrogen nuclear resonances in CF3H, CH3F, and
CH2F2• Using the same procedure as for the spherical
molecules, N mJ can be determined for these nonspherical
molecules, although the results are less reliable since
spherical-top assumptions in the theory for NmJ are
not completely valid. In particular, the assumption
TABLE III. Calculated N mJ values for spherical molecules at 5
and 10 atm and 300oK.A
Molecule NmJ (5 atm) NmJ (10 atm) NmJ(Av)b
CF4 2.80±0.8 2.37±0.6 2.59±0.8
SiF4 2.81±0.4 2.93±0.4 2.87±O.4
SF6 2.47±0.8 3.24±0.6 2.86±0.8
a These values are determined from Eq. (8) using the data in Tables I and II.
b Average of 5-and IO-atm NmJ values. Zo(5 atm) Zo(10 atm) (T
Y(10 atm) (secl X 10-1°) (secl X 10-1°) (1)
1.033 3.172 6.344 4.662
1.044 3.492 6.984 5.100
1.045 2.979 5.957 5.128
that the zero frequency width ovo equals the high
frequency width 0/' for nonspherical molecules is not
necessarily valid and in fact the gas-phase hydrogen
resonance measurements indicate that ovO~O,h. This
point is discussed in more detail below. Table IV gives
the experimental values Ov and (v2) as well as Zc and
N mJ for CF3H, CH3F and CH2F2•
It can be seen that the values of NmAAv) for the
nonspherical molecules are less than N mJ (A v) for the
spherical molecules by a factor of approximately 2.
This is very reasonable since collisions between non
spherical molecules should be more effective at reorien
tation than collisions between spherical molecules. The
details of the collision mechanism which changes J are
examined in the next section.
The gas-phase hydrogen resonances in CF3H, CH3F,
and CH2F2 are broadened also by the fluorine spin
rotation interaction because of the electron-coupled
spin-spin interaction hJHFIHIF. The hydrogen spin
rotation interaction is small compared to the fluorine
interaction and therefore does not contribute noticeably
to the hydrogen resonance linewidth. The molecular
beam hydrogen resonance curves have widths of ap
proximately 40 kc/sec with a natural broadening (the
apparatus resolution limit) of about 25 kc/sec. There
fore the interaction width is roughly between 15 and
40 kc/sec with a corresponding second-moment spread
of 100 to 300 (kc/sec)2 (computed by using the relation
ship between the second moment and half-width for a
Gaussian curve). These second moments are lower than
the fluorine resonance second moments by a factor of
10-100 and from Eq. (8) the gas-phase NMR line
width contributions are lower by the same factor.
Therefore the hydrogen spin-rotation interaction does
not contribute appreciably to the hydrogen NMR line
width.
The first-order energy levels of the nuclear spin sys
tem for CH3F, CF3H, and CH2F2 in a uniform magnetic
field Ho are given by
E= -hl'nHomH-'1i:yFHomF+hJHFmFmH
and the transition frequencies for LlmH= 1, LlmF=O are
(24)
In these equations I'H and I'F are the gyromagnetic
ratios for hydrogen and fluorine respectively, mF is the
projection of the total fluorine nuclear spin h along
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TABLE IV. Experimental gas-phase fluorine NMR Iinewidths 5, and molecular-beam fluorine resonance second moments (v") for CHaF,
CF3H, and CH2F2 at 300oK; calculated kinetic collision frequencies Zo and values of NmJ and tT are also included.
5,(5 atm) 5,(10 atm) (v2) Zo(5 atm)
Molecule (cps) (cps) (kc/sec)2 (secl X 10-10)
CH3F 1.8b 0.7b 1 110 3.77
CFaH 35.2 16.2 30000 3.62
CH2F2 22.0 12.5 27 000 3.51
• The molecular diameters are estimated by a comparison with CF. and CR.
diameters.
the direction of Ho, and JHF is a constant giving the
magnitude of the scalar part of the electron-coupled
spin-spin interaction between equivalent hydrogen and
fluorine spins. Equation (23) then predicts a sharp
spectral line for each value of mF. However, mF for a
given molecule changes on the average many times
during an absorption measurement because of the re
laxation of the fluorine nuclei by the spin-rotation
interaction. Abragam37 shows that as the rate of change
of mF increases from zero to a rate just less than JHF,
the spectrum changes from sharp lines to broadened
lines. For CF3H, CH3F, and CH2F2 the rate of change
of mF is related to the transition probabilities between
energy levels of the nuclear spin system which in turn
are dependent on the fluorine spin-rotation interaction
and molecular collision rate among other things. Flynn
and Baldeschwieler15,19 have considered this problem
in considerable detail and thus it is not pursued further
here. Table V gives the measured gas-phase hydrogen
resonance widths for CF3H, CH3F, and CH2F2 which
for each molecule are nearly inversely proportional to
pressure (and hence collision frequency) as well as
being correlated to the fluorine resonance widths (that
is, the hydrogen width appears to be about 20%-30%
less than the fluorine width for CF3H and CH2F2; the
fluorine signal for CH3F was too weak to allow a
comparison). The difference in these widths is most
probably due to the zero frequency contribution to
the fluorine resonance. If this is true then o.o~:::~40.h
and the corresponding values of NmJ for CF3H, CH3F,
and CH2F2 are lower by a factor of 2 from the values
calculated above. This correspondence between Hand
TABLE V. Experimental gas-phase hydrogen NMR widths 5, at 5
and 10 atm and 3000K for CH3F, CF,H, and CH2F2.··b
Molecule 5,(5 atm) 5,(10 atm)
(cps) (cps)
CH,F 2.3±0.2 1.1±0.2
CF,H 29.8±4.0 12.4±1.8
CH2F2 17.3±1.5c 9.2±0.5
a All measurements made with a Varian A-60 spectrometer.
b The electron-coupled spin-spin interaction between Hand F causes a split
ting of the resonance lines. Widths are taken from the individual split lines.
C Taken from Ref. 15.
37 See Ref. 31,~Chap.!I1. Zo(10 atm) u8
(secl X 10-10) (1) NmJ (5 atm) NmJ (10 atm) NmJ (Av)
7.54 4.00 2.43 1.89 2.16
7.24 4.70 1.69 1.56 1.63
7.02 4.30 1.14 1.29 1.22
b These values are inferred from the hydrogen resonance widths since the
fluorine signal was too weak to obtain accurate data.
F resonance widths allows an estimation of the fluorine
resonance width in CH3F since it could not be meas
ured accurately.
The spectrometer resoltuion limit for all of the gas
phase NMR measurements is 0.5 cps at best; and
therefore, in estimating the interaction width for the
CHaF fluorine resonance, 0.5 cps is subtracted from
the hydrogen width which renders the NmJ result for
CHaF relatively inaccurate. The error estimates in
Table V were made from the consistency of different
runs.
V. DISCUSSION
The number of collisions88 necessary to change mJ
for the spherical molecules CF4, SiF4, and SFs appears
to be approximately three. Existing theories and ex
periments concerned with rotational relaxation agree
remarkably well with the results quoted here, lending
further support to the present theory which relates
molecular-beam second moments to gas-phase NMR
linewidths.
Classically the change in the angular momentum J
of a molecule can be calculated from dJ/dt=N, where
N is the torque acting on the molecule. For radial
forces N = 0 and therefore no change in angular mo
mentum occurs. For perfectly spherical molecules the
long-range electric and dispersion interaction potentials
are isotropic, giving rise to radial forces only and conse
quently no rotational relaxation mechanism. However
in their classical calculations of the rotational relaxa
tion for spherical molecules, Wang Chang and Uhlen-
TABLE VI. Values of N J for CF4, SiF4, and SFs calculated from
Eq. (26); the moment of inertia I, mass m, and parameter K
are also included.
I m
Molecule [{l)2'amuJ (amu) K NJ
CF4 89.13 88 0.184 5.72
SiF4 123.0 104 0.182 5.76
SFs 187.3 146 0.195 5.49
38 Only the hard collisions which produce large changes in mJ
and consequently a significant motion narrowing modulation are
considered.
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:03COLLISION NARROWING 1449
TABLE VII. Values of N J for CF4, SiF4, and SF6 calculated from Eq. (27) at 300oK, also included are O'Neal and Brokaw N J deter
minations' as well as N mJ values from the present work.
E/k NJ NJ
Molecule (OK)b exp(E/kT) (theory) (experimental) a NmJo
CF4 152.5 1.66 3.44 3.03 2.59±O.8
SiF4 180.0 1.82 3.17 2.87±0.4
SF6 200.9 1.95 2.82 2.80 2.86±0.8
• See Ref. 43. b See Ref. 35, p. 1111. • Present determinations from Table III.
beck,39 and Sather and Dahler40•41 assume a frictional
force between molecules. This force occurs at the point
of contact of colliding molecules and thus at a molecular
separation where electron clouds begin to overlap; in
this sense the frictional force is a short-range inter
action and the friction arises from electron-cloud in
teraction. In the limit of perfectly rough spheres, finite
rotational relaxation times are obtained while for per
fectly smooth spheres (no friction) no relaxation occurs.
For colliding identical rough spheres Sather and
Dahler40 show that the rate of approach of the rota
tional temperature Tr to its equilibrium value T t (the
translational temperature) is given by
dTr/dt= -(Tr-Tt)/T,
where the relaxation time T is obtained from
I/T= [16n(r2j3(K + 1) ](1TkT t/m)lB(p). (25)
In Eq. (25) n is the number of molecules per cubic centi
meter, u is the molecular diameter, B(p) =pK/(I+K)
(p= 0 for perfectly smooth spheres and p= 1 for per
fectly rough spheres), and K =41/mu2 (I is the moment
of inertia and m is the molecular mass.) The average
number of collisions42 NJ which are necessary to estab
lish thermal equilibrium between the translational and
rotational degrees of freedom is then approximately
NJ=ZOT where Zo equals 4nu2(-lrkTt/m)! and is the
average collision rate. Combining this result with Eq.
(25) gives in the limit p= 1
NJ=![ (1 + K) / B(p)]=![ (1 +K)2/K]. (26)
Wang Chang and Uhlenbeck39 derive the same result
for rough rigid spheres. Table VI gives values of NJ
for CF4, SiF4, and SF6 computed from Eq. (26).
Equation (26) was derived from a model of perfectly
rigid (no attractive potential) rough spheres and gives
values of NJ about twice the values of NmJ from nu
clear relaxation. In a more realistic theory Sather and
Dahler41 have assumed an attractive square-well poten
tial of depth E (the Lennard-Jones 6-12 potential pa
rameter) with a repulsive rough inner core. For this
3Q c. S. Wang Chang and G. E. Uhlenbeck, Report CM-681,
Project NOrd 7924, University of Michigan (1951).
40 N. F. Sather and J. S. Dahler, J. Chern. Phys. 35, 2029 (1961).
41 N. F. Sather and J. S. Dahler, J. Chern. Phys. 37.1947 (1962).
.2 N J is roughly the average number of collisions for a molecule
to change its J state. model they derive
NJ=![(I+K)2/K][l/g(u)], (27)
where g(u) =exp(E/kT) is the value of the radial dis
tribution function at u in the limit of low density. The
frictional force still occurs only on the rough inner core
while the square well ensures that a greater fraction
of the total number of molecules will be involved in
short-range collisions since the molecules are attracted
by the potential well. Table VII gives the values of
NJ computed from Eq. (27) and these seem to agree
quite well with the N mJ determinations in Table III.
O'Neal and Brokaw43 have obtained experimental
values of NJ for CF4 and SF6 by thermal conductivity
measurements. Their NJ values are 3.03 and 2.80 for
CF4 and SFa, respectively; these are included in Table
VII.
The agreement between the values of N mJ in Table
III and NJ from the other determinations is certainly
close, perhaps even fortuitous. The quantity N mJ is
the average number of collisions for sizeable changes
in mJ whereas N J is the average number of collisions
to effect the transition J to J'. If the transition mJ
to mJ' occurs when J changes much more often than
it occurs when J does not change, then Nm/"'NJ.
The theories we have considered use rough-sphere
models to describe rotational relaxation. It is the rough
ness of the spherical surface which is responsible for
the exchange of translational and rotational energy
because a torque can be transmitted to the sphere and
thus changes in the angular momentum and rotational
energy can occur. It can be argued,44 using ~J =N~t,
that in the classical limit the number of collisions in
which J changes its orientation without changing its
magnitude is considerably less than the number of
collisions in which it changes both simultaneously. The
argument is not rigorous but it makes the assumption
N m/'" N J for the case of a classical collision analysis
of rough spheres seem reasonable.
Discussions with Purcell4 have brought to light the
fact that a much more realistic theoretical approach
to rotational relaxation of spheres could be made by
doing a Monte Carlo analysis of collisions between
43 C. O'Neal and R. S. Brokaw, Phys. Fluids 6, 1675 (1963).
44 L. M. Crapo, Ph.D. thesis, Harvard University, 1964 (un
published).
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132.248.9.8 On: Fri, 19 Dec 2014 21:24:031450 L. M. CRAPO AND G. W. FLYNN
frictionless tetrahedral dumbbells in which the amount
of dumbbell curvature could be varied. The theories
above are arbitrary with respect to the roughness pa
rameter of knobiness of the sphere and therefore do
not unambiguously describe the collision mechanics.
It is also worth mentioning that 1/T2 pulse experi
ment determinations can be made which are much
more accurate than the NMR linewidths reported in
this paper. Accurate temperature-dependent 1/T2 meas
urements could give further information about the colli
sion processes. It should be possible to investigate the collision prop
erties of other molecules, e.g., BF3, XeF4, PF6, PF3,
POF3, and the fiuorosilanes, by the method used in
this study.
ACKNOWLEDGMENTS
We wish to thank Professor Norman F. Ramsey,
Professor John D. Baldeschwieler, and Dr. C. H. An
derson for helpful discussions and continuing interest
throughout this work. In addition, we thank Professor
Myer Bloom for valuable criticisms.
THE JOURNAL OF CHEMICAL PHYSICS VOLUME 43, NUMBER 5 1 SEPTEMBER 1965
Charge-Transfer-Controlled Vaporization of Cadmium Sulfide Single Crystals. I.
Effect of Light on the Evaporation Rate of the (0001) Face
G. A. SOMOR]AI AND J. E. LEsTER*
Department of Chemistry, University of California, Inorganic Materials Research Division, Lawrence Radiation Laboratory
Berkeley, California
(Received 5 May 1965)
Light of greater-than-band-gap energy was found to change markedly the vacuum evaporation rate of the
(0001) face of cadmium sulfide single crystals. Evaporation temperatures of 680°-740°C and light intensities
of 5.0XlOL2.0X105 p.W/cm2 were used. The results were interpreted assuming that charge transfer is
the rate-determining step in the sequence of vaporization surface reactions. An evaporation mechanism in
terms of charge transfer has been proposed. Light (1) changes the free-carrier concentrations at the vaporiz
ing surface, and (2) under proper conditions, changes the composition of the crystals. High-resistivity
crystals showed a fivefold increase of their evaporation rate under illumination due to the increase by light
of both electron and hole concentrations. In low-resistivity crystals, in which illumination can only sig
nificantly increase the minority free-carrier concentration above the dark equilibrium value, effects, which
are due to the change of the crystal composition, dominated.
INTRODUCTION
THE evaporation mechanism of cadmium sulfide
single crystals has been studied recently by several
techniques. Dependence of the evaporation rate on
temperature/.2 surface concentration,! crystal orienta
tion,3 and minute excesses of the crystal constituents,
cadmium and sulfur, in the crystal lattice,4 has been
investigated. In most of the investigations the vacuum
evaporation rate of one face of the single crystal has
been studied. From these results, a tentative mecha
nism for the rate-determining surface reaction has been
proposed! and the bulk diffusion rate of sulfur vacancies
has been measured.41t was found that the evaporation
* National Science Foundation Graduate Fellow.
1 G. A. Somorjai and D. W. Jepsen, J. Chern. Phys. 41, 1389
(1964).
2 G. A. Somorjai, Condensation and Evaporation of Solids,
edited by E. Rutner, P. Goldfinger, and J. P. Hirth (Gordon and
Breach Science Publishers, New York, 1964).
3 G. A. Sornorjai and N. R. Sternple, J. Appl. Phys. 35, 3398
(1964) .
4 G. A. Somorjai and D. W. Jepsen, J. Chern. Phys. 41, 1394
(1964). rate is a sensitive function of small excesses of cadmium
or sulfur in the cadmium sulfide crystal lattice. The
solubility of either cadmium or sulfur in cadmium
sulfide is so low,6 however, that, by heating the crystals
at the highest temperature and dopant pressure
(1200°C, 20 atm cadmium or sulfur), less than 1 in
103 surface atoms/cm2 can be substituted.4 For the
presence of such small concentrations of surface sites
to have such a drastic influence on the evaporation
rate makes it improbable that any atomic-surface step
could playa rate-controlling role in the vaporization
process.6 A small excess of either cadmium or sulfur in
cadmium sulfide, however, changes the equilibrium
free-carrier concentration of the crystals by more than
10 orders of magnitude by shifting the Fermi level;
thus, it also drastically changes the free-carrier con
centration at the surface.7 It was then thought that the
Ii F. A. Kroger, H. J. Vink, and J. von der Boorngaard, Z. Physik
Chern. 203, 1 (1954).
6 W. K. Burton, N. Cabrera, and F. C. Frank, Phil. Trans. Roy.
Soc. (London) A243, 299 (1951).
7 G. A. Somorjai, Surface Sci. 2, 298 (1964).
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1.1702560.pdf | Electrical Properties of Single Crystals of Indium Oxide
R. L. Weiher
Citation: Journal of Applied Physics 33, 2834 (1962); doi: 10.1063/1.1702560
View online: http://dx.doi.org/10.1063/1.1702560
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to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 20:24:242834 F. H. HORN
what is going on. For those boules containing small
amounts of cobalt the oxygen treatment is, as expected,
too oxidizing and there is evidence for a small amount
of Fe203 formation. The CoO is clearly visible as a
second phase in metallographic specimens of sections
of boules as grown containing more cobalt than expressed
by the ratio 0.4, Fig. 3(a). Metallographically prepared
sections of these boules after the heat treatment in
oxygen show large sections of single phase, single crystal
[Fig. 3(b)]. These regions are shown to be single crystal
by their Laue patterns.
Regardless of composition, all boules as grown were
electrically n type. Following the heat treatment in
oxygen those crystals of Co/Fe composition less than
0.5 remained n type, but those for which Co/Fe is
greater than 0.5 converted to p type in agreement with
the results reported by Jonker.6
From the reported investigations it appears that
single-crystal cobalt ferrites can be prepared from two
phase material if the phases are grown coherent in
structure and if crystal preparation is followed by an
oxidizing (or reducing) procedure performed at a high
enough temperature to permit diffusion. This suggests
that the problem of nonuniformity of composition with respect to Co :F e as a function of length of the boules due
to CoO enrichment from the melt into the solid during
growth might be taken care of by a high temperature
oxidation of longer duration than used in these studies.
Uniformity of metal ion distribution may also be
achieved, in principle, by using a zone leveling procedure
for crystal growth. Control of the vacancy concentration
requires the use of the known appropriate temperature
oxidation conditions for the particular composition of
cobalt ferrite prepared.
It would appear that the observations reported may
be applied generally to the preparation of crystalline
oxides grown from metals in which the atmosphere
employed causes a dissociation of the solid. The require
ment for the possibility of recovering single-phase single
crystal is that the phases grown from the melt are
coherent and oriented with respect to the oxygen lattice
in the grown crystal and that conditions for an anneal
after growth permit diffusion of the metal ions in the
solid to the final phase desired.
This work is abstracted from a study on ferrites
performed in cooperation with G. A. Slack and W. E.
Engeler. I wish to acknowledge the work of P. Friguletto
in forming, preparing, and annealing crystals.
JOURNAL OF APPLIED PHYSICS VOLUME 33. NUMBER 9 SEPTEMBER 1962
Electrical Properties of Single Crystals of Indium Oxide
R. L. WEIHER
Central Research Laboratories, Minnesota Mining &-Manufacturing Company, St. Paul, Minnesota
(Received February 23, 1962)
An investigation of electrical properties of indium oxide single crystals has been made. Indium oxide has
been found to be a n-type excess semiconductor over a wide temperature range. The electrical conductivity at
room temperature is of the order of 10 fl cm-.I and the mobility is approximately 160 cm2 V-seCI• The tem
perature dependence of the mobility has been quantitatively interpreted in terms of lattice and ionized
impurity scattering. The donor ionization energy has been found to decrease with increasing impurity con
centrations. High "apparent intrinsic" conductivity with an activation energy of 1.55 eV has been observed
at elevated temperatures.
I. INTRODUCTION
ONE of the first investigations of indium oxide
(In203) as a semiconductor was made by
Rupprecht! on thin vapor-coated films. Because of the
polycrystalline nature of the films, most of the phe
nomena he observed can be attributed to surface and
barrier effects. Although studies as the above contribute
much to the over-all knowledge of a material, studies of
single crystals are required to observe true bulk effects.
It is the purpose of this paper to present an investiga
tion of some of the electrical properties of single crystals
of indium oxide recently grown in this laboratory. The
investigation consists of low temperature conductivity
and Hall effect measurements along with conductivity
I G. Rupprecht, Z. Physik 139, 504 (1954). measurements at elevated temperatures. Analysis of the
presented data reveals mechanisms of conduction
common among the many semiconductors previously
investigated.
II. MATERIAL PREPARATION
Single crystals of indium oxide were grown from the
vapor phase of indium metal and ambient oxygen.
Approximately equal amounts of carbon and indium
metal were mixed in a porcelain crucible, loosely
covered, and heated in a furnace at lOOO°C for 24 h.
Pale yellow, needle-shaped crystals approximately
O.SXO.SXS.O mm in size grew on the walls and cover
of the crucible. The crystals were identified as In20a by
x-ray diffraction. Square and hexagonal cross sections
were obtained corresponding to growth in the [l00J and
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to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 20:24:24ELECTRICAL PROPERTIES OF SINGLE CRYSTAL In203 2835
[111 J crystal directions, respectively. The crystal
structure of indium oxide is body-centered cubic. The
crystals of growth in the [l11J direction were used for
the electrical measurements discussed herein. Spectro
chemical analysis indicated that silicon and molyb
denum were the only impurities present in concentra
tions slightly greater than one part in 106• However,
because of the small quantity of crystals available the
absolute value of impurity content could not be ascer
tained for the four crystals discussed herein. Electrical
measurements indicate impurity concentrations of the
order of 1018 per cc which is much greater than the
estimate from spectrochemical analysis. It is on this
basis that we conclude that the electrically active
impurities are probably a stoichiometric excess of
indium or effective imperfections such as oxygen
vacancies.
III. LOW-TEMPERATURE ELECTRICAL
MEASUREMENTS
A. Experimental Apparatus
The sample holder used for electrical conductivity
and Hall effect measurements at low temperatures is
shown in Fig. 1. The top probe assembly is spring loaded
to simplify crystal removal, to hold the crystal in place,
and to maintain sufficient pressure for probe contacts.
The current leads were attached to the ends of the
crystals by electrolytically reducing the ends of the
crystals to indium metal, tying No. 38 copper wire over
the indium metal and then painting over the indium
copper contacts with silver paste. This method gave
strong, low resistance contacts.
The crystal holder was enclosed in a copper box to
maintain a uniform temperature and act as a radiation
shield. The complete assembly of apparatus is shown in
Fig. 2. The entire apparatus was maintained in a
vacuum of approximately 0.5 IJ. of mercury. The cooling
of the sample was accomplished by immersing the
cooling substage in liquid nitrogen until the desired
temperature was obtained. The magnet for Hall
measurements was raised and lowered by a pulley
system controlled outside the vacuum.
LEAD
LEA
THERI'IOCOUP
6-UAD D
Li LUtl"
: COI'PlRJ-r-1 : I r--r
-/MTAL :
: lEAD
L EAD
HU.I'IOCOUPLE
&" LEAD T
S PIUNG'" COPPEll COPPEll -SPill NG
c;:::~ "'ICA/ f::;~
l CO"11l I
I I
FIG.!. Low-temperature sample holder. O'~ERVIITIOM PORT
C;;=====dr:===:::;;;:J..--...:AWI'IINUM PLIITE
LUCITE
C'tLINDER.
FIG. 2. Complete low-temperature experimental apparatus.
A Keithley 200B electrometer was used to measure
the potential drop across the conductivity probes and a
Weston 430 ammeter was used to measure the current.
The Hall voltage was measured with a Cary 31 vibrating
reed electrometer. The temperature was measured with
copper-Constantan thermocouples and a Leeds and
Northrup potentiometer.
B. Electrical Conductivity
The largest variation in electrical conductivity at
room temperature for the crystals as-grown was from
approximately 5 to 50 Q cm-J• Figure 3 depicts the con
ductivity of four such crystals as a function of tempera
ture from room temperature to 90°K. Four approxi
mately parallel curves are obtained with maxima at
about -100°C and no specific activation energy in this
temperature range. The curves reflect the fast decreas-
1~r-_~~_~~~°r-~-~IOO~T~:~(~-~I~~~-~17~5 __ -~ln~ __ _ • • 7 •
4
10' • ,
7 •
4 ~~'H'~ /~~H' •
H-e
H-7
o I 1 S 4 , G 7 &.~ W " n ~ ~. IO'/T"K .
FIG. 3. Electrical conductivity vs temperature
for four single crystals of indium oxide.
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to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 20:24:242836 R. L. WEIHER
ing mobility in the near "saturation" range at higher
temperatures and the changing number of ionized
donors in the lower temperature range.
C. Carrier Concentration Analysis
Hall measurements on the single crystals indicate
n-type conductivity. It will be assumed that there is no
impurity band conduction so that all the conduction is
in the conduction band. If one assumes single band
conduction, spherical energy surfaces, noninteracting
carriers, etc., one can show that the Hall mobility
differs from the microscopic mobility by some factor
which depends upon the type of scattering mechanism
involved. Because the validity of the above assump
tions, along with other assumptions required for a direct
conversion from Hall mobility to microscopic mobility,
is not known, the more accepted conversion (3'71'/8) will
be used; that is, the carrier concentration was calculated
from
n= (311/8) (a/eJJ.H)' (1)
Shown in Fig. 4 are four curves of carrier concentra
tion as a function of temperature calculated from Eq.
(1). The donor ionization energies cannot be obtained
directly from these curves for two reasons: (1) Because
of the weak temperature dependence of the carrier
concentrations, the density-of-states (N c) cannot be
considered constant (2), degeneracy of the donor level
must be considered. The donor ionization energy can,
however, be determined with the type of analysis
~.'r-~~~~*~~O _____ T~~T_:_C_-~I~r-~-~lnL-~-~I'~i __ ~ , ,
1 ,
4
II-a
11'1
o III 1\ 1 11 14
FIG. 4. Carrier concentrations vs temperature
for four single crystals of indium oxide. _ * 0 -100 T:C. -ISO -17S -I'S' ~.r--T~r-~---T~--~---T~~~-' 1
1 ,
~
4
FIG. 5. Determinations of donor ionization energies
for four single crystals of indium oxide.
presented by Hutson2 in which degeneracy of the donor
level was considered. Assuming only one major donor
level (fd) in the vicinity of the Fermi level (f[) and
degenerate statistics for this donor level, the number of
un-ionized donors (nd) can be written as
nd=Nd/[1+g-1 exp(fd-~f)/kTJ, (2)
where JV d= nd+n. The factor g is the spin degeneracy of
the donor states. If nondegeneracy is assumed for the
carrier population in the conduction band, the concen
tration of conduction band electrons can be expressed
by,
n=Ne eXp(fj-fe)/kT, (3)
where Nc=2(27rmnkT/h2)! and mn=density-of-states
effective mass. The value of mn, which determines the
validity of the use of Eq. (3) in this case, was approxi
mated by the conjunctive use of Hall and thermo
electric power measurements to be ",-,O.SSm, where m
is the free electron mass. The effective density-of-states
using mn=O.SSm is depicted by the upper curve of
Fig. 4 from which values of Ne/n tend to validate
Eq. (3). Eliminating nd and Ej from Eqs. (2) and (3),
one obtains the expression
n2/(Nd-n)Bc= (mn/m)lg-l exp(~d-fc)/kT, (4)
where Be is the effective density-of-states for mn=m.
Therefore, if one plots the logarithm of the left-hand
side of Eq. (4) vs ljToK, one should obtain a straight
2 A. R. Hutson, Phys. Rev. 108, 222 (1957).
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line with a slope of (~d-~c)/k with an intercept, as
1fT ~ 0, equal to (mn/m)~g-I.
The results of this treatment on the four single
crystals of indium oxide using Eq. (4) are depicted in
Fig. S. The only parameter in n2/ (N d-n)B c not directly
known is N d, which was arbitrarily chosen to give the
best straight line fit at the lower temperatures. At
higher temperatures, when n becomes comparable in
magnitude to N d, small experimental errors in n have a
pronounced effect and values may deviate slightly from
a straight line. Attempts at introducing slight com
pensation resulted in curves which deviated from a
straight line greater than if one assumes no compensa
tion. A summary of the values obtained for iY d and
~d-Ec by this analysis is given in Table I.
The dependence of the ionization energy on the donor
concentration shown in Table I is an effect common
with semiconductors. When impurities are present in
sufficient quantities so that there are electrostatic
interactions between centers and other crystal imper
fections, the simple hydrogen-like model is not realized
and the ionization energy decreases with increasing
impurity concentration. This decrease in ionization
energy with increasing impurity concentration is usually
explained as a broadening of the impurity level due to
overlap of wave functions of the hydrogen-like centers.
Pearson and Bardeen3 have suggested a model for the
decrease in ionization energy with increasing donor
concentrations by considering the electrostatic attrac
tion of donors for an electron which has escaped from
its own donor. This consideration yields the expression
(5)
where Ed is the donor ionization energy for a given
concentration of donors, EdO is the donor ionization
energy when the number of donors approaches zero, a is
a constant, and N d is the number of donors. The data
in Table I plotted as shown in Fig. 6 indicate good
agreement with this model, at least in our very limited
range of donor concentrations. The donor ionization
energy as a function of donor concentration shown in
Fig. 6 is then calculated to be,
~d=0.093 eV-S.15XlO-sNa i eV. (6)
These data predict that the donor ionization energy
should effectively go to zero for a donor concentration
of 1.4SX 1018 cm-3•
T ABLE I. Summary of donor ionization energies and donor
concentrations for four single crystals of indium oxide.
Crystal No. (Ed-Ec) eV Nd cm-a
H-6 0.0085 1.09 X 1018
H-9 0.0166 8.40X1C17
H-8 0.0210 6.50XlO17
H-7 0.0280 4.95XlO17
a G. L. Pearson and ]. Bardeen, Phys. Rev. 75, 865 (1949). .~~----------~--------------~~
.011
.07
.06
.04
.01 €d(Nd). £01.. -cr Nd. Va
fd.o• .09~ .v
ex '" &.IS" 10·'.v em
O~O--~~--+4--~,~--71--~~~~~llr-~14 Nl1 .. 1o-I
FIG. 6. Donor ionization energies vs impurity concentrations.
It is seen in Fig. 5 that the quantity (mn/m)!g-I
given by the intercept is equal to 0.205. If one uses the
approximate value of effective mass (mn=0.55m),
previously discussed, a value of 2.0 is obtained for the
g factor. This suggests that the impurity level can be
described as a simple donor state with one electron of
either spin. This is, however, consistent with assump
tions implicit in Eq. (2) for which the excited states of
the impurities were assumed negligible. If one is to
consider the excited states, Eq. (2) should be rewritten
as given by Shifrin,4
'" na=NaL [1+gr-I exp(Er-~f)jkTJ-r, (7)
r=O
where Er and gr are the energy and degeneracy of the rth
state, respectively. The assumption in Eq. (2) is,
therefore, to neglect the terms other than the r=O term
for which g= 2 and Er= Ed for a simple donor with one
electron with either spin.
D. Mobility Analysis
The polarity of the Hall potential indicates n-type
conductivity, so the mobility under consideration in
this section is for the electrons in the conduction band.
The marks in Fig. 7 represent the experimental Hall
mobilities as a function of temperature, from room
temperature to 90oK, for four single crystals. Included
in Fig. 7 are four solid lines represented by the expres-
4 K. S. Shifrin, Zhur. Tekh. Fiz. 14, 43 (1944).
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to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 20:24:242838 R. L. WEIHER
.. 0,..-------------------.
'3
1\
o "' H-"
11. .. H-7
c 1-1-6
Il. = H-9
, 3 o
1~~~-----~~--.If.~~~~~~tS~O~300~'~~~~~
I'K
FIG. 7. Carrier mobilities vs temperature for four single
crystals of indium oxide.
sion,
/J-H=ABTJ/(A+BP), (8)
where A is a constant for all curves, and B is a constant
for a given curve but varies with the particular curve.
It is seen that the solid lines represented by Eq. (8) fit
the experimental mobility data extremely well.
Equation (8) can be interpreted as a combination of
acoustical mode lattice scattering and ionized impurity
scattering if one assumes that the total resistivity (p)
is simply a sum of the resistivities due the various
scattering mechanisms so that,
(9)
or,
(10)
Bardeen and Shockley5 have shown that for acoustical
mode lattice scattering, the mobility can be expressed
as,
(11)
where (22 is the average longitudinal elastic constant
and ~ln is the shift of the edge of the band per unit
dilation. (22 and ~)n are usually considered to be tem-
TABLE II. Summary of the experimentally determined
constants of Eqs. (12) and (14).
Crystal No. Ni cm-' B BNi A
H-6 l.09X 10'• 0.344 3.8X 1(}l7 8.52XlOD
H-7 4.95XI017 1.000 4.9XI017 8.52X10·
H-8 6.50X10 '7 0.758 4.9X1017 8.52X10·
H-9 8.40X1017 0.480 4.0X1017 8.52X10·
6 J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950). perature independent, thus
(12)
where A is a constant.
Conwell and Weisskopf6 have shown that for ionized
impurity scattering, the mobility can be expressed as,
27/2K2(kT)~ 1 -----X ,
7r~(mm)!e3N; In[1 + (3KkT /eW;l)2J (13)
where Ni is the concentration of impurities and K is the
dielectric constant. The logarithmic term is usually
considered to be a slow varying function of temperature
and therefore the mobility due to ionized impurities can
be approximated as,
(14)
where B is a constant. Summing the reciprocals of
Egs. (12) and (14), as previously assumed, yields
Eg. (8) for which the experimental mobilities fit.
The experimental value of the constant A in Eq. (12)
was found to be 8.52 X 105 and constant for all the
crystals. It is seen from Eqs. (13) and (14) that the
constant B should be approximately proportional to
N;-l or the product BN; should be approximately a
constant. The values obtained experimentally, shown
in Table H, show reasonable agreement between theory
and experiment.
The apparent agreement between experiment and
theory is probably better than should be expected from
the approximations and assumptions implicit in Egs.
(1), (10), and (14). For instance, ConwelF has shown
that simply summing the reciprocals of the individual
mobilities [Eq. (10)J is not accurate since the relaxation
times for the various scattering mechanisms are
dependent upon energy in different ways. It is therefore
concluded that although Eq. (8) fits the experimental
data extremely well, the interpretation of Eq. (8) could
be somewhat in error.
ELECTRICAL CONDUCTIVITY AT
ELEVATED TEMPERATURES
Measurements of the electrical conductivity of the
single crystals were extended from room temperature
to 1S00°C using a spring loaded, four-probe apparatus
similar to that shown in Fig. 1. No Hall measurements
were made at the elevated temperatures. The measure
ments were made in a normal atmosphere (P02",0.2
atm) in which no detectable sublimation took place.
A typical curve of the electrical conductivity as a
function of temperature, from approximately 1700° to
900K, is shown in Fig. 8. The lower portion of the
temperature range was discussed in Sec. HIB. What is
of greater interest here, is the dramatic increase in
6 E. Conwell and V. Weisskopf, Phys. Rev. 77, 388 (1950).
7 E. Conwell, Proc. IRE 40, 1327 (1952).
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to ] IP: 128.123.44.23 On: Sun, 21 Dec 2014 20:24:24E LEe T RIC ALP R 0 PER TIE S 0 F SIN G LEe R Y S TAL In, 0 , 2839
100 • T,·C -100 -I -I'll -185
CR.'mAl H -\ \
..,
FIG. 8. Electrical conductivity vs temperature from 90° to 1700oK.
conductivity at high temperatures, which resembles
intrinsic conductivity. The electrical conductivity of
various crystals coincide at high temperatures with a
common "activation energy" (~A""1.55 eV) as shown
in Fig. 9. The electrical conductivity can thus be
represented, in this temperature range, as
0'= 0'0 exp-1.55/ kT. (15)
It should probably be mentioned that crystal HZ-13 of
Fig. 9 is of lower conductivity due to the addition of a
small amount of zinc added during growth.
The interpretation of the "apparent intrinsic" con
ductivity is difficult at this time because at least two
different mechanisms are capable of giving plausible
explanations: (1) Band-to-band transitions, and (2)
dissociation of the compound. If one considers band-to
band transitions with the assumptions that the density
of-states is proportional to Ti, the mobility is propor
tional to T-i, and that the band gap varies linearly with
temperature, one can express the conductivity as,
0'= const exp~gol2kT, (16)
where ~gO is the band gap at absolute zero. This con
sideration yields a band gap of 3.1 eV for indium oxide
which is in close agreement with optical determinations.
Rupprechtl has reported the value of 3.5 eV for the band
gap of indium oxide as determined from optical trans
mission measurements on thin vapor coated films.
However, preliminary optical measurements being IIDD 1100 100 100 _ T:C '100 100
IO'r-----!~;::..-::;:....:=;:-....;:;:~-....::;:-----!:r_--.....,
B
•
4
10'
B
•
;:.... 4
E ...
cI ~
b
10' • •
4
1
10'
0 X .. H-I!
o • H-It •• H-13
A • H-14
u
FIG. 9. Electrical conductivity vs temperature from room
temperature to 17000K for four single crystals of indium oxide.
conducted in this laboratory on thin crystal plates
indicate a band gap nearer 3.1 eV_
Although the above consideration gives quite a
convincing argument for band-to-band transitions,
other facts remain which indicate that the "apparent
intrinsic" conductivity is due to a dissociation of the
compound. Electrical measurements on thin films of
indium oxide by Rupprechtl show that at a constant
temperature, above 500°C, the conductivity is depend
ent on oxygen pressure (0' cc P02-o·19). Rupprecht
suggests the dissociation reaction,
In20aP2 In+3+6e-+! O2, (17)
which, with the aid of the law of mass action, predicts
that the conductivity should be proportional to
P02-O·l875. From the agreement between the experi
mental and calculated oxygen pressure dependences, it
is evident that dissociation of the compound must at
least be considered as a probable mechanism for the
"apparent intrinsic" conductivity because the oxygen
pressure dependence seems unlikely for true intrinsic
conductivity.
ACKNOWLEDGMENTS
The author wishes to thank his colleagues in this
laboratory, especially G. K. Lindeberg, for helpful
discussions and suggestions. The author also wishes to
thank B. Gale Dick of the University of Utah for his
valuable consultation, especially in theory. He is also
indebted to F. A. Hamm for making this work possible.
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1.1735322.pdf | Nature of Bombardment Damage and Energy Levels in Semiconductors
J. H. Crawford Jr. and J. W. Cleland
Citation: Journal of Applied Physics 30, 1319 (1959); doi: 10.1063/1.1735322
View online: http://dx.doi.org/10.1063/1.1735322
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
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IP: 130.216.129.208 On: Sat, 06 Dec 2014 06:46:33DISCUSSION 1319
A. G. Tweet: This is a comment concerning interpretation of
extremely strong temperature dependence of mobility. Under
certain conditions such effects may be caused by inhomogeneities
in samples rather than a large density of defects. For example, in
material doped with only 1015 to 1016 nickel atoms per cm3, re
sults that are uninterpretable have been obtained. Further in
vestigation has shown that this is caused by an inhomogeneous
distribution of impurity.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. BEMSKI
M. Niseno.fJ: We have found spin-resonance absorptions in
many samples of neutron-irradiated silicon. Irradiation until the
Fermi level was near the center of the forbidden gap produced a
reproducible pattern, independent of the original donor or acceptor
concentration. Under annealing, while the Fermi level remained
at the center of the gap, two changes took place in the spin-reso
nance pattern. However, even after annealing at 500°C two
centers still remained, indicating that at least four types of cen
ters are introduced by neutron irradiation at room temperature.
We also found that the intensity of the lines found in these ma
terials increased with irradiation (the range of flux used was
1011-10'9), indicating that these resonances are produced by a re
distribution of electrons on defects rather than by electrons origi
nating from impurities. In samples whose post-irradiation Fermi
level was closer than 0.1 ev to the conduction band or 0.2 ev to
the valence band, different types of spin-resonance absorption
patterns were seen.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. D. WATKINS
L. Slifkin: I have a question about how the irradiation-induced
vacancies get to the oxygen atoms. For the case of germanium,
subtracting Logan's value for the creation energy of vacancies,
which is a little over 2 ev, from the self-diffusion activation energy
of less than 3 ev, one obtains a little less than 1 ev for the mobility
energy. This gives, for the temperature range in which the A
centers are produced, a frequency of vacancy motion of about 1
per day. It might be expected to be even less in silicon, which has
a higher melting point and, presumably, a higher activation
energy for vacancy motion. It is, therefore, a bit difficult to see
how the vacancy can diffuse to the oxygen atom in silicon at such
a low temperature.
H. Brooks: Is it absolutely excluded that the oxygen moves?
A. G. Tweet: Although oxygen is interstitial, it is bonded and,
therefore, probably does not move. I refer you to work of Logan
and Fuller.
G. D. Watkins: Is it possible that in self-diffusion measurements
one is dealing with aggregates of vacancies and that the only
place where you really form a single vacancy is in the radiation
experiment?
H. Brooks: The binding energy of the vacancies would have to
be pretty high.
A. G. Tweet: If diffusion proceeded by both vacancies and
divacancies, a break would appear in the activation energy curve
for self-diffusion. The fact that no such break has been observed
implies that diffusion is all going by divacancies with a binding
energy of 3 ev or more, or by single vacancies.
G. H. Vineyard: At least two things about these annealing
processes deserve comment. In your Fig. 5 two stages are visible
in the formation of the A center. No simple movement of one kind
of defect is going to explain that. Second is the fact that on anneal
ing to 3000K only 3% of the number of centers expected for room
temperature irradiation remain. Perhaps you have an explanation. G. D. Watkins: I think Dr. Wertheim will be able to say some
thing about the second of your comments.
(G. K. Wertheim presented some results concerning the tem
perature dependence of the introduction rate of the A center in
silicon for electron bombardment. These results have been sub
mitted for publication in The Physical Review.)
J. Rothstein: An oxygen atom coupled to a vacancy should have
a dipole moment. It is conceivable that this might be detected by
measuring dielectric loss. Moreover, one might pick it up by
measuring infrared absorption. It seems possible that if one
applied a polarizing field one might actually get a Stark-splitting
of the level.
Nature of Bombardment Damage and Energy
Levels in Semiconductors
J. H. CRAWFORD, JR., AND J. W. CLELAND
R. W. Balluffi: Would you expect a large range of sizes of the
damage spikes?
J. H. Crawford: Yes. A large range would be expected. This is
the reason that two different sizes were shown in the figure; the
large size is completely blocking, whereas the small size is not.
What we have done here, in essence, is to arbitrarily split the
damage into two groups. The first group is composed of isolated
defects or small clusters which produce the effects of isolated
energy levels, whereas the large groups are envisioned as affecting
almost entirely those properties requiring current transport.
W. L. Brown: I would like to comment on your p-type bombard
ment results. These are in contrast to what we have seen with
electron bombardment of specimens at liquid nitrogen tempera
ture. While you observe an increase in hole concentration we
observe only a decrease in hole concentration over the entire tem
perature range when the bombardment occurs at 77°K.
R. L. Cummerow: Our results with electron bombardment at
room temperature in which n-type material is converted to p type
seem to agree with your proposal of a limiting Fermi level .\*.
The question I would like to ask is, does the mobility determina
tion support your two-level scheme? That is to say, trapping with
the Fermi level halfway between these two levels would require
that mobility decrease with long bombardment, whereas if a
single level and annealing were responsible for the limiting con
ductivity. the mobility should show no further decrease.
J. H. Crawford: We have insufficient data on the mobility in
the impurity scattering range to decide this point.
H. Y. Fan: Do I understand that you assume that the energy
levels near the center of the gap that determine the behavior of
converted material are produced at about the same rate as the
0.2-ev level below the conduction band?
J. H. Crawford: Perhaps. The only way one can obtain a limiting
Fermi level value is by the introduction of a filled level near 0.2 ev
above the valence band and an empty state quite near the center
of the gap at the same rate.
H. Y. Fan: In the case of electron bombardment our results
indicate that the level near the middle of the gap is produced in
relatively low abundance.
J. H. Crawford: For the gamma irradiation we know simply
from the shape of the Hall coefficient vs temperature curve that
there is a deep vacant state present in concentration equal to the
0.2-ev state below the conduction band.
R. L. Cummerow: I have some evidence that there is a level in
the center of the gap, but it is somewhat indirect and was obtained
by analyzing the potential variation in the junction produced by
electron bombardment.
Precipitation, Quenching, and Dislocations
A. G. TWE>;T
W. L. Brown: It has been shown that electron irradiation altered
the precipitation of lithium in a manner that would indicate that
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IP: 130.216.129.208 On: Sat, 06 Dec 2014 06:46:331320 DISCUSSION
the vacancies, or some complexes of vacancies, served as precipi
tation sites.
A. G. Tweet: That is correct. The effect of vacancy aggregates
or of certain pre-precipitates on precipitation rates is a subject of
great interest.
M. S. Wechsler: Several descriptions have been given at this
Conference of the precise and informative way in which diffusion
controlled reactions may be studied in semiconductors. Since we
may anticipate experiments on the effect of radiation on such re
actions in semiconductors, it may be of interest to mention some
work of this type of metal alloy systems, done at Oak Ridge
National Laboratory in collaboration with R. H. Kernohan and
D. S. Billington.
Two alloys, in particular, have been studied. The first of these,
Ni-Be (14.5 at. % Be), exhibits a precipitation reaction in which
the intermetallic compound NiBe precipitates from the primary
substitutional solid solution. The amount of Be remaining in solid
solution during the aging can be determined rather conveniently
by measuring the ferromagnetic Curie temperature, which in
creases rapidly upon the reduction of Be in solid solution. The
second alloy whose behavior upon neutron irradiation has been
investigated is Cu-AI (15 at. % AI). It has been found that a
diffusion-controlled reaction is triggered or accelerated upon irra
diation at 35-45°C that causes a decrease of about 2% in the
electrical resistivity. Measurements on unirradiated samples at
elevated temperatures and upon quenching indicate that the
alloy is initially in a metastable state and the irradiation causes
the return to equilibrium at lower temperatures than is normally
possible. However, the nature of this metastability has not been
fully established.
Despite the fact that the reactions in these alloys are probably
rather different in their essential nature, striking similarities are
observed in the effect of neutron irradiation on them. This is
illustrated by two types of experiments. In the first of these,
samples are quenched and the diffusion-controlled reaction is
allowed to go on in the reactor at a given temperature. The re
action curve that results is compared with one obtained for an
experiment at the same temperature outside of the reactor. In the
case of both Ni-Be and Cu-AI, in-pile measurements show that
the reaction proceeds considerably more rapidly in the neutron
flux environment. This result had previously been established for
Ni-Be as a result of before-after measurements. [Kernohan,
Billington, and Lewis, J. App!. Phys. 27, 40 (1956)]. In the sec
ond type of irradiation experiment, the samples were irradiated
at temperatures considerably below the reaction temperatures;
and the annealing or aging was carried out by raising the tempera
ture after the irradiation was completed. Under these conditions,
the curious result obtained for both alloys is that, although the
reaction starts more rapidly for the samples that receive the in
termediate low-temperature irradiation, the reaction becomes
stabilized before its completion. Hence, the reaction curves for
the irradiated samples eventually lag behind those for the un
irradiated ones.
Thus, the experiments on these alloys show that when samples
are irradiated at reaction temperatures a clear acceleration of the
reaction takes place. This is perhaps not surprising. Regions of
local radiation damage should provide efficient centers at which
the reaction can be nucleated. Furthermore, in those cases in which
diffusion occurs by a vacancy mechanism, the radiation-produced
vacancies should enhance diffusion, thereby accelerating the
growth of the reaction product. However, the effect of prior irra
diation below the reaction range of temperatures is more com
plicated, due to a type of stabilization that takes place after the
reaction is underway. This stabilization may correspond to the
formation of metastable centers, similar to those discussed by
Tweet for the precipitation of Cu in Ge. In any case, experiments
conducted in this fashion indicate that the irradiation has a
unique effect on diffusion-controlled reactions, which is not equi
valent to simply raising the reaction temperature. The broad similarity in the observations for Ni-Be and Cu-Al
suggests that a similar behavior may be found for the effect of
radiation on reactions in semiconductors. A comparison of experi
ments on metals and semiconductors should lead to a better un
derstanding of radiation effects in both types of materials.
G. K. Wertheim (to M. S. Wechsler): Just one brief question.
Do you know to what extent the effects you observe are compli
cated by the similarity of a number of kinds of radiation damage
centers and by the precipitation of dispersed impurities?
M. S. Wechsler: We are not able to say anything as yet about
the detailed mechanism of the radiation effects. Thus, on the as
sumption that the radiation-enhancement occurs chiefly because
of more effective nucleation, little can be concluded concerning
the structure of the nucleation center. However, it is significant
that the low temperature irradiation itself has no effect on the
reaction; the temperature must be raised before its effect is felt.
H. Reiss: I should like to comment on the possibility that a
break in the curve of the type you (Wechsler) described may not
be due to metastable traps. This could occur if you had a fairly
inhomogeneous distribution of nuclei. Certain regions would pre
cipitate very quickly and other regions less quickly. But the re
sistivity measurements would average the effects of the separate
regions, so that it would seem as though the precipitation had
slowed down abruptly. So unless you have a uniform distribution
of damage you have to be careful about postulating traps.
