Some Theoretically Interesting Phenomena Observed at High Pressures *

P. Bridgman

Contents:

I. Introduction. II. Intra-atomic changes occurring under pressure. III. Volume changes and the “force law.” IV. Thermal expansion and entropy at infinite pressure. V. \(p, v, T\) relations in liquids: 1) compressibility, 2) compressibility and chemical composition, 3) thermal expansion, 4) pressure coefficient and mechanism of pressure, 5) small deviations from the law of regularity. VI. Periodic relations. VII. Compressibility of single crystals. VIII. Equilibrium of two phases: 1) melting curve, 2) polymorphic transitions in solids. IX. Irreversible transitions. X. Discontinuities—transitions of the second kind, etc. XI. Electrical resistance. XII. Thermoelectric phenomena. XIII. Thermal conductivity. XIV. Viscosity of liquids. XV. Conditions of rupture. XVI. Conclusion. XVII. Bibliography.

I. Introduction

Until very recently the condensed phases of matter—“solid and liquid”—seemed so complex that it appeared not even worth the effort to study them, especially since the much more easily investigated region of “rarefied” matter, for example in vacuum tubes, was still far from being completely studied.

But now our understanding of atomic—as distinct from nuclear—processes occurring in the rarefied state is becoming more and more complete; the number of unstudied bodies is steadily decreasing, and consequently the study of denser states is coming to the fore. And indeed, we can already take satisfaction in the considerable successes of theory in this field, for example in the rapidly developing theory of metals in general and of the electrical properties of solid conductors in particular.

The condensed state, par excellence, is precisely what is given to us by matter under high pressure, so that our ideas about the condensed state cannot be regarded as at all satisfactory until we have

* Reviews of Modern Physics, 7, 1, 1935. Translated from the English by I. Nelidov.

will we be able to clarify for ourselves the influence of pressure on each individual physical phenomenon, which at present is possible only for a very small number of cases.

There may be two points of view on phenomena occurring at high pressures: a broader one and a narrower one. From the more general point of view, the most essential task is to develop our theoretical explanations of all high-pressure phenomena and to be able to predict new phenomena at pressures not yet attained.

It seems to me that this task should not be regarded as being of no particular interest, nor should one think that this problem is reduced simply to overcoming computational difficulties, while our basic concepts are completely clear to us. Such, undoubtedly, is the point of view of many theoretical physicists. Indeed, apparently, the basis of theoretical physics in its consideration of matter as a whole is the assertion that there are no special “emergent” problems, or, in other words, that all the properties of atomic aggregates can be obtained from our complete knowledge of the properties of isolated atoms. In a certain sense this may lead to a tautology, since I think that no one would assert that it is impossible, by introducing a sufficient number of parameters into the equations of individual atoms or electrons, to reproduce, with the help of a more or less complex theory, all the properties of the combination of atoms as a whole.

But such a general view is justified if one assumes that, by experimenting on individual atoms or on atoms in rarefied conditions, it will be possible to determine all the necessary conditions for describing more condensed assemblages of atoms. Undoubtedly, many will think that in experiments connected with nuclear bombardment and the disintegration of atoms, we are dealing with individual atoms under the action of forces considerably greater than those we encounter in aggregates of atoms, and that therefore one cannot think that such experiments will fail to provide all the necessary parameters. But, on the other hand, it should be remembered that in strongly condensed phases the character of the forces acting on an atom differs greatly from those forces that occur in collision experiments, since the atom is under the action of more or less symmetrical forces acting continuously from all sides; therefore the possibility of effects of a new kind still remains.

As far as I know, no theoretical assumptions about the possibility of the existence of matter at densities of the order of 100,000—before this was experimentally shown by astronomy—were made by theory.

Our present difficulties in understanding the phenomenon of superconductivity of metals must also be taken into account. Thus it nevertheless seems to me that, even from the point of view of pure logic, this possibility of the appearance of “special” problems cannot in any case be rejected so long as the possibility

P. BRIDGMAN

theoretical study of all the properties of matter in the condensed state has not been fully proved. The question of the interest or usefulness of such thorough investigations is, to a considerable degree, a matter of taste and inclination.

From a narrower point of view, everything comes down to the limits within which we can still interpret the phenomena observed at high pressures from the standpoint of the latest ideas of the wave mechanics of atoms and atomic interactions.

It is precisely from this point of view that we shall consider the phenomena in this review. Of course, it would be very good if this article could be written by someone who has worked on the applications of the apparatus of wave mechanics to the behavior of condensed states; the author, however, is acquainted mainly with the experimental material. Therefore I shall have to confine myself merely to indicating the phenomena that most readily lend themselves to explanation and that probably constitute the most convenient field for theoretical investigation, as well as those phenomena which seem to me the most interesting and significant. It seems to me that our theoretical conceptions in this field are still far from perfect, and that a careful discussion of the qualitative significance of various general types of phenomena at high pressures will therefore be fruitful. This seems so to me because several years ago I arrived at various qualitative pictures, for example for the phenomena of electrical resistance or polymorphic transitions—pictures which at that time differed from the generally accepted ones, but which are now being more and more justified by wave mechanics.

The experimental material on which the whole edifice of our theoretical views is built relates almost entirely to the pressure interval between 10,000 and 20,000 kg/cm², although some measurements have also been carried out at higher pressures, and we have the possibility of moving toward increasing pressures in the future. These pressures are already of the same order of magnitude as the internal pressures which the most diverse theories ascribe to condensed matter. Internal pressures vary from 3,000 or 4,000 kg/cm² for organic liquids and 10,000 or 20,000 kg/cm² for the most compressible metals, and reach several hundred thousand for the least compressible substances such as iridium and diamond.

Moreover, the volume changes obtained in such experiments are noticeably greater than those changes which are obtained on cooling from room temperatures to 0° K. Therefore it will be quite legitimate to assert that an understanding even of those phenomena which we are able to reproduce now can already give us much for our general knowledge of matter in the condensed state. By using pressure, we use an instrument for the artificial production of a very large number of new dense states of matter.

In the following exposition I shall try to give only the briefest description of the experimental material and refer the read-

reader, for a more detailed acquaintance, to my book Physics of High Pressures.

II. Intra-atomic transformations occurring under pressure.

Perhaps the very first and most important question encountered in this field is the question of whether atoms may be regarded as unchanging systems, or whether the pressures attained already produce noticeable changes in the atoms themselves. An approximate answer is given by one very important theorem, unfortunately often neglected; this theorem was first stated by Schottky[^1] and proved by him on the basis of classical mechanics. Subsequently Born, Heisenberg, and Jordan showed that it is essentially valid also in wave mechanics. Recently Slater[^2] has used this same theorem for certain molecular problems.

