Theoretically Interesting Aspects of High Pressure Phenomena

REVIEWS OF

VOLUME 7 JANUARY, 1935 NUMBER 1

Theoretically Interesting Aspects of High Pressure Phenomena

P. W. BRIDGMAN, Harvard Un~eersity

TABLE, OF CONTENTS

I. Introduction.II. Atomic Changes Under Pressure. . . . . . . . . . . . . . .

III. Volume Changes and the "Law of Force". . . . . .IV. Thermal Expansion and Entropy at InBnite Pres-

sure.V. P—V-T Relations in Liquids

1. Compressibility2. Compressibility and chemical composition . .

3. Thermal expansion.4. The pressure coefficient and the mechanism

of pressure.5. Small scale irregularities. . . . . . . . . . . . . . . . .

VI. Periodic Relations.

91011

111213

VII. Compressibility of Single Crystals. . . . . . . . . . . . . 13VIII. Two Phase Equilibrium

1. The melting curve 132. Polymorphic transitions between solids. . . . 16

IX. Irreversible Transitions . 20X.Discontinuities —Transitions of the Second Kind, etc. 21XI. Electrical Resistance. 23XII. Thermoelectric Phenomena 27XIII. Thermal Conduction 28XIV. Viscosity of Liquids. 30XV. Conditions of Rupture. 30XVI. Speculations. . . . . . . . 31XVII. Bibliography . 33

I. INTRQDUcTIoN

T HE invitation of the Editors of Reviews ofModern Physics to write some account of the

theoretical aspects of high pressure phenomenawas one which I was glad to accept because itseems to me that the time is now approachingwhen theoretical physics may hope to investigatethe problems of this field with good prospects ofsuccess. Until very recently the condensedphases of matter, solid and liquid, have appearedtoo complicated to make it worth while tospend much effort in acquiring an understandingof them, particularly as long as the much moreprofitable field of the investigation of matter inits rarefied condition, as in vacuum tube phe-nomena of all sorts, had not yet been fully ex-ploited. But now our understanding of theatomic, as distinguished from nuclear, phe-nomena presented by matter in its rarefied states

is rapidly becoming satisfactory, and in a senseexhausted, so that an attack on the problem ofthe condensed states is obviously next on theprogram. Indeed already a very considerabledegree of success has rewarded theoretical at-tack in this field, as, for example, in our rapidlygrowing theory of the metallic state in generaland of the electrical properties of solid conduc-tors in particular. The condensed state, parexcellence, is obviously presented by matter under

high pressure, so that, to say the least, our under-

standing of the condensed state cannot be re-

garded as satisfactory until we can give an ac-count of the effect of. pressure on every varietyof physical phenomena. This we can at presentdo in very few cases indeed.

There are two aspects of theoretical concernwith high pressure phenomena: a broader and anarrower aspect. From the broad point of view

P. W. 8 R I DG MAN

the eventual problem is to work out theoreticalexplanations of all known high pressure phe-nomena, and to predict the result of fresh ex-tensions of experiment to pressures and groupsof phenomena not yet reached. It must not beassumed too easily, I think, that this is a taskof no particular interest, and that the problemis merely the problem of overcoming the compli-cations of an analysis, the fundamentals of whichare already completely understood. Such iswithout doubt the attitude of many theoreticalphysicists; in fact it seems to be a thesis oftheoretical physics in its treatment of matterin bulk that there are no "emergent" properties,or in other words, that all the properties ofaggregates of atoms can be found from anexhaustive knowledge of the properties of theisolated atoms. There is a sense in which thisthesis may be regarded as a mere tautology,because I suppose no one would deny that itwould be possible to put enough parameters intothe equations for the individual atoms or elec-trons to reproduce by some more or less compli-cated kind of theory all the properties of com-binations of atoms in bulk. But the thesis doeshave real content if one understands it to meanthat it is possible by experiments on isolatedatoms, or atoms in the rarefied condition, todetermine all the parameters necessary todescribe all the properties of condensed assem-blages of atoms. No doubt the impulse of manywould be to say that in experiments involvingnuclear bombardment and breakdown we aredealing with individual atoms under conditionsof much greater intensity of force than are everencountered in condensed aggregates of atoms,so that there is no reason to think that such ex-periments will not give all the effective atomicparameters. But on the other hand it must beremembered that in highly condensed phases thecharacter of the force to which the atom is sub-jected is different from that in collision experi-ments, the attack on the atom being now a moreor less symmetrical and simultaneous attackfrom all sides, so that there may be a possibilityof new kinds of effect. So far as I know no the-oretical intimation was given, before the experi-mental evidence of astronomy, of the possibilityof the existence of matter in conditions of densityof the order of 100,000. Our persistent difficulty

in understanding superconductivity of metalsmay also be significant. At any rate, I believethat it must be conceded as a matter of purelogic that the thesis of non-emergent character-istics cannot be securely established until atleast the possibility of a theoretical deduction ofall the properties of matter in the condensedcondition has been established; what one' sfeeling will be as to the interest or profitablenessof actually producing such detailed explanationswill be largely a matter of taste and temperament.

The narrower aspect of the high pressure prob-lem concerns the extent to which we can under-stand high pressure phenomena in terms ofrecent wave mechanics pictures of atomic be-havior and interaction. This is the aspect of theproblem which will serve as the background ofour present discussion. It would have been de-sirable if this paper could have been written bysomeone who has made actual contributions toour understanding in terms of wave mechanicsof the behavior of condensed phases, instead ofby one whose qualification is merely an acquain-tance with the nature of the experimental ma-terial. I shall have to content myself, therefore,with pointing out those aspects of the phenomenawhich are simplest and therefore where theoret-ical attack may most reasonably anticipatesuccess, or those aspects which seem to me mostsuggestive and of intrinsic interest. I believethat our theoretical mastery in this field is notyet so far advanced but that a careful ponderingof the qualitative significance of various generaltypes of pressure phenomena will'be profitable.I have this feeling because a number of yearsago I had arrived at various qualitative pictures,as of the phenomena of electrical conduction orof polymorphic transition, which were somewhatat variance with the pictures common at thetime, but which are becoming more and morejustified by wave mechanics.

The experimental material which must formthe basis of our present theorizing is almostentirely confined to the range from 10,000 to20,000 kg/cm', although a few data have beenobtained at somewhat higher pressures and thereis the possibility of more in the future. Thesepressures are of the order of magnitude of theinternal pressures which various theories haveagreed in assigning to ordinary condensed phases;

HIGH PRESSURE PHENOMENA

such internal pressures vary from 3000 or 4000kg/cm' for organic liquids and 10,000 or 20,000for the more compressible metals, to a few hun-dred thousand for the most incompressible sub-stances like iridium and diamond. Furthermore,the volume changes producible by pressures inthe experimental range are materially greaterthan the volume changes due to temperature oncooling from room temperature to O'K. Onewould appear to be justified therefore in antici-pating that an adequate understanding of eventhe effects of pressure known at present wouldreact beneficially on our general understandingof matter in the condensed state. By the applica-tion of pressure we have the tool, as it were, forproducing artificially a great variety of newcondensed states of matter.

In the following I shall not attempt to givemore than the very briefest possible indicationof the nature of the experimental material andrefer the reader to my book The Physics of HighPressure published by Bell in England and Mac-millan in this country, whenever he feels the needof more detailed acquaintance with the data.

I I. ATOMIC CHANGES UNDER PRESSURE

Perhaps the first and most important questionthat confronts one on entering this field is

d T= —dE+3d(pv),d V= 2dE —3d(pv).

(1)

(2)

Here T is the average internal kinetic energy andV the average internal potential energy, theaverages being taken over a time interval longenough to give constancy, Z is the total energyof the system and v the volume.

whether it is legitimate to treat the atoms asfixed units, or whether pressures in the experi-mental range produce important changes in theatoms themselves. A rough answer is suggestedto this question by means of an importanttheorem, which has been much neglected, origi-nally due to Schottky' and proved on the basisof classical mechanics, Later Born, Heisenbergand Jordan' showed that essentially the samesituation holds in wave mechanics, and thetheorem has recently been re-emphasized andapplied to some molecular problems by Slater. ~

Schottky's theorem states that in any systemcontrolled by internal electromagnetic forces,whether there are or are not in addition con-straints imposed by quantum conditions, as forexample, systems composed of molecules andatoms built of nuclei and surrounding electronicatmospheres, and on which in addition to theelectromagnetic forces an external hydrostaticpressure, p, acts, the following relations hold:

By thermodynamics we have:

dE= C„—p — d7- r — +p — dp.

Eliminating dB gives:

dT= 4P — —Cp dr+ 3v+~ — +4p — dp,Br p 87' p Bp

(3)

d V= 2C„—SP — d, —3v+2r — +5P — dP. (4)

Apply these equations to ordinary solidsubstances in the experimental range of pres-sure, and consider first the variation of T withpressure at constant temperature, paying atten-tion only to orders of magnitude. r(8v/8r)p isevidently small compared with 3v and may beneglected. The most compressible solid metal

is caesium; at 15,000 kg/cm' 4P(8v/BP), has thevalue 0.6 and 3v is equal to 2, so that (aT/ap),=1.4 for Cs at 15,000. For lithium, on the otherhand, the compressibility is much less, and(BT/BP), at 15,000 has the value 2.4, the deriva-tive referring in both cases to that quantity ofmatter which occupies 1 cm' at atmospheric

P. W. BRIDGMAN

pressure. Most solids are much less compres-sible even than this, so that we would not bemaking an error on the average much more than10 or 15 percent if we set BT/8&=3 in the ex-perimental range of pressure. Hence at 20,000kg/cm' we have at once T2p, ppp

—Tp 60,000kg/cm =3.5 X 10" electron volts. Expressedper atom, this becomes 3.5X10"X1.66X10 "Xat, wt. /dens. =0.06 at. wt. /dens. , or

Tpp, ppp—rp 0.06 Xat. vol.

in electron volts per atom. (5)

For elements of high atomic volume this ap-proximation is in general less good than for ele-ments of low atomic volume, because compres-sibility is highest for high atomic volumes, andthe actual value is less than the approximatevalue by a term proportional to the compres-sibility. In the ordinary range of temperaturethe specific heat of a solid is given approximatelyby Dulong and Petit's law, which ascribes threedegrees of freedom to the kinetic energy oftranslational motion of the atom as a whole, andthree degrees to the potential energy of position.Hence to this degree of approximation the kineticenergy of translation of the atoms is constant,independent of pressure, at constant tempera-ture, and the change of kinetic energy with pres-sure which is given by Eq. (5) is change of in-ternal kinetic energy of motion of the electronsinside the atoms. For lithium at 20,000 this is0,8 e.v. per atom, for bismuth 1.3, for aluminum0,6, and for iron 0.4. These energy changes arethus considerably smaller than the ionizationenergies of the atoms, but they are neverthelessof the same order of magnitude, being 18 percentfor bismuth, and lead to the conclusion, I be-lieve, that one should at least entertain the ideathat appreciable internal changes may be pro-duced by experimental pressures in the outerelectron orbits of the atoms.

It is to be noticed that an increase of the in-ternal kinetic energy of the electrons in theirorbits such as we have just found means a shrink-age of the orbits, that is, a compression of theatom, assuming the same relation to hold in thecondensed phase between atomic radius andelectronic energy as holds for the isolated atom.The orbital shrinkage is evidently the theoreticalversion of the "compressible atom" first insisted

upon by Richards and which has seemed to me tobe demanded by many qualitative aspects ofmy measurements of compressibility.

The order of magnitude of the changes ofinternal kinetic energy just calculated showsthat in the range of pressure at present realizableno very drastic rearrangements in the structureof the atom are to be anticipated, unless perhapsthere may be a few cases in which the atom isalready near some critical configuration. Drasticchanges may, however, perhaps be expected atpressures of the order of 100,000 kg/cm2. Con-sideration of what the nature of these effectsmay be will be postponed until the end of thepaper; in the meantime we shall be concernedwith the present experimental range in whichchanges in the atom itself may be expected to besmall, although appreciable.

III. VQLUME CHANGEs AND THE ' LAw QFFoRcE"

Doubtless the simplest of all the effects pro-duced by hydrostatic pressure is the uniformchange of volume of a fluid or an isotropic solid,and it is natural that theory should first attackthis problem. The simplest of all condensedphases from the theoretical point of view is anionic lattice of the NaC1 type, and as is wellknown, theoretical attack on this problem,largely at the hands of Born, ' has been successfulup to a certain point. The ionic lattice is heldtogether by the electrostatic attractions of theions and prevented from collapsing by a repul-sive force due to ionic interpenetration. Theattractive force can be completely dealt with interms of the known lattice structure and theknown magnitudes of the ionic charges. Therepulsive force is more difficult. In Born's origi-nal discussion an attempt was made to give someaccount of the repulsive forces in terms of thestructure of the atom as pictured at that timeby Bohr's theory, but this was unsatisfactorybecause it gave stability only for certain relativeorientations of the atoms. The final result wasthat the repulsive force had to be treated froman almost purely empirical point of view, and arepulsive force acting as some unknown inversepower of the distance between atomic centerswas assumed, with an unknown coefficient of


proportionality. Two conditions, one on the sizeof the lattice at O'K and the other on the com-pressibility, permitted a determination of thetwo parameters of the empirical law of repulsion.For most of the alkali halides the repulsive forceturned out to be approximately as the inverseninth power of the distance between atoms. Thecomplete law of force thus having been assumedand its parameters determined, it was possibleto carry the computation further and find howthe compressibility should vary with pressure.It was calculated in this way that the compres-sibility decreases with increasing pressure, andto this extent there was agreement with experi-ment, but numerically the agreement was sofar from satisfactory that it was obvious that arepulsive force of the form assumed couldserve as an approximation over only a verynarrow range when this type of phenomenon wasconcerned. The wave mechanics picture of theatom gives a more satisfactory basis than didBohr's theory for calculating the repulsive forcein terms of the mutual action of interpenetratingelectron atmospheres, and Born, in recent revi-sions of his theory, has given the repulsive forcean exponential form, which is the form thatresults most simply from the mathematics of thewave mechanics picture. But even this modifica-tion does not give the correct change of compres-sibility with pressure, so that again we have acomparatively short range approximation.

Experimentally, the compressibilities of allthe alkali halides except RbF, which does notcrystallize properly, have been determined up to12,000 kg/cm'. In this range the relation betweenpressure and volume may be represented by atwo power expression in the pressure, the twoparameters determining the initial compres-sibility and the variation of compressibility withpressure. In the case of such substances theorywould at present have the task merely of repro-ducing these two parameters. There are, however,many substances for which the volume changein this pressure range is very definitely not r0pre-sentable by a two power series in the pressure,or even by a three or four power series, as forexample, the alkali metals, and there are evensubstances whose compressibility increases withincreasing pressure instead of decreasing. Forsuch substances theory will eventually have to

reproduce a considerable number of parameters,or discover some type of function better adaptedto reproducing the volume than a power series.

