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1.1750721.pdf | Conductivity in Insulators and Its Interpretation
A. von Hippel
Citation: The Journal of Chemical Physics 8, 605 (1940); doi: 10.1063/1.1750721
View online: http://dx.doi.org/10.1063/1.1750721
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IP: 130.102.42.98 On: Sat, 22 Nov 2014 20:35:58AUGUST. 1940 JOURNAL OF CHEMICAL PHYSICS VOLU~E 8
Conductivity in Insulators and Its Interpretation*
A. VON HIPPEL
Electrical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts
(Received April 17. 1940)
The migration of charge carriers in insulators is discussed in this paper on the basis of a larf!:e
amount of experimental material on hand. Beginning with deviations from Ohm's law, the
trapping of ~harge carriers is considered and the formation of the" F" band in alkali-halide
crystals. Then the laws of electronic conductivity in solid dielectrics are formulated.
THE old conception of insulators as "non
conductors" of electricity is dead. It
became undermined by accumulating experi
mental evidence demonstrating that conductivity
can be created in insulators if only the proper
conditions are chosen. These conditions are
manifold and changing from case to case; time
and temperature, field strength and illumination,
structure and previous history, even the sur
rounding atmosphere and the contacting elec
trodes prove to be of importance. The definition
"insulator" has become vague; it can mean any
thing but a metal. Consequently theoretical phys
ics defined insulators directly as "nonmetals"
with the help of the zone structure of the elec
tronic levels.1 But this is not sufficient; detailed
knowledge is needed about the elementary proc
esses providing mobile charge carriers and about
the laws regulating their motion. These laws of
motion are discussed in the following pages.
DEVIATIONS FROM OHM'S LAW
The concept of conductivity is normally based
upon Ohm's law. It implies constant density of
the charge carriers and assumes that they move
in the field direction with an average velocity
v=bE (b=mobility) (1)
(E = field strength).
This assumption of a friction factor l/b inde
pendent of the driving force (1) has proved in
valid lately in several cases.
M. Wien.2 measuring the conductivity A of
* Invited paper presented before the Division of In
dustrial and Engineering Chemistry at the Spring Meeting
of the American Chemical Society, Cincinnati, 1940.
I See, for instance, F. Seitz and R. P. Johnson, J. App.
Phys. 8, 84, 186 (1937).
2 M. Wien, Ann. d. Physik 83, 327 (1927); 85, 795
(1928). electrolytes with high impulse voltages (105
volt/cm), found an increase of A with field
strength to a final value Aoo, which corresponds
to the equivalent conductivity for infinite dilu
tion. This fact was explained3 with Debye's con
cept of the ionic atmosphere: Each ion in solution
surrounds itself preferentially with ions of the
opposite sign, with a space-charge cloud char
acterized by a radius and a relaxation time. At
the high voltages used, the mobility increases be
cause the ions begin to move faster than their
atmosphere can form.
Another deviation from Ohm's law was found
by the author4 while studying the laws of electric
breakdown in ionic single crystals. In metals the
number of free electrons about equals that of the
atoms, local space-charge effects cannot develop,
and Fermi statistics result. Consequently the
electrons have a long free path between inelastic
collisions because the coupling with the lattice
ions is weak, the time of interaction is short, and
the number of free final states into which the
electrons can go is limited. In nonmetals such as
the alkali-halides the situation is quite different.
If surplus electrons exist, their concentration is
low; they are slow, because Boltzmann statistics
apply, and act as space charges distorting the
surroundings. The electrostatic coupling between
electron and structure is strong and far-reaching;
the electrons are in a state of permanent interac
tion with the lattice. The probability of exciting
lattice vibrations dominates the friction factor
l/b of Ohm's law. This probability may be
roughly described by an excitation function co
ordinated to the spectrum of vibrations of the
3 G. Joos and M. Blumentritt, Physik. Zeits. 28, 836
(1927); G. Joos, ibid. 29, 755 (1928).
4 A. von Hippel, Zeits. f. Physik 75, 145 (1932); J. App.
Phys. 8, 815 (1937).
60S
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IP: 130.102.42.98 On: Sat, 22 Nov 2014 20:35:58606 A. VON HIPPEL
BREAKDOWN EFFECT
FELO~H,E
FIG. 1.
material. The interaction time between electron
and surroundings decreases with increasing veloc
ity of the electron; therefore the function passes
over a maximum above a critical field strength,
the friction decreases, and acceleration, impact
ionization, and breakdown result. 5
TRAPPING OF CHARGE CARRIERS
The two deviations from Ohm's law discussed
are interrelated, as Fig. 1 illustrates. Debye's
ionic atmosphere can be pictured as a potential
minimum floating along with the ion under con
sideration. It represents a polarization of the
medium, which is normally slight because the
number and mobility of the charge carriers of
opposite sign are about equal. In extreme cases,
on the other hand, it can become very large, as
the movement of protons in palladium indicates;6
the swarm of electrons around the H+ in the
metal is so dense that the ion responds only with
-./" of its charge to an outside field. Wien's effect
indicates that the potential cup flattens out in
electrolytes when the ions reach velocities of
meters per second.
This final state of ionic migration is, thanks to
the small electron mass, more or less the initial
state of electronic motion. The charge carriers
move along, distorting temporarily the electron
clouds of the molecules; the nuclei tending to
swing into new equilibrium positions set up sound
waves.7 While the heavy ions stay practically at
the lower end of this friction barrier of vibrations,
6 The quantitative theory is being developed in the last
years by H. Frohlich, R. J. Seeger, and E. Teller, W.
Franz and F. Seitz, but has not yet reached a final form.
• J. Franck, Nach. Ges. der Wiss., Gottingen 44, 293
(1933); B. Duhm, Zeits. f. Physik 95, 801 (1935).
7 A calculation of the longitudinal vibrations excited is
given by E. Teller and his co·workers, Phys. Rev. 57.
1084A (1940). the electrons in high fields may pass it (break
down effect).
The space-charge poten tial around ions in
electrolytes is the first indication of ionic binding;
it comes to a full development in lattice struc
tures like rocksalt, where each charge carrier is
surrounded by six opposite ions and kept locally
trapped in a potential cup several electron volts
deep. Figure 1 suggests that also around electrons
a polarization of the medium should take place if
they are slow enough. In crystals with strong
binding forces they do not even need to be very
slow, because the ions respond quickly. Hence
also electrons should be trapped if they are
strongly coupled to locally bound particles.
Such a capture of a surplus electron in a NaCl
lattice, for instance, may be described quite
graphically: An electron traveling with the
kinetic energy of 1/10 e volt has a wave-length
}.,=h/mv=3.8 10-7 em, and acts therefore like a
space charge of one elementary charge spread
over a volume }.,3 containing about 2500 lattice
ions. The distortion produced is small but far
reaching, thanks to the slow decay of the Cou
lomb potential with distance. The ions respond
with a top frequency higher than 5.7 1012 vibra
tions per second ("Reststrahl"-frequency); there
fore the ·distant ions have time enough to be
somewhat displaced by the electron speeding
with a velocity of 1.9.107 em/sec. through the
structure. A small potential cup begins to form
around the electron; it slows down by dissipating
energy into lattice vibrations. But the Coulomb
field springing up around the electron, thanks to
the lattice distortion, increases the total kinetic
energy of the electron because it must oscillate or
rotate in this field with a kinetic energy equaling
half the potential energy.8 Therefore the wave
length shortens, space-charge density and inter
action time increase, the potential cup deepens
into a local trap, and the electron becomes
localized around a positive ion of the lattice.
THE "F" BAND OF THE ALKALI-HALIDES
If this picture of electron capture is correct,
an optical absorption band should appear, indi
cating by its location the position of the surplus
electrons in the material and by its color the
8 N. Bohr, Phil. Mag. 26, 24 (1913).
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IP: 130.102.42.98 On: Sat, 22 Nov 2014 20:35:58CONDUCTIVITY IN INSULATORS 607
energy needed for their photoelectric liberation.
Now Pohl and his co-workers9 have found an
absorption band in the alkali-halides, the "F"
band, which can be created by photo-effect, heat
treatment in alkali vapor, or electric current at
high temperature. This band moves in the electric
field like an electron cloud, as Stasiw10 first found,
and has been identified by the author with the
trapping band he expected.ll But there is a
strong controversy at the moment about the cor
rectness of this model; another possibility exists,
which we shall consider next, because the discus
sion of it introduces concepts of importance.
Landau12 first proposed a quantum-mechanical
trapping mechanism for electrons, taking the
standpoint that slow electrons cannot be trapped
on account of their long wave-length, but that
electrons might be captured if speeded up suffi
ciently by an activation energy. The author
came, without knowing of this paper, to the op
posite conclusion-that slow electrons will be
trapped by the mechanism outlined above. By
comparing both processes it can be seen that the
Coulomb field of the distorted surroundings per
forms the service of the activation energy to
shrink the wave-length of the electron. Gurney
and Mott13 took first our standpoint-that an
electron can be trapped in the ideal lattice-but
later they discarded this modeP4 in favor of one
Distance from missing ion_
+
FIG. 2. Potential energy of an electron in the neighbor
hood of a missing negative ion according to de Boer and
Mott. The dotted lines give the curve -e2/Kv.
9 See, for instance, R. W. Pohl, Proc. Phys. Soc. 49, 3
(1937).
10 O. Stasiw, Nach. Ges. Wiss., Gbttingen, No. 50 (1933).
11 A. von Hippel, Ergebnisse d. exakt. Naturwiss. 14,
113 (1935); Zeits. f. Physik 101, 680 (1936).
12 L. Landau, Physik. Zeits. Sowjetunion 3, 664 (1933).
13 R. \V. Gurney and N. F. Mott, Proc. Phys. Soc. 49,
32 (1937).
14 That the reason given at the Bristol Conference
(Proc. Phys. Soc. 49, 36 (1937» is invalid, has been pointed
out already (A. von Hippel, J. App. Phys. 8, 832 (1937». given by de Boer.15 De Boer assumes that an
electron is captured at a lattice point where a
negative ion is missing (Fig. 2). That such vacant
lattice points exist can be concluded from the fact
of ionic conduction. The ions in an ideal lattice,
like rocksalt, fill out the crystal structure by
very tight packing; their activation energy is
much too high to explain the large conductivity
observed. There must be defects in the structure,
such as ions in interstitial positions or transferred
N,' c,· No' c,-N,' c,-N,' CI-N,' CI-D D e c,-Na' el-Na+ CI-No' CI-D CI-N,' c,-Na'
N,' CI' D CI-N,' CI- No' CI-No' D Na' m-
CI-No' c,-No' c,-N,' c,-N,. c,-N,' CI-Na' e N(I+ CI-N,' CI-No' c,-N,' c,-N,' c,-D CI-
c,-No' c,-D CI-N,' c,-N" D N,' c,-No'
H,.I N"-oono_tdltllo,nI.,.HMII,o"IIOII. ,,..4. E ... I __ of N4< OIMI tl-__ in
FIG. 3. Lattice disorder.
to the surface, leaving vacant places (Fig. 3) for
ionic migration.16 In NaCl, in thermal equi
librium at 1000oK, about 10-5 of the lattice
points should be disordered to account for the
ionic mobility measured. Frenkel, Wagner, lost,
and Schottky mainly have developed this
theory, and Schottky and others17 have calcu
lated that in the alkali-halides the dominant kind
of disorder is his "Type 4" of vacant places
(see Fig. 3).
It is of importance to establish which of the
two models given above represents the" FH band,
because a deeper insight into the properties of
ideal and defective crystal structures is involved.
From the standpoint of our model, a decision
might be reached by the following cycle process
(Fig. 4): Given the ideal lattice of NaCl and a
migrating surplus electron in its conduction
band. Now (1) remove one Na+ ion into vacuum
by spending the lattice energy Uo= 183 kcal.
minus a polarization energy Up, because the lat
tice around the hole becomes distorted and settles
down into a new equilibrium position. Up, ac-
15 J. H. de Boer, Rec. Trav. Chim. Pays-Bas 56, 301
(1937).
16 J. Frenkel, Zeits. f. Physik 35, 652 (1926); C. Wagner
and W. Schottky, Zeits. f. physik. Chemie 11, 163 (1930);
W. Jost, J. Chern. Phys. I, 466 (1933); W. Schottky,
Zeits. f. physik. Chemie 29, 335 (1935).
17 W. Jost and G. Nehlep, Zeits. f. physik. Chemie 32, 1
(1936); N. F. Mott and M. J. Littleton, Trans. Faraday
Soc. 34, 485 (1938).
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IP: 130.102.42.98 On: Sat, 22 Nov 2014 20:35:58608 A. VON HIPPEL
+ + +
+ + +
FIG. 4. Cycle for calculation of release energy. (F band.)
cording to Mott and Littleton,17 amounts to
about 77 kcal. (2) Remove the electron from the
crystal by spending the energy of the work func
tion X = 11.5 kcaJ.ls (3) Combine electron and
ion, gaining the ionization potential J = 118 kcaJ.
(4) Place the sodium atom back into the hole; a
small energy gain n in the order of 37 kcal. re
sults.IS (5) Remove the electron by light absorp
tion back into the conduction band; the quantum
hVt necessary should represent the energy of the
"F" band. Result:
hVt= -Uo+ Up-x+I +n=37.5 kcal.
= 1.63 ev (2)
as compared with the experimental value 2.73
ev.19 This difference can be in the limits of error,
and the value calculated indicates that an elec
tron will be trapped in the ideal lattice, but a
decision between the two models can only be
made by new experimental evidence. We are
engaged in this study.
ELECTRONIC CONDUCTIVITY IN SOLIDS
Conductivity in solid insulators, as the preced
ing discussion shows, is normally not produced
by a free flow of charge carriers through a resist
ant medium but by a progression of the charges
in jerks. The carriers are trapped after an average
displacemen t distance 'II!; they stick un til the
heat vibration of the structure supply with
statistical probability the activation energy for
their release.
18 N. F. Mott, Trans. Faraday Soc. 34, 500 (1938).
19 E. Mollwo, Zeits. f. Physik 85, 56 (1933). For electrons in crystals this conduction
process can be fol1owed up step by step in direct
experiments. The liberation of the charges can be
produced photoelectrically by a narrow light
beam; if the insulator is kept cool, no second re
lease of the electrons after capture will take place.
The current measured as function of distance
between light beam and anode (Fig. 5) reflects
directly the decay of the number of charge car
riers with the distance of migration.20 An expo-
Pllolo Curt'nt
in 10-13 Amp. -Width of hgltbeom
10 1---.,-----r=::::=-,---r:'~312;--,
~
+ 208
104
I 2 •
X, di,tanc, from onode to li9hl beam, in mm.
FIG. 5. Measurement of the displacement distance of
electrons in AgCl at -186°C.
nential decay law has been found,
(3)
the displacemen t distance 'II! increases propor
tionaIly to the field strength applied and has a
value 'lI!o of 2.5.10-5 cm in AgCI, and of about
10-7 cm in NaCl at a field of 1 volt/cm.21 The
large difference between the two crystals of
identical lattice structure is to be expected from
the picture of electronic migration given in the
preceding paragraphs. While the alkali halides
are strongly ionic crystals coupling the electron
intensely to an extended volume of the material,
the silver halides have a large portion of atomic
binding, and a much weaker and more limited
interaction results. Values for the free path of
surplus electrons found by "Hal1-effect" meas
urements give additional evidence. No effect
could be found in NaCl22 indicating that the free
path was below 10-7 em, while in diamond
crystals a large free path of 10-6 cm has been
observed.23 It is quite possible that in atomic
structures like diamond the coupling is so weak
20 W. Flechsig, Physik. Zeits. 32, 843 (1931); K. Hecht,
Zeits. f. Physik 77,235 (1932).
21 G. Glaser, Nach. Ges. der Wiss. Giittingen, 3, No.2
(1937).
22]. Evans, Phys. Rev. 54,47 (1940).
23 H. Lenz, Ann. d. Physik 77, 449 (1925); 82, 775
(1927).
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that electrons are not trapped in the free volume
but only on boundaries or impurities; such
materials should also have a comparatively low
breakdown strength if they do not possess elec
tronic excitation levels as a second barrier
against impact ionization.
The velocity of the surplus electorns in alkali
halide crystals can be measured by observing the
motion of the color cen ters forming the" F" band
or by calculating the density of the centers and
the electronic current transferred.24 An exponen
tial temperature dependence
(4)
has been found in this way which can be easily
understood. The outside observer, integrating
over the intermittent motion of the charge car
riers records the average velocity
(5)
where tt marks the mean time of capture at one
point, and t, the mean time of free travel between
two capture acts. The time t"t is ended statisti
cally by the temperature motion of the crystal
supplying the activation energy U for release of
the electron
(6)
tl is normally «tt; therefore the form (4) results
v= (wla)· e-U/kT (7)
and the conductivity can be written
A= N·e· (wo/a)· e-U/kT
(N=number of surplus electrons per cm3). (8)
The factor a measures the degree of coupling
between the trapped electron and its surround
ings; 1/ a can be interpreted as the frequency of
oscillation produced by this binding. It should be
appreciably lower than the "Reststrahl" fre
quency of the lattice points because the size of
an alkali atom is large; therefore the electron will
be shared in the trapped state by one central ion
and a group of neighbors. With Smakula's
values for NaCl:24 U=0.94 ev, vo=20 cm/sec./
volt/cm, T=973°K; and Glaser's2l wo=1.10-8
24 A. Smakula, Nach. Ges. der Wiss. Gottingen, I, No.4
(1934). cm25) for a concentration of color centers 10l7/cm3
we find: a=5·1O-lO sec., tl=4.9·10-5 sec.;
pt=1/a=2·109 sec.-l as expected «p,=5.7·1012
sec.-I. With increasing temperature the lattice
structure widens, the oscillation frequency goes
down, and the maximum of the" F" band shifts
towards the red, as observed.26 The ratio between
thermal activation energy U and optical absorp
tion energy hpt is 1 : 2 in the limits of error; this
it should be, according to the Franck-Condon
principle, if the electron is trapped by lattice
distortion.