Schottky’s theorem asserts that in any system acted upon by internal electromagnetic forces—irrespective of whether or not there are, in addition, quantum restrictions (as, for example, in systems consisting of molecules and atoms, which are nuclei surrounded by electron shells)—if, in addition to the electromagnetic forces, an external hydrostatic pressure \(p\) acts, the following relations must hold:

\[ dT=-dE+3d(pv), \tag{1} \]

\[ dV=2dE-3d(pv), \tag{2} \]

where \(T\) is the mean internal kinetic energy, \(V\) is the mean internal potential energy (the mean values are taken over a sufficiently large time interval so that constant values are obtained), \(E\) is the total energy of the system; \(v\) is the volume.

Thermodynamics gives

\[ dE=\left[C_p-p\left(\frac{\partial v}{\partial \tau}\right)_p\right]d\tau-\left[\tau\left(\frac{\partial v}{\partial \tau}\right)_p+p\left(\frac{\partial v}{\partial p}\right)_\tau\right]dp; \]

eliminating \(dE\), we have

\[ dT=\left[4p\left(\frac{\partial v}{\partial \tau}\right)_p-C_p\right]d\tau+\left[3v+\tau\left(\frac{\partial v}{\partial \tau}\right)_p+4p\left(\frac{\partial v}{\partial p}\right)_\tau\right]dp, \tag{3} \]

\[ dV=\left[2C_p-5p\left(\frac{\partial v}{\partial \tau}\right)_p\right]d\tau-\left[3v+2\tau\left(\frac{\partial v}{\partial \tau}\right)_p+5p\left(\frac{\partial v}{\partial p}\right)_\tau\right]dp. \tag{4} \]

Let us apply these relations to ordinary solid bodies in the range of pressures attainable experimentally. Let us first consider the changes of \(T\) with pressure at constant temperature. Paying attention to orders of magnitude, we see that \(\tau\left(\frac{\partial v}{\partial \tau}\right)_p\) is obviously small in comparison with \(3v\), and that this term may be neglected. The most compressible solid metal is caesium: at \(15\,000\ \mathrm{kg/cm^2}\) we have \(4p\left(\frac{\partial v}{\partial p}\right)_\tau\) of order \(0.6\); \(3v\) of order \(2\), so that \(\left(\frac{\partial T}{\partial p}\right)_\tau=1.4\). On the oth-

on the other hand, lithium is considerably less compressible, and at \(15\,000\ \mathrm{kg}/\mathrm{cm}^2\)

\[ \left(\frac{\partial T}{\partial p}\right)_\tau = 2.4 . \]

In both cases the derivative refers to such an amount of substance which at atmospheric pressure occupies a volume equal to \(1\ \mathrm{cm}^3\). Most solids are compressed still less; therefore, without risking an error greater than \(10\%—15\%\), we may take \(\dfrac{\partial T}{\partial p} \simeq 3\) for all experimentally attainable pressures.

Hence, for a pressure of \(20\,000\ \mathrm{kg}/\mathrm{cm}^2\), we at once obtain

\[ T_{20000} - T_0 \simeq 60\,000\ \mathrm{kg}/\mathrm{cm}^2 \simeq 3.5 \cdot 10^{22}\ \mathrm{eV}. \]

Recalculating per atom, we have

\[ T_{20000} - T_0 \simeq 0.06 \times \text{atomic volume (in electron-volts per atom)} \tag{5} \]

This approximation is usually better justified for elements with small atomic volume than for elements with large atomic volume, since the compressibility is greater for larger atomic volumes; the actual value is smaller than the approximate one, since a term proportional to the compressibility enters. For ordinary temperatures the heat capacity of a solid approximately satisfies the Dulong and Petit law, according to which a solid should be assigned three degrees of freedom of kinetic energy (translational motions of the atom as a whole) and three degrees of freedom of potential energy of position. Consequently, with the same degree of accuracy, the kinetic energy of the translational motion of the atoms is constant and does not depend on pressure at constant temperature.

The change in kinetic energy with pressure given by equation (5) is the change in the internal kinetic energy of the electrons within the atom. For lithium at \(20\,000\ \mathrm{kg}/\mathrm{cm}^2\) it is equal to \(0.8\ \mathrm{eV}\) per atom. For bismuth it is \(1.3\ \mathrm{eV}\), for aluminum \(0.6\), and for iron \(0.4\ \mathrm{eV}\). Thus these changes in energy are considerably smaller than the ionization energy of the atom, but they are nevertheless of the same order of magnitude; for example, for bismuth this change amounts to \(18\%\) of the ionization energy. The magnitudes of these energy changes compel one to think that external pressure can cause appreciable changes in the peripheral electron orbits in atoms.

It should be noted that this increase in the internal kinetic energy of the electrons in the orbits means a contraction of the orbit (i.e. compression of the atom), if it is assumed that under pressure the same relation will hold between the radius of the atom and the energy of the electron as for an isolated atom.

The contraction of the orbit, first proposed by Richards, is precisely the theoretical description of the “compressible atom” to which I have repeatedly appealed in qualitative explanations of my measurements of compressibility.

The order of magnitude of the change in internal kinetic energy just calculated by us shows that, at those pressures,

which can already be realized in practice, one should not expect any sharp changes in the structure of the atom, except, of course, in a few rare cases when the atom is already close to some critical configuration. Sharp changes, apparently, may be expected at pressures of \(100\,000\ \mathrm{kg/cm^2}\). At the end of the article we shall consider what the nature of these phenomena should be. For the time being we shall consider those pressures which have already been attained and which cause small, but nevertheless noticeable, changes in the atom itself.

III. VOLUME CHANGES AND THE “FORCE LAW.”

Undoubtedly, the simplest of the phenomena caused by hydrostatic pressure is the homogeneous change in the volume of a liquid or of an isotropic solid; it is natural that theory should have begun its investigations precisely with this problem. From the theoretical point of view, the simplest condensed form is an ionic lattice of the NaCl type. It is known that the theory, developed mainly by Born \(^{3}\), has to some extent solved this problem. The ionic lattice is stable owing to the electrostatic attraction of the ions and does not contract owing to the repulsive forces arising from the mutual penetration of the ions. The force of attraction can be calculated from knowledge of the structure of the lattice and the known values of the ionic charges. The repulsive force is more complicated. In Born’s own works attempts were made to elucidate the nature of this force on the basis of Bohr’s model of the atom, but these attempts proved unsuccessful, since they gave stability only for certain relative orientations of atoms. Ultimately it became necessary to treat the repulsive force almost purely empirically and to introduce it in the form of a function of the distance between atomic centers in an unknown negative power with an unknown coefficient of proportionality.

Two conditions, the first—the lattice dimensions at \(0^\circ\mathrm{K}\)—and the other—the compressibility—made it possible to determine the two parameters of the empirical law of repulsion. For most halide compounds of the alkali metals, the repulsive force turned out to decrease inversely proportional to approximately the 9th power of the interatomic distance.