In addition to the work of Born and others onthe compressibility of ionic lattices there haverecently been several calculations of the com-pressibility of the simplest metals, particularlythe alkali metals. 4 Here again it has been foundpossible to reproduce the initial compressibilitywith some success, but the variations of com-pressibility with pressure are very wide of themark.

One aspect of the method of treatment fol-lowed by Born is conventional in practically allderivations of an equation of state, whether ornot intended to be applicable to high pressuresand condensed phases, namely the assumptionof a "law of force" between atoms, a functiononly of the distance of separation of atomiccenters. In view of the failure of all attempts toreproduce more than the first derivative ofvolume with respect to pressure the questionpresents itself as to how far the action betweenatoms in condensed phases under variations oftemperature and pressure can be represented by"a law of force." In assuming a law of force ofthe form, for example, —a/r'+b/r", we areevidently maintaining that the behavior of theentire assemblage of atoms can be found bypostulating that each atom of every pair ofatoms acts on the other with the force given,irrespective of the presence of other atoms, ,inall orientations (orientation is not a factor forthe type of ions just considered, which by wavemechanics have spherical symmetry), and at alldistances. The assumption of a law of force of theform given certainly corresponds to the factsfrom the point of view of one very importantfirst approximation. For two atoms acting oneach other with a force of this character may bebrought indefinitely close together by the actionof sufficiently large forces, that is, such atomsare not rigid but are electively deformable, as isdemanded by many lines of experimental evi-dence. But the question is, how much further isthe approximation represented by such a lawvalid. One may imagine oneself carrying throughthe detailed solution of the wave-mechanicalproblem of the NaC1 crystal, for example, find-

ing the complete f function, splitting this up

P. W. 8R1DGMAN

into parts corresponding to p, s, d, etc. , electronsassociated with the different atoms, and thencoalescing the results into a final law' of force.Into this law of force will certainly enter thedistribution of the electrons within the atoms.The law of force may remain good as long as thedistribution of electrons remains fixed or as longas the electron distribution depends uniquely onthe distance r, as when pressure is changed atconstant temperature. But if the electron dis-tribution is not fixed uniquely by r, a possibilitywhich we must recognize if both pressure andtemperature are allowed to vary, then we mustbe prepared to find that the interaction cannotbe described in terms of constants and r only,that is, we must be prepared to find that thereis no "law of force." Doubtless in a sufficientlynarrow range the assumption of a law of force isa valid approximation. We have to considerwhether it is a valid approximation in the rangeof pressure and temperature now open to experi-ment. We can obtain a qualitative answer tothis question by means of Schottky's theorem.

In Eqs. (3) and (4) write as an approximationthat the total kinetic energy T is the sum of twoparts: one, mass motion of the atom as a whole

(T,&) and the other kinetic energy of the electronsinside the atom (T,i), so that T= T,~+T,i. Wecould in the same way put V= U, &+ V, &, whereV, & is the part of the potential energy. given bythe "law of force" as in the discussion above, andU, i is the internal potential energy of the elec-trons inside the atom. This resolution into twoparts should be a fairly good approximation fora simple ionic lattice; for complicated molecularlattices the resolution would be more question-able. We assume further that the substance underconsideration approximately satisfies Dulong andPetit's law, which means that (8T,~/Bp), =0,(8T„/Br)~=C„/2, neglecting the difference be-tween C„and C„which is legitimate for con-densed phases. At high pressure it is highlyprobable that the specific heat becomes some-what'less because of the stiffening of the atomicconstraints, so that more exactly, (aT.,/aP), (0.This will have as a result that the conclusionsto follow are in the nature of an understatementrather than an overstatement of the effect ofpressure on T,i. Combining these results with

(3) and (4) now gives at once

=4p ——C

=4P+ 3v+r — —, . (7)

Let us compare these two derivatives at lowpressures, that is, compare —(3/2) C,/(Bv/Br)„with. [3v+ r(8v/87) „j/(Bv/8P), . For NaCI wehave the numerical values: C, =0.219 g cal. /g,(Bv/Br)„=0.00012, and (Bv/BP), =42X10, inkg/cm~ units. Reducing everything to kg/cm'units, the two derivatives become, respectively,2.5X10' and 7X10'. That is, the change ofelectronic energy internal to the atom is nearlythree times as great when a definite change ofvolume is brought about by a change of pressureas when brought about by a change of tempera-ture. The internal electronic energy may be takenas a rough measure of the internal condition ofthe atom, and the force with which one atomacts on another depends on its internal condition.It follows therefore that the force between atomschanges differently when the mean distance ofseparation is changed by a definite amount bythe application of pressure and when the distanceis changed the same amount by a change oftemperature, and that therefore the assumptionof a "law of force" to be used in writing out anequation of state, or in writing out an expressionfor the virial, can be only an approximation.To find just how good an approximation it is toassume a law of force might well be one of thefirst tasks of a serious attempt to derive an equa-tion of state valid for high pressures. The numer-ical values for Nacl show that the approximationis less good for changes of pressure than forchanges of temperature. Furthermore, the for-mulas show that the approximation becomespoorer as pressure increases in virtue of the 4pterm, which has an opposite sign from the other.However, even at 20,000 kg/cm' the 4P term hasbecome only 33 percent of (3/2) C„/(8v/Br)„, as-suming constancy of C„/(Bv/Br)„. Both (av/ar)„and (Ov/Bp), vary importantly with pressure,however, so that actually the variation in theother terms is the important effect at high pres-sures. As a matter of experiment (Bv/ap), gener-


ally drops off with increasing pressure much morerapidly than (Bv/B7)~, so that for this reason alsothe assumption of a "law of force" is a less validapproximation at high pressures than at lowpressures.

The important role played by the shrinkage ofthe atoms under some conditions is well broughtout by another line of argument. Consider thechange of total energy brought about by a changeof pressure at constant temperature

dE, = — v —+p — dp.

The two terms in this expression are of oppositesigns. The term P(Bv/BP), starts with the valuezero, so that at low pressures the sign of dB isdetermined by (Bv/Br)„. That is, the internalenergy of all normal substances at first decreasesas pressure is applied isothermally, more energyflowing out in the form of heat to counteract forthe rise of temperature produced by the compres-sion than is put in in the form of mechanical workby the external pressure. But at high pressuresit is a matter of experiment that the termP(Bv/BP), preponderates, so that at high enoughpressures the energy increases as pressure in-creases, the increase obviously being accountedfor mainly by the work done against the repul-sive forces of the atoms. The volume at whichdE vanishes is the volume at which the attrac-tive and repulsive forces balance. At O'K thelattice is in equilibrium with all the lattice pointsat rest (neglecting zero point energy), so that thevolume of O'K at atmospheric pressure shouldbe the volume at which the two sets of forcesbalance. This is indeed the case, because of O'K(Bv/B~)~=0 and dB= —P(Bv/BP), dP for the iso-thermal at O'. That is E is a minimum at O'K at0 pressure, and with increasing pressure con-tinually increases due to the action of the repul-sive forces. If now we assume that the mutualpotential energy is a function of volume only,independent of temperature, or amplitude ofatomic vibration, which is obviously an approxi-mation, and if we also assume that the transla-tional kinetic energy is a function of temperatureonly, we would expect that at higher tempera-ture (dZ), would become zero when the pressurehad increased sufficiently to reduce the volume

to the volume at O'K at atmospheric pressure.The pressure at which this occurs is obviously—r(Bv/B7)„/(Bv/BP), . This pressure I have com-puted for a number of solids at room temperature,and it is of the order of 10,000 or 20,000 kg/cm'for ordinary metals such as silver and iron. Thevolume corresponding to this pressure is approxi-mately the same as that at O'K and atmosphericpressure, although somewhat less. For helium,on the other hand, the relative changes of(Bv/Br) and (Bv/BP), under pressure are differ-ent, Bv/BP approaching 0 much faster relativelyto Bv/B~ than it does for solids, with the resultthat at room temperature the pressure at whichBZ/BP=O is higher than yet reached experi-mentally, that is, greater than 15,000 kg/cm'.But at room temperature at 15,000 the volume ofhelium is only one-half the volume at O'K atatmospheric pressure. In other words, the volumeat which the attractive and repulsive forces arein balance at 15,000 at room temperature is lessthan half the volume at which they are in bal-ance at O'K. So large a discrepancy surely can-not all be attributed to departures of the forcesfrom linearity, but there must be a change inthe structure of the atom itself, resulting fromsome sort of rearrangement of the electron dis-tribution. Under such conditions a "law offorce, " using as the only parameter the distanceof separation of the atomic centers, must beinadequate. It is interesting to notice that the.direction of the discrepancy with ordinarymetals is the same as with helium, only it isless extreme. That is, at room temperature thevolume at which BB/BP vanishes is less, in eithercase, than the volume at O'K at atmosphericpressure. In either case the internal kinetic energyof the atom increases with increasing pressure,accompanied by a shrinkage in the effective sizeof the atom, so that the sign of the effect iswhat would be expected.

A crude calculation of the effective change ofsize of the atom under pressure may be of in-terest. We have approximately at low pressure1/v(BT, &/Bv), =3/(Bv/BP), . For NaC1 this givesthe numerical value 1.4X10 "erg per atom perkg/cm~ external pressure. Making now the crud-est kind of an approximation, taking an atom of14 electrons as the average of Na and Cl, sup-posing the 14 electrons to rotate in a single orbit

P. W. BRIDGMAN

about the nucleus with a radius of 1.35)C10 ' cmwith such a kinetic energy that the radial acceler-ation is held in equilibrium by the electrostaticforce of attraction of the nucleus, one may cal-culate the change in radial distance when thekinetic energy in the orbit increases by 1.4)& 10—".The change of radius for this change of energyturns out to be of the order of only 1/100th ofthe change of distance between atomic centersbrought about by the external pressure of 1

kg/cm'. That is, the change in size of the atomwhen external pressure is applied probably ac-counts for only a small fraction of the totalchange of volume.

It is interesting to apply Schottky's theoremto a monatomic gas instead of to a solid ashitherto. Dulong and Petit's law. does not applyto a gas, and we now have (aT„t/ap), =0 and(aT,~/ar)„=C„—R to replace the former rela-tions. Substitution in 3 will now give

(aT„/aP), =3v+ ~(av/a. )„+4P(av/8P) „as before, and

(aT,(/ar) „=4P(av/ar) p—2C„+R.

For a perfect monatomic gas r(av/a7. )„=v,

p(av/ap), = —v, and C„=SR/2, so that (aT,~/

ap), =0, and (a T,~/ar) „=0, and there is no

change in the internal structure of the atom forchanges of either pressure or temperature. No

gas remains approximately perfect, however, be-

yond a few hundreds of kg/cm', and under afew thousand kg/cm' all distinction between agas and an ordinary liquid is lost. In this pres-sure range a little consideration shows that thechange of T,~ is of the same order of magnitudeas it is for substances originally in the condensedcondition (for example, for nitrogen pv rises from

unity at atmospheric pressure to 16.5 at 15,000kg/cm'), and that therefore the internal distor-tion of the atoms of a gas is not diRerent fromthat of other substances.

One might anticipate that it would be a simplertask for theory to calculate the variation withpressure of the volume of a substance like gas-eous hydrogen or helium than of a solid likeNaC1. This has not proved to be the case, how-

ever, and we have as yet no theoretical deriva-tion of an equation of state for any gas at highpressures.

IV. THERMAL EXPANSION AND ENTROPY AT

INFINITE PRESSURE

The most successful of the theoretical attackson problems presented by the equation of stateof solid bodies has been on the problem ofcompressibility at constant temperature; suc-cess has not been so great in dealing withthermal expansion. In nearly all cases thermalexpansion decreases with an increase of pres-sure; this of course is the same thing as anincrease of compressibility with rising temper-ature. As a general rule, the decrease of ther-mal expansion for a given increase of pressureis by a factor considerably smaller than thedecrease of compressibility. It has been knownfor some time that thermal expansion involvesa departure from linearity of the restoring forceson the atoms. At first glance, therefore, the de-crease of thermal expansion with increasingpressure (decreasing volume) is paradoxical,because the repulsive forces between atoms areordinarily represented as increasing very rapidlyat small distances of separation, so that thesmaller the volume, the greater would one ex-pect the deviation from linearity to be, andtherefore the greater the expansion. But theparadox is to a certain extent resolved if oneconsiders the state of afFairs at large volumes inthe perfect gas. Here the restoring force on amolecule is zero during free flight, and it abruptlyrises to a very large value during the collisionthat terminates free flight. This sort of thingis the extreme in the way of departure fromlinearity, so that a large thermal expansion maybe expected, as is indeed the case. This collisionalaspect of the interaction between atoms must beresponsible for a large part of the thermal ex-pansion of actual solids and liquids, and thefact that this aspect is becoming less and lessimportant at high pressures is doubtless themain factor responsible for the decrease ofthermal expansion with increasing pressure. Itis not perfectly clear what to expect at exceed-ingly high pressures, beyond reach of presentexperiment. To the extent to which the "law offorce" point of view is justified, one might per-haps expect that eventually the rapid increaseof repulsive forces would bring about a reversal,so that at very high pressures thermal expansion


might begin to rise again. The subject is an im-portant one, but very difficult experimentally;I have made repeated attempts to get good valuesof the thermal expansion of some of the moreinteresting substances, but without much success.Quite recently, however, results have been ob-tained for the alkali metals, and the results arein process of publication in The Proceedings ofthe National Academy of Sciences.