A. G. Tweet: Another possibility for the interpretation of the
enhanced precipitation observed (by Wechsler) when aging pro
ceeds in the pile serves to call attention to how intricate this sub
ject can be. Almost all of us have talked about the limitation of
precipitation by diffusion. However, there might be another factor
that can limit the precipitation. A strain field built up around the
precipitate particle may serve as a repulsive potential which re
tards diffusion toward the precipitate. It is not impossible that
the enhanced precipitation for the in-pile aging is caused by the
fact that this strain field is gotten rid of by the formation of the
small precipitate particles produced as a consequence of the
irradiation.
Radiation Defect Annealing
W. L. BROW'"'
G. J. Dienes: The following applies to the papers presented by
R. W. Balluffi and W. Brown. It has been generally assumed that
the lattice collapses around the vacancy in germanium, similar to
a metal. I once looked at diamond and argued that carbon-carbon
double bonds might actually produce a contraction away from the
lattice, that is, a lattice expansion around the vacancy. The calcu
lations were very crude and were not published.
H. Reiss: I think that enough evidence has been presented at
this Conference to indicate that the acceptor considered by W. L.
Brown in these annealing studies cannot possibly be a vacancy.
We know that, for many substances, the log of the diffusion co
efficient vs inverse temperature is a straight line for many decades.
We also have a rough idea of the self-diffusion activation energy
in germanium and silicon, and this predicts that the vacancy does
not move appreciably at very low temperatures. On the other
hand, a vacancy is not actually trapped by an atom such as Sb,
because the Sb will actually diffuse more rapidly by the vacancy
method than it would by self-diffusion. If the acceptor is an inter
stitial, however, it will be trapped. If it is paired in an ion-pairing
sense, it will pair less with Sb than As since the Sb has a larger
radius and a smaller binding energy. If the acceptor is a charged
interstitial, one may get an over-all reduction in the activation
energy for diffusion, and the interstitial may therefore diffuse
more rapidly. I therefore feel that the acceptor in question cannot
be ascribed to a vacancy under these conditions.
G. D. Watkins: I favor the mechanism of vacancy diffusion at
all temperatures. Is it possible that annealing under consideration
is not diffusion-limited, but trap limited? That is, the long-range
motion may still occur at low temperatures, but trapping results;
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1.1734535.pdf | Electron Mobility in Crystals
Marshall Fixman
Citation: The Journal of Chemical Physics 39, 1813 (1963); doi: 10.1063/1.1734535
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:10THE JOURNAL OF CHEMICAl. PHYSICS VOLUME 39, NUMIlEl<. 1 OCTOBER I~,d
Electron Mobility in Crystals*
MARSHALL FIXMANt
Imtitute of Theoretical Science, University of Oregon, Eugene, Oregon
(Received 20 May 1963)
A perturbation expansion of the mobility of independent electrons is given and is examined in detail
f~r w~ak scattering by impur.ities or intracellular vibrations, and for strong scattering by intracellular
VlbratlOns. In the latter case, If the lattice coordinates have a Gaussian distribution a cell-to-cell random
tunneling picture is shown valid for very strong scattering. For weak impurity scattering by random centers
of concentration ii, the correction to the Boltzmann-Bloch mobility is of order iilnii. The transition be
tween the weak scattering limit, where the Boltzmann equation holds, and the random tunneling limit
where the Boltzmann equation is invalid, is correctly given by the first approximation in which scatterin~
events do not interfere. The perturbations vanish at both extremes.
I. INTRODUCTION
THE recent extensive attention to the theory of the
mobility of independent electrons in crystals has
been mainly directed to impurity scattering in metals
and to a demonstration of the validity of the Bloch
scattering theory.l-8 The Bloch theory has been shown
valid for weak scattering, or for low concentrations of
strong scatterers. The perturbation theory at the basis
of these results was carried far enough that it could be
seen under what conditions corrections to the Bloch
theory would be small, but no attempt seems to have
been made to actually evaluate the perturbation even
for a simple model. One of the purposes of this work is
to correct the deficiency for a simple model of impurity
scattering.
The second and main purpose of this work is to con
sider the transition from a weak scattering situation,
where the Bloch states are long lived, to a strong
scattering situation, where an excess electron in a
localized state has a very long life, but occasionally
tunnels to an adjoining site. It is shown that the
theoretical transition is adequately handled in an inde
pendent scattering picture; deviations from the first
approximation vanish both for very strong and very
weak scattering. Of course for strong scattering the
Boltzmann equation becomes invalid, but it surprisingly
* Supported in part by the National Science Foundation
(G19518) and the Division of General Medical Sciences, Public
Health Service (09153). t Alfred P. Sloan Fellow.
1 J. S. Van Wieringen, Proc. Roy. Soc. (London) A67, 206
(1954) .
2 W. Kohn and J. M. Luttinger, Phys. Rev. 108, 590 (1957).
3 J. M. Luttinger and W. Kohn, Phys. Rev. 109, 1892 (1958).
4 S. F. Edwards, Phil. Mag. 3, 1020 (1958).
5 A. A. Abrikosov and L. P. Gorkov, Zh. Exsperim. i Teor.
Fiz. 35, 1558 (1959); 36, 319 (1959); [En.!(lish transl. Soviet
Phys.-JETP 8,1090 (1959); 9,220 (1959)J.
6 Neil Ashby in Lectures in Theoretical Physics, edited by W. E.
Brittin, B. W. Downs, and J. Downs (Interscience Publishers,
Inc., New York, 1961), Vol. III.
7 G. Rickayzen in The Many-Body Problem, edited by C.
Fronsdal (W. A. Benjamin, Inc., New York, 1962).
8 C. Herring in Proc. Intern. Conf. Semiconductor Phys.,
Prague 1960, 60 (1961). turns out that the independent scattering model does
not become invalid.
Several of the simplifications in the Hamiltonian
make the calculation most relevant to carrier mobilities
in molecular crystals, although the impurity model of
weak scattering does not exclude metals or semicon
ductors treated in an independent electron approxima
tion. It was for molecular crystals that the calculation
was devised in an attempt to understand tunneling
calculations of mobility in the solid aromatics. As is
well known, for any nonvanishing coupling of excess
e~ectron states between .neighboring molecules, the
eIgenstates of an electron m a perfect crystal are Bloch
states rather than localized molecular states. It is
rather difficult to reconcile this fact with one's intuitive
expectation that the localized states should be very
good first approximations for weakly coupled molecular
crystals, the excess electron only rarely jumping from
one site to a neighbor. One attempt9,lO has been made
to justify the localized state picture on the assumption
of strong external electrical fields which remove the
degeneracy between neighboring sites. However, if
calculated bandwidthsll,12 are correct in order of magni
tude, the electrical fields actually used are much too
small to provide the required localizations. (Only the
zero-field limit of the mobility is calculated here.) The
localized-state tunneling model is here found to be a
valid limit if the energy eigenvalue of the localized
state has a Gaussian distribution of possible values,
due to lattice vibrations. For the aromatic molecular
crystals a practical realization of the model (although
not necessarily the strong scattering limit) arises in the
low-frequency vibration of the two molecules in the
unit cell relative to each other. At least along the c-1
direction the band splitting seems to be large12 and the
energy of the lowest (symmetric) cell wavefunction
should be strongly perturbed by a classical vibration
gR. A. Keller,/. Chern. Phys. 38, 1076 (1963).
10 M. L. Sage, . Chern. Phys. 38, 1083 (1963).
II O. H. LeBlanc, Jr., J. Chern. Phys. 35,1275 (1961).
12 J. L. Katz, S. A. Rice, S. I. Choi, and J. Jottner, BuU. Am.
Phys. Soc. 8, 234 (1963).
1813
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:101814 MAH.SHALL FIXMAN
which can be reasonably isolated within the cell.
Another perturbation which might exemplify the
present theory stems from the polarization interaction
of an excess electron in one aromatic molecular orbital
with neighboring molecules. The neighbors could
reasonably be taken to be in their equilibrium positions
and orientations, and the perturbation would arise
from quasistatic vibrations and librations of the central
molecule. Even in the aromatics, electron propagation
is probably rapid enough to justify a dynamical calcula
tion based on an equilibrium distribution of phonons
(calculated in the absence of the excess electron).
No claim is made for the originality of the perturba
tion expansion, which is probably equivalent to diagram
expansions already given.13 Emphasis is instead given
to the detailed behavior of the expansion for various
scattering strengths.
II. HAMILTONIAN
In the absence of perturbations the system is one
electron in a perfect lattice, described by a Hamiltonian
H. Two matrix representations of H and other operators
will be usedl4: a Bloch representation with orthonormal
basis bk(x), and a Wannier representation with ortho
normal basis wr(x). By definition the bk(x) diagonalize
H, and are periodic in the electron coordinate space x
and the reciprocal lattice space k. The Wannier func
tion Wr(X) , for unit cell r, is a localized function of
x-r, and is related to the Bloch basis by
wr(x) =N-t 2:)k(X) exp( -ik·r), (1)
k
bk(x) =N-t LWr(x) exp(ik·r). (2)
r
The sums in (1) and (2) are over the first Brillouin
zone and the cells of the lattice, respectively. The
matrix elements of an operator A will be designated
Akl in the Bloch representation, and Am" in the Wannier
representation. From Eq. (1),
Amn=N-ILA u exp[i(k·m-l·n) J,
k.l
Akl=N-1LAmn exp[i(l·n-k·m)]. (3)
m,n
The Hamiltonian H is to represent a narrow band
system, with nonvanishing matrix elements Hmn only
between neighboring cells. The diagonal elements
Hmm= E have no effect on the dynamical properties of
the system, and are set equal to zero. For the molecular
crystals which form the most natural application of
this calculation, the wr(x) are very close to molecular
orbitals of the isolated molecules (or linear combina
tions of such orbitals if the unit cell has more than one
molecule). In the Bloch representation,
Hk/=Hkbk/o
13 No attempt has been made to prove an equivalence.
14 G. H. Wannier, Elements of Solid State Theory (Cambridge
University Press, Cambridge, England, 1959). The electron in the presence of a perturbation is still
describable, by supposition, as a linear combination of
single-band Bloch waves, the wavefunction being
lP(x,t)=LCk(t)bk(x). (4)
k
The Ck of course have an exp( -iHktln) time depend
ence in the absence of perturbations. Equation (4)
is an approximation, because bkex) from only one
band are allowed, and its validity requires that the
perturbation Hamiltonian P be in some sense a small
perturbation. Fortunately this requirement in no way
limits the effect of P as a source of scattering, that
being determined by the relative magnitude of the Pk!
and off-diagonal elements Hmn. In general P might
represent the interaction of the electron with optical
or acoustic modes or with a randomly located sub
stituent on the molecules, or some other kind of im
purity. Also P might be a function of time, and might
not be a function of a classical coordinate. However,
it is supposed that P is small enough to have negligible
effect on the equilibrium governing the behavior of the
lattice coordinates on which P does depend. The
equilibrium density matrix of electron coordinates and
lattice coordinates then factors into a direct productl5
and the Bloch or Wannier matrix elements of P need
only implicitly be taken as quantum mechanical
operators in the orthogonal space of lattice coordinates.
At the end of the calculation an appropriate average
over the elements of P must be taken.
A fundamental restriction on the nature of P will
now be imposed by the requirement that P does not
have any matrix elements between differing unit cells.
(5)
Moreover, pm(tl) and pn(t2) are taken to be independent
stoichastic variables for m ~n, and without further loss
of generality, (pm(t) )=0. Equation (5) and the inde
pendence seem quite reasonable for optical phonons or
random substituents. The conditions are not so reason
able for acoustical phonons except when the phonon
energy may be suppressed and P taken independent of
time, as for semiconductors at high temperature.16
III. DYNAMICS
A study of one of the components vet) of the electron
velocity, rather than of the density matrix, seems to
be the most oirect path to the mobility. vet) satisfies17
-idvldt= UV-2U
U=H+P, (6)
where the convention Ii= 1 has been adopted. The
effects of P are partially isolated in an interaction
15 Paul S. Hubbard, Rev. Mod. Phys. 33, 249 (1961).
16 J. M. Ziman, Electrons a.nd Ph(lnrms (Oxford University
Press, London, 1960). -
17 E. Montroll, Ref. 6.
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:10ELECTRON MOBILITY IN CRYSTALS 1815
representation:
vet) = exp(iHt)w(t) exp( -iHt) ,
-idw/dt=Qw-wQ,
Q= exp( -mt)p exp(iHt). (7)
(8)
(9)
The study of wet) can be reduced to a simpler study of
the unitary operator S(t).
wet) = S(t)w(O) S-l(t),
dS/dt=iQS; S(O) = 1. (10)
(11)
The assumption (5) causes Q to decompose in an inter
esting way which will be exhibited in a Bloch repre
sentation. Equations (3), (5), and (9) give
(12)
where Hkl=Hk-H I (the notation Hkl for a matrix
element of H is not used in the following equations).
Pkl=N-IL.Pm exp[i(l-k) ·m].
'"
Consequently Q has the form
(13)
m
where the matrix Am has Bloch components (Am)kl
(Amhl=N-l exp{ -i[Hklt- (l-k) ·m]}. (14)
Many procedures have been devised to carryon the
analysis from this or some equivalent point. The object
has been a precise examination of the Boltzmann equa
tion and to this end pm would be assumed small com
pared to the bandwidth. (The Boltzmann equation
has also been derived for strong but dilute random
scatterers.2,3) However, in this calculation large values
of pm may correspond to a simple and easily described
physical situation: namely a very rare tunneling of the
electron from one site to another, and it seems not to
have been noticed that Eq. (11) permits the same easy
hut valid solution for very large pm as for very small.
The easy solution is that for independent scattering
by the various pm. Also, when corrections to the Boltz
mann equation have been examined, only the order
of magnitude and not the functional form of the cor
rection has been reported.
An independent scattering model results when S is
approximated by a symmetrized product of operators
(Sr) which satisfy
dsr/dt=iArprSr; Sr(O) = 1 (15)
analogous to Eqs. (11) and (13). The physical motiva
tion is that a given electron state has a certain (high)
chance of surviving an interaction with one scatterer.
The chance of surviving all the interactions is the
product of the separate probabilities. The product So of the operators Sr, averaged over the pr, is easy enough
to get at least in the Born approximation, which is all
that will be meant by So. It turns out that So is diagonal
and has the form
So= exp[ -'A2b(t)], (16)
where 'A is a numberto be set equal to unity, and
measures the power of p that occurs. The diagonal
operator bet) will be displayed shortly; at this stage So
could be an arbitrary first approximation to S, chosen
to make the perturbation series converge rapidly.
Examination of the series then confirms the motivation
behind Eg. (15). Put
S=So(1+S), (17)
an equation which defines s. Equation (11) becomes
with the ordering parameter
dS/dt=i'AQS.
Put
Equations (17)-(19) give
s(l)(t) =i[dtoQ(t o)
o
lt 1/1 S(2)(t) =b(t) -dh dtoQ(tl)Q(to).
o 0 (18)
(19)
(20)
(21)
The condition (pm)=O makes (s(1»=O, and bet) is
taken as
1/ 1/1 bet) == dtl dto(Q(tl)Q(to».
o 0 (22)
Equation (22) makes (S(2» vanish. Now what is
eventually wanted is (S(t)v(O) S-l(t». It can be
demonstrated straightforwardly for the potential of
Eq. (5) that
(S(t)v(O) S-l(t) )= (S(t) )v(O) (S-I(t» (23)
for N--+oo, at least to the order of perturbation theory
that will be retained.
The next term in Eq. (19) is (S(3», and it does not
in general vanish. The condition (pm)=O means that
only terms like (pm3), that is, corrections to the first
approximation to the interaction of an electron with a
single scatterer, will contribute to (S(3». Moreover,
only the real part of (S(3» remains when the product of
Eq. (23) is taken, and this vanishes for all the explicit
calculations made here.
The last term considered is (S(4», and this is in gen-
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:101816 ~l A R S HAL L FIX MAN
eral the lowest-order term which shows interactions
between different scattering elements.
(s(4) (t) )
= {dt3jl'dh[{2dtl{'dtO(Q(ta)Q(t2) Q(tl) Q(to) )
o 0 0 0
-(Q(t3)Q(tZ) )b(t3) 1 (24)
Let
(25)
where ho=t1-to; ! is of course independent of m. perturbation Hamiltonian, the expansion is thus far
exact, and we have
(S(t) v(O) 5-1(t) )k!= Vk(O) okd exp[ -2CRbkk(t)]1
X[1+2<R(s(4)(t) )kk+"'], (34)
where <R indicates the real part. The operators in Eq.
(34) are diagonal, and therefore the general expression
for the static diffusion constant D and mobility J1.
takes the forml8
D= (kT/e)JJ.= fro (v(O) s(t)v(O) S-I(t) )dt
o
Equations (13), (14), and (22) give where (35)
(26) Ck= ["dt{exp[ -2<Rb kk(t)]I[1+2CR (S(4) (t) )kk+"'],
o
where
(27)
One has to note that
N-I2: exp[i(l-k) ·m]= Ok!, (28)
m
since 1 and k are both in the first Brillouin zone.
(S(4» has four contributions:
4
(s(4) )kl= Okl2:f3k(i)(t) , (29)
i=1
f3k(2) (t) = N-22: f dt exp{ -i[Hrqt31 +HkrtzOJ U(t31)!(tzO),
q,r
(31)
f3k(3) (t) = N-22: f dt exp{ -i[Hkrtao+Hrqt21]I!Ct30)!(t21),
q,r
f3,,(4)(t) =N-3 2: fdt
p,fl,r
X exp{ -i[Hkpt3+Hpqt2+Hqrtl+HrktO] I
X [(P(t3) P(t2) P(h) p(to) ) (32)
-!(t31)!(t 20) -!(t30)!Ct12) -!( t32)!(t10) ]. (33)
The time integrations have been abbreviated
f 11 1/3 112 1'1 dt-'> dt3 dt2 dt1 dlo.
o 0 0 0
It will be observed that (S) is diagonal to the order
of perturbation theory used, and an initially diagonal
density matrix will remain diagonal. For the given (36)
where Pk is the probability of the kth Bloch state being
occupied at time zero, and Vk is the velocity along an
assigned direction of an electron in the kth state. The
integration in Eq. (36) will be carried through for
two limiting situations: very weak scattering by a
scattering potential, in particular for random impurities,
and very strong scattering hy a potential with a
Gaussian clistribution.
IV. WEAK SCATTERING
Take !(tIO) in Eq. (25) to be independent of ito. Let
~-,>(V/87r3) fdHkak' (37)
where
(38)
dSk being an element of constant energy surface in k
space. Then Egs. (26) and (27) give
2 <Rbkk (t) = h!( V /8rN) t dH pap
o
X [(2/7rH k/) sin2( tHkpt)], (39)
where h is the maximum band energy. For large t the
quantity in square brackets becomes o (Hkp) , but the
deviations must be examined. Put
(40)
Then
exp[ -2CRb kk(t)]
= exp( -Vkt) [1-(vl/ak) f dHpapg+' • -J (41)
where
Vk= 27rak!(V /87r8N). (42)
'8 R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:10ELECTRON MOBILITY IN CRYSTALS 1817
Consequently, Ck in Eq. (36) is given by
pkCk=1-(pk?lak) fhdHpapfoo exp(-pkt)gdt
o 0
+2pkffi[O exp( -pkt) (S(4)(t) )dt+, , '. (43)
o
Equation (40) gives
PkDk= -(pk2Iak) jhdHpap foc exp( -pkt)gdt
o 0
= 1-(pk/Trak) fhdHpal'(pk2+Hkp2)-1.
o (44)
The integral in Eq. (44) has been evaluated only in
the effective mass approximation, for which H p ex p2
and apex Hpt. The result is
(45)
Obviously this is not a linear term in pk; a strict expan
sion in powers of Pic would approximate the arctan
function by
(46)
The last term in Eq. (43), the term in (S(4», will only
be evaluated through the linear term in pk, but it is
fairly easy to carry through the integral of Dk exactly,
and worthwhile to get a picture of the singularities
which occur. The expansion in powers of pk evidently
breaks down for electrons moving near the band edges,
but the worst behavior is logarithmic rather than the
inverse power singularity of (46). Consequently the
/irst few terms of the perturbation expansion are inte
grable over the initial equilibrium distribution of
states, even though a formal expansion in powers of pk
is not.
The remaining term in Eq. (43) is
pkEk=2pkffifoo exp( -pkt) (s(4)(t) )k~t (47)
o
4
=vkL.Ek(i), (48)
i=1
where
pkEk(i)=2pkffi fro exp(-pkt)13k(i) (t)dt
o (49)
and the 13k(i) are defined in Eqs. (29)-(32). Each Ek(i)
has been evaluated only in the limit of small pk. Since
the 13k(i) (t) contain a factor 1',,2, the desired limit is ob
tained from the linear term in t of each 13Lw at large t
(higher powers of t do not enter at large t, that is, limr-l13k(i) as t-H/:) is finite). The desired limits are most
readily obtained by going over to a generalized func
tion treatment.19 One example wiII be discussed in
detail, 13k(2), and the other results simply written down.
From Eqs. (31) and (37),
ffi/3k (2) (t)
= (V 18rN)?j2ffi jtdtajt3dft ff dHqdH.arll.(H kr)-2
o 0
X exp(-iH. qts1) [1-exp(-iH k.l1)J
X [exp( -iHk.tS I) -1J (50)
after the integrations over to and t2 are performed. The
part proportional to t at large t is wanted. This part
wiII be t times the limit of the tl integral as ta--*~. For
ts--*~, the tl integral receives contributions only from
th in the neighborhood of ta, and if we now adopt a
generalized function interpretation of the integrals, to
eliminate the apparent singularity at Hk.=O, we can
suppress the term exp( -iHk.t1). The real part of the
resultant integral over tal from 0 to ~ is then just a
delta function, and
ffi(3k(2)(t) = t( V 18rNFP1r ff dHqdH,.arllrHk.--~
X [0 (Hkg) -o(Hrq) J, (51)
=t(VI8rN)2j'21r
x {akfdH.arHkr-2- fdHrar2Hkr-+ (52)
With arex H,i, the second of the integrals in Eq. (52) is19
f dHrar2Hk,-2= ak2Hk-1[ln(h-Hk)Hk-1-h(h-Hk)-IJ
(53)
and the first isl9
f [1 hl-Hkl (hIHk)!]
dH.arHkr-2=ak 2Hk Inhi+Hki -h-Hk .
When Eqs. (52), (53), and (54) are combined
(3k(2) is substituted into Eq. (49)
{ 1 hi-Hki (hIHk)i pkEk(2)=Pk(21r)-1 -In--- ---'---
2Hk ht+Hkt h-Hk
__ 1 lnh-Hk+ (hIHk)}.
HIc Hk h-Hk
In a similar way one finds (54)
and
(55)
Vk (Ek(l) + Ek(a» = -pk(21r)-1 -+-In--- . (56) [ hl 1 hi-Hkl]
Hkl 2Hk hi+Hkl l~ .
The last term Ek(4) arises from multiple electron
19 M. J. Lighthill, Fourier Analysis and Generalized Functions
(Cambridge University Press, Cambridge, England, 1958).
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:101818 MARSHALL FIXMAN
scattering from a single center (a correction to the
Born approximation). We find
vkEk(4)= tVk2j-2[ (P4)-3f2]. (57)
For a Gaussian distribution of the p, Ek(4) vanishes.
This can be seen more generally from the definition of
/3k(4) in Eq. (33); if the time evolution of the p's follows
any stationary Gaussian process, iN4) vanishes. For a
random distribution of impurities, on the other hand,
P is proportional to (n-ii), where n is the actual num
ber and ii the average number of impurities per cell,
and P has a Poisson distribution. A computation of the
moments gives
and
(58)
For the random distribution of impurities, then, all
the VkEk(i) are proportional to ii, since (p2), j, and Vk
are proportional to ii.
In summary, the mobility is given by Eqs. (35)
and (36),
(59)
and the definitions of the Dk and Ek(i) in Eqs. (43),
(44), and (48) give
4
VkCk= l+v"D k+vkEE k(i)+.... (60)
i=l
The Dk and Ek(i) are displayed in Eqs. (45), (55),
(56), and (58), and there is little point in displaying
the sum in one equation. However, a few comments
may elucidate the nature of the various terms.
First, for electrons of small energy the dominant
perturbation to the Boltzmann mobility is the ht/Hk!
in (Ek(1)+E,,(3»; see Eq. (56). This term gives a
negative divergence from the Boltzmann mobility.
For metals this divergence is irrelevant, because the
conduction electrons enter the band above the Fermi
level. For excess electrons in semiconductors and in
sulators the states with small Hie are occupied, and the
Boltzmann equation is clearly inadequate for electrons
in these states. However, the V,,2 in Eq. (59) and the
k2dk in the integration over the band provide ample
convergence factors to make the total fractional cor
rection to the mobility finite and proportional to j.
For electrons near the boundary of the first Brillouin
zone, (Ek(1) + Ek(2) + Ek(3» diverges positively as
Hk-lln(h-H k). Equation (45) for Dk gives a similar
positive logarithmic divergence, and also the curious
behavior of the arctan function. If the latter were
simply expanded to the lowest power of Vk, that is, the
lowest power of (P2) or ii, the resultant contribution
to the mobility would diverge. However, the integral
of the arctan function as well as the logarithmic
divergence give fractional corrections to the Boltzmann
mobility which are finite. But the arctan contribution
has the form ii Inn. The contribution of the arctan term to the diffusion
constant for very small Vk is
Darctan = (V /81r3) 1r-lphVh2ah In (h/ Vh) ,
where the subscript h indicates the value of a function
on the zone boundary. Whether this nonlinear term in
ii would be observed for metals with a Fermi surface
near the Brillouin boundary or for narrow band semi
conductors, or whether it ought to be predicted con
sidering the use of an effective mass approximation,
are speculative questions.
V. STRONG SCATTERING
For a certain class of perturbations, as the interac
tion between electron and lattice is increased or as h is
decreased, the electron will most of the time be in a
simple localized Wannier state, and will occasionally
make transitions between such states. In this section
the perturbation series for D, Eq. (36), is shown to
converge rapidly for this kind of strong scattering.
The existence of this limit is most closely tied to the
certainty of the randomization of the phase of the
Wannier state into which an electron enters, rather
than to the strength of the scatterer. Whatever the
strength of impurity scattering, for example, the
Boltzmann equation will still be valid if the impurity
concentration is low. And even for a rather high con
centration of impurities (say ii= 1), the probability
of two or more adjoining unit cells having the same
number of impurities will be large, and the carrier will
pass through them without scattering. In these situa
tions the band structure or the cross section for scatter
ing by a single scatterer and not the validity of the
Boltzmann equation are the interesting theoretical
problems.
Turning now to Eq. (36), we first examine b(t). For
small h (or large p, since this makes small t important),
Eqs. (26) and (27) give
2<Rbkk(t) = 2 {(t-r)f( r)dr
o
(62)
For the present takej(r) independent of r again. Then
2<Rbkk(t) =ft2-jt4(V /96?rsN) tdHpapHkp2. (63)
o
(s(S» will again be suppressed, but it should be
noted here that it is not at all negligible for impurity
scattering when h is small. However, the failure of
impurity scattering to localize the states is best seen
from an exact solution of Eq. (11) in the limit of zero
h. Q is then independent of time and Eq. (11) can be
directly integrated with the result that (S(t» is a
unit operator given by
(S(t) )= (exp(ipt) ), (64)
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131.94.16.10 On: Sun, 21 Dec 2014 06:25:10ELECTRON MOBILITY IN CRYSTALS 1819
where p stands for any of the pm. With p proportional
to (n-ii) , and a Poisson distribution of n, (S(t»
just oscillates for large t. The following development
will therefore be restricted to a Gaussian distribution
of the p.
The various contributions to (S(4) are tabulated in
Eqs. (29)-(33). It is evident at once that (3k(4) vanishes
identically for the Gaussian process. The remaining
terms are each of the form f2t4 in the limit of vanishing
h, and at first it seems that the contribution from
(S(4) does not vanish at zero h. However, all the terms
cancel. In the limit of zero h,
,3k(l)=!f jtdt3[8dt2(t.}-ts2) = -(1/12)f2t4,
o 0
(3k(2)= (3k(8)= (1/4!)f2t4.
Consequently (S(4) (t) ) has a term proportional to h'lj'lt6
as the first nonvanishing one in an expansion in powers
of h.
When the time integral in Eq. (36) is computed,
both the second term in Eq. (63) and (S(4) make
fractional corrections to the mobility which are pro
portional to h2j-l, and will be suppressed. What remains
in Eq. (36) is
and
(65)
Since the localized state tunneling model can be ex
pected to hold exactly only for (h/kT)«1, the mean
square velocity should be computed for the purpose of
comparing Eq. (65) with that model as
(66)
The trace is most simply computed in the Wannier
representation as Na2h2, where a is the lattice spacing
and h=Hii for I i-j I = 1 along the given direction.
Consequently
D=!aW(7r/f)i. (67)
This is the same result as follows from D=!a2v, where
v is the frequency of electron transitions between
neighboring sites, when v is computed by treating h as a
perturbation which induces transitions between states
of differing (random) energies Pi and Pi, VI. DISCUSSION
The independent scattering approximation to the
mobility is given by
D= LPkVk2Ck, (68)
Ck':::::.j dtexp[ -2CRbkk(t)], (69)
o
where bkk is given generally by Eq. (26), or for the
special case of static scattering centers by Eq. (39).
Equation (69) gives the Bloch expression for the mo
bility when the scattering is weak (and the dominant
correction to it for weak scattering), and also reduces
to the localized state tunneling model for strong
(Gaussian) scattering centers, for which a Boltzmann
transport equation is completely inadequate. The cor
rections to Eq. (69) vanish for very weak and for very
strong (Gaussian) scatterers; presumably the correc
tions are small for intermediate scattering but it would
be very interesting to have a full numerical analysis
for the particular case of Gaussian scatterers.
Concerning the practical utility of the observations
made here, molecular crystals are the most natural
systems to examine for a deviation from the Bloch ex
pression for the mobility. The Hamiltonian used here
is most suitable when the perturbation arises from low
frequency intermolecular vibrations localized within
the unit cell. It is the localization of the vibrations
that is important here; the results could easily be gen
eralized to nonclassical vibrations and to nearly de
generate bands if only the lattice vibrations are in
equilibrium. The inclusion of fluctuations in the over
lap matrix elements Hii appears trivial only for very
weak scattering (where the Boltzmann equation holds
and the results are well known), and for a strong
scattering situation in which the tunneling model
holds (i.e., intracellular vibrations still dominate).
Reference should be made to the elegant develop
ment by Kub020 of a stoichastic theory of line shape
and relaxation, which became known to the author
after this work was done. Although his methods would
not have aided the treatment of the simple Hamiltonian
used here, they do appear to provide the most powerful
formalism available to begin the treatment of more
complicated perturbations.
20 R. Kubo in Fluctuation, Relaxation and Resonance in M ag
netic Systems, edited by D. ter Haar (Oliver and Boyd Ltd.,
Edinburgh, 1962), p. 23.
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1.1735320.pdf | Magnetic Susceptibility and Electron Spin Resonance—Experimental
G. Bemski
Citation: Journal of Applied Physics 30, 1319 (1959); doi: 10.1063/1.1735320
View online: http://dx.doi.org/10.1063/1.1735320
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Technique for magnetic susceptibility determination in the highly doped semiconductors by electron
spin resonance
AIP Conf. Proc. 1610, 119 (2014); 10.1063/1.4893521
Magnetic properties of nested carbon nanostructures studied by electron spin resonance and
magnetic susceptibility measurements
J. Appl. Phys. 80, 1020 (1996); 10.1063/1.362835
Absolute Determination of Change in Magnetic Susceptibility Due to Electron Spin Resonance
Rev. Sci. Instrum. 44, 1118 (1973); 10.1063/1.1686314
Magnetic Susceptibility and Electron Spin Resonance—Experimental
J. Appl. Phys. 30, 1319 (1959); 10.1063/1.1735321
Magnetic Susceptibility and Electron Spin Resonance
J. Appl. Phys. 30, 1318 (1959); 10.1063/1.1735319
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IP: 130.88.53.18 On: Mon, 24 Nov 2014 15:41:50DISCUSSION 1319
A. G. Tweet: This is a comment concerning interpretation of
extremely strong temperature dependence of mobility. Under
certain conditions such effects may be caused by inhomogeneities
in samples rather than a large density of defects. For example, in
material doped with only 1015 to 1016 nickel atoms per cm3, re
sults that are uninterpretable have been obtained. Further in
vestigation has shown that this is caused by an inhomogeneous
distribution of impurity.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. BEMSKI
M. Niseno.fJ: We have found spin-resonance absorptions in
many samples of neutron-irradiated silicon. Irradiation until the
Fermi level was near the center of the forbidden gap produced a
reproducible pattern, independent of the original donor or acceptor
concentration. Under annealing, while the Fermi level remained
at the center of the gap, two changes took place in the spin-reso
nance pattern. However, even after annealing at 500°C two
centers still remained, indicating that at least four types of cen
ters are introduced by neutron irradiation at room temperature.
We also found that the intensity of the lines found in these ma
terials increased with irradiation (the range of flux used was
1011-10'9), indicating that these resonances are produced by a re
distribution of electrons on defects rather than by electrons origi
nating from impurities. In samples whose post-irradiation Fermi
level was closer than 0.1 ev to the conduction band or 0.2 ev to
the valence band, different types of spin-resonance absorption
patterns were seen.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. D. WATKINS
L. Slifkin: I have a question about how the irradiation-induced
vacancies get to the oxygen atoms. For the case of germanium,
subtracting Logan's value for the creation energy of vacancies,
which is a little over 2 ev, from the self-diffusion activation energy
of less than 3 ev, one obtains a little less than 1 ev for the mobility
energy. This gives, for the temperature range in which the A
centers are produced, a frequency of vacancy motion of about 1
per day. It might be expected to be even less in silicon, which has
a higher melting point and, presumably, a higher activation
energy for vacancy motion. It is, therefore, a bit difficult to see
how the vacancy can diffuse to the oxygen atom in silicon at such
a low temperature.
H. Brooks: Is it absolutely excluded that the oxygen moves?
A. G. Tweet: Although oxygen is interstitial, it is bonded and,
therefore, probably does not move. I refer you to work of Logan
and Fuller.
G. D. Watkins: Is it possible that in self-diffusion measurements
one is dealing with aggregates of vacancies and that the only
place where you really form a single vacancy is in the radiation
experiment?
H. Brooks: The binding energy of the vacancies would have to
be pretty high.
A. G. Tweet: If diffusion proceeded by both vacancies and
divacancies, a break would appear in the activation energy curve
for self-diffusion. The fact that no such break has been observed
implies that diffusion is all going by divacancies with a binding
energy of 3 ev or more, or by single vacancies.
G. H. Vineyard: At least two things about these annealing
processes deserve comment. In your Fig. 5 two stages are visible
in the formation of the A center. No simple movement of one kind
of defect is going to explain that. Second is the fact that on anneal
ing to 3000K only 3% of the number of centers expected for room
temperature irradiation remain. Perhaps you have an explanation. G. D. Watkins: I think Dr. Wertheim will be able to say some
thing about the second of your comments.
(G. K. Wertheim presented some results concerning the tem
perature dependence of the introduction rate of the A center in
silicon for electron bombardment. These results have been sub
mitted for publication in The Physical Review.)
J. Rothstein: An oxygen atom coupled to a vacancy should have
a dipole moment. It is conceivable that this might be detected by
measuring dielectric loss. Moreover, one might pick it up by
measuring infrared absorption. It seems possible that if one
applied a polarizing field one might actually get a Stark-splitting
of the level.
Nature of Bombardment Damage and Energy
Levels in Semiconductors
J. H. CRAWFORD, JR., AND J. W. CLELAND
R. W. Balluffi: Would you expect a large range of sizes of the
damage spikes?
J. H. Crawford: Yes. A large range would be expected. This is
the reason that two different sizes were shown in the figure; the
large size is completely blocking, whereas the small size is not.
What we have done here, in essence, is to arbitrarily split the
damage into two groups. The first group is composed of isolated
defects or small clusters which produce the effects of isolated
energy levels, whereas the large groups are envisioned as affecting
almost entirely those properties requiring current transport.
W. L. Brown: I would like to comment on your p-type bombard
ment results. These are in contrast to what we have seen with
electron bombardment of specimens at liquid nitrogen tempera
ture. While you observe an increase in hole concentration we
observe only a decrease in hole concentration over the entire tem
perature range when the bombardment occurs at 77°K.
R. L. Cummerow: Our results with electron bombardment at
room temperature in which n-type material is converted to p type
seem to agree with your proposal of a limiting Fermi level .\*.
The question I would like to ask is, does the mobility determina
tion support your two-level scheme? That is to say, trapping with
the Fermi level halfway between these two levels would require
that mobility decrease with long bombardment, whereas if a
single level and annealing were responsible for the limiting con
ductivity. the mobility should show no further decrease.
J. H. Crawford: We have insufficient data on the mobility in
the impurity scattering range to decide this point.
H. Y. Fan: Do I understand that you assume that the energy
levels near the center of the gap that determine the behavior of
converted material are produced at about the same rate as the
0.2-ev level below the conduction band?
J. H. Crawford: Perhaps. The only way one can obtain a limiting
Fermi level value is by the introduction of a filled level near 0.2 ev
above the valence band and an empty state quite near the center
of the gap at the same rate.
H. Y. Fan: In the case of electron bombardment our results
indicate that the level near the middle of the gap is produced in
relatively low abundance.
J. H. Crawford: For the gamma irradiation we know simply
from the shape of the Hall coefficient vs temperature curve that
there is a deep vacant state present in concentration equal to the
0.2-ev state below the conduction band.
R. L. Cummerow: I have some evidence that there is a level in
the center of the gap, but it is somewhat indirect and was obtained
by analyzing the potential variation in the junction produced by
electron bombardment.
Precipitation, Quenching, and Dislocations
A. G. TWE>;T
W. L. Brown: It has been shown that electron irradiation altered
the precipitation of lithium in a manner that would indicate that
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IP: 130.88.53.18 On: Mon, 24 Nov 2014 15:41:501320 DISCUSSION
the vacancies, or some complexes of vacancies, served as precipi
tation sites.
A. G. Tweet: That is correct. The effect of vacancy aggregates
or of certain pre-precipitates on precipitation rates is a subject of
great interest.
M. S. Wechsler: Several descriptions have been given at this
Conference of the precise and informative way in which diffusion
controlled reactions may be studied in semiconductors. Since we
may anticipate experiments on the effect of radiation on such re
actions in semiconductors, it may be of interest to mention some
work of this type of metal alloy systems, done at Oak Ridge
National Laboratory in collaboration with R. H. Kernohan and
D. S. Billington.
Two alloys, in particular, have been studied. The first of these,
Ni-Be (14.5 at. % Be), exhibits a precipitation reaction in which
the intermetallic compound NiBe precipitates from the primary
substitutional solid solution. The amount of Be remaining in solid
solution during the aging can be determined rather conveniently
by measuring the ferromagnetic Curie temperature, which in
creases rapidly upon the reduction of Be in solid solution. The
second alloy whose behavior upon neutron irradiation has been
investigated is Cu-AI (15 at. % AI). It has been found that a
diffusion-controlled reaction is triggered or accelerated upon irra
diation at 35-45°C that causes a decrease of about 2% in the
electrical resistivity. Measurements on unirradiated samples at
elevated temperatures and upon quenching indicate that the
alloy is initially in a metastable state and the irradiation causes
the return to equilibrium at lower temperatures than is normally
possible. However, the nature of this metastability has not been
fully established.
Despite the fact that the reactions in these alloys are probably
rather different in their essential nature, striking similarities are
observed in the effect of neutron irradiation on them. This is
illustrated by two types of experiments. In the first of these,
samples are quenched and the diffusion-controlled reaction is
allowed to go on in the reactor at a given temperature. The re
action curve that results is compared with one obtained for an
experiment at the same temperature outside of the reactor. In the
case of both Ni-Be and Cu-AI, in-pile measurements show that
the reaction proceeds considerably more rapidly in the neutron
flux environment. This result had previously been established for
Ni-Be as a result of before-after measurements. [Kernohan,
Billington, and Lewis, J. App!. Phys. 27, 40 (1956)]. In the sec
ond type of irradiation experiment, the samples were irradiated
at temperatures considerably below the reaction temperatures;
and the annealing or aging was carried out by raising the tempera
ture after the irradiation was completed. Under these conditions,
the curious result obtained for both alloys is that, although the
reaction starts more rapidly for the samples that receive the in
termediate low-temperature irradiation, the reaction becomes
stabilized before its completion. Hence, the reaction curves for
the irradiated samples eventually lag behind those for the un
irradiated ones.
Thus, the experiments on these alloys show that when samples
are irradiated at reaction temperatures a clear acceleration of the
reaction takes place. This is perhaps not surprising. Regions of
local radiation damage should provide efficient centers at which
the reaction can be nucleated. Furthermore, in those cases in which
diffusion occurs by a vacancy mechanism, the radiation-produced
vacancies should enhance diffusion, thereby accelerating the
growth of the reaction product. However, the effect of prior irra
diation below the reaction range of temperatures is more com
plicated, due to a type of stabilization that takes place after the
reaction is underway. This stabilization may correspond to the
formation of metastable centers, similar to those discussed by
Tweet for the precipitation of Cu in Ge. In any case, experiments
conducted in this fashion indicate that the irradiation has a
unique effect on diffusion-controlled reactions, which is not equi
valent to simply raising the reaction temperature. The broad similarity in the observations for Ni-Be and Cu-Al
suggests that a similar behavior may be found for the effect of
radiation on reactions in semiconductors. A comparison of experi
ments on metals and semiconductors should lead to a better un
derstanding of radiation effects in both types of materials.
G. K. Wertheim (to M. S. Wechsler): Just one brief question.
Do you know to what extent the effects you observe are compli
cated by the similarity of a number of kinds of radiation damage
centers and by the precipitation of dispersed impurities?
M. S. Wechsler: We are not able to say anything as yet about
the detailed mechanism of the radiation effects. Thus, on the as
sumption that the radiation-enhancement occurs chiefly because
of more effective nucleation, little can be concluded concerning
the structure of the nucleation center. However, it is significant
that the low temperature irradiation itself has no effect on the
reaction; the temperature must be raised before its effect is felt.
H. Reiss: I should like to comment on the possibility that a
break in the curve of the type you (Wechsler) described may not
be due to metastable traps. This could occur if you had a fairly
inhomogeneous distribution of nuclei. Certain regions would pre
cipitate very quickly and other regions less quickly. But the re
sistivity measurements would average the effects of the separate
regions, so that it would seem as though the precipitation had
slowed down abruptly. So unless you have a uniform distribution
of damage you have to be careful about postulating traps.
A. G. Tweet: Another possibility for the interpretation of the
enhanced precipitation observed (by Wechsler) when aging pro
ceeds in the pile serves to call attention to how intricate this sub
ject can be. Almost all of us have talked about the limitation of
precipitation by diffusion. However, there might be another factor
that can limit the precipitation. A strain field built up around the
precipitate particle may serve as a repulsive potential which re
tards diffusion toward the precipitate. It is not impossible that
the enhanced precipitation for the in-pile aging is caused by the
fact that this strain field is gotten rid of by the formation of the
small precipitate particles produced as a consequence of the
irradiation.
Radiation Defect Annealing
W. L. BROW'"'
G. J. Dienes: The following applies to the papers presented by
R. W. Balluffi and W. Brown. It has been generally assumed that
the lattice collapses around the vacancy in germanium, similar to
a metal. I once looked at diamond and argued that carbon-carbon
double bonds might actually produce a contraction away from the
lattice, that is, a lattice expansion around the vacancy. The calcu
lations were very crude and were not published.
H. Reiss: I think that enough evidence has been presented at
this Conference to indicate that the acceptor considered by W. L.
Brown in these annealing studies cannot possibly be a vacancy.
We know that, for many substances, the log of the diffusion co
efficient vs inverse temperature is a straight line for many decades.
We also have a rough idea of the self-diffusion activation energy
in germanium and silicon, and this predicts that the vacancy does
not move appreciably at very low temperatures. On the other
hand, a vacancy is not actually trapped by an atom such as Sb,
because the Sb will actually diffuse more rapidly by the vacancy
method than it would by self-diffusion. If the acceptor is an inter
stitial, however, it will be trapped. If it is paired in an ion-pairing
sense, it will pair less with Sb than As since the Sb has a larger
radius and a smaller binding energy. If the acceptor is a charged
interstitial, one may get an over-all reduction in the activation
energy for diffusion, and the interstitial may therefore diffuse
more rapidly. I therefore feel that the acceptor in question cannot
be ascribed to a vacancy under these conditions.
G. D. Watkins: I favor the mechanism of vacancy diffusion at
all temperatures. Is it possible that annealing under consideration
is not diffusion-limited, but trap limited? That is, the long-range
motion may still occur at low temperatures, but trapping results;
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1.1735321.pdf | Magnetic Susceptibility and Electron Spin Resonance—Experimental
G. D. Watkins
Citation: Journal of Applied Physics 30, 1319 (1959); doi: 10.1063/1.1735321
View online: http://dx.doi.org/10.1063/1.1735321
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Technique for magnetic susceptibility determination in the highly doped semiconductors by electron
spin resonance
AIP Conf. Proc. 1610, 119 (2014); 10.1063/1.4893521
Magnetic properties of nested carbon nanostructures studied by electron spin resonance and
magnetic susceptibility measurements
J. Appl. Phys. 80, 1020 (1996); 10.1063/1.362835
Absolute Determination of Change in Magnetic Susceptibility Due to Electron Spin Resonance
Rev. Sci. Instrum. 44, 1118 (1973); 10.1063/1.1686314
Magnetic Susceptibility and Electron Spin Resonance—Experimental
J. Appl. Phys. 30, 1319 (1959); 10.1063/1.1735320
Magnetic Susceptibility and Electron Spin Resonance
J. Appl. Phys. 30, 1318 (1959); 10.1063/1.1735319
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 137.112.220.85 On: Sat, 13 Dec 2014 14:55:12DISCUSSION 1319
A. G. Tweet: This is a comment concerning interpretation of
extremely strong temperature dependence of mobility. Under
certain conditions such effects may be caused by inhomogeneities
in samples rather than a large density of defects. For example, in
material doped with only 1015 to 1016 nickel atoms per cm3, re
sults that are uninterpretable have been obtained. Further in
vestigation has shown that this is caused by an inhomogeneous
distribution of impurity.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. BEMSKI
M. Niseno.fJ: We have found spin-resonance absorptions in
many samples of neutron-irradiated silicon. Irradiation until the
Fermi level was near the center of the forbidden gap produced a
reproducible pattern, independent of the original donor or acceptor
concentration. Under annealing, while the Fermi level remained
at the center of the gap, two changes took place in the spin-reso
nance pattern. However, even after annealing at 500°C two
centers still remained, indicating that at least four types of cen
ters are introduced by neutron irradiation at room temperature.
We also found that the intensity of the lines found in these ma
terials increased with irradiation (the range of flux used was
1011-10'9), indicating that these resonances are produced by a re
distribution of electrons on defects rather than by electrons origi
nating from impurities. In samples whose post-irradiation Fermi
level was closer than 0.1 ev to the conduction band or 0.2 ev to
the valence band, different types of spin-resonance absorption
patterns were seen.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. D. WATKINS
L. Slifkin: I have a question about how the irradiation-induced
vacancies get to the oxygen atoms. For the case of germanium,
subtracting Logan's value for the creation energy of vacancies,
which is a little over 2 ev, from the self-diffusion activation energy
of less than 3 ev, one obtains a little less than 1 ev for the mobility
energy. This gives, for the temperature range in which the A
centers are produced, a frequency of vacancy motion of about 1
per day. It might be expected to be even less in silicon, which has
a higher melting point and, presumably, a higher activation
energy for vacancy motion. It is, therefore, a bit difficult to see
how the vacancy can diffuse to the oxygen atom in silicon at such
a low temperature.
H. Brooks: Is it absolutely excluded that the oxygen moves?
A. G. Tweet: Although oxygen is interstitial, it is bonded and,
therefore, probably does not move. I refer you to work of Logan
and Fuller.
G. D. Watkins: Is it possible that in self-diffusion measurements
one is dealing with aggregates of vacancies and that the only
place where you really form a single vacancy is in the radiation
experiment?
H. Brooks: The binding energy of the vacancies would have to
be pretty high.
A. G. Tweet: If diffusion proceeded by both vacancies and
divacancies, a break would appear in the activation energy curve
for self-diffusion. The fact that no such break has been observed
implies that diffusion is all going by divacancies with a binding
energy of 3 ev or more, or by single vacancies.
G. H. Vineyard: At least two things about these annealing
processes deserve comment. In your Fig. 5 two stages are visible
in the formation of the A center. No simple movement of one kind
of defect is going to explain that. Second is the fact that on anneal
ing to 3000K only 3% of the number of centers expected for room
temperature irradiation remain. Perhaps you have an explanation. G. D. Watkins: I think Dr. Wertheim will be able to say some
thing about the second of your comments.
(G. K. Wertheim presented some results concerning the tem
perature dependence of the introduction rate of the A center in
silicon for electron bombardment. These results have been sub
mitted for publication in The Physical Review.)
J. Rothstein: An oxygen atom coupled to a vacancy should have
a dipole moment. It is conceivable that this might be detected by
measuring dielectric loss. Moreover, one might pick it up by
measuring infrared absorption. It seems possible that if one
applied a polarizing field one might actually get a Stark-splitting
of the level.
Nature of Bombardment Damage and Energy
Levels in Semiconductors
J. H. CRAWFORD, JR., AND J. W. CLELAND
R. W. Balluffi: Would you expect a large range of sizes of the
damage spikes?
J. H. Crawford: Yes. A large range would be expected. This is
the reason that two different sizes were shown in the figure; the
large size is completely blocking, whereas the small size is not.
What we have done here, in essence, is to arbitrarily split the
damage into two groups. The first group is composed of isolated
defects or small clusters which produce the effects of isolated
energy levels, whereas the large groups are envisioned as affecting
almost entirely those properties requiring current transport.