Having thus introduced a complete definition of the force and determined its parameters, we obtained the possibility of going further and seeking the dependence of compressibility on pressure. It was calculated that with increasing pressure the compressibility decreases, and this agreed with experiment, but the numerical agreement was so unsatisfactory that the accepted dependence of the repulsive force on distance could evidently be admitted only over a very narrow interval of pressures. The wave-mechanical picture of the atom in the form of interpenetrating electron shells gives more satisfactory results than Bohr’s theory; therefore Born, in his new theory, adopted for the repulsive force an exponential

dependence. This dependence is obtained in the simplest way in a mathematical calculation of the atom by the methods of wave mechanics. However, even this innovation does not give the correct change of compressibility with pressure, so that again we have only a certain approximation, justified within a comparatively narrow interval.

The compressibilities of all halide compounds of the alkali metals, with the exception of RbF, which crystallizes differently, were determined experimentally up to a pressure of \(12\,000\ \mathrm{kg/cm^2}\). In this region the relation between pressure and volume can be represented as a quadratic dependence on pressure with two parameters that determine the initial compressibility and the dependence of compressibility on pressure. For these substances, theory has only to calculate the values of these parameters. However, there exists a large number of substances for which the volume changes with pressure definitely do not fit not only a quadratic dependence on pressure, but even dependences of the 3rd or 4th degree. Among them are, for example, the alkali metals. There are also substances for which the compressibility increases with increasing pressure instead of decreasing. For such substances, theory will evidently have to introduce a number of parameters or seek another function that corresponds to the conditions better than a power series.

In addition to the work of Born and others on the compressibility of ionic lattices, several calculations have recently appeared of the compressibility of the simplest metals, in particular the alkali metals. Here again it has been possible to determine the initial compressibility fairly correctly, but the data on the change of compressibility with pressure are very far from satisfactory.

One feature of Born’s method of calculation determines its applicability to practically all derivations of the equation of state, regardless of whether it is intended for high pressures and the condensed phase or not; this is the assumption of a “law of force” between atoms as a function only of the distance between the centers of the atoms. In view of the failure of all attempts to give more than the first derivative of volume with respect to pressure, it is natural to ask within what limits the interaction between atoms in condensed states of matter, when temperature and pressure change, can be represented by a “law of force.”

Assuming a “law of force,” for example, in the form

\[ \frac{a}{r^{2}}+\frac{b}{r^{n}}, \]

we obviously imply, as a postulate, that the behavior of the entire collection of atoms can be determined if each atom in any pair of atoms acts on the other with a specified force, at any distances and in any directions, independently of the presence of other atoms (for the given type of ions, orientation is of no significance, since according to wave mechanics we have here a case of spherical symmetry). The adoption of a law of force in the form given above corresponds to the facts in a first, but very important, approximation.

Two atoms, interacting in the indicated manner under the action of sufficiently large forces, can be brought arbitrarily close to one another. This means that the atoms are not rigid, but are substantially deformable, which agrees with many experimental data. But the question is how far the approximation given by such a law will remain valid.

One can imagine the solution of the wave-mechanical problem of a NaCl crystal, for example, in the form of finding the complete \(\psi\) function, then separating it into parts corresponding to the \(p\)-, \(s\)-, \(d\)-electrons associated with the various atoms, and, finally, obtaining from this the force law. This force law will undoubtedly include the distribution of electrons within the atom. The force law will be valid as long as the electron distribution remains unchanged or as long as the electron distribution depends exclusively on the distance \(r\), as, for example, under changing pressure in the case of constant temperature. If, however, the electron distribution is not determined by \(r\) alone, as occurs if both temperature and pressure change, then we must be prepared for the fact that the interaction cannot be described by means only of constants and \(r\), and this essentially means that there is no force law.

Of course, in a sufficiently narrow interval the assumption of a force law is a legitimate approximation. We must only consider whether this approximation will hold at all those pressures and temperatures which can already now be realized experimentally. This question can be answered qualitatively by using Schottky’s theorem.

In equations (3) and (4) one may approximately assume that the total kinetic energy \(T\) consists of two parts: 1) the energy of motion of the mass of the atom as a whole (\(T_{\mathrm{at}}\)) and 2) the kinetic energy of the electrons inside the atom (\(T_{\mathrm{el}}\)); then \(T = T_{\mathrm{at}} + T_{\mathrm{el}}\). Similarly, one may take \(V = V_{\mathrm{at}} + V_{\mathrm{el}}\), where \(V_{\mathrm{at}}\) is the part of the potential energy determined by the “force law” in accordance with the considerations set forth above, while \(V_{\mathrm{el}}\) is the internal potential energy of the electrons composing the atom. Such a decomposition of the energy into two parts gives a good approximation for the case of an ionic lattice; for more complex molecular lattices the decomposition is not so obvious. Moreover, we assume that the substance under investigation satisfies, at least approximately, the Dulong and Petit law, i.e.

\[ \left(\frac{\partial T_{\mathrm{at}}}{\partial p}\right)_{\tau}=0 \quad \text{and} \quad \left(\frac{\partial T_{\mathrm{at}}}{\partial \tau}\right)_{p}=\frac{C_{p}}{2}, \]

neglecting the difference between \(C_{p}\) and \(C_{v}\), which is quite legitimate for condensed states. It is very probable that at high pressures the heat capacity becomes somewhat smaller owing to the strengthening of atomic bonds, so that it would be more accurate to write

\[ \left(\frac{\partial T_{\mathrm{at}}}{\partial p}\right) < 0 . \]

This reduces to the fact that in what follows

in the presentation we shall be more likely to underestimate than to overestimate the role of pressure on \(T_{\mathrm{el}}\). Substituting the values obtained for \(\left(\dfrac{\partial T}{\partial p}\right)_{\tau}\) and \(\left(\dfrac{\partial T}{\partial \tau}\right)_{p}\) into (3) and (4), we obtain

\[ \left(\frac{\partial T_{\mathrm{el}}}{\partial v}\right)_{p} = 4p-\frac{3}{2}\, \frac{C_{p}}{\left(\dfrac{\partial v}{\partial \tau}\right)_{p}}, \tag{6} \]

\[ \left(\frac{\partial T_{\mathrm{el}}}{\partial v}\right)_{\tau} = 4p+ \left[ 3v+\tau\left(\frac{\partial v}{\partial \tau}\right)_{p} \right] \frac{1}{\left(\dfrac{\partial v}{\partial p}\right)_{\tau}} . \tag{7} \]

Let us compare these two derivatives with one another at low pressures, i.e., let us compare

\[ -\frac{3}{2}\, \frac{C_{p}}{\left(\dfrac{\partial v}{\partial \tau}\right)_{p}} \quad \text{and} \quad \frac{\left[ 3v+\tau\left(\dfrac{\partial v}{\partial \tau}\right)_{p} \right]} {\left(\dfrac{\partial v}{\partial p}\right)_{\tau}} . \]