Equation (8) takes care of the conductivity
produced by surplus electrons, but this is only
one side of the problem. If these movable
charge carriers are created by transferring elec
trons from a lower filled zone into a conduction
band, normally two possibilities for conduction
are created in each case-one excess electron
and one hole (Fig. 6). By treating the hole as
positron of electronic mobility, the anomalous
Hall effect has been explained,27 and since that
time the theoretical discussion has normally dis
tinguished between the two modes of conduction
only by the sign. But the processes may be very
different, as the case of the alkali-halides demon
strates. If an electron is transferred from the halo
gen ion into the conducting state, it represents the
excess electron moving along distances w between
trapping, as discussed; the observer sees a
sodium atom moving relatively fast towards the
anode. The missing electron at the halogen ion
represents the hole; it may also move and trans
fer the chlorine atom towards the cathode, but
it can do so only in jerks w h of atomic distances
by electron exchange from neighbor to neighbor.
Keeping this difference in mind, the hole conduc
tivity can be represented by an equation like (8)
---) J:L .. IN .. ,,ml
BAND PICTURE OF'
ELECTRON AND HOkE
FIG. 6. ELECTRON AND HOLE
MIGF(ATION IN NaCI
25 It is hard to see how such small distances can be
explained with the de Boer-Mott model of trapping.
26 U and Vo in (4) are therefore dependent on T.
27 W. Heisenberg, Ann. d. Physik 10, 888 (1931).
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IP: 130.102.42.98 On: Sat, 22 Nov 2014 20:35:58610 G. B. KISTIAKOWSKY AND W. W. RICE
and semiconductors with excess-and defect
conductance can now be treated as individuals
characterized by N, W, a, and U.
CONCLUDING REMARKS
The discussion of conductivity in the preceding
paragraphs has limited itself to the laws of mo
tion. But charges have not only to be transported, they have to be generated; they have to enter
the material and to leave it; they have to be
balanced by counter-charges providing the neces
sary electroneutraIity. Only this more general
picture, which includes the space-charge effects
and field distortion, can explain the peculiar
reactions of insulators observed in rectifiers and
barrier-layer photo-cells. The author will come
back to these questions elsewhere.
AUGUST, 1940 JOURNAL OF CHEMICAL PHYSICS VOLUME 8
Gaseous Heat Capacities. IF
G. B. KISTIAKOWSKY AND W. W. RICE
Department of Chemistry, Harvard University, Cambridge, Massachusetts
(Received May 8, 1940)
The present paper describes a continuation of the work on the heat capacities of lower hydro
carbons by the Lummer-Pringsheim adiabatic expansion method. The apparatus and the pro
cedure described in the first paper of this series, henceforth to be denoted as Part I, was used in
the present research without important modifications,
SINCE an absolute calibration of the resistance
thermometer forms the most important
feature of the present work, Fig. 1 shows the
resistance temperature coefficients of the Wol
laston wire which was used in most of the meas
urements here described. To appreciate the re
liability of these coefficients it may be pointed
out that in a year's time the resistance of the
wire at the ice point has changed by only 0.02
ohm out of a total of 280 ohms. Determinations
of the thermal resistance coefficients which were
made with every gas studied gave points which
fell perfectly on the curve given in Fig. 1.
As discussed in Part I the temperature of the
wire does not remain constant after expansion
but changes linearly with time. This was tenta
tively explained as due to absorption of thermal
radiation from the warmer walls of the expansion
vessel by the gas and it was proposed to mini
mize the effect by using a vessel with surfaces of
very low emissivity. With this idea in mind the
present research was conducted with the same
expansion vessel and thermometer wires as in
Part I except that the inside of the expansion
1 Part I, G. B. Kistiakowsky and W. W. Rice, J. Chern.
Phys. 7, 281 (1939). vessel was gold-plated and polished to a mirror
like surface. Unfortunately it was found that the
constancy of the wire temperature after expan
sion was thereby only slightly improved.
The failure of the gold plating to improve the
appearance of the experimen tal records and to
reduce experimental errors is very disappointing.
However, this failure does not eliminate radiation
as the possible source of the temperature in
constancy because at the low temperatures of
the experiments rather long wave-lengths may
•. 01>00-, I I (I I I I
10 .. 0 )0 40 50 60 10 80 gO 100
FIG. 1. The resistance-temperature coefficient of the
Wollaston wire thermometer plotted against the tem
perature in DC.
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1.1712704.pdf | Physics in 1939
Thomas H. Osgood
Citation: Journal of Applied Physics 11, 2 (1940); doi: 10.1063/1.1712704
View online: http://dx.doi.org/10.1063/1.1712704
View Table of Contents: http://aip.scitation.org/toc/jap/11/1
Published by the American Institute of PhysicsPhysics in 1939
THOMAS H. OSGOOD
University of Toledo, Toledo, Ohio
I
DURING the decade which is just closing,
three new fundamental particles, the
neutron, the positron and the meson have been
discovered. In addition, theoretical considera
tions seem to demand the existence of two others,
the neutrino and the neutretto, although decisive
experimental confirmation is still lacking. In
these few years the inventory of particles has
more than doubled. Since much of our present
knowledge of all primary particles has been
acquired with much ingenuity, in the best manner
of detective fiction, by the sifting of conflicting
evidence obtained from a study of fog-trails, we
celebrate the occasion by printing in Fig. 1 a
representative collection from the Rogues Gal
lery of the Cloud Chamber. Three of these items
are shown completely masked, although in
appropriate company (such as a hydrogen-con
taining substance) one of these, the neutron, will
betray its passage by producing recoil nuclei.
II
Pride of place among new phenomena in the
field of atomic physics must this year be given
to the discovery of the fission of the heavy
nuclei, uranium, protoactinium, and thorium,
which occurs when they are bombarded with
neutrons. It would hardly be true to say that the
discovery was Quite unexpected, for although the
effects of neutrons on nearly all the elements in
the periodic table were investigated in an ex
ploratory manner several years ago, yet the
details of individual processes were far from being
accurately known. One of the commonest proc
esses which takes place when a neutron, especially
a slow neutron, interacts with a nucleus, is the
temporary addition of the neutron to the nucleus,
and the emission of radiation. The new nucleus is
unstable. A typical example is summarized in the
equation
Agl09+nL~AgllO+1'.
The synthetic nucleus of mass 110 and charge 47
2 units still has the chemical properties of silver.
But it is unstable, and decays rapidly in ac
cordance with the scheme
AgllO----7Cd llO+e-.
The emission of a negative electron increases the
positive nuclear charge by one unit, without
appreciably diminishing the mass, so that the
final nuclear product must have the chemical
properties of element number 48, that is, cad
mium, with a mass 110. If, however, as happens
in other reactions, an alpha-particle, or a proton,
or a neutron were emitted in place of, or accom
panying the gamma-ray of the first equation,
then the final product would, of course, belong
to a different atomic species, lower in the periodic
table.· So well-established were processes of this
type that it was practically taken for granted
that the interaction of neutrons with nuclei,
followed only by the emission of beta-particles,
would transmute those nuclei into other species
one unit higher in the atomic series.
Several years ago during their survey of the
effects of neutrons, Fermi and his colleagues
bombarded uranium and observed the subse
quent emission of beta-rays. It was natural to
conclude that the uranium had been transformed
into an element of atomic number 93. Thus were
born the "transuranic" elements. It happened
that the beta-rays emitted during the formation
of this and other "transuranic" elements were
divisible into many classes, each characterized
by its own decay constant. Unfortunately the
half-lives were all so short that conclusive
evidence as to the nature of the products, such
as only chemical separation could give, was well
nigh impossible to obtain. Then in 1937 Curie and
Savitch discovered a product of period 3.5 hours,
and proceeded forthwith to separate it by
chemical means. It turned out to be a substance
with the chemical properties of lanthanum,
although its mode of production entitled it to be
classed as a transuranic element. This was the
beginning of the end of these mythical elements,
JOURNAL OF APPLffiD PHYSICS Recoil K uc1eus
Alpha-Particle
Proton
Neutron (apart from the 23-minute U239) for soon other
products were discovered which had the chemical
properties of barium.
Early in 1939, l\Ieitner and Frischl pointed out
that most of the inconsistencies among the ob
servations dealing with the nuclear reactions of
uranium could be eliminated by supposing that
the nucleus formed after capture of a neutron
split into two parts of comparable masses. In
arguing their case they placed great reliance on
the liquid-drop model of the nucleus developed
by Bohr, and we quote their own words on this
point. "On account of their close packing and
strong energy exchange, the particles in a heavy
nucleus would be expected to move in a collective
way which has some resemblance to the move
ment of a liquid drop. If the movement is made
sufficiently violent by adding energy, sllch a drop
may divide itself into two smaller drops."
"In the discussion of the energies involved in
the deformation of nuclei, the concept of surface
tension of nuclear matter has been used and its
value has been estimated from simple considera
tions regarding nuclear forces. It must be
remembered, hO\\'ever, that the surface tension
of a charged droplet is diminished by its charge,
and a rough estimate shows that the surface ten
sion of nuclei, decreasing with increasing nuclear
charge, may become zero for atomic numbers of
the order of 100." ~Ieitner and Frisch predicted
that the total kinetic energy of the two receding
fragments from the uranium reaction should be
about 200 ~Iev. They remarked also that under
neutron bombardment, thorium was already
l\leson Neutrino
Positron
Electron
Xcutrctto
FIG. 1. The Rogues Gallery of the Cloud Chamber. The meson track, noticeably heayier than the electron track in the
same picture, is by E. ]. Williams, University College, Aberystwyth, and is reproduced from Nature 141, 684 (1938).
The other tracks are from the collection of H. R. Crane, University of :\Iichigan. The range of the recoil nucleus is so
small that its track appears merely as a spherical cluster of droplets. The droplets which it has formed here have actually
condensed on neutral molecules. Had the ions not been removed beforehand, the droplets would have been far more
numerous,
VOLUME 11, JANUARY, 1940 3 FIG. 2. Fission of uranium. The uranium oxide is de
posited on a semi-circular strip backed by a section of
paraffin. Fast neutrons are slowed clown by the paraffin
ancl bomuard the uranium. Two thin tracks clue to natural
alpha-particles of range about 2.4 cm from li I are seen,
and in addition a very heavy track of somewhat smaller
range is seen resulting from a uranium fission fragment.
The range here is a little more than 2 cm. It happens that
there arc no alphas frol11 the li II isotope. These have a
longer range, about 2.75 cm, and are present in equal
auundance. (Photograph by G. L. Weil, Colul1luia Uni
versity.)
known to give rise to products, some of whose
decay periods were apparently the same as those
of the products resulting from the fission of
uranium. It was reasonable, then, to suppose
that the thorium reaction \yas a fission process
like that of uranium.
I t was not long before unambiguous confinna
tion of these suggestions came from several other
laboratories, and we are privileged to reproduce
here (Fig. 2) a cloud-chamber photograph made
recently at Columbia University \yhich shows the
intense ionization along the track of one of the
fission products of uraniuIll. As an indication of
the interest which this ne\v phenomenon has
aroused we may point out that about t\Yenty
percent of the Letters to the Editor in the
Physical Review since February have been con
cerned \vith it. To follow these in historical order
would probably be confusing, so we shall attempt
to give a brief account of the details of the
process as they are nmy knmY11.
Since the masses of atoms in the atomic
sequence increase in general faster than twice the
nuclear charge, a very heavy nucleus, if split,
will divide into two fragments which include
4 more neutrons in proportion to protons than are
found in normal atoms of about the same masses
as the two fragments. For example, if uranium
were to divide so that one immediate product
were "BBal39, the equation
shows that the other fragment (if there be only
two) must be krypton, and a very peculiar
krypton, with mass 100. This equation must be
construed merely as indicating the kind of process
involved in fission, because chemical analyses
show the presence of nearly a dozen elements,
solid and gaseous, among the products. These
may be in a variety of high energy states. After
the splitting of uranium, the question naturally
arises as to the ultimate fate of these excess
neutrons. Some at least, might be expected to be
emitted during the fission process, others might
be given out later, as the radioactive fragments
decay. Experiments designed to count approxi
mately the numbers of neutrons emitted have
been made by Szilard and Zinn,2 who find that
about hyo neutrons per fission are given out
"instantaneollsly." In addition there is a
delayed emission of neutrons3 with periods of
about 12 seconds and 45 seconds. The emission
of these neutrons indicates that the first products
of fission are probably nuclei in high energy
states, and that these decay, some with the emis
sion of neutrons, to more stable forms which are
in many cases still radioactive.
As a result of their own experiments Heyn,
Aten and Bakker4 suggest the following sequence
for the gaseous products after fission has oc
curred:
0.5 min. 10 min. 87 min.
Xe139 ___ --+CS139 __ --+ Ba 139 ___ --+ La 139
(stable)
0.5 min. 30 min. long
Xe ----+Cs ------+Ba ----+La
17 min.
Kr88 ----+Rb88-----+Sr88 (stable).
This evidence supports the idea that the primary
process of fission may occur in more than one
way; that is, the xenons in the table above are at
least in different states of excitation and may well
have different masses. Later experiments' from
JOURNAL OF ApPLIED PHYSICS this side of the Atlantic will probably cause a
revision of some of the decay periods just quoted.
Neutrons, slow or fast, are able to initiate the
uranium-splitting reaction, and during its sub
sequent course more neutrons are emitted with
energies apparently in the range which is effective
in causing the primary fission. It is an interesting
game therefore, to speculate upon the possibility
of a chain reaction6 going on in a mass of ura
nium, which might continue with explosive
violence once it has been started. Explosive,
indeed, would be a mild word to use, for the
reaction is, mass for mass, perhaps ten million
times as energetic as the explosion of hydrogon
and oxygen to form water. To date, there seems
to have been no report of such an occurrence.
\Vhy not, if two neutrons are given out for every
original uranium nucleus which divides? The
explanation depends on the effective cross sec
tions of the atoms for the different processes
which may occur. If slow neutrons are used to
cleave uranium, it turns out that they are also
able to transform the U238 to em by radiative
capture, and since the latter reaction is very
probable, that is, characterized by a com
paratively large cross section, it would become
operative and quickly use up the slow neutrons
which accompany the original breaking down of
uranium. On the other hand, if fast neutrons are
used to cleave uranium, the other fast neutrons
which are emitted during the subsequent reac
tions have a much greater chance of being scat
tered (i.e. the uranium atoms have a larger
scattering cross section than cross section for
fission) than of initiating a new fission process.
Hence there is no possibility here of a chain reac
tion unless many more than two additional
neutrons are liberated as a result of the step-by
step decay of the fragments of the primary
uranium nucleus. If there actually turns out to
be a danger of such a cumulative process in large
masses of uranium, it could be prevented by
mixing uranium with hydrogen-containing sub
stances. Then, the fast secondary neutrons would
very quickly be slowed down so that they could
not initiate the fast neutron splitting of uranium,
and more slow neutrons would be available for
the harmless process of creating the unstable
isotope U239. At the present time, the general
opinion seems to be that no dangerous chain
VOLUME 11, JANUARY, 1940 reaction can occur unless there is an extraordi
narily high concentration of the rare isotope
U235. And it is comforting to think that during
the accumulation of such a single isotope, an
explosion would probably occur spontaneously
(that is, be initiated by cosniic rays) before the
quantity collected reached lethal proportions.
FIG. 3. The ionization causred by the fission fragments
was measured in a parallel plate ion chamber, 6 mm deep,
containing argon at 100 Ib./in.2. The two major groups of
fragments stand out clearly, one having about 100 Mev
maximum energy, the other about 7S Mev. The alpha
particles cause so insignificant an ionization that they do
not stand out from the background. (Courtesy of ]. R
Dunning, Columbia Cniversity.)
The enigmatic behavior of this U239 is an im
portant problem at the moment. It is definitely
formed by the resonance capture of neutrons of
about 2S electron-volt energy, and thereafter it
emits beta-particles with a period of 23 minutes.
A true lransuranic element of atomic number 93
therefore seems to be formed. But here the clues
end. No one has eyer detected the emission of
alpha-particles from it, nor is any member known
of the permanent radioactive family which it
would be expected to sire. \\Then Segrc states7
that "transuranic elements have not yet been
observed," his sentence must be interpreted as
meaning that the apparent formation of a
transuranic element of number 93 must not be
taken as final proof that it exists.
Alreadv some aspects of this new disintegration
process have been investigated in some detail.
For example, Booth, Dunning and Slack8 have
shown that the fragments fall into two distinct
energy groups, as illustrated in the oscillograph
record in Fig. 3, where each of the long black
S vertical lines represents the ionization-due to one
fragment. This was done by absorbing individual
recoiling particles completely in an ionization
chamber and measuring the ionization produced.
The groups had maximum energies of about 7S
and 100 Mev, from which the authors calculate
that the ratio of the masses of the two fragments
would be about 96/140. This ratio is in good
agreement with the chemical evidence which
(for Sr/La) would be about 90/140. However, the
total energy measured falls short of that pre
dicted by Meitner and Frisch by some 2S Mev,
which may well be taken up by secondary
processes such as excitation of the nuclear frag
ments, or else the true energy of recoil may be
greater than that which is measured on account
of the difficulty of collecting all the ions from
tracks of such great ionization density as these
heavy nuclei form. The ranges of the particles
are quoted by Booth, Dunning, and SIack9 as 1.S
cm and 2.2 cm, measured in air. It would be
wrong to make the assumption that these two
groups are homogeneous, that is, that uranium
always divides into exactly the same two parts,
for Ba, Xe, Kr, I, Te, La, and other elements
have been identified among the products of disin
tegration; yet some authorities seem to think
that the primary fission process is comparatively
simple,lo and that the apparent complexity of the
products is brought about by their subsequent
behavior.