W. L. Brown: I would like to comment on your p-type bombard
ment results. These are in contrast to what we have seen with
electron bombardment of specimens at liquid nitrogen tempera
ture. While you observe an increase in hole concentration we
observe only a decrease in hole concentration over the entire tem
perature range when the bombardment occurs at 77°K.
R. L. Cummerow: Our results with electron bombardment at
room temperature in which n-type material is converted to p type
seem to agree with your proposal of a limiting Fermi level .\*.
The question I would like to ask is, does the mobility determina
tion support your two-level scheme? That is to say, trapping with
the Fermi level halfway between these two levels would require
that mobility decrease with long bombardment, whereas if a
single level and annealing were responsible for the limiting con
ductivity. the mobility should show no further decrease.
J. H. Crawford: We have insufficient data on the mobility in
the impurity scattering range to decide this point.
H. Y. Fan: Do I understand that you assume that the energy
levels near the center of the gap that determine the behavior of
converted material are produced at about the same rate as the
0.2-ev level below the conduction band?
J. H. Crawford: Perhaps. The only way one can obtain a limiting
Fermi level value is by the introduction of a filled level near 0.2 ev
above the valence band and an empty state quite near the center
of the gap at the same rate.
H. Y. Fan: In the case of electron bombardment our results
indicate that the level near the middle of the gap is produced in
relatively low abundance.
J. H. Crawford: For the gamma irradiation we know simply
from the shape of the Hall coefficient vs temperature curve that
there is a deep vacant state present in concentration equal to the
0.2-ev state below the conduction band.
R. L. Cummerow: I have some evidence that there is a level in
the center of the gap, but it is somewhat indirect and was obtained
by analyzing the potential variation in the junction produced by
electron bombardment.
Precipitation, Quenching, and Dislocations
A. G. TWE>;T
W. L. Brown: It has been shown that electron irradiation altered
the precipitation of lithium in a manner that would indicate that
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 137.112.220.85 On: Sat, 13 Dec 2014 14:55:121320 DISCUSSION
the vacancies, or some complexes of vacancies, served as precipi
tation sites.
A. G. Tweet: That is correct. The effect of vacancy aggregates
or of certain pre-precipitates on precipitation rates is a subject of
great interest.
M. S. Wechsler: Several descriptions have been given at this
Conference of the precise and informative way in which diffusion
controlled reactions may be studied in semiconductors. Since we
may anticipate experiments on the effect of radiation on such re
actions in semiconductors, it may be of interest to mention some
work of this type of metal alloy systems, done at Oak Ridge
National Laboratory in collaboration with R. H. Kernohan and
D. S. Billington.
Two alloys, in particular, have been studied. The first of these,
Ni-Be (14.5 at. % Be), exhibits a precipitation reaction in which
the intermetallic compound NiBe precipitates from the primary
substitutional solid solution. The amount of Be remaining in solid
solution during the aging can be determined rather conveniently
by measuring the ferromagnetic Curie temperature, which in
creases rapidly upon the reduction of Be in solid solution. The
second alloy whose behavior upon neutron irradiation has been
investigated is Cu-AI (15 at. % AI). It has been found that a
diffusion-controlled reaction is triggered or accelerated upon irra
diation at 35-45°C that causes a decrease of about 2% in the
electrical resistivity. Measurements on unirradiated samples at
elevated temperatures and upon quenching indicate that the
alloy is initially in a metastable state and the irradiation causes
the return to equilibrium at lower temperatures than is normally
possible. However, the nature of this metastability has not been
fully established.
Despite the fact that the reactions in these alloys are probably
rather different in their essential nature, striking similarities are
observed in the effect of neutron irradiation on them. This is
illustrated by two types of experiments. In the first of these,
samples are quenched and the diffusion-controlled reaction is
allowed to go on in the reactor at a given temperature. The re
action curve that results is compared with one obtained for an
experiment at the same temperature outside of the reactor. In the
case of both Ni-Be and Cu-AI, in-pile measurements show that
the reaction proceeds considerably more rapidly in the neutron
flux environment. This result had previously been established for
Ni-Be as a result of before-after measurements. [Kernohan,
Billington, and Lewis, J. App!. Phys. 27, 40 (1956)]. In the sec
ond type of irradiation experiment, the samples were irradiated
at temperatures considerably below the reaction temperatures;
and the annealing or aging was carried out by raising the tempera
ture after the irradiation was completed. Under these conditions,
the curious result obtained for both alloys is that, although the
reaction starts more rapidly for the samples that receive the in
termediate low-temperature irradiation, the reaction becomes
stabilized before its completion. Hence, the reaction curves for
the irradiated samples eventually lag behind those for the un
irradiated ones.
Thus, the experiments on these alloys show that when samples
are irradiated at reaction temperatures a clear acceleration of the
reaction takes place. This is perhaps not surprising. Regions of
local radiation damage should provide efficient centers at which
the reaction can be nucleated. Furthermore, in those cases in which
diffusion occurs by a vacancy mechanism, the radiation-produced
vacancies should enhance diffusion, thereby accelerating the
growth of the reaction product. However, the effect of prior irra
diation below the reaction range of temperatures is more com
plicated, due to a type of stabilization that takes place after the
reaction is underway. This stabilization may correspond to the
formation of metastable centers, similar to those discussed by
Tweet for the precipitation of Cu in Ge. In any case, experiments
conducted in this fashion indicate that the irradiation has a
unique effect on diffusion-controlled reactions, which is not equi
valent to simply raising the reaction temperature. The broad similarity in the observations for Ni-Be and Cu-Al
suggests that a similar behavior may be found for the effect of
radiation on reactions in semiconductors. A comparison of experi
ments on metals and semiconductors should lead to a better un
derstanding of radiation effects in both types of materials.
G. K. Wertheim (to M. S. Wechsler): Just one brief question.
Do you know to what extent the effects you observe are compli
cated by the similarity of a number of kinds of radiation damage
centers and by the precipitation of dispersed impurities?
M. S. Wechsler: We are not able to say anything as yet about
the detailed mechanism of the radiation effects. Thus, on the as
sumption that the radiation-enhancement occurs chiefly because
of more effective nucleation, little can be concluded concerning
the structure of the nucleation center. However, it is significant
that the low temperature irradiation itself has no effect on the
reaction; the temperature must be raised before its effect is felt.
H. Reiss: I should like to comment on the possibility that a
break in the curve of the type you (Wechsler) described may not
be due to metastable traps. This could occur if you had a fairly
inhomogeneous distribution of nuclei. Certain regions would pre
cipitate very quickly and other regions less quickly. But the re
sistivity measurements would average the effects of the separate
regions, so that it would seem as though the precipitation had
slowed down abruptly. So unless you have a uniform distribution
of damage you have to be careful about postulating traps.
A. G. Tweet: Another possibility for the interpretation of the
enhanced precipitation observed (by Wechsler) when aging pro
ceeds in the pile serves to call attention to how intricate this sub
ject can be. Almost all of us have talked about the limitation of
precipitation by diffusion. However, there might be another factor
that can limit the precipitation. A strain field built up around the
precipitate particle may serve as a repulsive potential which re
tards diffusion toward the precipitate. It is not impossible that
the enhanced precipitation for the in-pile aging is caused by the
fact that this strain field is gotten rid of by the formation of the
small precipitate particles produced as a consequence of the
irradiation.
Radiation Defect Annealing
W. L. BROW'"'
G. J. Dienes: The following applies to the papers presented by
R. W. Balluffi and W. Brown. It has been generally assumed that
the lattice collapses around the vacancy in germanium, similar to
a metal. I once looked at diamond and argued that carbon-carbon
double bonds might actually produce a contraction away from the
lattice, that is, a lattice expansion around the vacancy. The calcu
lations were very crude and were not published.
H. Reiss: I think that enough evidence has been presented at
this Conference to indicate that the acceptor considered by W. L.
Brown in these annealing studies cannot possibly be a vacancy.
We know that, for many substances, the log of the diffusion co
efficient vs inverse temperature is a straight line for many decades.
We also have a rough idea of the self-diffusion activation energy
in germanium and silicon, and this predicts that the vacancy does
not move appreciably at very low temperatures. On the other
hand, a vacancy is not actually trapped by an atom such as Sb,
because the Sb will actually diffuse more rapidly by the vacancy
method than it would by self-diffusion. If the acceptor is an inter
stitial, however, it will be trapped. If it is paired in an ion-pairing
sense, it will pair less with Sb than As since the Sb has a larger
radius and a smaller binding energy. If the acceptor is a charged
interstitial, one may get an over-all reduction in the activation
energy for diffusion, and the interstitial may therefore diffuse
more rapidly. I therefore feel that the acceptor in question cannot
be ascribed to a vacancy under these conditions.
G. D. Watkins: I favor the mechanism of vacancy diffusion at
all temperatures. Is it possible that annealing under consideration
is not diffusion-limited, but trap limited? That is, the long-range
motion may still occur at low temperatures, but trapping results;
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IP: 137.112.220.85 On: Sat, 13 Dec 2014 14:55:12 |
1.1735323.pdf | Precipitation, Quenching, and Dislocations
A. G. Tweet
Citation: Journal of Applied Physics 30, 1319 (1959); doi: 10.1063/1.1735323
View online: http://dx.doi.org/10.1063/1.1735323
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/8?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Dislocations and precipitates in gallium arsenide
J. Appl. Phys. 71, 620 (1992); 10.1063/1.351346
Interaction of a dislocation with a misfitting precipitate
J. Appl. Phys. 53, 8620 (1982); 10.1063/1.330459
Precipitation of Helium along Dislocations in Aluminum
J. Appl. Phys. 32, 1045 (1961); 10.1063/1.1736157
StressAssisted Precipitation on Dislocations
J. Appl. Phys. 30, 915 (1959); 10.1063/1.1735262
Copper Precipitation on Dislocations in Silicon
J. Appl. Phys. 27, 1193 (1956); 10.1063/1.1722229
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 141.100.75.156 On: Tue, 25 Nov 2014 10:41:31DISCUSSION 1319
A. G. Tweet: This is a comment concerning interpretation of
extremely strong temperature dependence of mobility. Under
certain conditions such effects may be caused by inhomogeneities
in samples rather than a large density of defects. For example, in
material doped with only 1015 to 1016 nickel atoms per cm3, re
sults that are uninterpretable have been obtained. Further in
vestigation has shown that this is caused by an inhomogeneous
distribution of impurity.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. BEMSKI
M. Niseno.fJ: We have found spin-resonance absorptions in
many samples of neutron-irradiated silicon. Irradiation until the
Fermi level was near the center of the forbidden gap produced a
reproducible pattern, independent of the original donor or acceptor
concentration. Under annealing, while the Fermi level remained
at the center of the gap, two changes took place in the spin-reso
nance pattern. However, even after annealing at 500°C two
centers still remained, indicating that at least four types of cen
ters are introduced by neutron irradiation at room temperature.
We also found that the intensity of the lines found in these ma
terials increased with irradiation (the range of flux used was
1011-10'9), indicating that these resonances are produced by a re
distribution of electrons on defects rather than by electrons origi
nating from impurities. In samples whose post-irradiation Fermi
level was closer than 0.1 ev to the conduction band or 0.2 ev to
the valence band, different types of spin-resonance absorption
patterns were seen.
Magnetic Susceptibility and Electron Spin
Resonance-Experimental
G. D. WATKINS
L. Slifkin: I have a question about how the irradiation-induced
vacancies get to the oxygen atoms. For the case of germanium,
subtracting Logan's value for the creation energy of vacancies,
which is a little over 2 ev, from the self-diffusion activation energy
of less than 3 ev, one obtains a little less than 1 ev for the mobility
energy. This gives, for the temperature range in which the A
centers are produced, a frequency of vacancy motion of about 1
per day. It might be expected to be even less in silicon, which has
a higher melting point and, presumably, a higher activation
energy for vacancy motion. It is, therefore, a bit difficult to see
how the vacancy can diffuse to the oxygen atom in silicon at such
a low temperature.
H. Brooks: Is it absolutely excluded that the oxygen moves?
A. G. Tweet: Although oxygen is interstitial, it is bonded and,
therefore, probably does not move. I refer you to work of Logan
and Fuller.
G. D. Watkins: Is it possible that in self-diffusion measurements
one is dealing with aggregates of vacancies and that the only
place where you really form a single vacancy is in the radiation
experiment?
H. Brooks: The binding energy of the vacancies would have to
be pretty high.
A. G. Tweet: If diffusion proceeded by both vacancies and
divacancies, a break would appear in the activation energy curve
for self-diffusion. The fact that no such break has been observed
implies that diffusion is all going by divacancies with a binding
energy of 3 ev or more, or by single vacancies.
G. H. Vineyard: At least two things about these annealing
processes deserve comment. In your Fig. 5 two stages are visible
in the formation of the A center. No simple movement of one kind
of defect is going to explain that. Second is the fact that on anneal
ing to 3000K only 3% of the number of centers expected for room
temperature irradiation remain. Perhaps you have an explanation. G. D. Watkins: I think Dr. Wertheim will be able to say some
thing about the second of your comments.
(G. K. Wertheim presented some results concerning the tem
perature dependence of the introduction rate of the A center in
silicon for electron bombardment. These results have been sub
mitted for publication in The Physical Review.)
J. Rothstein: An oxygen atom coupled to a vacancy should have
a dipole moment. It is conceivable that this might be detected by
measuring dielectric loss. Moreover, one might pick it up by
measuring infrared absorption. It seems possible that if one
applied a polarizing field one might actually get a Stark-splitting
of the level.
Nature of Bombardment Damage and Energy
Levels in Semiconductors
J. H. CRAWFORD, JR., AND J. W. CLELAND
R. W. Balluffi: Would you expect a large range of sizes of the
damage spikes?
J. H. Crawford: Yes. A large range would be expected. This is
the reason that two different sizes were shown in the figure; the
large size is completely blocking, whereas the small size is not.
What we have done here, in essence, is to arbitrarily split the
damage into two groups. The first group is composed of isolated
defects or small clusters which produce the effects of isolated
energy levels, whereas the large groups are envisioned as affecting
almost entirely those properties requiring current transport.
W. L. Brown: I would like to comment on your p-type bombard
ment results. These are in contrast to what we have seen with
electron bombardment of specimens at liquid nitrogen tempera
ture. While you observe an increase in hole concentration we
observe only a decrease in hole concentration over the entire tem
perature range when the bombardment occurs at 77°K.
R. L. Cummerow: Our results with electron bombardment at
room temperature in which n-type material is converted to p type
seem to agree with your proposal of a limiting Fermi level .\*.
The question I would like to ask is, does the mobility determina
tion support your two-level scheme? That is to say, trapping with
the Fermi level halfway between these two levels would require
that mobility decrease with long bombardment, whereas if a
single level and annealing were responsible for the limiting con
ductivity. the mobility should show no further decrease.
J. H. Crawford: We have insufficient data on the mobility in
the impurity scattering range to decide this point.
H. Y. Fan: Do I understand that you assume that the energy
levels near the center of the gap that determine the behavior of
converted material are produced at about the same rate as the
0.2-ev level below the conduction band?
J. H. Crawford: Perhaps. The only way one can obtain a limiting
Fermi level value is by the introduction of a filled level near 0.2 ev
above the valence band and an empty state quite near the center
of the gap at the same rate.
H. Y. Fan: In the case of electron bombardment our results
indicate that the level near the middle of the gap is produced in
relatively low abundance.
J. H. Crawford: For the gamma irradiation we know simply
from the shape of the Hall coefficient vs temperature curve that
there is a deep vacant state present in concentration equal to the
0.2-ev state below the conduction band.
R. L. Cummerow: I have some evidence that there is a level in
the center of the gap, but it is somewhat indirect and was obtained
by analyzing the potential variation in the junction produced by
electron bombardment.
Precipitation, Quenching, and Dislocations
A. G. TWE>;T
W. L. Brown: It has been shown that electron irradiation altered
the precipitation of lithium in a manner that would indicate that
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ]
IP: 141.100.75.156 On: Tue, 25 Nov 2014 10:41:311320 DISCUSSION
the vacancies, or some complexes of vacancies, served as precipi
tation sites.
A. G. Tweet: That is correct. The effect of vacancy aggregates
or of certain pre-precipitates on precipitation rates is a subject of
great interest.
M. S. Wechsler: Several descriptions have been given at this
Conference of the precise and informative way in which diffusion
controlled reactions may be studied in semiconductors. Since we
may anticipate experiments on the effect of radiation on such re
actions in semiconductors, it may be of interest to mention some
work of this type of metal alloy systems, done at Oak Ridge
National Laboratory in collaboration with R. H. Kernohan and
D. S. Billington.
Two alloys, in particular, have been studied. The first of these,
Ni-Be (14.5 at. % Be), exhibits a precipitation reaction in which
the intermetallic compound NiBe precipitates from the primary
substitutional solid solution. The amount of Be remaining in solid
solution during the aging can be determined rather conveniently
by measuring the ferromagnetic Curie temperature, which in
creases rapidly upon the reduction of Be in solid solution. The
second alloy whose behavior upon neutron irradiation has been
investigated is Cu-AI (15 at. % AI). It has been found that a
diffusion-controlled reaction is triggered or accelerated upon irra
diation at 35-45°C that causes a decrease of about 2% in the
electrical resistivity. Measurements on unirradiated samples at
elevated temperatures and upon quenching indicate that the
alloy is initially in a metastable state and the irradiation causes
the return to equilibrium at lower temperatures than is normally
possible. However, the nature of this metastability has not been
fully established.
Despite the fact that the reactions in these alloys are probably
rather different in their essential nature, striking similarities are
observed in the effect of neutron irradiation on them. This is
illustrated by two types of experiments. In the first of these,
samples are quenched and the diffusion-controlled reaction is
allowed to go on in the reactor at a given temperature. The re
action curve that results is compared with one obtained for an
experiment at the same temperature outside of the reactor. In the
case of both Ni-Be and Cu-AI, in-pile measurements show that
the reaction proceeds considerably more rapidly in the neutron
flux environment. This result had previously been established for
Ni-Be as a result of before-after measurements. [Kernohan,
Billington, and Lewis, J. App!. Phys. 27, 40 (1956)]. In the sec
ond type of irradiation experiment, the samples were irradiated
at temperatures considerably below the reaction temperatures;
and the annealing or aging was carried out by raising the tempera
ture after the irradiation was completed. Under these conditions,
the curious result obtained for both alloys is that, although the
reaction starts more rapidly for the samples that receive the in
termediate low-temperature irradiation, the reaction becomes
stabilized before its completion. Hence, the reaction curves for
the irradiated samples eventually lag behind those for the un
irradiated ones.
Thus, the experiments on these alloys show that when samples
are irradiated at reaction temperatures a clear acceleration of the
reaction takes place. This is perhaps not surprising. Regions of
local radiation damage should provide efficient centers at which
the reaction can be nucleated. Furthermore, in those cases in which
diffusion occurs by a vacancy mechanism, the radiation-produced
vacancies should enhance diffusion, thereby accelerating the
growth of the reaction product. However, the effect of prior irra
diation below the reaction range of temperatures is more com
plicated, due to a type of stabilization that takes place after the
reaction is underway. This stabilization may correspond to the
formation of metastable centers, similar to those discussed by
Tweet for the precipitation of Cu in Ge. In any case, experiments
conducted in this fashion indicate that the irradiation has a
unique effect on diffusion-controlled reactions, which is not equi
valent to simply raising the reaction temperature. The broad similarity in the observations for Ni-Be and Cu-Al
suggests that a similar behavior may be found for the effect of
radiation on reactions in semiconductors. A comparison of experi
ments on metals and semiconductors should lead to a better un
derstanding of radiation effects in both types of materials.
G. K. Wertheim (to M. S. Wechsler): Just one brief question.
Do you know to what extent the effects you observe are compli
cated by the similarity of a number of kinds of radiation damage
centers and by the precipitation of dispersed impurities?
M. S. Wechsler: We are not able to say anything as yet about
the detailed mechanism of the radiation effects. Thus, on the as
sumption that the radiation-enhancement occurs chiefly because
of more effective nucleation, little can be concluded concerning
the structure of the nucleation center. However, it is significant
that the low temperature irradiation itself has no effect on the
reaction; the temperature must be raised before its effect is felt.
H. Reiss: I should like to comment on the possibility that a
break in the curve of the type you (Wechsler) described may not
be due to metastable traps. This could occur if you had a fairly
inhomogeneous distribution of nuclei. Certain regions would pre
cipitate very quickly and other regions less quickly. But the re
sistivity measurements would average the effects of the separate
regions, so that it would seem as though the precipitation had
slowed down abruptly. So unless you have a uniform distribution
of damage you have to be careful about postulating traps.
A. G. Tweet: Another possibility for the interpretation of the
enhanced precipitation observed (by Wechsler) when aging pro
ceeds in the pile serves to call attention to how intricate this sub
ject can be. Almost all of us have talked about the limitation of
precipitation by diffusion. However, there might be another factor
that can limit the precipitation. A strain field built up around the
precipitate particle may serve as a repulsive potential which re
tards diffusion toward the precipitate. It is not impossible that
the enhanced precipitation for the in-pile aging is caused by the
fact that this strain field is gotten rid of by the formation of the
small precipitate particles produced as a consequence of the
irradiation.
Radiation Defect Annealing
W. L. BROW'"'
G. J. Dienes: The following applies to the papers presented by
R. W. Balluffi and W. Brown. It has been generally assumed that
the lattice collapses around the vacancy in germanium, similar to
a metal. I once looked at diamond and argued that carbon-carbon
double bonds might actually produce a contraction away from the
lattice, that is, a lattice expansion around the vacancy. The calcu
lations were very crude and were not published.
H. Reiss: I think that enough evidence has been presented at
this Conference to indicate that the acceptor considered by W. L.
Brown in these annealing studies cannot possibly be a vacancy.
We know that, for many substances, the log of the diffusion co
efficient vs inverse temperature is a straight line for many decades.
We also have a rough idea of the self-diffusion activation energy
in germanium and silicon, and this predicts that the vacancy does
not move appreciably at very low temperatures. On the other
hand, a vacancy is not actually trapped by an atom such as Sb,
because the Sb will actually diffuse more rapidly by the vacancy
method than it would by self-diffusion. If the acceptor is an inter
stitial, however, it will be trapped. If it is paired in an ion-pairing
sense, it will pair less with Sb than As since the Sb has a larger
radius and a smaller binding energy. If the acceptor is a charged
interstitial, one may get an over-all reduction in the activation
energy for diffusion, and the interstitial may therefore diffuse
more rapidly. I therefore feel that the acceptor in question cannot
be ascribed to a vacancy under these conditions.
G. D. Watkins: I favor the mechanism of vacancy diffusion at
all temperatures. Is it possible that annealing under consideration
is not diffusion-limited, but trap limited? That is, the long-range
motion may still occur at low temperatures, but trapping results;
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IP: 141.100.75.156 On: Tue, 25 Nov 2014 10:41:31 |
1.1725627.pdf | Spectroscopy of Silicon Carbide and Silicon Vapors Trapped in Neon and Argon
Matrices at 4° and 20°K
William Weltner Jr. and Donald McLeod Jr.
Citation: The Journal of Chemical Physics 41, 235 (1964); doi: 10.1063/1.1725627
View online: http://dx.doi.org/10.1063/1.1725627
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/41/1?ver=pdfcov
Published by the AIP Publishing
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Spectroscopy of 3, 4, 9, 10-perylenetetracarboxylic dianhydride (PTCDA) attached to rare gas samples:
Clusters vs. bulk matrices. II. Fluorescence emission spectroscopy
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Spectroscopy of 3, 4, 9, 10-perylenetetracarboxylic dianhydride (PTCDA) attached to rare gas samples:
Clusters vs. bulk matrices. I. Absorption spectroscopy
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J. Chem. Phys. 121, 7195 (2004); 10.1063/1.1787492
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J. Chem. Phys. 108, 1836 (1998); 10.1063/1.475561
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45THE JOURNAL OF CHEMICAL PHYSICS VOLUME 41, NUMBER 1 1 JULY 1964
Spectroscopy of Silicon Carbide and Silicon Vapors Trapped in Neon and Argon Matrices
at 40 and 200K
WILLIAM WELTNER, JR., AND DONALD McLEOD, JR.
Union Carbide Research Institute, Tarrytown, New York
(Received 13 March 1964)
The numerous molecules vaporizing from silicon carbide at 26000K and from silicon at 23000K have
been trapped in neon and argon matrices at 4° and 200K and studied spectroscopically in the infrared,
visible, and near-ultraviolet regions. The Siz and SiCz molecules have been observed, and less definitely,
also SizC, SizCs, Sis, and Si4• In the case of silicon carbide vaporization, the absorption spectrum of SiCz
appears strongly at 4963 A in neon and 4993 A in argon as compared with 4977 A in the gas. The spectrum
agrees with the gaseous observations of McKellar and Kleman, but with the addition of three weak but
distinct bands. It is interpreted as a 1IIu+---X 1~+ transition where the vibrational assignments in these
states are now as follows: 1~+, 111"=853, liZ" = 300, liS" = 1742 cm-1; III, 11,'=1015, liZ' = 230, 1Is'=1461 cm-1•
The vibrational structure in the upper state is anomalous in that it requires a large positive value for XIs'.
The spectrum of symmetrical SizC is believed to occur at 5300 A in argon with 111'=500 cm-1• An extensive
absorption spectrum between 6500 A and 5300 A is attributed to Si.:!Cs and has been partially analyzed. Tbe
infrared spectra of these matrices confirm the ground state frequencies of SiCz and lead to a tentative
assignment of the SizCs frequencies. In the case of silicon vaporization, 17 vibrational levels in the upper
state of the 3~u-+---X 3~"-transition of Siz at 4000 A, formerly seen in the gas by Douglas and by Verma
and Warsop, have been observed. Two other weak systems of bands at 4660 and 5700 A have been tenta
tively attributed to Sis and Si4, respectively. Other regularities in the carbon, silicon, and silicon-carbon
molecular series are discussed.
INTRODUCTION
THIS research is a continuation of our study of the
molecules found in the saturated vapor over high
temperature solids and in stellar and cometary atmos
pheres.!,2 The technique of matrix isolation in neon
and argon has again been utilized to allow a spectro
scopic study of these molecules at 4°K.
Silicon carbide is the solid material of principal in
terest here. It vaporizes at about 20000K to yield
molecules similar to those observed over carbon, the
subject of an earlier matrix study.2 The mass spectro
metric work of Drowart, de Maria, and Inghram3
has shown that the vaporization products of silicon
carbide are indeed numerous: they find Si (constitut
ing ",-,83 mole % of the vapor), SiC2 (",-,9%), Si2C
(",-,8%), Si2 (",-,0.2%), and Si2C2, SisC, Sb, SiC, Si2Cs
in detectable amounts. The many species are, of course,
highly undesirable for matrix isolation studies, and in
order to eliminate some uncertainties, it was advan
tageous also to trap silicon vapor since it contains a
significant proportion of Si2 and Sia.4
The SiC2 spectrum, seen in stars by several workers,,6
and first produced in the laboratory and identified by
Kleman,1 has been observed strongly in the matrices
1 W. Weltner, Jr., and J. R. W. Warn, J. Chern. Phys. 37, 292
(1962) .
2 W. Weltner, Jr., P. N. Walsh, and C. L. Angell, J. Chern.
Phys. 40, 1299 (1964); W. Weltner, Jr. and D. McLeod, Jr.,
ibid., p. 1305.
s J. Drowart, G. de Maria, and M. G. Inghram, J. Chern. Phys.
29, 1015 (1958).
4 R. E. Honig, J. Chern. Phys. 22, 1610 (1954).
6P. W. Merrill, Pub!. Astron. Soc. Pacific 38, 175 (1926);
R. F. Sanford, ibid., p. 177 (1926); C. D. Shane, Lick Obs. Bul!.
13, 123 (1928). in absorption and emission. The low-temperature spec
tra generally corroborate the gaseous observations but
require reinterpretation of the vibrational analysis. The
spectrum of Si2 (31;u ---X 31;g -transition) has been
observed, and, less positively, the spectra of Si2C, Si2C3,
Sis, and Si4•
Finally, we have discussed the similarities which
one expects between the properties of these molecules
and the carbon species C2 and C3 in the framework of
simple molecular orbital considerations.
EXPERIMENTAL
The same experimental apparatus and techniques
were used here as in the previous studies on boric
oxide and carbon vapors.!,2 For silicon, the only modi
fication was to align the axis of the tantalum resistance
heated cylinder in a vertical direction so that the liquid
silicon was retained in the volume below the effusion
hole. In the experiments with silicon carbide about
one-half of the vaporizations took place from a tantalum
effusion cell (t in. diameter by lr\ in. long), with car
bon liner, in the induction-heating unit. The remainder
utilized the resistance-heated tantalum cell, with a car
bon liner. Both black industrial-grade and pure straw
colored solid SiC were used, but no difference was
found in the observed absorption spectra. The fluores
cence spectrum of the matrix produced from the in
dustrial grade did show the presence of impurities.
The temperature measurements, made as previously
described,2 were not consistent with the observed spec
tra since the intensity of the bands varied under what
was often considered to be identical operating condi
tions. For this reason f numbers (oscillator strengths) 6 A. McKellar, J. Roy. Astron. Soc. Canada 41, 147 (1947).
7 B. Kleman, Astrophys. J. 123, 162 (1956). could not be reliably obtained. The temperature during
235
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45236 W. WELTNER, JR., AND D. McLEOD, JR.
3638
I
WAVELENGTHS IN A
FIG. 1. The H 3~u-+-X 3~g-transition of Si2 observed in a
neon matrix at 4°K.
the various SiC vaporizations varied from about 2500°
to 2850°C, and during the Si vaporizations from about
1900° to 2400°C.
Some broad tantalum atom lines appeared in ab
sorption and emission, but these could be identified
from the "blank" runs previously made.2
A run was also made with excess silicon added to
the solid silicon carbide in order to simulate the con
ditions of Drowart and de Maria,s but the silicon
apparently reacted rapidly with the carbon liner, the
spectrum being unchanged from that in which no sili
con was added.
Only the 0.5-m Ebert grating spectrophotometer2
was used for the visible work, so that the accuracy in
that region is limited to about ±1 A.
VAPORIZATION OF SILICON
The mass spectrometric work of Honig4 on the va
porization of silicon has established that atomic silicon
is the predominate vapor species and that Si2, Sia, and
Si4 molecules are present in lower but comparable
amounts. Our absorption spectrum of silicon vapor
trapped in a neon matrix at 4 OK exhibits at least
three distinct band systems beginning at about 3970,
4660, and 5700 A and going toward the violet. The
first system is definitely produced by Si2• The other
two are relatively weak but are probably to be at
tributed to Sia and Si4, respectively.
Si2 Absorption
A long progression of bands (each about 140 cm-1
wide) was observed beginning at 3974 A in neon and
increasing in intensity to a maximum at about 3700 A
(see Fig. 1). The position of these bands and the
associated frequency of about 260 cm-1 identify this
as the H 3~" -t-X 3~D -system of Si2 studied by Douglas9
and by Verma and Warsop.lO Table I shows the ap-proximate band positions observed for 17 vibrational
levels and the frequency differences.
The 0-0 band of the H-X system of Si2 appears to
occur at 25 156 cm-1 in a neon matrix, but it could be
that weaker bands at 24900 cm-1 or below have not
been observed. The position of the 0-0 band in the
gas is not definitely known, but Verma and WarsoplO
have given it as 24582.64 cm-1 if m=O applies in
Douglas' Table IV.9 (They have already shown that
n should be set equal to one in that table.) Hence for
Si2 the gas-matrix spectral shift for this transition can
not be definitely specified.
It is clear that the observation of the 3~u-t-3~g
transition in the matrix supports a 3~g-ground state
for the Si2 molecule. 9.10
Sia and Si4 Absorption
The vaporization of silicon produced a series of weak
bends between about 4700 and 4200 A in a neon
matrix. The maximum intensity among these bands
occurred near 4600-4400 A indicating a change in the
dimensions of the molecule upon excitation. Table II
gives the observed band positions which were, in some
cases, difficult to specify because of their rounded
shape. The bands appeared to be in groups of three
separated by about 110 em-I, with the individual
groups separated by about 310 em-I, as indicated in
the table.
As mentioned above, the mass spectrometer results4
indicate that these bands are most likely due to Sia
or Si4• Since only one vibrational frequency is observed
(if we attribute the groups of bands to matrix effects)
it seems likely that the observed molecule is Sia. The
frequency of 310 cm-1 would then be expected to be
vI', the symmetric stretching frequency. A frequency
of about this magnitude in the excited state is rea-
TABLE I. H 3~,,-+-X 3~"-transition of Si2 in a
neon matrix at 4°K.
V'· A(!) v (cm-1) .:lG'.+I(cm-1)
0 3974 25 156 260
1 3933.5 25 416 257
2 3894 25 673 257
3 3855.5 25 930 251
4 3818.5 26 181 249
5 3782.5 26 430 247
6 3747.5 26677 241
7 3714 26 918 237
8 3681.5 27 155 227
9 3651 27 382 227
10 3621 27609 231
11 3591 27 840 210 12 3564 28 050 211
13 3537.5 28 261 209 14 3511. 5 28 470 216?
15 3485? 28 686 195?
16 3461.5? 28 881 8 J. Drowart and G. de Maria, Silicon Carbide: A High Tem
perature Semiconductor, Proceedings of the Conference on Silicon
Carbide (Boston, Massachusetts, April 1959), edited by J. R.
O'Connor and J. Smiltens (Pergamon Press, Inc., New York,
1960), p. 16.
9 A. E. Douglas, Can. J. Phys. 33, 801 (1955). • This numbering is arbitrary since the 0-0 band has not been established (see
10 R. D. Verma and P. A. Warsop, Can. J. Phys. 41,152 (1963). text).
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45SPECTROSCOPY OF SILICON CARBIDE AND SILICON 237
TABLE II. Analysis of She?) absorption spectrum in a neon matrix at 4°K,
v' x(A) v (cm-1) x(A) v (cm-1) x(A) v (cm-1)
0 4658.5 21 460
313
1 4591.5 21 773 4566.5 21 893 ~4550 21 974
311 311 339
2 4527 22084 4502.5 22 204 ,...,,4480 22 313
306 307 ~318
3 4465 22 390 4441 22 511 ~4418 22 631
,...,,321 308 ~306
4 ,...,,4402 22 711 4381 22 819 4358.5 22 937
~311 309 304
5 ~4343 23 022 4322.5 23 128 4301.5 23 241
310
4245 23 551
• This vibrational level designation is arbitrary since the 0-0 band has not been identified.
sonable on the basis of VI" = 358 cm-l calculated by
Drowart, et al.3 for the ground state. Then we infer
that a different transition is occurring here than the
lIlu+-X ll;g+ observed for Cs,2·l1 and this is to be ex-pected from a comparison of the properties of Si2 and C2.
The molecular orbital configurations and electronic
states appropriate to linear Sis are probably the fol-
10wing12:
(3sl) 2(3s2) 2( 3pu g) 2( 3puu) 2 (3po-g) 2( 3P7ru) 2 ll;g+, Sl;g-, lLlg
••• (3po-g)2(3pu) (3P7rg) l.sl;u+, 1,3l;u -, 1,3Ll"
••• (3po-g) (3pu)2(3P7r g) 1,sIlg, etc .
••• (3po-g) (3pu)3 1,sIl".
[A configuration analogous to that of the ground state
of Cs results if the 3po-g orbital (which is outer-atom
outer-atom binding but inner-atom-outer-atom anti
bonding12) were at a higher energy than the tru orbital.]
These orbitals are similar to those in the Si2 molecule,9.lo
and as in that case, one would expect a Sl;g-ground
state. If a bonding 3P7ru electron is excited to an anti
bonding 3p7rg orbital, one expects an increase in inter
atomic distance in Sis in the ensuing sl;u-+-sl;g- transi
tion. This seems to be the most likely assignment for
the observed band system.
Another even weaker system of bands with a head
at about 5707 A is also observed in neon. The bands
are broad and rounded, and it was again difficult to
measure their positions. Progressively weaker bands
occurred at 5707, 5625, 5534.5, 5440, 5373, 5267, and
5192 A. A satisfactory analysis of these measured bands
could not be made, and it may be that they are at
tributable to the Si4 molecule with several vibrational
modes excited in the transition.
VAPORIZATION OF SILICON CARBIDE
The mass spectrometry work mentioned earlier indi
cates that the spectra of SiC2, Si2C, and Si2 might be
11 L. Gausset, G. Herzberg, A. Lagerqvist, and B. Rosen, Dis
cussions Faraday Soc. 35,113 (1963). observable in matrices formed by trapping the vapors
over silicon carbide. These predictions are borne out
quite well, although our identification of the Si2C spec
trum is still in doubt. The most intense spectrum is
that of SiC2, and the H-X system of Siz, although
comparatively weak, is almost always present. Sur
prisingly, there also appears a system of bands at
tributable to SizCs, which is present to the extent of
0.004% in the equilibrium vapor at 2300°K. Appar
ently, our higher vaporization temperatures have caused
an increase in the relative concentration of the mole
cule in the vapor.
Visible A bsorption SPectrum
Figure 2 illustrates the absorption spectrum observed
in a neon matrix at 4°K beginning at 4970 A and ex
tending to about 4000 A. This is the region in which
the blue-green stellar bands appeared that were sub
sequently produced in the laboratory by Kleman.7 The
broad matrix bands have a distinctive shape with a
sharp spike appearing at the top of each band so that
their positions can be easily measured to ± 1 A. It is
12 See A. D. Walsh, J. Chern. Soc. 1953,~2266.
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45238 W. WELTNER, JR., AND D. McLEOD, JR.
4963 I 4626
I
4527 4413 r-r-
WAVELENGTHS IN A
FIG. 2. Absorption spectrum of SiC2 in a neon matrix at 4°K.
important to emphasize that, because of the character
istic shape of these bands, one can easily distinguish
them from bands belonging to other progressions. (This
is very often true in matrix studies and points up an
advantage of environmental Rerturbations.) The most
intense band occurs at 4963 A in neon and, less accu
rately, at 4993 A in argon. These values may be com
pared wi~h the most intense band in the gas spectrum
at 4977 A.
The argon bands are very rounded with a shoulder
on the long wavelength side. It was quite difficult to
obtain accurate vibrational spacings because of the
need to measure shoulder-to-shoulder distances and
the shoulders gradually disappeared from the bands
at shorter wavelengths. However, the stronger bands
in neon also appeared in argon matrices and the spac
ings generally confirmed those in the lighter gas (see
Fig. 6). The spectra of SiC2 provide a good example of
the advantage of use of neon matrices in preference
to argon, as noted in our previous work on carbon
vapor.2
Figure 3 shows a schematic diagram of the neon
matrix bands attributed to this system, and Table III
makes a direct comparison of them with the observed
gas bands of McKellar6 and Kleman? Subtracting about
14 A from the strongest gas bands yields the position
of the strongest matrix bands, but there are also three
bands observed in the matrix (at 4725, 4413, and
"-'4147 A) which are not listed by those authors.
McKellar divided his observed bands into two series,
"strong" and "weaker" bands, and it is interesting
to note that all of the former and none of the latter
appear in the matrix spectrum. The relevant columns
of his Table II are reproduced in our Table IV for
reference in the discussion below.
There is a band at 4893 A in the neon matrix spec
trum which approximately corresponds, with the 14-A
shift, to the 4906-A weak band in the gas, but its
shape (see Fig. 2) indicates that it is not a member
of the 4963-A matrix system. It is attributed to Si2C
which is believed to absorb at slightly longer wave
lengths (see below). The rounded shoulder appearing
on the long wavelength side of the 4413-A band in
Fig. 2 has been observed to almost disappear on some
traces, but there is a small spike, characteristic of the SiC2 bands, remaining. Similar shoulders at 4302 and
4264 A do disappear entirely on occasion.
Since SiC2 is a major species in the vapor evolved
from solid silicon carbide, the observation of this band
system in the matrix substantiates Kleman's conclusion
that the spectrum is to be attributed to such a mole
cule. The matrix work also demonstrates that McKellar
and Kleman were correct in their vibrational assign
ments of bands originating from the (0, 0, 0) level of
the ground state, since in the matrix absorption spectra,
"hot" bands would not be expected to occur. More
over, it shows that the 4540.9-and 4261-A gas bands
(see Tables III and IV) which Kleman did not assign,
must be considered as part of the same series, as
McKellar assumed.
It is expected that the molecule will be linear in
both the ground and excited states, as for Ca,ll,2 and
transitions can be expected to occur in which .:1vI and
.:1va take any integral value, and where .:1V2=0, 2, 4,
etc., but only weakly, with .:1V2=0 predominating.
Here VI is essentially the Si-C stretching frequency,
V2 the bending frequency, and Va the C-C stretching
frequency.7 Then the relatively strong 4963-, 4626-,
4339-, and 4091-A progression in the neon matrix
spectrum yields va' values of 1465, 1432, and 1394
cm-I between successive vibrational levels in the ex
cited state. Corresponding gas values are 1461.6 and
1420.8 cm-I as derived from the three observed tran
sitions.7 Beyond this point our assignment of the ex
cited state frequencies diverges from that of Kleman
since we no longer accept 460 cm-I as the frequency
of the other stretching mode, vr'.
It is expected that the intense band at 4963 A in a
neon matrix is the 0-0 band of the transition, and
this is supported by the fluorescence spectrum to be
discussed below. Hence it is clear that the molecular
structure is not greatly changed by the electronic exci
tation, and the transition is basically similar to that
(0,0,01
(0,2,0)
463
1015 (1,0,0)
(0,0,1)
1463 (0,2,1)
475
1048
o
5000 (1,0,1)
AVICM")
4000 >.(;.)
FIG. 3. Schematic diagram of observed neon matrix spectrum
of SiC2• The bands are assigned to various vibratbnal levels in
th~ upper Il stat\! (see text).
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45S P E C T R 0 S COP Y 0 F S I L I CON CAR BID E AND S I L I CON MOL E C U L E S 239
TABLE III. Comparison of matrix and gaseous bands of SiC2 and their assignments.
Neon matrix bands Matching gas bandsb
>-.(!) ,,(cm-l) Int.- >-.(!) Int. Assignment·
(Vl, V" Va) II
4963 20 144 20 4977 .44 20 (0,0,0)
4893 20 432 Si.C
4851 20609 8 4866.99 4 (0,2,0)
4776 20 932 Si.C
4745 21 069 0 (0, 4,0)
4724.5 21 160 2 (1, 0, 0) 4629.5 21 595 0 ?
4626 21 611 14 4639.79 5 (0,0, 1)
4527 22084 4 4540.9* 2 (0, 2, 1)
4413 22 654 1 (1,0, 1) 4338.5 23 043 8 4350.87 1 (0,0,2) 4251.5 23 515 2 4261* 0 (0,2,2) ,...,4147 24 110 0 (1,0,2)
4091 24 437 3 (0,0,3)
• Intensities are approximate peak height relative to (0, 0) band taken as 20.
b Taken from Kleman' except for two starred bands which were taken from McKellar,' (see Table IV).
e Since all the bands listed arise from a transition from the (0, 0, O)~ level of the ground electronic state, only the upper level assignment is given. The observed
transition is assumed to be 'II+-'~.
observed in C3 and not to that described above for
Si3• In other words, it is considered that the transition
is most probably III+--X I~ where the molecular orbital
picture is similar to that for C3:
•.. (<TI) 2 (11'1) 4 I~+
... (<Tl) (11'1) 4 11'2 III, alI
TABLE IV. Stellar spectrum of SiC •. -
>-.(!) " (em-l) Ll" (cm-l)
Strong bands
4977 .1 20 086
456
4866.8 20 542
1460
4639.9 21 546
470
4540.9 22 016
1426
4352 22 972
490
4261 23 462
Weaker bands
4906.0 20377
434
4804.1 20811
1489
4572 21 866
439
4482 22 305
1416
4294 23 282
(436)
(4215) (23 718)
& Except for omission of a column listing the number of plates on which each
band was measured, this is a copy of Table II of McKellar' entitled "Average
wavelength and wavenwnbers of heads of the red-degraded unidentified bands." where 11'1 and 11'2 are analogous to the 1I'u and 1I'g orbitals
of the BAB molecules of Walsh.I2 Another possibility,
that another <T2 level has been lowered in energy below
the 11'2, may also be considered. Then a transition of
an electron in the <Tl orbital, in the I~+ ground state,
to the <T2 orbital could give a I~++--I~+ transition.
Since the <T2 orbital is antibonding one expects in that
case, however, an increase in interatomic distance not
indicated by the observed spectrum.
Neglecting at first any possible Renner effect in the
excited II state which occurred so strongly in Ca,ll,2
one could assign the 4851-and the 4725-1 matrix
bands as (0, 2, 0)-(0, 0, 0) and (1, 0, 0)-(0, 0, 0)
transitions, and thereby derive 2V2' = 463 cm-1 and
vI' = 1015 cm-I at the va' = 0 level. This is fine until
the 4527-and 4413-1 bands are considered as the cor
responding transitions when va'= 1, and then there is
obtained 2V2'=475 and v/=1048 cm-I (see Fig. 3).
Note the increase rather than decrease in these fre
quencies. For va'=2, 2V2' is 474 cm-I and the (1,0,2)
(0, 0, 0) transition was observed less accurately at
,..,.,4147 to give v/"'1067 em-I. Supporting this hy
pothesis, there is a very weak band at 4745 1 which
may be assigned as the (0, 4, 0)-(0, 0, 0) transition
and gives 4V2' = 925 cm-I. However, another very weak
band at 4630 1 does not fit into the assignment.
These observations imply that both xu' and X23' are
positive (,,",+30 and +5 cm-I, respectively) in the
excited state of SiC2, whereas in most triatomic mole
cules, at least in the ground state, these quantities
are negative.la Increasing values, in the higher va' levels,
of what we have called 2V2', were also listed by McKellar
(see Table IV). Kleman undoubtedly disregarded the
13 Examples of molecules in which Xi; are positive are CO. and
NCO where Xl. is +3.65 and +2.66, respectively. [R. N. Dixon,
Phil. Trans. Roy. Soc. A252, 165 (1959).J These and other
examples were supplied by D. A. Ramsay to whom we owe our
thanks.
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45240 W. WELTNER, JR., AND D. McLEOD, JR.
FIG. 4. Infrared absorption bands
of SiC2 in a neon matrix at 4°K.
two stellar bands at shorter wavelengths (4540.9 and
4261 A) when making his vibrational assignment be
cause of this anomalous behavior. The matrix: results,
however, now substantiate McKellar's inclusion of
these bands in his table. [Fermi resonance between
4v2' and v!, must be discounted as a possible cause
of the increase in v/ in successive vs' levels. For reso
nance to be occurring one would expect two bands, of
perhaps unequal intensity, to appear at the (1, 0, 1)
and at the (1, 0, 2) positions, and that is not observed
to be the case.]
This unusual anharmonic behavior leads to the con
sideration of an alternative interpretation, prompted
by the properties of CS,1l.2 in which these irregularities
appear because of vibronic interactions in the excited
state involving the bending frequency V2" If the levels
which are 463 and 1015 cm-1 above the zero level
(see Fig. 3) are assigned as vibronic (0, 2, O)n-and
(0, 2, O)n+ levels, one can derive a vibrational fre
quency v2'=376 cm-1 and a parameter ~= +0.52 from
Renner's equations.14 Then the' spectrum can be inter
preted as showing transitions to the first four vibronic
levels with vs' = 0 in the n electronic state, where the
highest level is the (0, 6, O)n-and appears as a small
spike at 4629.5 A. Again at each higher value of va'
this pattern could be considered as essentially repeated
but with less intensity and with small changes in the
Renner parameters to account for the above anomalies.
An approximate fit to the observed levels can then be
made. The exclusion of transitions involving Vl' is the
serious objection to this assignment since, as is seen
below, quanta of vt" seem to appear quite readily in
the matrix and gaseous emission spectra.
1 ... ·~----1745CM·I ______ --t ... 1
5490 r--
WAVELENGTHS IN A
FIG. 5. Fluorescence spectrum of SiC2 trapped in a neon matrix
at 4°K.
14R. Renner, Z. Physik. 92,172 (1934). Further discussion of the excited-state spectrum of
SiC2 is deferred until after the "hot" bands in the gas
and the matrix infrared and emission spectra have
been considered.
I nfrared Spec/rum
An extensive series of bands was observed in the
infrared under the usual conditions since there were
several absorbing molecules present in the matrix.
However, on occasion and for reasons which are not
entirely clear, only the spectrum of SiC2 appeared with
any intensity in the visible spectrum, and then it was
found that two bands at 1751 and 835 cm-1 (in neon)
were relatively strong in' the infrared. Figure 4 shows
these bands; the 1750 band also appeared with several
TABLE V. Assignment of "hot" bands in gaseous
spectruma of SiC2.
Assignment
X I' Relative Upper Lower
(1) (cm-1) intensi tiesb state state
4294 23 282 ° (0,4,2) CO, 2, 0)
4482 23 305 0 (0, 2,2) (0,4,0)
4573.82 21 857.5 1 (0, 4, 1) (0, 2,0)
4802.57 20 816.4 1 (0, 2, 1) (0,4,0)
4905.51 20 379.6 3 (0,0,1) (0,4,0)
4909.35 20 363.6 3 (0, 0, 1) (0,4,0) ?
ShC?
5048.00 19 804.3 1 (0,0, 1) (0,0, 1)
5128.19 19 494.6 3 (0,0,0) (0,2,0)
5198.05 19 232.6 3 (0, 0, 0) (1, 0, 0)
5317.60 18 800.3 1.5 (0, 2,0) (0, 0, 1)
5450.04 18 343.4 3 (0,0,0) (0,0, 1)
5527.45 18 086.5 1 (0, 0, 1) (0,0, 2)
5632.41 17 749.5 1 (0,0,0) (0, 2, 1)
a Observed bands are taken from Table I of Ref. 7.
b These numbers are relative to the ()-{) band which was given an intensity of
20 in Ref. 7.
branches in argon, presumably due to multiple sitesH'
or similar matrix effects.Is No bands in the bending
region could be definitely assigned to SiC2•
These two stretching modes also appear in the ma
trix emission and gaseous emission spectra, as we shall
show. Kleman first assigned 1742 cm-1 as V3" in the
gas, and the remaining Si-C stretching mode, v/', is
then assigned to the 835-cm-1 band in neon or, as it
will turn out, to 852 Cln-1 in the gas.