For NaCl we have the following numerical values: \(C_{p}=0.219\ \text{g cal/degree}\); \(\left(\dfrac{\partial v}{\partial \tau}\right)_{p}=0.00012\) and \(\left(\dfrac{\partial v}{\partial p}\right)_{\tau}=42\cdot 10^{-7}\ \text{kg/cm}^{2}\). Reducing everything to \(\text{kg/cm}^{2}\), we obtain values of the derivatives equal respectively to \(2.5\cdot 10^{5}\) and \(7\cdot 10^{5}\). This means that the change in the energy of the electrons inside the atoms, caused by a definite change in volume under changing pressure, is almost three times greater than the change which we obtain if the same change in volume is produced by a change in temperature. The internal electronic energy may be regarded as an approximate measure of the internal state of the atom. The force with which an atom acts on a neighboring atom depends on this internal state. Hence it follows directly that the force of atomic interaction changes differently for a given change in the mean distance between centers, depending on whether this change has occurred owing to a change in temperature or to a change in pressure. Consequently, the force law can be used for establishing the equation of state or in the expression of the virial only approximately. To determine exactly how justified such an approximation is constitutes one of the first tasks of any serious attempt to derive an equation of state valid also at high pressures. The numerical values obtained for NaCl show that this approximation satisfies changes of pressure worse than changes of temperature. Moreover, the formulas show that with increasing pressure the approximation becomes worse and worse, owing to the presence of the term \(4p\), which has the opposite sign relative to the second term. However, even at a pressure of \(20\,000\ \text{kg/cm}^{2}\), the term \(4p\) represents only 33% of

\[ \frac{3}{2}\, \frac{C_{p}}{\left(\dfrac{\partial v}{\partial \tau}\right)_{p}}, \]

if one assumes that

\[ \frac{C_{p}}{\left(\dfrac{\partial v}{\partial \tau}\right)_{p}}=\mathrm{const}. \]

But

\(\left(\dfrac{\partial v}{\partial p}\right)_{\tau}\) and \(\left(\dfrac{\partial v}{\partial \tau}\right)_{p}\) vary strongly with pressure; these changes in the other terms produce noticeable effects at high pressures. Experiment shows that usually \(\left(\dfrac{\partial v}{\partial p}\right)_{\tau}\) decreases with increasing temperature much more rapidly than \(\left(\dfrac{\partial v}{\partial \tau}\right)_{p}\), and therefore from this point of view the adoption of the “force law” for high pressures is considerably less justified than for low ones.

The important role played by the “drawing together” of atoms under certain conditions is clearly shown by the following argument. Let us consider the change in the total energy caused by a change in pressure at constant temperature:

\[ dE_{\tau}=-\left[\tau\left(\frac{\partial v}{\partial \tau}\right)_{p}+p\left(\frac{\partial v}{\partial p}\right)_{\tau}\right]dp. \]

Both terms in this expression have opposite signs. \(p\left(\dfrac{\partial v}{\partial p}\right)_{\tau}\) has the initial value zero, so that at low pressures the sign of \(dE\) is determined by the term \(\left(\dfrac{\partial v}{\partial \tau}\right)_{p}\). This means that the internal energy of all ordinary substances, upon isothermal application of pressure, at first decreases, i.e. that more energy leaves in the form of heat (counteracting the rise in temperature caused by compression) than enters from the external pressure in the form of mechanical work. At high pressures, however, as experiment shows, the term \(p\left(\dfrac{\partial v}{\partial p}\right)_{\tau}\) predominates, so that at sufficiently high pressures the energy increases with increasing pressure. Obviously, this increase of energy occurs entirely at the expense of work done against the repulsive forces between atoms. The volume at which \(dE\) vanishes is that volume at which the forces of attraction and repulsion balance one another.

At \(0^\circ\mathrm{K}\) the lattice is in equilibrium, and all its points are at rest (we neglect the zero-point energy). Consequently, the volume at \(0^\circ\mathrm{K}\) and atmospheric pressure is also the volume at which the forces balance.

This is indeed so, since \(\left(\dfrac{\partial v}{\partial \tau}\right)_{p}=0\) at \(0^\circ\mathrm{K}\), and \(dE=-p\left(\dfrac{\partial v}{\partial p}\right)_{\tau}dp\) for the isotherm \(0^\circ\). Hence it follows that \(E\) is minimal for \(0^\circ\mathrm{K}\) at zero pressure and increases with increasing pressure owing to the action of the repulsive forces. If we assume that the mutual potential energy is a function of the volume alone and does not depend on the temperature or on the amplitude of atomic vibrations—which, of course, is an approximation—and if, moreover, we assume that the kinetic energy of translational motion is a function only of the temperature, then we must expect that at high temperatures \((dE)_{\tau}\) should vanish when the pressure has increased so much that the volume is reduced to the volume occupied ...

possessed by a body at \(0^\circ\mathrm{K}\) and atmospheric pressure. The pressure at which this can occur is equal to

\[ -\tau \frac{\left(\dfrac{\partial v}{\partial \tau}\right)_p}{\left(\dfrac{\partial v}{\partial p}\right)_\tau}. \]

It lies between 10,000 and 20,000 kg/cm\(^2\). The volume corresponding to this pressure approximately corresponds to the volume at \(0^\circ\mathrm{K}\) and atmospheric pressure, being, however, somewhat smaller.

On the other hand, for helium the relative changes \(\left(\dfrac{\partial v}{\partial \tau}\right)_p\) and \(\left(\dfrac{\partial v}{\partial p}\right)_\tau\) under pressure are different: \(\dfrac{\partial v}{\partial p}\) tends to zero considerably faster, in comparison with \(\dfrac{\partial v}{\partial \tau}\), than is the case for solids. Hence it follows that the pressure at which, for room temperatures, \(\dfrac{\partial E}{\partial p}=0\), has not yet been reached experimentally, i.e. it is greater than 15,000 kg/cm\(^2\). But at room temperature and a pressure of 15,000 the volume of helium is only one half of the volume at \(0^\circ\mathrm{K}\) and atmospheric pressure. In other words, the volume at which the attractive and repulsive forces balance at 15,000 kg/cm\(^2\) and room temperature is less than half the volume at which they balance at \(0^\circ\mathrm{K}\). Such a strong discrepancy, without any doubt, cannot be attributed to a departure of the forces from linearity; apparently, some change takes place in the atom itself, occurring as a result of redistribution of the electrons. Under such conditions the “law of force,” which uses only a single parameter, namely the distance between atomic centers, cannot be valid. It is interesting to note that the discrepancy observed for ordinary metals has the same sign as for helium, but is only quantitatively less pronounced. This means that at room temperatures the volume at which \(\dfrac{\partial E}{\partial p}\) vanishes is, in both cases, less than the volume at \(0^\circ\mathrm{K}\) and atmospheric pressure. In both cases the internal kinetic energy of the atom increases with increasing pressure, and this is accompanied by a compression of the effective dimensions of the atom, so that the sign of the effect is precisely the one that should have been expected.