Consideration of the thermal expansion sug-gests another interesting line of speculation onthe probable limiting behavior of substances atvery high pressures connected with the thirdlaw of thermodynamics. In some respects anincrease of pressure is equivalent to a decrease oftemperature, the volume effects are the same andthere is also a stiRening of the constraints towhich the atoms are subjected, resulting in anincrease of the characteristic temperature witha displacement of specific heat, thermal expan-sion and similar properties to lower values,corresponding effectively to lower temperatures.This is very probably an important factor inthe decrease of thermal expansion of solidswith increasing pressure. Now at O'K entropyvanishes in most cases according to the thirdlaw, and this led Lewis4' to suggest that it isplausible to expect that along an isothermal athigher temperatures the entropy will approachzero at infinite pressure. Since (BS/BP), = —(trav/

Bv-)~, integration givesp

S„,,—Sp, ,= — (Bv/8 r),dP.p

If we assume that Sp, p=O by the third law, it isobvious that the thermal expansion must goto zero at infinite pressure, for if (cia/Br)~ re-mained finite S„, , would decrease without limitas pressure increases indefinitely. It can be seenat once that if 1'(elm/Br)QP is to remain finite,Bo/Br must decrease with increasing pressure atleast as rapidly as 1/p. In the experimental rangeof pressure this condition is very definitely notmet, either by ordinary solids, metals* or ioniccrystals, or by the gases nitrogen and argon, theexpansion in no case decreasing as rapidly as1/p. If then the condition lim S=O holds' there

*The recently completed experiments on the alkalimetals indicate that they are exceptional in that they havea very large decrease of thermal expansion at high pressure.

must be a reversal in the trend of Bv/Br at pres-sures beyond those yet reached. It is to beseriously considered, however, whether theexpectation that the entropy at infinite pressuremust vanish is a necessary one. In the argumentwhich represents the third law as resulting fromthe complete vanishing of all randomness at 0'abs. , the atom is treated as the inviolable unitwith respect to which randomness is calculated.But at infinite pressure the atom certainly isnot inviolate, for there are all sorts of penetrationeffects and interactions between electrons. Ifthe electron or some other subgroup in the atombegins to acquire individuality at high pressures,there would seem to be no reason why entropyshould not have negative values with respect toa system in which the atom is the unit. This isperhaps merely another way of saying that asystem under very high pressures is one of thoseexceptional cases to which Fowler and Sternehave shown that the third law does not apply.

V. P—U—T RELATIoNs IN LIQUIDs

1. Compressibility

Thus far we have considered mainly the effectof pressure on the volume of solids, and havealso brieAy considered gases. Theory has hadsome partial success in these fields, but has upto the present been almost powerless when con-fronted by the complications of an ordinaryliquid. All that I can do here is to make some'qualitative observations suggested by the be-havior of liquids at high pressures. The experi-mental material is fairly extensive, includingmeasurements up to 12,000 kg/cm' in the tem-perature range between 0 and 100'C on some60 liquids, all of them organic except water andmercury. Mercury is in a class by itself, its com-pressibility being about one-tenth that of water,and of the same order magnitude as that of thesolid metals, SO percent more compressible thanlead, for example, and much less compressiblethan the alkali metals. It would be interestingto know the compressibility of other liquidmetals, but there is no reason to anticipateanything very striking. The other liquids fallroughly into three groups with regard to thetotal compression under 12,000 kg/cm'. Glycerineis in a group by itself with a total volume decre-

10 P. W. BRI DGMAN

ment of 1.4.5 percent; in the second group iswater, C6H5Br, C6H~Cl and a number of theglycols with a volume decrement varying from18 to 20 percent, and in the third group are allthe others with a volume decrement of the orderof 30 percent. The division into groups is roughand probably to a certain extent fortuitous.Doubtless it would be possible to find liquidsfitting into all the gaps.

The order of magnitude of the compressionis not particularly characteristic of the liquid asdistinguished from the solid phase, but is rathe~a characteristic of the chemical nature of thematerial, for a number of organic solids havecompressions of 15 percent at 12,000. The notablecharacteristic of the compressibility of liquidsas distinguished from that of solids seems to bethe relatively large decrease of compressibilityat high pressure; the compressibility at 12,000kg/cm' is in many cases only 1/15th of that atatmospheric pressure. Organic solids do notshow the stage of very rapid decrease of com-pressibility at low pressures shown by the liquids.The decrease of compressibility of liquids is notuniform with pressure, but is by far the mostrapid at low pressures; for many substances therelative decrease in the first 1000 kg of the rangeis as much as the relative decrease in the last6000. Qualitatively an adequate picture is thatat low pressures there is a large amount of "slack"between the molecules; increasing pressure re-moves this slack rapidly, and during this processthe compressibility is high. When the slack hasbeen removed there remains the compression ofthe molecules themselves, part of which mayarise from a closer atomic grouping within themolecule, and part from a shrinkage of theatoms, as demanded by Schottky's theorem. Alarge part of the high initial compressibility ofordinary liquids is connected with the nearnessof the critical point liquid-gas, for compres-sibility in the gas phase is high and at the criticalpoint itself compressibility is infinite.

The persistence of the compressibility of themolecules at high pressures is a factor whichhas seldom been adequately taken into accountin the various theories of liquids which havebeen proposed. Most of these theories have beenmore or less empirical in character and many ofthem have agreed in assigning a finite limiting

volume to the liquid at infinite pressure. Thislimiting volume, extrapolated from measure-ments made in a range of 3000 or 4000 kg/cm',is sometimes higher than the volume actuallyreached at 12,000, and in any event the equa-tions almost always give too small a compres-sibility at high pressures. Another aspect of thissame phenomenon is that compressibility variesvery much less from liquid to liquid at highpressures than it does at low pressures; among theliquids measured there is a tenfold variation ofcompressibility at atmospheric pressure, whereasat 12,000 the factor of variation is only 1.8.

2. Compressibility and. chemical composition

There are various rough connections of thekind that might be expected between the com-pressibility and chemical composition. Thus athigh pressures the volume of isomers tends morenearly to equality than at lower pressures, pro-vided the volumes at low pressures are markedlydifferent. That is, at high pressures the tendencyis for structural differences to become obliter-ated and for the volume to approach morenearly to the sum of the volumes of the compo-nent atoms. An example is ether and n-butylalcohol, both of which have the same composi-tion, C4HioO, but which structurally are sodifferent in character that the chemist oftendoes not think of them as isomers. The ratio ofthe volumes of equal weights of these two sub-stances at atmospheric pressure is 1.096 and is1.037 at 12,000. On the other hand, if the isomersare not greatly different in structure, as n- andi-butyl alcohols, then the tendency to greaterequality of volume at high pressures is not somarked, and there may even be slight changes inthe opposite direction, 12,000 kg/cm~ not beinga sufficiently great force to assume complete con-trol in these cases of slight difference.

If the molecule is bound together by intenseforces, the compressibility is small. The —OHgroup appears to exert such a consolidating effecton the molecule; the abnormally low compres-sibility of glycerine is to be attributed to thefact that the molecule contains three —OHgroups. The comparatively low compressibilityof the glycols and of water is evidently due to aneffect of the same sort. In general the additionof links of the hydrocarbon chain, CH~, seems to

H IGH PRESSURE PHENOMENA

favor high compressibility, and the presence ofan oxygen atom anywhere in the molecule lowcompressibility.

3. Thermal expansion

The thermal expansion of liquids is mucheasier to measure than that of solids, and it ispossible to obtain both the variation of expansionwith pressure at constant temperature and thevariation with temperature at constant pres-sure. The effect of increasing pressure at constanttemperature is to decrease the thermal expansion,as is also the case for solids, but by a much largerfactor. The variation with temperature is interest-ing. Normally, at atmospheric pressure, thermalexpansion increases with increase of temperature,or (8'v/8~'), &0. At a pressure between 3000 and4000 kg/cm', however, this effect reverses fornearly all organic liquids, so that at pressureshigher than this 8'v/8r'&0. An apparent ex-planation suggests itself in terms of the failureof linearity of the restoring forces. At lowertemperature at constant pressure the volume isless, and therefore the non-linearity in the re-storing forces arising from the intense repulsiveforces is greater and so the thermal expansionis greater. This suggestion I have published inseveral papers, but it evidently cannot be right,for the same argument would demand that theexpansion increase with increasing pressure atconstant temperature, which is not the case. Itappears to me now that a possible factor in thesituation is the Schottky change of dimensionsof the molecules already discussed. We haveseen that the molecules increase in size as tem-perature increases at constant pressure. At smallvolumes (high pressures) this may mean animportant curtailment of the free Hight aspectof molecular motion, an increasing degree oflinearity, and therefore a decreasing thermalexpansion. An exact working out of what is tobe expected here obviously demands a carefulbalancing against each other of several factors;it is an interesting and important problem fortheoretical attack.

Since (8C~/Bp), = —r(8'v/8r')„, the fact that8'v/Bv' becomes negative between 3000 and 4000kg/cm~ means that C„ increases with pressurebeyond this point. As a rough average, C„de-creases to about 0.9 its initial value at 3000 or

4000, increasing from here on at such a rate thatat 12,000 it has not quite recovered its initialvalue. The increase in C~ is not what one mightat first expect, since part of the effect of pressureis to increase the stiffness of the constraints,increasing the characteristic temperature, anddisplacing the effective temperature to lowervalues and so decreasing the specific heat. Butit must be remembered that the reversal of8'v/Br' has been established experimentally onlyfor organic substances with rather complicatedmolecules. Under ordinary conditions the specificheat of such substances is known to be consider-ably less than that corresponding to the fullnumber of internal degrees of freedom of themolecule. If the effect of pressure is to make theatom rather than the molecule the individualunit of structure, an increase of specific heatwould be expected. Such a tendency is consistentwith the increasing approach to equality of iso-mers at high pressures. If these considerationsare correct, one would expect that metals andsimple ionic lattices would not show the reversalin sign of 8'v/Bv', it is unfortunate that av/a~cannot at present be measured accuratelyenough for these substances to permit an evalua-tion of yv/gr'.

4. The pressure coefBcient and. the mechanismof pressure

The ratio (Bv/Br)~/(Bv/Bp)„which is mathe-matically identical to —(8p/Br) „, has played animportant part in various theories of liquids.It has received the name "pressure coefficient"and obviously gives the rise of pressure for onedegree rise of temperature at constant volume.Various arguments have been advanced forsupposing that the pressure coefficient is a func-tion of volume only, and equations of state havebeen deduced on this basis. The physical meaningof this assumption is easy to see. If we put(8p/87)„=f(v), integration gives at once for theequation of state

where q is an arbitrary function of integration.The perfect gas equation and van der Waalsequation are bot'h special cases of this generalform. The physical meaning of such an equation

12 P. W. BRI DGMAN

is obviously that the pressure exerted by thesubstance can be regarded as arising from twodifferent effects, each acting independently ofthe other. The part given by p(v) is the same atall temperatures as at O'K and obviously arisesfrom the forces between molecules, as for ex-ample the force in Born's analysis for ioniccrystals. The part ~f(v) is proportional to thetemperature, and is evidently the part arisingfrom kinetic bombardment against the walls.An elementary argument of kinetic theory showsthat the kinetic pressure is of this form if weassume that the size of the molecules is inde-pendent of temperature and that the kineticenergy of temperature agitation is proportionalto temperature, as in classical statistics.

When we make comparison with experiment,we find that in many cases (Bp/Br) „ fails to be afunction of volume only by an amount farbeyond experimental error. In general thetendency is for 8p/87 to decrease with increasingtemperature at constant volume. The generalreason for the discrepancy is clear; if the mole-cules are deformed by changes of pressure andtemperature, as we have seen they must be,then the two mechanisms by which pressure isexerted do not act independently, but there mustbe interactions between them. Furthermore,there is always a contribution to the q(v) termarising from terms of higher order when theamplitude of vibration increases; just how im-

portant this is compared with the other effectis a question for detailed theory.

The change of internal energy of liquids underpressure presents the same general features asthat for solids, that is, at low pressures it de-creases with increasing pressure and then athigher pressures there is a reversal. The physicalinterpretation is also doubtless the same; at low

pressures and large volumes the attractive forcespreponderate, whereas at higher pressures andsmaller volumes the repulsive forces preponderate.The pressure at which reversal occurs is markedlylower in the case of organic liquids than forsolids, being in the neighborhood of 7000 kg/cm'.This is what would be expected on the mostgeneral grounds, pressure being in general moreeffective in changing the properties of a liquidthan of a solid.

5. Small scale irregularities

So far we have been concerned with com-paratively large scale effects in which all organicliquids are more or less alike. Superposed onthese large scale phenomena there is a be-wildering amount of small scale behavior or finestructure, which varies with the individual liquid.Examples can be found of nearly every con-ceivable type of abnormal behavior. Thus thecompressibility may increase with rising pressureor decrease with rising temperature, and thermalexpansion may increase with increasing pressureand increase or decrease with rising temperature.Such a variety of behavior must mean that theindividual differences of the different kinds ofmolecule are coming into play and becomingaccentuated at high pressure, as is indeed mostnatural when one considers how the moleculesare pushed into more intimate contact bypressure and forced to conform to each others'irregularities. The picture of the molecule thatwave mechanics is developing seems to providethe possibility of just the sorts of complicationthat are required, valencies directed in spacegiving the possibility of shapes much morecomplicated than spherical, and regions in theouter parts of the atoms pulled together by thepairing of electrons of opposite spins giving thepossibility of local centers of attractive force—infact just the sort of complication that seemeddemanded by qualitative considerations beforethe development of wave mechanics. Thecomplications involved in the detailed workingout of the possibilities will doubtless be pro-hibitive for a while, so that probably at firsttheory will confine itself to the broad featurescommon to all liquids, where the possibilitiesare interesting enough.

From the existence of the great amount ofindividual detail one may draw the conclusionthat there is no such thing as "an" equation ofstate, as has been tacitly assumed so often. Itwould obviously be hopeless to attempt toreproduce in a single type of equation withvariable parameters all the possibilities of all

types of molecule. The most that can be meantby "an" equation of state is a description of ageneral method, not a general result.

HIGH PRESSURE PHENOMENA 13

VI. PERIQDIc RELATIoNs

One of the tasks of theory in dealing with thecompressibility of the elements is to reproducethe strikingly periodic character of the com-pressibility, a feature first emphasized byRichards. ' In Fig. 1 is plotted the logarithm to

8f" f::~. ~I

La+g t "' '+" +"":C'0; ': t':+

+I+

Ld+ Ih

. Nsl. +"t.g~h+ j..

10 " ' '"'+ "+'++

+. ":::4:':v:if.:f.@f:s"~N "~ '. N + ' ~

~ hh+h .''' '' I++ ~ ~ + Ih+ ~ +I+ ~ + - ~ ~ ~ ~ hh Ih+H + lhI+10 QO 80 40 50 dO 70 8U AO

FIG. 1. The logarithm to the base 10, plus 7, of the com-pressibility of the elements plotted as ordinate against theatomic number as abscissa.

the base 10 plus 7 of the cubic compressibility ofall known elements against the atomic number.There is good reason to think that the maximumpositions in the figure would be occupied by thecondensed phases of the rare gases, which havenot yet been measured. This might be expectedon general grounds, because of the nature of theforces holding the rare gases together, namely vander Waals forces, which are the weakest of thevarious recognized types of atomic force. Passingto the alkali metals, their high compressibility isevidently to be correlated with the single electronin the outer shell, making for a very deformableatom.