III
Studies of the artificial disintegration of ele
ments are continued with unflagging zeal. New
reactions are catalogued, new unstable nuclei are
detected, energy levels of nuclei are mapped, new
isotopes are discovered. Rather than present a
mere condensed catalog of such findings, we
prefer to deal with one topic in which the present
knowledge of nuclear reactions has been applied
to solve the problem of stellar energy. It must
come as a keen pleasure to all who have worked
during the last decade in the field of nuclear disin
tegration, to find so direct an application of their
researches as this. The information which we now
have concerning the conditions which exist in the
interior of a star form an outstanding example of
deductive logic, for no direct observation can be
made at all. The solution of the problem of the
6 origin of stellar energy is restricted at the outset
by several well-tested generalizations which have
been found from astrophysical observations, and
by theories which are based upon them. Any
acceptable solution must account for the evolu
tion of a star according to the scheme of the
Russell-Herzsprung diagram; it must account for
Eddington's mass-luminosity relation; it should
show why some stars follow the main sequence,
and some are giants; it should offer some plaus
ible reason for the occurrence of white dwarfs;
it should be consistent with the known abun
dance of the different chemical elements in stars;
and should explain why stars have their masses
grouped in so small a range. While it is too much
to hope for a thoroughly satisfactory theory at
first, considerable progress has been reported in
papers by Bethe,ll and by Gamow and Teller.12
Normally we think of nuclear reactions as
occurring when a fast-flying particle interacts
with a nucleus. The terrestrial difficulty of
accelerating particles to the requisite speeds is
absent in the interior of stars, where the thermal
velocities (due to a "temperature" of the order
of ten million degrees C) are great enough for
the purpose. Bethe shows that "the most im
portant source of energy in ordinary stars is the
reactions of carbon and nitrogen with protons."
In essence these reactions amount to the creation
of helium out of protons according to the follow
ing scheme:
C12+HL·c>N13+1'~C13+e++1',
C13+Hl ~N14+l',
N14+Hl~015+1'~N15+e++1',
N15+Hl ~C12+He4.
The sums of the first and last columns lead to a
condensed summary of the reactions, in the form
which shows that the process is essentially one
of building helium out of hydrogen. For each
intermediate atom, C12, C13, Nl4, N15 which is
transmuted, another identical one appears later,
so that the stock of atoms of these elements
remains constant. This cycle, according to Bethe,
is the chief one in hot stars. In cool stars a
straightforward building up of protons into
deuterons, and then further into helium, appears
JOURNAL OF ApPLIED PHYSICS to predominate, and in average stars, like the
sun, the two types of helium building are equally
likely. One of the essential points of the theory is
the exclusion of reactions involving protons with
atoms either heavier or lighter than carbon and
nitrogen. Such reactions, of course, could provide
the requisite energy, but at the expense of a
degradation of the atoms downwards in the
periodic table, so that the proportions of most
elements would suffer tremendous changes as the
star proceeded on its evolutionary course. Since,
in all cases, helium is being formed at the expense
of hydrogen, it is to be expected that old stars
contain a much smaller percentage of hydrogen
than young ones. This is well borne out by spec
troscopic observations.
Bethe finds that central temperatures of
typical stars throughout the main sequence, cal
culated according to his theory. agree within a
few percent with those derived from Eddington's
theory which employs observational data to
arrive at these temperatures. There is, however,
a glaring discrepancy if Bethe's theory is applied
to giant stars, so that it must be inferred that
some other process is operative in them. Gamow
suggests that gravitational condensation, long
toyed with as a general source of stellar energy
before much was known about the transmutation
of elements, is mainly responsible for the energy
production in these very diffuse giants.
IV
There is no evidence yet for the failure of the
law of conservation of energy or of the law of con
servation of momentum in atomic collisions or
explosions. These are always assumed to be valid,
and if an apparent violation occurs, it is deemed
to be due to some cause yet undetected. During
the last year or two, Crane and HalpernlS at the
University of Michigan have been measuring
carefully the energies and momenta of the par
ticles observed in the /:1-disintegration of radio
active chlorine (CI3S). If the particles observed,
the nucleus and the beta-particle, were the only
ones involved, then the momentum acquired by
the recoiling atom should be the same as that
carried away by the smaller particle. Theoretical
treatment of the problem requires the presence
of a third particle, a neutrino, in order that
VOLUME 11, JANUARY, 1940 energy and angular momentum be conserved,
but there are two current variations of the
theory, one due to Ferni, the other to Konopinski
and Uhlenbeck. They differ principally as to the
relative direction of ejection of the neutrino and
electron, and as to the proportions of low energy
electrons which they predict, the Fermi theory
giving more than are found by experiment. It is
to be remembered that beta-rays of low energy
are much more likely to be missed14 for reasons
including scattering and absorption in a solid
source, so that although the Konopinski
Uhlenbeck version seemed to agree closely with
experimental results a few years ago, the correct
ness of that version could hardly be accepted
without qualification. Crane and Halpern now
find from their cloud-chamber experiments that
the momentum of the recoiling argon nucleus
(from CPS) is often much greater than the mo
mentum of the beta-particle. It seems very likely
that the neutrino should be burdened with the
difference. The next step, of course, would be to
discover if possible from the momentum relations
in what direction the neutrino escapes with
respect to one of the observed momenta. The
two investigators tried to do this, but "found
that if the results indicate anything at all in this
respect, they favor slightly the predictions of the
Fermi theory."
V
The literature on cosmic rays is so confusing
that it is impossible at the present time to pick
out those experiments which will, in later years,
prove to be most significant; but before attempt
ing to summarize the conflicting interpretations
which have been placed on them, we shall first
describe a new multiple counter which represents
the latest type of electrical detecting apparatus.
This new multiple counter has been developed by
Swann and his associates for the study of shower
phenomena produced by the penetrating com
ponent. The recording devices are hQused in the
lowest 12-foot section of a vertical cylinder
(Fig. 4) whose upper part is filled with water.
Beneath this water absorber is a set of six or
more tube-counter trays (Fig. 5) each containing
18 units. Everyone of these is connected electric
ally to its own electroscope (Fig. 6) which is
provided with a mirror whose movement records
7 FIG. 4. Tank in which the appar;llus is installed. Th,'
upper part is filled with water. The counters are in the
lowest 12-ft. sectiun. (Courtesy of \V. F. G. Swann, Bartol
Research Foundation.)
the passage of an ionizing particle through the
counter. Above the complete unit is a block of
lead 18 cm thick to make sure that the counters
record only phenomena associated with the
penetrating rays. Furthermore, no electroscope
will react unless at least one ray passes through
all of the trays of counters. If now a shower is
produced ill the llpper lead block, the flying
particles will trip only the counters through
which they pass, and the actual paths of these
particles are recorded, via the electroscope
mirrors, on a photographic plate. The apparatus
has been used to investigate the relativt' numbers
of multiple events accompanying the passage of
one penetrating ray. As might 1H:' expected, the
proportion falls off fast as the multi plici t y of the
events increases. For example, for every twelve
events in which a single secondary particle is
created, there are only two in \\'hich three
secondary particles occur. This is in rough agree
ment with Bhabha's theory of the secondary
effects of mesons.
\Vhile the soft component of cosmic rays is
fairly satisfactorily explained as consisting of
electrons produced in showers, the progress of
this year's research concerning the penetrating
component seems merely to have increased the
uncertainty regarding its source, and its behavior
is interacting \\'ith matter. For a time, however.
there appeared to be a hope that some of the
phenomena associated with the hard component
would become much more simply understandable
8 011 the assumption that this hard component
consists of heavy electrons or mesons. But this
early promise has not been fulfilled. In the first
place it has been knO\\'n for some time that the
apparen t mass absorption of penetrating cosmic
rays is greater in air than in dense materials.
This curious fact receives a simple explanation
if the meson, which is without question an
unstable particle, disintegrates and loses its
identity in a mean lime of about 2 X 10-6 sec.
FIG. 5. Ass(,Illbh' of six counter tran. Slabs of lead are
seen at the top. '(Courtesy of \\'. F'. G. Swann, Bartol
J.I.esearch Foundation.)
after it is born. Usually it is assumed that the
meson disintegrates into a negative electron and
a neutrino, though recently l\Iajumdar and
Kothari!7 suggest that its products may be a
neutron and a proton.* It is clear that the time
* This would account readily for the presence of heavy
particles which have been found in cosmic rays, but would
require that the lllC'son haye an initial energy which is at
JOURNAL OF ApPLIED PHYSICS required by a fast moving particle to traverse
(say) 10 cm of lead must be very much less than
the time the same particle would take to travel
through an equivalent harrier of the atmosphere.
10 cm of lead are equivalent to something of the
order of 3 km of air of the lower atmosphere. It
would take a particle, even one moving at pl-ac
tically the speed of light, aoout 1.7X10 6 sec. to
travel this distance, so that a meson has a good
chance of decaying during this time, while the
probability would be much greater than it would
emerge intact from the 10 cm of lead. Hence the
apparent absorption in air must be greater than
in a substance such as lead (or even water) for
the simple reason that in the atmosphere many
of the original mesons disappear on the way to
the recording device.
An argument of the same kind shows \\"hy (he
aosorption of penetrating cosmic rays in a ver
tical direction is less than that of rays \yhich have
come in obliquely; the particles \yhich come ill
obliquely have traveled for a longer time, and
fewer of them are left. On the assumption that
mesons are produced by primary cosmic rays
after the latter have traveled one-tenth of the
way through the atmosphere from the top (i.e.,
at 16 km height, where the barometer reads 7.5
cm of mercury), Blackett18 calculates, \yith the
aid of some meteorological data, that the mean
range of mesons before decay is about 25 kl11.
This calculation applies to mesons of the energy
group which are likely to die at abou t sea level.
For more energetic mesons, such as can penetrate
60 meters of \Yater, the mean range is much
larger. Blackett also suggests (hat the seasonal
variation in cosmic-ray intensity may he due to
this characteristic behavior of mesons: in winter
the atmosphere is colder, and therefore less thick
than in summer (for an equivalent atmospheric
pressure). The mesons \\"ould therefore traverse
it in less time before decaying, \\"ith the result that
more mesons should reach sea le\-el in winter
than in summer, and cause greater currents in
ionization chambers there.
Several routes which involve assuming the
mass to be about 150 electron masses are avail
able for calculating the decay times of these
particles. Blackett gives 1.7X10-6 sec., Rossil9
least equal to the sum of the rcst mass energies of these two
massive particles.
VOLUME 11, JANUARY, 1940 about 2 X 10-6 sec., and other determinations
run a little higher. Nevertheless Nordheim20 is of
the opinion that the strict application of Fermi's
(heory of {3-clecay would lead to a theoretical
lifetime of the meson some 10:1 times smaller than
the ooserved values just quoted. His suggestion
is (hat the theory requin-'s reformulation, or else
FJ(;. 6. A tray of counters. The horseshoe-shaped ele
ments to the left indic;]te the positions of the electroscopes.
The transformer ;]nd rectifier elements are seen in the
center of the picture, leading; to;] row of 18 counters shown
bv the cylindrical tubes immcdiately above the center.
(Courtesy of \\'. F. G. Swann, lbrtol Rl"Sl~ar("h Founda
tion.)
(hal the differences in conditions (e.g. whether
(he meson is in a nucleus or free) may affect its
disintegration in a \yay \yhich has not yet been
taken into account. In this connection Yuka\va
and Sakata~l point out that theoretical considera
tions lead them to believe that the lifetime of the
meson would be much longer if its mass were
slightly smaller, alld that satisfactory agreement
\vith cosmic-ray ubsen-ations can be obtained
only if the mass is taken somewhat lower than
the 170-or-so electron masses which cloud
chamber observations suggest. It is interesting
to couple this \,"ith the theoretical possibility~2
that mesons may occur \\"ith masses which vary
by perhaps 30 percent. Furthermore, investiga
tions carried out by the Bartol Research Founda
tion of the Franklin Institute23 do not seem to be
capable of direct interpretation on the basis of
the known behavior of mesons as outlined above,
and to take care of this difficulty several possible
modifications of current ideas are suggested. For
instance, the work just quoted could be reason
ably understood if there happen to be two types
of mesons, a light variety associated directly with
9 incoming cosmic rays, the other heavier, anslng
from nuclear disintegration. If this be so, then
the mesons whose masses have been determined
with some certainty from cloud chamber photo
graphs must be nuclear mesons, detectable
because they have much smaller energies than
their lighter brethren w-hich fly with cosmic rays.
If, however, the particles are of only one type,
the experiments are explicable on the supposition
that low energy mesons are so strongly absorbed
by nuclei that they never come to rest before
losing their identity in a capturing nucleus.
Either of these hypotheses would require a
drastic modification of the accepted ideas of
mesons. They are mentioned here mainly to
emphasize the complexity and uncertainty of the
present body of knowledge.
A summary of the difficulties and an analysis
of possible explanations has very recently been
given by Nordheim21 and by Nordheim and
Hebb25 from \>"hose papers most of the informa
tion below is taken. There appears to be no
reasonable doubt that mesons are of secondary
origin, for if they were originally a component of
the primary radiation, practically all of them
would decay on their long passage through space
on the way to the earth, and there is no process
by which the original supply might be re estab
lished in the absence of fairly dense matter. Those
which are found above ground mllst therefore be
produced in the earth's atmosphere. The out
standing problem at the moment is to determine
the nature of the primaries to which mesons (m-e
their brief existence. The possibility that they
may be produced by the soft comporwnt is a
simple but difficult hypothesis, for it prt'dicts,
apparently logically", certain details of the process
of absorption and creation which are at variance
with observations. Nordheim and Hebb are of
the opinion that it would not be "impossible to
overcome this difficulty by varying the (assump
tions as to the) primary distribution, or intro
ducing numerical factors or the like," but they
imply that such a solution of the problem would,
at present, be unnecessarily artificial; and SLlggt'st
that the hypothesis of the soft-componen t origin of
the hard radiation be shelved at least temporarily
while other possible ayenut's are explored.
Nordheim then proceeds to discuss the con
sequences of assuming that protons or neutral
10 particles are the source of mesons. The proton
hypothesis is well supported by Johnson's inter
pretation of his expt'rillwnts on the east-west
effecl, but is undermined in an equally telling
fashion by other observations 011 the geomagnetic
etTect and on the proportion of slow protons found
at sea leyel. \Vith regard to neutral particles as
the primaries which might be responsible for the
production of mesons, the most likely hypothesis
from a theoretical point of view seems to be that
the neutrettos or neutral mesons postulated last
year by Arley and Heitler play an important
role. The experimental workers may rebel at this
suggestion, 011 the grounds that the existence of
neutrettos has hardly been conclusively demon
strated; and it might be maintained, without
undue seriousness, that in the present state of
our ignorance it is scarcely possible to deny the
presence of any undetected particle as a C0111-
ponent of the primary rays, espt'cially as the
origin of the rays is still a free field for specula
tion.
Although to date there appears to be no con
tlrmatiol1 of \fajumdar and Kothari'sl7 suggestion
that the meson breaks up into a proton and a
..
FIG. 7. Stereoscopic pictures of a burst of heavy par
ticles, caused bv cosmic ra\-s, in the emulsion of a photo
graphic plate. JVIagnification about 120 times. Reproduced
from Kature 143, 682 (1939).
neutron, nevertheless there is a grmving body of
experimental evidence showing that heavy par
ticles in small numbers are found in cosmic rays.
For example, Froman and Stearns26 believe that
non-ionizing heavy particles are produced in
cosmic-ray showers.
They give as evidence the results of experi
ments of the following type. A triple coincidence
JOURNAL OF ApPLIED PHYSICS counter system is set up. At the bottom is a pair
of counters a little distance apart on the same
horizontal leveL These are well shielded all round
by 10 em of lead. A little above this lead is the
third of the group of counters. A few em above
this is a lead scattering plate. A slab of paraffin
is introduced, first between the lead scatterer and
the top counter and later between the top counter
and the massive lead shielding. Assuming that
there are neutrons among the shower particles
coming from the lead, it is to be expected that
ionizing particles projected by them from the
paraffin will trip all three counters, provided, of
course, that the paraffin is above the highest
counter. The results showed that the ratio of
counts per hour with the paraffin above the top
counter to the counts per hour with the paraffin
below the top counter was about 1.30. With
additional support from other like experiments
the authors reasoned that non-ionizing particles
must occur in showers, and the natural supposi
tion at the present time is that they are neutrons
or neutrettos.
Quite recently, Heitler, Powell and Ferte127
have examined photographic plates exposed (to
cosmic rays) both at sea level in England and at
a high altitude in Switzerland. They find, of
course, that far more long tracks occur at the
higher altitude, but by analyzing the results
obtained with different amounts of lead round
the plates, they conclude that the longest tracks
(which are the most significant) are not produced
by electrons nor by mesons. There remain
neutrons as a probable origin, especially as the
tracks have the characteristics28 of fast moving
protons which would be expected to originate in
the emulsion.
Spurred by the report of Jesse and Gill29 that
a 30 percent latitude effect is found for very large
cosmic-ray bursts, Vallarta30 speculates upon the
occurrence of much heavier particles than those
of unit atomic mass. If each burst is caused by
one primary particle, then its energy must be so
large that if it were an electron or proton its
course would be inappreciably affected by the
earth's magnetic field. But since there is a geo
magnetic latitude effect, these particles must be
slower than electrons or protons would be, and
to carry the requisite energy they must be fairly
heavy. Vallarta suggests that nuclei even as
VOLUME 11, JANUARY, 1940 massive as oxygen might behave in a manner'
which would explain this effect. But he points
out that other hypotheses might fit the facts just
as well, so that the presence of particles heavier
than a neutron remains speculative at present.
Nuclear disintegrations caused by the rays are
seldom observed in cloud chambers, but they may
be observed much more frequently in thick
emulsioned photographic plates which have been
exposed for a long time. A first class example of
such an occurrence is given by Jdanoff.31 His
stereoscopic pictures are reproduced in Fig. 7,
and show (according to the author) 100 heavy
particles originating from a single focus. Electron
tracks may also have been present, but are not
recorded as the emulsion is not sensitive to them
individually. In this burst there are twleve
tracks of length equivalent to 18 em of air, all
in roughly the same direction. The total energy
liberated must have been of the order of 200 Mev.
An interesting feature is the presence of several
minor disintegrations within the field of view of
the microscope. The frequency of their occur
rence, per unit area of the emulsion, turns out to
be about seventy times what is normally found,
sO that there is a strong probability that they
represent secondary processes associated with the
main large burst. The burst recorded in these
pictures is presumably of the same kind as those
investigated by Carmichael and Chang-Ning
Chou32 using a thin-walled ionization chamber.