Emission Spectrum
By exciting the neon matrix with light from an AH-6
high-pressure mercury arc (with a glass filter cutting
off light at longer wavelengths than about 4800 ft.),
the emission spectrum of SiC2 was obtained and is
shown in Fig. 5. Although a complete interpretation
of this spectrum is not clear, one does detect frequen-
16 K. B. Harvey and J. F. Ogilvie, Can. J. Chem. 40. 85 (1962).
16 G. C. Pimentel and S. W. Charles, Pure and Appl. Chem. 7,
111 (1963).
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45S P E C T R 0 S COP Y 0 F SILl CON CAR BID E AND SILl CON MOL E C U L E S 241
cies of 836 cm-1 and ",,1745 cm-1 to be assigned to the
ground electronic state of SiC2. The (0,0, 0)-(2, 0, 0)
band probably lies under the broad band at 5490-
5460 A.
Presumably the appearance of bands at 5010 and
4975 A is due to matrix effects since a duplicate of
this same band structure appears at 5229 and 5193 A,
i.e., 836 cm-1 away.
"Hot" Bands
The matrix work has allowed the "hot" bands in
the gaseous spectrum to be distinguished from those
originating from the lowest vibrational level. These
"hot" bands have been selected from Kleman's Table I
and are listed in our Table V. There are more bands
which could be included but could not be satisfactorily
measured by him because of an overlapping continuum.
All of McKellar's "weaker" absorption bands, except
the dubious 4215-A one, are included in Table V.
Our assignment of these bands is given in Column 3
of Table V. Most of the assignment of Kleman has
been carried over to this Table by now recognizing
that the 456 and 591 cm-1 frequencies are associated
with bending frequencies in SiC2 rather than Si-C
stretching. The strong 5198.05-A band has been as
signed to yield 852.5 cm-1 as /lI", and as mentioned
above, McKellar's two weak bands at 4294 and 4482 A
have also been assigned.
Using 853 cm-1=/lI" and 1742 cm-1=/la", the stretch
ing force constants in SiC2 can be calculated on the
basis of valence forces17:
k(Si-C) = 7.44X105 dyn/cm,
k(C-C) = 7.98X105 dyn/cm.
k(Si-C) is now considerably larger than the value of
2.9X1OS dyn/cm obtained when using Kleman's assign
ment of /l1"=591 cm-I. However, as Drowart et al.
have pointed out, one does expect a force constant
greater than 3.1 X 105 dyn/ cm, which is that of a single
Si-C bond. Our value is therefore more realistic since
the bond must certainly be considered as Si=C. On
the other hand, the k(C-C) value has dropped appre
ciably from the value of 10.34X 106 dyn/ cm for Ca,2
FIG. 6. AbsorRtion spec
trum near 5300 1 of silicon
carbide vapor trapped in an
argon matrix at 20oK.
17 G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand
Company, Inc., New York, 1945). FIG. 7. Absorption spectrum near 5200 A of silicon carbide
vapor trapped in a neon matrix at 4°K.
or 12.2X 100 dyn/cm for C2.18 This may be rationalized
by assuming some loss of stabilizing electron delocali
zation in going from Ca to SiC2• Of course, one must
also remember that k12 is not really zero in the correct
derivation of the above constants.
Si2C Absorption
In both neon and argon matrices a strong band
system has been observed on the long wavelength side
of the SiC2 system and somewhat overlapping it. The
remarkable thing is the great difference in the shapes
and positions of the bands in this wavelength region
in argon and neon matrices. In argon, three distinctly
shaped broad bands (""160 cm-1 wide) are found be
ginning at 5303 A and decreasing in intensity at shorter
wavelengths (see Fig. 6). From these three bands one
finds a vibrational frequency of about 490 cm-I• In
neon, there is an entirely different appearance to the
bands adjacent to the SiC2 spectrum (see Fig. 7);
they begin at 5171 A, i.e., at much shorter wavelengths,
and are much more complex.
The similar shape of the bands at 4893 A (also
shown in Fig. 2) and 5016 A in neon suggests that
they are associated with the same band system, and
their difference is 500 cm-I as in argon. Another weaker
band of this system lies at 4776 A in Fig. 2. It may be
that the 0-0 band lies under the very strong absorption
at 5171-5146 A in neon but it cannot lie at longer
wavelengths since no bands were found in the 5350-
5200 A region. Whether it appears at 5170 or 5016 A
in neon, the 0-0 band at 5303 A in argon must then
exhibit an exceptionally large matrix shift.
The supposition is that Si2C is the source of the
assignable bands with /l""500 cm-t, and that another
molecule, which is formed by diffusion in the less-rigid
neon matrix, provides the remaining bands between
5171 and 5016 A. The variable intensities of these
latter bands in neon relative to the SiC2 spectrum is
considered as evidence of this. Also the simplicity of
the analysis of the argon bands would be in favor of a
symmetrical Si2C molecule. Crude bond energy con-
18 E. A. Ballik and D. A. Ramsay, Astrophys. J. 137, 61, 84
(1963).
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45242 W. WELTNER. JR., AND D. McLEOD, JR.
5617 .-
5496 r-
Wavelengths in A
FIG. 8. A portion of the absorption spectrum of Si.C3 in a neon
matrix at 4°K.
siderationsl9 lead to the suggestion that this molecule
is symmetrical, and one should therefore expect only
single quanta of the symmetric stretching frequency
to appear with any strength in the spectrum, so that
1'1' = 500 cm-l• This is reasonable, since by using the
above derived Si=C bond force constants, a value of
1'/'=672 cm-I in the ground state is obtained.
An analysis of the 5171-1 system in neon could not
be made except to note that the three peaks at 5079,
5072, and 5069 A have an appearance similar to the
intense 5171-, 5164-, and 5159-1 bands, and all three
pairs yield differences of 350 cm-l•
It is also interesting to note that McKellar6 observed
a square absorption feature in the stellar spectrum ex
tending from 5192.2 to 5211.7 1 (observed by Kleman
at 5192.25 and 5198.05 1). It may be that the same
molecule producing the matrix bands in this region is
also responsible for these gaseous bands.
Si2C3
A series of distinctively shaped bands appears be
tween 6498 and 53501 in a neon matrix (6612-56001
in argon) when silicon carhide is vaporized at about
26000K (see Fig. 8). The intensity variation indicates
19 One may derive from the heat of atomization of Si2(g) ,3 a
value of 75 kcal for an approximate S1=Si bond energy. A value
of 103 kcal for the Si=C bond energy is obtained from the !lHoo
for the dissociation of SiC {g)3, and the C=C bond energy is
taken to be 145 kcal [K. S. Pitzer, J. Am. Chern. Soc. 70, 2140
(1948)]. Using these values one may calculate the energy of dis
sociation of symmetrical and unsymmetrical forms of the silicon
carbon molecules. In all cases, there is a discrepancy from the
observed values,3 which is largely due to neglect of energy of
delocalization. We assume that the molecule will take that con
figuration for which the required delocalization energy is a mini
mum. For example, for Si=C=C one calculates 248 kcal, for
C=Si=C, 206 kcal, and the observed value is 303 kcal. The un
symmetrical molecule is favored, as the spectra indicate. In a
similar fashion it is found that symmetrical Si.C, Si2C2, and
SiC.Si are favored and that SiaC would have the configuration
Si=Si=C=Si. The evidence appearing later in this paper indi
cates that the unsymmetrical ShCa molecule is probably more
stable than SiCaSi. In larger molecules such as this where the
calculated difference in energy is relatively small (28 kcal) as
compared to the dissociation energy (556 kcal), the approxima
tions involved become more serious. that the 0-0 band lies at 6498 1 in neon. At least two
progressions, shown schematically in Fig. 9, are found
to occur among these bands and involve differences of
393 and 1997 cm-I• The higher frequency indicates
the presence of a Ca unit in the molecule, and one then
infers from the mass spectrometry results3 that the
observed molecule is Si2Ca. Apparently this species has
become important at the higher vaporization tempera
ture, and indeed, if the two vapor pressure data given
by Drowart et at. are extrapolated to 2600oK, the
vapor pressure of Si2Ca is found to be of the same
order of magnitude as that of SiC2 at that tempera
ture.
A series of weaker bands is also found which have
been analyzed as shown in Table VI. It may be that
the 616 and 887 cm-1 frequencies are the first and
second overtones of the 305 cm-1 value, but if so, the
anharmonicity behavior is irregular and the intensities
do not decrease toward the shorter wavelengths. If
Si2Ca remains linear in the excited state, then the
many observed progressions are more in accord with
an unsymmetrical molecule, Si=Si=C"'C=C, than a sym
metrical one, Si=C=C"'C=Si.
From Fig. 9, it can be inferred that a third weaker
series of bands might be expected to begin at about
19370 cm-I, and there is, in fact, a series of bands
beginning at about that frequency which was illus
trated in Fig. 7. Although the Si2Ca absorption may
contribute to the strength of the latter bands, it can
be stated quite definitely that the bands at 5171-51591
cannot be attributed to Si2Ca• These bands were ob
served weakly, but clearly, in one run in which silicon
carbide was vaporized at 2500oK, although no absorp
tion occurred at all between 6500 and 5300 1. Only
SiC2 and weak Ca bands were otherwise observed in
the spectrum.
A linear and symmetric (Deok) molecule would have
only four infrared-active frequencies (two 1:,,-and two
IIu) whereas, if unsymmetrical (Ceo.), all seven would
be allowed. The matrix infrared spectra exhibit many
infrared bands, even if allowance is made for addi-
...... --- 1997cm-I __ _
15000 19000
V (em-')
FIG. 9. Schematic diagram of the SizCs absorption spectrum in
a neon matrix at 4°K.
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45SPECTROSCOPY OF SILICON CARBIDE AND SILICON MOLECULES 243
TABLE VI. Spectrum of Si2Cs in neon matrix at 4°K (6498 to 5350 A). Frequencies in reciprocal centimeters.
15 87
393
15407 t 15692 616
887
15780.
1997
16003 15901
1(274
17384 t t t 304 390 t t 613 17688
17774
18091
18266
TABLE VII. Observed infrared frequencies (in reciprocal
centimeters) and tentative assignment of SizCs vibrational modes.
Vibrational
Infrared bandsa,b frequencies
in excited
Neon Argon electronic
matrix matrix state Assignment
120} 305 Si,C. bending
523 w br
600 w} 612 w 595 w 393 Si2C.(Si;:Si str)
654W}
656w 657 w br 616 Si2C.(C=Si str)
680 vw br
792 w
840w
914w 902 m} ?
910w
956 m
9825
9875 994 S 890 Si2C.(C=C str)
12055 1187 S Si2C(?)
1875-} 2048 m 1967 s 1997 SizC. (C=C str)
a Intensities and band shapes are indicated hy s=strong, m=medium, w= 397 t 604
16099 875
16177
16384 397
16503 !
16574 T 16655
3f
396 16969
t 605
18170 t 880 ! 330 r 18379 t
18500
18654? 18568
tional structure due to matrix effects.IS,Is These bands
are enumerated in Table VII along with a possible
assignment of some of the bands based on a COO" mole
cule. An indication of the expected stretching frequen
cies has been obtained by a valence-bond calcula
tion17 using k1(Si=Si) 2.1XlOs, k2(Si=C)=7.44X105,
ka(C1=C2)=7.98X10s, k4(C2=Ca)=10.34XlO s dynl
cm. kl is from Drowart et al} k2 and ka from the above
discussion of SiC2, and k4 from the Ca molecule.2 The
calculated stretching frequencies are 391, 736, 1043,
and 2027 cm-I. The doubly-degenerate bending fre
quencies have been previously estimated.3 [For a sym
metrical molecule, using k2 and ka, one calculates the
stretching frequencies to be 459, 1557 (2:1/+)' 1003,
1971 (2:u-).]
DISCUSSION
There is a difference in the bonding of carbon and
silicon which is best illustrated by beginning with the
diatomic molecules C2 and Si2• C2 has a 12:q+ ground
state18 resulting from a configuration of molecular or
bitals20:
weak, vw=very weak, br= broad. 20 G. Herzberg, Spectra of Diatomic Molecules (D, Van Nos-
b The two SiC, bands have been omitted. trand Company, Inc., New York, 1950).
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45244 W. WELTNER, JR., AND D. McLEOD, JR.
In Si2 a (j orbital is lowered in energy relative to the , 0 •
11"" so that a 31::0-ground state results wIth the con-
figuration9•lo:
(Unfortunately, the SiC molecule has not been identi
fied spectroscopically.)
In the corresponding triatomic series, Ca has a 11::0+
ground state 2.11 with outer orbitals as in the C2 con
figuration, so that the 4050-1 system of ~3 m~tches
the Phillips IIIu~I1::0+ system of C2. Our dlscusslOn of
Si3 in an earlier section has considered it to be an
analog of Si2 with similar molecular orbitals and a
31::0-ground state.
In general, a Cn molecule with n odd has completely
filled shells . with a bonding 11" orbital lying highest,
so that its 'ground state is 11::0.21 Each time a carbon
atom is added to give an even numbered molecule two
valence electrons go into a (j orbital and two i~t? a
11" orbital resulting in a 31:: ground state of less stabIlIty.
Now we see that the opposite is true in the Sin series
where the lower members contain another (j orbital
which lies lower than the bonding 11" orbital so that
Si2 and Sia have 31::0-ground states. Each higher mem
ber with n even then adds two (j electrons and two 11"
electrons to a 31:: -molecule to give filled shells and a
11::0 ground stat:. These statements are .in ~ssential
support of the general belief that a relatIve mcrease
in (j bonding, or decrease in 7r bonding, resul~s when
Si replaces C. This variation of stability of SIn mol~
cules with n is apparent in the mass spectrometnc
results.4
From this discussion it follows that the substitution
of silicon for carbon atoms in C3 would tend to lower
the (j orbital energy relative to the strongly bonding
11" orbital and result in a gradual transition from singlet
t~ triplet ground states. Then, for example, if SiC2 is
singlet, Si2C might be triplet. A brief review of wh~t
we think is now known about these two molecules IS
appropriate at this point. .
Our analysis of the 4977-1 system of SlC2 leads us
to believe that it is probably a transition from a ground
11::+ to an excited III state. In view of the evidence
from the infrared and emission (both gaseous and ma
trix) spectra, the Si-C stretching frequency in the
ground state (VI") very probably lies a~ 853 cm-I. ~s
mentioned earlier the appearance of thIS frequency m
emission indicate~ that the corresponding stretching
frequency, vI', in the excited state would be expected
to appear in absorption. This is also favored by the
appearance of the intense bands associated ~ith quanta
of va' the C-C stretching frequency. On thIS basIs, the
vibr;tional assignments for the two electronic states
21 K. S. Pitzer and E. Clementi, J. Am. Chern. Soc. 81, 4778
(1959) . are as follows:
III:
v/= 1015 cm-I, V2'= 230 cm-I, va'= 1461 cm-I.
To within the accuracy of our observations, no Renner
effect is observed in the upper state. Of course, the
remarkable thing is the increase in VI in the excited
electronic state and its continued increase with the
higher vibrational quanta of va'. Whether this inter
pretation of the spectrum of SiC2 is correct or not, it
is clear that the properties of the excited state are
quite different from those of Ca. The ground state
differs principally in the higher bending frequency for
SiC2 which may be rationalized by attributing increas
ing bending resistance to increasing strength of (j rela
tive to 11" bonding when Si replaces C. It is hoped that
recent gas spectra22 will confirm some of these findings.
Our results on Si2C are very tentative of course,
since the intensity of the 5300 1 system of bands in
argon is our only real clue that they belong to that
molecule. Accepting this, then the strong band at
5303-5257 1 must be the (0-0) band and again the
transition is assumed to be IIIuf-X 11::0+, We have ex
cluded other transitions2a which require a lengthening
of bonds in the molecule during excitation because the
intensities of the bands decrease regularly from 5300 1
toward the violet. As mentioned earlier, our derived
value of vI' = 500 cm-I is about what one would expect
for Si2C. An oversimplified rationalization of the lack
of a triplet ground state for this molecule, even though
it now contains two silicon atoms, is to say that it is
symmetrical and contains no Si-Si bonds. The substi
tution of another Si in SiC2 apparently still does not
lower the (j orbital below the 7r.
The appearance of the relatively strong Si2Ca spec
trum in the matrix suggests a re-examination of the
importance of this molecule in silicon carbide vapor
at high temperatures. Although the transition proba
bility may be high and account for the strength of
the Si2Ca bands in the visible region, a high concentra
tion in the vapor is expected from the meager mass
spectrometry data, as mentioned previously. Hence,
Si2Ca should probably be included in the calculation
of the total pressure over silicon carbide as carried
22 R. D. Verma has informed the authors that he has obtained
a good spectrum of SiC2 in the gas by flash photolysis of diphenyl
silane. It is now being analyzed.
23 Another possibility is t~at the eqt;iv~lent 0'£ the Swa~ bapd
transition in C2 is observed m the matnx, mvolvmg the eXCitatIOn
of a <F electron to a <Fu orbital, i.e., a 3IIu<-3IIu transition. This
has beuen excluded on the basis that sITu is not a likely ground
state and that it is improbable for ShC to be trapped in an ~x
cited electronic state, in spite of the evidence of that occurrmg
for C2•
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132.174.255.116 On: Wed, 17 Dec 2014 23:37:45S P E C T R 0 S COP Y 0 F SILl CON CAR BID E AND SILl CON MOL E C U L E S 245
out by Drowart et at., especially at the higher tem
peratures.
The weight of evidence is in favor of an unsymmet
rical Si'=Si=C=C=C molecule, but the vibrational
assignment leaves much to be desired when a com
parison is made between the lower infrared frequencies
and those found by a valence-bond calculation. This
is unfortunate in view of the possible role that this
molecule might play in the vaporization of silicon
carbide.
Aside from Si2C3, if one asks how the results of this
research will alter the thermodynamics of the vapori
zation of silicon carbide, the answer lies in the effect
of the reassignment of the SiC2 vibrational frequencies
upon the calculated properties of the other silicon
carbon vapor species. The force constants for the
THE JOURNAL OF CHEMICAL PHYSICS Si==C and C==C bonds as derived above from the
SiC2 data lead to the following revised stretching fre
quencies for these molecules (comparison may be made
with Drowart et al.3) : SiC, v= 1226 cm-I; symmetrical
Si2C2, VI = 542, V2= 1862, and V3= 1226 cm-I; symmet
rical Si2C, VI = 672 cm-I, Va= 1559 cm-I; Si==Si==C=Si,
vI=739, v2=388, v4=1602 cm-I. We would take the
ground electronic states to be I~+ for the first three
of these molecules with SiaC (and Si2C3) probably
being a~-.
ACKNOWLEDGMENTS
The authors are particularly indebted to W. D. Bird
for experimental assistance. Dr. C. L. Angell, Dr. D. A.
Ramsay, and Dr. V. Shomaker have also contributed
by helpful discussions of the spectroscopic results.
VOLUME 41, NUMBER 1 I JULY 1964
Gamma-Induced Divalent Dysprosium in Calcium Fluoride*
FRANCIS K. FONG
RCA Laboratories, Princeton, New Jersey
(Received 6 February 1964)
Experiments dealing with photochemical and thermal reactions of dysprosium in calcium fluoride are
described. Trivalent dysprosium ions doped in calcium fluoride have been reduced by gamma irradiation
at room temperature. The efficiency of reduction of Dy3+ in CaF2 is described in terms of the dosage of
gamma irradiation and concentration of the dopant ions. The optimum conditions for the photoreduction are
given. The absorption band responsible for the reoxidation of the gamma-reduced dysprosium has been
determined. It has been established that high densities of gamma-induced defect sites are necessary for
the conversion of the majority of divalent ions to the trivalent state, which may be indicative of the tun
neling of Dy2+ electrons to hole centers via the gamma-induced defect centers. The tunneling process is
also believed to be responsible for the saturation of local and macroscopic concentrations of DyH ions.
The thermal reoxidation of Dy2+ ions is accompanied by bright luminescence, the spectrum of which
resembles that of the photoluminescence of Dy3+ ions. The decay of the thermoluminescent emission bands
is nonexponential, and the reaction kinetics fit the description of a second-order reaction. The evaluated
activation energy (0.33 eV) is indistinguishable from the energy required for the thermal ionization of holes
from Vl centers in alkali halides.
I. INTRODUCTION
THE lanthanide elements are characterized by the
uniform stable trivalent oxidation state which would
be expected of members of Periodic Group IlIa. Devi
ation from this state usually results from the tendency
of the element to attain or approach the electronic con
figurations of the ions La3+( 4.f5s25p6), GdH( 4f5s25p6),
and LuH(4f45s25p6), where the 4f orbitals are empty,
half-filled, and completely filled, respectively. Although
well-characterized only with samarium, europium, and
ytterbium, the more general existence of the divalent
oxidation state has been suggested by the isolation of
* This research has been sponsored by the Aeronautical Systems
Division, U.S. Air Force Systems Command, Wright-Patterson
Air Force Base, Ohio, under Contract Number AF33 (657) 11221. a series of carbides of composition MC2, which is diffi
cult to reconcile by any postulate other than the pres
ence of a 2+ state.1 More recently, methods have
become known for the partial reduction of SmH and
Dy3+ by ionizing radiation to make CaF2(Sm2+)
and CaF2(Dy2+) lasers.2,3 Trivalent thulium in stron
tium chloride has also been reduced by gamma ir
radiation at room temperature.4 In view of the con
siderable interest in the chemistry and physics of
1 T. Moeller, Inorganic Chemistry (John Wiley & Sons, Inc.,
1955), p. 698.
2 J. R. O'Connor and H. A. Bostick, J. Appl. Phys. 33, 1868
(1962) .
3 Z. Kiss and R. Duncan, Proc. IRE 50, 1531 (1962).
4 F. K. Fong, Presented at the Fourth Rare Earth Conference,
Phoenix, Arizona, 1964.
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1.1733850.pdf | Collision Lifetimes and the Thermodynamics of Real Gases
Felix T. Smith
Citation: The Journal of Chemical Physics 38, 1304 (1963); doi: 10.1063/1.1733850
View online: http://dx.doi.org/10.1063/1.1733850
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128.42.202.150 On: Tue, 25 Nov 2014 15:02:31THE JOURNAL OF CHEMICAL PHYSICS VOLUME 38, NUMBER 6 15 MARCH 1963
Collision Lifetimes and the Thermodynamics of Real Gases*
FELIX T. SMITH
Stanford Research Institute, Menlo Park, California
(Received 21 November 1962)
The perfect gas law p/kT=n="Zm, (where n, is in molecules per cm3) is inadequate for describing
real gases because of the interactions during collisions. By a simple intuitive argument, these interactions can
be taken into account exactly if you know the collision lifetimes. The product of collision rate and lifetime
gives the concentration of transient collision complexes, which must be considered in the perfect gas law
along with the stable species. As a result, the complete virial expansion is obtained, in both quantal and
classical mechanics. The argument leads further to a new form for the partition function which includes the
continuum as well as bound states. From this all the thermodynamic functions can be obtained.
A. EQUATION OF STATE
THE equation of state of real gases involves devia
tions from the perfect-gas law,
p/kT=n= Lni,
i ( 1)
where n is measured in molecules per cma and the ni
refer to different species in the gas. The deviations
caused by the formation of bound molecules from the
parent gas can be handled by introducing the proper
equilibria, and the various excited states of the parent
species in the gas can be taken into account similarly.
Each of the excited states can then be given a separate
concentration ni in the sum in Eq. (1).
Another class of interactions which also contribute
to deviations from the perfect gas law are the inter
actions between two or more unbound molecules of
the gas-these include the repUlsive interactions that
in the simplest case appear as the excluded volume in
the van der Waals equation of state, and the attractive
interactions that may result in the formation of
metastable clusters ranging from a transient orbiting
pair to the long-lived but unstable molecules that
participate in unimolecular reaction processes. The
effects of these interactions appear in the virial expan
sion of the gas law, and various devices have been used
to compute the virial coefficients. In this note I try to
carry through a simple intuitive approach that con
nects the gaseous equation of state with the lifetime
matrix of collision theory. Truly bound molecules and
excited states can be ignored and attention focused on
the collisional interactions.
The lifetime matrix! provides a tool for analyzing
each collision into a free-flight portion and a collision
lifetime, which incorporates all the effects of the inter
action. The free-flight portion corresponds to continued
motion of the particles as if the interaction had not
occurred (in the case of an inelastic collision, an in-
* Supported by the National Aeronautics and Space Adminis
tration and by the National Science Foundation.
1 F. T. Smith, Phys. Rev. 118,349 (1960). See reference 2 for
the correction of an error. stantaneous switch from an incoming to an outgoing
free-flight path is assumed). The collision lifetime is
always uniquely defined-it may even be negative, if
the colliding particles separate sooner than they would
have in free flight without interaction. The collision
lifetime is well defined in the classical limit as well as
quantally, but I use the quantal form with its relation
to the scattering matrix since quantal effects are often
important in molecular collisions. The collision life
time is well defined for interactions of short range; it
diverges for the Coulomb interaction, but may be
defined for a shorter range interaction superimposed on
the Coulomb, leaving the contribution of the Coulomb
part to be dealt with otherwise. The gas is assumed
dilute enough so that ordinary Boltzmann statistics
can be used.
The basic idea of this article is to assume that a pair
of molecules involved in any collision act like a single
bound molecule for the duration of the collision life
time and are completely free the rest of the time. The
effective concentrations Cj of these collision complexes
can be computed if their lifetimes are known, and they
can be inserted in the sum on the right-hand side of
Eq. (1) along with the stable species,
n= Lni+ LCj. (2)
i i
In doing this it must be remembered that the concen
trations of the free stable species are reduced to the
extent that they are transiently tied up in the com
plexes. Since the collision complex may have a negative
lifetime, its concentration may be negative-but in
this case the concentrations of its parents are effectively
increased; this is, in fact, the result to be expected
classically because of the excluded volume when the
interaction is repulsive.
For example, if the collision occurs between molecules
A and B, with initial concentrations nAG and nBo, the
reaction is
A+B~(AB)complex,
and the concentrations are (3)
(4)
1304
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As a result the right-hand side of Eq. (1) becomes
n=nA+nB+cAB=nAo+nBo-cAB, (5)
which will be larger than nO = nAo+nBo it CAB is negative.
Clearly there is no chance for n itself to become
negative.
We now wish to relate the concentrations Cj to the ni
and the lifetimes. What is needed is an equilibrium co
efficient connecting Cj and the ni:
(6)
The equilibrium coefficient Gik(T) may of course be
negative, and will be so in general if the interaction is
predominantly repulsive. Gik (T) is just the equilibrium
concentration of complexes in the presence of unit
concentrations ni, nk of the parent species.
The connection with the collision lifetimes comes
about because the concentration of an unstable com
plex is just the product of its rate of formation and its
lifetime. The relationship has already been used in
connection with a discussion of reaction rates and
collision rates.2 The collision lifetime for any colliding
pair Ai, Ak depends not only on their internal states
(which must be assumed to be completely specified
by i, k) but also on the angular momentum quantum
numbers I, mz and the energy E of relative motion:
(7)
Assuming unit concentrations of the colliding partners,
the rate of production of this complex with energy in
the range (E, E+oE) and in the given angular mo
mentum state I, mz is2
k(E, I, ml)oE= (h2/27rjJ.kT)!h-1 exp( -E/kT)oE. (8)
The concentration of the complex in the range
(E, E+oE; I, ml) is the product of this with the
lifetime:
gik(E, 1, ml)oE=Qik(E, I, mz)k(E, I, ml)oE. (9)
The total concentration of this complex is then
Gik(T) = f""'L-gik(E, I, ml)dE
o l.mz
= (~)\-l 'L-f'" Qik( E, I, mz)
27rjJ.kT l.mz 0
X exp( -E/kT)dE. (10)
It is easy to see that the infinite sum in I converges,
because the interaction and the lifetime vanish at large
values of I (or large impact parameters).
The lifetime Qik(E, I, ml) that enters into Eq. (10)
is actually one of the diagonal elements of the lifetime
matrix of reference 1, and it can be derived from the
scattering matrix for the system. In the special case
2 F. T. Smith, J. Chern. Phys. 36, 248 (1962). where Ai and Ak are spherically symmetric atoms, Q
becomes simply the energy derivative of the phase
shift and is independent of mz:
Q(E, I, ml) =2h(do 1/dE). (11)
When this is inserted in Eq. (10) we encounter an ex
pression which was first derived (by a different argu
ment) by Kahn and Uhlenbeck3 for the second virial
coefficient for an atomic gas.
As the density of a gas is increased, higher-order
complexes become important. These can be handled in
just the same way as the two-body complexes, by intro
ducing the collision lifetime and the collision rate for
n-body collisions. The lifetime matrix for these cases
has just been discussed elsewhere,4 and the collision
rate expressions are to be found in reference 2. By an
argument similar to that followed in deriving Eq. (10)
we can find its generalization to an n-body collision
equilibrium:
Gij.jn)*( T) = C7r::T rn-l)h-l~""~Qij"'I(nl(E, 'Y)
X exp( -E/kT) dE. (12)
(The reason for writing G* will become apparent later.)
Here 'Y is a set of 3n-4 quantum numbers for the gen
eralized orbital angular momentum of the n-body
collision5 and jJ. is the n-body reduced mass,
jJ.n-l= IIm;/'L-mi. (13)
i
From G(n) * we can now obtain an apparent concentra
tion for n-body complexes,
Ci.j .•• ./n)*=Gi,j .... /n)*(V)n;nj"· -n/. (14)
However, caution must be adopted in using this expres
sion, because of a contribution of lower-order collision
complexes to the lifetime Q(n) that went into the defini
tion of G;i ... /n)*(T). This is most easily seen in the
three-body case, which is now examined.
In Reference 4, it is shown that the three-body life
time Q(3l includes contributions due to pure two-body
interactions such as the lifetime of AB while C is far
away. This can be taken care of by a subtraction pro
cedure described in that paper, but it is just as legiti
mate to postpone the subtraction until after all the
averages have been taken. After that has been done, it
is easy to see that Cijk (3) * includes three spurious con
tributions represented by the products ciPlnk, Cjk(2)ni,
and Cik(2lnj. These can be eliminated by using the cor-
3 B. Kahn and G. E. Uhlenbeck, Physica 5, 399 (1938); B.
Kahn, doctoral dissertation, Utrecht, 1938. See also J. O. Hirsch
felder, C. F. Curtiss, and R. B. Bird , Molecular Theory of Gases and
Liquids (John Wiley & Sons, Inc., New York, 1954), pp. 404 ff.
4 F. T. Smith "Collision Lifetimes in Many-Body Processes,"
Phys. Rev. (to be published).
5 F. T. Smith, Phys. Rev. 120, 1058 (1960).
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rected equilibrium expression
Gijk(3)(T) =Giik(3)*(T) -Gi/Z)-Gik(2)-Gjk(2). (15)
The true concentration of the three-body complex is
then
(16)
Similar subtractions are needed for the higher-order
equilibria:
GijkZ(4) = Gijk/4) *-Gijk(3) -GikZ(3) -GjkZ(3) -GiP)
-Gjk(2) -Gd2)-Gi/(2)-G jZ(2) -Gik(2). (17)
Using these equilibrium coefficients it is now possible
to set down the equilibrium equations for all the colli
sion complexes CiP), Cijk(3l, CijkZ(4), etc., as well as for all
the stable species ni. These must be combined with
the chemical conservation conditions, for instance
n,o=ni+2ciP)+ LCiP)+···. (18)
jr"i
Finally, these are to be supplemented by the perfect
gas law in the form
p/kT=N= Lni+ LCiP)+ L Cijk(3)+.... (19)
i i?,J i?,i?,k
Equation (19) is a complete cluster expansion for
the equation of state of real gases. It is entirely equiva
lent to the virial expansion, and reduces the problem
to the evaluation of the collision lifetimes of the various
clusters. These clusters are defined not by their spatial
extent but by their lifetimes. Since the lifetime has a
classical meaning, the expansion is also valid in the
classical limit; the classical lifetimes may in fact give a
useful approximation to the quantal ones. In the
quantal case the effect of the statistics of the particles
(Bose-Einstein or Fermi-Dirac) has not been explicitly
included in the above argument, but it can readily be
introduced in evaluating the equilibria.3
It is gratifying to find that this approach reduces to
the Kahn-Uhlenbeck form in the case of binary atomic
encounters. Their result was derived by considering
the quantal density of states3
1 dOL 1ik Pl=--=-Q(E, t).
7r dk 27rJ.l. (20)
This formula may also be used now to give the density
of states where inelastic collisions are possible.
It is of interest to examine one of the quantal effects
that enter into the evaluation of the equilibria G(n).
This is the existence of metastable levels inside a po
tential barrier. If the barrier is sufficiently thick and
the levels are well separated, their effects will es
sentially not overlap and they can be treated as isolated
Breit-Wigner resonances with a narrow half-width rm
and a center at Em. The shape of Q near Em is deter
mined by this resonance,! and its contribution to the integral in Eq. (10) can be
evaluated separately. It is
Gm(T) = (h2/27rJ.l.kT)!(21+1) exp( -Em/kT). (22)
Thus, the metastable states with long lives behave
just as bound ones, a result that is intuitively obvious
but reassuring. Except near the top of the barrier,
where leakage broadens the levels unduly, it is often
possible to consider these resonances separately in the
statistics, leaving the rest of the integral in Eq. (10)
to be evaluated by using the smooth, elastic, contribu
tion to Q( E). This elastic portion may sometimes be
usefully evaluated classically. The importance of de
viations from the classical behavior in the region near
the top of the barrier remains to be evaluated.
B. THERMODYNAMIC FUNCTIONS
The procedure of the last section can readily be ex
tended to give other thermodynamic properties of the
gas. Of these the most accessible is the internal energy
U per unit volume.
The internal energy of the gas can be considered to
be divided into a perfect gas portion
(23)
a portion deriving from the internal energy of the
stable molecules
U(l)= "E-n·o ~ t t,
i (24)
and a portion due to the interactions of the various
collision complexes. To obtain the average energy of a
typical complex Ai; we must start with the differential
concentration
OCij= ninjgij(E, t, mz) oE. (25)
It is then convenient to write
OCij gijoE 0 Yij (26) Cij Yij ,
where
oYii=h-1 exp( -E/kT) Qij(E, t, mz)oE, (27)
and
Yij= !dYiJ=h-IL to exp( -E/kT) Qij(E, t, mz)dE.
l,mz 0
Then the average energy of the complex is
Eij= Yirl! EdYii, (28)
(29)
~nd the total internal energy due to binary complexing
IS
U(2) = LEijCij.
ij (30)
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Similar expressions apply to the higher order com
plexes. The total internal energy is then
U= U(0)+U(1)+U(2)+... (31)
(Note that the same result would be obtained had we
written
and U(1)*= LE,n,
i
U(2)*= L(Eii+E.+Ej)Cih
ij
etc., and summed over these.)
To get the other thermodynamic functions we ob
serve that Yil( T) is just the internal partition function
for the collision complex, and that the average internal
energy per complex can be expressed as
Eij=kT2(a InYiijaT). (32)
In integrating the equation
ea~j)v =~(aa~j)v
(33)
to get the entropy, it is convenient to write
Sij= T-IEii+k In I Yii I +So, (34)
in order to take care of the cases where the collision
partition function Yii may be negative.
It will be observed that Yij is in fact constructed
just like other partition functions if we construe
oWij=h-1Qii(E, I, ml)oE (35)
as the statistical weight associated with the complex
at the energy E. Bound states then are a special case
in which Q(E) becomes a 0 function [r~ in Eq. (21) ]
and w= 1. This indeed allows us to treat in a unified
way all the possible situations including transient
colliding pairs, long-lived metastables inside a potential
barrier, excited states below the dissociation limit (with
a finite radiative lifetime), and the ground state of the
molecule.
It can now be seen that the equilibrium coefficient
G'i( T) can be expressed as a ratio of partition func
tions. The partition function for the free species A i is
the familiar expression
Zi= (21t'mikT/h2)lwi exp( -E/kT), (36)
where Wi is the weight factor in case of degeneracy. The
complete partition function for the complex A ii is the
product
Zij= (27f'MiikT/h2)!UirlWiWj exp[ -(Ei+Ej)/kT]Y i;,
(37)
where Mij=mi+mj and (fij is the symmetry number
«(f ii= 2, (f i;o!j= 1). If A if is a stable state the same ex
pression holds since Yii reduces to exp( -Eii/kT) (Eij is measured relative to Ai and Aj at infinite sepa
ration, and is negative for a bound state). The equilib
rium relation can now be expressed simply as
The generalization to higher order collisions is obvious.
We can now include all the states of an atomic or
molecular species in a single partition function. For
molecules this will include the unstable complexes
along with the bound states. (The proper continuum
spectrum of the molecule is now to be defined as the
difference between the gross continuum and the con
tinuum due to atoms in free fiight.) In the case of the
atomic species A, comprising all the states A i, we can
define the various angular momentum and spin states
by a set of quantum numbers "a, so that any degener
acies are taken care of by a summation over 'YG. The
effect of the radiative lifetime of the levels will appear
in the lifetime function QA(E, "a), which takes the
form of Eq. (21) near an isolated level and becomes a o function for nonradiating states (collisional broaden
ing can also be taken into account, of course). The
internal partition function of the atom is then
ZAint= L f exp( -E/kT) h-1QA (E, "a)dE
'Ya
~Lw>"exp( -E;/kT). (39)
i
The approximate equality at the end holds only if
ionized states can be neglected.
In the case of molecules a similar expression applies,
and the integration includes bound state energies as
well as the continuum. For diatomic molecules AB the
set of quantum numbers" includes "a, "b, and 1, ml for
their relative motion. The energy E must now be taken
from some common origin such as the ground states of
A and B. The result is
ZABint= Lf exp( -E/kT)h-1QAB(E, 'Yab)dE. (40)
~ab
It will be remembered that the components Q(E, 'Y)
appearing in Eqs. (39) and (40) are actually the
diagonal elements of the matrix Q. These are summed
over all the indices 'Y, thus forming the trace of the
matrix. Thus we can express the internal partition
functions in the compact form:
In the case of triatomic and larger molecules like ABC
that may split into three or more fragments, the neces
sary subtractions discussed in Sec. A are already taken
care of in the definition of the complete lifetime matrix
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QABC of reference 4. QABC includes all the bound states
of ABC, and all the collisions involving the complex
ABC, both the binary processes such as AB+C and
the ternary ones A+B+C. Consequently the complete
internal partition function for a triatomic or poly
atomic species has just the same form as for the smaller
species, for example
ZABCinb! exp( -E/kT)h- I TrQABc(E) dE. (43)
Using these partition functions the concentrations
Clf all the molecular species and clusters can be expressed
in a concise form. The thermodynamic functions of the
mixture follow immediately.
C. AN EXAMPLE: CLASSICAL HARD SPHERES
In its quantal form this version of the gaseous equa
tion of state reduces correctly to the Kahn-Uhlenbeck
formula. The theory can also be applied classically,
and the hard-sphere gas provides an example of the
treatment. Consider a gas of atoms of diameter u. The
collisions can be classified by their relative energy E
and by the magnitude L of the angular momentum.
Collision only occurs if
(44)
where
(45)
The collision lifetime is negative in this range and zero
elsewhere; its magnitude is just the time the inter
action-free trajectory would have taken to pass through
the sphere r~u,
Q(E, L) = -(2p./ E)!u(1-V/ Lm2)!(L~ Lm) I
(46)
=0 (L"C.Lm).
In converting the quantal expression Eq. (28) for
the collision partition function to classical form, we
can take advantage of spherical symmetry (inde
pendence of ml) and make the substitution
Then we find
4-n-2 '" Lm2
YU=-3j exp( -E/kT)j Q(E, L)d(V)dE
h ° 0
(48)
In writing the expression for the equilibrium coefficient
Gu we must remember that the two collision partners
are identical, so that a symmetry factor of t is needed
to avoid counting the collisions twice:
Gll( T) = t(h2/27rp.kT) !Yll (T) = -i{7r(3). (49) If we confine our attention to binary collisions, we have
the conditions
and consequently
n=nI+cU=nIo-cll=nlo+(27ru3/3) (n1o) 2+ " "". (51)
This reproduces the well-known second virial coefficient
for the hard-sphere gas.
Similarly we can compute the average energy of the
collision dimers,
Eu= Yu-I! EdYl1=!kT" (52)
The internal energy density is then
U=!kTn+cuEu=!kT(n+cu) =!kTno. (53)
This is again the correct result, showing that the in
stantaneous hard-sphere collisions do not affect the
energy density or the specific heat.
D. TRANSFORMATION TO THE CLASSICAL
CLUSTER FORMULATION
The partition function we have derived here, Eqs.
(41) to (43), has a very different appearance from the
classical expression in terms of the configuration inte
gral. Nevertheless, it is not hard to derive the usual
expression from the integral over the collision lifetimes.
It will suffice to show the transformation explicitly for
a gas of atoms interacting through the spherically
symmetric potential V (r).
Classically the collision lifetime can be expressed in
terms of the relative kinetic energy E and angular mo
mentum L by the equation
Q(E, V) = 2 lim( lR {~[E-V(r) -~J}-ldr
_'" ~~ p. 2~
-lR {~[E- ~J}-tdr). (54)
Rmiu' p. 2p.r2
Here the first integral represents the time taken to get
from R to the point of closest approach, Rmin, and the
second is the corresponding time for an interaction-free
collision. The square roots represent the radial veloci
ties; Rmin and Rmin' are defined by the condition that
the radial velocity vanishes there,
2P.Rmin2[E- VeRmin) J= V,
(55)
Using Eq. (47), the internal part of the collision
partition function becomes classically, in this case:
Zint=47r2jj,--3j'" j'" Q(E, V)d(V) exp( -E/kT)dE.
° °
(56)
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The integral over d(L2) corresponds to taking the trace
of Q in the quantal formulation. Q itself, by Eq. (54),
involves integrals over the radial coordinate r, in which
the lower limit of integration depends on £2 and E.
Nonetheless, the order of integration can be changed if
the limits are changed appropriately; the condition to
be observed is that the square roots must remain real.
Integrating first over £2, from 0 to a maximum de
pending on E and r, we find
00 ( RJLuun2 J Q(E, L2)d(£2)=2p.lim { {2p.r2[E-VCr)]
o R .... OO JRo 0
where we have used the conditions
(58)
and
V(Ro) =E.
Now we can integrate over dE in Eq. (56), setting the
lower limit in the first integral in E to keep [E-V(r)]!
real:
Zint= 87r2h-S(2p.) tJ{ rro [E-V(r)]! exp( -E/kT)dE
o JV(r)
_ ~ro Et exp( -E/kT)dE}r2dr
=47rC7r~:'0i~OO[exp( -V(r)/kT) -1]r2dr. (59)
If the colliding atoms are identical, the symmetry factor
! must be included. The final form is the familiar one
involving the two-atom configuration integral.
A similar development can clearly be used to obtain
the configuration integral from the lifetime expression
in more general cases such as many-body collisions or
collisions of molecules without spherical symmetry. In
these cases the lifetime Q will depend on additional
variables besides E and £2, but it can always be written
in an integral form analogous to Eq. (54). The quantal
sum forming Tr Q becomes a classical multiple integral
over several variables. I now briefly sketch the gen
eralization of the derivation of Eq. (59) to cover more
general cases.
First, consider an encounter of two molecules with
out spherical symmetry and with a potential depending
only on r, 0, q,. Q may still be written in the form of Eq.
(54), with VCr, 0, q,), but the integration in dr must be carefully taken over the full path from the first passage
into the sphere at r= R to the last passage out. Instead
of Eq. (47), we need now the equivalences
M.mrllL.= o(L cosO) = L sinOM,
hM-'>oL. (60)
L defines only the plane in which the motion occurs,
and we must still average over the various orientations
in that plane represented by the angle q,. As before we
integrate explicitly over LdL and dE, and we are left
with the configuration integral over the volume element
dT=r2dr sinOd8dq,.
Next, consider the three-body collision lifetime Q(S).
In this, as in higher order collisions, r must be replaced
by the generalized distance p which can be defined in
terms of the trace of the inertia tensor for the in
stantaneous configuration of the three-body system
with respect to the center of mass:
(61)
All the other relative coordinates can be written as
angles. Now Q(S) may classically be written in a form
just like Eq. (54), but using the coordinate p and its
related velocity Vp, The integrals in Q are of the type
fRvp-1dp with
!p.vl=E- V(q) -A2/2p.p2, (62)
where A2 represents the generalized angular momentum
in the six-dimensional space of the relative motion.5 To
form the trace of Q we must sum over five angular
momenta 1', which may be taken in the regular repre
sentation defined in reference 5. In this representation
we define the angular momenta in successive subspaces
of the six-space, getting the equivalences
hA-'>A,
hA.-'> A. = A COS04,
hA3-'> Aa = A. COS03,
M\2-'> A2 = As COS02,
hA1-'>A1 = A2 COS01. (63)
The sum over l' becomes an integral over the angular
momenta
dAdA.aAsdA~A1 = A tdA cos30. sinB.d8. COS203 sin03d08
X COS02 sinO~02 sin01d01. (64)
To this must be added an integration over the angle q,
in the remaining 2-space defined by the A's. The inte
grations over dA and dE can now be carried out,
leaving a configuration integral in p and 5 angles,
representing the 6 coordinates of relative motion of the
three particles.
In this way, a hierarchy of configuration integrals
can be obtained. They are a classical form of the ex-
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pressions G(n)* of Eq. (12). The first two of them are
G(2)=j{ex p[ -V(r)/kT]-1}d3r,
G(3)*= j {exp[- V(~)/kT]-1}d6~. (65)
To get the equilibrium coefficients we must perform
the subtractions of Eqs. (15) and (17) ; for the case of
identical particles we must also introduce the symmetry
factor, so that the final equilibrium coefficient becomes
bn=G(n)/n!. The quantity bn is in fact identical with
the cluster integral designated bn in Mayer's classical
theory6-this can easily be shown by approximating
the potential by a sum of pair potentials and developing
the integrand as a polynomial in jij as in Mayer's
treatment. Thus the clusters defined by the n-body
lifetimes turn out to be a physical realization of Mayer's
clusters.
The clusters used in this treatment differ from the
"physical clusters" considered by Hill,7 which are
defined by setting some boundary in phase space, and
which are more easily defined when the intermolecular
forces are strong. Hill's treatment requires the residual
consideration of collisions between clusters, which
therefore cannot be treated as a perfect gas.
E. CONCLUSIONS
Starting from the simple assumption that gas im
perfections can be attributed to the formation of
transient complexes that can be treated just like stable
molecules, we have arrived at results of unexpected
generality. The lifetime matrix Q for each atomic or
molecular species can be extended to include the bound
6 J. E. Mayer and M. G. Mayer, Statistical Mechanics (John
Wiley & Sons, Inc., New York, 1940).
7 T. L. Hill, J. Chern. Phys. 23, 617 (1955); T. L. Hill, Statistical
Mechanics (McGraw-Hill Book Company, Inc., New York,
1956), pp. 152 £I; N. Davidson, Statistical Mechanics, (McGraw
Hill Book Company, Inc., New York, 1962), pp. 337 £I. states as well as the collisional continuum for which it
was originally introduced. A new, more general form
for the molecular internal partition function has been
discovered,
Zinb j exp( -E/kT)h-1 TrQ(E) dE, (66)
which reduces, for bound states, to the familiar form.
For an atomic gas, it reduces to the Kahn-Uhlenbeck
expression for the quantal second virial coefficient;
in the classical limit it can be transformed to the usual
configuration integral. All possible molecular combina
tions, including nonbonding systems that may even
have negative partition functions and concentrations,
must be considered as present in the gas. Using the
partition functions, the concentrations and the thermo
dynamic functions can be immediately expressed. The
result is equivalent to an exact quantal cluster expan
sion, and the transition to the classical limit follows
simply and directly by introducing the classical colli
sion lifetimes.
The principal formal deficiency of the theory in its
present form is that it does not cope with the long
range Coulomb interaction.
The effectiveness of the lifetime matrix in dealing
with the thermodynamic properties of gases leads to
the hope that it may also be useful in connection with
transport properties. It is well known that the usual
derivation of the Boltzmann equation neglects the
duration of the collisions.
It now becomes important to seek practical methods
for calculating the lifetime matrix or its trace. Percival's
work8 in this direction is welcome.
ACKNOWLEDGMENT
I wish to record here the stimulus of a recent con
versation with Dr. Adolf Hochstim, who provoked
this work by asking how the Kahn-Uhlenbeck formula
could be generalized to molecular encounters.
8 I. A. Percival Proc. Phys. Soc. (London) 80, 1290 (1962).
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1.1728905.pdf | Enhanced Photoemission and Photovoltaic Effects in Semitransparent Cs3Sb
Photocathodes
Frederick Wooten
Citation: Journal of Applied Physics 33, 2110 (1962); doi: 10.1063/1.1728905
View online: http://dx.doi.org/10.1063/1.1728905
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IP: 69.166.47.134 On: Wed, 03 Dec 2014 06:35:55JOURNAL OF APPLIED PHYSICS VOLUME 33, NUMBER 6 JUNE 1962
Enhanced Photoemission and Photovoltaic Effects in Semitransparent
Cs3Sb Photocathodes*
FREDERICK WOOTEN
Lawrence Radiation Laboratory, University of California, Livermore, California
(Received December 11, 1961)
Measurements were made of photoemission and photovoltaic effects in thin (300-1250 A) films of CSaSb,
on a glass substrate, by scanning with a 20 J£-diam light spot. Enhanced photoemission and photovoltaic
effects are obtained in regions within 1 to 3 mm from an Al electrode contact. Enhancement is about 25%
with white light, and as much as a factor of 2.5 at 6200 A. A band model is proposed in which the top of the
valence band is fixed with respect to the Fermi level at the vacuum surface, but bends up at the glass
substrate and, even more, at the electrodes.