VII. CoMPREssIBILITY QF SINGLE CRYsTALs

Theory must ultimately account for the differ-ence of compressibility in different directions ofsingle crystals of the non-cubic substances. Thedifferences in different directions are often verymarked; thus zinc is eight times more com-pressible along the axis than at right angles to it,and tellurium actually has a negative com-pressibility in one direction. Qualitatively thebehavior is roughly as one would expect, thecompressibility being greatest in the direction ofgreatest atomic separation, and in the case of

tellurium there has been some success in con-necting the behavior qualitatively with thefilamentary structure of spirally arranged atomsin the direction of the axis. But quantitatively,practically. nothing has been done, and one canhardly expect much before the more fundamentalproblem is solved of reproducing the latticestructure itself with its difference of spacing indifferent directions.

VIII. Two PHAsE EQUILIBRIUM

We now' pass from consideration of the effect ofpressure on a single condensed phase, solid orliquid, and consider the change in the equilibriumbetween two condensed phases, either solid andliquid or two solids, produced by pressure. Thefundamental equation governing the phaseequilibrium is of course the equation of Clapeyron

dr/dp = re/L,where dv is the change of volume during thetransition and L is the latent heat.

1. The melting curve

Clapeyron's equation applies to any kind oftwo phase equilibrium. With regard to themelting equilibrium between solid and liquid theproblem which has attracted the most attentionis that of the shape of the melting curve.Clapeyron's equation obviously allows dr/dp tobe any function of pressure, if only bv and L haveappropriate values, and in fact thermodynamicsin general has no restriction to place on the shapeof the melting curve. The question reduces to oneof experiment, therefore. Expectation has been toa large extent controlled by the correspondingsituation with regard to equilibrium between aliquid and its vapor. Here it has been establishedthat all sorts of substances show the same sort ofbehavior; there is a critical point at which thedistinction between liquid and vapor ceases,which means that it is possible by propermanipulation of pressure and temperature topass around the critical point, and thus passwithout discontinuity from the vapor to theliquid state. This general result seems to havebeen often taken over for the solid-liquidequilibrium without any very self-conscious

P. W. BRI DGMAN

realization that a special assumption was beingmade, for it has been generally assumed thatthere is such a thing as "a" melting curve, thesame in general character for all substances,whether the substance is held together by vander Waals forces as in an inert gas, or by ionicforces as in a salt, or by exchange forces as in ametal or by valence forces as in some molecularcompounds. This assumption did not appear tobe so serious before the existence of these differentsorts of force was clearly recognized, but now itwould appear that to assume that there is such athing as "a" melting curve is of itself con-siderable of an assumption.

Assuming that there is such a thing, there havebeen various expectations as to the character ofthe melting curve. The earliest was that therewould be a critical point just as in the case ofliquid-vapor, at which liquid and solid wouldbecome identical. Other expectations have beenthat the melting curve would rise to a maximumand then fall again (Tammann), that it wouldrise to an asymptotic temperature at infinitepressure (Schames), or merely that it wouldcontinue to rise indefinitely with pressure andtemperature. In the experimental range ofpressure neither of the first two expectationshave been realized, while obviously the last twodemand an infinite pressure range. The questionhas to be settled then on the basis of some sort ofan extrapolation. Such an extrapolation can bemade in terms of the behavior of dv and L inClapeyron's equation. Thus if there is a criticalpoint, Av and J must both tend to vanish at thesame pressure (and temperature); if there is amaximum, b,v must tend to vanish at some finitepressure (and temperature) and L must tend to afinite value at the same pressure, etc. It is notnecessary to go into greater detail here becausethe question has been freshly discussed in arecent paper in the Physical Review. ' It seems tome personally that all the present experimentalevidence is unequivocally in favor of the ex-pectation that the melting curve rises indefi-'

nitely; this holds for substances as diverse asargon, nitrogen, liquid metals, and many organiccompounds. This conclusion is, of course, re-stricted to a pressure range in which nothingoccurs drastically new, like an atomic break-down, and definitely does not apply to the

extreme pressures considered in the Section XVIof this article.

The tacit assumption of "a" melting c'urve

therefore seems to be justified. The explanationof it must rest on some broad general property ofsolid and liquid and not depend on the particulartype of force binding the molecules together.Such a broad feature is obviously the regulararrangement of the crystal as opposed to thehaphazard arrangement of the liquid. A meltingcurve which rises indefinitely merely means thatno matter how high the temperature, it is possibleto apply to any substance enough pressure tomake it assume the regular arrangement of thecrystal which it has at some lower temperature(except for polymorphic changes). Such a be-havior appears most natural and quite in linewith the experimental fact of the universality ofthe existence of the crystalline state of aggrega-tion. But theoretically the explanation has notyet been given for the necessity of the existenceof a crystalline phase. It is doubtless our inabilityas yet to understand theoretically this funda-mental fact that is responsible for the persistencewith which a number of physicists still maintain,as it seems to me against the experimentalevidence, the idea that the melting curve will endin a critical point if only pressure can be carriedhigh enough. It would be surprising if theuniversal tendency to crystallize could be sup-pressed by pressure, particularly in view of thefact that many substances are already in theirnatural state under a very high internal pressure.

Some theoretical discussions have been givenof melting, mostly from a roughly empirical pointof view, as, for example, that of Lindemann, ~ whopostulates that melting occurs when the ampli-tude of atomic vibration has become about 10percent of the distance of separation of atomiccenters at O'K. One characteristic of thisargument of Lindemann has been common tomost of the other arguments also, namely thatthe solid will melt when some critical condition isreached in the solid phase, irrespective of thenature of the liquid, as if melting were like thefalling over of a row of dominoes. This of course,is not the case, but the melting point, which isthe point at which solid and liquid are inequilibrium, is that point at which there is acertain relation between the properties of solid


and liquid, as is shown by the thermodynamicformulation in terms of the equality of thethermodynamic potentials of liquid and solid.Any valid theory of melting must properlyconsider this aspect of the situation. The com-plete theory of melting must also take intoaccount the still more detailed consideration ofthe precise crystal system of the solid.

There is room for the collection of considerablymore experimental material which would be ofvalue in formulating a theory of melting. It sohappens that in spite of the large number ofliquids whose properties have been studied underpressure, very few are liquids whose meltingcurves have also been investigated, The reasonfor this is connected almost entirely withquestions of technique; thus the sylphon withwhich the compressibility of a large number ofliquids has been measured is likely to be ruined ifthe liquid is allowed to freeze in it, so that liquidshave mostly been picked out for measurementthat do not freeze in the experimental range.The result is that we have entirely inadequateexperimental knowledge of how the properties ofliquid and solid vary together along the meltingline. For example, the following simple questionscannot at present be answered: does the volumeof the liquid in general increase or decrease alongthe melting line; or, does the energy content ofthe liquid in general increase or decrease alongthe melting line?

The normal melting curve rises to highertemperature with rising pressure, a consequence,according to Clapeyron's equation, of the factsthat in nearly all cases the volume of the solidphase is less than that of the liquid, and thelatent heat of transition from the low to the hightemperature phase is of necessity positive. Theonly known exceptions have been water, bismuthand gallium. The abnormal falling melting curveof water is, however, only a temporary episode,for beyond 2200 kg/cm' a new modification ofice, denser than the liquid, replaces ordinary ice,and from here on the melting curve of waterrises, as is normal. The expectation would bethat this must also be the eventual state of affairsfor bismuth and gallium, and a number ofattempts have been made to find a new modifi-cation of bismuth. I have recently been successfulat last in finding the second modification of

bismuth, ' the transition taking place at 25,000 kgat room temperature. It has not been actuallyshown experimentally that the melting curve ofthis new modification rises, but the magnitude ofthe change of volume leaves no doubt of it. Onesuspects that the new bismuth will prove not tohave many of the abnormal properties ofordinary bismuth and that the general e8ectof the action of pressure on bismuth is to force itinto normality, as is known to be the case withwater. No serious attempt has been made to finda new modification of gallium beyond 12,000 kg;the slight departure of the melting curve ofgallium from linearity leads one to be prepared tofind that the high pressure modification ofgallium occurs at higher pressure than that ofbismuth. But with the example of bismuthfreshly before us, I think that theory need haveno hesitation in proceeding on the assumptionthat the rising melting curve is entirely normal.

There is another universal feature which allmelting curves have in common that theory mustultimately explain, namely they are all concavedownward, or d'v/dp'(0, whether they rise orfall.

Two remarks of a negative character can bemade with regard to a theory of melting. In thefirst place, if melting temperature is plotted as afunction of the atomic number of the elements,very much greater irregularities will be foundthan for such atomic properties as atomic volumeor compressibility or electrical conduction; thereare even places of unexpected reversal, as forexample mercury between gold and thallium, andgallium between zinc and germanium. Theseabnormalities cannot be explained by the naturalassumption that there may be missing modifi-cations of the solid, which might be forced toappear by high enough pressure, like the newmodification of bismuth. For these abnormalmelting points are already too low, and themelting point of any as yet undiscovered highpressure modification would be even lower. Thesituation could be saved only by a new highpressure phase of the liquid, and two liquidphases are not known for any pure substance,unless one wants to count liquid crystals.

The second remark is that the melting ofdifferent sorts of substance is not governed by alaw of corresponding states like the vaporization

P. W. B R I DGM AN

of the liquid phase, If there were such a law ofcorresponding states, then the temperature atwhich the difference of volume between liquidand solid phase has fallen to a definite fraction ofits value at the normal melting point would beara definite ratio to the normal melting tempera-ture, the same for all substances. This is verydefinitely not the case. There may, however, besomething corresponding to the law of corre-sponding states for restricted groups of sub-stances. Thus the alkali metals, sodium, po-tassium, rubidium, and caesium, satisfy veryapproximately the condition just described onthe volume differences.

The alkali metals show another interestingphenomenon with regard to melting; the meltingcurves of sodium and potassium cross in theneighborhood of 9000 kg/cm', and the slope ofthe other mel'ting curves in the experimentalrange is such that it seems probable that belowperhaps 30,000 kg/cm' there will be a completereversal of the normal order of melting of all thealkali metals, lithium becoming the lowest andcaesium the highest melting. A phenomenon likethis I believe is very plausibly associated with themodification in the internal structure of the atomwhich we have already discussed with the help ofSchottky's theorem. Caesium obviously hasmuch greater latent possibility of internal re-adjustment than lithium, because the electronicstructure of its atom is so much more complicated.

Schottky's theorem demands a special internalchange in the atom on melting apart from thepressure and temperature effects in the homo-geneous phase. Written for melting, Eq. (1)becomes:

AT = —b,E+3pb,v.

But dE L —pd, v. Hence

AT = —L+4phv.

To the degree of approximation that the kineticenergy of mass motion of the molecules is thesame in the liquid and solid phase, this equationmeans that at low pressures (neglecting the tvterm) the internal kinetic energy of the moleculeis less by the latent heat in the liquid phase thanin the solid, which would indicate a swelling ofthe molecule in the liquid phase. But at highpressures this is counteracted by the 4phv term,

so that at high pressures the internal structure ofthe molecules of liquid and solid tends to becomemore nearly equal. In fact, it will be found thatfor most of the liquids whose melting curve hasbeen. studied 4pbv becomes greater than L atpressures below 12,000, so that there is areversal. Of course in order to get a precisecharacterization of what occurs, some precise

way of dealing with the kinetic energy of massmotion in the two phases must be devised, but atany rate it is indicated that at high pressuresthere may be important differences as comparedwith low pressures,

2. Polymorphic transitions between solids

Thermodynamically, as far as the applicationof Clapeyron's equation and of Schottky'stheorem goes, there is no difference between aliquid and a solid (crystalline) transition and atransition between two solid phases. As a matterof experiment, however, the transition phe-nomena between two solid phases are very muchricher and more varied than between a liquid anda solid. Melting phenomena all conform to onesimple pattern; the melting curve rises withdownward concavity, the difference of volumebetween liquid and solid decreases with risingtemperature along the melting curve, and thecompressibility of the liquid phase is alwaysgreater than that of the solid even in the ab-normal case of ordinary ice. Transition curvesbetween solids, however, may either rise or fallwith either upward or downward curvature, thevolume difference may either increase or de-crease along the curve, and the compressibility ofthe phase of larger volume may be either greateror less than the compressibility of the phase ofsmaller volume. Not only may solid-solid trans-itions show very great variability with regard tothe behavior of the thermodynamic parametersof the transition, but there may be the greatestvariation with regard to the dynamic charac-teristics. The transition may run rapidly orslowly; the speed of the transition may beunsymmetrically affected by small displacementsof pressure and temperature, positively ornegatively, away from the equilibrium values;the transition may support subcooling only orsuperheating only or both or neither; the newmodification may have a definite orientation with


regard to the original orientation, or it may be atrandom; and finally the transition may run onlywhen the parent form is a single crystal in a stateof high purity and under other circumstances beentirely suppressed.

Thus far there appears to be only one general-ization applicable to a solid-solid transitioncurve, namely that it never ends in a criticalpoint. This of course means that one type ofcrystal lattice never changes continuously intoanother type. Such a continuous change is notimpossible as a matter of pure geometry; thus acubic lattice might change continuously into atetragonal lattice by a gradual preferentialexpansion along one of the cubic axes. But such achange would involve a distinction between thecubic axes, and if the structure was originallytruly cubic there can be no such distinction.Certainly from a physical point of view con-tinuous transition from one type of lattice toanother must seem highly improbable, and thefact that it is never found is a gratifying check onthe validity of our picture of the constitution of acrystal, at least in this respect.

But beyond the impossibility of a critical point,apparently any type of behavior which might atfirst strike one as abnormal is possible. The mereexistence of certain types of transition is signifi-cant. Thus there are transition lines which runvertically, which means that there is no latentheat. .There is however in such cases a change ofinternal energy of amount pAv at the transition.If the transition takes place at comparatively lowpressures, where the forces in both phases are onthe average attractive, this means that althoughthe atomic or molecular centers are pushed closertogether the attractive forces receive work. Thisis not understandable if the centers of attractiveforce coincide with the geometrical centers of themolecules, but it does become understandable ifthe attractive centers are situated on what areeffectively projections on the molecules. If thelow pressure phase is one in which the projectionson the different molecules are in register, heldtogether by the attractive forces, it is easy tosee that these projecting centers may be pulledout of register by high pressure, thus doing workagainst the attractions, but at the same time thevolume may become less, the projections slippingpast each other and interlocking in the high

pressure modification. Such a picture is quite inaccord with the possibilities now allowed by wavemechanics.