They state that the distribution of particles
which they find "is due, not to the existence of
showers of two kinds, but to the fact (already
noted by Auger) that each extensive shower has
a core of closely spaced particles surrounded by
a relatively wide fringe of much more thinly
spaced particles able to produce bursts of small
size."
VI
The weigher of atoms is at present content to
express his results on a relative scale, taking the
mass of the 016 isotope to be 16.0000. During the
last few years he has been able to attain incredible
accuracy in his work, mainly because he is able
to measure the mass differences between various
atoms in terms of the kinetic energy of disin
tegration products. This gives him, as it were,
an extremely fine adjustment in his weighing, far
11 exceeding that which can be reached in the
absolute measurements such as those which de
termine the electronic constants. This year
notable work has been done at the University
of Chicago by AIlison and his colleagues. Their
experiments are primarily concerned with the
determination of the masses of light atoms, and
the general method of attacking the problem is
well known and need not be reviewed here. One
of the key equations which AlIison38 uses is
3Li6+1D2~22He4+22.08±0.07 Mev,
in which the letter symbols mean that when a
deuteron reacts with a Li6 nucleus, two alpha
particles are produced. The numerical term
means that the two alpha-particles emerge from
the reaction with kinetic energy equivalent to
22.08 Mev, or 0.02372 atomic mass unit. These
figures, based on a four-year old experiment of
Rutherford, Kempton and Oliphant, were be
lieved to be accurate except in the last place. On
repeating the experiment with some refinements,
however, Smith34 found that they were too low;
his new measurements gave 22.20 Mev in place
of 22.08. With this adjustment, the masses of
four of the light elements turn out to be
LiB = 6.01682±0.OOOll,
Li7 = 7.01814±O.0009,
Be8= 8.00766±0.00015,
Be9= 9.01486±0.00013.
It is interesting tD compare these and other
masses similarly determined with the masses cal
culated by Barkas35 on the basis of an empirical
theoretical formula derived from the theory of
the nucleus developed in the last two or three
years by \Vigner and others. I t is not possible
yet to carry the comparison much beyond atomic
number 40, but an inspection of the table which
Barkas gives shows a remarkable agreement. It
should be noted, however, that on account of
uncertainty regarding some of the constants
occurring in the theory, the calculated values are
not unique, though these constants can be chosen
arbitrarily to give good accord with experi
mental results. Determination of other nuclear
masses will therefore be immediately useful in
working out further details of the theory.
In 1936 Oliphant pointed out that the mass of
Be8 appeared to be almost exactly twice that of
12 the alpha-particle, with nothing to spare for
binding energy, so that it could scarcely be ex
pected to be stable; indeed, Livingston and
Bethe's table (1937) quotes its mass as a trifle
more than that of two alpha-particles, although
the difference was easily covered by the uncer
tainties in the data. Then Skaggs reported that
his experiments implied that Be8 was stable by
0.000174 mass unit, but the latest dispatches
throw doubt on this, and leave the question
entirely open.
Heavy atoms cannot yet be handled by the
methods which are used for the light ones, for a
tremendous chain of accurately known nuclear
reactions would be required to link each to its
predecessors in the periodic table. Masses can,
however, be determined by the mass spectro
graph by the method of bracketing. Dempster
has used this method very successfully to deter
mine the masses of many heavy elements.
Usually the masses of the heavy atoms are not
quoted directly on the scale 016= 16.0000, but
are expressed in terms of the packing fraction
which is essentially a measure of the relative
deviation from the simple whole number rule.
With Dempster's instrument, Graves36 has f/Jund
the packing fractions for a number of elements
in the range of atomic weights between 100 and
200. These substantiate the general trend of the
packing fraction curve and confirm the gentler
slope which has already been reported in the
neighborhood of platinum and gold. When the
relative abundances of the various isotopes are
taken into account the agreement with recent
chemical determina tions of atomic weigh ts is very
sa tisfactory.
VII
The tentacles of spectroscopy are so far reach
ing that we shall be content here to point to three
notable landmarks which represent the comple
tion of new work. The first of these is the publica
tion of the Massachusetts Institute oj Technology
Wavelength Tables, compiled under the direction
of Professor Harrison. This new book contains
over 100,000 entries "giving the wavelength, the
intensity in arc, spark, or discharge tube, the
stage of ionization of the parent atom when
the line has been classified in a term array, and
the wavelength authority, for each of the most
JOURNAL OF ApPLIED PHYSICS importan t known spectrum
lines emitted between 10,000
and 2000 angstroms by
atoms in the first two stages
of ionization." l\lore than
75,000 of these lines are the
results of recent measure
ments made in the ~I.I.T.
laboratories and have re
placed older and less reliable
values already in the litera
ture. Such a tremendous
undertaking could hardly
have been carried through
before its results began to be
obsolete without a machine
which measures the spec
trum lines and computes and
records their \,"ave-Iengths
and intensItIes automati
cally. An instrument which
performs these functions,
and which can he attached
to a moving plate compara
tor, was built in the early
years of this decade and de
scribed37 in the literature in
1935. A general view of the
machine is giyen in Fig. 8
wi th Professor Harrison at
the can trois. It gi ves a 20-
fold increase in speed over
the usual manual methods,
wi th an accuracy as grea t
as that which can other
wise be obtained without FIC;. 8. l\lachin(· for measuring and computing wave-lengths of spectrum lines.
Professor George R. Harrison is seen opcraling the machine. (Courtesy of News
S('rvice, lVlassachusetls Inslitute of Technology.)
resorting to interferometer methods of determin
ing the waye-Iengths of indiyidual lines.
As a second example of progress in spectro
scopy we reproduce in Fig. 9 rotational spectra of
water and of heavy water to illustrate the work
which is heing carried out under the direction of
Professor Rand,1I138at the Universityof ~lichigan.
Of the yarious types of radiation emi tied by a
polyatomic molecule, the rotational spectrum in
volves the smallest energy changes, and therefore
lies in the most extreme part of the infra-red, at
wave-lengths of the order of 50ft. Such a spec
trum can be studied successfully only in absorp
tion, with special echelette gratings in a vacuum
VOLUME 11, JANUARY, 1940 spectrographa9 The spectra in the figure coyer
the range 31.6ft to 38.3,u, and are parts of the
rotational spectra of H20 and of D20 between
30,u and 150,u. An elementary interpretation can
be ohtained by imagining one of the figures to be
cu t along its axis of symmetry; then considering
the profile of the top half only, it may be regarded
as a microphotometer tracing of the absorption
spectrum. The symmetry of the records is caused
by the nature of the recording and amplifying de
vices. Dips in the profiles correspond to absorp
tion lines whose positions can he determined \,,-ith
an accuracy of abou t 0.5 cm -); and a pair of ab
sorption lines of similar intensity can be detected
13 FIG. 9. Photographic reductions of the recordings of the
far mfra-red spectra of D20 (above) and H20 (below)
between 31" and 38" by a vacuum spectrograph using a
grating with 600 lines per inch. The grating has a ruled
surface 10X20 inches. The maximum deflection in these
records is 290 mm. The records represent parts of the
rotational spectra of D,O and H20, obtained hy l\". Fuson
and H. ;\1. Randall at the University of ;\lichigan. The
two records differ noticeably in structural uetail. (Courtesv
of H. M. Randall.) . -
as double if they are not closer than O.OS cm-I.
The rotation frequencies given by the spectrum
of D20 have been analyzed ill terms of energy
levels of the molecule, in the fashion used two
years ag"o for H20. It is interesting to note that a
molecule stretches as it rotates (oE course, the
stretching will depend 011 the frequency of rota
tion) and that corrections can be applied success
fully so as to express the energies of the lines in
terms of a common standard.
Turning for the third example to atomic
spectra we call attention to the work of Jenkins
and SegdAO on the quadratic Zeeman effect. This
is an unsymmetrical splitting and displacement
of the lines of an atom by a magnetic field; the
effect is small and should be most easily detected
for high series members, since it depends on the
fourth power of the total quantum numb!:'r.
Large enough values of the quantum number are
14 brought into play only in absorption spectra,
which in turn requires a magnetic field of large
extent. The authors took advantage of the large
field available in the new cyclotron magnet of the
Crocker Radiation Laboratory of the University
of California, which was available while the
vacuum chamiwr for the cyclotron itself was
being constructed. The magnet with spectro
scopic equipment in place is shown in Fig. 10.
Good spectrograms were obtained showing the
effect both in sodium and potassium. The photo
graph shows, in the background, the hydrogen
lamp for producing the continuous spectrum;
and, in the Jeft foreground the 3-meter Littrow
quartz spectrograph "which gave a dispersion of
O.8A per mm in the region used. By the ap
propriate placing of extra iron, a field of 27,000
oersteds was obtained over a space 5 X IS X 60 cm.
This enabled the series lines to be studied up to
the total quantum number 30.
VIII
It has been on record in the literature for
several decades that the reflection of light from
clear glass could be diminished markedly by
certain chemical treatments of the surface; but
although experiments were done to measure the
change in reflectivity there was no appreciation
of the reasons for the phenomenon, nor any
thorough-going analysis of it. It was known, of
course, that the partial extinction of the reflected
light was due to some kind of film on the surface,
so that when the technique of building up very
thin films was developed, as it has been in the
last few years, the time was ripe for a systematic
study of their action in subduing undesirable
reflections. Such a study was published in
February by Biodgett,41 and its most striking
fea t ures are probably \\·ell known on account of
the publicity given to it in the daily press. In
the first place Blodgett recognized, as Strong had
done a few years earlier, that the presence of a
thin ftlm on a piece of glass would affect the re
flection by a process of interference. Her paper
shows that the intensity of light so reflected de
pends on the refractive index of the glass, the
wave-length of the light, and the factors which
affect the equivalent optical path of the incident
light in the film. The particular case which is
JOURNAL OF ApPLIED PHYSICS most important from a practical point of view is
that in which the light is incident normally. In
this case the reflected beam should be ex
tinguished entirely for one particular wave
length, if the refractive index of the film is equal
to the square root of the refractive index of the
glass, and provided furthermore that the film
thickness is an odd multiple of one-eighth of the
wave-length. The film must be of unusually low
refractive index, therefore, to be most effective.
The well-known thin films of barium stearate
have a refractive index of the order of that of
glass (ca. 1.5), but this can be reduced materially
by soaking in benzene, which removes the excess
of stearic acid, leaving the stearate as a skeleton
film of very nearly the same thickness as before.
Blodgett chose cadmium arachidate* as the film
material, for after a treatment of a similar type
it gives a skeleton film of very nearly the correct
refractive index. How well the reflection from
glass is suppressed may be judged from the
accompanying Fig. 11, ,,·hich was specially pre
pared for this journal by Dr. Blodgett. l.1n
fortunately these films are sensitive to mechanical
and actinic damage, but if protected from such
FIG. 10. The new cyclotron magnet of the Crocker Radi
at.ion Laboratory of the University of California arranged
for studying the quadratic Zeeman effect. (Courtesy of
F. A. Jenkins.)
treatments there seems to be no reason why they
should not retain approximately the same prop
erties for a long time.
Reflection-destroying films can also be pre
pared by evaporating in vacuum the right
* Stearic acid, CH3(CH2)16 COOH; arachidic acid
CH,(CH 2)'B COOE.
VOLUME 11, JANUARY, 1940 FIG. 11. One-half of t.he glass face-cover of this abrm
clock has been covered (both sides) with a nonreflecting
film. (CourtC'sy of K. B. Blodgett, Research Laboratory,
The General Elt'ctric Company.)
amounts of certain substances \\·hich have lmv re
fractive indices. Several experimenters have used
this technique with J\IgF2 and other fluorides
"which are the most readily available substances
with about the right indices. These films, like
cadmium arachida te films, reduce the reflected
light to one percent or less, but 110 films have yet
been made which possess the permanence of the
glass surfaces which they cover.
IX
I t is Ii ttle more than six years ago that the
magnetic moment of a nucleus was first measured
by Stern and his colleagues. Before that time,
indeed, the magnetic moments of complete atoms
had been measured, but there the ovenvhelming
contribution is made by the electrons which form
part of the normal structure. To measure the
nuclear moment it was necessary somehow to
get rid of the disturbing electronic contribution,
and also to make the measurements more delicate
and the dispersion greater. There have been, of
course, many calculations of nuclear moments
based upon hyperfine structure measurements,
but it is desirable to have an experimental
method of comparable precision if only to justify
the theoretical values.
An important new experimental technique was
described in some detail early this year by Rabi,
15 Millman, Kusch and Zacharias.42 They show that
the principle upon which the method is based is
applicable to any system which has angular mo
mentum and magnetic moment. If some details
are omitted, the general scheme may be sum
marized as follows. Molecules are projected first
through a magnetic field, inhomogeneous in the
two directions at righ t angles to the general direc
tion of projection. This acts as a selector. Then,
a little farther on, they go through another in
homogeneous field whose gradient is opposite to
that of the first. This field acts as a kind of
analyzer. Regarding an atom as a tiny magnet,
it is clear that it will be pushed one way by the
first field and back again by the second field.
Some typical molecules which start out at a
sligh t angle to the axis of symmetry of the mo
lecular beam will be deflected just the right
amount and in the right direction to bring them
back to the axis after passing through each field.
But between the two inhomogeneous fields is a
steady field of smaller extent and also a small
oscillating field. Here some of the molecules
undergo a process of reorientation if their Larmor
precessions are in resonance with the oscillating
field; they acquire magnetic properties which
distinguish them from their fellows, and prevent
them from returning to the axis of the system
along with the group with which they started.
Hence a fraction of the molecules which would
otherwise reach a collector do not. This effect is
dependent upon the frequency of the oscillating
field, and in principle a measurement of the
frequency jJ leads to an evaluation of the mag
netic moment J.I. in the equation jJ = J.l.Ho/lh,
where I is the angular momentum of the atom
and Ho is the strength of the field in which
reorientation occurs.
If a study is to be made of nuclear magnetic
moments the most convenient subjects for ob
servation are atoms in a state with electronic
angular momentum equal to zero, or molecules
in which the electronic angular momentum of one
constituent is approximately balanced by that of
the other constituents. It happens that the nuclei
Li6, Li7 and p9 can be investigated in the mole
cules Liel, LiF, NaF and Li2• The values turn
out to be 0.820, 3.250 and 2.622 in nuclear mag
netons. The possible error of these values is given
by the authors as about 0.3 percent, and in view
16 of the fact that the values obtained from the best
hyperfine structure calculations differ from the
experimental ones by about two percent, the
question is raised whether the accuracy of the
assumptions which go into the theoretical calcu
lations is as great as is generally believed. In later
papers by the same workers, many more nuclear
magnetic moments have been determined experi
mentally in the way described above, and in
general the evidence seems to point to the trust
worthiness of the indirect calculations made on
the basis of hyperfine structure measurements.
x
VVe have had space here to touch on only a
few of the recent developments in fundamental
physics. The new contributions to applied phys
ics which keep pace with these advances are
probably well known to readers of this J ourna!.
Muskat and Morgan43 have continued their im
portant studies in the lubrication of journal
bearings both theoretically and experimentally,
generalizing the earlier theory of Sommerfeld
to include bearings of length as short as half the
bearing perimeter. Another excellent series of
papers44 comes from Stanford Upiversity. They
deal with klystron oscillators, which are devices
for the generation of electromagnetic oscillations
at wave-lengths of 10 cm or less. Such oscillations
can, of course, be produced by the classical
Hertzian methods and by triodes, but in both
cases there are great difficulties from the points
of view of convenience, stability and efficiency.
The new klystron osciIIator employs a tube (it
really is the tube) in which a beam of electrons is
sent through a pair of grids between which a
small oscillating field is maintained. The emer
gent beam therefore consists essentially of a
fluctuating electron current superimposed upon
a direct electron current. When this particular
cathode-ray stream is sent through another pair
of grids between which is another appropriate
alternating field, a considerable fraction of the
power originally supplied can be withdrawn in
the form of high frequency oscillations. VVebster4S
calculates that the maximum attainable effi
ciency is 58 percent.
These two series of papers on applied physics
are worth at least a cursory inspection by the
general reader if only for the reason that they
JOURNAL OF APPLIED PHYSICS bring out clearly one of the essential differences
between pure and applied physics. On the more
practical side of the science there is a rapid and
harmonious matching of experiment and theory;
experiment is usually in the lead, for it often
deals with some common detail of modern engi-neering equipment. On the academic side, as in
the study of the fundamental particles, there are
theories in abundance, and the most recent de
velopments may take on a speculative aspect,
since crucial experiments are frequently beyond
the range of present-day technique.
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17 |
1.1750657.pdf | Physics of Stressed Solids
Roy W. Goranson
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IP: 129.120.242.61 On: Sat, 22 Nov 2014 13:03:10APRIL, 1940 JOURNAL OF CHEMICAL PHYSICS VOLUME 8
Physics of Stressed Solids
Roy W. GORANSON
Geophysical Laboratory, Carnegie Institution of Washington, Washington, D. C.
(Received December 1, 1939)
The internal energy of a system is subdivided into a
work or potential function and a thermal or kinetic func
tion, the former expressed in terms of the current electro
static theory of intercrystalline bonding, and these
functions then examined for variations of temperature,
hydrostatic pressure, unidirectional stress and combined
hydrostatic and unidirectional pressure. From these con
siderations a theory is evolved which not only seems
satisfactorily to explain and correlate phenomena of defor
mation, creep or plastic flow, cold working, elastic after
working, rupture, shear and certain other phenomena
hitherto described as "anomalous" effects but has been
corroborated experimentally in some of its predictions, in
particular for the effect of hydrostatic pressure on deforma
tion and compressive strength. The mechanism evolved
consists of two processes-one an elastic deformation
which is a function of the strain or potential energy of the
system. Failure occurs here by "brittle" rupture wherein
. THE thesis presented in this paper is that the
phenomena of deformation, flow or creep,
and rupture may be interpreted and correlated
from a study of the internal energy stored up in
the lattice as a result of deformation.