INTRODUCTION
CESIUM antimonide is one of the most efficient
photoemitters known. It is also a p-type semicon
ductor. Indeed, it has been found that the best photo
emitters in the family of alkali-antimony compounds are
all p-type semiconductors.!,2
There has been some speculation that the greater effi
ciency of the p-type alkali-antimony compounds may
arise in part from favorable band bending, produced at
both the semiconductor-vacuum interface and the semi
conductor-backing interface.a A diagram which illus
trates favorable band bending is shown in Fig. 1, from
which it is seen that photoexcited electrons are acceler
ated toward the vacuum by the space-charge field.
Experiments to be reported here include studies of
photoemission and photovoltaic effects in semitrans
parent CsaSb films. These experiments indicate that the
valence band in CsaSb bends up at the glass substrate
and, to a greater degree, at the electrode contacts. En
hanced photoemission, observed in the vicinity of the
electrodes, thus supports the view that band bending is
an important factor in photoemission from thin films.
By thin films, it is meant that the film thickness (us
ually 300-600 A) is of the order of the mean escape
depth4 (250 A) for photoelectrons. No studies of the in
fluence of band bending at the vacuum surface were
made.
METAL SEMICONDUCTOR FIG. 1. Band bending
in a p-type semiconduc
tor with n-type surface
states and a metal sub
strate. Bands also tend
to bend up for other sub
strate materials (glass)
if the semiconductor has
the lower work function.
For a fuller discussion,
see reference 3.
* Work performed under the auspices of the U. S. Atomic
Energy Commission.
1 A. H. Sommer, J. App!. Phys. 29, 1568 (1958).
2 W. E. Spicer, Phys. Rev. 112, 114 (1958).
3 W. E. Spicer, RCA Rev. 19, 555 (1958).
~ J. A. Burton, Phys. Rev. 72, 531A (1947). EXPERIMENTAL
A typical phototube used in these experiments is
shown in Fig. 2. It consists of a Cs.Sb cathode, nickel
anode, and a Corning 7740 borosilicate glass en
velope.
Thin films of CsaSb were prepared by standard tech
niques5 on the flat, polished end windows of the glass
envelopes. The films were not sensitized with oxygen nor
was any conducting substrate deposited. These films
were rectangular (2 mm wide and 15 mm between elec
trodes) and 300 to 1250 A thick. Films of uniform thick
ness were produced by evaporating antimony from a
nickel ring (see Fig. 2). The thicknesses of the antimony
layers first deposited were estimated from transmission
measurements. An expansion factor of 6.3, calculated
from the crystal structure of CsaSb, was used to obtain
the CsaSb film thickness.6 A nickel shield, movable by
gravity or with a magnet, was used to produce the rec
tangular shape of the CsaSb film. The shield served also
as the anode when measuring photoemission. Aluminum
electrodes provided electrical contacts at the ends of the
CsaSb films.
Dark resistivity of the CsaSb was about 200 ohm-cm,
but sometimes varied from this value by as much as a
factor of 4. A rough measurement of thermoelectric
power indicated p-type conductivity with Seebeck co
efficient of 0.6 mv;oK at 320oK, which is in agreement
with previous determinations of conductivity type.7 Av
erage luminous sensitivity was 40 ~a/lu, on the basis of
a tungsten-filament light source operated at a color tem
perature of 2870oK.
The white-light source was a tungsten bulb in series
with a pinhole (200-~ diameter) and a lOX microscope
objective. The image of the pinhole, focused on the
CsaSb film, thus produced a 20-~-diam light spot. Meas
urements near 6200 A were made by utilizing a No. 29
Kodak-Wratten gelatin filter. The latter is a feasible
method since the efficiency of a CsaSb photocathode
. drops rapidly at wavelengths greater than 6200 A, and
6 A. H. Sommer and W. E. Spicer in Methods of Experimental
Physics, edited by K. Lark-Horovitz (Academic Press Inc., New
York, 1959), Vo!' 6, Part B, p. 385.
6 K. H. Jack and M. M. Wachtel, Proc. Roy. Soc. (London)
A239, 46 (1957).
7 Toshimichi Sakata, J. Phys. Soc. Japan 9, 1030, 1031 (1954).
2110
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IP: 69.166.47.134 On: Wed, 03 Dec 2014 06:35:55SEMITRANSPARENT Cs3Sb PHOTOCATHODES 2111
transmission of the Wratten filter drops sharply below
6200A.
RESULTS
When the CsaSb photocathode was scanned with
white light from one electrode to the other, photoemis
sion was found to be uniform over nearly all the cathode.
Within a millimeter or so from either electrode, however,
photoemission was greater by about 25%. Repeating
the experiment with red light, it was found that photo
emission near the electrodes increased by a factor of as
much as 2.5. The photocurrent was directly proportional
to light intensity for intensities ranging up to a factor
of 100 greater than that used in most experiments.
Photovoltaic effects were also observed in the vicinity
of the electrodes. These photovoltages were most pro
nounced over just that distance along the cathode from
which enhanced photoemission was observed. The maxi
mum photovoltage obtained, by illuminating the entire
junction at the electrode, was 0.75 v. The sign of the
photovoltage indicated a positive space-charge region
in the CsaSb and a negative charge on the aluminum
electrode. That is, the bands in CsaSb bend up at the
electrode.
Typical results of photoemission and photovoltage
measurements near an electrode are given in Fig. 3. Also
included in Fig. 3 is a simplified picture of band bending
which will be discussed later in more detail. Note,
though, that there are really two components of band
bending: Both are of interest. One component is per
pendicular to the vacuum surface. It is this component
which can yield enhanced photoemission. The other
component is parallel to the cathode and occurs because
of the electrode contact. It is this latter component of
band bending which results in the photovoltaic effects
reported here.
Enhanced photoemission and photovoltaic effects
were observed over distances of about 1-3 mm from the
electrodes in seven out of eight phototubes. For one
phototube, with cathode thickness of 1050 A and resis
tivityof 120 ohm-cm, a photovoltage was observed over
a distance of only 75 p,. The magnitude of the photo
voltage in this case was only 0.3 mv with maximum light
ALUMINUM FILM
CATHODE ELECTRODE
Sb RING EVAPORATOR
FIG. 2. Phototube utilizing:antimony ring evaporator to produce
uniform cathode film thickness. b
d
mm I
Z
lJ.J
0::
0::
:::J
U o
~ lOll AMP
~ I -T-"----
+4mv
o 3
FIG. 3. (a) Idealized cross section of electrode-Cs 3Sb junction.
Also indicated is an electrometer for measuring photovoltage. (b)
Band bending in Cs3Sb near electrode. Experiments indicate that
the bands bend both parallel and perpendicular to the Cs3Sb film.
(c) Photoemission with white light. (d) Photoemission at 6200 A.
(e) Photovoltaic effect. (f) Scale of distance along photocathode.
intensity, and no enhancement of photoemission was
observed. In general, enhanced photoemission and
photovoltaic effects were more pronounced with thinner
cathodes (300-500 A) and higher resistivities (400-800
ohm-cm).
DISCUSSION
The photovoltaic effects observed in the vicinity of
the electrodes clearly show the presence of band bending
parallel to the plane of the CsaSb cathode. The sign of
the effect, as can be seen from Fig. 3 (a, e), shows that
holes move toward the electrode, indicating that the
bands bend up at the e1ectrode-Cs aSb interface and form
an ohmic contact for holes. Since CsaSb has a work func
tion ( ~ 1.8 ev) much less than that of the aluminum elec
trode, this is the type of bending which would be ex
pected.a What is surprising is the range of these effects.
Solving Poisson's equation in one dimension,8 and as-
8 C. Kittel, Introduction to Solid State Physics (John Wiley &
Sons, Inc., New York, 1957),~2nd ed.,_p. _388.
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] IP: 69.166.47.134 On: Wed, 03 Dec 2014 06:35:552112 FREDERICK WOOTEN
ALUMINUM GLASS
ELECTRODE SUBSTRATE
-=~=--- --==~-----
FIG. 4. Two-dimensional model of valence band of CsaSb near
an electrode. Refer to Fig. 1 for a cross section which shows how
band bending results in a reduced barrier for photoemission.
suming a carrier density of 1016 cm-3 (reference 7), one
expects band bending over a distance of the order of
only lOa A. However, from the arguments presented
below, it will be apparent that the band bending is
two-or three-dimensional in nature. Thus, one should
not be surprised that a simple one-dimensional analysis
of the problem is grossly inadequate.
The enhanced photoemission in the vicinity of the
electrodes can only be understood by assuming that
there is a bending of the bands perpendicular to the
film in such a manner as to yield an effectively reduced
surface barrier for photoemission and, further, that the
degree of bending perpendicular to the film must de
crease with increasing distance from the electrodes.
Since the work function of materials can be determined
entirely by the surface,9.lo it seems most probable that
the bands are fixed at the emitting surface, but that the
position of the bands at the glass-CsaSb interface varies
with distance along the film. Such a situation is shown
schematically in Fig. 4.
Since the work function of glass is greater than that
of CsaSb, one should expect the bands in CsaSb to bend
up at the glass-CsaSb interface. Thus, some band bend
ing at the glass-CsaSb interface may well be present over
the entire film. This band bending may extend into the
CsaSb for a distance of 50 A.ll
9 J. Bardeen, Phys. Rev. 71, 717 (1947).
10 F. G. Allen and A. B. Fowler, J. Phys. Chern. Solids 3 107
(1957). '
11 w. ~. ?pice~, J. App!. Phys. 31, 2077 (1960). Spicer points
out that It IS unhkely that the fit between calculated and experi
mental curves for photoemission in the alkali-antimonides would
be possible if there were considerable band bending (normal to the
cathode) over more than about 50 A. He further notes it is doubt
ful if Taft and Phillip [E. A. Taft and H. R. Phillip, Phys. Rev.
115, 1583 (1959)J could have seen valence-band structure in the
velocity distribution of photoemission if there were considerable
band bending. The work of Taft and Phillip did differ from the
present work, however, in that a nickel substrate was used. GLASS
(a) = t+ I ......... +.+ '\.+ ... + .... t;+ + ...... * ............... * ... + ++
ALUMINUM ~ + + + + C
:: + + + 53Sb
GLASS
+ + ....... ot •••••• 1-.................... +-•• (b)
ALUMINUM il:+ + (e)
FIG. 5. (a) Schematic of total charge distribution. (b) That part
of the total charge which establishes an electrical double layer at
the glass-CsaSb interface. The space-charge field is such as to ac
celerate photoelectrons toward the vacuum, away from the glass
substrate. (c) That part of the total charge which leads to band
bending parallel to the plane of the Cs3Sb cathode. Because of
competition from the glass substrate for electronic charge the
field strength parallel to the cathode is greatly reduced. '
Although no attempt has been made to perform a de
tailed analysis of the band bending observed here, it is
probably worthwhile to point out some of the phenom
ena which must take place. First, negative charge must
be transferred from the CsaSb film to both the glass
substrate and the electrodes. The competition for the
available charge near the electrodes may contribute to
the extreme range over which band bending parallel to
the film takes place. This is illustrated in Fig. 5. Also,
because of the ease with which CsaSb is electrolyzed the
density of defect centers may vary with position i~ the
cathode.I2•la This, again, may affect the band bending
and the associated phenomena reported here.
ACKNOWLEDGMENTS
I wish to thank G. A. Condas, for his all-round sup
P?rt and intere.st. To A. M. Portis, A. Rose, and espe
CIally W. E. SpIcer, I am grateful for helpful discussions.
Thanks are due W. F. Lindsay for his contributions
during the early stages of this work, A. L. Greilich and
C. W. McGoff for tube construction, and D. R. Dalgas
and C. M. Howard for design and construction of the
light source and associated equipment. Finally, I wish
to acknowledge the continuing support and encourage
ment of L. F. Wouters.
12 W. E. Spicer (private communication).
1a H. Miyazawa and S. Fukuhara, J. Phys. Soc Japan 7 645 (1952). . ,
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1.1732853.pdf | Effects of Helium Buffer Gas Atoms on the Atomic Hydrogen Hyperfine Frequency
George A. Clarke
Citation: The Journal of Chemical Physics 36, 2211 (1962); doi: 10.1063/1.1732853
View online: http://dx.doi.org/10.1063/1.1732853
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132.174.255.116 On: Mon, 22 Dec 2014 05:50:22THE JOURNAL OF CHEMICAL PHYSICS VOLUME 36, NUMBER 8 APRIL 15, 1962
Effects of Helium Buffer Gas Atoms on the Atomic Hydrogen Hyperfine Frequency* t
GEORGE A. CLARKE
Department of Chemistry, Columbia University, New York, New York
(Received August 18, 1961)
Calculations are made for the pressure, mass, and temperature dependence of the atomic hydrogen
hyperfine frequency shift arising from the perturbing influence of helium buffer gas atoms. The calculated
fractional pressure shift is found to be +1.73XIQ-9 mm Hg-l compared with the experimental value of
+3.7 X 10-9 mm Hg-l. From the standpoint of the calculation this discrepency may be related to a failure
of the simple wave function employed to adequately characterize the unpaired electron spin density at the
hydrogen nucleus. The results of the calculations for the temperature variation and mass dependence
indicate that these are very small effects. The form of the mass dependent quantum statistical correction,
used to elucidate the mass effect, suggests that the relatively large variations experimentally observed with
other buffer gases, due to hydrogen isotope variation, are in error.
Finally, a calculation is made for the pressure shift from a kinetic theory point of view and the result ob
tained (+1. 79X lQ-9 mm Hg-l) is in fair agreement with the above mentioned quantum statistical
calculation.
INTRODUCTION
RECENT experiments by Anderson et al.1 on the
determination of the atomic hyperfine frequencies
of the hydrogen isotopes in the presence of various
buffer gases by an optical pumping method have re
vealed a dependence of the experimentally determined
fractional pressure shifts on the nature of the hydrogen
isotopes and on the buffer gases employed. This latter
dependence has been shown by Anderson et al, to
parallel the polarizability of the buffer gases and is
characteristic of the effect of buffer gases on other
optically pumped atoms2-5 (see Table I). The ordering
reflects the predominating perturbation of the relevant
hyperfine levels, but this net effect is in general difficult
to predict without recourse to detailed calculation.
The purpose of this note is to report on calculations
for the fractional pressure shift and the temperature
variation of the frequency shift for hydrogen atoms in
the presence of helium buffer gas atoms. Similar calcu
lations6-8 have been presented for more complicated
systems; we present here a nonparametric approach to
the problem. In order to elucidate the isotopic mass
dependence for this favorable system a quantum
statistical correction to the classically averaged frac
tional pressure shift has also been calculated.
* This research was supported by the U. S. Atomic Energy
Commission and the U. S. Air Force. t Preliminary results were reported previously [Bull. Am. Phys.
Soc. 6, 248 (1961)].
1 L. W. Anderson, F. M. Pipkin, and J. C. Baird, Jr., Phys.
Rev. Letters 4, 69 (1960); Phys. Rev. 120, 1279 (1960); 122,
1962 (1961).
• M. Arditi and T. R. Carver, Phys. Rev. 109, 1012 (1958).
3 E. C. Beaty, P. L. Bender, and A. R. Chi, Phys. Rev. Letters
1, 311 (1958).
4M. Arditi and T. R. Carver, Phys. Rev. 112,449 (1958).
6 E. C. Beaty, P. L. Bender and A. R. Chi, Phys. Rev. 112,
450 (1958). FORMALISM
In a dilute hydrogen-helium (H-He) gas mixture
in which the gas atoms are only weakly coupled, the
interactions at anyone atom may be considered inde
pendent at any instant. Furthermore, with an extreme
dilution of hydrogen atoms in the helium gas we may
neglect those effects that arise from hydrogen atoms in
close proximity to one another. In the gas mixture,
then, we consider that the hydrogen atoms, in a given
hyperfine level of the ground state, experience adiabatic
interactions with the helium gas atoms such that only
level shifts occur at any instant. It is assumed that the
sole source of disorientation of these hydrogen atoms
would be an externally applied radio-frequency field.
In order to calculate some average property (e.g., the
hydrogen atom fractional pressure shift) associated
with the hydrogen atoms in the gas mixture, it is re
warding if we immediately simplify a formally more
difficult problem by imagining the gas mixture to be
one in which the distribution of helium gas atoms
about each hydrogen atom is independent of the other
hydrogen atoms in the gas mixture. By so doing we
need only focus attention on a single hydrogen atom
in the gas mixture (together with its helium atom dis
tribution) and can readily employ the statistical-
mechanical concept of an ensemble average to this
situation to calculate the probable value of the property
of interest. This result we place in correspondence with
an experimentally determined value for the gas mixture.
We can employ another approach to the calculation
and we do so in the appendix. There we consider
specifically the details of the individual H-He collision
processes and relate them to an observed property in
a manner familiar to kinetic theory. These two ap
proaches, the statistical-mechanical and the kinetic
theory, will be seen to lead to similar results.
6 F. J. Adrian, J. Chern. Phys. 32, 972 (1960).
7 H. Margenau, P. Fontana, and L. Klein, Phys.
87 (1959).
8L. B. Robinson, Phys. Rev. 117, 1275 (1960). Since the interactions experienced by a hydrogen
Rev. 115, atom in a helium gas may be considered to be pairwise
additive, we need only consider the perturbation of the
2211
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132.174.255.116 On: Mon, 22 Dec 2014 05:50:222212 GEORGE A. CLARKE
TABLE 1. Fractional pressure shifts (mm Hg-IX1(9) observed
for several paramagnetic 2S! state atoms in the presence of buffer
gases.1•3•4
Pumped atom&
Buffer gas H D T Rb87 Cs133
He +3.7 +105.3 +174.0
Ne +2.1 +2.5 +3.7 +57.4 +70.7
Ar -4.7 -3.6 -2.4 -7.5 -27.2
Kr -84.9 -141.4
Xe -261.1
H2 -0.31 +96.6 +206.7
D2 +98.0
Nt +76.1 +101.2
CH4 -73.2
n-C.H12 -409.7
n-C7H16 -614.5
& Results for the hydrogen isotopes were obtained at 50'C; those for CS'33
and Rb87 were obtained at 30°C and "near room temperature," respectively.
hydrogen atom hyperfine level arising from a single
helium atom at any instant (the nuclear distribution
function relative to the hydrogen atom may be ob
tained from statistical considerations). In order to do
so we may consider the electronic wave function for
this or any other interacting subsystem to be given
formally as
(1)
where fH and fHe are the set of electron coordinates
associated with each atom and R denotes the implicit
dependence of the wave function on the internuclear
coordinates. As we will assume that each of the wave
functions of the type given by Eq. (1) is a solution of
a spin-independent Hamiltonian, each will serve as a
valid description in a first-order perturbation theory
calculation for the spin-dependent interactions arising
from a hydrogen atom in the hyperfine state F=O or
F= 1 merely by a proper choice for the electron spin
associated with the hydrogen atom in a basis set. By
virtue of the Franck-Condon principle, then, we can
examine with equal facility the instantaneous hyper-fine level separation of a hydrogen atom in the presence
of a perturber helium atom.
The spin-independent Hamiltonian (in atomic units)
for each of the independently interacting subsystems
from which the wave function of Eq. (1) would result
is of the form
3
Ho= (2/R)-:L [tV?+(l/riH)+(2/riHe)- (l/n;)J,
1>i=1
(2)
such that
and where E(R), when taken relative to the energies
of the infinitely separated atoms, is the intermolecular
potential energy VCR) for a given set of intermolecular
coordinates. We shall assume for the moment that this
three-electron problem is completely solved, and will
return to it at a more convenient point in the discussion.
Under the experimental condition of a weak applied
magnetic fieldl we need only consider as a perturbation
correction to Eqs. (2) and (3) that term which arises
from the nonzero interaction of the magnetic moments
of the individual electrons in the subsystem with the
magnetic moment of the hydrogen nucleus, i.e., the
Fermi contact term,
3
HF= (8·11/3)gNg.fJNfJ.:LIH'Sia(r,rI). (4)
;=1
In Eq. (4) gN and g. are nuclear (N) and electronic (e)
g factors, /3N and /3e the nuclear and Bohr magnetons,
IH the nuclear spin vector for the hydrogen nucleus,
S .. the electronic spin vector for the ith electron, and
a( riR) the Dirac delta function for the ith electron
which vanishes everywhere except at the hydrogen
nucleus.
The first-order perturbation theory correction to
E(R), the variation of a hyperfine level as a function
of the internuclear coordinates, is given simply by the
expectation value of. the Fermi contact term:
3
EF(1)(R) = (871'/3) gNg.fJNfJ. (iYH_He(rH, rHei R) I L:IHoS,a(r,rI) I \}fH-He(rH, rHe; R». (5)
i=I
The fractional variation of the hyperfine frequency follows immediately from an evaluation of Eq. (5) for the
relevant hyperfine levels and may be obtained directly by an evaluation of Eq. (6) for a particular hyperfine level.
8
(\}fH_He(rH, rHe; R) I L:IH'Sia(riR) I \}fH-He(rH, fHei R) )-vo
v(R) -Vo Av(R) i=1 ---=
Vo Vo 3
(\}fH-He(rH, fHe; (0) I L:IH"S;o(rm) I \}fH-He(fH, rHe; (0»
;=1 (6)
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132.174.255.116 On: Mon, 22 Dec 2014 05:50:22ATOMIC HYDROGEN HYPERFINE FREQUENCY 2213
In Eq. (6) Vo is obtained by an evaluation of Eq. (5)
with the hydrogen and helium atoms at infinite sepa
ration; Vo is, apart from a constant, the atomic hyper
fine frequency for an isolated hydrogen atom. The ratio
.:lv(R) /vo expresses the fractional variation of the
hydrogen hyperfine frequency arising from a single
interacting helium atom; the quantity of physical
interest, the ensemble average fractional frequency
shift, from classical statistical considerations can be
obtained by multiplication of Eq. (6) by the classical
configurational distribution function for a single sub
system times the number of helium atoms in the system
and summing over the internuclear coordinates, i.e.,
(.:lV/vo)=PHej [.:lv(R)/voJ exp[ -V(R)/kTJdR. (7)
In Eq. (7) PHe is the number density of helium atoms,
VCR) the intermolecular potential energy for a sub
system and k is Boltzmann's constant. Substitution of
the equation of state for an ideal gas in Eq. (7) and
differentiation with respect to pressure, at constant
temperature and volume, gives immediately a quantity
desired-the ensemble average fractional pressure shift.
(a/ap) «.:lv/vo»)r.v= (l/kT) j [.:lv(R)/voJ
X exp[ -V(R)/kTJdR. (8)
The temperature variation of the ensemble average
fractional frequency shift, at constant buffer gas den
sity, is obtained by differentiation of Eq. (7) with
respect to temperature, i.e.,
(a/a T) ( (.:lv/vo») PH.,V= (PHe/k'J'2) j [.:lv(R) /voJ
X exp[ -V(R)/kTJV(R)dR. (9)
A more convenient quantity, the temperature variation
of the frequency shift, is obtained simply by multipli
cation of Eq. (9) by the atomic hydrogen hyperfine
frequency.
In order to exhibit a mass dependence for the frac
tional pressure shift we may argue that the relative
nuclear motion of a H-He subsystem is not sufficiently
described classically and should be described instead in
quantum mechanical terms-the de Broglie wave
length associated with the relative motion of a H-He
subsystem at room temperature is on the order of the
distance over which these atoms effectively interact
with one another (A"-' 10-8 cm).
In the quantum statistical description the sub
system configurational distribution function employed
in the ensemble average [Eq. (8) J is no longer correct;
in the near-classical limit, however, it has been shown
that the distribution function can be described by the
classical distribution function modified by correction
terms arising from the noncommuting terms in the TABLE II. Variation of the fractional frequency shift !:>.v(R)/vo
and intermolecular potential energy VCR) with internuclear
separation.
R VCR)
(a.u.) !:>.v(R) /vo (a.u.)
1.00 0.017075 0.61420
2.00 0.120229 0.10510
3.00 0.032847 0.01963
4.00 0.006730 0.003455
5.00 0.001196 0.000568
6.00 0.000195 0.000088
7.00 0.000030 0.000013
8.00 0.000005 0.000002
9.00 0.000001 0.000000
10.00 0.000000 0.000000
Hamiltonian for relative mass motion.9-11 The con
figurational distribution function in this limit, with
only the first nonzero corrections (of order 'A2) included,
is proportional to
exp[ -V(R)/kTJ(1+('A2/6kT) {(1/2kT)[VV(R)J2
-V2V(R)}). (10)
The thermal de Broglie wavelength ('A) associated
with a molecule of reduced mass J.L is given by Eq. (11),
(11)
The quantum-statistical mass dependent correction to
the classical ensemble average fractional pressure shift
follows then from Eq. (8):
HVkT)2j [.:lv(R)/voJ exp[ -V(R)/kTJ
X ((1/2kT) [vV(R) JL V2V(R) }dR. (12)
QUANTUM MECHANICS OF A SUBSYSTEM
It is now clear that to actually compute the effects
of helium buffer gas atoms on the atomic hydrogen
hyperfine frequency we must first specify explicitly the
intermolecular potential function [VCR) J and the
electronic wave function for the H-He subsystem. To
simplify this quantum-mechanical problem [Eq. (3) J
we choose for the subsystem wave function (suitably
normalized) the antisymmetrized product of one
electron spin orbitals, i.e.,
'l'H_He(rH, rHe; R) = A I uH(1)uH.(2)UHe(3) I. (13)
In this first approximation we restrict the basis set to
Is hydrogen and hydrogen-like orbitals centered on
the hydrogen and helium atoms, respectively, with
orbital exponents ZH= 1.0000 and ZHe= 1.6875. The
intermolecular potential energy for this spherically
symmetric three-electron problem is obtained from
9 J. G. Kirkwood, Phys. Rev. 44, 31 (1933).
10 M. L. Goldberger and E. N. Adams, ]. Chern. Phys. 20,
240 (1952).
11 J. E. Mayer and W. Band, J. Chern. Phys. 15, 141 (1947).
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TABLE III. Effects of helium buffer gas atomso on the atomic
hydrogen hyperfine frequency at 323 K.
Fractional pressure shift (mm Hg-l) X 1()9
Classical average + 1. 67
Quantum average + 1.73
Experimental +3. 7 ±O. 71
Temperature variation of frequency shift (cps/mm Hg-OK) X1()3
Classical average +5.64
Eq. (3) by an evaluation with the single determinant
wave function and is given formally by Eq. (14) as a
function of the internuclear separation:
E(R) -E( (Xl) = VCR)
= ('l1H_He(rH, rHe; R) \ Ho \ 'l1H_H.(rH, rHe; R)
-('l1H-He(rH, rH.; O()) \ Ho \'l1H-He(rH, rH.; (Xl). (14)
Ho is the Hamiltonian given by Eq. (2), and E( (Xl) is
the energy of the H-He sybsystem at infinite separa
tion, i.e., the energy of the isolated atoms. The calcu
lation for the intermolecular energies with the single
determinant wave function has been considered pre
viouslyI2 (R~5 a.u.) and for present purposes we
have recalculated some values and extended these
results to include larger values of R. As observed pre
viouslyI2 the results of such a calculation yield a set of
energies that increase monotonically with decreasing
internuclear separation; the variation of VCR) with R
is given in Table II.
The fractional frequency shifts, computed by an
evaluation of Eq. (6) with the single determinant
wave function, also yield a monotonically increasing
variation with decreasing R (to about 2 a.u.) and are
tabulated in Table II for several values of R.
In order to facilitate numerical calculations we have
expressed [Eq. (15)J the intermolecular energies of
Table II in terms of two analytical expressionsI2 that
satisfactorily characterize these results.
VCR) =4.606e-1.76OB+ (3.666X 10--4/ R) -(2.94/ R6)
2~R~5,
VCR) = +5542.9688/ RIO R?5. (15)
RESULTS
The results of numerical evaluation of the integrals
for the classical and quantum-statistical ensemble
average fractional pressure shift and the temperature
variation of the frequency shift, at constant buffer gas
density, are given in Table III together with the experi
mentally determined fractional pressure shift. The
calculated temperature effect indicates that the H-He
system can be expected to be a rather stable one to
modest temperature variations-more so than the
12 E. A. Mason, J. Ross, and P. N. Schatz, J. Chern. Phys. 25,
626 (1956). alkali atom systems where observed temperature vari
ations have been found to lie approximately between
± 1 cps/mm Hg-°C.3-6 This result is not surprising
in view of the relatively weak distortions of the hydro
gen atom in the presence of several different buffer
gases (Table I) .
The rather poor agreement between the experi
mentally determined pressure shift and that calculated
with the single determinant wave function is indicative
of the inadequacy of the wave function. This situation
is not significantly altered if we employ, instead, a
molecular orbital function formed from the 1s hydrogen
and helium atomic orbitals [which differs from Eq.
(13) by the incorporation of an additional determi
nant, A I J.LHJ.LHeJi.H I , with a coefficient at any value of
R determined from a variational calculationI2]' Further
more, any attempt at a recognition of a van der W~als
polarization,6.13 leads naturally to a further reductlOn
in the calculated shift. The basic difficulty of the present
calculation appears to be the failure of the subsystem
wave function to realistically account for the distor
tions that arise in the free helium atom and, especially
important here, the hydrogen atom electronic orbitals
as a result of mutual interaction. For a small atom inter
action system it is to be expected, and the experimental
result indicates, that the principal interaction experi
enced by the hydrogen atom electron arises from the
Pauli exclusion effect which, with significant orbital
overlap, leads to an increase in the unpaired spin den
sity at the hydrogen nucleus and a positive pressure
shift. The function given by Eq. (13) is partially defi
cient in this respect but the calculated shift will not
improve measurablyI2 unless the basis set of functions
is extended to include excited state configurations.
Preliminary results obtained by Matsen and BrowneI4
for the unpaired electron spin density at the hydrogen
nucleus with a more detailed wave function offers
much encouragement in view of the experimental result.
Their wave function includes the hydrogen and helium
atomic orbitals through to and including the 2p Slater
orbitals with the nonlinear coefficients obtained from a
variational procedure.
The quantum statistical mass correction for the
H-He system is small (,..",4%); since distortions in the
electronic charge distribution (with resultant variation
in the potential energy of interaction) of a subsystem
are not expected to vary significantly from one hydro
gen isotope to another, it follows from the form of the
quantum correction that somewhat smaller corrections
-proportional to the ratio of reduced masses
JLH-H./ J.LD-He and JLH-He/ J.LT-He for the deuterium
helium and tritium-helium systems, respectively
would result for the D-He and T-He systems. With
an experimental procedure sensitive to these small
variations one could hope to investigate this effect and
13 See, for example, T. P. Das and R. Bersohn, Phys. Rev. 102,
733 (1956).
14 F. A. Matsen and J. C. Browne (private communication).
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its consequences, however, the present experimental
uncertainty is such that it renders these mass effects
inaccessible. The fractional pressure shifts reported1
for the hydrogen isotopes in the presence of neon and
argon buffer gases appear to exhibit an isotopic mass
dependence, but these variations are not at all con
sistent with a quantum mass effect. If the alterations
in the potential energy and electronic charge distribu
tion are indeed negligible, as we should expect, then
there is, in the present context, no other apparent
mechanism available through which such observed
variations in the fractional pressure shifts may arise.
We can only conclude, therefore, that these results are
m error.
ACKNOWLEDGMENTS
The author gratefully acknowledges his indebtedness
to Professor Richard Bersohn for suggesting this
problem and for many informative discussions in its
pursuit. The author would also like to express his
appreciation for the use of the computing facilities
made available by the IBM Watson Scientific Com
puting Laboratory, Columbia University.
APPENDIX
In this section we present an alternative approach
to the calculation of the hydrogen atom fractional
pressure shift from a kinetic point of view. We shall be
concerned here with the details of the collision act,
i.e., a H-He encounter.
The instantaneous mean fractional frequency shift
of a collection of hydrogen atoms in a unit volume of a
H-He gas mixture in which the hydrogen atoms are
sufficiently dilute so that they experience at any in
stant predominantly binary (adiabatic and inde
pendent) encounters with the helium atoms in the unit
volume is given in terms of a sum of the contributions
from the several hydrogen. atoms in the unit volume of
number N and density PH. Each of these contributions
will be dependent upon an instantaneous relative
separation Ri for a particular hydrogen atom and
helium atom that constitute one of the collision pairs
at any given instant. The instantaneous mean frac
tional frequency shift for a hydrogen atom in the unit
volume may be expressed by the following:
illl( R1, R2, .;., RN) ~ ..(...illl( R.).
L.J (A1)
110 PH ;"'1 110
The function for the ith collision pair, illl (Ri) /110, as
previously defined, is the fractional frequency shift of
a particular hydrogen atom due to a perturbing helium
atom with the instantaneous relative separation Ri.
The summation extends over all of the hydrogen atoms
in the unit volume although only a fraction of these
atoms contribute effectively at any given instant. This
is of course due to the fact (Table II) that the shift
differs from zero only over a very limited range of R.
The instantaneous mean fractional frequency shift is not, however, of interest as it corresponds to a condi
tion other than that physically attainable. A more
meaningful quantity that will be associated with an
experimentally determined mean shift is the time
average of this instantaneous shift over an interval
ilt which is long compared with the effective interaction
time T. but short compared with the mean free time,15
i.e.,
1 1 j'o+6.'illl(Ri) =-L- --dt.
PH j ilt to 110 (A2)
In Eq. (A2) the sum is understood to extend over the
probable number of H-He encounters that occur in
the unit volume within ilt. (A typical ilt under experi
mental conditions would be on the order of 10-10 sec
with Tc on the order of 10-l2-10-13 sec.) In this way a
large number of binary encounters are included and
the instantaneous fluctuations due to these encounters
are smoothed over the interval. Each hydrogen atom
can, however, suffer at most a single collision in ilt. An
implicit assumption of Eq. (A2) is that the time aver
age is independent of further extensions in the interval
ilt.
In order to facilitate the evaluation of the mean shift
which is dependent upon the behavior of the hydrogen
atom hyperfine levels (F=O and F= 1) in the individual
collision pairs over their respective collision paths we
note that the effective interaction interval T. is very
much smaller than ilt. If we regard for the moment that
the encounters are instantaneous, i.e., if we ignore the
finite extent of the interactions, then each of the
encounters in ilt may be considered as occurring in
terior to this interval. For convenience we shall proceed
with such an assumption so that for any encounter
we have only to evaluate an integral
1 j'o+6.tilll(Ri) ---dt
ill to 110 (A3)
over the interval ilt that includes the actual collision
act. It follows immediately that we can extend the
limits of Eq. (A3) to ± 00 without error since these
included times correspond to conditions prior and sub
sequent to the particular collision. If the collision,
furthermore, is symmetric in the time we then have
for the time average fractional frequency shift of a
single collision pair the expression
!l"'illl(Ri) dt.
ilt 0 110 (A4)
16 With the relatively long time associated with the measure
ment we consider that the time average of a single hydrogen atom
over a large number of encounters in a time much greater than
At to be equivalent to this time average of a large number of
H-HE encounters in the short time At.
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Recalling from the equations of motion for pair-wise
interacting particles that the time t may be expressed
in terms of the several collision constants, we may con
veniently replace the integration over t by one over R"
and thus obtain for the time average shift for a collision
pair an equation of the form
110 2 jOOllll(R i)
llt Eo 110
dRi
X {(2e/JL)[1- (b/R)L V(R)/eJli' (AS)
Ro is the internuclear separation at t= 0, and it corre
sponds to the distance of closest approach. The reduced
mass of the collision pair is p., b is the impact parameter,
and e, obtained from the Hamiltonian for the relative
mass motion, is the relative energy of the collision pair.
Equation (AS) may be considered to be the time
average shift for a collision pair of the collision type
(b, e). Given a weighting function that represents the
probable number of encounters in the unit volume
within llt of type (b, e) with an impact parameter
between band b+db and relative energy between E
and e+de or simply (b, db) and (e, de), the sum over
the individual collision pairs in Eq. (A2) may be trans
formed to a summation of the time average shift with
the weighting function over band E consistent with
the set of collisions. If we neglect the effect of external
fields of force on the H-He gas mixture, and further
more assume it to be a random gas (composed of
elastic spheres), then from a fundamental assumption
in the kinetic theory of gases we obtain for the probable
number of H-He encounters per unit volume that occur
within llt with relative energy (e, de) and impact
parameter (b, db) the expression16
lltPHPHe[3211"/p.(kT)3Ji exp( -e/kT)ebdbde. (A6)
PH and PH. are, respectively, the density of the hydrogen
and helium atoms in the gas mixture. It follows im
mediately from Eqs. (A2), (A4), (AS), and (A6) that
the mean hydrogen atom fractional frequency shift in
18 R. H. Fowler, Statistical Mechanics (Cambridge University
Press, London, 1929), p. 421. the H-He gas mixture is given by
(1l1l/1I0)= 8PH.[1I"/ (k T) 3J1lOO
bdb tOe exp( -Elk T) de
o 0
Xl oollll(R) dR
Eo 110 {e[l-(b/R)2_ V(R)/eJli' (A7)
The range of b over which an effective encounter occurs
should, in principle, be given with greater precision
but because of the finite range of the shift we need not
bother to specify it and we may formally extend the
upper limit to + 00 without incurring error.
In the assumed ideal gas condition such that PHe
may be replaced by P/kT, the mean hydrogen atom
fractional pressure shift is obtained directly by differ
entiation of Eq. (A7) with respect to this pressure at
constant temperature and volume:
(a/ap) «IlIl/1I0)kv=8[1I"/(kT)5J!j'°bdb
o
100 100 IlIl(R) X exp( -e/kT)ede --
o Eo 110
dR
X (e{1-(b/R)2_[V(R)/eJl)f (A8)
It is to be noted that this classical expression is inde
pendent of the reduced mass JI. of the colliding H-He
atoms.
Using the analytical expressions [Eq. (15)] for the
intermolecular energies and the condition on the rela
tive energy e such that at R = Ro, (dR/ dt) E=Ro = 0, and
therefore
e= V(Ro)/[1- (b/Ro)2].
Equation (A8) has been evaluated and the value of + 1. 79X 10-9 mm Hg-I obtained for the mean hydrogen
atom fractional pressure shift.
Although the approach of this calculation for the
shift is not equivalent to that previously given, the
fair agreement in the two results, which would prob
ably not be altered by an improved electronic wave
function for a collision pair (subsystem), nevertheless
indicates that the hard sphere interaction approxima
tion employed here is an adequate one.
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1.1732140.pdf | Calculation of Interaction Matrix Elements for Asymmetric Rotors with
Resultant Electronic Spin and Nuclear Spin
R. F. Curl and James L. Kinsey
Citation: J. Chem. Phys. 35, 1758 (1961); doi: 10.1063/1.1732140
View online: http://dx.doi.org/10.1063/1.1732140
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Downloaded 20 Aug 2013 to 128.118.88.48. 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_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 35, NUMBER 5 NOVEMBER, 1961
Calculation of Interaction Matrix Elements for Asymmetric Rotors with Resultant
Electronic Spin and Nuclear Spin*
R. F. CURL, JR., AND JAMES L. KINSEyt
Department of Chemistry, Rice University, Houston, Texas
(Received December 22, 1960)
The application of Racah or vector recoupling coefficients to the calculation of interaction matrix ele
ments for asymmetric rotors with resultant electronic and nuclear spin is outlined. This approach is com
pared with Van Vleck's method of reversed angular momenta.
INTRODUCTION
THE system to be considered is an asymmetric rotor
molecule with one or more unpaired equivalent
electrons and one nuclear spin. Two stable molecules
in this category for which extensive spectroscopic data
are available are N02 and Cl02. The present work
formulates a unified method for calculating the matrix
elements needed in fitting the microwave spectra of
these molecules.
Such molecules have three angular momenta which
can interact: N (molecular rotation); S (resultant
electronic spin) ; and I (nuclear spin).
Because of the interaction between the momenta,
they couple, giving a total resultant angular mo
mentum F. There are three limiting coupling schemes
which result if one considers one interaction much
stronger than the others. These schemes are the J
scheme, N+S=J, J+I=F; the G scheme S+I=G
G+N=F; and the E scheme N+I=E, E+S=F.
The relationship between these schemes is analogous
to that betweenj-j and L-S coupling in atoms. Linl and Baker2 use Van Vleck's reversed angular
momenta3 for this calculation. The matrix elements
may also be found in a quite straightforward manner
by use of Racah4 or vector recoupling coefficients.
Under certain circumstances there appears to be an
advantage to the latter approach.
RELATIONSHIP BETWEEN MATRIX ELEMENTS IN
THE COUPLED AND UNCOUPLED SCHEMES
The calculation of the interaction terms is easiest in
the case of complete uncoupling or strong Paschen
Back effect. By the use of vector recoupling coefficients
the matrix elements in any coupled scheme may be
related to those in the uncoupled scheme. In the next
section the calculation of matrix elements in the un
coupled scheme will be discussed.
Each of the interaction terms conceivable for the
system (including those with external fields) can be
written5
(1)
The interactions in systems such as this have been A tensor of rank kl' [T (kl)], couples with a tensor of
discussed by Linl and by Baker.2 The procedure they
follow is first to derive the effective Hamiltonian, then rank k2, [U (k2)], giving a resultant tensor W (k12) ;
to calculate its matrix elements in a coupled scheme, W (k12) then couples with V (k3) giving a resultant
and finally to diagonalize the Hamiltonian matrix. tensor X(k) of rank k. T(kl) is assumed to act on
The portion of this process which is of primary concern angular momentum jl, U(k2) on j2, and V(k3) on j3'
here is the calculation of interaction matrix elements in jl, j2, and j3 all commute.
a coupled scheme. It can easily be shown [see reference 6, Eq. (7.1.5.)]
('Y'HHjI2'ja'j' II X(k) II 'YjlM12jaj) = [(2j+1) (2j'+1) (2k+1) (2jI2+1) (2jI2'+1) (2k12+1)]!
X ~::' ~~ : ~~ ~: : I L: (-y'H II TIl 'Y"jl)(-y"N II U II 'Y"'j2) ('Y"'ja' II V II 'Yj3), (2)
"1""1'"
j' j k j12' j12 k12
where the reduced matrix element is defined by
(j'll Xllj)(!~, : ~)=(_l)m'-i'(j'm'IX(kq) Ijm) (3)
* This work was supported by a grant from the Robert A. Welch Foundation. t Pre-Doctoral Southern Fellow, 1958-59. Present address: Department of Chemistry, University of California, Berkeley 4,
California.
1 C. C. Lin, Phys. Rev. 116, 903 (1959).
2 J. G. Baker, thesis, Cambridge University, 1958.
3 J. H. Van Vleck, Revs. Modern Phys. 23, 213 (1951).
4 G. Racah, Phys. Rev. 62, 438 (1942).
5 Spherical tensors will be shown in boldface sans serif T(k). Cartesian vectors will be shown in boldface as usual. In Eq. (6)
a Cartesian second rank tensor is shown in boldface German capital, U. The relationship between a spherical tensor and the
corresponding Cartesian tensor is indicated by an arrow from the spherical to the Cartesian. (T(1)->T). The direction of the
arrow implies the direction of one to one mapping.
6 A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1957).
1758
Downloaded 20 Aug 2013 to 128.118.88.48. 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_permissionsTABL.E I. Notation for interaction parameters. a
Name of interaction Van Vleckb Line Hendersond Baker" This work
1. Spin rotation interaction -ao ! (e,,+fy"+<zz) -2 (lL-lA) -a. (0).
-2a -~ (2."-<,, -EUY) 4A/3 -2b. (aa).
+a+b +!(2Eyy- Exz-e .. ) -(2A/3) -(41}/b) +e.+b. (bb) •
------~----- -------------- --------------+a-b +k (2Exx-Eyy-Ezz) -(2A/3H (41}/b) -e.+b. (ee). -------e ! (Ey,+E,y) + (ab). ------d ! (Exz+Ezx) +(ae).
-e HExy+Eyx) +A'(1/B+l/A)o+ +(be).
(a).
(b).
(e) , Molecule fixed Hamiltonian terms
(O).N·S
a .b,e
~ (ij).N,S;
ii
(ij). symmetric and traceless
2: i;kEijkNiSj (k),
+ih2:i(i),Si
------------~----------------
2. Fermi interaction q a1 (0) 1 (O)[(I·S)
(O)r= (16?r/3)gIILBPN[.p(0)]2
PB>O
3. Magnetic dipole dipole -2A 2b1 (aalr "2i;(ij) IS;!j (ij) 1 symmetric and
interaction X+2T -br-eI (bb) I traceless
X-2T -br+eI (cell (ij) [= -g,gIPBlLN[ (Oij-3riYj) /r3]Av
(ablr
(ae) r
(be) r
4. Nuclear quadrupole [eQ/21(21-1) 1 2ba (aa) a 2: (ij) aI;!; (ij) a symmetric and
interaction (c12V / aa2) traceless
etc. -ba-eQ (bb) Q (ij) Q= [eQ/21 (21 -1) II
(c12V/axi·aXj)
etc . -bQ+cQ (cc) 0
• Considerable confusion arises from naming of axes. Van Vleck's Hamiltonian has x-->a, y->b, <-->c. Henderson, Lin, and Baker make the same correspondence, but then in applying the derived equations make a the
smallest rotational constant and c the largest. The correspondences in the table above are based on x-->c, y-b, Z""4 in everyone's formulas.
b See reference 3.
• See reference t.
d See reference 16.
• See reference 2. E:::
:.
,..;j
?:I
......
;..:
i:>j
t"'
tt:I
E:::
i:>j z
,..;j
if)
"!j
o
?:I
:.
if}
><
E:::
E:::
i:>j
,..;j
?:I ....
(')
?:I o
,..;j
o
?:I
if}
.....
-.J
<.n \0
Downloaded 20 Aug 2013 to 128.118.88.48. 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_permissions1760 R. F. CURL, JR. AND J. L. KINSEY
TABLE II. Phase factors of (N'T' II DIll NT) tabulated in the form itJ.N dK_ldK+l tJ.N=N'-N, tJ.K_l=K_l'-K_ 1, tJ.K+l=K+l'-KT,.
tJ.N=O
aQ±2.'fl aQO.+l "QO,-, bQ+l._l bQ_l.+1 'Q+I.O 'Q-I,O 'Q'fI,±2
+1 +1 +1 -i +i +1 +1 +1
tJ.N=+1
"Ro.+1 aR+2._1 bR+I,+1 bIL'.+1 b~'._1 bR_l.+3 bR+3._1 cR-rI,o 'R_l.+2
+1 -1 +i +i +i +i +i -1 +1
tJ.N=-1
aPO._I ap_2.+1 bp_l._. bP+ •. _l bp_I.+, bPt-I.-3 bp_,.+. cp_ •. o cP+l._2
-1 +1 +i +i
and the symbols in curly brackets are the Wigner 9j
symbols. The symbol in parentheses in the definition
of the reduced matrix element is the Wigner 3j symbol.
The 9j symbol is related to the Fano X coefficient and
the 3j symbol to the Clebsch-Gordan or vector coupling
coefficients. If anyone of the five k's (kl, k2, ka, k12, k)
is zero, a 9j symbol in which it appears reduces to a 6j
symbol.
In using Eq. (2) several points should be remem
bered. These are expressed in the equations below.
(/'y' I[ 1 [[h)=Ojj'0'Y'l',(2j+1)! (4)
[T(k) XU(k) Jo= (-1)k(2k+1)-i(T- U) (S)
[T(l) X U(2) Jt~-(i)IT-U (6)
[T(l) X U (1) Jt~-i(2)-lTXU. (7)
To apply this to the problem in hand, jl, j2, and ja
mu"t be identified with N, S, and I. If jl corresponds
with N, j2 with S, and ja with I ("( is identified with the
asymmetric rotor quantum number T), the matrix
elements to be considered are for the J scheme. That is,
j12 = J, j = F. What remains is consideration of the
particular forms of the interaction and evaluation of
the uncoupled scheme matrix elements.
HAMILTONIAN TERMS
The Hamiltonian consists of 6 terms:
(8)
The origin of the terms and the calculation of their
matrix elements will now be discussed. The subscripts
do not refer to the order of the terms as in a perturba
tion expansion. However, the size of the interaction
does decrease with increasing subscript.
The notation of other workers for interaction param
eters is compared with the present notation in Table I.
1. Asymmetric Rotor Hamiltonian, Ho
(9)
Na, Nb, and Nc are components of N along the principal
axes of inertia of the molecule. Although the matrix +i +i +i +1 -1
elements of Ho may be found by use of Eq. (2), this
approach is inconvenient as much cancellation occurs.
The basis is chosen so that Ho is diagonal, and may be
obtained from the asymmetric rotor tables.7
2_ Electron Spin-Rotation Interaction,3 HI
HI =TlN(l) -Uls(l) +T'lN(2)' U'ls(2). (10)
Both of these terms arise as the result of a Van Vleck
perturbation treatment with inclusion of excited
electronic states. If S=! all matrix elements of U'ls(2)
are zero. If S>! the second term will ordinarily be
greater than the first. Equation (2) in J scheme re
duces for these interactions to a 6j symbol times a
simple function of the quantum numbers. In order to
apply Eq. (2) to the first term (N'T' [I TlN(1) [[ TN) and
(S [[ Uls(l) [[ S) are required.
U1s(1) =5
(S' [[ 5 I[ S)=oswh[S(S+l) (2S+1)J!
TlN(1) = (O)sN+N-T lN"(2) -iv2[NXT lN"'(l) Jl
(1)
(N'T' I[ N-TIN"(2) [[ NT )=yS( _l)N'+N
j 1 2 1) X .L: (N'T' I[ N [[ N"T")
N"T" N N' N"
X (N"T" [[ TIN" I[ NT). (11)
T"lN(2) has constant components in the frame
rotating with the molecule.
TlN"(2, q) = .L:D2qQ,(a, (3, "()'Jllv(2q') (12)
q'
'JIN(2)=*(ij)s.
'JIN(2, q') are the constant components of the tensor in
the molecular frame. a, {3, and "( are the Eulerian
angles locating the molecular axes in the space-fixed
coordinate system. (D2qq' is an element of the five-
7 C. H. Townes and A. L. Schawlow, Microwave Spectroscopy
(McGraw-Hill Book Company, Inc., New York, 1955), Appendix
IV.