The following rather rough calculation of the changes in the effective sizes of atoms under the action of pressure is of some interest. For low pressures we have approximately

\[ \frac{1}{v\left(\dfrac{\partial T_m}{\partial v}\right)_\tau} \approx \frac{3}{\left(\dfrac{\partial v}{\partial p}\right)_\tau}. \]

For NaCl we obtain a numerical value equal to \(1.4\cdot 10^{-11}\) erg per atom for each kg/cm\(^2\) of external pressure. Let us now make a simplifying assumption: take an atom with 14 electrons as an average for Na and Cl; assume further that all 14 electrons rotate around the nucleus in a common orbit of radius \(1.35\cdot 10^{-8}\) cm.

with such kinetic energy that the centrifugal force is balanced by the force of electrostatic attraction of the nucleus. Introducing all these simplifications, we can calculate the change in the radius length when the kinetic energy in the orbit is increased by \(1.4\cdot 10^{-11}\) erg. The change in the magnitude of the radius proves to be only one-hundredth of that change in the distances between atomic centers which is caused by a pressure of \(1\ \mathrm{kg}/\mathrm{cm}^2\), i.e., the change in the volume of the atom itself upon the application of an external pressure can hardly amount to more than a small part of the total change in volume.

We have applied Schottky’s theorem to a solid body, but it is also interesting to apply it to the case of a monatomic gas. The Dulong and Petit law is not applicable to gases, and therefore we have \(\left(\dfrac{\partial T_{\mathrm{at}}}{\partial p}\right)_\tau = 0\) and \(\left(\dfrac{\partial T_{\mathrm{at}}}{\partial \tau}\right)_p = C_p - R\) instead of the relations derived earlier. Substitution into relation (3) now gives

\[ \left(\frac{\partial T_{\mathrm{el}}}{\partial p}\right)_\tau = 3v + \tau\left(\frac{\partial v}{\partial \tau}\right)_p + 4p\left(\frac{\partial v}{\partial p}\right)_\tau, \]

as before, and

\[ \left(\frac{\partial T_{\mathrm{el}}}{\partial \tau}\right)_p = 4p\left(\frac{\partial v}{\partial \tau}\right)_p - 2C_p + R. \]

For an ideal monatomic gas

\[ \tau\left(\frac{\partial v}{\partial \tau}\right)_p = v;\quad p\left(\frac{\partial v}{\partial p}\right)_\tau = -v \]

and

\[ C_p = \frac{5R}{2}, \]

so that

\[ \left(\frac{\partial T_{\mathrm{el}}}{\partial p}\right)_\tau = 0 \]

and

\[ \left(\frac{\partial T_{\mathrm{el}}}{\partial \tau}\right)_p = 0, \]

i.e., there are no changes in the internal structure of the atom under changes either of temperature or of pressure. But no gas remains even approximately ideal at pressures greater than several hundred \(\mathrm{kg}/\mathrm{cm}^2\), and at pressures of several thousand \(\mathrm{kg}/\mathrm{cm}^2\) every distinction between a gas and an ordinary liquid disappears. In the region of such pressures, as is evident from a simple argument, the change in \(T_{\mathrm{el}}\) is a quantity of the same order as for substances in the condensed state; for example, for nitrogen the quantity \(pv\), from a value equal to unity at atmospheric pressure, reaches a value of 16.5 at a pressure equal to 15 thousand \(\mathrm{kg}/\mathrm{cm}^2\). Consequently, the internal distortions in the atoms of gases are the same as in the atoms of other substances.

It might have been thought that it would be easier to calculate the volume changes as a function of pressure for gaseous substances such as hydrogen or helium than for solids such as NaCl. However, we still do not have a theoretically derived equation of state for any gas at high pressures.

IV. Thermal expansion and entropy at infinite pressure

The greatest successes in attempts at a theoretical investigation of questions connected with the equation of state of solids have been achieved for compressibility at constant temperature. The success has not been so great in considering thermal expansion. Thermal expansion almost always decreases with increasing pressure; of course, this means the same thing as an increase of compressibility with increasing temperature. As a general rule, one may accept that the decrease of thermal expansion for a given increase of pressure is always considerably less than the decrease of compressibility. It has been known for a comparatively long time that thermal expansion is connected with the deviation from linearity of the action of attractive forces on atoms. Therefore, at first sight the decrease of thermal expansion with increasing pressure (with decreasing volume) seems paradoxical, all the more since the repulsive forces between atoms are always considered to increase strongly at small distances, so that the smaller the volume, the stronger one would expect the deviation from a linear course, i.e. the greater the expansion. This paradox is resolved to a certain extent if one considers the conditions, in large volumes, of an ideal gas. Here the attractive force acting on a molecule is equal to zero throughout the entire length of the free path, and then suddenly increases to a very large value in the collision that ends the free path. This is, of course, an extreme example with respect to deviations from linearity, so that here we should expect strong thermal expansion. Such a consideration (from the point of view of collisions) of the interaction between atoms should explain a large part of the thermal expansion of real liquids and solids, and the fact that these collisions become less significant at high pressures is undoubtedly responsible for the decrease of thermal expansion with increasing pressure. It is not yet quite clear what should be expected at exceptionally high pressures lying beyond the achievements of modern experiment. In that region where the “force law” is applicable, it may perhaps be expected that the rapid increase of repulsive forces will produce a counteraction, so that at very high pressures thermal expansion may begin to increase again. Clarifying this question is very important, but is connected with extraordinarily great experimental difficulties. I made many repeated attempts to obtain good data on thermal expansion for na-

THEORETICALLY INTERESTING PHENOMENA AT HIGH PRESSURES

more interesting substances, but without particular success. Quite recently some data were obtained on the alkali metals. They have been published in Proc. of the Nat. Acad. of Sc.

The consideration of thermal expansion provides the possibility of still another interesting approach to the study of the behavior of bodies at very high pressures. This approach is connected with the third law of thermodynamics. In certain respects, increasing the pressure is equivalent to lowering the temperature—identical volume changes and identical strengthening of the bonds acting on the atoms take place. The result will be an increase in the characteristic temperature and a displacement of the values of the heat capacity, thermal expansion, and similar properties to values corresponding to lower temperatures. At \(0^\circ\mathrm{K}\), according to the third law, the entropy in most cases vanishes, and this led Lewis to the idea that, if one moves along an isotherm toward an infinite increase of pressure, then the entropy will tend to zero at higher temperatures. Since

\[ \left(\frac{\partial S}{\partial p}\right)_\tau = -\left(\frac{\partial v}{\partial \tau}\right)_p, \]

then, integrating, we obtain

\[ S_{p,\tau}-S_{0,\tau} = -\int_{0}^{p} \left(\frac{\partial v}{\partial \tau}\right)_p\,dp. \]

If, in accordance with the third law, it is assumed that \(S_{0,0}=0\), then it is obvious that the thermal expansion must fall to zero under an infinite increase of pressure, since if \(\left(\frac{\partial v}{\partial \tau}\right)_p\) remained finite, then \(S_{p,\tau}\) would have to decrease without bound under an infinite increase of pressure. If \(\int \left(\frac{\partial v}{\partial \tau}\right)_p\,dp\) remains finite, then \(\frac{\partial v}{\partial \tau}\) must decrease with increasing pressure proportionally at least to \(\frac{1}{p}\). Experiment shows quite definitely that this is not so, at any rate for ordinary solids, metals* or crystals with an ionic lattice, nitrogen and argon. For all these substances the expansion in no case falls as rapidly as \(\frac{1}{p}\).