The energy of deformation may be subdivided
into energy of work and energy of heat, that is,
into an energy of position and an energy of
motion. If this energy of deformation exceeds a
certain critical limit, determined by the physical
characteristics of the material and the amount of the maximum extension or maximum internal tension is
the criterion. The other is a deformation by means of a
two-phase transfer mechanism and is a function of the
thermodynamic potential relations of the system. This
latter type is also a function of time and therefore a func
tion of the rate of application of load. When both processes
of this mechanism are operative failure occurs by shear;
the criterion for this type of failure is given by a function
of time, the strain or potential energy and the thermo
dynamic potential relations of the system. Expressions are
derived for creep or plastic flow of polycrystalline sub
stances from the thermodynamic potential relations which
not only satisfy the well-known phenomena of creep
in metals but also express recent empirical creep data of
some substances immersed in liquids in which they are
somewhat soluble. An expression is also derived for the
"brittle" potential type of rupture under combined thrust
and hydrostatic pressure.
effect in phenomena of creep and plastic de
formation.
In general, deformation and failure are effected
by a combination of these two mechanisms and
observed as shear and as "gliding along shear
planes."
Many of the ideas incorporated in this dis
cussion have been tentatively put forth by
other writers but the picture as a whole with its
correlations and the theory of rupture presented
appears to be new and is therefore presented
with the hope that it will prove of interest.
This subject needs stimulus for further work on
deformation phenomena under hydrostatic con
fining pressure because it is believed by the
writer, perhaps also by others, that further
advances in our knowledge of crystal lattice
forces will come from such studies.
EMPIRICAL HISTORY externally applied compensation, a release of the
strain energy takes place in such a manner that
the system does the least work. If the energy
relations are such that release is effected through
the potential component, i.e., work function, the
system exhibits potential or brittle rupture and
the system relieves itself of its strain energy by
doing work in the direction of greatest extension
which is the direction of least external con-Volume compressibility, like density or heat
straint. If the stress and energy relations are capacity, is insensitive to variations in structure.
such that release may be effected through the By that we mean the volume compressibility of
kinetic component, i.e., the work done on the a single crystal is the same as that of a poly
substance is largely dissipated by heat transfer, crystalline aggregate of the same material. Shear
the specimen deforms by means of a "two-strain, creep, and other related phenomena on
phase" flow mechanism. This is the dominant the other hand, will vary with crystallographic
323
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Q)
>
en :;
Q.
Q)
0: A
FIG. 1. Force vs. extension.
orientation, the geometry of grain boundaries,
and other structural factors.
Under certain conditions the material may
react like a "perfectly elastic body" whereas
under other conditions it may react like a
"viscous liquid." In general our material will be
found to behave in some intermediate manner
for which several words have been coined such
as "elastico-viscous," "firmo-viscous," and even
"elastic flow."
Some other phenomena which must also be cor
related are: Strain hardening (work hardening)
an effect for which yield stress increases with
strain; and elastic afterworking. The stress is
also found to vary with rate of deformation;
for liquids the stress-flow rate slope is a constant
related to the coefficient of viscosity. The type
of deformation may also vary with the speed of
loading and a "ductile" or "malleable" sub
stance may, under certain conditions of rapid
loading, behave as a "brittle" substance. Again
"aging," which is presumably a slow transition
toward a more stable state, is observed as a
change of yield stress with time. A similar
phenomenon may be observed in glass which
although breaking readily along a fresh scratch
does this with more difficulty after a lapse of
time and eventually will no longer break cleanly
along the scratch.
THE ENERGY FUNCTIC)NS
The current electrical theory of intercrystalline
bonding forces is based essentially on an electro
static model. The electrostatic potential <p at a distance r will include the sum of charge poten
tials proportional to r-l, of dipole potentials
proportional to r-2, of quadrupoles proportional
to r-3, of octupoles proportional to r-4, and so on.
There are also additional terms arising from
interactions, induction and dispersion effects.
For example the character of the van der Waals
forces has been correlated with polarization
forces (always attractive) produced by quadru
poles on molecules which are regarded as de
formable distributions of charge; the potential
from this effect is proportional to r-8• These
forces have also been correlated with the polariza
tion of one molecule by the time varying dipole
moment of another; these interaction potentials
have been computed as proportional to r-6, with
also terms of higher order as r-8 and r-lO•
From quantum-mechanical considerations the
repulsive potential is given by the encroachment
energy from overlapping wave functions of the
atoms and expressible, to fairly high pressures,
by an exponentiall term. The Thomas-Fermi
atom model has been suggested and applied2 to
calculations of densities at very high pressures,
i.e., exceeding 106 bars (1 bar = 106 dynes cm-I).
The force field is given by the negative gradient
of the electrostatic potential. Our present pur
pose will be satisfied by noting that we can write
the attractive force F on a particular element in
the form
(1)
where the Ci and Ci are, in general, complicated
functions of space and direction. The derivative
of Fwith respect to r/ro will denote the reciprocal
of the linear compressibility, assuming linear
symmetry. Volume cannot be arbitrarily sub
stituted for r3 because of asymmetry with respect
to the space coordinates.
The coefficients C; and Cj cannot, in general,
be directly determined or computed; we may,
however, expect to learn something of their
respective importance from studies of deforma
tion under high hydrostatic pressure. Initial
compressibilities have been computed for some
1 P. M. Morse, Phys. Rev. 34, 57-64 (1929).
2 J. C. Slater and H. M. Krutter, Phys. Rev. 47, 559-568
(1935). H. Jensen, Zeits. f. Physik 111, 373-385 (1938).
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ionic cubic crystals.3 Approximate correlations4
of structure with certain physical properties such
as viscosity have been obtained from an assumed
two term expression. Although these simplified
arbitrary expressions cannot be other than rough
approximations they are able to extend the
range of elastic theory· and also give us a very
useful qualitative picture. Some explicit account
may be taken of thermal energy pressure by
adding a third term.6 From such an assumed
relation it is then possible to calculate the
relative stability of certain types of lattices as
functions of r.
Schematic plots of these two functions F7 and
¢ are given in Figs. 1 and 2 for an arbitrarily
assumed two-termed expression wherein i=3 and
j=9. The horizontal scales are the same but the
vertical scales differ.
The internal energy may be divided into two
portions, a work function and a heat function,
and expressed as
(2)
where e denotes the internal energy, 1/; the
"maximum work" function, T the absolute tem
perature and 1'/ the entropy. Differentiation of
this expression gives us
where 7r is a -compressional stress (a tensional
stress will be expressed as a negative pressure).
The integral
ir7rdr= 1/;(r) -1/;( (0) =¢(r) (4)
'"
is the portion used herein as the potential
function, and e=1/;=¢ for T=O.
The curve of Fig. 2 will also denote 1/; as a
function of r if the zero ordinate is replaced by
the value 1/;( (0) where d1j;( 00 )/dT= -k, k denot
ing Boltzmann's constant.
I M. Born and J. E. Mayer, Zeits. f. Physik 75, 1-18
(1932). A. May, Phys. Rev. 52, 339 (1937).
'References in R. H. Fowler, Statistical Mechanics
(Cambridge University Press, 1936), Chapter 10.
6 F. D. Murnaghan, Am. J. Math. 59, 235-260 (1937),
has used an arbitrary function similar in form to Ar-ll-Br-t
for expressing pressure volume relations.
6 J. Bardeen, J. Chern. Phys. 6, 372-378 (1938).
7 The term -F will hereafter in this article be spoken of
as internal pressure and F as internal tension. The terms
pressure and tension without qualifying terms will refer to
the external forces. Similarly the integral of Td1'/ from T=O to
T= T may be considered the thermal or kinetic
function. A harmonic oscillator at temperature T
has an average thermal energy of kT according
to classical statistical mechanics. In summing up
these energies for a system of oscillators, how
ever, the mutual coupling terms are neglected.
Some calculations made for metals using Debye
functions wherein the Debye temperature de
pends on the elastic constants have indicated the
usefulness of this method but further develop
ment is needed.
The mean internal energy curve will have a
shape similar to the potential curve but with a
minimum which lies at a higher level and is
displaced in the direction of smaller r. Conse
quently the internal energy will at first decrease
on application of pressure at constant tempera
ture by loss of thermal energy exceeding that
gained from the work function 1/;. Although in a
large system the fluctuations in energy of the
oscillators tend to cancel out, across any surface
a certain fraction of them will have larger
energies than the mean and, under favorable
circumstances, can migrate from their positions.
It might be anticipated that Fm, the maximum
internal tension, of the F-r plot should bear
some relation to the cohesive tensile strength
and rm -ro, where rm is the separation distance
for F= Fm and ro the distance for F=O, to the
maximum elastic strain. This must however be
of the nature of an upper limit because of
(a) localized high stress regions resulting from
inequalities of load distribution and from dis
tortions in the lattice structures caused by
imperfections resulting from holes, fissures and
inclusions of foreign elements, and (b) localized
high energy levels resulting from thermal energy
o "'0 r.
, , , , , ,
1 ;
-~----- ~-~
--------- b c
FIG. 2. "Strain energy function" vs. extension.
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gradients and fluctuations. The calculated tensile
strengths, Fm, are all much higher than the
observed values which are indicated roughly by
So in the plot. In the following discussion Fm is
used as the ideal elastic tensile strength and
r m -r a as the ideal maximum safe elastic exten
sion. The actual values may be only about a
tenth or less of these maxima.
It should be noted that these considerations
are made for the case where the "two-phase"
transfer mechanism does not operate. Care must
therefore be exercised in drawing comparisons
between theoretical and empirical results and
this will be shown more clearly later.
TEMPERATURE
If the temperature of the system is increased,
at constant pressure thermal energy is absorbed
with increase in the internal pressure. The
system expands doing work against the external
surroundings until a new positional equilibrium
is attained for the new F. In this way we obtain
a family of curves for different T. Ignoring for
the time being the change in slope of these
curves and considering only the large scale
differences, we note that the effective zero
ordinate of the F-r plot moves up and ro
increases to some larger (roh. Similarly the mean
minim um potential (CPo) T moves to a larger
(roh and higher up the right limb of the potential
energy well which is therefore becoming shallower
and shallower in the direction of increasing r.
Eventually the right wall of this well would
vanish and no further readjustments yield a
stable configuration; the system actually breaks
down long before this point is reached. If in this
breakdown a reorganization in potential .and
kinetic functions of the components can take
place for a new minimum (cpoh the system melts,
and if such a redistribution cannot be effected
the system sublimes. The size and nature of the
elemental units in this breakdown will depend
on the variations in the interatomic bonds, the
structure splitting up across bonds of least
cohesive strength.
In this connection it might be of interest to
note that if the thermal energy be assumed as
proportional to v-f, or to r-2, the inte~nal pressure
contribution from this source will be propor-tional to r-3• Under increasing hydrostatic pres
sure the other internal pressure terms of higher
power, here assumed as r-9, eventually dominate
over the r-3 term and at an increasing rate. The
thermal expansion should therefore decrease
under hydrostatic pressure and approach zero at
very high pressures. This conclusion, in con
junction with the third law, means that, for this
model, the entropy must be zero at infinitely
high pressure for all temperatures.8 At moderate
pressures the smaller powers of ifr will play
more effective roles and the initial rate of de
crease of dilatation should, as for compressi
bility, be less for simple ionic lattices since here
the attractive r-2 term is initially more effective
than the r-3 term.
HYDROSTATIC PRESSURE
The same type of reasoning applies to systems
under hydrostatic pressure at constant tempera
ture but in this case the increase in potential is
equal to the work done on the system. The
system decreases in volume until an equilibrium
between internal and external pressure has been
achieved. The most stable pressure or phase
configuration is the one best able to withstand
the external pressure and therefore the one of
higher density. A change of phase under pressure
may take place either by reorganization of form
(potential or work energy) or in part by change of
form and in part by change of momenta (con
version to thermal energy). The pressure at
which the latter type of phase change occurs
will be a function of temperature; the former
should be independent of temperature.
Silicate glasses should in general behave some
what abnormally because these glasses are in a
metastable energy state, i.e., the configuration
stable for the high temperature kinetic and
potential energy relationships has been frozen
into the system. The internal forces should
therefore be tensions but the configuration is
presumably prevented from collapsing by the
randomness of orientation inherited as a result
of the previous thermal kinetic energy, that is
to say, the glass is not only metastable from
considerations of heterogeneous or phase equi-
8 This was also suggested by G. N. Lewis, Zeits. f.
physik. Chemie, Cohen Fest. Band 130, 532-538 (1927),
from other considerations.
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librium but is also metastable from considera
tions of homogeneous equilibrium.
This is suggested as the reason why viscosity
bears a relation to the previous thermal history
and why anomalous heat capacity effects are
observed in the annealing range.
Resistance to motion and reorientation is a
function of viscosity so we should expect
anomalous effects in highly viscous substances
quenched from high temperatures. In fact we
might expect a possible increase of compressi
bility with pressure since the system is in a
state which is up on the right limb of the energy
curve plotted in Fig. 2 and should be under
internal tension.
Bridgman 9 observes anomalous effects in the
compressibilities of quartz glass and of basalt
glass but he suggests a different explanation for
these behaviors. The results of Birch and Dow,lo
however, support the conclusion arrived at here
because at higher temperatures, where rearrange
ments for a more stable configuration will take
place more readily, they find the "abnormal"
observed increase of compressibility with pres
sure becomes less pronounced and eventually
disappears.
UNIDIRECTIONAL PRESSURE
If, instead of a hydrostatic pressure, a uni
directional pressure load 7f" is applied to the
system no external constraints will exist in the
plane at right angles, resulting in an unbalanced
energy distribution in the system. The spacing r
along the line of thrust decreases by an amount
r-ro and the strain energy increases, climbing
up the left limb of the <i> function plotted in
Fig. 2.
I t is possible to set up hypothetical unsym
metric arrangements of attractive and repulsive
forces such that on contraction of the longi
tudinal elements the effect laterally is an in
creased net attraction with consequent contrac
tion of these elements also. Such a condition
yields a negative Poisson's ratio, (Y, the ratio of
lateral elastic extension to longitudinal elastic
contraction under compression. A few examples
of this phenomenon exist, but in general (Y is
9 P. W. Bridgman, Am. ]. Sci. 237, 7-18 (1939).
10 F. Birch and R. B. Dow, Bull. Geol. Soc. Am. 47,
1235-1256 (1936). positive and varies in magnitude between 0.25
and 0.50. As a rule then, the symmetry will be
such that, as the longitudinal elements are
contracted, the effective lateral r for Eq. (1)
decreases and thus the system extends in the
plane at right angles to the compression. The
effective left limb of the strain energy curve <i> of
Fig. 2 moves to the right a distance (Y(r,,-ro).
When this extension reaches the value (rm-ro),
the strain energy increasing to <i>m, the system
becomes unstable and, on further extension,
ruptures by moving into the region of no stress.
This "brittle" potential rupture, observationally
called breaking along tension cracks, will occur
across surfaces that tend to parallel the axis of
compressive thrust.
The ideal "potential" rupture condition, ac
cording to this presentation, is given by
(5)
and the elastic or poten tial strength by
(6)
Thus the elastic strength in compression
should be 1/(Y, or roughly three, times the elastic
strength in tension. In general there is evidence
of more or less "plasticity" in compressional and
tensional test pieces of steel. Glass hard tool
steel, in which a minimum of plasticity is ob
servable, has a compressive strength of about 30
kilo bars but even here there is some evidence of
shear in the rupture. Direct comparisons for
potential rupture cannot therefore be made but
the tensile strength of this steel is not very
different from the expected value. Such tests
should be carried out at low temperatures.
In considering the effect of hydrostatic pres
sure alone we had only one region available to
the system, i.e., the entire region was subjected
to the hydrostatic pressure p. Here, however,
there are two regions available to our system
one subjected to compressive stress 7f" and one to
zero compressive stress. The system therefore,
if free of internal and external constraint, will
move into and occupy the region of lower energy.
If then, before the· lateral strain energy can
attain the value <i>m, the longitudinal energy has
increased to a value such that its thermodynamic
potential becomes equal to that of_the liquid for
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the region corresponding to zero compressive
stress, i.e.,
~(T,7r,p) = (T,p), (7)
the prime referring to the liquid phase, then the
solid would melt to the liquid phase at (T,p)
provided the stressed region is permeable to, i.e.,
not be active on, the liquid.
The thermodynamic potential is defined by the
expression
(8)
and a~ ja7r=r.
A schematic plot of ~ as a function of r at the
stressed face is given in Fig. 3. r will be observed
to increase with increasing compression but to
decrease with increasing tension until it reaches
a minimum, ~o, at rm. At the free face (r)FF
= (l/;ohF and thus at this face ~ increases with
1/;0 for both tension and compression. It should
perhaps be emphasized that in the direction at
right angles to the stress I/; is always at its
minimum value on the curve which however
changes by virtue of the change in our expression
for F.
The use of the terms "liquid" and "melting"
may seem confusing and should perhaps be am
plified. The process of the two-phase transfer
type of mechanism suggested here involves the
transition from an immobile solid phase to a
mobile phase and thence back again to the solid.
The mobile phase should more aptly be called
the "fluid" phase' and includes not only the
condition ordinarily accepted as melting but also
any other similar mechanism such as the one
known as "migration of lattice points" and
diffusion. To make this point somewhat clearer
the creep relations will be derived later for a
solid in contact with a liquid in which it is
slightly soluble and, for this case, the fluid phase
is the solution.
We have then two possible mechanisms to
transport our system into the region of lower
internal energy-one by snapping the cohesive
bonds, determined by the strain energy poten
tial; 'and the other by transition to, and flow of,
a fluid phase, solid~fluid~solid; this is deter
mined by the thermodynamic potential relations.
The "fundamental strength" of our material is
therefore not only a function of the strain energy ,
I
!;o --------, -I
, I
_1'"
FIG. 3. Free energy for the stressed face vs. extension.
potential but also of the thermodynamic po
tential.
The internal energy of the system is merely
a statistical mean of the lattice point energies
and, on any surface, elements will be found with
energy levels exceeding the mean value. The
lattice structure is also more or less distorted
because of impurities and other irregularities
with resulting non-uniform stress distribution.
Phase change is initiated at such regions of
localized high energy values and results in still
further localization of high stress areas. The
process therefore tends to accelerate until the
concentration of strain energy becomes too high
to be borne by the remaining bonds and the
specimen fails by shear, a combined "fluid" and
"brittle" release.