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dimensional irreducible representation of the rotation
group.) Matrix elements of D2qq'(a, fj, 'Y) may be
evaluated in several ways. For the symmetric top
(N'K'M'I D2qq, I NKM)
= (-1)K'-M'[(2N+1) (2N'+1) J!
x( N' 2 N)( N' 2 N). (13)
-M' q M -K' q' K
The transformation coefficients relating the asymmetric
rotor wave functions for given asymmetry parameter K
to the symmetric rotor wave functions were tabulated
by Schwendeman and Laurie.8
N
if;NT(K) = L: aNTK(K)[if;N,K-I+ ( -1)N+T+Kif;N,_K_J. (14)
IKI-6
Another method for evaluating (N'T'II TlNl/(2) II NT)
which is usually more convenient is based on the
asymmetric rotor line strengths.9
(N'T' II TlNl/(2) II NT)
a,b,c
= L: (ij)s(N'T'11 02ij II NT) (15)
i}
(N'T' II 02ij II NT)
1
= (-1)N+N'(S)l L: J 1
NI/TI/ IN N'
x (N'T' 1101; II NI/TI/) (NI/T', I OIj II NT)
The lPN"N,T"T are phase factors and are listed in
Table II for the types of transitions which contribute
significantly. These phases have been made consistent
with Edmonds6 and the choice of axes made in Table 1.
The SNT'N"T" (i) are the asymmetric rotor line strengths
and (ij)s are components of a traceless symmetric
second rank tensor.
Because of molecular symmetry Tl/lN(2) may have
only a few independent components. For example in
the case of CI02 with C2v symmetry, TI/1N(2) has only
two independent components. The number and type of
independent components of these tensor operators for
given molecular symmetry may be found by the usual
methods of group theory.
The axial vector TI/'lN(1) has constant components
8 R. H. Schwendeman and V. W. Laurie, available on request
from Dr. Schwendeman, Department of Chemistry, Michigan
State University, East Lansing, Michigan.
9 R. H. Schwendeman and V. W. Laurie, Tables of Line Strengths
(Pergamon Press, New York, 1958). in the molecular frame.
(N'T' \I TlNlII(1) II NT)
= L:(i)s(N'T' II Oli II NT)
i
= L:(i)siPN'N,T'T[SNT'NT(i)Jt. (17)
i
The second term in HI has T'lN(2) a constant in the
molecular frame and U'ls(2) = (5X5h- The same
methods may be applied to its evaluation.
3.'Electron Spin-Nuclear Spin (Fermi) Interaction
The electron spin and nuclear spin interact through a
contact term.IO
H2 = + (167r 13) glJLNJLB I if; (0) 12(S· I)
JLN>O JLB>O
(O)r= + (167r/3)gIJLNJLB I if; (0) \2. (18)
The calculation of uncoupled matrix elements is very
easy. Since kl and k are zero, Eq. (2) contains the
product of two 6j symbols in J scheme.
4. Magnetic Dipole-Magnetic Dipole Interaction
The dipole-dipole interaction between the electronic
and nuclear moments takes the form
Ha=TaN(2) 'Tas(l) ·T31(1)
= -2gIJLNJLB ([ (S· I) I,-3J-[3 (I· r) (S· r) Ir"J )Av:r
(19)
T3s(1) =5, T31(l) =1
TaN(2, q) = L:D2qq' (afj'Y):JaN(2, q')
q'
:J3N(2) =} (ij) I.
The matrix elements of TaN(2) are evaluated in the
same way as those of TI/1N(2). Since k=O, Eq. (2)
reduces, and contains the product of a 9j symbol times
a 6j symbol. If S = t, the 9j symbol may be found as the
product of two 6j symbols divided by another 6j
symbol [see reference 6, Eq. (6.4.8.) J or by explicit
formulas [see reference 6, Eqs. (6.4.15), (6.3.3), (6.3.4)]'
5. Quadrupolar Interaction
The electric quadrupole moment of the nucleus
interacts with the molecular electric fields
H4=-(5l/6)T4N(2)·T41(2). (20)
T4[(2) =nuclear quadrupole moment tensor; T4N(2) =
field gradient tensor (constant in the molecular frame);
(II I T41(2) I II)=6!/2eQ; and Q=nuclear quadrupole
moment. Matrix elements of T4N(2) can be evaluated
as TlNl/(2), or it can be evaluated as in Townes and
10 E. Fermi, Z. Physik. 60, 320 (1930). See also L. D. Landau
and E. M. Lifschitz, Quantum Mechanics Non-Relativistic Theory,
(Addison-Wesley Publishing Company, Inc., Reading Massa
chusetts, 1958), p. 486.
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Term
[IldIl(l)
[SldS](I)
[NhrI]{l)
+[Ih[N](l)
[NJdS] (l)
+[S]z[N] (I)
[Sh[Il(I)
+[IlzlS] (I)
Schawlow,7 p. 161. TABLE III.
Effect
Slight change in the effective moments of
inertia.
Pseudo-quadrupole term. (2nd rank ten
sor) additive constant (scalar).
Pseudo-quadrupole term
[T'IN(Z), U'IS(Z) J
additive constant (scalar).
Nuclear spin-rotation interaction (see
reference 11). Usually too small to be
observed. Same form as electron spin
rotation interaction.
Electron spin-rotation interaction.
Pseudo-dipole-dipole and pseudo-Fermi
interactions. Rotation-electron spin
nuclear spin interaction
-6![ 2N+1 ]!
(NT II T4N(2) II NT)=-2- (2N-1) (2N+3)N(N+1)
+ OZV[N(N+1) -ENr(K)+(K-l) O€NT(K)]}.
Oc2 OK (21)
This formula can be used analogously without the
minus sign for expectation values of T"lN(2) or T3N(2).
Since k2 and k are zero, Eq. (2) contains the product
of two 6j symbols.
6. Rotation-Electron Spin-Nuclear Spin Interaction
The perturbation treatment of Van Vleck3 did not
include nuclear spin terms. If these are includedll it is
possible (Appendix II) to get an interaction term of
the form
H6= (TON(1) X S) 1,1, (22)
where TuN(l) is constant in the molecular frame.
ToN(l) will have three components, one along each of
the principal axes. The matrix elements of components
of T6N(1) will be proportional to the square root of the
asymmetric rotor line strength9 for the transition cor-
11 See G. R. Gunther-Mohr, C. H. Townes, and J. H. Van
Vleck, Phys. Rev. 94, 1191 (1954). responding to the matrix element. Equation (2) will
contain a 9j symbol and a 6j symbol.
7. Stark and Zeeman Effect
The Stark interaction is given by Hx= l" X where l'
is the electric dipole moment, and X is the electric
field. l' is of the form T6."I'(l). Matrix elements of t' are
proportional to the square root of the asymmetric
rotor line strengths.9 The field direction is usually
chosen along the external z axis so that
(N'T'SJ'IF'M 1 jJ.z I NTSJIFM)
is required.
The Zeeman interaction is given by Hz= -2jJ.BS·H.
In both cases Eq. (2) will contain the product of two
6j symbols.
NUMERICAL CALCULATION
For some molecules sufficient accuracy can be ob
tained for most levels by consideration of matrix
elements diagonal in N. Explicit formulas have been
derived by Bakerz and Lin1 for the matrix elements
diagonal in N in a symmetric rotor basis. We have
rewritten these in the asymmetric rotor basis and listed
them in Appendix I. By the use of these formulas, the
hyperfine splitting of a rotational level for given
interaction parameters can be computed by hand in
about thirty minutes.
On the other hand it is often necessary to consider the
second order perturbation off-diagonal in N arising
from the interactions. Although it is possible in prin
ciple to give explicit formulas for the matrix elements,
calculation by their use probably will be inconveniently
laborious. For hand calculation there exist a number of
useful tables of Racah coefficients.12-11) Programs have
been devised for the calculation of these coefficients
on electronic computers and the computation may thus
be done automatically.
COMPARISON WITH VAN VLECK'S METHOD OF
REVERSED ANGULAR MOMENTA
A description of the calculation of matrix elements
by the use of reversed angular momenta has been given
by Van Vleck,3 and will not be repeated. It seems likely
that Van Vleck's method may be applied to any
matrix element of any interaction of this type no matter
how complex. For the simpler matrix elements, explicit
12 L. C. Biedenharn, Tables of the Racah Coefficients (Oak Ridge
National Laboratory, 1952).
13 Kenneth Smith and John Stevenson, A Table of Wigner 9j
Coefficients for Integral and Half-Integral Values of the Para/n
eters (Argonne National Laboratory, ANL-5776, 1957); Kenneth
Smith, Supplement to the Table of Wigner 9j Symbols (Argonne
National Laboratory, 1958), ANL-5860 Parts I and II.
14M. Rotenberg, R. Bivens, N. Metropolis, and]. K. Wooten,
The 3j and 6j Symbols, (The Technology Press, Cambridge,
Massachusetts, 1959).
16 K. M. Howell, Tables of the Wigner 6j Symbols, University
of Southampton Research Rept. U.S. No. 58-1, June 1958.
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ACKNOWLEDGMENTS fonnulas may be derived more rapidly than by the
method outlined in this paper.
On the other hand, in derivations by Van Vleck's
method a great deal of reasoning is involved. Almost
every step in the derivation requires some thought.
This is not the case for the method outlined here.
Most of the derivation is algebraic or numerical mani
pulation. We feel this constitutes the real advantage
of this approach. In addition tabulated values of
Racah coefficients may be used. The preparation of this paper was helped considerably
by discussions with Professor L. C. Biedenharn. We
wish to thank Dr. John Baker for giving us a copy of his
thesis. It must be emphasized that the equations in
Appendix I are based primarily on Dr. Baker's thesis
and are not original with us, although we have intro
duced the asymmetric rotor basis and checked Dr.
Baker's results.
APPENDIX I. EXPLICIT FORMULAS FOR SOME OF THE SIMPLER MATRIX ELEMENTS
1. Spin-rotation interaction
(Nr1F I HII NT1F>=_[(O)s+ Ls ]h2[N(N+1)+S(S+1)-1(1+1)J 2 2(2N+1)
+ (!CJ(CJ+1) -S(S+1)N(N+1»).z:2'" f
(2N-1)(2N+3)(2N+1) Ib £..is for S>!
L.= L(ii)sSNTNT,(i) =[(2N+1)/2N(N+1)J{ (aa)s[N(N+1) +EN~(K) -(K+1) (aENr/aK) J
T'.i
+2 (bb)S[<1ENr(K) /aKJ+ (cc)s[N(N +1) -ENr(K) + (K-1) (<1ENr/<1K) JJ
L.'= L(ii)s' SNTNT,(i)
'1" .i
The (i). terms have no nonzero resultant terms diagonal in either N or r.
2. Fermi interaction
(NT1F I H I Nr1F)= fi2(O)r[F(F+1) -1(1+1) -I(1+1)J[1(1+1)+S(S+1) -N(N+1) J
2 41(1+1)
for S=!
(NT1+1F I H I Nr1F)= + (O)r[(N+1-F+!) (N+1+F+!) (1+F-N+!) (N-1+F+!)JW
2 2(2N+l) .
3. Dipole-dipole for S=!
(NT1F I H I NT1F) h2[1(1+1) +1(1+1) -F(F+l)J[N(N+1)+S(S+1) -1(1+1)JL1
3 41(1+1) (2N+1)
Lr= L(ii)rSNTNT' (i).
T'.i
(Nr1+1F I H I NT1F) -n2[(N+1-F+!) (N+1+F+!) (1+F-N+t) (N-1+F+!)J'Lr
3 4(2N+l)2 .
4. Quadrupole for S = t
(NT1F I H I NT1F) h2(3C F(CF+1) -41(1+1) 1(1+1)JLQ
4 81(1+1) (2N+1)
CF=F(F+1) -1(1+1) -1(1+1)
LQ= L(ii)QSNTNT,(i)
T'.i
(NT1+1F I H41 Nr1F) -3[F(F+1) -1(1+1) -N(N+1)+!Jh2LQ
(2N-1) (2N+3) (2N+1)2
x [(N+I-F+!) (N+1+F+!) (I+F-N+!) (N-1+F+!)Jl.
5. Rotation-electron spin-nuclear spin interaction. No nonzero matrix elements diagonal in N or r.
Downloaded 20 Aug 2013 to 128.118.88.48. 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_permissions1764 R. F. CURL, JR. AND J. L. KINSEY
APPENDIX II. DERIVATION OF EFFECTIVE HAMILTONIAN ARISING FROM SECOND-ORDER PERTURBATION
The consideration of the electronic energy levels of a rigid molecule with only Coulombic interactions and in the
absence of rotation, neglects certain terms in the Hamiltonian which must be considered for hyperfine structure.
These terms3,U,16 are
a,b,c
Hr= -LA.(L.N.+N.L.) + Ln,j'Sj+nr I,
i
where the A. are the rotational constants of the molecule
L=m.L(rj-ro) X (Vj-Vo)
; (A.l)
n.j= (ge~plc) L(ZKe/l rj-rK 13) (rj-rK) X (!Vj-V K) + (ge~plc) L[( -e) II rj-rk 13J(rj-rk) X (!Vi-Vk)
K,i k>i
nr= (egr~Nlc) L I rj-rr 1-3(rj-rr) X {Vj-[1+(Zr MplgrMr) JVr}.
,.
(A.2)
The first three terms arise because the rotational energy of the molecule is given by L.A.R.2, where R=N-L
since N is the total mechanical angular momentum. The last two terms are magnetic dipole interactions.
Hr is nondiagonal in the electronic energy and therefore its effect will be manifested through second order
perturbation.
(ito I Heff I i'lo)= L (EIO-Ez)-l(il o I Hr I i"l)(i"lI Hr I i'lo)
Z"Zo,i"
= L (EIO-Ez)-rcL(lo I A.L.ll)(i I N.I i")+ L(lo Iw.' Il)(i I I., Ii")
Z"lo,i",8', .'
+ L(lo I 'YJ"i'" Il)(iS I Sj." I is')J
;vll
X [L (II A .L.llo) (i" I N. I i')+ L (II 'YJIv' 110> (i" I I., I i' >+ L (II 'YJ8i'" 110) (i" S' I Sj." I i" S)]. (A.3)
vI iv"
[Compare with Eq. (A9), reference 11.J The separation of i and I is made possible by restriction to the molecular
frame coordinate system. Let us schematically write the equation above as
(ilo I Heff I loi') = L([NJz+[IJI+[SJZ) ([NJ(l)+[IJ(l)+[SJ(l»),
l,i"
where (A.4)
there will be six types of terms [NJz[NJm, ([NJI[IJ(l)+[IJI[NJ(l)) , etc. Table III lists the various types of
terms and their effect.
Let us consider first ([NJI[SJ(l)+[SJI[NJ(l)) and then ([SJz[IJm+[IJz[SJ(l)) in more detail. We note that
A .L. does not connect electronic states with different total spin. Therefore S' = Sand
(i" S I Sj." I i'S)=kj(i" I S." Ii');
then
L[NJz[SJm+[SJz[NJ(l)= L M •• , (i I N.I i")(i" I S., I i')+M' •• , (i I S., I i")(i" I N.I i') (A.S)
l,ill i" ,1''111
where
M •• ,= L (EIO-Ez)-l[(to I A.L.IZ)(11 'YJ8i" 11o)kjJ
lr'Zo.i
M'pp'= L (EIO-Ez)-l[kj(lo I 'YJw' 11)(11 A.L.llo)]. (A.6)
lr'Zo.i
If JJ~JJ',N. does not commute with S., since N = J +8r• It is necessary to go to the space-fixed axis system to com
mute Nand 8. Therefore
(i I H" I i')= L M .. ,( (i I A.p.Np.A.,1" Sp.' I i')+(i I Sp.,A.,p.'Np.A.p.l i'»)
"V' ,illS'
= L M •• ,(2(i I Np.A.p.A.,p.'Sp.' I i')+(i I ifiep.'p.l'"A/'A.,!'''S!" Ii'») (A.7)
JlV',JlP/JJII
11 R. S. Henderson, Phys. Rev. 100, 723 (1955).
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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.
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1.1731880.pdf | ESR Studies on the Bonding in Copper Complexes
Daniel Kivelson and Robert Neiman
Citation: J. Chem. Phys. 35, 149 (1961); doi: 10.1063/1.1731880
View online: http://dx.doi.org/10.1063/1.1731880
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v35/i1
Published by the American Institute of Physics.
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Downloaded 30 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://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS VOLUME 35, NUMBER 1 JULY, 1961
ESR Studies on the Bonding in Copper Complexes t *
DANIEL KIVELSON AND ROBERT NEIMANt
Department of Chemistry, University of California, Los Angeles 24, California
(Received December 12, 1960)
ESR spectra of copper complexes have been interpreted by means of molecular orbital theory, and the
"covalent" character of both <T and 1r bonds have been discussed for a variety of compounds. Overlap
integrals have been considered in a consistent manner in treating <T bonds. Particular attention has been
given to Cu phthalocyanine and several of its derivatives. The in-plane 1r bonding may be as important
in determining the properties of a eu complex as is the in-plane <T bonding.
THEORY
ELECTRON spin resonance studies have been used
to investigate the covalent bonding of transition
metals in a variety of compounds.1-4 The work on copper
complexes was most highly developed by Maki and
McGarvey in their analysis of the ESR spectra of the
copper salicylaldehyde imine and acetylacetonate.4
In the present study the theory is modified slightly and
applied to a number of copper complexes.
Van Vleck· has shown that an appropriate linear
combination of central-ion atomic orbitals and ligand
atomic orbitals represents a satisfactory molecular
orbital for the complex. The molecular orbital approach
has proved most successful in explaining the complex
hyperfine structure that is observed in the ESR spectra
of covalently bonded metalsl-4•6; it will, therefore, be'
used here. The notation of Maki and McGarvey4 for a
square planar complex will be followed and it will be
assumed that all complexes considered have approxi
mately square planar symmetry. It will further be
assumed that each of the four ligand atoms has avail
able 2s, 2pz, 2PI/' and 2p. orbitals for the formation of
the molecular orbitals with the copper 3d orbitals.
Since a square planar complex has D4h symmetry, the
following antibonding molecular orbitals for the "hole"
configuration, labeled according to symmetry species,
can be formed from the central atom 3d and the ligand
2s and 2p orbitals. The four ligands are placed on the
±x and ±y axes and are labeled by superscripts start-
* Presented at the American Chemical Society Symposium on
Molecular Structure, University of Washington, Seattle, Wash
ington, June, 1960. t Supported in part by the National Science Foundation and
the Research Corporation. t Present address: Central Research and Engineering, Texas
Instruments Inc., Dallas, Texas.
1 K. W. H. Stevens, Proc. Roy. Soc. (London) A219. 542
(1953) .
2 J. Owen, Proc. Roy. Soc. (London) A227. 183 (1955); Dis
cussions Faraday Soc. 19. 127 (1955).
3 M. Tinkham, Proc. Roy. Soc. (London) A236, 535, 549
(1956).
4 A. H. Maid and B. R. McGarvery, J. Chern. Phys. 29. 31,
35 (1958).
6 J. H. van Vleck, J. Chem. Phys, 3. 807 (1935).
6 For a complete discussion see doctoral thesis by R. Neiman,
University of California, Los AngeJes (1960). ing with one on the +x axes and proceeding counter
clockwise,
B1g= adx2-yz-cl (-ox(1)+01l(2)+o}3)-0/4»/2
B2g = f3ldZl/-f31' (P1Y) + px (2) -PlI (3) -px (4) ) /2
AIg=aIdaz2_r2-al' (0",(1)+01/(2)-0",(3)-01/(4»/2
jf3dx.-f3' (P.(1)_ P.(8) )/21
Eg=
f3dllz-f3' (p.(2)_ p,<4) )/21,
where C1a)
(1b)
(1c)
(ld)
(le)
Here 0::; n::; 1 ; the plus sign applies to the ligand atoms
on the positive x and y axes, the minus sign to those on
the negative x and y axes.
The overlap between the copper and ligand orbitals
has been considered in Eq. (1). Normalization of the
Big orbital yields
a2+a'2-2aa' S = 1,
where S is the overlap integral
S= (d:;;2-y21 -0,,(1)+0
11(2)+0".,(3)-0'11(4) )/2
2 (dz2_1/21-0':t(I}). (2)
(3)
Similar relations hold between the other coefficients
but the other overlap integrals are small and so can be
neglected.
The orbitals in Eqs. (la)-(le) are given in order of
increasing energy although the position of the Alo
level is rather uncertain; in any case, the Ala state does
not effect the magnetic parameters in second order and
so is not relevant to the present discussion. Here BIg
represents in-plane 0' bonding, B2g represents in-plane
7r bonding, and Eg represents out-of-plane 7r bonding.
Bonding orbitals could be constructed by replacing the
unprimed by the primed and the primed by minus the
unprimed coefficients in the antibonding orbitals de
scribed above.
Figure 1 gives an approximate representation of the
energy levels for 0 bonding. The sequence of levels has
not been unambiguously assigned but the unpaired
electron is almost certainly in the Big level. Note that
there are three Aig levels that interact with each other
149
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although one will almost certainly be strongly bonding,
one antibonding, and the third will be almost nonbond
ing. A similar scheme for 71' bonding is shown in Fig. 2.
For the case of a 3d9 copper ion, the appropriate
crystal-field Hamiltonian is7
W'=X(r)L,S+i3 oH· (L+2.0023S)
+2'Yi3oi3N{ [(L-S)· l/r3J+[3 (r. S) (r. 1)/r5J
-(S7I'/3)o(r)S·11, (4)
where X (r) is the spin orbit operator, L is the orbital
angular momentum operator, S is the spin angular
momentum for the electron, 130 the Bohr magneton,
H the applied magnitic field, 'Y the gyromagnetic ratio
of the nucleus in question, i3N is the nuclear magneton,
1 the spin angular momentum operator for the nucleus,
r the distance from the central nucleus to the electron,
and oCr) is the Dirac delta function operator. Quad
rupole effects and effects arising from the interaction
of the nuclear moment with the external field have been
neglected in this expression since they may be expected
to be of little importance here.
The wave functions of Eqs. (la)-(le) can be used
to solve the Hamiltonian given above and to convert it,
through second order, to a spin Hamiltonian.7 The
resulting spin Hamiltonian assuming BIg is the ground
state, is
H =i3o[g"H zSz+g-L (HxSx+HlISlI) J+ A I.Sz
+B(lxSx+llISII)' (Sa)
where
g,,-2.0023= -Sp[ai3l-a'i3 IS-a' (l-i312)IT(n )/2J
(Sb)
g-L -2.0023= -2J.![ai3-a'i3S-a' (1-i32)tT(n)/2tJ (Sc)
A = P[ -a2(t+Ko)+ (g,,-2)+-.¥-(g-L -2)
-Sp{a'i3 IS+a' (1-i312)!T(n)/2}
-i-J.!{ a'i3S+a' (1-i32)tT (n )j2i} J (Sd)
B=P[a2(f-Ko)+H(g-L -2)
-HJ.!{a'i3S+a' (1-i32)!T(n )/2i} J, (Se)
and the following definitions are used
p=Xoai3r/AExy; J.!=Xoai3/AExz (6a,b)
T(n) =n-(1-n2)!RS (ZpZ.) 5/2 (Z.-Zp)/ (Z.+Zp)5 ao
(6c)
P= 2'Yi30i3N (dx2_y21 r-31 dX2_y2). (6d)
The Xo is the spin-orbit coupling constant for the free
eu(II) ion, i.e., (3dIX(r) 13d); Ko is the Fermi-contact
term for the free ion; AExy and AExz are, respectively,
Exy-Ex2-y2 and Exz-Ex2-y2; R is the metal-ligand dis-
7 A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. (London)
A205, 135 (1951). stance; hydrogenlike radial functions have been used;
Z. and Zp represent the effective nuclear charges on
the sand p orbitals, respectively; and ao is the Bohr
radius.
These equations differ from those of Maki and
McGarvey in that the contribution of the overlap in
tegral S is included; this term is of the same order of
magnitude as the T (n) terms. The equations also differ
from those of Roberts and Koski8 in that these authors
treated X (r) as a constant and not as a rapidly decreas
sing function of r. The Fermi contact interaction would,
of course, vanish for 3d electrons since these have no
density at the eu nucleus. We have assumed that the
required s character is present in the eu atomic or
bitals9•10 and that the ratio of s to d character is un
changed by the presence of the ligands. Furthermore,
it can be shown that the density of ligand orbitals
at the eu nucleus is very small. These arguments allow
us to write the Fermi-contact interaction as a2KO'
The parameter K introduced previously3.4.8,10 can be
equated to the present quantity a2Ko.
Finally it should be mentioned that in the evaluation
of the spin Hamiltonian all integrations over the ligand
orbitals that involve an inverse dependence upon r
were neglected according to the prescription given by
Maki and McGarvey.
If the ligand possesses a nuclear spin, the above spin
Hamiltonian contains an additional term which gives
rise to an extra hyperfine interaction characteristic
of the ligand. It is this hyperfine interaction which
requires the use of molecular orbitals rather than the
crystal-field approach to the problem. The extra hyper
fine interaction arises from dipolar interactions between
the nuclear and the electronic moments. The aniso
tropic contribution is sma1l4 and the isotropic contribu
tion arises from the s character Fermi interaction of
FIG. 1. Energy levels of Cu++ complexes with IT bonding.
8 E. M. Roberts and W. S. Koski, J. Am. Chern. Soc. 82, 3006
(1960) .
9 A. Abragam and M. H. L. Pryce, Proc. Roy. Soc. London
A230, 206 (1951).
10 A. Abragam, M. H. L. Pryce, and J. Horowitz, Proc. Roy. Soc.
(London) A230, 169 (1955).
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the ligand orbitals. The isotropic interaction energy is
W L= (%/9 h L,60,6Na'21 PN (0 )21 SzI/L), (7)
where L refers to the ligand nucleus, PN (0) is the value
of the ligand 2s function at the ligand nucleus. It is
assumed that the copper orbitals do not make an appre
ciable contribution to the spin density at the ligand
nucleus. Note that if a'2= 0, i.e., if the unpaired elec
tron is not even partly on the ligand, the extra hyper
fine structure vanishes. Only in the molecular orbital
approach is a' nonvanishing.
APPLICATIONS
In order to use the expressions obtained in the last
section the overlap integral S must be evaluated.
Hydrogenlike functions have been assumed and the
following effective charges have been chosen from
Hartreell:
for nitrogen: Z2s=4.50, Z2p=3.54
for oxygen: Z2.=5.25, Z2p=4.06
for copper: Z3d= 11.86.
Only nitrogen and oxygen ligands will be considered.
The ligand-to-metal distance R has been taken as 3.62
ao; this value differs slightly from compound to com
pound but is probably about 1.9 A within experi
mental error. The overlap integrals are then, Snitrogen=
0.093, Soxygen=0.076. In obtaining these values n has
been chosen as (i)i which implies Sp2 hybridization
in the ligands. The corresponding values of T (n) are
T (n ) nitrogen = 0 .333, T (n ) oxygen = 0.220.
The value of AO is chosen to be -828 cm-1 while P
is 0.036 cm-1.7,9 The estimated error in these quantities
is about 3%. The Ko is about 0.43 with an estimated
uncertainty of about 5%.9 The orbital excitation
energies t!..Exy and tJ.Exz can often be measured from the
visible spectrum; however, this is not the case in the
phthalocyanine and porphyrin complexes in which the
intense 7r transitions mask the eu transitions. Generally,
t!..Exy is about 15 000 cm-1 while tJ.Exz is about 25 000
cm-1•
If t!..Exy and tJ.Exz as well as gil, g.1., A, and Bare
known it is, in principle, possible to calculate a, ,61,
and ,61'. However, it is usually not possible to obtain
a well-defined set of solutions for these equations and
for this reason an iterative procedure is preferred.
Since a is often of the order of 0.9 and ,61 and ,6 are
close to unity, Eq. (5c) can be written in the following
approximate form
a2= -(AI P)+ (gll-2)+ (3/7)(g.1. -2)+0.04. (8)
Note that (4/7)+Ko= 1.0. Equation (8) can be used
to obtain a first approximation of a2• If tJ.Exy and t!..Exz
are known one can then obtain ,61 and ,6 from gil and
11 D. R. Hartree, The Calculation of Atomic Spectra (John
Wiley & Sons, New York, 1957). A2I1 \ I By
I ' \\ I B.
II / ,\~
~p. (1/ '-.'\ ,~
00000000 '\;
~ go IY
'-/.. "\ '.u ....
\ 00 ......... //
3d,.. oooo! \, B2C //
3ct,. 00 // "-.... .... B, 0000/ ] ADUboD41111
00000000 lIon-
00000000 Bond-
"'-Orbital. 1D5
J_4101
C~l.z.
FIG. 2. Energy levels of Cu++ complexes with 11" bonding.
g.1.; these can then be used to obtain better values for
a2• We have found that the determinations of Bare
generally not accurate enough to add information to
the calculations. Furthermore, in the cases where
tJ.Exy and tJ.Exz are not known they can be estimated
because a 20% error in tJ.Exy and tJ.Exz results in only
about a 5% error in ,61 and ,6, respectively.
The results of the calculations for a2 and ,612 for several
copper complexes are summarized in Table I; the ex
perimental data is taken from several sources. The
values of ,62 are probably so close to unity that the
uncertainty in the calculations does not allow us to
distinguish them from unity. It is quite clear that, ex
cept for laccase and ceruloplasmin, a2 is in the range
0,75 to 0.90 for most copper complexes [a, (3, II, 0 tetra
phenyl porphin eu (II) should probably lie within this
range; the discrepancy is thought to arise from experi
mental uncertainties]' This relatively small range of
a2 explains the relative constancy of the values of
K(or aho) as determined by others.4,8,12 These values of
a2 indicate that the (]" bonding is quite covalent in nature
since it shows that the bond is delocalized over both the
eu and the ligand orbitals. If a2= 1, the bond would be
completely ionic. If the overlap integral were vanish
ingly small and a2= 0.5, the bond would be completely
covalent. However, because the overlap integral is
sizeable we cannot speak strictly of covalent versus
ionic bonds, but we can say that the smaller the value
of a2 the greater the covalent nature of the bond. The
trend is in the expected direction; salicylaldehyde
imines and phthalocyanines are more covalent than
oxalates, The two enzymes, lac case and ceruloplasmin,
have a very particular behavior; they have been dis
cussed by Vanngard.13
By measuring the extra hyperfine splitting a2 can
also be determined. Maki and McGarvey have esti
mated Ip(O) 12 to be 33.4XI024 cm-3. Then by means
of Eq. (7), a'2 can be determined directly and it can in
turn be used to obtain a2 by means of Eq. (2). Extra
hyperfine structure due to nitrogen has been reported
in three eu (II) complexes. Table II gives the com-
12 J. F. Gibson, D. J. E. Ingram, and D. Schonland, Discussions
Faraday Soc. 26, 72 (1958).
13 B. G. Malmstom and T. Vanngard, J. Mol. BioI. 2, 118
(1960) ,
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TABLE I.
gJl gl.
Copper complex -2.0023 -2.0023 t1Exu (em-I)
Acetyl-acetonate 0.2638 0.0512 15 000
Salicylaldehydeimine 0.1981 0.0427 16 300
Phthaloeyanine 0.1723 0.0427 assumed
17 000
Phthalocyanine 0.1627 0.0427 assumed
17 000
Laccase 0.1947 0.0457 16 400
Ceruloplasmin 0.2067 0.0528 16500
Denatured Lacease 0.2277 0.0527 assumed
16 400
Denatured 0.2547 0.0528 assumed
Ceruloplasmin 16 500
Histidine 0.2277 0.0607 15 600
Imidazole 0.2647 0.0607 16 800
2-2' Dipyridyl 0.2677 0.0797 14 900
1-10 Phenanthroline 0.2777 0.0857 15200
Oxalate 0.3137 0.0757 15 400
EDTA 0.3347 0.0877 13 900
Citrate 0.3467 0.0717 13 700
Etioporphyrin II 0.1670 0.0593 assumed
17 000
t-phenylporphin 0.1677 0.0477 assumed
20000
pounds, the observed splittings, the values of 0/', 0/'2,
and 0/2 calculated from Eq. (Sd) and from Eqs. (7)
and (2). The agreement between the two different
methods of calculating 0/2 is quite good and is probably
within the uncertainty limits; this lends credence to
the assumption of Sp2 bonding for the ligands. The 0/2
calculated from Eq. (Sd) is smaller than that obtained
from the extra hyperfine structure from Eq. (7); this is
in accord with previous results.3 A different hybridiza
tion of the (j orbitals on the nitrogen atoms might
reduce the discrepancy. If n2 in Eq. (le) were about 0.6
instead of 0.667, the 0/2 calculated from Eq. (7) would
be reduced by about 0.05 while the 0/2 calculated from
Eq. (5d) would remain almost unchanged. The iso
tropy of the extra hyerfine splitting in the phthalo
cyanine is discussed in the next article.
Since the overlap integrals in the B2y function are
negligible, fN is a direct measure of the covalency of
the in-plane 7r bonding. Therefore, the values of fN
listed in Table I also furnish a pronounced trend in
in-plane covalent 7r bonding. The differences in {N are
much more pronounced than are those for 0/2• (312 is,
therefore, probably a better indication of "covalent
character" than is 0/2• The marked in-plane 7r bonding
covalent nature of the phthalocyanines may account
for their great stability. A (em-I) a2 {j12 Comments on {j12 Reference
-0.0160 0.75 0.87 4
-0.0185 0.76 0.72 4
-0.0202 0.78 0.65 assuming t1Exy = 17 000 cm-I this
work
-0.0220 0.82 0.59 assuming t1Exu = 17 000 cm-I 12
-0.009 0.49 1.0 tl2 assumed to be one to get 13
agreement with A, gil and
-0.008 0.47 1.0 t1Exu 13
-0.020 0.83 0.76 assuming t1Exu= 16 400 em-I 13
-0.018 0.80 0.88 assuming t1Exu= 16500 cm-I 13
-0.018 0.78 0.78 13
-0.018 0.82 0.89 13
-0.017 0.80 0.84 13
-0.015 0.76 0.95 13
-0.017 0.84 0.92 13
-0.016 0.84 0.89 13
-0.015 0.82 0.93 13
-0.0188 0.74 0.68 assuming t1Exy = 17 000 cm-I 8
-0.025 0.90 0.62 assuming t1Exy=20 000 em-I 14
PHTHALOCYANINES AND PORPHINS
Copper phthalocyanine is extremely stable; it can be
sublimed at 550°C and it can be dissolved only in
concentrat~d H2S04• Its chemistry has been described
by Linstead14 and its structure elucidated by Robert
son.15 Previous ESR studies were carried out by Ingram
et al.12,16 who worked on single crystals diluted with
diamagnetic Zn phthalocyanine. The present work
was carried out on a dilute frozen solution of Cu
phthalocyanine in concentrated H2S04, i.e., in a glass.
The details are given in the following article but both
hyperfine structure and extra hyperfine structure were
observed; the results are tabulated in Tables I and II.
The difference between the values of 0/2 and {N for
copper phthalocyanine which were calculated from the
data of Ingraml2 (but using the present theory) and
those values calculated from the experimental data
in this work (see the following article) is probably
just a reflection of experimental error or a slight differ-
14 G. J. Byrne, R. P. Linstead, and A. R. Lowe, ]. Chern. Soc.
1934,1017; C. E. Dent and R. P. Linstead, ibid. 1934, 1027; P. A.
Barrett C. E. Dent, and R. P. Linstead, ibid. 1936, 1719; J. S.
Anders~n, E. F. Bradbrook, A. H. Cook, and R. P. Linstead,
ibid. 1938, 1151.
15 ]. Robertson and R. P. Linstead, ]. Chem. Soc. 1936, 1736.
16 D. J. E. Ingram, J. E. Bennett, P. George, and J. M. Gold
stein, ]. Am. Chern. Soc. 78, 3545 (1956).
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TABLE II.
from Eq. (7) from A and Eq. (5c)
Splitting
Compound (gauss) a' a12
bis-salicylaldehyde-imine 11.1 0.49 0.24
Phthalocyanine 14.6 0.55 0.30
Etioporphyrin II 14.3 0.55 0.30
ence arising from the fact that the magnetic environ
ment was somewhat different in each case. Ingram,12
on the other hand, finds no hyperfine (hfs) structure
due to nitrogen in his single crystal work; this may be
due to two causes: (1) The crystals lack sufficient mag
netic dilution so that exchange and dipolar effects
wash out the nitrogen hfs and (2) large amplitude
modulation of the magnetic field might also wipe out
the nitrogen hfs. Reason (1) is more likely since the
concentration figures givenl2 would imply that the
copper phthalocyanine concentration in the single
crystal work is about 2 M. A 0.354 M solution of
copper phthalocyanine in 96% H2S04 shows no hyper
fine structure from the nitrogens; this supports reason
(1)-Ingraml2 also made an estimate of a2 based only on
first-order theory with neglect of any ligand contribu
tions. In this way flExy was estimated as 31 700 cm-l
and flExz as 29 000 cm-l which would of course place
the B20 level below the Ea levels. In the present work,
ligand contributions were included; flExy was estimated
to be about 17 000 cm-l and flExz to be about 26 000
cm-l if ,622:0.95 for this compound.
Some comment must also be made about the studies
on (a, fl, 'Y, 0-)-tetraphenyl porphin (TTP) and its
p-chloro derivative (TPPCI).14 The copper hyperfine
splitting constant A was the same for both compounds
(0.025 cm-l) which gives a value of a2 equal to 0.90;
this value of a2 is quite high but is probably within the
limits of error quoted for a2• The most striking result,
however, was that the spectrum of TPPCI contained
a great number of hyperfine lines (more than those
expected just from copper) so that this extra hyperfine
splitting was attributed to the chlorine atoms on the
periphery of the molecule. These findings are somewhat
surprising for several reasons: (1) If the Blo orbital is
as covalent as in other compounds, one might expect
that the interactions with the nitrogens (which are
only about 2 A distant) would be much stronger than
the interaction with the chlorines (which are about 9 A
distant) and (2) hyperfine splitting due to chlorine
has been observed in very few compounds; the most
outstanding example is IrC16-2 in which the chlorine
metal distance is much shorter.2,17
With these results in mind, spectra were run of tetra-
4-p-chloro-copper ph thalocyanine (CuPc), tetra-4-
nitro-CuPc, and tetra-4-sulfo-CuPc in 96% H2S04•
17 J. Owen, Discussions Faraday Soc. 26, 53 (1958). a! a' a'2 a2 Reference
0.84 0.56 0.31 0.76 4
0.79 0.55 0.30 0.78 this work
0.79 0.59 0.35 0.74 8
All three compounds gave identical spectra and, since
these spectra were also identical to that of the plain
CuPc, the assumption can be made that indeed the 3d
electron of the copper spends a negligible amount of
time on the phenyl rings of the phthalocyanine. If this
same assumption can be made for the porphins, another
explanation must be sought for the observed spectra of
these compounds. We feel that the splitting in the case
of TPPCl is due to the four nitrogens bound to the cop
per; failure to observe any further hfs in the unchlo
rinated TPP is probably again due to lack of magnetic
dilution.
FURTHER COMMENTS ON THE THEORY
Thus far the discussi<;1ll of bonding has been confined
to square planar bonding. It is possible, however, for
Cu (II) to assume octahedral bonding in some com
pounds18; this of course would invoke the dz2 orbital of
the copper ion. Two cases will be discussed: (1) small
perturbations along the z axis and (2) complete de
generacy along the x, y, and z axes: a cubic system.
For small perturbations along ihe z axis gil, A and B
remain unchanged but g-L increases slightly.6
For the cubic case, the Ala and Bla levels would be
degenerate, as would the B20 and Eo levels. Then,
gil wduld still be the same as in the square planar
case but g-L would be raised since now the (Alo I L", I Eo)
matrix elements contribute; in fact, gll=g-L and A=B.
No completely cubic compounds are known for
Cu (II),19 presumably because of the Jahn-Teller effect,19
although the fluorosilicates (M"SiF 6-6H20, where
M" is a divalent cation) are trigonal and have gll""g-L
(at certain temperature, this equality disappears and
the symmetry is more tetragonaP9). The Tutton salts
[M2'M" (S04h-6H20] (where M' is a monovalent cation
and M" a divalent cation) are more nearly tetragonal
in their symmetryl9; in effect, this means that one can
consider them to a first approximation, as a square
planar complex with perturbations along the z axis.
Therefore, one might expect that gil> g-L; this is borne
out by the available ESR data.19
The spectra of tetra-4-sulfo copper phthalocyanine
(TSCP) in H20 consists of a single, very broad line
18 E. Cartnell and G. W. A. F ow les, Valency and M olecldar
Structure (Butterworths Scientific Publications, Ltd., London,
1956) .
19 K. D. Bowers and J. Owen, Repts. Progr. in Phys. 18,304
(1955). .
Downloaded 30 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://jcp.aip.org/about/rights_and_permissions154 D. KIVELSON AND R. NEIMAN
("'" 75 gauss) with no hfs at both room temperature
and 77°K. (The spectrum in 96% H2S04 is similar to
the plain CuPc in H2S04.) A plausible explanation for
this spectrum is that the H20 approaches the copper
along the z axis and the crystal field becomes tetrag
onal (distorted cubic). The Aig level is now a fairly low
lying excited state and the hydrated TSCP can relax
quickly through a spin-orbit mechanism; this gives rise
to line broadening and the hyperfine structure is ob
scured.2o,21 No copper hyperfine structure has been
reported in aqueous copper solutions,21 presumably for
the same reason. TSCP in H2S04 gives rise to sharp lines
with hyperfine structure; presumably the H2S04 does
not interact strongly with the dz2 orbital of the copper.
The interpretation of the results on TSCP in H20 are
supported by the work on copper Tutton salts.19
Crystals of the Cu Tutton salts give rather broad ESR
lines at room temperature but these lines sharpen and
the hfs becomes evident at low temperatures.19 This
again is probably due to the fact that the crystal field
is that of a distorted cubic field, the dz2 level lies rather
close to the d,,2_y2 level, and the spins relax rather
rapidly through modulated spin-orbit interactions.
This mode of relaxation is reduced at low temperatures.
If the square symmetry does not exist in the plane,
instead of gl., one must write gx and gy where gx~gy(gx,
gy, and gz are the three components of a diagonalized
tensor). Furthermore, instead of B, one must write
Bx and By where Bx~By (Bx, By, and A are the three
components of a tensor; this tensor cannot necessarily
be diagonalized simultaneously with the diagonaliza
tion of the g tensor bulo the off-diagonal elements are
small and have not been detected as yet22). Differences
between gx and gy and between Bx and By have been
determinedl9 but for the cases listed in Table I, no such
differences were noted within the experimental error.
It is noteworthy that Maki and McGarvey4 in their
work on the salicylaldehyde-imine did not detect such
differences since they worked with single crystals and
the ligand structure
N 0 '" / Cu
/ '" o N of this orbital would predict that gil would be 2, which
is not the case. The B2g orbital could be the ground
state but again this seems unlikely since the nitrogen
hfs is so pronounced and B2g contributes no s character
at the nitrogen nucleus. Therefore, the BIg orbital is
taken as the ground state since it would be expected to
give the largest electron density at the ligands. Further,
placing of the unpaired electron in this orbital is in
agreement with the modern ligand-field theory.23
The 4s and 4p can also be ruled out on the basis that the
use of these orbitals would predict that gil = 2. In
particular, use of the 4pz orbital, as in the valence-bond
theory, for the ground state would be in error; it is for
this reason that the valence-bond theory is not satis
factory for the interpretation of the ESR results.
Furthermore, the 4s and 4p states do not in second
order contribute to the magnetic parameters.
INTERRELATION OF MAGNETIC PARAMETERS
As the coefficients a and {3 decrease the bonds become
more "covalent," gil and gl. decrease and A and B
increase. It must also follow that go= (gll+2gl.)/3
decreases and a = (A + 2B) /3 increases with increasing
"covalency." Because gll-2> gl. -2 and {31 varies more
than {3, we expect Llg=gll-gl. to follow the trend of
gil. Similarly b= A -B should follow the same trend as
A. The most sensitive of these functions appears to be
gil; consequently, it will be chosen as the best indication
of "covalent" character. From Table I it can be seen
that compounds considered to be covalent, such as the
phthalocyanine and etioporphyrin complexes, have
lower values of gil than do the "ionic" oxalate and
citrate complexes. A number of the more precisely
determined and significant magnetic parameters are
plotted against gil in Fig. 3. This figure may be useful
in correlating the magnetic properties with the bonding
character.
All the magnetic parameters can be obtained from
studies on single crystals and often by means of investi
gations on polycrystalline samples; a and go can be ob
tained directly from studies of solutions in liquids of low
viscosity.22 The Llg and b can be obtained, in principle,
from a careful analysis of the line shapes obtained in
low viscosity liquids.22
1.0..------------..176
is not obviously square symmetric. 0.9 ~
k-, .160
+ Throughout this discussion, it has been assumed that 0.8 "
the BIg level is lowest in energy. It is unlikely that the ~ Z>',
Eg levels could be the ground state since these orbitals ;:c 0.7
have no appreciable (]' character and therefore would 'g 0.6
probably not give enough electron density at the nitro-.. (),5 + + +
+ .144
.128 § FIG. 3. Plots of a
oJ and go vs gil; to
.112 :"0 represents a, +
.096 represents go.
gen nucleus to account for the nitrogen hyperfine 0.4+---~----,.---~----'-,.oeo
structure. The Ala orbital has some (]' character but use .1600 .2000 .2400 .2800 .3200
20 H. M. McConnell, J. Chern. Phys. 25, 709 (1956).
21 B. R. McGarvey, J. Phys. Chern. 61, 1232 (1957).
22 D. Kivelson, J. Chern. Phys. 33,1094 (1960). g,,-2.0023
23 J. S. Griffith and L. E. Orgel, Quart. Revs. (London) 11, 386
(1957) .
Downloaded 30 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://jcp.aip.org/about/rights_and_permissionsBONDING IN COPPER COMPLEXES 155
The compounds bis-N -methylsalicylaldehydeimine
Cu (II), called MSAI, and bis-N-t-butylsalicylalde
hydeimine Cu (II), called t-BSAI, were studied in
benzene solution. Here go= 2.116 and 2.127 for MSAI
and t-BSAI, respectively, while a=0.075 cm-l and
0.042 cm-l for MSAI and t-BSAI, respectively. It can
be seen from Fig. 3 that the values of these two param
eters for MSAI correlate well with a value of about 2.21
for gil' The correlation for t-BSAI is less satisfactory
and, furthermore, the a curve is not yet determined in
the region near 0.04 em-I. The construction of curves
similar to those in Fig. 3 should be useful in predicting
unmeasured magnetic parameters and relating them
to the "covalent" character of the bonds.
ACKNOWLEDGMENTS
We wish to thank Fred Bauer and Dr. William
Drinkard for supplying the samples of MSAI and t
BSAI. We would also like to express our appreciation
to the National Science Foundation and the Research
Corporation for their generous financial assistance
throughout this work. Thanks are also due to Donald
Myers and Dennis Silverman for carrying out some of
the computations. We are grateful to Professor B.
McGarvey for his useful comments and for pointing out
a critical error in our evaluation of T(n), an error
which affected all our numerical results.
Note added in proof: Interesting results may be ob
tained if one investigates the relationship between (32,
gJ., and I::,.Exz. Maki and McGarvey4 estimatel::,.E xz= 1.5
I::,.Exy to 1.6 I::,.EXY for bis-acetylacetonate Cu (II) and
bis-salicylaldehyde-imine Cu (II). A choice of I::,.Exz =
1.6 I::,.EXY yields (32=0.98 for bis-salicylaldehyde-imine
Cu(II) and (32=0.93 for copper phthalocyanine; for TABLE III. Ratio of Crystal Field Splittings: g, fj,Exy, IX, and {31
are taken from Table I; {32 = 1.
Copper Complex
Acetylacetonate
Salicylaldehyde-imine
Phthalocyaninea
Phthalocyanineb
Laccase
Ceruloplasmin
Denatured Laccase
Denatured Ceruloplasmin
Histidine
Imidazole
2-2' Dipyridyl
1-10 Phenanthroline
Oxalate
EDTA
Citrate
Etioporphyrin II
t-Phenylporphin
a This work.
b Reference 4. 1.54
1.67
1.68
1. 78
0.97
0.80
1.51
1.44
1.29
1.27
1.06
0.89
1.15
1.10
1.07
1.14
1.51
all the other compounds in Table I I::,.Eyz = 1.61::,.Exy
yields (32) 1, an unrealistic result. In Table III, (32 has
been taken equal to unity, gJ. has been combined with
the values of a, (31, and I::,.Exy given in Table I, and the
ratio I::,. Exz/ I::,.Eyz has been calculated for several com
pounds. It is interesting to note that for those com
pounds for which one generally assumes "strong
covalent bonds," I::,.Ex./ I::,.Eyz is large; for those com
pounds for which one does not assume "strong covalent
bonds," I::,. Exz/ I::,.Eyz= 1, a condition characteristic of
molecules with cubic rather than square planar sym
metry. Note that a 20% error in I::,.Eyz results in a 5%
error in (32. It would be interesting to check these pre
dicted values of I::,.Exz by means of optical spectroscopy.
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1.1703937.pdf | On the Development of Nonequilibrium Thermodynamics
S. R. De Groot
Citation: Journal of Mathematical Physics 4, 147 (1963); doi: 10.1063/1.1703937
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IP: 132.174.255.116 On: Fri, 28 Nov 2014 13:58:29JOURNAL OF
MATHEMATICAL
PHYSICS
VOLUME 4, NUMBER 2 FEBRUARY 1963
On the Development of Nonequilibrium Thermodynamics*
S. R. DE GROOT
Lorentz Institute of Theoretical Physics, University of Leiden, Leiden, Netherlands
(Received 5 July 1962)
A general view of the history of nonequilibrium thermodynamics shows how two main lines of
development have recently fused into a single branch of science. The field theoretical formulation
of thermodynamics (leading to a balance equation for the entropy) constitutes the framework of a
theory in which the Onsager reciprocal relations form the piece de resistance, the fundamental impor
tance of which is outlined in this paper.