The change in the internal structure of theatom at such a transition demanded by Schottky'stheorem may be considerable. Since L =0 for sucha transition, we have d, T=4phv. Consider thehigh pressure transition of bismuth, which takesplace at 25,000 kg/cm' with a volume change of 8percent. ET=4X25,000X0.08=8000 kg cm percm'. There are 2.8X10" atoms of bismuth percm' and 1 electron volt=1.62X10 " kg cm.Hence AT = 8000 X 10' /2. 8 X 10~X1.62 =0.18electron volt per atom. The sign of dT is suchthat the high pressure atom has the smallerinternal kinetic energy. This under normal con-ditions would mean an expansion of the atom.This result is so highly paradoxical that obviouslyone of the first tasks of an exact theory is toexamine to what extent our fundamental as-sumptions are valid, namely that the kineticenergy of atomic motion as a whole is the same inthe two phases (that is, kinetic energy constantat constant temperature), and that the samesimple connection apprdximately holds betweeninternal kinetic energy and mean atomic radiusfor highly compressed atoms that holds for thefree atoms of elementary theory.

At a vertical tangent the transition line ingeneral is not perfectly straight, but has per-ceptible curvature. It can be shown by a simplethermodynamic consideration that this meansthat under the point of vertical tangency thephase which is stable at the lower temperaturehas the higher specific heat, an abnormal state ofaffairs.

An example has been found of a transition withmaximum temperature, and it is probable thatthere is at least one example of a transition with aminimum temperature, which however could berealized in practise only with extreme difficultybecause of the viscous resistance to the transitionat low temperatures. At a maximum point thetransition runs with no change of volume, butwith a nonvanishing latent heat. On the trans-ition curve, to the high pressure side of such amaximum point, the phase with the smallervolume has the higher compressibility, again aparadoxical result. It turns out, however, thatmore than half the solid transitions for which the

measurements have been made are paradoxical inthis respect, namely that the phase of smallervolume has the higher compressibility. Such astate of affairs is not inconsistent with the picturepresented above, namely that in many crystalsthe attractive centers are effectively located onprojections and are not situated at the geo-metrical centers of the molecules. A phase inwhich the attractive centers are so lined up as tobe in register, that is, the phase of higher volume,would be expected to have a higher rigidity or asmaller compressibility, than a phase in whichthey are out of register, that is, a phase of smallervolume.

Chemical similarity of different substances isvery much less likely to result in similarity oftransition phenomena than in, similarity ofnearly every other physical property, and thereare very few examples of chemically relatedgroups which are polymorphically similar. Oneof the best examples of such a group is NH4CI,NH4Br and NH4I, all of which have a bodycentered and a face centered modification, thetransition line running very nearly vertically, atpressures below 1000 kg/cm', and with very littlelatent heat. Furthermore, the pressure of thetransition varies systematically in the series.RbC1, RbBr and RbI are another similar family,which is very similar to the family of the am-monium halides, there being also body centeredand face centered modifications with nearlyvertical transition lines, and a systematic varia-tion of pressure through the series, which, how-ever, is in the opposite direction from that in theammonium family. The mean pressure of thetransition for the rubidium family is in theneighborhood of 5000 kg/cm'. Very recently Ihave found that the potassium family shows thetransition in the neighborhood of 20,000, whereasin the sodium family the transition cannot occurmuch below 50,000.

The nitrates of the alkali metals are a familyin which polymorphism is very common, andthere are a certain number of resemblances in thebehavior of members of the family, but there arealso marked divergences. But there are alsoother examples in which there is no resemblancein the polymorphic behavior in spite of theclosest chemical resemblance, as CC14 and CBr4.

Theory has not advanced very far as yet in

understanding the reasons for polymorphictransition, much less is it able to predict when atransition may be expected and to calculate thethermodynamic parameters of the transition. Infact, one cannot expect very much in thisdirection as long as the fundamental problem ofunderstanding why a substance crystallizes intoa lattice is not solved. One cannot discuss theproblem of polymorphic transition until theproblem not only of understanding why thelattice exists is solved, but one should also be ableto calculate the lattice type. However, certainaspects of the problem of polymorphic transitionhave been subjected to theoretical attack. Thedifference between the body centered and the facecentered lattices has been put on the basis ofdifferences of minimum energy, and Hund' hasattempted to show that when the exponent of therepulsive forces passes a certain critical value thetype of lattice changes, and to explain in this waythe difference of crystal system between thecaesium halides and the other alkali halides.However, the attempt was not particularlysuccessful, because the critical value for theexponent occurred at values very much higherthan allowed by the compressibility, Hund's

method of attack has been modified by Born andMayer, "who have considered in addition to theattractive and repulsive forces the van der Waalsforces arising from the polarization of the ions.The NaCl type of lattice proves to be the morestable, but if the van der Waals forces aredoubled, a procedure which at first was purelyempirical but which was later given theoreticalsupport by Mayer, the CsCI type of structureturns out to be the more stable for the caesiumhalides (except the fluoride), and furthermore thedifference of energy between the two types. oflattice in the case of the rubidium salts proves tobe of the same order of magnitude as the valuesactually found for the transition under pressure.Additional considerations, however, would seemto be necessary to show why the heat of trans-ition is approximately zero, although the energychange is appreciable. For the potassium salts theenergy values of Born and Mayer would lead tothe expectation that it should be possible toforce the transition of these salts also from theNaC1 to the CsCl type of structure by pressures

H1GH PRESSURE PHENOMENA

within the experimental range and indeed I havejust found the transition near 20,000.

Even this simplest of all possible cases, thetransition of the alkali halides from the NaCI tothe CsC1 type of structure, is affected bycomplications. RbC1 undergoes a transition withvery small volume change" at a pressure of about2000 kg, half the pressure necessary to force theprincipal transition from what is supposedly theNaCl to the CsC1 type. There seems no place forsuch a transition, and the nature of the newlattice is entirely in doubt. It seems to meprobable that the lattice type does not change inthis transition, but that there may be within theatom some comparatively minor readjustment inthe energy levels of the electrons.

A knowledge of the lattice structure of the highpressure modifications would be most helpful informulating a theory. Unfortunately no experi-mental determinations have as yet been made byx-rays while the material is actually underpressure, .so that our knowledge of the structureof the high pressure forms is restricted to thosecases in which the high pressure form can berealized at atmospheric pressure under properconditions of temperature. Fortunately a numberof such structures are known. The result is some-what paradoxical in that in more than half thecases the form stable at higher pressure, that is,the form which has the smaller volume, is theform with lower symmetry. The crystal formswith .highest symmetry are the close packedarrangements of spheres; one would expect theform of smaller volume, that is, the form stableat the higher pressure, to be a close packedarrangement, and therefore a form of highersymmetry. The conclusion must be that at highpressures the atoms or molecules do not on theaverage have spherical symmetry; sphericalsymmetry is a property of the atom at lowpressures, where it has more free space at itsdisposal, but when the available space is less it isforced into less symmetrical form. In the case ofsuch phenomena as directed valence in the watermolecule, for example, the detailed proof has beengiven that wave mechanics demands a sphericallyunsymmetrical solution.

Any theory of polymorphic transition musttake proper account of one very importantrespect in which a transition between solids may

differ from the transition liquid-solid. If a solidand a liquid phase a.re in contact with each otherat equilibrium, and the equilibrium is disturbedby a displacement of pressure or temperature,equilibrium is always exactly restored by anautomatic change in the system, produced eitherby melting or freezing. That is, the system comesof itself to a sharply defined equilibrium point,the same from whichever direction it is ap-proached, provided the two phases are in contact.The ordinary explanation is that equilibrium is adynamic affair, that there are always moleculesleaving the liquid phase and crystallizing into thesolid, and also always molecules leaving the solidand melting to the liquid. At equilibrium thevelocity of the two streams in opposite directionsis the same; when equilibrium is disturbed thevelocity of one or the other stream preponderates,and this excess acts in such a direction thatequilibrium is presently restored automatically.In the case of two solids, on the other hand, thestate of affairs may be quite different, in that it isoften possible to displace the temperature orpressure on two solid phases which are in contactand in equilibrium with no resulting reaction atall. If, however, pressure or temperature issufficiently displaced, the transition will run insuch a direction as to tend to restore the initialcondition. That is, solids may show the phe-nomenon of a "region of indifference. " Theextent of this region of indifference may bemeasured with some precision, and will be foundto vary with pressure and temperature. Thereare no obvious regularities of behavior, but thefact of the existence of a region of indifference,and the'manner of its variation with pressure andtemperature are entirely independent matters,depending on the particular substance. In thosecases where a region of indifference exists, it isevident that equilibrium between two phasescannot be a dynamic affair of the equality of twovelocities in opposite directions, but the mecha-nism must be more static in character; probablythere is something of the nature of a hill ofpotential between the two phases which can besurmounted only when the condition of equilib-rium is violated by a sufficient amount. It isevident, I think, that in these cases, whatever itis that pa,sses from one potential valley to theother over the intervening hill, cannot be

20 P. W. B RI DGMAN

endowed with a Maxwellian distribution ofvelocities, for if this were so there would alwaysbe some molecules with velocity sufficientlygreater than the average to leap the potentialbarrier.

The very great variety in polymorphic trans-itions found. experimentally is understandablenow that the very great variety of theoreticalpossibilities is becoming appreciated. Thus it isrecognized that there are lattices held togetherby various types of forces: ionic lattices, latticesheld together by valence forces, or by van derWaals forces, layer lattices, and molecular lat-tices. Furthermore, the distinction between thetypes just enumerated is not always clean cut.Polymorphic transitions may be expected corre-sponding to the change from one to another ofthese types of lattice, or to partial changes intype. In addition we may be prepared fortransitions involving internal electron rearrange-ments in the atoms; such changes might beexpected particularly in those atoms with incom-pleted inner shells of electrons. Perhaps thetransitions under pressure of cadmium andcerium are of this type.

The ultimate theory of polymorphic transitionand of crystallization in general will doubtlessincorporate one consideration which has receivedlittle attention up to date. It seems to me thatbroad considerations, such as for example thecondition of minimum energy, are not enoughhere, but one must be sure in every special casethat it is possible to actually realize the detailedsteps necessary to build up the crystal. This sortof consideration has proved not to be necessary intreating certain classes of phenomena, since ourexperience has been that nature will often some-how find a way to adjust itself to the require-ments of broad principles, as for example, insystems in which there is a Maxwellian distribu-tion of velocities there are always some moleculeswhich are able to settle down into a position ofminimum energy even though an intermediatestage may be necessary in which the molecule ismoving toward higher energy. That this sort ofthing plays a definite role in polymorphictransitions is shown by the phenomenon of the"band of indifference" already considered. Thereis an analogous consideration with regard to thebuilding up of the crystal from the liquid or

vapor phase or from solution. Only recently havephysicists been considering the details of the wayin which the crystal grows. It appears that on thesurfaces of deposition there are conditions ofreversibility and growth to be satisfied quitedifferent from the conditions at interior points.These growth conditions, as well as conditions onthe completed structure, must obviously be metif the crystal is to exist. The thermodynamicconditions, expressed perhaps in terms of theequality of two thermodynamic potentials, areonly a small part of the story.

IX. IRREvERsIBLE TRANsITIQNs

The transitions thus far considered have allbeen reversible thermodynamically and me-chanically. There is one irreversible transitionproduced by high pressure, that from white toblack phosphorus, which I think contains possi-bilities of great theoretical interest. This trans-ition is remarkable for the very large increase ofdensity, 46 percent, and for the very great changeof properties, from a good insulator to a fairlygood conductor of electricity, and for the detailsof the way in which the transition takes place.The transition cannot be hastened in the usual

way by inoculating with a nucleus of the othermodification, but some preliminary process ofpreparation is necessary which apparently takesplace homogeneously throughout the volumewith slight increase of density, and which pro-ceeds at an accelerated pace for perhaps ten orfifteen minutes until some critical condition isreached which causes the entire system tocollapse suddenly into the black modification.Just what this process of preparation may consistin seems at present quite obscure. It would besurprising if phosphorus, were the only element,ordinarily non-metallic, which was capable ofassuming irreversibly a metallic form. I havesearched for a similar transition of sulfur, whichstands next to phosphorus in the periodic table,but without success. A pressure of 50,000 kg/cm'was applied at room temperature; it would bedesirable to much increase the pressure range forinvestigations of this character.

Ag~O affords an example of a transition inter-mediate between the completely irreversibletransition of white to black phosphorus, and the


clean cut perfectly reversible transition of manysubstances. If the volume of AgsO is determinedas a function of pressure, increasing pressureuniformly to a maximum and then decreasing itto its initial value, a very broad hysteresis loopwill be found, similar in shape to the magnetiza-tion loop of many ferromagnetic substances.Small amounts of hysteresis in solid-solid trans-itions are not uncommon, and would be expectedif there are impurities forming solid solutions, sothat phenomena of diffusion in the solid state areinvolved, but this seems to be something differ-ent, and the explanation does not yet appear.

X. DISCONTINUITIES —TRANSITIONS OF THE

SECOND KIND, ETC.

direction, abrupt within experimental error, of thepressure-volume isotherm, which of course meansa discontinuity in compressibility and thermalexpansion. The pressure of this discontinuity wasdisplaced toward higher values at higher temper-atures, just like the displacement of an ordinarytransition, and in fact at 9500 kg/cm' thetemperature of the discontinuity is +30'C. Ishowed that there is a purely geometrical relationconnecting the discontinuity in compressibilityand thermal expansion with the slope of the lineon which the discontinuity occurs, namely

d7/dp= —A(av/aP), /a{av/ar) „as indicated in Fig. 2. By utilizing the thermo-dynamic relation

The phase changes thus far considered, towhich the conventional Clapeyron's equationapplies, have involved discontinuities in thevolume and the energy content. Recently anothertype of change has begun to attract attention, inwhich there are discontinuities in the derivativesof volume and energy .content, that is, dis-continuities in thermal expansion and com-pressibility and specific heat. This sort ofphenomenon was first recognized at low tempera-tures, and perhaps the best known example isthe case of NH4C1 investigated by Simon, "whofound that there is a region only a few degreeswide centering around —30'C in which there areenormous variations in the specific heat and thethermal expansion. Somewhat later at Leyden,Keesom" found that anomalous effects which hehad previously discovered in liquid helium at lowtemperatures and which he had been at firstinclined to explain as showing two modificationsof the liquid, actually involved discontinuities inonly the derivatives of volume and energy, thelocus of these discontinuities being a curve in thep —t plane. Ehrenfest" published a paper on thethermodynamic aspects, and proposed that suchdiscontinuities should be called "transitions ofthe second kind, " a name which has apparentlybeen accepted.

Some time before the paper of Ehrenfest I hadmade an experimental study of the effect ofpressure" on the anomalies of NH4C1 andNH4Br found by Simon. For NH4Cl it appearedthat the anomaly manifests itself by a change of

this can be thrown into a form exactly analogousto Clapeyron's equation, namely

dr/dP= rdv/dQ,

where dv/dQ is the ratio of volume change to heatabsorbed in making the change corresponding toAB of the figure. Ehrenfest used also the equiva-lent relation

d r/dp = —r68v/Br/hC, .