An increase in compressive load thus acts
similarly to an increase in temperature with
respect to our "two-phase" mechanism. Phase
change is initiated at, and proceeds from, loci of
high energy levels at a finite rate, and not as an
instantaneous disintegration of the lattice con
figuration, The time gradient of energy inter
change set up by change in the external condi
tions is also an important factor. The transfer of
thermal energy will lag behind that of strain
energy. This is evident from explosive phenomena
wherein the rotational and vibrational energies
can be observed to lag behind the translational
energy.
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The fact that this phase-change type of
deformation is a function of time, and the elastic
type is not, has important consequences. Failure
as a result of a rapidly applied large compressive
load is by brittle fracture across surfaces parallel
to or making small angles with the axis of load;
as the rate of application is decreased failure will
tend to occur as shear along surfaces which are
at larger angles to the axis of load; if the appli
cation is slow enough and the thermodynamic
potential relations are favorable the specimen
will deform by "gliding along 45° planes," or by
"flow. "
If the load is not too great the shear may
proceed in steps-"melting" at high energy
points, shear of the remaining cohesive bonds,
slipping, recementation by solidification, the
cycle repeating itself as conditions again become
favorable. The most favorable surface for this
process is the 45°-plane along which the resultant
shear stress is a maximum and equal to one-half
the compressive load.
These statements need some qualification
because the force and potential functions are not
symmetric in space and therefore vary with
orientation of the system. The curve of Fig. 1 is
thus a function of a vector r and the lattice will
therefore, if conditions are favorable, rupture or
shear along surfaces across which the cohesive
bond, Fm, is the least such as cleavage, parting,
and twinning planes.
"Melting" can occur in specimens under ten
sion, viz. along shear planes, but the phenomenon
should not be so evident here because it will take
place only at the free surfaces (see Eqs. (llb),
(12b), (14b».
DERIVATION OF EXPRESSIONS FOR "CREEP" OR
"PLASTIC FLOW"
The creep relations will be derived first for a
solid under compressive load and immersed in a
liquid in which it is somewhat soluble.
Assume an initially "ideal solid" in which the
thermodynamic potential is the same for all the
faces and equal to the potential of the solid in
solution (solute), i.e., assume that the solid is in
equilibrium with the saturated solution at tem
perature T and hydrostatic pressure p. If now
the solid is loaded by a longitudinal compressive force 7r the thermodynamic potentials at the
stressed and free faces of the solid and of the
solute will no longer be equal and the system no
longer in equilibrium so long as the stress exists.
In order to derive our thermodynamic equa
tions the system is first divided into hypothetical
isolated parts, namely the regions at the stressed
surface, at the free face, and of the solution bulk.
We also assume any coexistence of phases neces
sary for our derivations. The physical interpreta
tion follows readily from this procedure and
moreover we avoid any confusion that might be
introduced from a compromisell between the
thermodynamic and the physical picture.
The following formulae were derived by ordi
nary thermodynamic methods applied to stressed
systems12 and have been somewhat simplified
for purposes of clarity.
We have
(9a)
and
where Eq. (9a) refers to the stressed face (sub
script SF) and (9b) to the free face (subscript
FF). a2 denotes the activity of the solute, 7r the
compressive stress, M the mole weight, p
the density of the solid, R the gas constant
(R=83.156 bar cm), and T the absolute tem
perature. E is Young's modulus of elasticity in
compression, i.e., E=dXjde where X is the
stress (negative for pressure) and e the extension
per unit length in the direction of the stress. For
small stresses E may be approximated by a
constant, generally of the order of 106 bars.
On integrating (lla) we have, at the stressed
face,
(lOa)
where p is the mean value for the integration
limits.
11 P. W. Bridgman, Phys. Rev. 7, 215 (1916), derives an
expression which may be correlated with (14a) and (14b).
E. D. Williamson, Phys. Rev. 10, 275 (1917). H. C.
Boydell, Ec. Geo!. 21, 1 (1926). Boyden derives Poynting's
expression which applies to a solid under hydrostatic
pressure PI and liquid at P2 where PI >P2. Poynting's
expression has the same form as (14a) of this paper.
12 R. W. Goranson, Thermodynamic Relations in Multi
component Systems (Carnegie lnst. Washington. Pub!. No.
408, 1930). For usage of activity see R. W. Goranson, J.
Chern. Phys. 5, 107-112 (1937).
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If the system is impervious to liquid and the
load contact is a "perfect surface" then only the
external free face can be in contact with the
solution. Actually the system is, in general,
never impervious to the liquid and perfect con
tact surfaces non-existent. Penetration occurs
along crystal boundaries, across crystal cleavages
and cracks. The stressed surfaces carry the load
non-uniformly and contain high points of
localized stress. As these high poin ts are dis
solved the stress distribution shifts to other high
points and the process is repeated.
Thus at the stressed face solubility is increased
by a compressive load and lowered by a tensile
load. At the free face solubility is increased by
either a compression or tension but by a much
smaller factor. It should be noted however that
the integral expression for (9b) is
(lOb)
which may become large for large 7r. p and E are
here mean values for the range of integration.
This relation is the plastic flow factor for tension.
According to this interpretation it is found
that, whereas under compressive load the solute
concentration tends to increase at both stressed
and free face, diffusion of solute away from these
faces would leave the bulk of the solution super
saturated. Consequently deposition of solute
should occur at unstressed places. If, however,
the solution can supersaturate by an amount in
excess of the relatively small increased solubility
at the free faces, then crystallization will take
place on these free surfaces which, under these
conditions, act as nuclei for deposition. Our
mechanism can thus be considered as a diffusion
process along crystal grain boundaries wherein
the solution acts as a transfer medium. This dif
fusion rate will approach a steady state when a
dynamic equilibrium between rate of solution
and rate of deposition has been established, and
will be governed by the mobility of the solute in
the solution, the path length, and the concen
tration head between the stressed and free faces.
If, by changing the load, we change only the
concentration head of the polycrystalline sub
stance then the change in the steady creep rate,
EB, should be directly proportional to the change in activity, or
din a2 din e. ---=B---,
d7r d7r (11)
whereB is a constant. From (lOa) and (11) we
obtain
(12)
where K is a physical constant. This is the rela
tion obtained empirically13 for the steady creep
rate of compressively loaded alabaster in contact
with water.
As compressive load is increased the solubility
at the stressed surface eventually becomes
larger than the amount by which the solution
can supersaturate. Under these conditions the
solution will drop the excess solute in any avail
able unstressed space. Consequently, the texture
of the substance may thus become so loosened
that it crumbles under the load.
The remainder of the discussion parallels that
for plastic flow in polycrystalline metals. As
before, we divide our system into hypothetical
isolated portions and derive expressions for the
stressed face and free face. We assume also the
coexistence of two phases-the bound atoms of
the solid and the free migrating atoms with
energies exceeding those necessary to break the
cohesive bonds of the crystal lattice. At the
melting point this will be equivalent to the heat
of melting.
The analogous relations to (9a) and (9b) con
necting compressive stress 7r and "melting"
temperature, T m, in degrees absolute are readily
derived as
(~n Tm) 1
(l2a)
d7r SF ph
and
(d In Tm) 7r
, (12b)
d7r FF pEh
where h is the heat of "melting" and the other
quantities are as before.
The melting point at the stressed face is thus
depressed for compression and raised for tension.
This may be observed graphically from Fig. 3
13 The experimental data were obtained by Griggs. To be
offered for publication, w!th Gor'.lnson, in a f~ture issue of
the Bulletin of the Geologtcal Soctety of Amertca.
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where the thermodynamic potential ~ is seen to
increase with pressure and decrease with tension
at the stressed face. The melting point at the
free face is depressed for both compression and
tension. Here (~OhF = (thF always increases.
The ratio of lowering at the two faces is the
same as that found for (9a) and (9b).
The number of atoms with Maxwellian ener
gies in excess of that needed to break the lattice
bonds, i.e., the number of free migrating atoms,
increases with increase in temperature. At the
melting point this fraction of the total number
becomes unity. The factor
(13)
where T denotes the constant experimental and
T m the melting temperature, has been used in
correlating temperature versus creep data for
different metals. The behavior of a low melting
point metal is thus considered as equivalent in
behavior to a high melting point metal at a cor
respondingly higher temperature to give the
same IJ, other factors remaining unchanged. The
argument should be carried out in terms of
energy but, as compensation is here effected
through the h term this expression is a sufficiently
satisfactory approximation.
On combining (12a) and (13) and integrating
we have
7r -7ro = ph In (IJ I lJo), (14)
where p and h are here the mean values over the
range of integration.
For a sufficiently large load, i.e., for relatively
large creep rates, we may in the same manner as
before write
(15)
where K' is a physical constant and e. the steady
state or minimum creep rate. This is the ex
pression that has been used for about thirty
years to express the empirical relation between
yield stress and minimum creep rate Es in metal
mosaics.
The expressions (12) and (15) were derived for
relatively large creep rates and thus for com
pressive loads large enough to iron out the initial
inequalities over a short period of time. Let us
assume now that we are operating with either
very small stresses or with a substance that does not "migrate" readily. For these conditions the
number of migrating units will be relatively few
and their paths much shorter. Thus, instead of
being able to picture a statistical streaming
action, we are slowed down to a hop-skip process
and can no longer set up a steady mean flow.
The crystallization process must be explicitly
considered here and for these cases we have for
mobility along the stressed surfaces.
K(IJIlJo) = Ke(7r-7rol/ph (16a)
and for crystallization along the same s1,lrfaces
(16b)
The creep rate will be given by the difference
between these two quantities or
7r -7ro
(EI Eo) =K sinh --.
ph (17)
The same reasoning and therefore a similar sinh
expression also replaces (12) for very small creep
rates.
An expression of this type was obtained em
pirically14 by combining analytically the empirical
logarithmic relation for high creep rates (see Eq.
(15)) with the linear relation between stress and
creep rate observed for very low creep rates.
So far we have considered only the relation
between steady creep rate and stress. When
compressive load is first applied the irregularities
in the structure of the polycrystalline material
set up an initial localization of stress at the
raised points and thus, for large enough loads,
in an initially high creep rate. This rate gradu
ally diminishes toward a steady state as the
original inhomogeneities of texture become
ironed out and the stress redistributes itself over
larger and larger surface areas. The effective
stress, for constant load, thus decreases with
time. Finally the pore spaces in the texture
become filled and the grains more or less reori
ented into their most stable crystallographic
configuration for an axial load distribution. If
now a series of such creep-time tests is made for
varying compressive loads on initially identical
specimens this last state should be reached at
14 For example see H. Mussmann, Ann. d. Physik 31, 130
(1938).
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IP: 129.120.242.61 On: Sat, 22 Nov 2014 13:03:10332 ROY W. GORANSON
100 200 300 400 sao 600 700 BOO 900 1000 1100
TilliE. DAYS
FIG. 4. Tensile creep tests on coarse-grained lead rods at
room temperature. By suitable changes in coordinates this
diagram can be made to express, to a fair approximation,
the compressive creep tests of alabaster in water.
about the same longitudinal contraction Em
provided none of the material has precipitated
elsewhere. At this point the specimen has
stabilized itself and from this point on should
therefore behave more or less as a unit, i.e.,
approximate to the condition for a single crystal.
Solution (or migration) then begins to work
along crystallographic surfaces and will be most
effective on surfaces across which the cohesive
bonds are smallest, i.e., have the largest t, in
the general 45° trend to the stress axis. If the
load is large enough the specimen will then
deform by a gliding action along 45° planes and
the creep rate, measured by the rate of con
traction, will appear to accelerate until the
specimen "fails." This is graphically illustrated
in Fig. 4.16
Plastic flow may also occur in single crystals
but the effect in general is a gliding along
cleavage, twinning, or parting planes from com
bined "melting" and snapping of bonds. Local
ized high stress regions may be set up in crystals,
as was mentioned earlier, and thus an initially
single crystal may, under a compressive load,
finally become a mosaic of reoriented crystalline
grains.
The energy at the external crystal faces cannot
be denoted by the energy of the interior because
of loss of symmetry at these external faces; the
difference is known as surface tension. For ionic
crystals this difference will be greatest along
edges and at corners and least at the centers of
16 Data taken from J. McKeown, J. lnst. Metals 60, 207
(1937). faces; this is the reason for the initial skeleton
growth of such crystals. With a high symmetry
type of structure this difference might become
greatest at re-entrants. For our system solution
or melting will be most rapid at the stressed
surfaces of highest t and growth by crystalliza
tion most rapid on the free faces of lowest t.
These directions are, in general, indicated by the
crystalline form and cleavage. For example, mica
and related minerals will tend to grow with two
crystal axes (cleavage planes) perpendicular to
the axis of compressive load, asbestos and
related minerals to elongate (one crystal axis)
in the plane perpendicular to the stress. The best
examples we have are the metamorphic rocks
which have recrystallized according to this
mechanism under high confining pressure (see
below).
ELASTIC AFTERWORKING
There is a type of deformation which on load
ing is a contraction and on unloading a recovery
by extension, according to the expression
where E denotes the strain rate, El the purely
elastic portion of the strain, B a constant, t the
time and T a time constant. This type of re
covery, after unloading, is called "elastic after
working" and has been observed in glass fibers,
in steels, and in rocks.
This is the kind of·deformation to be expected
from materials which are relatively strong elas
tically. For these materials the strain is mainly
an elastic one but, for a long continued applica
tion of a moderate load, a small amount of
migration of lattice points will take place if the
load is left on long enough. On release of load
the material tends to recover elastically, i.e.,
instantaneously, but it cannot recover com
pletely because under load lattice elements have
migrated and then solidified in conformity with
the equilibrium conditions existing while under
load. These therefore formed bonds tying in this
equilibrium state so that when load is released
an opposing stress distribution is consequently
initiated and the initial equilibrium conditions
are attained only by a backward migration of
these same elements in retracing their paths.
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WORK HARDENING
The mechanism of work hardening should not
require particular comment. There is a further
consequence of this phenomenon: The reorien
tation and reorganization of a polycrystalline
aggregate for greater elastic stability to a com
pressive load, for example, means that the ma
terial should then be weaker for a tensional load
along the same axis. This is observable in push
pull load-extension diagrams of "overstrained"
steels.
EFFECT OF HYDROSTATIC PRESSURE
ON STRENGTH
If hydrostatic pressure is applied to a crystal
lattice, r decreases by an amount ro-rp= Ep
and if the electrostatic force field is symmetric
in space the decrease will be uniform in all direc
tions. If now an additional stress is applied by
means of a superimposed unidirectional com
pressive load 7r then r along this axis will decrease
to r" by the additional amount E .. =rr-rp.
The lattice, assuming "normal" behavior,
expands laterally doing work against the con
fining pressure until a new equilibrium is estab
lished between the repulsive, attractive, and
external forces.
It might be anticipated that Poisson's ratio
should bear a functional relation to r. The
evidence from seismic data indicates, however,
that within the earth (j remains approximately
constant, at about 0.27, independent of pressure
(depth).
The lateral extension will increase as the
longitudinal compressive load is increased and"
thus r will gradually move back through ro to
rm at which point the specimen becomes unstable
and ruptures. The load for this condition will
represent the compressive strength of the
specimen.
Rupture, here the "brittle" potential type of
fracture, will occur then for the condition
fF~-P dr
-dF-(rm-rO)
fF~-" dr F=O dF -dF=----------------
F~-p dF (J ,
where [(7r)m-PJ denotes the "elastic" com-pressive strength of the specimen under confining
pressure p.
The slope of the" r -F curve,
dr ri+l
flattens rapidly as r decreases and approaches
zero as a limit. This initial change in slope may
be observed graphically in the curves of com
pressibility plotted as a function of pressure.
Hence the smaller the r, and consequently the
higher the hydrostatic confining pressure, the
larger must be the unidirectional thrust for the
same rm-rp but this distance, as a matter of
fact, is itself also increasing by virtue of increas
ing p. The compressive strength of the iipecimen
should therefore accelerate rapidly with increas
ing confining pressure and become infinitely
strong elastically as the confining pressure con
tinues to increase indefinitely. If, therefore, the
compressive strength be plotted as ordinate and
hydrostatic confining pressure as abscissa the
curve should be found to rise gently at first and
then to steepen rapidly, eventually becoming
infinitely steep.
The above conclusions from this theory, de
pending entirely on seismic data, have been held
in abeyance until they could be further confirmed.
Recently confirmation has been obtained from
the experimental work of Griggs.16 Experimental
work is also being conducted at the Geophysical
Laboratory to verify the calculated strengths for
steel. The initial pressure effect is complicated by
the geometry of cracks but these cracks close up
at a few thousand atmospheres pressure.
The curve of compressive strength versus
hydrostatic confining pressure for substances
which have polymorphic pressure modifications
should exhibit discontinuities at such trans
formation points. In fact if such transformatio"ns
took place rapidly while the substance was under
unidirectional load the structure should break
down. These points represent therefore loci o(
instability, so that the above conclusions would
be applicable only to the homogeneous regions
above and below such loci.
18 D. T. Griggs, J. Geol. 44, 541-577 (1936). D. T.
Griggs and J. F. Bell, Bull. Geol. Soc. Am. 49, 1723-46
(1938). .
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IP: 129.120.242.61 On: Sat, 22 Nov 2014 13:03:10334 BENEDICT, WEBB AND RUBIN
The energy levels rise very steeply and even
tually for some value 7r>P a state will be reached
for which
r(T; 7r, p) = nT, p),
where these quantities are as before.
If this condition becomes fulfilled failure will
then take place by plastic deformation or "plastic
shear." In other words, a substance under axial
compression will become not only stronger elas
tically but also more malleable with increasing
hydrostatic confining pressure. This is shown
very clearly from the experimental results of
F. D. Adams and Coker.17 Their results were not
duplicated by Griggs but the reason for it should
be fairly obvious. Adams and Coker's confining
pressur~ was obtained by means of a pliable steel
wall, Griggs' by means of a thin liquid. As soon
as incipient melting takes place along, say, a
17 F. D. Adams and E. G. Coker, An Investigation into the
Elastic Constants of Rocks (Carnegie Inst. Washington, Pub!.