If the condition \(\lim_{p\to\infty} S=0\) exists\(^8\), then on the curve \(\left(\frac{\partial v}{\partial \tau}\right)\) there must be a point of inflection at pressures lying beyond the limits attainable by present-day experiment. However, it is necessary to weigh carefully whether the proposition that the entropy at infinite

* Experiments on the alkali metals, completed quite recently, have shown that for them there is a very strong decrease of thermal expansion at high pressures.

pressure disappears, is already so necessary. In the arguments leading to the third law, it is assumed that at \(0^\circ \mathrm{K}\) chaoticity completely disappears and the atom is regarded as some unchanging system, relative to which chaoticity is then computed. But at infinite pressure this is, of course, not so: in the atom there will probably occur all possible interactions of electrons and their interpenetration. If, however, an electron or some subgroup in the atomic system begins, at high pressures, to act individually, then there would seem to be no grounds for considering that the entropy cannot take negative values with respect to a system in which the atom acts as a whole. In other words, one may suppose that a system under very high pressure apparently belongs to the class of exceptions for which, as Fowler and Stern have shown, the third law is inapplicable.

V. \(p, v, T\)-Relations in Liquids

1. Compressibility. Up to now we have considered the influence of pressure on the volume of solids, and have also touched briefly on gases. Theory has succeeded in doing something in these directions, but it proves completely inadequate in attempts to approach ordinary liquids. All that I can do here is to present several qualitative observations concerning the behavior of liquids at high pressures. The experimental material here is fairly extensive, including measurements up to \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) at temperatures from 0 to \(100^\circ \mathrm{C}\) for more than 60 liquids, chiefly organic ones, and for water and mercury. Mercury stands somewhat apart—its compressibility is approximately \(1/10\) that of water and of the same order as that of solid metals, in particular 50% greater than the compressibility of lead, but considerably less than the compressibility of the alkali metals. It would be interesting to test the compressibility of other metals in the liquid state, although there is no special reason to expect anything extraordinary. All other liquids, with respect to their behavior under a pressure of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\), may be roughly divided into 3 groups: the first—glycerin—constitutes a special group with a total volume change of 14.5%. The second group includes water, \(\mathrm{C}_6\mathrm{H}_5\mathrm{Br}\), \(\mathrm{C}_6\mathrm{H}_5\mathrm{Cl}\), and a series of glycols; for this group the volume change amounts to from 18 to 20%. The third group includes all the remaining liquids, with volume changes reaching up to 30%. Such a division into three groups is, of course, very rough and probably quite arbitrary. Undoubtedly, liquids satisfying all intermediate values can be found.

The order of magnitude of compression is a less characteristic feature for a liquid than for a solid substance. It rather characterizes the chemical nature of the substance, since for most solid organic compounds as well the compression reaches 15% at \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\). The distinguishing feature of the compressibility of liquids as compared with solids, apparently, is a very strong

... decrease in compressibility with increasing pressure: the compressibility at \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) in many cases amounts to only \(1/15\) of the compressibility at atmospheric pressure.

Solid organic compounds do not exhibit this noticeable drop in compressibility at low pressures: the decrease in the compressibility of liquids with pressure is nonlinear; thus, for example, the relative change in compressibility over the first \(1000\ \mathrm{kg}/\mathrm{cm}^2\), for a number of substances, is of the same order as over the last \(6000\ \mathrm{kg}/\mathrm{cm}^2\). The qualitative explanation is that at low pressures there are many voids between the molecules in a liquid; increasing pressure rapidly destroys these voids, and therefore at first the compressibility is large. When the voids have been filled, the compressibility of the molecules themselves begins to appear, which is partly due to the densification of atoms within the molecule and partly to the compression of the atoms themselves, according to Schottky’s theorem. A significant part of the large initial compressibility of ordinary liquids is connected with the proximity of the liquid–gas critical point, since the compressibility in the vapor phase is large, and at the critical point itself the compressibility is infinite.

Most theories of the liquid state have paid too little attention to the constancy of the compressibility of molecules at high pressures. At their foundation these theories are more or less empirical, and most of them ascribe a finite limiting volume to a liquid under infinite pressure. This limiting volume, obtained by extrapolating experimental data for pressures of \(3\)—\(4\) thousand \(\mathrm{kg}/\mathrm{cm}^2\), has sometimes proved to be larger than the volume actually attained at a pressure of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\).

The theories usually give values of compressibility at high pressures that are too small. Moreover, a very important circumstance is that compressibility varies much less from liquid to liquid at high pressures than at low pressures: among the liquids measured one can name those for which, at atmospheric pressure, the compressibilities differ by a factor of 10, whereas at \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) the difference is only a factor of 1.8.

2. Compressibility and chemical composition. A number of rather rough considerations can be offered concerning the presumed relation between compressibility and chemical composition. Thus, for example, at high pressures the volumes of isomers noticeably tend toward equality, even if at low pressures they differ greatly. This means that at high pressures a tendency is observed toward the elimination of structural differences, and the volume approaches the sum of the volumes of the constituent atoms. Ether and \(n\)-butyl alcohol both have the same composition—\(C_4H_{10}O\)—but structurally they are so different that chemists often even forget that they are isomers. The ratio of their specific volumes at atmospheric pressure is 1.096, while at a pressure of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) this ratio is 1.037. On the other hand, if the isomers do not differ very strongly in structure, such as, for example, \(i\)-butyl alcohol and \(n\)-butyl alcohol, then this tendency toward the same volume at high pressures is not so...

clearly, and there may even be deviations in the opposite direction; \(12000\ \mathrm{kg/cm^2}\) is not a sufficient force to ensure complete control in the case of such a small difference in structure.

If a molecule is bound by very large forces, then its compressibility is small. Apparently, the OH group acts precisely in this way. The abnormally small compressibility of glycerin may be attributed to the presence of three OH groups in the molecule. The comparatively small compressibility of glycols and water may be attributed to the same reason. The addition of the bonds of the hydrocarbon group \(\mathrm{CH_2}\), in general, apparently favors strong compressibility; the presence of oxygen in the molecule decreases compressibility.

1. Thermal expansion. The thermal expansion of liquids is much easier to measure than the expansion of solids; we can measure both the change of expansion with pressure at constant temperature and the change of expansion with temperature at constant pressure.