The behavior of NH4Br was unlike that ofNH4C1 in several respects. The volume anomalyis of the opposite sign, so that at high pressures itis displaced to lower temperatures instead of tohigher. Because of the inconvenience of pressuremanipulations at low temperatures, the anomalywas studied only at —72', at which it occurs at1600 kg/cm'. Furthermore, the anomaly is much

Pressure

Fio. 2. Indicates the geometrical relations satisfied by adiscontinuity of the derivatives of volume which experi-ences a temperature displacement when pressure is dis-placed.

P. W. BRI DGMAN

more abrupt than for NH4CI, so that withinexperimental error it appeared like a discon-tinuity of the volume itself rather than of thederivative, and therefore had the characteristicsof an ordinary transition.

Later I found a large number of anomalies" inthe behavior of solid substances in the tempera-ture interval between 0' and 100'C and up to12,000 kg/cm'. A number of these are, withinexperimental error, sharp and reversible discon-tinuities in the first derivatives, and wouldtherefore be described as "transitions of thesecond kind. " Phenomena of this sort are com-mon among certain classes of alloys. There arealso examples in which the discontinuities are notperfectly reversible, but there are hysteresiseffects. There are also cases in whioh the anomalyhas no sharply defined edges, but there is aregion of anomalous curvature, which is displacedto other pressures when the temperature ischanged. An example of this sort of anomaly isalso afforded by NH4Cl, this anomaly havingapparently no connection with the one alreadydiscussed; another very striking example isafforded by metallic chromium in a state of high

purity.The great variety of phenomena offered by

these small scale anomalies shows that a classi-fication of transitions into "first, second, third"etc. kinds (for obviously Ehrenfest's scheme ofclassification and nomenclature can be continuedindefinitely) is to a certain extent at present amatter of convenience, reflecting more or lessclosely the experimental accuracy. NH4Br is acase in point; in Simon's measurements thereappeared only very rapid variations in thethermal expansion, but under pressure, at atemperature 40' lower, the anomaly had been sosharpened that it appeared as a discontinuity inthe volume itself. What may appear to roughmeasurements as a discontinuity in the firstderivative may appear as a discontinuity in thesecond derivative to more refined measurements.It is to be remembered furthermore that anordinary phase change would not appear to beperfectly discontinuous if measurements couldbe made accurately enough to detect the effect ofthe varying ratio of surface energy to volumeenergy as the transition progresses. It would

appear to me that the physical value of the

concept of the transition of the "second kind"will depend on the discovery of some character-istic physical process corresponding to such atransition, analogous to the change from onelattice to another which may characterize anordinary transition. As far as I know this has notyet been done, but I can see no reason why itshould not be done, or why the transition of thesecond kind should not be found to correspond tosome significant type of physical process. In factthere may be several such kinds of significantprocess, just as we know that a transition of thefirst kind may mean a change from one amor-phous phase to another or a change from anamorphous to a crystalline phase, or a changefrom one crystalline phase to another. It is, Ibelieve, very probable that we already under-stand the mechanism of at least one type ofprocess corresponding to a transition of thesecond kind; when certain of the gold-coppersystem of alloys are warmed there occurs achange from a type of lattice in which gold andcopper atoms are arranged regularly to a latticein which they are scattered about in haphazardpositions, and the change from one to the othertype of lattice seems to be initiated with adefinite break in the derivative, according to thework of Bragg and Williams. "

Some progress has been made toward theexplanation of these anomalies; probably thebest known is that of Pauling" for the anomaliesof NH4C1. Pauling's explanation is that there is atemperature range in which the molecules rapidlyacquire rotational motion. This is a process whichto a certain extent catalyzes itself, since theattaining of a small amount of rotational motionopens up the structure and makes the attainingof additional rotational energy increasingly easy.A similar element of autocatalysis is contained inthe explanation of Bragg for the change in thegold-copper system, and it may be that this is animportant characteristic of the transition of thesecond kind. But the details of Pauling's expla-nation have not been worked out; it does notappear whether there is a sharp break in aderivative or whether the transition is initiatedmore or less gradually, and the explanation of thedifference in sign of the volume effects in NH4C1and NH4Br has not been given in detail, althoughthere would appear to be no unsurmountable


difficulty here. In general, one may anticipate anumber of different types of thing that mightgive rise to these anomalies, just as we havealready seen that there must be different types ofpolymorphic change. Particularly does thenumber of possibilities appear very large whenone reflects that the solid is not merely built upof a, number of independent atoms, but that thewhole solid is one structure, with an enormousnumber of energy levels and different types ofwave function, much more complicated than theinterior of a single atom.

XI. ELECTRICAL RESISTANCE

We next consider the effect of pressure onelectrical resistance, and in particular the re-sistance of metals. Theory has not yet beenparticularly successful in dealing with thisphenomenon, and in fact the only attempt whichendeavors to base the explanation on the mostrecent pictures of the conduction process andwhich can be regarded as promising is that ofKroll. "The reason for this comparative failure oftheory can be appreciated when it is realized thatwave mechanics treatments of electrical con-ductivity in metals give an expression for theconductivity itself only as a third approximationin the method of calculation, * and we are hereconcerned with the effect of pressure on thefactors in this third approximation. The diffi-culties are suggested by a consideration of thesimple formula for electrical conductivity givenby the elementary Sommerfeld theory, in whichthe electrons are approximately treated as a gasobeying the Fermi statistics:

0. = e'ln/mv

where l is the mean free path, ti the limitingvelocity of the Fermi distribution, and the otherletters have conventional meanings. The ele-mentary theory sets 1/8 proportional to the meandistance of separation of atomic centers, so thatfrom this point of view the effect of pressure on1/t~ should be proportional to the compressibility

* The first approximation treats the atomic nuclei asstationary, the second approximation is that the atomsexecute elastic vibrations independent of the electrons,and the third approximation is that the coupling betweenatomic vibrations and the motion of the electrons resultsin electrical resistance. See Bethe, Handbgck d. Physik,Vol. XXIV, 2, 2nd Ed. , p. 369.

But we know from Schottky's theorem that theaverage kinetic energy of the electrons increaseswith pressure by a factor which is not simplyrelated to the compressibility, so that from thispoint of view it is not probable that the con-nection between pressure end 1/8 can be assimple as the elementary theory would suggest.The calculation of the variation of / with pressurewould obviously be very complicated, since it hasto be left unknown in the elementary Sommerfeldtheory, and can be computed only inversely byassuming the formula and substituting numericalvalues for all the other factors. l will obviouslydepend in a complicated way on the amplitude ofatomic vibration, and the distance of separationof the atoms and on the frequency, that is, thestiffness of the restoring forces. The situationwith regard to the pressure coefficient is evenmore complicated if one considers the expressionfor the conductivity given by the more exacttheory, as in Bethe's article in the IIandbuch onpage 523. This contains four or five factorswhich may vary with pressure and which are notsimple.

In spite of the lack of perspicuousness of thepresent theoretical picture with regard to thepressure effects, I believe that the experimentalphenomena themselves are important enough sothat we must demand that theory eventually givea satisfactory account of them. For the effects arenot small; they are almost always greater by afactor of several fold than the changes of volume.Thus the volume of potassium under 15,000kg/cm' is about 70 percent of its initial value,while its resistance is only 25 percent of its initialvalue; the resistance of strontium under 12,000kg is 84 percent greater than initially whereas itsvolume is only 9 percent less. Among the non-metallic elements there are even more extremeexamples; the conductivity of black phosphorusat 18,000 is 100 times its initial value, that oftellurium at 20,000 is 100 times initial, and AgsSat 3000 is about 20 times as good a conductor asinitially.

Confining ourselves now to the metals it wouldappear at first glance that there are greatvarieties of behavior. Most of the commonmetals decrease in resistance under pressure, andthe rate of decrease becomes less at high pres-sures. However, the resistance of something like

24 P. W. B R I DGM AN

one-fifth of all the metals investigated increasesunder pressure; included here are bismuth andantimony, which would be expected to beabnormal, but there are other metals which one isless likely to think of as abnormal such as lithiumand strontium. There seems to be no significantcorrelation between position in the periodic tableof the elements and a positive coefficient. Thepressure coefficient of resistance of all metals withpositive coefficient increases with increasingpressure, that is, the plot of resistance againstpressure is convex toward the pressure axis.Finally there are a few metals whose resistanceat first decreases, but then passes through aminimum, and increases at higher pressures;these are caesium with a minimum at 4000kg/cm', rubidiUm with minimum at 1'7,800, andbarium with minimum at 9000. According to thepresent pictures of metallic conduction, a de-crease of resistance with increasing pressurewould seem to be the effect naturally to beexpected. For electrical resistance is thought toarise essentially from interference with theelectron waves by departures of the atomiclattice from perfect periodicity, and these de-partures are simply connected with the amplitudeof atomic vibration. At high pressures theamplitude of atomic vibration becomes lessbecause of increased stiffness of the forces ofconstraint and increase of the characteristictemperature, and hence the scattering of theelectron waves becomes less and the resistanceless. This wave mechanics picture is in somerespects much like the "gap" theory of electricalresistance to which I was led by my pressureexperiments, in which resistance to electronmotion was pictured to arise from the gapsbetween atoms produced by temperature agita-tion. The existence of positive coefficients forsome metals, however, would seem to demand asecond type of mechanism; in the theory of Krollsuch a second mechanism is provided by theeffect of the distance of separation of atomiccenters. By a sort of scattering effect theresistance becomes greater if the distance ofseparation becomes less, any inequalities in thestructure becoming relatively accentuated atsmall distances. This gives Kroll a term pro-portional to the compressibility, tending to

increase resistance with pressure. * In my gaptheory I was similarly drawn to provide a secondmechanism to account for some of the abnormalcases, particularly the phenomena of tensioncoefficient of resistance when combined withpressure coefficient. The second mechanism whichI imagined was that some of the electrons passthrough open channels between the atoms, andthat these channels become contracted when theatoms are pushed together by hydrostatic pres-sure. This sort of thing was therefore very muchlike the second mechanism of Kroll. By thusproviding two different mechanisms, oppositesigns for the pressure effect can be provided for indifferent metals, and also the reversal of sign andthe minimum of resistance, if we suppose thatat first one of the effects preponderates and athigher pressures the other becomes larger. Inthis way Kroll provides qualitatively for theminimum of resistance of caesium; quantitatively,however, his results are rather wide of the markin many cases.

Consideration of the complete body of ex-perimental evidence, many features of whichhave been discovered since I last worked on myown theory ten years ago, makes me much lesssatisfied with this picture of two differentmechanisms, one or the other of which maypreponderate. The reason is that the effect ofpressure on the resistance of nearly all metals(elements) yet measured can be described quali-tatively in terms of a single family of curves,shown in Fig. 3. Here the resistance of any metalat constant temperature as a function of pressurehas a minimum with respect to pressure, theminimum occurring at higher pressures as thetemperature rises. The location of the origin ofpressure in the diagram depends on the metal; formetals normally having a positive pressurecoefficient of resistance the pressure origin mustbe taken beyond the minimum, so that for suchmetals a negative coefficient would be anticipatedif high enough negative pressures could be

* Kroll also has a term involving the variation withpressure of Poisson's ratio, which he ignored because hehad no experimental values for it. I may remark paren-thetically that I have determined the effect of pressureon rigidity/' and the change of Poisson's ratio underpressure may be calculated from this and the effect ofpressure on compressibility. In general Poisson's ratioincreases by about the same percentage amount as theincompressibility.

HlGH PRESSURE PHENOMENA 25

T'ressvre

FxG. 3. The resistance of all metals as a function ofpressure and temperature is consistent with a family ofcurves of this character. The curves are arranged in orderof increasing temperature, that is, r2)r&, etc.

realized. For normal metals, on the other hand,the pressure origin is as shown, and the pressureof the minimum is beyond the present experi-mental range. For only a few metals is theminimum within the range of experimentallyrealizable pressures. This diagram suggests thatmetals with positive and with negative coeffi-cients both display aspects of essentially onephenomenon. Furthermore, the ultimate state ofaffairs with all metals woufd thus appear to beone in which resistance increases with pressure.What is still more significant, the curvature ofresistance against pressure is always upward.Many phenomena vary with pressure in theopposite way, becoming less affected by equalincrements of pressure at higher pressures; this iswhat might naturally be expected by a sort oflaw of diminishing returns. The fact that thecurvature is like that shown suggests that thereis no limit to the possible resistance of a metal,and that the tendency is to become a perfectinsulator at infinite pressure. This would suggestthat possibly as the atomic centers are pushedindefinitely close together a new type of solutionof the wave equation will appear, by which theouter electrons, which are normally free andprovide for the conduction, will become bound inthe same way that the inner electrons are nor-mally bound, the entire metal becoming as itwere a single complicated nucleus.

It would be most important if the minimum ofresistance could be established by actual experi-ment for elements of all situations in the periodictable. It has actually been established only forcaesium, rubidium and barium, and indicatedwith high probability by a short extrapolation for

potassium and sodium. Extrapolation for othermetals does not give much hope of actuallyrealizing the minimum for them, the probablepressure of the minimum being in all cases higherthan 40,000 kg/cm'. However, the existence of aminimum is made more probable in these cases ifone plots log R against pressure instead of R.The reason is that the curve of resistance againstpressure of those metals with a negative coe%-cient must almost inevitably be convex towardthe pressure axis since the resistance can notbecome less than zero, and convexity has littlesignificance, but log R on the other hand may goto minus infinity, there is no such restriction on .

the curvature, and convexity is significant. As amatter of fact, log R of all metals with a negativecoefficient is convex toward the pressure axis,which is the direction of curvature demanded bythe existence of a minimum.

It is an experimental fact that the pressurecoefficient of resistance is little affected bytemperature, or what is the same thing, thetemperature coefficient of resistance is littleaffected by pressure. What slight change there isis consistent with Fig. 3. One would expect thetemperature coefficient to be little affected bypressure because the temperature coefficient ofall metals is roughly the same, 1/r, and the samemetal under pressure would not be expected todiffer more from the same metal not underpressure than two different metals. The tempera-ture coefficient 1/r is accounted for by the presenttheory, so that this aspect of pressure phenomenawill be taken care of automatically in a theory ofpressure effects based on present pictures.

There are a few abnormal metals which do notfit into the scheme of Fig. 3. But bismuth, whichmight be expected to be abnormal, does fit intothe scheme. This may to a certain extent beaccidental. It is highly probable that theabnormalities of bismuth are connected with itscrystal structure, because the pressure effects ofliquid bismuth are quite normal. This view withregard to bismuth becomes more plausible in thelight of the recent discovery of the high pressuremodification of bismuth, with normal volumerelations to the liquid. It would be of greatinterest to determine the sign of the pressurecoefficient of resistance of this high pressuremodification of bismuth.