No. 46, 1906). F. D. Adams and J. A. Bancroft, ]. Geo!.
25, 597-658 (1917). shear plane the substance is relatively free to
shear in Griggs' case because the only work
involved is (a) that of breaking the remaining
cohesive bonds across this plane and (b) that of
exchanging regions with the pressure fluid, since
the resistance to flow of the pressure fluid, for a
finite rate, is a simple function of its viscosity.
For Adams and Coker's case we have, in addition
to (a), the work of pushing aside the supporting
steel wall which is here the total lateral support.
Release of strain energy can take place rapidly
in the former case but only at a slow rate in the
latter case.
Thus, although Griggs has expressed the belief
that his results are more applicable to problems
of geological deformation in depth and that
rocks do not flow in the dry state, the conclusions
arrived at herein do not support this contention
as a general conclusion. These two apparently
dissimilar experimental results are special cases
of the same physical hypothesis described in
this paper.
APRIL, 1940 JOURNAL OF CHEMICAL PHYSICS VOLUME 8
An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons
and Their Mixtures
1. Methane, Ethane, Propane and n-Butane
MANSON BENEDICT, GEORGE B. WEBB AND LOUIS C. RUBIN
Petroleum Research Laboratory, The M. W. Kellogg Company, Jersey City, New Jersey
(Received December 23, 1939)
An empirical equation is given for the isothermal variation with density of the work content
of pure hydrocarbons in the gaseous or liquid state. From this fundamental equation are de
rived (a) an equation of state, (b) an equation for the fugacity, and (c) an equation for the
isothermal variation of the enthalpy. These equations summarize P-V-T properties of the
gaseous or liquid phase, critical properties, vapor pressures, and latent heats of evaporation.
A procedure is suggested for determining numerical values of the parameters in the equation.
Such values are given for methane, ethane, propane, and n-butane. A comparison is made
between observed properties of these hydrocarbons and those predicted by the equations.
A. INTRODUCTION
RECENT experimental studies of pure light
hydrocarbons by Sage and Lacey and co
workers and by Beattie and co-workers are useful
in developing an equation to represent the ther
modynamic properties of these substances. An
equation for this purpose has several advantages: it permits interpolation of experimental data; it
facilitates thermodynamic calculations involving
integration and differentiation; it provides a
concise summary of a large mass of data; and it
provides a point of departure for the treatment
of the thermodynamic properties of mixtures.
Concurrent with this advance in experimental
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1.1750882.pdf | The Fundamental Frequencies of Certain Halomethanes. II. The Raman Spectrum
of Fluoroform
G. Glockler and W. F. Edgell
Citation: The Journal of Chemical Physics 9, 224 (1941); doi: 10.1063/1.1750882
View online: http://dx.doi.org/10.1063/1.1750882
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Published by the AIP Publishing
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07MARCH, 1941 JOURNAL OF CHEMICAL PHYSICS VOLUME 9
The Fundamental Frequencies of Certain Halomethanes. II.
The Raman Spectrum of Fluoroform
G. GLOCKLER AND W. F. EDGELL*
Chemical Laboratory, State University of Iowa, Iowa City, Iowa
(Received December 11, 1940)
The Raman spectrum of CHF 3 has been determined and the assignment to the six funda
mentals has been made by considering the intensities and by comparison with the other
haloforms. Checks have been obtained by two independent methods based on the theory of
small vibrations. The fundamentals in CHCIF, have been revised to harmonize with the data
of CHF 3 and they have been empirically correlated in several spectral sequences. Molar heat
capacities have been calculated for CHF3, CHCIF" and CHCI,F in the range T=250oK to
T=650oK and compared with experimental values where available. The heat capacities have
been fitted to empirical equations and a term added to correct to finite pressures by means of
the modified Berthelot equation of state. It has been shown that these equations are accurate to
5 percent and are far more reliable than the small amount of thermal data available at present.
I. INTRODUCTION
THE haloforms are an important group of
the halogen derivatives of methane. The
complete understanding of their vibrational
spectra is necessary before one can accurately
correlate the data for the other halogen deriva
tives of methane which possess less symmetry.
The fundamentals of CHCla and CHBra are well
known and their assignments are certain.!
Recently a number of halogen substituted
methanes which contain fluorine as one of the
substituents have been studied by means of the
Raman effect.2 A knowledge of the fundamentals
in fluoroform would be very helpful in the
correlation of these data. Glockler and Leader,a
in making such a correlation, were led to a
prediction of the values for CHF 3. Moreover
* This article is based upon excerpts from a thesis to be
presented to the faculty of the Graduate School of the
State University of Iowa by Walter F. Edgell in partial
fulfillment of the requirements for the degree of Doctor
of Philosophy.
1 The data on the fundamental vibrations of the mole
cules discussed in this paper are from the following sources
unless otherwise specified: CHCIa, CHBr3-M. Magat,
"Effet Raman" No. 15 of Tables Annuelles de Constantes et
Donnees Numeriques (Paris, 1937). CH4, CF4, CH3F,
CHD3, CHIa-Ta-You Wu, Vibrational Spectra and
Structure of Polyatomic Molecules (China Science Corp.,
Shanghai, 1939). CHCI,F-G. Glockler, W. F. Edgell and
G. R. Leader, J. Chern. Phys. 8, 897 (1940). CHCIBrF,
CHCIBr" CHCIF,-G. Glockler and G. R. Leader, J.
Chern. Phys. 7, 553 (1939); 8, 125,699 (1940).
'G. Glockler and co-workers, J. Chern. Phys. 7, 278,
382, 553 (1939); ibid. 8, 125, 699 (1940); Phys. Rev. 54,
970 (1938); C. A. Bradley, Phys. Rev. 40, 908 (1932).
3 G. Glockler and G. R. Leader, ]. Chern. Phys. 8, 699
(1940). they used these predicted values to eliminate
several lines from the Raman spectrum of
CHClF2 by attributing them to CHFa.
The purpose of this investigation was to check
these predicted values. It will be seen that .they
are not nearly as accurate as one would expect
from the method used in obtaining them.
II. EXPERIMENTAL DETAILS
The fluoroform used in this study was supplied
by Dr. A. F. Benning of the Jackson Laboratory
of E. I. du Pont de Nemours and Company and
was specified as "practically pure." No lines were
observed that could be attributed to any likely
impurity. The Raman spectrum was determined
in the liquid state at -95°C using the usual low
temperature apparatus described by Glockler
and Renfrew.4 A stream of dry air, precooled
successively by a dry ice-acetone mixture and
liquid air, was the cooling medium. The source
T ABLE I. The Raman lines of fluoroform.
RELATIVE EXCITING DEGEN-
~v INTENSITY LI"SES* ASSIGNMENT ERACY
508.1 5 a, b V45 2
696.7 6 a, b V6 1
936.8 0 a, V23 2
1116.5 8 a, b VI 1
1376.2 3 a, b V89 2
3062.0 10 a, b V7 1
* a =4358 ,b =4046A.
4 G, Glockler and M. M. Renfrew, Rev. Sci. lnst. 9, 306
(1938).
224
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07FREQUENCIES OF CERTAIN HALOMETHANES 225
of exciting radiation was eight Ne-Hg discharge
tubes arranged cylindrically around the liquid
sample. Eastman spectroscopic plates, type 1-J,
were used. Six lines were observed and they are
given in the Table I. Microphotometer tracings
of a portion of the CHF a spectrum, showing all
six lines, are reproduced in Fig. 1.
III. VIBRATIONAL ANALYSIS
The haloforms belong to the symmetry class,
Cav. There are three parallel vibrations of the
type A and three perpendicular vibrations of the
type E. The predictions of group theory with
regards to the symmetry properties of the normal
modes of vibration are given in Table II.5a The
notation is that employed by Kohlrausch for
CHCIa.5b
Intensity considerations
The symmetric vibrations, V6 and VI are very
strong in CHCla and CHBral while the symmetric
vibration, V7, is somewhat weaker. The perpen
dicular vibration, V45, is moderately strong while
the other two perpendicular vibrations, V2a and
V89, are weaker. In CHCla and CHBra the order
of increasing values for the Raman shifts
expressed in cm-1 is: V45, V6, VI, V23. V89, and V7.
By comparison of the data in Table I with the
above, it is evident that the CHF a lines 508,
697, 1376,and 3062 cm-I correspond, respectively,
to V45, V6, V89, and V7. This leaves a weak line at
937 and an intense one at 1117 cm-I yet to be
TABLE II. Symmetry properties of the halo form vibrations.
STATE OF
POLARIZA- No. OF
TYPE c~ (II nON LINES NOTATIO-:-J ---
A s s P 3 V1, V6, V7
E e e dp 3 Vn, V4S, V89
assigned to VI and V2a. The vibrations VI and V2a
lie at 539 and 656 cm-I in CHBra with relative
intensities 10 and 5. The corresponding CHCla
fundamentals are 668 and 760 cm-I with relative
intensities 9 and 4.
Thus one expects VI to be strong in CHF a and
to lie .80 to 100 cm-1 lower than the weaker V2a.
5 K. W. F. Kohlrausch, Dey Smekal Raman Effekt,
Ergiinzungsband (J. Springer, Berlin, 1938), (a) p. 45;
(b) p. 150. ~~(508)
FIG. 1. Microphotometer tracing of Raman spectrum of
CHF 3. 3062 cm-I is excited by 4046 while all others by
4358A.
However, the opposite order of intensities is
actually observed. It is hard to conceive of any
logical reason for the symmetric "breathing"
vibration to be so weak. The weak line observed
at 937 cannot be an overtone unless we assume
an abnormally large anharmonicity. It is also
impossible to assign it to any likely impurity.
On the basis of the above considerations 1117
cm-I is assigned to VI and 937 cm-I to V2a. A
systematic study of the haloform vibrations by
means of the theory of small vibrations shows
that this is actually the case. The assignment is
given in Table I.
Application of the theory of small vibrations
The factors which determine the region of the
spectrum in which a particular vibration will be
are in the order of their influence: (1) mass;
(2) force constants; (3) interatomic dimensions.
By far the most dominant of these is the mass.
Thus, one may assume a reasonable potential
function with reasonable force constants and
trace the influence of mass upon the various
vibrations in a spectral sequence of the same
symmetry, the same force constants being used
throughout the sequence. The one chosen here
was CHXa where X=H, D, F, Cl, Br, and I.
':fhis same method has been applied with notable
success to the methyl halides by Wagner.6
It is known7 that the methane derivatives
with several heavy substituents (as CCI4) are
best described by a potential function of the
central force field type rather than one based on
a valency force field. The general quadratic
6 J. Wagner, Zeits. f. physik. Chemie B40, 36 (1938).
He has also applied the same method to the methylene
halides (Zeits. f. physik. Chemie B45, 69 (1939)), but here
the reduced symmetry calls for aU nine frequencies, six of
which lie somewhat close together in the region below 1500
COl-I. This leads to some question as to the assignment of
the fundamentals to the particular modes of vibration,
7 D. M. Dennison, Rev, Mod. Phys. 12,175 (1940).
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07226 G. GLOCKLER AND W. F. EDGELL
, , "'" "''''' "'" """ JOOO em-I
·f' / / , ~/ , /
/ /
/ I ~~~ / I ,jif--! .• / / /~~
j I , /' If I I /'
/ I /
/ I ,
/
m Z ~ \
V4~'V. VI V:a U" U,
FIG. 2. Theoretical trends in the fundamental vibrations
of the molecule CHX 3 under influence of change in mass
of X atom. --parallel vibrations; ---perpendicular
vibrations.
potential as proposed by Rosenthal and VogeS
was used in these calculations. The force con
stants were taken as intermediate between those
of CH4 and CHCIa (as determined by Voge and
Rosenthal9) and were closer to the values for
the latter molecule. They are as follows: G1l,
2.90XI05 dynes/em; G22, 4.92; Gaa, 0.82; G12
and G2a, 0; Gal, -1.11; G44, 1.50; G66 and G66,
0.56; G46 and G46, -0.20; G66, -0.30. The inter
atomic dimensions used were C -Hand C - D
=1.09A, C-F=1.40, C-CI=1.71, C-Br
= 1.90, C - I = 2.20. Tetrahedral angles were
assumed for the valence directions.
The parallel vibratons gave no difficuty. In
the case of the perpendicular vibrations there is
a large change in the force constants in passing
from CH4 to CHCla. If the values were taken
too close to those for CHela, the solutions for
the perpendicular vibrations in CH4 and CHDa
were complex. If, on the other hand, the values
for CH4 were taken, the trend in the haloform
vibrations was not followed too closely. The
perpendicular force constants, G44 to G66, are a
compromise between these two difficulties. They
yield good values for CHCla, but do not quite'
give real values for the perpendicular vibrations
in CHDa and CH4• However, the value taken
for 1'46 in CHDa, 826 em-I, satisfies the cubic
equation for the perpendicular frequencies to
4 percent. The values for 1'2a and I'S9 of CHDa
are the solutions of the quadratic equation which
results from the perpendicular cubic when the
above value of V46 is substituted into its solutions.
8 J. Rosenthal and H. H. Voge, J. Chern. Phys. 4, 134
(1936).
9 H. H. Voge and J. Rosenthal, ]. Chern. Phys. 4, 137
(1936). This procedure is justified since values derived
from force constants which yield real solutions
(i.e., closer to the perpendicular CH4 constants)
do not lie far from those determined by the
above method. It is obvious in which CH4
fundamentals these perpendicular frequencies
have their origin, and the corresponding curves
were extrapolated back to these values. The
calculated trends in the fundamental vibrations
of the spectral sequence CH4-CHDa-CHFa
-CHCla-CHBra-CHl a under the influence of
mass and interatomic dimensions are given in
Fig. 2. The frequencies, expressed in em-I, are
plotted versus v(1/m) where m=mass of X
atom in the molecule CHXa.
The salient features of Fig. 2 are the wide
sweep of VI from a high value in CH4 to a
comparatively low value in CHla and the slight
effect of change in mass upon V23. Thus VI and
V23 must cross. In predictions of the fundamental
frequencies of CHF a it has been tacitly assumed
that this crossing took place in molecules of
smaller mass than CHF a if indeed this sequence
was ever considered. Fig. 2 also has this property.
However, this reasoning (as well as the calcula
tion) fails to allow for the comparatively high
values of the force constants involving the
motion of the F atom. For the purpose of
comparison the values of the force constants in
the methyl halides as determined by LinnettiO
may be consulted. Both the C - F stretching
and the deformation constants are high. The
force constants and the masses enter into the
solutions for the classical vibrations as ratios
(i.e., kim). The frequency, PI, which is most
dependent upon a change in mass of the X atom,
is therefore most sensitive to a change in force
constants involving the motion of the X atom
in particular the C-X stretching constant. For
the same reason the vibration, V23, which is
most independent of a change in the mass of
the X atom, is also least sensitive to changes in
these same constants. Thus it is not surprising
to find that VI is displaced far enough to higher
values to delay the experimental crossing of VI
and V23 until after CHF 3 is passed in this sequence.
During the course of the preparation 0'£ this
paper it was found that Wagnerll had recently
10 J. w. Linnett, ]. Chern. Phys. 8, 91 (1940).
11 ]. Wagner, Zeits. f. physik. Chemie B45, 341 (1940).
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07FREQUENCIES OF CERTAIN HALOMETHANES 227
made a similar calculation for this same sequence
in reference to the assignment of fundamentals
in trimethyl methane. His calculations were
based upon a hypothetical molecule where all
stretching force constants were set equal to
3.6X 106 dynes/cm and all deformation constants
equal to 1/10 of this value. A simple valency
force field was assumed. His theoretical curves
give essentially the same trends as those found
in Fig. 2. Inasmuch as the haloforms are more
closely expressed by a potential of the central
force field type rather than a valency field, and
since the force constants chosen in this investi
gation are based on those actually found in this
sequence, the theoretical curves, 111 and 112a, in
Fig. 2 follow the experimental curves somewhat
better. For instance, 111 and 112a cross at m-! = 0.35
in Wagner's curves, 0.30 in Fig. 2, and 0.24 in
the experimentally determined curves (Fig. 3).
The trends in the fundamental vibrations of
this sequence as experimentally determinedl are
shown in Fig. 3. It should be pointed out that
the force constants play a larger role in the
haloform sequence than in the methyl halides,
and hence there is not as close agreement
between the calculated curves (which take into
consideration the change in mass only) and the
experimental curves as in the latter case.6
Nevertheless there is no difficulty in establishing
the trends in the experimental values. In fact
it would be quite surprising if 112a in CHF a came
at any other place than the experimentally
determined one.
Wagner'sll results for (CHa)aCH confirm the
conclusions drawn above. The experimental
value for his 112, the vibration least dependent
upon mass and force constants, is 965 cm-l• As
can be seen from our Fig. 3, this value lies
10 cm-l from the experimental curve for 112a at
the place where the mass of the X atom is fifteen.
The values for the other similar vibrations all
lie between those of CHCla and CHF a-the
largest shift being in 111. This is in harmony with
the fact that the C - C stretching constant is
about 4.5XI05 dynes/cm.12
The theory of small vibrations gives yet
another, independent indication of the correct
ness of the assignment made in CHFa. Voge and
12 See for instance Ta-You Wu, (reference 1), p. 303. Rosenthal9 postulating constancy in the CHa
and CCla groups and the C -Hand C -CI bonds,
have determined values for the twelve force
constants in CHCla by means of the sequence
CH4, CHaCl, CCI4 and CHCla. They found that
if one sets G12=G23=0 in CHCIa and if they used
the values of the other constants from CCI 4, CH4
and CHaCl then the values for the parallel
vibrations (116, lilt and 117) were exceptionally
close to the experimental values. In applying
the same method to the corresponding fluorine
molecules the solution for the force constants of
the parallel vibrations in CHaF is complex. This
is not surprising inasmuch as Slawsky and
Dennison,13 using a very similar method applied
to the methyl halides only, encountered the same
difficulty. This results from the assumption of
constancy in the CHa group, neglect of anhar
monicities, etc. It is not serious as long as the
complex part of the solution is small in com
parison to the real part. This is the case here.