The effect of increasing pressure at constant temperature, as in the case of solids, reduces to a decrease of thermal expansion, only in this case it manifests itself much more strongly. The change with temperature is of special interest. Usually, at atmospheric pressure, thermal expansion increases with increasing temperature, i.e.,
\[ \left(\frac{\partial^2 v}{\partial \tau^2}\right)_p > 0. \]
However, at pressures of about \(3\text{—}4\) thousand \(\mathrm{kg/cm^2}\) this effect changes sign, and for almost all organic liquids at pressures greater than \(4000\ \mathrm{kg/cm^2}\)
\[ \left(\frac{\partial^2 v}{\partial \tau^2}\right)_p < 0. \]
The following explanation seems to suggest itself: a deviation from linearity occurs in the law of attractive forces; at low temperatures and constant pressure the volume is small, and therefore the nonlinearity of the attractive forces, arising as a result of strong repulsion, is great; hence the thermal expansion is also large. I gave such an explanation in several of my papers. However, it is incorrect, since it should give an increase of expansion with increasing pressure at constant temperature, which does not agree with experiment. It now seems to me that the answer should be sought in changes in the sizes of the molecules themselves, as follows from Schottky’s theorem, already considered by us. We saw that, as the temperature rises, molecules increase in volume at constant pressure. At small volumes (large pressures) this may mean a considerable restriction of free molecular motion, i.e. an increase in linearity, and therefore a decrease in thermal expansion. A detailed investigation of what we should expect here will evidently require careful consideration of many factors. This is an important and necessary topic for theoretical investigation.

Since
\[ \left(\frac{\partial C_p}{\partial p}\right)_\tau = -\tau\left(\frac{\partial^2 v}{\partial \tau^2}\right)_p, \]
the fact that
\[ \frac{\partial^2 v}{\partial \tau^2} \]
becomes negative for pressures above \(3\text{—}4\) thousand \(\mathrm{kg/cm^2}\) means that \(C_p\) increases with pressure, beginning from this point. Approximately one may say that \(C_p\) decreases to \(0.9\) of its initial-

of its minimum value at 3–4 thousand kg/cm² and then begins to increase, but in such a way that by 12,000 kg/cm² it still does not reach its initial magnitude. This increase of \(C_p\) is not so easy to understand, since at first glance it seems that the effect of pressure reduces to an increase of bonds. By raising the characteristic temperature and shifting the effective temperature toward lower temperatures, the increased pressure decreases the value of the heat capacity. But it should be remembered that the change of sign of \(\dfrac{\partial^2 v}{\partial \tau^2}\) has been observed experimentally only in organic liquids consisting of complex molecules. Under ordinary conditions the heat capacity of such liquids, as is known, is considerably lower than the value given by the full number of internal degrees of freedom. If pressure tends to make the atom, rather than the molecule, the basic unit of structure, then an increase of the heat capacity should be expected. Since the tendency is suggested by the tendency of the isomers at high pressures toward equalization of the specific volumes. If these considerations are correct, then for metals and simple ionic lattices one should expect the opposite sign for \(\dfrac{\partial^2 v}{\partial \tau^2}\); unfortunately, up to the present time it has not yet been possible to measure the quantity \(\dfrac{\partial v}{\partial \tau}\) with sufficient accuracy to make it possible to calculate \(\dfrac{\partial^2 v}{\partial \tau^2}\).

4. Pressure coefficient and the mechanism of pressure

The ratio
\[ \frac{\left(\dfrac{\partial v}{\partial \tau}\right)_p}{\left(\dfrac{\partial v}{\partial p}\right)_\tau}, \]
mathematically identical to \(\left(\dfrac{\partial p}{\partial \tau}\right)_v\), has played a major role in various theories of liquids. It was called the pressure coefficient and, obviously, gives the change of pressure per degree of rise in temperature at constant volume. The assertion was made that the pressure coefficient depends on the volume, and on this basis equations of state were constructed. The physical meaning of this assertion is as follows: if one puts
\[ \left(\frac{\partial p}{\partial \tau}\right)_v=f(v), \]
then integration immediately gives \(p=\tau f(v)+\varphi(v)\), i.e. an equation of state. Here \(\varphi(v)\) is an arbitrary constant of integration. The equation of state of an ideal gas and the Van der Waals equation are special cases of this general form.

The physical meaning of such an equation consists in the fact that the pressure exerted by a given substance may be regarded as arising from two different causes, independent of one another; the part given by \(\varphi(v)\) is the same at all temperatures, and therefore also at \(0^\circ\text{K}\), and, obviously, arises from intermolecular forces, as, for example, in Born’s theory of ionic crystals. The part \(\tau f(v)\) is proportional to the temperature and arises from the kinetic bombardment of the walls. Elementary considerations of kinetic theory show that the kinetic pressure is precisely of this kind, if we assume that the sizes of the molecules do not depend on temperature and that

the kinetic energy of thermal motion is proportional to the temperature, as is assumed in classical statistics.

Comparison with experiment shows that in many cases \(\left(\dfrac{\partial p}{\partial \tau}\right)_v\) is not a function of the volume alone; the discrepancy obtained is many times greater than the experimental errors. Generally speaking, \(\dfrac{\partial p}{\partial \tau}\) shows a tendency to decrease with increasing temperature at constant volume. The general cause of this discrepancy is quite clear: if the molecules themselves are deformed with changes of pressure and temperature—and we have seen that they must be deformed—then the two mechanisms producing pressure do not act independently, but there is an interrelation between them. Moreover, there is always a dependence of \(\varphi(v)\) on higher-order terms when the amplitude of the vibrations increases. How important this dependence is in comparison with the second effect must be shown by a more detailed theoretical analysis.

The change in the internal energy of liquids under pressure has the same character as for solids, i.e., at first, at low pressures, it decreases, and then, at high pressures, changes sign. The physical meaning is undoubtedly the same: at low pressures and large volumes the chief role is played by attractive forces, while at high pressures and small volumes it is played by repulsive forces. The pressure at which the change of sign occurs is, for organic liquids, appreciably lower than for solids; it usually lies around \(7\,000\ \mathrm{kg}/\mathrm{cm}^2\). This is what may be expected on the basis of the most general considerations; pressure, generally speaking, changes the properties of a liquid more strongly than those of a solid.

1. Small deviations from the general regularities. Up to now we have considered substantial changes of properties, which are more or less the same in all organic liquids. But superposed on these changes is an innumerable number of less significant ones (fine structure), which vary from liquid to liquid. Examples of abnormal behavior of almost any type may be found: compressibility may increase with pressure or decrease with temperature; thermal expansion may increase with pressure and may either decrease or increase with increasing temperature. Such diversity must mean that individual features of various types of molecules begin to exert an influence; this is, of course, quite natural if one takes into account that high pressure brings molecules into closer contact and forces them to accommodate themselves to mutual irregularities. The picture of the molecule given by wave mechanics seems to make it possible to introduce precisely the complications that are required. Valences acquire spatial directions, which leads to considerably more complex forms than spherical ones; regions appear in the outer parts of atoms that attract one another owing to the pairing of electrons with opposite spins, which makes it possible for...

nuclei to local centers of attraction; i.e., in essence, here we have precisely those complications which were required by qualitative considerations even before the development of wave mechanics. The complications connected with a detailed working out of all the possibilities will, of course, be set aside at first; probably the theory will initially concern itself with a broad range of properties common to all liquids, which is of quite sufficient interest.