P. W. BRI DGMAN

There have been comparatively few measure-ments of the effect of pressure on the resistance ofliquid metals, and the theory of the resistance ofliquid metals has been little considered. It seemsto me that an attack on the problem of theresistance of liquid metals should not be too longdeferred. Qualitatively, liquid metals show manyof the same phenomena as solids, particularlywith regard to the order of magnitude of theresistance itself. The present theory of electricalresistance, on the other hand, starts from theassumption of the lattice, and this assumptionenters essentially into all the discussion. Butresistance phenomena in liquids make it appearthat these assumptions are not necessary to someof the conclusions, which could probably bededuced on more general grounds. In this con-nection, one pressure phenomenon is probably ofsignificance. The resistance of the liquid phaseseems to bear an approximately constant ratio tothe resistance of the solid phase at the reversiblefreezing point, irrespective of whether freezingtakes place at low temperature and low pressureor is displaced to higher temperature by highpressure. Thus the ratio of the resistance of liquidto solid potassium is 1.56 at the normal freezingpoint at 62.5' at atmospheric pressure, and is 1.55at the reversible freezing point at 165' at 9700 kg.

It is probable that an ultimate increase ofresistance with pressure is as characteristic ofliquids as of solids. In the case of caesiumextrapolation makes it highly probable, althoughthe actual measurements could not be carried out,that the resistance of the liquid will pass througha minimum at about the same pressure as theresistance of the solid. Lithium in the liquidstate has a positive pressure coefficient ofresistance, and the plot of resistance againstpressure is convex toward the pressure axis, as itis for solids. Liquid bismuth„on the other hand,has a negative coefficient, although the coefficientof the solid is positive. It may be, therefore, thatthere are special mechanisms capable of giving apositive pressure coefficient at low pressure aswell as the universal mechanism giving theultimate positive coefficient of all metals atextremely high pressures.

The effect of pressure on the resistance of non-metallic substances is of interest. A non-metallicconductor or a semiconductor is supposed to

differ from a metallic conductor in that the al-lowed energy bands in some semiconductors areseparated by wider intervals than in metallicconductors. One would expect that the effect ofpressure might be to crowd the energy bandsmore closely together, giving the semiconductormore the characteristics of a metal. The fewsemiconductors that have been measured do haveenormously large negative coefficients of re-sistance, so that at high pressures their resistancedoes as a fact approach that of a metal. Telluriumis a substance which is ordinarily classed as onlypartially metallic in character. Under 20,000kg/cm' its resistance drops to one percent of itsinitial value; the plot of log R against pressure isconvex toward the pressure axis, so that a mini-mum of resistance at considerably higher pres-sure would be consistent with the effect in theexperimental range. Black phosphorus is, ofcourse, more like the metals than ordinary yellowor red phosphorus, because it does conductelectricity appreciably whereas yellow and redphosphorus are complete insulators, but blackphosphorus would be said to be much lessmetallic than tellurium. A non-metallic character-istic of black phosphorus is that its temperaturecoefficient of resistance is negative. The effect ofpressure on the resistance of black phosphorus isalso very large. Initially Iog R is concave towardthe pressure axis, which means an increasinglyrapid approach to the metallic condition as pres-sure increases; this is not unnatural because blackphosphorus at atmbspheric pressure is furtherfrom the metallic condition than tellurium. Inthe neighborhood of 12,000 kg/cm', however,the curvature of log R of black phosphorus re-verses, becoming convex toward the pressure axisabove 12,000, which means that at high pressureblack phosphorus has become metallic in charac-ter at least to this extent. Furthermore, between15,000 and 20,000 the sign of the temperaturecoefficient reverses, so that in this respect alsoblack phosphorus becomes like the metals athigh pressure.

One of the successful features of the newtheories of conduction, which has been mademuch of, is the account which it gives of theresistance of alloys. When the effective free pathis long, as it is in the wave mechanics picture, asingle misfit atom may obviously have a great

H I GH PRESSURE PHENOMENA

effect in introducing resistance. This picture isconsistent with a generalization which may bemade from all measurements to date, withoutexception, on the resistance of alloys: this is thatthe initial effect of adding a foreign substance toa pure metal is to make the pressure coefficient ofresistance more positive (algebraically). Thereason for this is purely geometrical; the relativemisfit of a foreign atom which is a little too large,let us say, becomes greater the closer togetherthe atomic centers of the other atoms of thelattice. But increasing misfit means increasingresistance, and increasing pressure pushes theatoms closer together. Hence pressure in suchsubstances tends to increase the resistance morerapidly than in the pure substance, or the pres-sure coefficient of resistance is displaced in thepositive direction.

Phenomena in single crystals of non-cubicmetals have not as yet received much theoreticalattention. There are, however, important phe-nomena to be considered here, and one may ex-pect that this may be one of the next problemsattacked. Such uniformities as, for example, thatthe resistance in a crystal is almost always great-est in the direction of greatest atomic separation,which is also the direction of greatest compressi-bility, must be significant. There are also uni-formities in the behavior under pressure whichmust be significant. Thus the effect of pressure onthe three "normal" non-cubic metals, zinc, cad-mium and tin, is to make the crystal more nearlyisotropic with respect to resistance, which isalso the effect on the lattice spacing. The effecton the two abnormal metals bismuth and anti-mony is the opposite, however, pressure heremakes the lattice spacing more nearly equal indifferent directions, but accentuates the non-isotropy of resistance.

Theory can usually derive some profit fromconsidering the abnormal cases, which playsomewhat the same role as pathological cases inmedicine. Bismuth and antimony have alreadybeen mentioned, in which the pressure effect onresistance is probably accounted for by a differentmechanism from that responsible in more normalmetals such as lithium and calcium. There isanother highly abnormal metal which I haverecently found in a place in the periodic tablewhere one would little expect it, namely,

chromium. The resistance of chromium at at-mospheric pressure is an abnormal function oftemperature; there is a minimum of resistance at10' and a maximum at O'C, the shape of thecurve being much like that of the curve for thevolume of water as a function of temperature. Asa function of pressure the resistance is highlyabnormal; the coefficient is throughout negative,but the curvature of an isothermal of resistanceagainst pressure may be of either sign, or theremay be a reversal of sign on the same isothermalwith a point of inflection. The resistance iso-therms cross and recross in a complicated way,which means both positive and negative tem-perature coefficients at constant pressure. Notonly is the resistance abnormal, but the relationbetween pressure and volume is also abnormal;there is, however, no obvious correlation be-tween the abnormalities of resistance and volume.

Another abnormal metal is arsenic; this showsabnormalities only when very pure and onlywhen in the single crystal condition, and theabnormalities are almost entirely connected withthose directions in the crystal perpendicular tothe principal axis. The abnormality seems to con-sist in deFinite pressure ranges within which therelation between pressure and resistance is linear,but with different coefficients in the differentranges, and no discontinuity of resistance itselffrom one range to the next. That is, there is a sortof transition of "the second kind" in the re-sistance. There are also some very unusual sea-soning effects in single crystal arsenic. *

XII. THERMQELEcTRIc PHENQMENA

Another group of electrical phenomena is thatof thermoelectricity. The rigorous theory of theseeffects is complicated; the simple Sommerfeldtheory, in which the electrons are treated as free,gives acceptable results only for the alkalimetals, for other metals there being discrepancieseven of sign. The more rigorous theory takesspecific account of the way in which the electronsare distributed among the allowed energy bands,and finds different signs for the Thomson effect

* Since this paper was written, a very promising theoryof the effects of pressure on resistance of the metals hasbeen made by N. H. Frank, and submitted for publicationin the Physiicat Renew. This theory accounts for theminimum resistance of the alkali metals,

28 P. W. BRI DGMAN

according as the bands are almost empty or al-most full. As far as I know, no attempt has beenmade to deduce the effect of pressure on thermo-electromotive force according to the morerigorous theory, although Houston" has givenan expression on the basis of the simple Sommer-feld theory which fails in the case of copper by afactor of ten. One would expect that the results ofthe more rigorous theory would be complicated.This expectation agrees with the experimentalfindings, for the effects of pressure on thermale.m. f. are perhaps the most complicated of anypressure eff'ect; there are great departures fromlinearity, curvature in both directions, points ofinfection, maxima and minima, crossing of thecurves and reversals of sign. As a rough averagefor the some twenty metals examined, the sign ofthe effect is such that in more cases than not theelectrons absorb heat in flowing from compressedto uncompressed metal. In this respect the elec-tron gas is like an ordinary gas which also ab-sorbs heat on expanding. There are so manyexceptions to this rule, however, that the generalresult must not be given much significance. It is,however, significant that the magnitude of theeffect is large; the Peltier heat when an electronRows from a metal compressed to 12,000 kg/cm'to the same metal under no pressure is on theaverage of the same order of magnitude as whenan electron flows from one to another differentmetal. In view of the theoretical suggestion thatthe thermal e.m.f. depends importantly on theway in which the electrons are distributed in anenergy band it would therefore appear that pres-sures in the experimental range are capable ofaff'ecting important redistributions within theenergy bands. Hence in spite of, or rather becauseof, the complications, it may well be that a care-ful theoretical study of the effects of pressure onthermoelectric effects will give a deeper insightinto the details of the electron distribution thanother simple phenomena.

XIII. THERMAL CONDUCTION

Thermal conduction of metals also receives atreatment in the new theories, and in fact one ofthe most important results of Sommerfeld'ssimplified theory was that it still provided anexplanation for the Wiedemann-Franz ratio,

which had been the most important result of theclassical theory of Drude and Lorentz, whilemeeting the specific heat difficulty which theclassical theory could not avoid. But the newtheory applies only to the part of the conductionperformed by the electrons; so far as I know therelatively small part played by the atoms inconducting heat has not yet been fitted into thepicture. Experimentally, of course, the contribu-tions of the two agencies are not separated, sothat a measurement of the pressure coefficient ofthermal conductivity is a measurement of thepressure coefficient of both mechanisms. It turnsout that there is no uniformity from metal tometal; for example, the thermal conductivity oflead and tin increases under pressure, while thatof copper, silver and nickel decreases. Further-more, in all cases except lead and tin the changeof thermal conductivity under pressure is lessthan the change of electrical conductivity, so thatexcept for lead and tin the pressure coefficient ofthe Wiedemann-Franz ratio is negative. Theexperiments are difficult; it was not possible tomeasure the more compressible metals for whichlarge effects would be expected, and in practicallyevery case it was not possible to find in which waythe effect departs from linearity. Theoreticallyone would be prepared to find a zero coefficientfor the Wiedemann-Franz ratio by the argumentused before for the variation of temperature coef-ficient of resistance with pressure, namely thatthe Wiedemann-Franz ratio of all metals is nearlythe same, and since in general a metal underhigh pressure would be expected to be more likeitself under zero pressure than another metal, onewould expect the effect of pressure on theWiedemann-Franz ratio to be small. It is to beremembered, however, that for a single metalthe variation with temperature of the Wiede-mann-Franz ratio is in many cases not exactlythat demanded by theory, and pressure doubtlessaffects the characteristic temperature of a metal.In view of the smallness of the eff'ect in the metalswhich could be subjected to measurement it isobvious that a comparatively small shift in theparts played by electronic and atomic conductioncould account for the experimental results.Furthermore, this shift may be a small rangeeffect and be reversed at high pressures as far asany evidence goes which we have at present. An


explanation of the pressure effects must thereforewait for the completion of the theory of theatomic part of thermal conduction; such atheory must be prepared to explain why in manycases atomic conduction becomes poorer at highpressures. It is perhaps not hopeless to anticipatethat such an effect will be found in view of thefact that thermal conduction involves the scat-tering of elastic waves, and that any lack of per-fect fit, which is conducive to scattering, may beaccentuated by high pressure. But it must be asomewhat difficult point to show why it is thatin the majority of cases pressure diminishes thescattering of electron waves but increases thescattering of elastic waves.

The effect of pressure on the thermal con-ductivity of non-metallic solids or of liquids ismuch easier to measure, and there is considerableexperimental material of much greater accuracythan for the metals. The thermal conductivityof organic liquids increases under pressure byan amount which to a first approximation is thesame for all liquids, the increase under 12,000kg/cm' being roughly by a factor of two. Thecurve of conductivity against. pressure is concavetoward the pressure axis, so that the effect ofequal increments of pressure dimin'ishes at highpressures, as is usual. It turned out from themeasurements that there is a rather close corre-lation between the effect of pressure on thevelocity of sound, which can be calculated interms of the compressibility, and the effect onthermal conductivity. This correlation suggesteda very simple expression for the thermal con-ductivity of a liquid, namely

X=2av8 ~,

where X is thermal conductivity, n the gasconstant, 2.02 X10 ", v is the velocity of sound,and 8 is the average distance between centers ofmolecules, calculated approximately by assumingthe liquid piled up with the molecules in simplecubical array as in a simple cubic crystal. Onecan derive this expression in an elementary wayby supposing that the mean difference of thermalenergy in the direction of the thermal gradientbetween adjacent molecules, which is 2abdr/dx(~ is temperature) gets handed along in thedirection of the gradient with the velocity ofsound. Jeffreys~ has shown that a better way

of arriving at the same result is to suppose thatin a liquid the molecular irregularity is so greatthat the energy of an elastic wave is completelyscattered in the minimum possible distance, thedistance between centers of molecules. Thederivation suggests that the formula should alsobe applicable to amorphous solids, and in fact itis known to be applicable to hard rubber and tosome glasses.

The very simple expression 2nvb—' is applicable

with an error of 10 or 15 percent to normalorganic liquids, and is also applicable to water,the thermal conductivity of which is four timesgreater than that of the normal organic liquid.It is therefore somewhat surprising that theformula does not give very accurately thepressure coefficient of thermal conductivity; itwould predict an increase of thermal conductivityat 12,000 between three and fourfold on theaverage, whereas the actual increase is only bytwofold. Any lining up of the molecules of theliquid under high pressure by which the distanceof complete scattering becomes greater wouldincrease the thermal conductivity by a newfactor and would therefore increase the discrep-ancy. On the other hand, if there is a tendencyfor the atom to become the unit of structure athigh pressure instead of the molecule, which isconsistent with the known tendency of thespecific heats, centers of scattering would developinside the molecule, and the too small pressurecoefficient would be accounted for. The exactanalysis of any such effect is obviously compli-cated; this much is at least evident, that anadequate theory of the effect of pressure on ther-mal conductivity in liquids will have to wait untila theory is developed competent to account atleast for the simple volume relations.