Algebraic manipulation makes the quartic in
G12 easiest to obtain. The value G12 = 1.55 X 105
dynes/cm is the real value which most nearly
satisfies this equation. This leads to G22 (the
C - F force constant) = 6.4. Slawsky and Denni
son, using a valency force field with cross terms.
arrive at the value 6.09 X 105 dynes/cm for the
same quantity. Using the above value and the
most general quadratic potential function for T d
symmetry as proposed by Rosenthal,14 the
constants for CF4 are: A=9.25, B=0.671,
C=0.532, D=0.863, E=0.569. The set with the
negative values for D has been discarded. When
these values, together with the C - H force
, .00 'zoo 1100 '" s~o ~m'
HCH
:./. / ~~/I / I
/ I
HCD /' /
l~/ t // /1 "'/'
V45 /~ IVn V ........ 4, v, . 7 I / {;k I I /
HC' ( VI //
HCC!, ,--..----- / /
HC", '/ / .' HCI i.-I
FIG. 3. The spectral sequence CHX 3, --parallel
vibrations; - - -perpendicular vibrations; II estimated
values.
13 Z. I. Slawsky and D. M. Dennison, J. Chern. Phys. 7,
522 (1939).
14 J. Rosenthal, Phys. Rev. 45, 538 (1934).
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07228 G. GLOCKLER AND W. F. EDGELL
CH
CH F
CHF.
CF """
I
I •
FIG. 4. The spectral sequence CH.-CF •. --connects
non degenerate vibrations; ---connects degenerate vi
brations.
constant=4.92X10· dynes/em, are applied to
the CHF 3 molecule, the values for liS, III, and IIi
are 748, 1163, and 3019 em-I. These are to be
compared to the experimental values 697, 1117,
and 3062 em-I. It can further be shown that
any change in the CF4 constants caused by a
change in the C - F constant (using this method
of evaluation) affects liS and III almost identically
and 117 in the opposite sense. Thus a change that
would bring liS from 748 to 697 would also bring
III from 1163 very near to the experimental
value, 1117, at the same time raising the value
of 117 above that of 3019 em-I. To summarize
the above it may be stated that, using force
constants based entirely upon the molecules
CH4• CH3F, and CF4 with the assumption of
constancy in the CH3 and CF 3 groups and C - H
and C - F bonds, it is possible to calculate values
for the parallel frequencies of the CHF 3 molecule
which differ by only 50 em-I from the experi
mental values. Furthermore any change which
tends to bring anyone of these calculated
frequencies closer to its experimental value also
tends to bring the other two calculated fre
quencies closer to their own respective experi
mental values.
These two independent calculations based on
the theory of small vibrations, neither of which
is dependent in any way upon any experimental
data for CHF 3, point definitely to the assignment
given in Table I. Nevertheless, this assignment
must be considered as tentative until polarization
data are available for this molecule.
Empirical correlation
An important property of a set of correctly
assigned fundamentals for the halomethanes is
that it be consistent with any spectral sequence
that may 'be devised for it. It is apparent that this consistency has been achieved in the
sequence given in Fig. 3. The assigned funda
mentals in CHF 3 also correlate nicely with other
sequences. For instance, the spectral sequence
CH4-CH3F-CHF 3-CF 4 is shown in Fig. 4.
No other assignment would be nearly as
sa tisfactory.
IV. THE FUNDAMENTALS IN CHClF 2
As mentioned above, Glockler and Leader3
have estimated the positions of the CHF 3
fundamentals. On the basis of these estimated
values they have attributed the lines 460, 667,
and 831 cm-I, observed by theml• in the Raman
spectrum of CHClF 2, as being due to CHF 3.
In view of the experimental data presented
above this interpretation can no longer be made,
and hence the question of the fundamentals of
this molecule must be reopened.
The values for the Raman lines reported by
them are as follows: 369.1 (4); 410.5 (3); 415.4
(10); 459.5 (1); 595.1 (9); 667.3 (2); 793.8 (4);
804.7 (4); 830.8 (2); 1088.4 (1); 1109 (1);
1310.2 (2); 1350.0 (2); 3035.0 (6). The values in
parenthesis are the relative intensities. From
these fourteen values nine fundamentals must
be chosen. Their interpretation that 410 and
415 cm-I are due to the same vibration in
CHCl37F 2 and CHCpsF 2, respectively, seems
reasonable and is made here also. These two
lines ar.e only 5 cm-I apart, yet they are clearly
seen as two lines of different intensities on their
plates. This raises some question as to their
treatment of each of the b!,oad lines at ca. 800
cm-I and ca. 1100 cm-1 as a pair of lines. Here
these lines shall be treated each as only one line,
and hence measurements were made on the
center of these lines. The values are given in
TABLE III. The fundamentals in CHClF, and CHCl,F.
As-I As- SIGN- sIGN-
MENT CHCIF, CHCJ,F MENT CHClF, CHCJ,F ---------- ------ ----v. 369 cm-I 277 cm-I va 1099 cm-I 1067 cm-I v, 415 366 V8 1310 1255
V6 595 457 V9 1350 1310
VI 799 732 Vl 3035 3020
V2 831 795
i
16 Two of these lines were previously observed by G.
Glockler and ]. Bachmann, Phys. Rev. 55, 669 (1939)
using the same sample.
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07FREQUENCIES OF CERTAIN HALOMETHANES 229
Table III. The lines at 460 and 667 cm-I are now
attributed to CCI2F 2 as an impurity. The two
strongest lines of this molecule, as reported by
Bradley,2 are 455 and 664 cm-I with intensities
7 and 10. These are the relative intensities
observed for the lines in question. Moreover
values reported by this laboratory are on the
average about 3 cm-I higher than those observed
by Bradley for the same substances (d. data for
CHCI2F as determined by Glockler, Edgell, and
LeaderI6). Dr. Benning, who also supplied the
CHCIF 2 used by Glockler and Leader, found
evidence of an impurity in his sample of CHCIF 2
which he was unable to removeY Vapor density
measurements at low pressures indicated a
molecular weight of 87.25 instead of the theo
retical value of 86.46. This corresponds to 2.3
percent of CCI2F2 as the impurity.
The above elimination leaves nine lines and
they are attributed to the nine fundamentals of
CHCIF 2. Six of these are the same as reported
by Glockler and Leader. Two more come from
the treatment of the two broad lines as having
only one component. The ninth fundamental is
the line at 830.8 cm-I. This might be interpreted
as the overtone of 415.4, but this postulates no
anharmonicity. Moreover it appears too intense
for this classification. It is evident from a
consideration of several sequences that a funda
mental should lie in this region, and hence we
assign it to 1'2.18 The values of the fundamentals
selected for CHCIF2 and their assignments are
given in Table III. These fundamentals are
correlated with those of CHCbF, CHCIBrF,
and CHCIBr 2in Fig. 5.1 It would appear at
first glance that Va is inconsistent. However, it
has been showna that trihalomethanes show a
characteristic C - X vibration whenever X is
the lightest halogen present. It can also be
shown that the presence of more than one of the
lightest halogen atoms modifies this vibration
only slightly. In CHCIBr2, Va has the value
characteristic of the C-Cl vibration, 750 cm-I.
In passing to CHCIBrF and CHCI2F fluorine
16 G. Glockler, W. F. Edgell, and G. R. Leader, J. Chern.
Phys. 8, 897 (1940).
17 A. F. Benning and R. C. McHarness, Ind. Eng.
Chern. 32, 698 (1940).
18 A private communication from Dr. Ta-You Wu has
recently been received in which he also makes the same
assignment. becomes the lightest halogen present and Va has
the characteristic C - F vibration, ca. 1065 cm-I.
The addition of a second fluorine atom to
form CHCIF 2, of course, only slightly changes
the value of Va to 1099 cm-I. The consistency of
Fig. 5 is a strong argument for the correctness
of the assignment given to CHClF 2 in Table III.
V. HEAT CAPACITIES OF CHCl 2F, CHClF2,
AND CHFa
These compounds are of importance in the
refrigeration industry, hence accurate thermo-
1400-___ ""-1
c"",o.
\\ \ I I , , I I I I \ I , I
\ \ \ \ \ , , I ,
I I \ I I , \ \
I I \ \ \ , , \ I
\ , \ \ \ , \ \
I I , I , I I CHCIBoT I \ , , ,
I, I
I \ , I I I I I , I \ \ \ \ \ I ,
\ \ \ I I I
\ I ,
\ \ I I \ I I I
\ \ I I I : I I
CHC I I I , I , I
I \ \ \ I I \ I
\ \ \ I I \ I I \ \ I \ I \ I \ , , \ \ \ \ I \ I \ , \ \ \ \ I I
CHelF ' I II I , I
".115 V, VrVz " 1)6 Vg V,
FIG. 5. The spectral sequence CHCIBr2-CHCIF2.
dynamical data are of considerable importance.
Unfortunately no thermal data are available by
which one might evaluatel::..Hoo of formation.
It was therefore impossible to obtain the free
energy or heat of formation foe them. The heat
capacities were calculated by the well-known
harmonic oscillator-rigid rotator approximation.
It has been shown by a number of investi
gatorsl9 that for this type of molecule the
rotational heat capacity has very nearly its
classical value by the time room temperature
has been reached. Moreover the assumption of
rigid rotators and simple harmonic oscillators
causes little error.20 The values for the funda
mental frequencies of CHCI2F are given in
Table III from the data of Glockler, Edgell, and
Leader.16 They have interpreted the two lines
at 727 and 738 cm-I as being due to Fermi
resonance, so 732 was taken as the value of the
unperturbed fundamental. The use of the mean
19 D. P. MacDougall, Phys. Rev. 38, 2074 (1931); D. S.
Villars, ibid. 38, 1552 (1931), and others.
20 A. R. Gordon and C. Barnes, J. Chern. Phys. 1, 692
(1933); ibid. 2, 65 (1934), and others.
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07230 G. GLOCKLER AND W. F. EDGELL
leads to little error in this case since the two
lines are so close together. The values used for
CHClF 2 and CHF a are from this paper. In both
CHClF 2 and CHCl2F each fundamental corre
sponds to just one degree of vibrational freedom.
In CHF3 there are nine degrees of vibrational
freedom and six lines. Three of these, the
perpendicular vibrations, have a statistical
weight of two.
As is customary, the values of all constants
were taken from the International Critical
Tables. Recently a paper has appeared21 in which
the newer values for the fundamental constants
were used. In the case of heat capacities this
makes little difference-changing the value of
hc/k from 1.432 to 1.435 if the values of Dun
nington22 are used. Some change, however, would
be introduced into the values for entropy and
free energy. Since thermodynamics is primarily
interested in the change in these quantities for
certain reactions, it is felt that greater inner
consistency will be achieved if all calculations
are based on the same set of constants. Giauque,23
in a quaint parable, has shown the difficulties
that arise when two sets of values are used in
the case of temperature.
The calculated values of Cpo appear in Table
IV. The standard state is the usual one of unit
fugacity and zero pressure.
The correction to finite pressures is given by
Benning and McHarness17 have determined data
of state for CHCl 2F and CHClF2 which they
TABLE IV. Cpo for some trihalomethanes.
T"K CHF, CHClF, CHCbFt T"K CHF, CHClF, CHCbFt -- --cal./"Kmole cal./"Kmole
250 11.37 12.50 13.41 450 16.24 17.12 17.74
273.1 12.01 13.14 14.01 473.1 16.68 17.53 18.11
298.1 12.68 13.79 14.64 500 17.15 17.95 18.51
300 12.73 13.85 14.68 550 17.95 18.67 19.17
050 14.02 15.07 15.85 600 18.65 19.30 19.75
373.1 14.58 15.59 16.32 650 19.27 19.85 20.26
400 15.19 16.17 16.85
t CpO for this substance was calculated by G. R. Leader, Studies in Molecular
Structure (Ph.D. thesis), University of Minnesota (1940), at different temperatures
and using hc/k= 1.435.
21 E. H. Eyster and R. H. Gillette, ]. Chern. Phys. 8,
369 (1940).
22 F. G. Dunnington, Phys. Rev. 55, 683 (1939).
23 W. Giauque, Nature 143, 623 (1939). express by means of the equation
Although their equations reproduce their data,
they lead to unreasonable values for (a2v/ap)p.
This is quite understandable since only several
measurements were made at anyone pressure.
The modified Berthelot equation of state was
used in evaluating this second derivative. This
gives
The critical temperature and pressure of CHCI 2F
and CHClF 2 used were determined by Benning
and McHarness.24 They are 451.6°K, 51.0 atmos.
and 369.1 °K,48.7 atmos., respectively. Booth and
Swinehart25 have also determined Tc=369.5°K
and Pc=48.48 atmos. for CHClF 2• The critical
temperature of CHF 3 has been estimated by the
Guldberg-Guye26 rule, Tc=KTb, where Tb is the
temperature of the boiling point. Bachmann27
has shown that K = 1.6 for the fluorochloro
methanes. This gives To = 306°K. He has also
devised a means of correlating Pc with molecular
weights for these same molecules. This leads to
an estimated value of Pc=47 atmos. for CHFa
which cannot be too much in error since this
method leads to a value of Pc=49.7 atmos. for
CHClzF. Should the fluoroform value be in error
by 3 atmos., it would lead to an error of only
6 percent in the correction which is only 0.1
caI.;oK mole at 298.1 OK and 0.05 cal. at 373.1 oK.
Benning and co-workers28 have determined the
heat capacities of CHCIzF and CHCIF2 at one
atmosphere in the temperature range 35°C to
135°e. In view of the small number of determi
nations no conclusions can be drawn with regard
to the correctness of our assigned fundamentals.
The values of Cpo have been fitted to the
usual empirical equation by the method of least
squares. To each of these equations have been
.. A. F. Benning and R. C. McHarness, Ind. Eng. Chern.
31,912 (1939).
26 H. S. Booth and C. F. Swinehart, ]. Am. Chern. Soc.
57, 1337 (1935).
28 P. Fugassi and C. Rudy, Ind. Eng. Chern. 30, 1029
(1938).
21]. H. Bachmann, Studies in Molecular Structure (Ph.D.
thesis), University of Minnesota, (1939).
28 A. F. Benning, R. C. McHarness, W. H. Markwood,
and]. W. Smith, Ind. Eng. Chern. 32, 976 (1940).
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129.120.242.61 On: Sat, 29 Nov 2014 00:29:07SPECTRA OF ALDEHYDES AND KETONES 231
added (the last term) the correction to finite
pressures given by the equations and constants
of state discussed above. They are
CHCI 2F: Cp=5.334+0.03815T
-23.48X 1O-6T2+9.081 X 106P IT",
CHCIF 2: Cp=3.929+0.04034T
-24.51 X 10-6T2+5.189X 106P IT",
CHF 3: Cp=2.699+0.04037T
-22.92 X 1O-6T2+3.066X 106P IT".
Error
The largest error in these calculations is that
due to the use of the Raman frequencies deter
mined in the liquid state. This on the whole
should not exceed 2 percent. The error due to
the assumption of rigid rotators and simple
harmonic oscillators is somewhat less than 1 percent. Thus the tabulated values of Cpo
should be accurate to 3 percent. The empirical
equation for Cpo of CHCl 2F reproduces the
tabular values to within 0.04 cal.;oK mole; for
CHClF 2, 0.03; for CHF 3, 0.02. The use of the
modified Berthelot equation of state in correcting
to finite pressures introduces further error. All
in all, the empirical equations above should give
the heat capacities of the gases at one atmosphere
to 5 percent. Even so they are more reliable
than the fragmentary experimental data that is
available at present.
We wish to thank Dr. Benning of the Jackson
Laboratory of E. I. du Pont de Nemours and
Company for the loan of the fluoroform used in
this investigation.
MARCH, 1941 JOURNAL OF CHEMICAL PHYSICS VOLUME 9
The Long Wave-Length Spectra of Aldehydes and Ketones
Part I. Saturated Aldehydes and Ketones
HENRY L. McMURRY
Ryerson Physical Laboratory, University of Chicago, Chicago, Illinois
(Received October 21, 1940)
Intensity calculations are used to help in deciding
which of two theoretically predicted transitions should be
identified with the weak longest wave-length absorption
"'" characteristic of the unconjugated C =0 group. In terms
/
of the localized LCAO MO and the AO approximations
used, one of these transitions is forbidden while the
calculated intensity for the other, allowed, transition is
much too large to be compatible with the low intensities
observed for the absorption. The observed absorption
should, therefore, be ascribed to the forbidden transition.
The Q and S integrals needed for the calculations, involving
AO's of the C and 0 atoms, are tabulated. The intensity
for the allowed transition, when expressed using semi
localized LCAO MO's depends somewhat on the nature of
I. INTRODUCTION
WHENEVER the ca,bonyi (~c~ 0 )gmup
is present in a molecule a weak absorption
region appears at relatively long wave-lengths.t-5
1 Cf. E. Eastwood and C. P. Snow, Proc. Roy. Soc.
AI49, 434 (1935). the atoms attached to the C = 0 group. The absorption
regions appearing at shorter wave-lengths in ketones and
aldehydes are discussed. It is concluded that the A1900
region in ketones is characteristic of the C = 0 group and
probably is due either to a Rydberg transition or most
likely to the allowed transition for which intensity calcu
lations have been made. If the latter is true the upper level
of the A1900 absorption could be largely responsible for the
perturbations which cause the appearance of the long wave
length forbidden transition. It is shown that the long wave-
"'" length absorptions from the C = S group can be explained
/
in the same manner as those for the C = 0 group.
/
In saturated aldehydes and ketones (molecules of
the type RR'CO, where Rand R' are alkyl
2 G. Scheibe and W. Fromel, Eucken-Wolf Hand-und
Jahrbuch der chem. Physik 9, Part IV (1936); especially
p. 164 on.
a F. O. Rice, Proc. Roy. Soc. A9I, 76 (1914-1915): pure
liquid ketones.
4 J. Bielecki and V. Henri, Ber. d. D. Chern. Ges. 46,
3627 (1913): aldehydes and ketones in alcohol solution.
6 Int. Crit. Tab. Vol. 5; Landolt-Bornstein, Physikalisch
Chemische Tabellen.
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Subsets and Splits