I. INTRODUCTION
THIRTY one years ago, two articles appeared
in the Physical Review, in which Lars Onsager,
then of Brown University, derived the celebrated
reciprocal relations between irreversible processes,
which bear his name. I This work formed the culmi
nation of a theoretical development started seventy
seven years before, when thermodynamic considera
tions were first applied to the treatment of irrever
sible phenomena. That was done by William
Thomson who gave an analysis. published in the
Proceedings of the Royal Society of Edinburgh,2 of
the various thermoelectric effects. He established
two relations between them, of which the first
followed simply from the conservation of energy.
The second Thomson relation, which connects the
thermoelectric potential of a thermocouple to its
Peltier heat, was obtained from the two laws of
thermodynamics and an additional assumption
concerning the so-called reversible contributions to
* Paper read at the Conference on Irreversible Thermo
dynamics and the Statistical Mechanics of Phase Transitions,
Brown University, Providence, Rhode Island, 11-16 June
1962.
1 L. Onsager, Phys. Rev. 37, 405 (1931); Ibid. 38, 2265
(1931).
2 W. Thomson (Lord Kelvin), Proc. Roy. Soc. Edinburgh
3, 225 (1854); Trans. Roy. Soc. Edinburgh part I 21, 123;
(1857); Math. Phys. Papers 1, 232 (1882). the process. Later, Boltzmann3 attempted to justify
the Thomson hypothesis, but he was unable to find
a basis for it. We now know that this hypothesis
cannot be justified. Thomson's second relation was
finally proved correctly by Onsager who showed that
it was an example of his reciprocal relations, which
are themselves a consequence of microscopic
reversibility, i.e., ultimate invariance of the micro
scopic equations of motion under time reversal.
II. THE ONSAGER RELATIONS
Since 1931, the Onsager relations have played an
essential role in all thermodynamic treatments of
coupled irreversible phenomena. Let us write, to
fix the ideas, the following linear laws for two
irreversible phenomena and their interference effects:
J! = LuX! + LI2XZ,
J2 = L21XI + L2ZXZ, (1)
(2)
where J I and J 2 are called fluxes, such as, for ex
ample, the heat flow and the electric current (in
Thomson's thermoelectric case), and Xl and X2
are called thermodynamic forces, such as the tem
perature gradient and the electric field. The diagonal
3 L. Boltzmann, Sitzber. Math. Naturwiss. Akad. Wiss.
Wien II 96, 1258 (1887); Abh. 3, 321 (1909).
147
Copyright © 1963 by the American Institute of Physics
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coefficients Lll and L22 are then related to the heat
and electric conductivities, and the off-diagonal
coefficients La and L2l to the Peltier heat and the
thennoelectric power. Onsager proved quite generally
for an arbitrary set of two or more coupled ir
reversible phenomena, that (with an appropriate
choice of fluxes and thermodynamic forces) the
scheme of coefficients L is symmetric, i.e., for the
example considered,
(3)
which is Thomson's second relation. These reciprocal
relations reflect, on the macroscopic level, the time
reversal invariance of the microscopic equations of
motion. Onsager's proof was based on two starting
points, (in addition to a few well-known concepts
of statistical mechanics), viz., (a), microscopic re
versibility, and (b), linear laws of the type (1) and
(2). I would like to discuss both of these points in
a little more detail.
A. Microscopic Reversibility
The microscopic state of a system of N particles
without internal structure is described in the classical
theory by a point rl, r2, ••• , rN, Pl, P2, .. , , PN
in phase space, where the r i are the position vectors
of the particles and Pi their momenta. (For particles
with an internal structure more parameters are
needed, but the following reasoning is not essentially
changed.) The dynamics of an adiabatically in
sulated system of this kind is given by a Hamiltonian
H(r, p) which depends on the coordinates and mo
menta (written here symbolically as r, p), but not
explicitly on the time. The Hamiltonian of such a
system of molecules is invariant under time re
versal, i.e., in the classical case, for the transforma
tion p ~ -p,
H(r, p) = H(r, -p). (4)
This property expresses the time-reversal invariance
of the mechanical equations of motion of the N
particle system.
We assume that the behavior of the system can be
studied by means of classical statistical mechanics.
In particular, the time-dependent average value aCt)
of any dynamical quantity a(rp) is obtained by
mUltiplying a(rp) with a probability density p(rpt)
of a representative ensemble of systems, and inte
grating over phase space. The time behavior of
p(rpt) follows from the knowledge of H by means of
the Liouville equation.
For the macroscopic description of the system,
one is not interested in the complete set of me-chanical variables describing its microscopic state,
but only in a much more restricted number of vari
ables. One may choose for these variables, the ex
tensive properties (such as the energies, masses,
electric charges) of macroscopically infinitesimal sub
systems. These subsystems should still contain
enough particles so that the concepts of statistical
mechanics may be applied to them. Let us denote
this restricted set of variables by al, a2, ... , an
(or a, to abbreviate the notation), where n is much
smaller than N. We nonnalize these variables in
such a way that their mean values in equilibrium
[which is described by p = Pm(rp), the micro
canonical ensemble] is zero. We can introduce a
probability density I(a, t) which gives the proba
bility of finding the system at time t in a state for
which a = a. This quantity I(a) will be independent
of time in a stationary ensemble p",(rp). We can
also introduce a joint probability density I(a, t;
a', t + T) for finding the system in a state a = a
at time t, and in a state a = a' at time t + T. In
a stationary ensemble this quantity will be inde
pendent of t, and will be denoted by tea, a', T).
Finally, one can define a conditional probability
density
pea, tj a', t + T) = I(a, tj a', t + T)/I(a, t), (5)
which in the stationary ensemble, can be written as
pea, a', T) = I(a, a', T)/t(a). (6)
Let us suppose for the moment that the a variables
are even functions of the particle velocities. We can
then fonnulate the property of microscopic reversi
bility which follows from time-reversal invariance (4)
by means of statistical mechanics as the equality
I(a, a', T) = tea, a', -T), (7)
or alternatively, making use of the stationarity of
the system, as
tea, a', T) = I(a', a, T), (8)
which expresses the property of detailed balance in
the a space of the macroscopic variables. It is usually
written, using (6), as
f(a)P(a, a', T) = !(a')p(a', a, T). (9)
In this form the property of detailed balance is
written for the stationary ensemble (equilibrium)
quantities. If, however, a system not in equilibrium
has been found by measurement to be in a epecified
state a at a given initial time to, then it can be shown
that its conditional probability density is equal to
the equilibrium quantity, i.e.,
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Pea, to; a', to + T) = Pea, a', T). (10)
This relation between a nonequilibrium and an
equilibrium quantity is only true for t = to, and
not for all times.
A direct corollary of (8) or (9) can be obtained by
introducing the time-correlation matrix Rik which
is defined as the expectation value of ai(t)ak(t + T)
in the stationary ensemble. It can also be written as
Rik = II aia~f(a, a', T) da da' (i, k = 1,2, ... ,n),
(11)
where we have used the joint probability density
f(a, a', T). Then it follows from detailed balancing
(8) that
(i, k = 1,2, ... ,n). (12)
On sager used this equality as a starting point,
and called it the principle of microscopic reversibility.
The proof of the property of detailed balancing
(9) from time reversal invariance (4) was first given
by Wigner,4 who used classical statistical mechanics.
It can also be obtained if the motion of the particles
is governed by quantum-mechanicallaws.5•6
Let us now make a few additional remarks on
the preceding.
1. The property of time-reversal invariance of
the microscopic motion, i.e., the invariance of H
under the operation t ~ -t, was the fundamental
symmetry which lead to detailed balancing. One
might alternatively call this property "invariance
under reversal of the motion," because it means,
classically, that if one were to reverse all velocities,
all particles would retrace their former paths; (a
similar statement could be made in quantum me
chanics). In this way, the wording is such that one
could, at least in principle, perform the reversal
operation, which is not the case for t ~ -t.
2. In the usual irreversible phenomena the
systems consist of particles which have "molecular"
interaction, i.e., one must deal ultimately with a
system consisting of an electromagnetic field in
interaction with electrons and nuclei. The time-re
versal invariance of the equations of motion for
such systems is well established. In recent years
much attention has been paid to time reversal, and
the other noncontinuous symmetry operations in
nuclear phenomena, especially in view of the non-
, E. P. Wigner, J. Chern. Phys. 22, 1912 (1954).
& N. G. van Kampen, PhyslCa 20, 603 (1954); Fortschr.
Physik 4, 405 (1956).
I J. Vlieger, P. Mazur, and S. R. de Groot, Physica 27,
353, 957, 974 (1961). conservation of parity in weak interaction. It seems
today that in both strong and weak interactions,
time-reversal invariance is valid. If nuclear forces
really influence irreversible phenomena, then it might
be reassuring to know that at least the foundation
of the Onsager relations, namely time-reversal in
variance, is still valid in such cases.
3.1f an external magnetic fieldB acts on the system,
the Hamiltonian will only be invariant for time
reversal if one includes reversal of the magnetic
field B in this operation. (Similarly one must reverse
the angular velocity vector if the system is subjected
to Corio lis forces.) One derives then, along the same
lines as before,
f(a)P(a, a', T,B) = f(a')P(a', a, T, -B), (13)
instead of (9).
4. Casimir7 remarked in 1945 that it may occur
that one needs for the macroscopic description of the
system not only variables a, which are even functions
of the particle velocities, but also variables {3, which
are odd functions of the particle velocities (e.g.,
momentum densities). Then time-reversal invariance
leads to a form
f(a, b)P(a, b, a', b', T)
= f(a', b')P(a', -b', a, -b, T) (14)
for detailed balancing, where band b' indicate values
of (3, just as a and a' indicated values of a.
B. Linear Laws
In order to find the effect of microscopic reversi
bility on the macroscopic properties of irreversible
phenomena, one must make a statement about the
time behavior of the "coarse-grained" variables a
and b introduced above. In particular, it is now
postulated that the conditional averages aCt) obey
linear first-order differential equations of the form
di(t) = -.L M ikak(t) (i = 1, 2, ... ,n), (15)
Ie
where the conditional averages are defined by
ai(t) = I aiP(a O, a, t) da (i = 1,2, ... ,n), (16)
and M'k is a matrix of phenomenological coefficients
(which are independent of time). The conditional
probability P(ao, a, t) refers to a nonstationary
ensemble, which corresponds to a system which at
time t = 0, is in a specified state ao, as indicated by
measurement. This nonstationary probability den-
7 H. B. G. Casimir, Rev. Mod. Phys. 17,343 (1945).
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sity is, according to (10), identical with the station
ary probability density.
Before investigating the influence of microscopic
reversibility on the properties of the linear laws
(15), we must specify how these laws are related to
the phenomenological laws mentioned in the begin
ning, and which were written as
(i = 1,2, ... ,n). (17)
This can be achieved in the following way: The
deviation of the value of the entropy Sea) =
k In tea) + const. from its maximum value will be
a quadratic expression in the ai, which can be
written as
fj.S = -! :E gihaka.,
ik (18)
where the gik are certain equilibrium properties
(second derivatives of S with respect to the ai)'
If we now define the fluxes Ji as the time derivatives
of ai, and the forces Xi as linear combinations of the
ai in the following fashion:
Ji = iii, (19)
then the regression laws [Eq. (15)1 take the form
of Eq. (17) with
(20)
In this way the connection of the linear laws with
thermodynamics is established, and a prescription
is given for finding the proper thermodynamic forces
Xi conjugate to the ai variables. One of the problems
of nonequilibrium thermodynamics is precisely to
arrive at a proper form of the linear laws (17) for
which the Onsager relations will be valid.
Now from (10), and (15)-(20), and statistical me
chanics, it can be shown that the phenomenological
coefficients Lik are directly related to the correla
tion matrix Rik (11) in the following way:
L -_k-1 l' aRk;
ik - 1m !it ' '_0 u (21)
where k is Boltzmann's constant. From detailed
balancing in the form R.k = Rki (12) the Onsager
relations
(i, k = 1,2, ... ,n) (22)
now follow immediately. A number of remarks on
this derivation can be made.
1. It is known empirically that linear laws of the
form (15) are valid for a large class of irreversible
phenomena, if the initial values ao lie in the macro-scopic region, i.e., far outside the region of average
equilibrium fluctuations. In the course of the deriva
tion however the regression law (15) is assumed to
hold also for small values of ao, i.e.,for values lying
in the region of equilibrium fluctuations, since the
main contributions to Rik are due to small values of
ao. This assumption is in agreement, for instance,
with Svedberg's and Westgren's experiments on
colloid statistics. Their results show that the
average behavior of density fluctuations is in perfect
agreement with the macroscopic law of diffusion.
2. The derivation of the Onsager relations is
based solely on microscopic reversibility and in the
assumption of the validity of linear regression laws.
The problem of establishing these linear laws from
first principles is not approached here, and the
existence of irreversible behavior is taken for
granted.
3. If an external magnetic field acts on the
system, we have the expression (13) for the property
of detailed balancing. This leads to Onsager rela
tions of the form
(23)
instead of (22).
4. If odd variables (3 must be taken into account,
we have formula (14) for detailed balancing. We
then obtain reciprocal relations in Casimir's form,
which show a minus sign on one side of the equality,
if an a-variable i is coupled with a (3-variable k.
For the coupling of two a variables or of two (3
variables, (22) or (23) remains valid.
5. Macroscopic equations describing irreversible
phenomena are often partial differential equations
containing derivatives of state variables with respect
to space coordinates. This is, in particular, the case
for vectorial phenomena such as heat conduction,
diffusion and electric conduction, and also for
tensorial phenomena such as viscous flow. The point
is namely that for such phenomena, the fluxes are
not direct time derivatives of state variables, as is
required in the proof of the Onsager relations.
Casimir7 was the first to show how one can cast
the macroscopic equations for these phenomena into
the form (15) or (17), in order to find the proper
Onsager relations. Subsequently one can then find
the effect of these Onsager relations on the properties
of the measurable phenomenological coefficients
which occur in the differential equations. It may be
stressed that this program has been worked out
for all irreversible processes. In particular formulas
of the type (15), (18), and (19) have been given
either explicitly (for instance for heat conduction,
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diffusion and electric conduction) or in such a form
that it is immediately clear how these expressions
are to be obtained.s
6. If independent state variables other than the
set ai, occurring in (IS) are chosen, one can of course
derive Onsager relations in exactly the same way.
One method of obtaining a new set of independent
state variables is to take linear combinations of the
original variables.
7. The phenomenological equations may be of a
more general type than (15) in still another way than
that discussed above in the fifth point. This is the
case in systems in which some properties (described
by the variables ai) respond to external driving
forces Fi• We can now study the influence of micro
scopic reversibility without making assumptions on
the mean regression of fluctuations. Let us assume
that the Hamiltonian is of the form
H(rpt) = Ho(rp) -L ai(rp)Fi(t) , (24)
where Ho is the Hamiltonian of the system in the
absence of driving forces. If one supposes that at
t = - CXJ the system was in equilibrium before the
forces F i were switched on, then one can derive
from (24) and statistical mechanics for the linear
response of ai to the forces
where, for macroscopic systems)
Kik(r) = 0
Kik(r) = -(kT)-1 fJRk;/iJr (r < 0),
(r 2:: 0),
a matrix which fulfills a causality condition. (26)
The microscopic reversibility property, [Eq. (12)]
leads here to the following symmetry property for
the Fourier transform Kik(W) of Kik(r), namely
(27)
which constitutes a generalization9 of the Onsager
relations for the laws (25). These laws are themselves
more general than the laws (15). (If magnetic fields
and odd variables play a role, then modifications
similar to those discussed before are necessary.)
One can also derive for this system a connection
between the equilibrium correlation matrix Rik(r)
8 Reference 7; Chap. VI, Sec. 4 of the first book of the
Bibliography and papers quoted therein.
9 H. B. Callen and R. F. Greene, Phys. Rev. 86, 702 (1952).
R. F. Greene and H. B. Callen, Phys. Rev. 88, 1387 (1952).
H. B. Callen, M. L. Barash, and J. L. Jackson, Phys. Rev.
88, 1382 (1952). R. Kubo, Lectures Boulder Summer School,
(1958). and the imaginary part K~~(W) of Kik:
kT fa> K~'(W) Rik( r) = - CP cos wr --dw.
~ _~ W (2S)
This formula represents the fluctuation dissipation
theorem, due to Callen and Greene, and discussed
also by Kubo.9 It connects the correlation function
matrix Rik' which characterizes the time behavior
of fluctuations in an equilibrium system, to the
imaginary part K~~(W) of the "susceptibility" matrix,
which is a measure for the dissipation of energy in
the system.
S. Finally we remark that linear laws of the type
(15) or (17) can only hold on a "macroscopic" time
scale, i.e., for times larger than some characteristic
microscopic time ro (but still small compared to the
relaxation time M~l). Indeed, the limit in formula
(21), which for a single a variable is equal to the
microcanonical average of aa, would vanish due to
the stationarity of the microcanonical ensemble.
However for sufficiently long times, where the limit
discussed is to be considered as difference quotient
instead of a differential quotient, one obtains a
finite result for (21). Regression laws of the type
(15) or (17) can then hold.
III. THE CONSERVATION AND BALANCE EQUATIONS
A second main line in the development of non
equilibrium thermodynamics is the "field theo
retical" formulation of the laws of thermodynamics.
Indeed in nonequilibrium situations, the state
variables are field quantities in the sense that they
are continuous functions of space coordinates and
of time. One must formulate the basic equations
of the theory in such a way that they contain quanti
ties referring to a single point in space at one time,
i.e., in the form of local equations. This should be
done in the first place for the various conservation
laws of mass, momentum, angular momentum, and
energy. Then if one uses these results along with the
thermodynamic Gibbs relation-which connects the
rate of change of entropy in each mass element to
the rate of change of energy, the rate of change of
composition etc.--one can establish a balance equa
tion for the entropy. This balance equation expresses
the fact that the entropy of a volume element
changes with time for two reasons. First it changes
because entropy flows into the volume element,
second because there is an entropy source due to
irreversible phenomena inside the volume element.
This is the local formulation of the second law of
thermodynamics. It is found that the entropy source
is a sum of products of fluxes and thermodynamic
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forces, the latter being related to the non uniformity
of the system or to the deviations of some internal
state variable from its equilibrium value. The
entropy source strength can thus serve as a basis for
the systematic description of irreversible processes
occurring in a system.
While several authors, beginning with Clausius,
had attempted to obtain entropy balance equations,
a systematic treatment along the lines just mentioned
was completed in the early forties by MeixnerlO
and by Prigogine,11 whose work will be discussed
in the next section.
Just as was done for the Onsager relations in the
preceding section, one can discuss the microscopic
basis for the second law of thermodynamics outside
equilibrium. This discussion has mainly been per
formed for two models, in both of which the ir
reversibility itself is already contained in the funda
mental equations, viz., the Gaussian Markoff process
and the kinetic theory of gases. Onsager and Mach
lup12 contributed to the study of the first model,
while Prigogine13 started to derive the thermody
namic laws from the kinetic theory of gases. It is
possible to justify the use of the second law outside
equilibrium in both cases. In particular, it can be
proven that for macroscopic initial states, both the
Boltzmann and the Gibbs definitions of entropy
lead to the macroscopic entropy law
I1S = -! L gi,jiiCt.k (29)
ik
for the Gaussian Markoff process. Such a law, and
incidentally, also the Onsager relations, may be
derived from the kinetic theory of gases.
IV. THERMODYNAMICS OF IRREVERSIBLE
PROCESSES
A consistent phenomenological theory of irre
versible processes incorporating both Onsager's
reciprocity theorem and the explicit calculation of
the entropy source strength was set up first by
Meixner10 and by Prigogine.11 Thus, by the system
atic amalgamation of the two lines of develop
ment treated in Secs. I and II, a new field of "thermo
dynamics of irreversible processes" was created. The
set of conservation laws, together with the entropy
balance equation and the equations of state, must
10 J. Meixner, Ann. Physik 39,333 (1941); 41 409 (1942);
43,244 (1943); Z. Physik. Chern. (Frankfort) B 53, 235 (1943).
11 I. Prigogine, Etude thermodynamique de8 phenomenes
irrweT8ible8, (Dunod, Paris, and Desoer, Liege, Belgium,
1947); see also Bibliography.
11 L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953).
S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953).
,. I. Prigogine, Physica IS, 272 (1949). be supplemented by the linear laws which relate
the fluxes and thermodynamic forces appearing in
the entropy source strength. One then has at one's
disposal, a complete set of partial differential equa
tions for the state parameters of a system, which
may be solved with the proper initial and boundary
conditions.
It is one of the main aims of nonequilibrium ther
modynamics to study the physical consequences
of the Onsager reciprocal relations in applications
of the theory to various physical situations. In
addition to the reciprocity theorem, possible spatial
symmetries of the system may further simplify the
scheme of phenomenological coefficients. This re
duction of the number of independent coefficients,
which results from invariance under special ortho
gonal transformations, goes under the name of the
Curie principle, but should more appropriately be
called Curie's theorem. Pierre Curie14 devotes only
a few lines to his statement, which make apparent
however, that he clearly understood the basis of
his theorem. An explicit proof can be given by
performing the relevant orthogonal transformations,
as mentioned above. It is then possible to find out,
for systems with arbitrary symmetry elements, which
fluxes are coupled to which thermodynamic forces.
There are a few additional theorems of nonequilib
rium thermodynamics which determine the trans
formations of fluxes and thermodynamic forces
under which the Onsager relations remain valid,
and other theorems which determine the special
properties of the entropy source strength at me
chanical equilibrium and in nonequilibrium station
ary states.
The theory has found a great variety of applica
tions in physics and chemistry, which can be clas
sified according to their tensorial character. First,
one has scalar phenomena. These include chemical
reactions and structural relaxation phenomena. On
sager relations are of help in this case, in solving the
set of ordinary differential equations which describe
the simultaneous relaxation of a great number of
variables. A second group of phenomena is formed
by vectorial processes, such as heat conduction,
diffusion and their cross effects (e.g., thermal dif
fusion). (Recently, Onsager relations were found
experimentally for ternary diffusion in a very non
ideal system.) Viscous phenomena (shear, bulk and
rotational viscosity) and the theory of sound ab
sorption and dispersion have been consistently de
veloped within the framework of nonequilibrium
If P. Curie, Oeuvres (Gauthier-Villars, Paris, 1908) p. 129.
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thennodynamics.15 Completely new aspects arise
when an electromagnetic field acts on a material
system. Then the continuity laws for electromagnetic
energy and momentum must also be taken into
account. Applications include electric conduc
tion, thermomagnetic and galvanomagnetic effects,
electro-kinetic processes, effects in polarized media
(e.g., the problem of ponderomotive forces). A great
number of membrane and similar effects have also
been studied.
BIBLIOGRAPHY
The material of this lecture is also dealt with in:
de Groot, S. R and Mazur, P., Non-Equilibrium
Thermodynamics (Interscience Publishers Inc.,
New York, and North-Holland Publishing Com
pany, Amsterdam, 1962).
11 J. Meixner, Ann. Physik. 43, 470 (1943); Acustica 2,
101 (1952). For general references, see also:
de Groot, S. R, Thermodynamics of Irreversible
Processes (Interscience Publishers Inc., New
York, and North-Holland Publishing Company,
Amsterdam, 1951).
de Groot, S. R, Rendi Scuola Intern. Fis. "Enrico
Fermi" Varenna, (1959).
Mazur, P., Proc. Intern. Summer Course Funda
mental Problems in Statistical Mechanics, Nijen
rode, Netherlands, (1961).
Mazur, P., Rendiconti Scuola Intern. Fis. "Enrico
Fermi" Varenna, (1959).
Meixner, J., and Reik, H. G., Encyclopedia of
Physics, (Julius Springer-Verlag, Berlin, 1959),
Vol. III, Part 2, p. 413.
Prigogine, I., Etude thermodynamique des phenomenes
irreversibles (Dunod, Paris, and Desoer, Liege,
Belgium, 1947).
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1.1696113.pdf | GaussianType Functions for Polyatomic Systems. I
Sigeru Huzinaga
Citation: J. Chem. Phys. 42, 1293 (1965); doi: 10.1063/1.1696113
View online: http://dx.doi.org/10.1063/1.1696113
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Published by the American Institute of Physics.
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maXImIze the binding energy of the inner electrons
should essentially maximize the total binding energy.
The former, in turn, should be expected to be hydrogen
like in character.
That nothing fundamental in terms of a quantum
mechanical description has been rendered by the
present theory is underlined by the fact that no shell
structure made its appearance in any of the densities
which were obtained. Nevertheless, the improvement
over Thomas-Fermi theory is considerable. Thus, the
latter theory yields a divergent mean value of 1/r2 in
marked contrast to the values obtained here.
We may draw attention to a similar statistical
theory of Barnes and Cowan,5 in which a "pseudo-
6 J. F. Barnes and R. D. Cowan, Phys. Rev. 132, 236 (1963).
See also P. H. Levine and O. Von Roos, ibid. 125. 207 (1692);
T. L. Schwartz and S. Borowitz, ibid. 133, A122 (1964). potential" which varies inversely as the square of the
radius is added to V (r). Such an inclusion, while ad hoc,
makes a truly significant improvement in the calculated
atomic binding energies. The effect of this modification
is to eliminate states of zero angular momentum, a
feature of some arbitrariness. The theory reported here
introduces no ad hoc exclusions of this sort, although
the physical effect of reducing the charge density at
the nucleus is similar.
ACKNOWLEDGMENTS
We wish to acknowledge the following grants in
support of this research: GSH-AT(30-1)-2968 and
SG-Nonr-1677(01). We are also grateful for a grant
of access to the 7090 data processing equipment at
the MIT Computation Center.
THE JOURNAL OF CHEMICAL PHYSICS VOLUME 42, NUMBER 4 15 FEBRUARY 1965
Gaussian-Type Functions for Polyatomic Systems. I
SIGERU HUZINAGA
IBM San Jose Research Laboratory, San Jose, California
(Received 2 October 1964)
In view of rapid progress of computer capability, it is very desirable to have a reliable assessment of the
usefulness of Gaussian-type orbitals as basis functions for large-scale molecular calculations. In the present
paper several attempts are made to answer this question mainly at the level of atomic Hartree-Fock calcu
lations. The necessary number of terms of Gaussian-type basis functions in the analytical Hartree-Fock
expansion calculation is apparently more than twice as much as the number of terms needed in the ex
pansion with Slater-type basis functions. However, this fact does not necessarily suggest a definite choice of
Slater-type orbitals. Discussions pertinent to this point are presented in the latter part of the present paper.
I. INTRODUCTION
THE advent of high-speed computers has encour
aged us to launch a major programming effort on
quantum-mechanical calculations of polyatomic sys
tems. If we are to proceed with concepts and methods
available at present, the most difficult problem is the
calculation of necessary molecular integrals.
Slater-type orbitals have been widely used for atomic
and molecular calculations. The effort to reach Hartree
Fock solutions by analytical expansion with Slater
type basis functions has been rewardingly successful
in lighter atoms and diatomic molecules. However,
attempts to evaluate general many-center molecular
integrals with adequate accuracy in reasonable time
have met with great difficulties. There has been no
conspicuous change or breakthrough on the side of
mathematical analysis. It is the spectacular advance
of computer capability that brings prospects of future
success. McLeanl was the first to tackle the calculation
1 A. D. McLean, J. Chern. Phys. 32, 1595 (1960). of many-center integrals over Slater-type orbitals with
considerable success in practical applications, but the
programming effort was confined to linear molecular
systems. McLean's scheme is essentially a direct nu
merical integration method supplemented with sophis
ticated program structure. The present limitation to
linear systems can be relaxed if one is prepared to
lengthen the calculation by an order of magnitude.
Given the present status of computing equipment, this
should turn out to be reasonable in the very near
future. Recent work of Shavitt and Karplus2 repre
sents the most successful effort to date in handling
general molecular integrals over Slater-type orbitals.
The central idea is the application of the integral
transform from exponentials to Gaussians:
r foo (r2) exp( -s-r) =-r= a-J exp --exp( -ar2)da (1.1) 2v7r 0 4a
It is obvious that by doing this additional integrals
are introduced on top of the already multidimensional
2 I. Shavitt and M. Karplus, J. Chern. Phys. 36, 550 (1962).
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integrals, but the reason for the success of this trans
formation lies in the fact that integrals over Gaussian
functions exp ( -ar2) are so simple in comparison to
those over exp ( -tr), that one regains here the loss
which is created by additional integrals over transfor
mation variables. Because of this contrasting simplic
ity of evaluation of molecular integrals over Gaussian
functions, several workers have hopefully proposed the
use of Gaussian and associated functions, which we
call Gaussian-type orbitals, instead of Slater-type or
bitals in the quantum mechanical calculations of poly
atomic systems.
The first systematic consideration of integrals over
Gaussian-type integrals is that due to Boys.s Since
then several papers with applications have been pub
lished. For instance, the treatment of the methane
molecule by Nesbet4 and later by Krauss5 is worthy
of mention, because they adopted Gaussian-type or
bitals to carry out calculations without any dubious
approximation of many-center integrals. At the time,
this was something which could not be done with a
reasonable amount of computer time with Slater-type
orbitals. This is, the present author believes, a very
important point to be taken into account when one
contemplates using Gaussian-type orbitals in molecular
calculations.
In the present paper several important findings on
Gaussian-type orbitals are presented. In the following
section we report results of the attempt to approximate
a single Slater-type orbital in terms of a linear combi
nation of Gaussian-type orbitals. Then, the effort to
reach the Hartree-Fock solutions by Roothaan's ex
pansion method with Gaussian-type orbitals is de
scribed.
It is the author's judgment that, in future calcula
tions on polyatomic systems, both Slater-type and
Gaussian-type functions will be used for several years
to come, and that work using one type will comple
ment that using the other type. A discussion pertinent
to this point of view is presented in the last section.
II. GAUSSIAN EXPANSION OF SLATER-TYPE
ORBITALS
Various methods may be used to obtain approximate
expansions of Slater-type orbitals in terms of Gaussian
type orbitals. The method being used in the present
work is due to McWeeny.6 We first describe the method
and then present the numerical results. Methods based
on least squares have also been under consideration
and are discussed toward the end of this section.
1. McWeeny's Variational Method
It is to be noted first that the Slater-type orbitals
are themselves exact eigenfunctions of a certain central-
3 S. F. Boys, Proc. Roy. Soc. (London) AlOO, 542 (1950).
4 R. K. Nesbet, dissertation, The University of Cambridge,
1954; J. Chern. Phys. 32, 1114 (1960).
6 M. Krauss, J. Chern. Phys. 38,564 (1963).
6 R. McWeeny, Acta Cryst. 6, 631 (1953). field problem. Once the Hamiltonians are established,
they may be used to obtain approximate eigenfunctions
as a linear combination of Gaussian-type orbitals by
means of variational procedures. McWeeny6 applied
this method to is, 2s, and 2p Slater-type orbitals.
Let us first formulate the scheme in a general form
and then apply it to several cases. The definition of
the normalized Slater-type orbitals (STO's) is
t/;.= Rn.(r) Y1m(ll, cf»,
Rn,(r) = [(2n.) !]-i(2Z/n.)n.H rn.-l exp[ -(Z/n.)r].
(2.0
Here, the parameter Z is not restricted to integer val
ues. It is easily verified that these STO's satisfy the
following equations (in atomic units),
(2.2)
where
H.= -t~-(Z/r) -(1/2r2) [Z(l+ 1) -n.(n.-l)],
(2.3)
E= -t(Z/n.)2. (2.4)
The first term of the" additional potential"
-(1/2r2) [1(1+ 1) -n.(n.-l)]
cancels out the angular dependent part of the kinetic
energy operator when the polar coordinate system is
introduced. Thus,
1 a2 !a Z n.(n.-1) H.= ----r----+ . (2.5) 2 ar2 ar r 2r2
For later convenience, we introduce a variable p= Zr
and write the eigenvalue operator as
Z-2H.= _! ~_! ~+ n.(n.-O ~_!. (2.6)
2 fJp2 pdp 2 p2 P
Now let us introduce the normalized Gaussian-type
orbitals (GTO's) :
Xg,;=Rn.(r) Y1m(l:l, cf»,
Rn.(r) = Nt-rna-1 exp( -tt-r2)
= NiZ-(no-l)pn"l exp( -aiP2), (2.7)
(2.8)
(2.9)
(2.10)
In the calculations of the present section we do not
mix different values of ng just for simplicity when we
set up linear combination of GTO's. This restriction
is not inherent in the calculations of Sec. III. For
convenience, we use the labels (nol)o for GTO's and
(n.l). for STO's to avoid possible confusion. We must
note that if the power of r is written as rn,,! then odd
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TABLE I. (a) opt from Eq. (2.13).
n. na=1 3 5 2 4 6
1 0.2829
2 0.02105 0.05016 0.04527
3 0.003493 0.01478 0.01842 0.01060 0.01714
4 0.0009790 0.005202 0.007587 0.003367 0.06570 0.008341
5 0.0003696 0.002187 0.003475 0.001346 0.02889 0.003963
no goes with even 1 and even no with odd 1. Thus we
may name GTO's as (1s)0, (3s)0, (5s)0, ••• ; (2p)0,
(4p) 0, ••• ; (3d) 0, (5d) 0, ••• ; etc. It is only under
this restriction that various atomic and molecular inte
grals over GTO's can be performed with great ease.
A straightforward calculation gives the following
results:
(Xo,; 1 Z-2H.I Xu,;)
=[(211
0+1) aia; +[n.(n.-1)-n o(no-1)](ai+aj)
(ai+a;) (2no-1)
(no-1) !2n
g 1J
- (2no-1) !!,,;;:(a;+a;)' (Xo,i 1 Xg,;), (2.11)
. . _[2(a iaj)l Jna+t
(Xo,.1 Xo,,)-(a;+aj) . (2.12)
From these general formulas it is readily seen what can
be done with a single Gaussian-type orbital to approxi
mate a Slater-type orbital by applying the variational
principle. The optimal single a value is
[ (no-1) !2nuV1 1 J2
(a)opt= ( r ( , 2no-3) !!"1I"4no+4n 8 n.-1)-1 (2.13)
and the corresponding approximate eigenvalue is
(Z-2H.)opt= _~ (2no-1) [(no-1) !2nuV1J2
2 4no+4n8(n.-1) -1 (2no-1) !!v,;:
(2,14)
Numerical values from these two formulas are col
lected in Tables I and II. It is interesting to note that,
for instance, a (2p) 0 GTO works better than a (4p) 0
GTO to approximate a (3p). STO and a (4p). STO
as far as the energy is concerned. We return to this
point later. A computer program in FORTRAN II language has
been used to obtain various approximate expansions of
STO's in terms of GTO's:
(2.15)
As mentioned earlier, Xo,; is normalized, and the ex
pansion coefficients {C;j are also normalized to give
(1/;.11/;.)= 1. An advantage of using p=Zr and a with
normalized GTO's is that it makes it clear how to
rescale a GTO expansion for arbitrary values of expo
nent of STO's without recalculating the expansion coeffi
cients {C;j. Suppose we want an approximate expansion
of a Slater-type orbital, rn.-1 exp(-r.r)Ylm(8, cp), in
terms of GTO's, rn..-1 exp( -ror2) Y1m(8, cp). First we
decide on a value of no and the number of terms to be
used in the expansion. Then the optimization of the
quantity (Z-2H.)(Z=n 8r8) by using (2.11) and (2.12)
gives us a set of {Ci} and {a;j.
To determine the optimum set of {ro,;) we use the
formula,
(2.16)
which follows from (2.10). The fact is that we need to
work out a best expansion with given length of expan
sion only once for each Slater-type orbital.
Numerical results so far obtained are shown in Tables
III to VII. Some of them overlap the results published
by McWeeny,6 Singer,? Whitten,8 and Reeves,9 but
usually with different values of parameters. This is
due to the existence of multiple minima of approximate
eigenvalues in the space spanned by variational param
eters. Difficulty in achieving true minimization, because
of these multiple minima, grows rapidly with increase
in the number of variational parameters. Because of
this difficulty it is not claimed that we have obtained
and listed true optimum values of {a;} in the tables.
The existence of multiple minima is very annoying in
TABLE II. (Z-2H,) from Eq. (2.14).
n, na=1 3 5 2
1 -0.4244
2 -0.1157 -0.09531 -0.11318
3 -0.04716 -0.05174 -0.04400 -0.05476
4 -0.02497 -0.03069 -0.02824 -0.03087
5 -0.01534 -0.01990 -0.01911 -0.01951
7 J. V. 1.. Longstaff and K. Singer, Proc. Roy. Soc. (London) A258,421 (1960).
8 J. 1.. Whitten, J. Chem. Phys. 39, 349 (1963).
9 C. M. Reeves, J. Chem. Phys. 39, 1 (1963). 4 6
-0.04776
-0.02956 -0.02692
-0.01961 -0.01856
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TABLE III. (Z-21I,) in a.u.: 1s, 2p STO's.
N, number of terms in GTO expansion
N
1
2
3
4
5
6
7
8
9
10 (1s),-(1s).
Presen t calc. Singer-
-0.424413
-0.485813 -0.486
-0.496979 -0.49689
-0.499277 -0.49928
-0.499809 -0.49976
-0.499940 -0.49988
-0.499976
-0.499991 -0.49992
-0.499997
-0.499999
Exact value = -0. 5
" See Ref. 7.
h Sec Ref. 9. (2p).-(2p).
Presen t calc. Reevesb
-0.113177 -0.113177
-0.123289 -0.123289
-0.124728 -0.124728
-0.124952 -0.124952
-0.124991
-0.124998
Exact value = -0.125
applying the variational procedure. Different sets of
{ad can give essentially the same energy. The set
actually chosen can be the one which is most suitable
for a specific application. Such an example will be
presented in the second paper of this series when we
evaluate many-center integrals between STO's by using
Gaussian expansion approximations.
We now turn to the point raised in the brief com
ment on Tables I and II. McWeeny6 approximated
Slater-type (2S)8 orbitals in terms of r2exp(-a,.y2),
not exp ( -a,.y2). However, as far as the energy is con
cerned, the results in Table IV clearly show that
exp( -a,.y2) works better than r2 exp( -a,.y2). Even
for a (3S)8 STO, exp( -a,.y2) is still quite adequate
as shown in Table IV. A similar situation can be seen
in the case of a (3p). STO.
This fact seems to be a mixed blessing as regards
the use of Gaussian-type orbitals. On the one hand,
it is an indication of slow convergence in approximat
ing STO's by GTO's. On the other hand, the adequacy
of using GTO's with lower ng values should be very
helpful to reduce complications in molecular integral
calcula tions.
2. Method of Least Squares
This is the method most frequently adopted for vari
ous kinds of curve fitting. Applications to the Gaussian
expansion are discussed by Boys and Shavitt.lO A com
puter program for least-squares fitting has been written
for the present purpose but we have found that the
program based on McWeeny's method is more con
venient to use. However, there is no denying that the
method of least squares is inherently more versatile,
especially with -a choice of a weighted function, than
the straightforward variational method. But if one tries
to find optimum values of parameters by using a least-
10 S. F. Boys and I. Shavitt, Proc. Roy. Soc. (London) A254,487
(1960). squares method, one has to deal with similar problems
of multiple minima, and these could well be worse than
in the minimum energy method.
Optimally we would like to have a method of curve
fitting which is not marred by the trouble of multiple
minima. We present here a possible mathematical de
vice which could sidestep such trouble. This is an
adaptation to Gaussian's of a method originally devised
for (and successfully applied to) exponential functions.ll
It consists of a set of well-defined mathematical pro
cedures, although it has not been put into a numerical
test.
The objective is to find a suitable Gaussian expan
sion of a well-behaved arbitrary functionj(x):
j(x),-""CI exp(aIx2) +C2 exp(a2x2) + ... +Cm exp(a mx2).
(2.17)
Assume that with proper choice of {ad and {C;I the
following n(n"22m) equations are satisfied:
m
jk= j(Xk) = LCj exp(aixk2) ,
j=I k=1,"',n, (2.18)
where Xk= Ckw)i, w being a constant increment. This
requirement cannot generally be satisfied, but let us
proceed on the basis that it can for the moment. Later
we introduce least-squares procedures to correct this
point.
It is convenient to use a new set of parameters
(2.19)
in place of aj. Now suppose that m real values of Vi
are known. Then it is always possible to construct an
mth-order algebraic equation.
(2.20)
which yields VI, V2, "', Vm as its m real roots. Of course,
these {Vj I are the very quantities we wish to deter
mine. However, if there is any method to determine
the coefficients, SI, S2, "', Sm, the above equation
will give us VI, V2, "', tim as its roots. This may be
achieved in the following way.
From (2.18), (2.19), and (2.20) it is easily verified
that
1= 1,2, "', n-m. (2.21)
Here we have n-m"2m equations from which SI,
Sz, "', Sm can be determined. In principle, m equa
tions should be enough to determine m unknown {SII.
However, this is totally inadequate as a curve-fitting
11 R. A. Buckingham, Numerical Methods (Pitman Publishing
Corporation, New York, 1957).
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TABLE IV. (Z--2H.) in a.u.: 2s, 3s, 3p STO's.
N, number of terms in GTO expansion
(2s),-(1s). (2s).-(3s). (3s) ,-(Is) g (3s),-(3s). (3p).-(2p). (3p),-(4p).
1 -0.11575 -0.09531 -0.047157 -0.051738 -0.054763 -0.047758
2 -0.12380 -0.11610 -0.055461 -0.055080 -0.055420 -0.053877
3 -0.12441 -0.12230 -0.055515 -0.055493 -0.055519 -0.055189
4 -0.12493 -0.12415 -0.055549 -0.055544 -0.055549 -0.055472
Exact value for 2s= -0.125. Exact value for 3s, 3p= -(1/18) = -0.055555.
TABLE V. Parameters of GTO expansion. (ls).-(ls) •. procedure. Thus the reasonable procedure to be taken
N C; here would be the following: (1) Pick wand 1Z. (2)
Cl; Computejk=j(xk) with Xk= (kw)t. (3) Solve a set of
2 0.201527 0.82123 n-m equations (2.21) by means of a standard least-
1.33248 0.27441 squares method to obtain Sl, S2, "', Sm. (4) Find
3 0.151374 0.64767 the m roots Vl, V2, "', Vm of the equation (2.20). (5)
0.681277 0.40789 Calculate aj from (2.19) . (6) Apply once again a
4.50038 0.07048 least-squares procedure to determine the coefficients
4 0.123317 0.50907 Gl, G2, "', Gmin (2.17) by using Ukl. 0.453757 0.47449 It is easy to extend the present scheme to the 2.01330 0.13424
13.3615 0.01906 Gaussian-type functions, rn"l exp( -ar2) , with no larger
5 0.101309 0.37602 than one.
0.321144 0.50822
1.14680 0.20572 III. ATOMIC SELF-CONSISTENT-FIELD 5.05796 0.04575 CALCULATIONS WITH GAUSSIAN-TYPE 33.6444 0.00612
6 0.082217 0.24260 BASIS FUNCTIONS
0.224660 0.49221 In Roothaan's expansion method relatively few STO's 0.673320 0.29430
2.34648 0.09280 are needed to closely approximate Hartree-Fock solu-
10.2465 0.01938 tions of atoms. For example, six s functions and four p 68.1600 0.00255 functions are enough to give seven-figure accuracy in 7 0.060738 0.11220
0.155858 0.44842 total energy for the ground states of the first-row
0.436661 0.38487 atoms.I2 It should be instructive, especially with pro-
1.370498 0.15161
4.970178 0.03939
22.17427 0.00753 TABLE VI. Parameters of GTO expansion. (2p),-(2p)u. 148.2732 0.00097
8 0.0525423 0.06412 N C, 0.123655 0.35846 Cl,
0.315278 0.42121
0.886632 0.21210 2 0.032392 0.78541
2.765179 0.06848 0.139276 0.32565
9.891184 0.01694 3 0.024684 0.57860 43.93024 0.00322 0.079830 0.47406 293.5708 0.00041 0.337072 0.09205
9 0.0441606 0.03645 4 0.020185 0.41444 0.106151 0.29898 0.055713 0.53151 0.250988 0.40433 0.174211 0.18295 0.618330 0.25781 0.733825 0.02639 1. 714744 0.10769
5.478296 0.03108 5 0.017023 0.28504
19.72537 0.00720 0.042163 0.52969
87.39897 0.00138 0.111912 0.27049
594.3123 0.00017 0.346270 0.06550
10 0.0285649 0.00775 1.458369 0.00833
0.0812406 0.20267 6 0.015442 0.21705
0.190537 0.41300 0.035652 0.49334
0.463925 0.31252 0.085676 0.32224
1.202518 0.14249 0.227763 0.10439
3.379649 0.04899 0.710128 0.02055
10.60720 0.01380 3.009711 0.00241
38.65163 0.00318
173.5822 0.00058
1170.498 0.00007 12 E. Clementi, C. C. J. Roothaan, and M. Yoshimine, Phys.
Rev. 127, 1618 (1962).
Downloaded 02 Dec 2012 to 132.210.244.226. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1298 SIGERU HUZINAGA
TABLE VII. Parameters of GTO expansion.
(2s).-(Is). (2s).-(3s). (3s).-(1s). ----------- --------------
N 0', C, N (Xi C; N (Xi C
2 0.026725 1.0078 2 0.045936 0.81372 2 0.0065103 1.05120
0.10456 -0.04872 0.20563 0.33894 0.076432 -0.15798
3 0.014660 0.44492 3 0.037585 0.66009 3 0.0066851 1.06712
0.037634 0.60335 0.13675 0.44967 0.056599 -0.15411
0.98413 -0.05385 0.61246 0.11033 0.22040 -0.02115
4 0.016500 0.54627 4 0.031669 0.53789 4 0.0047430 0.47178
0.042726 0.50899 0.10053 0.49646 0.0086456 0.61886
0.58274 -0.05708 0.35008 0.17988 0.057156 -0.16770
4.6935 -0.00843 1.54205 0.03814 0.22271 -0.01895
(3s),-(3s). (3p),-(2p). (3p).-(4p) 0
2 0.010769 0.76147 2 0.0064787 0.55899 2 0.014413 0.78986
0.036358 0.35875 0.016457 0.50670 0.050378 0.36321
3 0.0082807 0.53553 3 0.0071691 0.65491 3 0.011149 0.59414
0.022528 0.51193 0.018954 0.41337 0.032400 0.49513
0.075079 0.10596 0.14851 -0.01769 0.11271 0.12343
4 0.00721080 0.40977 4 0.0051045 0.31376 4 0.0091999 0.495203
0.017891 0.55618 0.011265 0.57213 0.026107 0.52730
0.049976 0.17847 0.023277 0.18931 0.067819 0.19064
0.18578 0.01918 0.23747 -0.01371 0.23712 0.04176
lation in Roothaan's expansion method are spective molecular calculations in mind, to see how
many GTO's would be necessary to achieve more or
less the same accuracy. For this purpose the computer
program written by Roothaan and his collaborators13
has been modified to use Gaussian-type basis functions
in the expansion method instead of Slater-type basis
functions. Necessary modifications and changes in the
program were cut down to less than 300 words in FAP
language for the IBM 7090 computer. This has been
accomplished by preparing the whole mathematical
setup for the Gaussian-type basis functions to resemble
as closely as possible that of the original Slater-type
basis functions. S)..pq= f"'(du)u2RXp(U)R)..q(U) , (3.3)
a
The atomic orbitals are expanded in terms of basis
functions according to
cf>iXa= LXp~aC.')..p.
p (3.1)
The symmetry species is represented by X, the sub
species by a, i labels the orbitals which cannot be
distinguished any more by symmetry, and p has the
same role for the basis functions. Let the basis func·
tions be given by
(3.2)
The matrices and supermatrices which enter the calcu-
13 C. C. J. Roothaan and P. S. Bagus, Methods in Computational
Physics (Academic Press Inc., New York, 1963), Vol. 2, p. 47. U)..pq= f'" (du)uR)..p(u) R)..q(u), (3.4)
a
TXpq= ~ f'" du[u2R\p(u) R\q(u)
o
+X(X+ 1) R)..p(u) R)..q(u)], (3.5)
x [Rxp(u) R~q(u) Rp.r(v) Rp.s( v)
+ Rl'r(u) Rp.s(u) R)..g(v) R)..q(v)], (3.6)
x [R)..p(u) Rp.r( u) R)..g( v) Rp.s (v)
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TABLE VIII. Calculated total and orbital energy TABLE IX. Orbital parameters for He(ls)2 IS.
for helium (ls)2 IS.
N to Co
N E 2 0.532149 0.82559
4.097728 0.28317 1 -2.3009869 -0.65638 3 0.382938 0.65722 2 -2.7470661 -0.858911 1.998942 0.40919 3 -2.8356798 -0.903577 13.62324 0.08026 4 -2.8551603 -0.914124
5 -2.8598949 -0.916869 4 0.298073 0.51380
6 -2.8611163 -0.917688 1.242567 0.46954
7 -2.8614912 -0.917895 5.782948 0.15457
8 -2.8616094 -0.917931 38.47497 0.02373
9 -2.8616523 -0.917946 5 0.244528 0.39728 10 -2.8616692 -0.917952 0.873098 0.48700
Hartree-Fock value 3.304241 0.22080
14.60940 0.05532
E=-2.861680 £=-0.91795 96.72976 0.00771
6 0.193849 0.26768
0.589851 0.46844
1.879204 0.29801
The basis functions are now taken to be normalized 6.633653 0.10964
Gaussian-type functions, namely, 28.95149 0.02465
192.4388 0.00330
Rxp(r) =lV(nxp, '\\p)rnxp-1 exp( -rxpr2), (3.8) 7 0.160274 0.18067
0.447530 0.43330
1. 297177 0.34285 N(nxp, rxp) = [(2/1r)!22nxp+lrxpnxpH/(2nxp-l) !!]!. 4.038781 0.15815
14.22123 0.04727
(3.9) 62.24915 0.00971
414.4665 0.00127
It is to be noted that the parameter rxp has the dimen- 8 0.137777 0.11782
sion of (length)-2 instead of (length)-l. As noted 0.347207 0.36948
0.918171 0.36990 before, odd nxp goes with even A and even nxp with 2.580737 0.21021
odd A. For these basis functions the integrals defined 7.921657 0.07999
by Eqs. (3.3) through (3.7) assume the following 28.09935 0.02134
124.5050 0.00415 forms: 833.0522 0.00053
SXpq= [V2n)..p (rxp) V2nxq(rXq) ]-W nxp+n)..q[Hrxp+rxq)], 9 0.129793 0.09809
0.308364 0.31570
0.725631 0.34783
(3.10) 1.802569 0.24466
UXpq= (2/1r) 12 [V2n)..p (rxp) V2n)..q(rXq) ]-! 4.951881 0.11748
15.41660 0.03844
55.41029 0.00939
X Vn)..p+nx.-1[Hrxp+rxq)], (3.11) 246.8036 0.00178
1663.571 0.00023
Txpq= trxprXq[V 2nxp(rxp) V2nxq(rXq) ]-! 10 0.107951 0.05242
0.240920 0.24887
X Vn)..p+nx.+2[Hrxp+rxq)] 0.552610 0.36001
1.352436 0.28403
-{WX,n)..p(rxp) + W"nxq(rXq) I VnXP+n)..q[Hrxp+rxq)] 3.522261 0.14909
9.789053 0.05709
30.17990 0.01721
+ WX,n)..p(rxp) WX,n)..q(rXq) VnXP+n)..q-2[Hrxp+rxq) J, 108,7723 0.00412
488.8941 0,00076
( 3.12) 3293.694 0.00010
TABLE X. Total energy (in a.u.) for the Li to Ne atoms. Comparison between GTO and STO calculations.