In working out a more exact theory of thermalconduction in liquids a significant point to bekept in mind is that the temperature coefFicientof thermal conductivity of nearly all liquids re-verses at high pressures; at atmospheric pressurethe liquid is a poorer conductor at high tempera-tures (greater elastic scattering) but at high pres-sures the liquid conducts better at higher tem-peratures. The pressure of reversal is roughly thesame as the pressure of reversal of the sign of(pp/g7. &.

The effect of pressure on the thermal conduc-

30 P. W. BRI DGMAN

tivity of simple crystals has been measured onlyfor NaC1; there is room here for much moreexperimental work.

XIV. VISCOSITY OF LIQUIDS

There is extensive experimental material forthe effect of pressure on the viscosity of liquids,mostly organic. The general characteristic of thisphenomenon is the great magnitude of the effect,which is much larger than that of any otherpressure effect. Thus eugenol increases in viscos-ity at 12,000 by a factor of 10'-fold. At highpressures the increase of viscosity is roughly ex-ponential, the plot of log (viscosity) against pres-sure being approximately linear; at the low pres-sure end this plot usually shows curvature whichmay be in either direction. There is a very markedcorrelation between the pressure effect and thesize of the molecule; for monatomic mercury theincrease of viscosity at 12,000 is only 33 percentagainst the 10' already mentioned for eugenol.There is no theory of the viscosity of liquidswhich adequately reproduces the effects of pres-sure. There have been elementary theories whichdemand that viscosity be a function of volumeonly; this was a fairly good approximation for therange of pressure experiments at that time avail-able, which reached to only 3000 kg/cm', butover the extended range up to 12,000 the relationfails by a very large factor. An improved theoryof Brillouin recognizes that viscosity need not bea function of volume only, but gives for the tem-perature coefficient of viscosity at constant vol-ume a numerical result which is in error by afactor of 5000. By far the most successful theoryof the viscosity of liquids is the recent one ofAndrade. "This gives the temperature coefficientof viscosity at atmospheric pressure with con-siderable success, and also in most cases gives afair account of the variations with pressure upto a few thousand kilograms, but at higher pres-sures it, goes badly wrong. Andrade remarks thatthe pressure at which the theory begins to gowrong is also the pressure at which 8'v/8~' re-verses sign; apparently at this pressure someimportant reconstruction begins to take place inthe structure of the liquid. From a qualitativepoint of view it has seemed to me that these highpressure effects are to be understood only in

terms. of an effective interlocking of the mole-cules; from this point of view the very largeeffects, particularly with the complicated mole-cules, and the failure of viscosity to be a purefunction of volume, are understandable. Suchinterlocking effects demand that in any mass mo-tion of the liquid such as is encountered in meas-uring viscosity the molecules retain their indi-viduality, for one can hardly conceive that theforces required to tear the molecules apart wouldnot be very much greater than are involved ineven the high viscosity at high pressure. Thehypothetical loss of individuality by the mole-cule at high pressure which we have discussed inconnection with other phenomena is a less drasticthing, for these other phenomena have involvedonly pure volume compressions. The less drasticsort of loss of individuality may perhaps corre-spond to an internal quivering of the atoms inthe molecule, with more relative freedom thanthey possess at lower pressures —the sort of thingthat would result in a higher specific heat.

XV. CQNDITIQNs oF RUPTURE

We now consider a topic connected with highpressure effects a little more remotely than thosewhich we have discussed hitherto. In the veryextensive discussions of the last few years withregard to the so-called "structure sensitive" and"structure insensitive" properties of crystals,considerable attention has been paid to the ten-sile strength. It is well known that the actualstrength is always very much less than the calcu-lated strength, and discrepancies by a factor of100 are not unusual. The "calculated" strengthis found in terms of the assumed law of forcebetween the molecules. The method alwaysadopted is to find the distance of separation atwhich the total cohesive force is a maximum, andthen to postulate that rupture occurs when theextension under external force becomes greatenough to result in this amount of separation.There are complications because the lateral con-traction and longitudinal extension are different,but this does not alter the main argument. Thefact. that rupture occurs in practise long beforethe extension reaches the calculated value is putdown to internal imperfections.

Entirely apart from the question of whether


there are or are not internal imperfections whichmight result in premature rupture, I do not be-lieve that a correct criterion of rupture has beenadopted in the theoretical calculation, and I be-lieve that a simple eRect observed at high pres-sure makes this absolutely certain. If a ring ofhard rubber is provided with a closely fitting steelcore, and if the whole assembly is then immersedin a liquid and subjected to hydrostatic pressure,the hard rubber ring will be found to split at apressure of a few thousand kilograms, just as ifa conical wedge has been driven into it. In fact,this is actually the nature of the eRect. For ifthere had been no steel core the rubber ringwould have shrunk by a very appreciable amountbecause of the high compressibility of hard rub-ber, but the steel core, the compressibility ofwhich is very small, prevents the natural con-traction of the rubber under pressure, so that wehave eRectively a steel wedge forced into thecontracted ring. A simple calculation shows thatalthough the rubber ring at the rupture point islarger than it would have been without the steelcore, it is nevertheless smaller than it was initiallyunder no pressure. In fact every strain in the hardrubber ring is a strain of compression, but nev-ertheless a clean tensile rupture takes place. Itis therefore obvious that the critical distance ofseparation criterion of rupture cannot possiblybe correct under these conditions. A little con-sideration shows, I believe, that the requirementthat the molecular cohesive forces be a maximumat the critical extension is simply not pertinent,this being a criterion taken over rather uncrit-ically from the method of function of a specialtype of testing machine. One may grant that themaximum force gives a sufficient condition forrupture, but not a necessary one. It seems to methat a stability criterion is much more pertinent;rupture certainly takes place when such an ex-tension has been reached that the structure isunstable. The stability condition must also be asu%cient condition, and it would seem to be closeto a necessary condition. A suggestion of thischaracter has been made by Born on pages 768—769 of his Handbuck article, but this is the onlysuggestion I have seen in print that the conven-tional criterion of rupture may not be correct.Conditions of stability are notoriously difficult toformulate, and in fact the ordinary theory of

ionic crystals does not yet deal satisfactorily withthis fundamental point, but can show only thatthe lattice is stable for certain restricted types ofdisplacement. It would probably therefore notbe easy to work out exactly the criterion of rup-ture demanded by the stability condition, but itis easy to see qualitatively, I think, that thecritical deformation would be less than that de-manded by the maximum force criterion, for thislatter is also a stability condition for a distortionuniform in every direction, which is itself a veryunstable kind of distortion. The stability condi-tion for uniform distortion doubtless sets only anupper limit. In view of the complexity of a gen-eral condition of stability, I do not believe thata rigorous application of the condition wouldresult in any such simple condition of rupture asis often assumed in practise, such as a maximumextension, or maximum tensile stress, or maxi-mum shearing strain criterion. This expectationalso agrees with experiment, for there are typesof rupture characteristic of high pressure whichshow that none of these criteria can in general bevalid; further details will be found in The Physicsof High Pressure.

XVI. SPECULATIONS

Finally we may indulge in a few perfectly frankspeculations as to what sorts of effects may beexpected at pressures very much higher thanthose yet reached in the l.aboratory. There is nonatural upper limit to pressure, nor is there anylimit to the amount of energy which can be im-parted to a substance by compressing it; in thestars there are perfectly stupendous pressures ofthe order of billions of atmospheres, and we knowthat sometimes under such conditions matter isconsolidated to densities of the order of 100,000—the field thus offered for speculation is a fasci-nating one.

A remark must first be made by way of correc-tion. A number of years ago I published specu-lations on this topic in two papers. '4 Some aspectsof the argument of the first paper were based toa certain extent on the experimental values of thecompressibilities of the alkali metals, and in par-ticular on the abnormal persistence of the com-pressibility of potassium at high pressures. I haverecently found that there was a serious error in

32 P. N. BRI DGMAN

the reduction of linear to cubic compressibility;the abnormalities now disappear, and part of theargument of the first paper is thereby vitiated.The corrected details will be given in a forth-coming paper, in which the pressure range isextended from 12,000 to 20,000. Furthermore,the principal point of those two papers was thatat high pressures the solid may be expected tobreak down into a gas of electrons and nuclei; theargument was made both from thermodynamicsand from Schottky's theorem. However, no ac-count could be taken at that time of the exclusionprinciple of Pauli, and it now appears that sucha breakdown is probably inconsistent with theexclusion principle, for it is difficult to see howan increase of pressure would increase the num-ber of energy levels in the neighborhood of -', «7.

sufficiently to allow all the electrons to find placeswith this mean energy. It is therefore probablethat the idea of a decomposition into a perfectelectron gas at high pressures must be aban-doned, although it is perhaps still legitimate toask whether the quantum relations must neces-sarily hold under such extreme conditions.

Intimations as to possible behavior at veryhigh pressure have appeared incidentally fromtime to time in the course of this paper. Thusthere has been the suggestion that the smallerunits in the structure may come to play moreimportant parts at high pressure; the compressi-bility of isomers and the variation of the specificheat with pressure suggests that in liquids at highpressures the atom may come to play part of therole of the molecule at lower pressures, and inmetals the behavior of thermal expansion at highpressures and the consequent dilemma with re-spect to entropy at infinite pressure suggests thatthe electron may be assuming some of the roleof the atom. In semiconductors the very largeincrease of electrical conductivity under pressuremeans essentially a freeing of the electrons, whichis the same sort of thing. On the other hand, theprobable ultimate increase of resistance of allmetals means a closer binding of the electrons.

The irreversible change from yellow to blackphosphorus raises the question of whether theremay not be many other such changes possible ifthe pressure is only raised high enough. Couldheavy matter of density 100,000 continue to existat atmospheric pressure if it had once been forced

into existence by stupendously high pressure?One property of such heavy matter is indicatedby the uncertainty principle, which suggests thatsuch matter cannot exist in the condition of aregular space lattice. The reason is very muchthe same as the reason why hydrogen cannotexist as a lattice of the NaC1 type composed ofelectrons and protons. The mass of the electronis so small that the uncertainty principle does notallow to it the definiteness of location that wouldbe demanded by a lattice of the density of solidhydrogen. As mass increases the possible definite-ness of location increases, so that ordinary atomscan assume positions in space lattices of the ordi-nary density, but if the density is very high, themass of the ordinary atom may impose restric-tions. For example, consider the case of sodium,and suppose it exists in a simple cubic lattice ofdensity 100,000. The distance between atomiccenters would be 7.3X10 " cm. Assumingiforthe energy of temperature agitation the classicalvalue, mv'/2=(3/2)ar the uncertainty in posi-tion demanded by the uncertainty principle isdl=h/(3amr)'*=3. 0X10 ', at ordinary tempera-tures, and is thus four times greater than thelattice spacing. If instead of assuming classicalenergy one assumes that, because of the enor-mous restoring forces and high characteristictemperature at these high densities, the atomsare completely in the condition of a Fermi de-generate gas with an upper kinetic energy limitgiven by the Sommerfeld expression,

it will be found that the uncertainty in positiondemanded by an uncoordinated motion of thisamount is 2.1 times the lattice spacing. It istherefore meaningless to attempt to describe theatoms as situated on a lattice, and the state ofaggregation must be a more or less amorphousjelly. Matter at such extreme densities must beelectively in a new state, as different fromordinary matter, as for example, an ordinary gasis different from the "fourth state of matter" ofCrookes. One may suspect "emergent" proper-ties.

The value for hl given by the uncertainty prin-ciple is seen to become smaller at higher tem-

HIGH PRESSURE PHF NOMENA 33

peratures. The pressure state of matter is, then,characteristic of low temperatures. There is there-fore still some plausibility in one feature of thesuggestion in my previous paper, namely that theextended pressure-temperature plane is crossedby a diagonal band, rising from low tempera-tures and low pressures to high temperatures andhigh pressures. Within this band matter exists inthe state ordinarily known; on the high tempera-ture side of the band it dissociates to a gas ofelectrons and nuclei according to ideas first em-phasized by Saha, and on the high pressure side

we now see that it collapses to a "pressure-squash. " Whether quantum principles in theirpresent form apply to this "pressure-squash"can be told better when it is found to what extentthey apply to the interior of the nucleus. Fur-thermore, the possibility must be recognized thatthe pressure-squash is composed of neutrons,electrons and protons being forced to form closelycoupled pairs by the extreme pressure. Such asystem would be an electrical insulator, and inthis respect would fulfil the tendency found inmetals in the experimental range.

XVII. BIBLIoGRAPHY

1. W. Schottky, Phys. Zeits. 21, 232 (1920).2. M. Born, W. Heisenberg and P. Jordan, Zeits. f.

Physik 35, 557 (1925—26); J. C. Slater, Phys. Rev.44, 953 (1933).

3. See especially the article by Born and Goppert Mayerin the second edition of Vol. XXIV, 2, of theHandbuch der Physi&.

4. O. K. Rice, J. Chem. Phys. 1, 649 (1933).4'. G. N. Lewis, Zeits. f. physik. Chemic, Cohen Fest-

band, 130, 532-538, 1927.5. T. W. Richards, J. Frank. Inst. , July, 1924, 1—27.

A complete bibliography of Richards' other paperson this subject will be found here.

6. P. W. Bridgman, Phys. Rev. 46, 960 (1934).7. F. A. Lindemann, Phys. Zeits. 11, 609 (1910).8. P. W. Bridgman, Phys. Rev. 45, 844 (1934).9. F. Hund, Zeits. f. Physik 34, 833 (1925).

10. M. Born and J.E. Mayer, Zeits. f. Physik 'l5, 1 (1932).

11. P. W. Bridgman, Proc. Am. Acad. 67, 363 (1932).12. F. Simon, Ann. d. Physik 68, 241 (1922).13. W. H. Keesom and K. Clusius, Comm. Leiden, No.

216, 1931.14. P. Ehrenfest, Proc. Roy, Acad. Amst. 36, 153 (1933).15. P. W. Bridgman, Phys. Rev. 38, 182 (1931).16. P. W. Bridgman, Proc. Am. Acad. 68, 27 (1933).17. W. L. Bragg and E. J. Williams, Proc. Roy. Soc. 145,

699 (1934).18. L. Pauling, Phys. Rev. 36, 430 (1930).19. W. Kroll, Zeits. f. Physik 85, 398 (1933).20. P. W. Bridgman, Proc. Am. Acad. 64, 39 (1929).21. W. V. Houston, Zeits. f. Physik 48, 449 (1928).22: H. Jeffreys, Proc. Camb. Phil. Soc. 24, 19 (1928).23. E. N. daC. Andrade, Phil. Mag. 1/, 495, 698 (1934).24. P. W. Bridgman, Phys. Rev. 27, 68 (1926); 29, 188

(1927).

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Theoretically Interesting Aspects of High Pressure Phenomena