GTO GTO STO STO
Atom State 9-(Is)., 5-(2p). 10-(Is)., 6-(2p). best double" accurateb
Li 2S -7.4322794 -7.4325033 -7.4327184 -7.4327257
Be IS -14.572068 -14.572579 -14.572368 -14.573020
B 2p -24.527130 -24.528282 -24.527890 -24.529052
C 3p -37.685247 -37.687324 -37.686677 -37.688611
N 's -54.395336 -54.398909 -54.397873 -54.400911
0 3p -74.800289 -74.806295 -74.804180 -74.809360
F 2p -99.395586 -99.404870 -99.401164 -99.409284
Ne IS -128.52674 -128.54094 -128.53480 -128.54701
a E. Clementi, J. Chern. Phys. 40, 1944 (1964).
b See Ref. 11.
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TABLE XI. Orbital exponents of the Gaussian basis set: 9-(ls)a. 5-(2P)a'
Li(2S) Be(1S) B(2P) N(4S) --.-----------------------------------------
1s
Is
1s
ls
ls
1s
1s
Is
Is 1.15685 2.18473 3.40623 5. 14i73 7.19274 9.53223 12.2164 14.9060
132.463
432.759
1821. 39 9.35329 17.6239 28.0694 42.4974 59.8376 81.1696 104.053
31. 9415 60.3255 96.4683 146.097 204.749 273.188 350.269
138.730 262.139 419.039 634.882 887.451 1175.82 1506.03
0.44462 0.85895 1.30566 1.96655 2.68598 3.41364 4.36885 5.12741
1.49117
43.7659 0.076663 0.18062 0.32448 0.49624 0.70004 0.93978 1.20775
3.15789 5.93258 9.37597 14.1892 19.9981 27.1836 34.8432
921.271 1741.38 2788.41 4232.61 5909.44 7816.54 9994.79
0.028643 0.058350 0.10219 0.15331 0.21329 0.28461 0.36340 12102.2
0.44676
2p
2p
2p
2p
2p 0.21336 0.35945 0.53136 0.71706 0.93826 1.20292
3.86542 0.68358 1.14293 1. 70740 2.30512 2.99586
2.43599 3.98640 5.95635 7.90403 10.0820 12.9187
56.4511
0.34440 11. 3413 18.1557 26.7860 35.1832 44.3555
0.070114 0.11460 0.16537 0.21373 0.27329
where
and Vi(X) = (i-1) !!/[(X)lJi+I,
Wij(x) =2xl(j-i-1), (3.13)
( 3.14)
= (2/1r) 12[V2nI.pG-xp) V2nXqG\q) V2n~r(r"r) V2n~.(r"8) J-i
X {VnhP+"hq-v-I[Hrxp+tXq) JV n~r+n~.+v[Htl'r+t"8) J
X Cn"p+nhq-v-I,n"r+n",+V[ (txp+tXq) / (r"r+t".) J
+ Vn"r+n",-V-I[! (tl'r+t".) JVnAP+nh'+v[! (txp+rXq) J
X Cn"r+n",-v-I,nhP+nh'+v[ (t"r+tl's) / (txp+tXq) J I, (3.15)
= (2/1r)t[V 2nhP(rXp) V2nhq(tXq) V2n"r(tl'r) V2n", (t".) J-1
X { VnAP+n"r-V-I[! (tXp+t"r) JVnhq+n~.+v[! (rXq+t".) J
X CnhP+n",.-v-I,nhq+n",+v[ (tXp+t"r) / (tXq+t"s) J
+ V nh.+n",-V-I[Htxq+r"s) JVnhP+n"r+v[Htxp+r"r) J
X CnA.+n",-v-l,nhP+n~r+v[ (tXq+t"s) / (tXp+t"r) J
+ V"h"-tn",-v-I[Htxp+r,,s) JVnh.+n~r+'[HtXq+t"r) J
X C"hP+n~,-v-l,nA.+n~r+v(tXp+t"s) / (rXq+t"r) J
+ V nhq+n",.-v_I[!(tXq+t"r) JV nh,,-tn".-tv[Hrxp+rl's) J
X CnAq+n",-v-I,nhP+n~,+v[ (tXq+r"r) / (txp+r".) J I, (3.16)
where
a (X+{3-2) !! ( t )(A-I)/2
Ca.~(t)=(1+t)-(a+~)/2t;(X-1)!!({3-1)!! l+t .
( 3.17)
A very close resemblance is apparent between the
formulas listed above and those used in the original
STO version of the program described in detail by Roothaan and Bagus.13 Tables VIII and IX show nu
merical results of a series of calculations designed to
reach the Hartree-Fock solution in terms of Gaussian
expansion for the ground state of helium. With 10
GTO's we have obtained almost six figure accuracy
in total energy, but this appears to be a rather dis
appointing finding if we recall that a linear combina
tion of only two STO's gives a better result.
Results of calculations for the first-row atoms are
shown in Tables X through XIV. We have used vari
ous sizes of basis sets but only two sets, one with nine
(1s)u and five (2p)u and the other with 10 (1s)u and
six (2p)u are shown here. Because of the existence of
multiple minima, it is not claimed that the values of
orbital exponents are truly optimized, although a con
siderable amount of machine time has already been
consumed in optimization, A set of mixed basis func
tions consisting of seven (Is) 0 and two (3s) u orbitals
was also used for the ground state of helium but the
result turned out to be poorer than that of nine (1s) u
basis functions. Similar attempts have been made also
for lithium, carbon, and nitrogen; but so far nothing
has come out to encourage the inclusion of (3s)0 orbit
als to describe the Is and 2s atomic orbitals of the
first-row atoms.
IV. DISCUSSIONS
The purpose of the present paper has been primarily
to present several numerical facts about Gaussian-type
orbitals. To some people, the results presented here may
be a confirmation of their belief that Slater-type orbit
als should be the choice for molecular and solid-state
calculations. There is no denying that the Gaussian
type orbitals are far inferior basis functions to the
Slater-type orbitals in representing atomic orbitals. If
the true Hartree-Fock solution with seven or eight
figure accuracy is one's objective, Slater-type orbitals
should definitely be one's choice. To other people, how-
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TABLE XII. Orbital exponents of the Gaussian hasis set: 10-(ls)o, 6-(2p)g.
Li('S) Be(tS) B(2P) Crap) N('S) o(ap) F(2P) Ne(1S)
Is 1.90603 .~.66826 6.25286 9.40900 13.4578 17.8966 2.,. 370S 29.1672
Is 16.7798 32.6562 55.8340 84.5419 120. 89() 160.920 209. In 261.476
Is 60.0718 117.799 202.205 307.539 439.998 585.663 757.667 946.799
Is 267.096 532.280 916.065 1397.56 1998.96 2660.12 3431. 25 4262.61
ls 0.71791 1. 35431 2.31177 3.50002 4.99299 6.63901 8.623i2 10.7593
1.1' 0.26344 0.38905 0.68236 1.06803 1.56866 2.07658 2.69163 3.3425S
Is 0.077157 0.15023 0.26035 0.40017 0.580017 0.77360 1.oon:; 1.24068
Is 0.540327 10.4801 17.8587 26.9117 38.4711 51.1637 66.7261 83.3433
Is 1782.90 3630.38 6249.59 9470.52 13515.3 18045.3 23342.2 28660.2
Is 0.028536 0.052406 0.089400 0.13512 0.19230 0.25576 0.33115 0.40626
2p 0.15033 0.24805 0.37267 0.48209 0.62064 0.78526
2p 0.39278 0.65771 0.99207 1.32052 1.73193 2.21058
2p 1.06577 1. 78730 2.70563 3.60924 4.78819 6.2187i
2p 3.48347 5.77636 8.48042 11. 4887 15.2187 19.7075
2p 15.4594 25.3655 35.9115 49.8279 65.6593 84.8396
2p 0.057221 0.091064 0.13460 0.16509 0.20699 0.25665
TABLE XIII. Orbital energies and expansion coefficients": 9-(1s)o, 5-(2p)g.
Li('S) Be(IS) B(2P) Crap) N('S) o (3P) F(ZP) Ne(IS)
is Is Is Is Is Is Is Is
-2.47761 -4.73230 -7.69485 -11. 3249 -15.6283 -20.6663 -26.3784 -32.7650
0.42505 0.42643 0.43331 0.43809 0.44611 0.46137 0.46223 0.47268
0.16064 0.15845 0.16002 0.15459 0.15043 0.14389 0.14291 0.13791
0.04984 0.04799 0.04763 0.04534 0.04411 0.04287 0.04240 0.04133
0.01042 0.00995 0.00983 0.00934 0.00909 0.00897 0.00887 0.00909
0.16878 0.16037 0.14005 0.14581 0.14553 0.14017 0.14063 0.12994
0.00253 0.00265 O.OOlU 0.00199 0.00127 -0.00058 -0.00035 -0.00212
0.34455 0.35122 0.36273 0.35867 0.35658 0.35555 0.35527 0.36255
0.00137 0.00130 0.00129 0.00122 0.00119 0.00118 0.00117 0.00120
-0.00013 -0.00045 0.00036 0.00041 0.00080 0.00139 0.00143 0.00183
2s 2s 2s 2s 2s 2s 2s 2.1'
-0.1930 -0.30906 -0.49441 -0.70506 -0.9440 -I. 24216 -1.56878 1.9245,'i
-0.10956 -0.14223 -0.16661 -0.17699 -0.18556 -0.19590 -0.20032 -0.20769
-0.02679 -0.03141 -0.03530 -0.03606 -0.03633 -0.03574 -0.03624 -0.03537
-0.00786 -0.00878 -0.00968 -0.00974 -0.00978 -0.00979 -0.0(l987 -0.00978
-(l.00164 -0.00184 -0.00201 -0.00202 -0.00203 -0.00207 -0.00208 -0. (l0217
-(l.10761 -0.07969 -0.05960 -0.05267 -0.04544 -0.03740 -0.03201 -0.01923
0.55797 0.54191 0.55856 0.57408 0.58434 0.59566 0.60464 0.61429
-0.06067 -0.07274 -0.08535 -0.08938 -0.09227 -0.09508 -0.09721 -0.10138
-0.00021 -0.00024 -0.00026 -0.00026 -0.00026 -0.00027 -0.00027 -0.00028
O.544Z3 0.57355 0.56245 0.54768 0.53747 0.52576 0.51555 0.50212
2p 2p 2p 2p 2p 2p
-0.30920 -0.43248 -0.566.\3 -0.62941 -0.72586 -0.84405
0.51687 0.50734 0.50679 0.49376 0.48636 0.48583
0.30565 0.30611 0.31026 0.31066 (\.31063 0.30927
0.08803 0.09150 0.09258 (1.09774 0.10199 0.10164
0.01435 0.01469 0.01452 0.01541 0.01636 0.01632
0.30567 0.31735 0.31773 0.33604 0.3442-1 0.3·1961
3. The first entry in each column is the orbital energy (in atomic units) and the following are the expansion coefficients corre:::.ponding to the basis set given in
Table XI.
Downloaded 02 Dec 2012 to 132.210.244.226. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions1302 SIGERU HUZINAGA
TABLE XIV. Orbital energies and expansion coefficients": 10-(Is)., 6-(2p) •.
Li(2S) Be(1S) B(2P) C('P) N(4S) O('P) F(2P) Ne(IS)
Is Is Is Is Is Is Is Is
-2.47765 -4.73223 -7.69503 -11. 3252 -15.6284 -20.6680 -26.3817 -32.7711
0.41995 0.43211 0.42870 0.42695 0.42369 0.42385 0.42181 0.42200
0.09236 0.08689 0.08051 0.07736 0.07389 0.07284 0.07080 0.07011
0.02425 0.02239 0.02038 0.01934 0.01833 0.01801 0.01754 0.01732
0.00467 0.00422 0.00381 0.00358 0.00399 0.00333 0.00327 0.00324
0.33028 0.33942 0.35377 0.35790 0.36706 0.36853 0.37564 0.37656
0.04546 0.03710 0.04397 0.04877 0.05356 0.05441 0.05670 0.05721
-0.00170 -0.00791 -0.00806 -0.00756 -0.00664 -0.00674 -0.00673 -0.00675
0.24636 0.24152 0.23088 0.22679 0.21952 0.21809 0.21300 0.21212
0.00060 0.00053 0.00047 0.00045 0.00042 0.00042 0.00041 0.00041
0.00090 0.00183 0.00207 0.00213 0.00202 0.00213 0.00218 0.00226
2s 2s 2s 2s Zs 2s 2s 2s
-0.19630 -0.30919 -0.49463 -0.70551 -0.94500 -1.24386 -1.57175 -1.92939
-0.08090 -0.10274 -0.11441 -0.12134 -0.12455 -0.12923 -0.13126 -0.13358
-0.01473 -0.01628 -0.01676 -0.01701 -0.01681 -0.01709 -0.01696 -0.01704
-0.00385 -0.00414 -0.00417 -0.00418 -0.00409 -0.00414 -0.00411 -0.00411
-0.00073 -0.00077 -0.00077 -0.00076 -0.00075 -0.0076 -0.00076 -0.00076
-0.13167 -0.15719 -0.17008 -0.17554 -0.18034 -0.18361 -0.18708 -0.18903
-0.04738 0.04809 0.07448 0.08502 0.08213 0.09512 0.09710 0.10831
0.56761 0.59099 0.60399 0.60689 0.60379 0.60828 0.60756 0.61435
-0.04270 -0.04911 -0.05202 -0.05399 -0.05400 -0.05525 -0.05497 -0.05560
-0.00009 -0.00010 -0.00010 -0.00010 -0.00009 -0.00009 -0.00009 -0.00010
0.54248 0.47194 0.44484 0.43809 0.44676 0.43379 0.43415 0.41924
2p 2p 2p 2p 2p 2p
-0.30964 -0.43305 -0.56708 -0.63119 -0.72892 -0.84904
0.43629 0.43276 0.42652 0.42475 0.42268 0.42027
0.35746 0.35871 0.35881 0.35801 0.36114 0.36527
0.17925 0.18263 0.18131 0.19028 0.19216 0.19244
0.05366 0.05479 0.05480 0.05797 0.05799 0.05676
0.00895 0.00875 0.00907 0.00891 0.00880 0.00857
0.19799 0.20347 0.21524 0.21977 0.22402 0.22777
" The first entry in each column is the orbital energy (in atomic units) and the following are the expansion coefficients corresponding to the basis set given in
Table XII.
ever, the results summarized in Table X should offer
much encouragement to proceed further with Gaussian
type orbitals. To pursue very accurate Hartree-Fock
solutions in the calculation of large polyatomic systems
could be unrealistic, because of economy and also
possibly because of the approximate nature of the
Hartree-Fock solutions. If the required accuracy is
of a few tenths of an electron volt per atom measured
by the Hartree-Fock standard, the number of neces
sary Gaussian-type basis functions is not prohibitively
large, as is shown in Table X. Let N. and Ny be the
number of Slater-and Gaussian-type basis functions
which give roughly comparable accuracy and a be
their ratio: Ny=aN •. If the required time to perform
molecular calculations using the expansion method in
creases in proportion to N4, N being the number of
basis functions used, then the time factor between com
parable calculations based on Slater-type and Gaussian
type basis functions would roughly be a4• Recently,
Harrison14 expressed a rather optimistic opinion about
this point by taking a"'2. The present author inclines
to think that a has to be larger than 2 and only if a
single molecular integral over GTO's can be computed
14 M. C. Harrison, J. Chern. Phys. 41, 499 (1964). 103 times faster than over STO's (as an average),
would the GTO basis have a definite edge over the
STO basis.
After these and other considerations in the course
of the present work, the author has come to the conclu
sion that is stated in the first section, namely that both
classes of functions, STO's and GTO's, can be comple
mentary to each other for various kinds of molecular
and solid-state calculations at least for several years
to come.
ACKNOWLEDGMENTS
I wish to thank E. Clementi, R. K. Nesbet, A. D.
McLean, and M. Yoshimine at IBM Research Labora
tory in San Jose for stimulating and useful discussions.
During the debugging stage of the GTO atomic SCF
program, Professor C. C. J. Roothaan provided me
every convenience, and Mr. A. Peterson rendered in
valuable help in programming details at the University
of Chicago Computation Center. Finally it is my pleas
ure to express sincere gratitude to Dr. J. D. Swalen
and to Dr. A. G. Anderson for their very generous
hospitality and support given to me at the IBM Re
search Laboratory in San Jose.
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1.1729239.pdf | Investigation of the Patch Effect in Uranium Carbide
George A. Haas and Richard E. Thomas
Citation: Journal of Applied Physics 34, 3457 (1963); doi: 10.1063/1.1729239
View online: http://dx.doi.org/10.1063/1.1729239
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/12?ver=pdfcov
Published by the AIP Publishing
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IP: 129.24.51.181 On: Fri, 28 Nov 2014 02:42:09THE R M ION I CST U DIE S 0 F V A RIO U SUR A N I U M COM P 0 U N D S 3457
The surface film of uranium, while having a strong
effect on the emission, does not exclude characteristics
of the individual uranium compounds on which it forms,
since slight variation in emission and decomposition
temperatures of these are noted. Because of its high
emission level and decomposition temperature, UC
appears to be the compound best suited as a thermionic
cathode, although the close similarity of the others
indicate that only a slight improvement in any of them
is sufficient to make them worthwhile for consideration.
In the inactive state, UC is quite patchy in nature and
can be described by the empirical constants cp= 3.8 eV and A = 120 A/cm2deg2. In the active state, the emission
is quite uniform and can be described by the previously
published value2 q,= 2.94, A = 33.
ACKNOWLEDGMENTS
The authors are deeply indebted to R. Woltz and J.
Klebanoff for the inert atmosphere construction of the
fifty-odd experimental diodes used in the course of this
study. Gratitude is also expressed to O. J. Edwards for
his aid with much of the mathematical calculations
needed in processing the data.
JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 12 DECEMBER 1963
Investigation of the Patch Effect in Uranium Carbide
GEORGE A. HAAS AND RICHARD E. THOMAS
Naval Research Laborat(}1'Y, Washington, D. C.
(Received 17 May 1963)
An electron beam scanning technique has been devised to measure variations in the surface work function.
These measurements indicate a wide patch distribution in the range of 3.25 to 4.5 eV for inactive uranium
carbide, which upon thermal activation becomes covered with a surface having a work function in the narrow
band of 3.0 to 3.25 eV. The active surface is very sensitive to ambient gases and can be easily poisoned to a
value of work function higher than was the case for the inactive state. These results support the hypothesis
that a uranium film is responsible for the activation as was suggested by previous thermionic measurements.
An analysis of the effect of a nonuniformity in work function on thermionic measurements, shows that
experimental Richardson plots or effective work function plots can give highly erroneous results. The magni
tude of this error can be determined and is expressed in terms of the spread of the patch distribution.
INTRODUCTION
PREVIOUS thermionic results! on uranium com
pounds and specifically those of UC have indicated
the presence of a nonuniformity of the surface work
function and a dependence of this nonuniformity on the
state of activation. This implies that the surface is not
composed of a single value of work function, but more
likely, is composed of many small regions having dif
ferent and individual values of work functions called
"patches." This so called "patch effect," which changes
for different surface conditions, manifests itself in
thermionic measurements, for example, in the effect on
the low-field Schottky characteristics. However, little
more than its presence can be determined by thermionic
techniques such as used in the preceding paper since the
total emission is measured, and it is impossible to dis
cern from which patch an electron originated. Conse
quently, because only an average value of the work
function is obtained from these measurements, no infor
mation is provided regarding the distribution in work
function of the surface, i.e., relative abundance of re
gions having any given work function.
This information regarding patch structure is very
desirable in the understanding of the thermionic prop-
1 See preceding paper, J. App!. Phys. 34, 3451 (1963), and Ref.
2 cited therein. erties of an electron emitter, but is of particular im
portance here because of the possible use of these emit
ters in the low-field application of thermionic energy
conversion. Here, the value of the emitter work func
tion is used for such computations as electron emission,
ion generation, and contact potential difference between
emitter and collector. If an appreciable distribution in
work function exists, it is possible that significant errors
might arise by assuming a single value for all of these
computations.
There are a number of ways currently available by
which the patch structure of a surface can be investi
gated. Three of the more commonly employed methods
are (1) electron emission microscope,2.2a (2) electron
mirror microscope,3 and (3) analysis of the anomalous
retarding potential region.4 However, because of limita-
2 See, for example, E. Briiche and H. Johannson, Z. Tech. Phys.
14, 487 (1933); R. F. Hill and S. R. Rouze, J. App!. Phys. 33,
1607 (1962); W. Heinze and S. Wagener, Z. Tech. Phys. 20, 16
(1939); R. D. Heidenreich, J. App!. Phys. 26, 757 (1955).
2a Note added in proof, See also recently published works of
B. Devin, G. Gayte, L. Koch, and L. Sondaar, Advanced Energy
Conversion (Pergamon Press Inc., New York, 1963), Vol. 3, p. 287;
A. V. Druzhinin, Radio Eng. Electron. Phys. 9, 1446 (1962).
3 L. Mayer, J. App!. Phys. 26, 1228 (1955); G. V. Spivak, 1. A.
Pryamkova, and V. N. Lepeshinskaya, Dokl. Akad. Nauk
USSR, 130, 751 (1960) [English trans!.: Soviet Phys.-Doklady
5, 110 (1963)].
4 D. G. Bulyginskii, Bull. Acad. Sci. USSR 20, 975 (1956).
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IP: 129.24.51.181 On: Fri, 28 Nov 2014 02:42:093458 G. A. HAAS AND R. E. THOMAS
tions inherent in these methods, a new approach was
de,:eloped c~lled the electron beam scanning technique
whIch combmes most of the good features of these three
methods while eliminating many of the disadvantages.
Among the disadvantages of the first method is that
the electro-optically magnified image of the emitter
surface which appears on the phosphor screen must be
first translated to electrical current before values of
work function can be determined. This requires the use
o~ a Faraday.cage and associated difficulties of viewing
dIfferent portIOns of the surface with it. Even if this is
d?ne; it ~ould be extremely difficult to obtain a patch
dlstnbutlOD plot5 for all regions of the surface. Another
constraint is that the surface to be studied must be hot
enough to emit electrons. The second method eliminates
this latter constraint by flooding the surface to be
studied with low-velocity electrons and then viewing
the reflected electrons on a phosphor screen similarly
as in the first case. Besides the same disadvantage of not
providing a patch distribution plot, as was encountered
by the first method, an additional difficulty is that the
region of best contrast for recording reflected electrons
is near the saturation region where most electrons are
accepted by the surface. However, the only ones that
are reflected in this region (and consequently measured),
are the low-energy ones which are strongly affected by
patches on the flooding emitter. The third method meas
ures the patch distribution by means of a detailed
analysis of the retarding-to-accelerating characteristics
of a diode. It has the disadvantage that one of the elec
trodes mus~ be patch free (which stipulates a single
work functIOn type surface) and assumes no localized
patch fields. Furthermore, since the total current is being
measured, no information can be obtained about in
dividual patches.
Of the three methods just described, the first two pro
vide a visual manner of describing the surface but do not
give a patch distribution curve while the third method
gives a patch distribution curve but gives no visual in
formation regarding shapes and configuration of the
patches. The electron-beam scanning technique, de
veloped for the type of measurements described here
provides not only visual pictures as well as patch dis~
tribution curves, but also avoids many of the type of
complications just noted which arise from patch effects
inherent in the measuring system. This paper describes
this technique and gives results obtained by it in the in
vestigation of the patch structure of the same type UC
emitter investigated by thermionic methods described
in the previous paper.
EXPERIMENTAL TECHNIQUES AND RESULTS
The surfaces investigated were prepared in a manner
identical to the indirectly heated pressed-powder
• 5 A plot of the area distribution in work function, i.e., the frac
tIon of the total area having work function between 4> and 4>+d4>
plotted as a function of 4>. ' ELECTRON BEAM
SCANNING TUBE
FIG. 1. Block diagram showing electron-beam scanning
tube and associated measuring circuitry.
samples in Ta cups as described in the previous paper.
The same precautions involving the use of the inert
atmosphere chamber, etc., were also used in this study.
The resulting pressed pellet with its attached heater
was mounted as a target (or anode) in an electron beam
scanning tube (Fig. 1).6 Incident on this target is an
electron beam 25 J.I. in diameter which can be scanned
over the entire target, a small portion of it, or it can be
made stationary to probe a particular spot on the sur
face. The electrons leave the cathode of the electron
gun, are accelerated to ~300 V, magnetically focused
and deflected and then decelerated just before striking
the target. This causes the electrons in the beam to
strike the target with near zero energy but still provides
a strong electric field in front of the target to reduce the
effects of patch field interaction. The energy of the beam
in relation to the target can be changed by adjusting the
voltage V as shown in Fig. 1. When the value of this
voltage7 is added to the work function of the cathode
(t/>c),8 the beam energy (EB)9 is given in relation to the
Fermi level of the target, by
(1)
As the beam strikes a certain spot on the target, the elec
trons are accepted or reflected depending on whether
their energy is larger or smaller than the work function
of that particular spot.
For dc measurements, where the beam probes the re
tarding potential characteristics of a particular spot on
the target, the amount of electrons that reach the tar
get can be measured by an electrometer. Figure 2 rep
resents retarding potential plots using this electrometer
technique made on four different spots on a UC sur-
6 This tube and associated circuitry is described in greater de
tail in a forthcoming NRL Report.
7 Actually V is multiplied by the electronic charge e. However,
in the units of energy used throughout this paper (eV) this factor
is just unity and is dropped.
84>. was determined from contact potential measurements using
Mo, Cu, and W targets and employing the published values of
work function of these.
9 EB represents the lowest energy of an electron in an idealized
Maxwellian distribution. This distribution is rarely found eiperi
mentally at low-electron energies and methods used to correct
for this are given.
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] IP: 129.24.51.181 On: Fri, 28 Nov 2014 02:42:09PAT C II E F FEe TIN IT l{ A ~ I U 1\[ CAR n IDE 3459
face.1o These spots were chosen to represent regions of
low to high work-function patches on the surface. It
was observed that below some threshold current Ith
(which occurs about one order of magnitude below the
saturated target current) a linear retarding region was
generally obtained in which the contact potentials
between various spots remained constant.
The beam energy required to give 110 saturated target
current EB(IIl,) can, therefore, be used to accurately
determine the work function <p of the spot on which the
beam is incident. This results in the relation,
(2)
where m is the slope of the retarding potential plot, and
the quantity 11m represents the difference in beam
energy in extrapolating one order of magnitude from
the threshold current to the saturated target current.
Since the threshold current was used to trigger the
various measuring systems to be described, all potentials
used in the following figures have been adjusted accord
ing to Eqs. (1) and (2) to read directly in terms of the
work function <p.
The method of obtaining all of the information in the
linear portion of the retarding region allows one to effec
tively probe the surface with electrons having a Max
wellian distribution in energy, since the low-energy elec
trons which are affected by patches are all reflected at
the beam energies (EB(lth) where the measurements
are made.
A visual picture of the work function variation is ob
tained by scanning the surface with the electron beam
and amplifying the ac output developed across the resis
tor R (Fig. O. This signal is then applied to the grid of
a Kinescope tube whose bias is adjusted so that the
transverse from the black level to the white level occurs
within a space of approximately t V in the region of Ith.
Since the sweep circuits of the Kinescope are syn
chronized with those of the scanning tube, a picture is
observed showing as white areas all regions of the
scanned portion of the target having work functions <p
or less.
Figure 3 shows such Kinescope pictures of a 2 Xl 0-3
cm2 portion of the UC surface taken at four different
beam energies. This surface represents a carbon-rich
unactivated sample identical to those described for the
thermionic studies in Fig. 6 of the previous paper. It is
seen that less than lo of the surface is covered by regions
having work functions <p:S3.25 eV, "'t with regions of
<P:S 3.5 e V, "'! with regions of <P:S 3.75 e V, and "'i with
regions of <P:S 4.0 e V. This is in good agreement with re
sults obtained for this same inactive state with the
thermionic measurements. The spots A, B, C, and D
identified in Fig. 3 (c) are those on which the retarding
potential plots of Fig. 2 were made. Since all Kinescope
10 Figure 2 represents a UC surface in a poisoned state and this
aspect is discussed in detail later. The plots are included here for
illustra tive purposes. FIG. 2. Retarding poten
tial plots obtained on four
different spots· of a UC
surface, three days after
activation. -8
-9
-12 EB leV)
40 5.0 6.0
pictures of this paper are of the same region of the UC
surface, spot A, B, C, etc., for any of the Kinescope pic
tures represents the same point on the surface.
Figure 4(a) shows this same surface taken at a beam
energy corresponding to <P:S 3.25 eV two months later.
The similarity between this and picture 3(a), which
was taken under identical conditions, indicates the
stability of the inactive UC surface. Figure 4(b) shows
this same surface after heating to 1l00°C for 2 min.n
The surface is now nearly half covered by regions having
<P:S 3.25 eV. Figures 4(c) and (d) show how the surface
becomes progressively covered with regions of <P:S 3.25
eV after a 2-min heat treatment of 1250° and 1400°C,
respectively. Figure 4(d) indicates that after the 1400°C
heat treatment the surface is almost entirely covered
with regions of work function <P:S 3.25 eV. Figure 4(e),
which was taken immediately after 4(d), but with a
beam energy t eV lower, shows that only a very small
fraction of the surface is covered by regions having
<p:S3.0 eV. Hence, during heating, the UC surface
appears to change from a very patchy surface that
ranged between 3.25 eV to over 4.0 eV, to an activated
surface which has a work function distribution largely
in the region of 3.0 to 3.25 eV. This is in very good agree
ment with the thermal activation phenomena observed
by the thermionic measurements previously described.
Figure 5 shows the UC surface taken 3 days after the
activation illustrated in Fig. 4. Unlike the case of the
inactive UC, it appears that once the UC is activated,
it is readily poisoned by residual gases in the vacuum
tube.12 It is seen that the surface now ranges in work
11 All surfaces illustrated here were scanned at room temperature.
The elevated temperatures represent heat treatments conducted
between scannings.
12 The pressure in this electron-beam scanning tube was in the
10-7 mm of Hg range and can be attributed largely to insufficient
outgassing of the metal parts. Rigorous rf outgassing was not un
dertaken in order to prevent premature activations of the UC.
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(h) </> =3.5 eV
(e) </> =3.75 eV Cd) </> =4.0 eV
FIG. 3. Kinescope pictures obtained at four different beam energies of a 2X 10-' cmZ portion of an unactivated UC
surface. The white areas are regions which have a work function of q, or less.
function between approximately 3.75 to 4.5 e V which is
even higher than it was in the inactive state. This state
of deactivation (which is still not the most "poisoned"
state as is seen later) is the same state for which the re
tarding potential plots of points A, B, C, and D in Fig.
2 were made. Note the agreement between the values of
work function obtained from the extrapolated retarding
potential curves and those indicated in the Kinescope
picture of Fig. 5. While the Kinescope pictures show a
good spatial configuration of the patches, it is desirable
to obtain more detailed and accurate information re
garding the distribution of these various patches accord
ing to work function since this is the important param
eter that determines the emission properties.
The fractional area covered by patches of work func
tion cf> or less can be obtained from the photographs by
measuring the fraction of the photograph that is covered
with light areas. Subtracting, for example, this value for
Fig. S(a) from S(b) gives an indication of the contribu-tion to the total area caused by patches having a work
function between 3.7 Sand 4 e V. By continuing this proc
ess for photographs taken at progressively higher beam
energy, the area distribution in work function can be
ascertained. A more accurate and less complicated
method of obtaining this information can be accom
plished electronically in the following manner.
The fractional area is first obtained by making a single
scan of the target at a given beam energy and measuring
the fraction of the time that the threshold current Ith
is reached or exceeded. This can be accomplished by
having a constant current source charge the capacitor
C (Fig. 1) each time Ith has been reached or exceeded
as the beam scans over the target area. The voltage to
which the capacitor is charged at the end of the sweep
is then recorded for different beam energies. Such a plot
for the surface described in Figs. 2 and 5 is given in Fig.
6(a). Here the fractional area covered by regions of cf>
or less is represented as a continuous curve and would
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IP: 129.24.51.181 On: Fri, 28 Nov 2014 02:42:09PATCH EFFECT IN URANiUM CARBIDE 3461
Ca) Initial, 1> :3.25 eV (b) llOOce, 1> :3.25 eV (e) 1250ce. 1> =3.25 eV
FIG. 4. Kinescope pictures showing the
thermal activation of a GC surface.
(d) 1400oe, 1> :3.25 eV (e) 1> =3.0 eV
correspond to measuring a particular shade of grayness
of over 200 photographs of the type shown in Fig. 5.
[The actual curve is much more detailed than that pro
duced in Fig. 6(a).J
The area distribution in work function is then ob
tained by taking the derivative with respect to cp of the
fractional area curve. The resulting curve (which is
called a patch distribution plot) is given in Fig. 6(b)
and shows a definite reproducible detail concerning
various peaks at certain work functions. Note, for ex
ample, that three times as much of the surface is covered
by regions having work functions in the vicinity of 4.25
eV than is covered by regions having work functions
near 4.0 eV.
It is also of interest to compute how the distribution
in thermionic emission from such a surface would look.
For strong-field thermionic emission, this can be done by
multiplying the curve in Fig. 6(b) by the quantity
e--1>/kT. When this is done the curve of Fig. 6(c) is ob
tained for temperatures in the region of 1500°K. It is
seen that the majority of the emission is obtained from
4.0-eV regions rather than the 4.25-eV regions even
though these latter ones occupy much more of the sur
face. Perhaps, the most pertinent fact is that the ma
jority of all of the emission comes from regions which
occupy only a small fraction of the surface. For example,
the electrons coming from the 4-eV regions originate
from less than 110 the total area of the cathode [Fig.
6(a)]. Consequently, the effective electron emitting
area can be in reality much smaller than the actual sur-face area. This can cause the constant A as measured
from a Richardson plot to appear appreciably smaller
than 120 AI cm2deg2 even wi thou t considering a tempera
ture coefficient of the work function. These problems are
discussed in greater detail in the following section as well
as in the Appendix,
Figure 7 shows how the patch distribution can change
by activation and deactivation phenomena. Note that
the broad distribution in work function for the inactive
statel3 decreases to a very narrow band in the 3.0 to
3.25-eV region when measured within 2 min after activa
tion at 1400°C. After about 15 min, the whole pattern
has already shifted ~~ eV to the higher work function
region and appears to have broadened slightly. This
shift to higher work function continues, and after 18
days the patch distribution has increased to a value sub
stantially higher than it was prior to the initial activa
tion. It is furthermore observed that the over-all char
acteristics of the distribution curve are retained in that
all peaks increase at nearly the same rate and their rela
tive order of height seems to stay the same.
Although the over-all distribution in work function
does not appear to change appreciably, there are slight
variations noted in the activation-deactivation phe
nomena of various individual spots. For example, prior
13 The capacitor integrating circuit ,vas not completed at the
time the inactive state was investigated. The dashed line, which
is inclnded for comparison, was obtained by the earlier technique
of analyzing photographs and, therefore, lacks the detail seen in
the other curves.
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IP: 129.24.51.181 On: Fri, 28 Nov 2014 02:42:093462 G. A. H A A SAN]) R. E. THO 1\1 ,\ S
(a) q, =.3.75 eV (b) q,=4.0eV
(e) q, =4.25 cV Cd) '" =4.5 eV
FIG. 5. Kinescope pictures at four different beam energies showing a DC surface three days after activation.
to activation, spots A and E both have a work function
of cp~3.2S eV [Figs. 3(a) and 4(a)] while spot F has a
work function of cp"-'3.S eV [Fig. 3(b)]. After activa
tion however [Fig. 4(e)], spots A and F have both been
(a)
(b)
(e) I
3.5 rL:"'%
1°%·
4.0 <p(el/) 4.5 FIG. 6. Distribution plots
obtained from a DC sur
face three days after activa
tion showing (a) the frac
tional area having work
function if> or less (b) the
area distribution in work
function, and (e) the emis
sion distribution in work
function. The arrows labeled
A, .H, C, and]) represent
the pusitiulls uf the fuur
*uts on these distributiun
curve::::., reduced to 3.0 eV, but spot E does not seem to have been
activated quite to this extent and is not observed at this
beam energy. After poisoning, however [Fig. 5 (a)],
spots A and E again have the same work function with
spot F just slightly higher. The rate of activation also
seems to be different for different spots. Spots F and G,
for example, are both somewhat higher in work function
than A and E prior to activation [Fig. 4(a)]. However,
spot G quickly activates at lloooe to ~3.25 eV
[Fig. 4(b)], while spot F is not observed at this beam
energy until after the 12500e heating [Fig. 4(c)]. After
heating to 14oooe, both spots F and G on the other
hand have gone from a state that is Jess active than
spot E, to one that is more active [Fig. 4(e)].
In aJl of the Kinescope pictures amI patch distribu
tion pluts presented so far, the same portiun of the tar
get was always scanned. It is of interest, however, to
see what the patch distribution might look like at the
very edge of a patch. For this study, the surface was
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scanned about a 2X 10-5 cm2 region at the lower right
corner of spot A. This area takes in a portion of a high
work-function patch adjacent to spot A as well as a
portion of spot A itself. Figure 8 is a graph of the dis
tribution in work function from the edge between these
two patches. This curve, which was obtained when the
surface was in a contamihated state, was taken directly
from a recorder and was obtained bv electronic differ
entiation of the type curve shown in 'Fig. 6(a). The fine
structure o~ the curve is the result of the finite intervals
at which the scannings took place. These intervals rep
resent successive beam energy steps of ~7 MV. The
threshold sensitivity (voltage change required to acti
vate the threshold circuit) was 1 MV. From the appear
ance of the two distinct peaks of work function, it is
seen that the edge is well defined rather than smeared
out in a continuous distribution. These adjacent patches
having abrupt changes in work function of ~0.6 eV can
easily give rise to the type patch fields observed in the
thermionic studies.
DISCUSSION AND CONCLUSIONS
The electron beam scanning technique used for the
measurements described in this paper represents a new
approach designed to study the nonuniformity in work
function of a surface. Errors arising from patchiness in
herent in the measuring system as well as patch field
interaction have been minimized by obtaining all in
formation from the deep-retarding region, by looking at
only a 25-J.I. region of the surface at a time, and by apply
ing a field to the surface which is large with respect to
most of its patch fields.
While the resulting resolution in work function is
probably better than 0.1 eV, the absolute value should
not be considered accurate to better than 0.2 or 0.3 eV
because of attendant difficulties associated with measur
ing the work function of the reference target. Further-
FIG. 7. Plots of
the area distribu
tion in work func
tion for inacti ve, acti
vated, and various
poisoned states of
DC. z o
1= u
~
l.L
" a: o 3
~
z o
i= m
~
is
<t W
!a 3.0
i
3.0 3.5 '" (eV)
4.0 4.5
_--------' INACTIVE - '-
"'2 MIN
15 MIN
30 MIN 4.0 4.4 4.6 4.8 5.0 5.2
¢ (eV)
FIG. 8. Plot of the area distribution in work function obtained
from ~ 2XlO~5 em' region ~t the patch edge between spot A and
an adjacent high work functIOn patch. (Plot obtained three months
after last activation.)
more, because of the finite beam size, patches smaller
than ~25 J.I. are not discernible except as an average
with their neighbors in a 25 J.I. region. Despite these
limitations, however, the resulting data are in very good
agreement with those obtained on identical samples of
UC studied by thermionic emission measurements, and
in addition provide far more information regarding the
nonuniformity of the surface than can be obtained by
the thermionic approach.
The results show that inactive carbon-rich UC has a
wide distribution of work function in the 3.25 to 4.5 eV
range and is relatively insensitive to poisoning. When
the DC is heated to 1400°C, it becomes covered with a
surface having a work function in the narrow band of
3.0 to 3.25 eV and is now easily poisoned by ambient
gases at pressures in the 10-7 mm of Hg region. This
activation is in agreement with the formation of a ura
nium film on the surface as postulated from the thermi
onic measurements. The distribution in work function
is not of a simple Gaussian type but rather appears to
have peaks for several work function values. The simi
larity of these peaks to those obtained when other por
tions of the surface were scanned suggests that these
peaks probably represent various crystal faces of UC
rather than specific patches located within a given
scanned area.
Upon continuous exposure to ambient poisoning
gases (e.g., oxygen), the average value of the work
function of the surface increases but the distribution re
mains fairly constant. If, as is often assumed, the change
in work function is dependent on the number of ab
sorbed atoms, the results imply that the uranium film
picks up oxygen at a rate which is not too dependent on
the underlying UC crystal face.
The final work function for the oxygen-covered sur
face is substantially higher even than the pure UC sur
face before activation. Upon reheating this poisoned
surface, an active surface can again be reached which
poisons in a similar fashion as before. Such a reactivated
surface, however, is not quite as low in work function
and does not seem to show the pronounced peaks as ob
served earlier. The apparent differences in activation -
deactivation effects noted for various individual regions
(spots A,E,F,G) might, among other considerations, be
also influenced by geometrical factors such as being
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located near pores where U effuses more rapidly than at
other places. These differences, however, tend to
statistically average out and their resultant effect on
the overall distribution is too slight to be interpreted.
A brief synopsis of the effect that a nonuniform work
function distribution has on the data obtained by
Richardson plots (or effective work-function plots) is
given in the Appendix.14 The results indicate that an in
terpretation from such plots is not valid without some
knowledge of the distribution in work function of the
surface. Using the results of the patch distribution
plots of this paper, one can see that for active DC
(which exhibits a half width in the patch distribution
curve of about 0.15 eV), the Richardson constants q,R
and A R as obtained in the previous paper are well
within the experimental error of those that would be
obtained from a single work function surface. The same,
however, cannot be said for Richardson plots (or
effective work-function plots )15 for the more patchy
state using either low field or strong field methods of ob
taining the current density. For such cases, these plots
still provide a good indication of various properties
like the emission capabilities or activation effects, etc.,
of a surface but cannot give an accurate determination
of the surface work function.
APPENDIX. EFFECT OF A PATCH DISTRIBUTION
ON MEASURED RICHARDSON CONSTANTS14
The evidence for a distribution in work function
(rather than a singular value), which is presented in
this paper, gives rise to the speculation of the accuracy
obtainable by thermionic measurements (like those of
the previous paper or other measurements) in de
scribing the average electron barrier of the surface.
There are two regions of applied field for which a
theory can be applied in analyzing the thermionic emis
sion of a surface, namely, the zero field region (case II
collecting fields16) or the strong field region where the
external field is much higher than the localized patch
fields (case I collecting fieldsI6). The intermediate re
gion (case III collecting fieldsl6) is difficult to interpret
since a detailed knowledge of the type of patches (size,
shape, work function distribution, etc.) is required.
The first of these approaches assumes that at zero
fields, the barrier maximum is so far from the surface
that all patches having a work function below the area
averaged work function see a barrier maximum in front
of them equal to the area averaged work function, while
the rest see a barrier maximum equal to their individual
work function. Among other simplifications, this model
neglects a localized space-charge effect in front of low
work function patches, and that, especially for small,
low work-function patches, the barrier maximum can
14 A complete analysis of this work is given in a forthcoming
NRL Report.
15 E. B. Hensley, J. Appl. Phys. 32, 301 (1961).
16 C. Herring and~M. H. Nichols, Rev. Mod. Phys. 21, 185
(1949). I
4
1.0
"" "-
0.9 ~ --
i
I 0.8
0.7
0.5 ;--;---f------~ rt-I-t+--It f---
-'--C-I-+-- ~'--~-
·-;-r ~~ .. rclt_ft~ RJ -tr- .~ +-Li-' . ++--tt-t-t-
I ~Itl.j.+r
-------~ . +-.. t- 1--.--+----i--h-,--~,I I
-I-
1.5 -~tt-t-----'--1. Oz,
I '1IF~--i -rf::-~
2.5
A ¢h ---.r 3.5 4 4.5
FIG. 9. Plot of gSF and gZF as a function of Mh/kT. The quantity
gSF is the ratio of the extrapolated strong field current to the cur
rent expected with no patch distribution. The quantity gZF is the
similar ratio for the zero field case and A<I>h is the Gaussian half
width of the patch distribution. Note the change in scale of the
ordinate.
be substantially larger than the area averaged work
function. Both of these factors cause the actual current
near zero field to be appreciably smaller than that pre
dicted by this theoryP
The strong field necessary for the second approach
eliminates many of the problems of patch-field inter
actions and permits the use of a model which as
sumes that the barrier maximum in front of each patch
is equal to the work function of that particular patch
(modified by a Schottky lowering of the barrier). Since
this assumption can be verified to some degree by the
observation of a linear region of proper slope in a
Schottky plot (see, for example, Fig. 1 of the preceding
paper), the second approach was chosen for the experi
mental measurements described in the previous paper.
The effect of a patch distribution in altering the
measured Richardson constants q,R and AR or the effec
tive work function q,eff15 for both of these cases can be
computed for certain simplified distributions in work
function. For a Gaussian distribution about some value
¢, the zero field emission can be written as
J ZF= (120e~"lk]'2e-~olkT)
(Al)
= JOgzF,
where ct is the temperature coefficient (i.e., ¢=¢o+aT)
which is also assumed to be constant for all patches, k
is the Boltzmann constant in eV jdeg, and ACPh is the
Gaussian half-width. The value Jo= 120e-"lk]'2e~~olkT is
17 This fact has also been suggested experimentally on DC
wherein the actual low field current appeared sUQstantially lower
than that computed from the measured patch distribution.
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the theoretical emission current for a surface having a
work function ¢ and gZF, the term in the brackets, rep
resents the correction caused by a distribution of patches
about 4'>. Similarly, the strong field emission extrapolated
to zero field gives
J SF= 120e-aW['2e-<I>olkT{ e(0.306<1> k1kT)2} (A2)
= JogSF•
A plot of gZF and gSF is given in Fig. 9 as a function of
Aq,h/kT.
These curves are accurate within 3% for values of
for the extrapolated strong field case and all values for
the zero field case (neglecting of course negative values
of q,). From these curves, it is seen that the extrapolated
strong field emission is always larger than the theoretical
single work function value, while the zero-field emission
is always smaller. For example, at 11600K and a half
width spread in work function of 0.2 V, the extrapolated
strong-field emission is "" 1.510, whereas the zero-field
emission is ""0.781 0• However, as mentioned previously,
one would expect other effects to give an experimental
value of the zero field current much lower than this.
From Eqs. (Al) and (A2) and the curves of Fig. 9, it
is possible to see what effect a distribution in work func
tion might have on the various values measured by
empirical thermionic methods. The effective work func
tion15 is now given by
(A3)
where g stands for either the strong-field or zero-field
value. The extrapolated strong field Richardson con
stants q,R(SF) and AR(SF) are given by:
q,R(SF)=¢O- (2.11· lO3/T) (Aq,,,)2, (A4)
and
AR(SF)= 120e-a/ke-(0.306<1>hl kT)2. (AS)
It should be noted that both of these extrapolated
strong-field Richardson constants increase with increas
ing temperature, and should, therefore, be evaluated at
some mean experimental temperature if a comparison
is to be made with empirically determined Richardson
constants. The zero-field Richardson constants are not ~
'\
'\
'\
\;
\
FIG. 10. Plot of a double Gaussian distribution in work function
and single Gaussian distribution (dashed line) which gives the
same emission values within 5% for T = 12500 to 2000°K.
so easily represented by an analytical value but can be
obtained graphically by use of Fig. 9.
In the calculations presented so far only a simple
Gaussian-type distribution has been discussed, whereas
the empirical patch distributions observed by the elec
tron beam scanning technique generally showed a non
Gaussian behavior. In Fig. lo, on the other hand, is
plotted a double Gaussian-type distribution curve which
is fairly representative of the type assymmetry noted
experimentally, and a corresponding single Gaussian
curve (dashed line) which gives the same emission
characteristics (within ",,5% between 1250o-20000K).
It is, therefore, seen that the more complex type dis
tribution curves measured in this paper can be analyzed
without appreciable error in terms of an equivalent
simple Gaussian distribution.
An analysis of Eqs. (A4) and (AS) shows that for
practical temperatures, an appreciable error arises in
interpreting results by means of a Richardson plot if a
half-width spread in work function of ""1 eV or larger
exists. For example, for a temperature of "" 15000K and
a half-width of 0.15 eV, which corresponds to the active
UC case, the measured Richardson work function is
only 0.0315 eV below the area averaged work function
and the apparent A value is about 11% too low. How
ever, for the poisoned case when the half width can be
0.35 eV (or larger), q,R is about 0.17 eV lower and AR
is ""t the value expected without a distribution in the
work function.
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