Considering the large number of individual properties of substances, one might perhaps think that an “equation of state” does not exist at all. Of course, it is hopeless even to try to obtain a single equation with variable parameters for describing all the possibilities of all kinds of molecules. The most that an equation of state can count on is to describe a general method, but not a general result.

VI. Periodic Relations

One of the principal tasks of the theory in developing the question of the compressibility of the elements will be to explain the astonishing periodicity of compressibility—a phenomenon first noted by Richards.^5 In Fig. 1 the logarithm of the volume compressibility to base 10, plus 7, is plotted for all investigated elements as a function of atomic number.

Fig. 1. On the ordinates are plotted the logarithms to base 10 of the compressibility plus 7; on the abscissae, atomic numbers.

We have every reason to believe that the maxima must be occupied by the noble gases in the condensed phase; this, however, has not yet been shown experimentally. This follows from general considerations about the nature of the forces holding noble gases together, i.e. van der Waals forces, which belong to the weakest of the known interatomic forces. Passing to the alkali metals, we must connect

their high compressibility by the fact that in their outer shell there is only one electron, which clearly favors easy deformation of the atom.

VII. Compressibility of Single Crystals

Theory must clearly explain the reason for the difference in compressibility along the various directions of single crystals of non-cubic substances. These differences are sometimes manifested very distinctly; for example, zinc is compressed 8 times more strongly along the axis than across it, while tellurium even has negative compressibility in one direction. This is precisely the behavior that we would expect; qualitatively, compressibility has its greatest value in the direction in which the interatomic distances are greatest. In the case of tellurium, some successful attempts were made to connect qualitatively its behavior, strange at first sight, with the helical structure of the arrangement of atoms relative to the axis. Quantitatively, however, nothing has yet been done, and one can hardly hope to achieve anything here before the fundamental problem of reproducing the structure of the lattice itself, with its differing distances in different directions, has been solved.

VIII. Equilibrium of Two Phases

Let us now pass from the consideration of pressure effects in one phase—liquid or solid—to the question of the change, caused by pressure, of the equilibrium between two phases—either between a liquid and a solid, or between two solids. The fundamental equation of phase equilibrium will, of course, as always, be the Clapeyron equation

\[ \frac{d\tau}{dp}=\frac{\tau \Delta v}{L}, \]

where \(v\) is the change in volume during the transition, and \(L\) is the latent heat of transition.

1. Melting curve. The Clapeyron equation is applicable to any case of equilibrium of a two-phase state. In the study of equilibrium during the melting of a solid body, the greatest attention of researchers has been directed to the form of the melting curve. The Clapeyron equation allows \(\frac{d\tau}{dp}\) to be any function of pressure, provided only that \(\Delta v\) and \(L\) have the corresponding values; indeed, thermodynamics does not restrict the form of the melting curve. Consequently, the question is reduced to experiment.

Assumptions were to a considerable extent limited by the corresponding state of affairs in the case of equilibrium between a liquid and its vapor. Here it was shown that the behavior of different substances is in general the same—there is a critical point at which the distinction between liquid and gas is effaced, which means that it is always possible

so select the pressures and temperatures as to bypass this critical point, i.e., to pass from vapor to liquid while avoiding discontinuities.

This general result is often accepted as given also for the case of the equilibrium liquid—solid body, apparently without giving oneself a clear account that this is a certain assumption. It was usually accepted that there exists some melting curve of a general form for all substances, irrespective of the fact that some substances are stable owing to van der Waals forces (for example, inert gases), others are stable owing to forces of ionic origin (salts), and, finally, in metals exchange forces act, while in some molecular compounds valence forces act.

This assumption did not seem so serious until the existence of these different forces had been distinctly shown. Now, however, the adoption of some general melting curve is an assumption requiring special consideration.

In assuming the existence of a curve, many suppositions were made concerning its character. The very first consisted in the supposition of the existence of a critical point (by analogy with the liquid—vapor case), at which the liquid and the solid body would become identical. Suppositions were also made that the curve, having reached some maximum, would begin to fall again (Tammann), or that it would asymptotically rise to some temperature at infinite pressure (Simon), or else that it would rise without bound as pressure and temperature increase. Experiment has shown that the first two suppositions are incorrect; the latter two, however, require infinite pressure, and therefore the question had to be decided by means of a certain extrapolation. Such an extrapolation can be carried out on the basis of the behavior of $\Delta v$ and $L$ in the Clapeyron equation. Indeed, if a critical point exists, then $\Delta v$ and $L$ will tend to zero at one and the same pressure (and temperature); if there is a maximum, then $\Delta v$ will tend to zero at some finite pressure (and temperature), while $L$—at the same pressure—will tend to some finite value, etc. We need not go into details here, since this question has recently been fully analyzed.^6 Personally, it seems to me that all the experimental material speaks in favor of the melting curve rising without bound. This is justified for such different substances as argon, nitrogen, liquid metals, and many complex organic compounds. Of course, this conclusion is valid only for the region in which nothing entirely new occurs, such as, for example, the disintegration of the atom, and it is not valid in the extreme cases examined in Section XVI.

Thus the tacit assumption of a “single melting curve” is, after all, as if legitimate. The explanation, apparently, must be based on some general property of the liquid and the solid body and must not depend on the kind of forces holding the molecule together. Such a general property is the regular structure of crystals as opposed to the chaotic structure of a liquid. Unboundedly in-

A descending curve means simply that, however high the temperature may be, it is always possible, by applying sufficient pressure, to force any substance to assume that regular crystalline structure which it has at lower temperatures (including polymorphic transitions). This is quite natural and is in complete agreement with the experimental fact of the universality of the crystalline state. However, to this day we still have no theoretical justification for the necessity of the existence of the crystalline phase, and I think that it is solely to our theoretical inability to justify this fundamental proposition that we owe the persistence with which many physicists, contrary to experiment, hold to the idea that the melting curve will end at a critical point, if only we should manage to raise the pressure sufficiently high. It would be extremely surprising if it turned out that the tendency of all substances to crystallize could be destroyed by pressure, especially if one bears in mind that many substances in their natural state are under colossal internal pressure.

Some theoretical investigations of melting proceed from crudely empirical points of view. For example, Lindemann postulates that melting occurs when the amplitude of atomic vibrations reaches 10% of the interatomic distance at \(0^\circ \mathrm{K}\). Most other investigations have one idea in common with Lindemann’s work—that a solid will begin to melt upon attainment of certain critical conditions for the solid phase, independently of the nature of the liquid. Of course, this is incorrect. At the melting point, i.e. at the point at which the liquid and the solid are in equilibrium, there will be satisfied a certain relation between the properties of the liquid and of the solid. According to the formulas of thermodynamics we shall have equality of the thermodynamic potentials of the liquid and the solid. Any real theory of melting must take this aspect of the phenomenon into account. A complete theory of melting must, in addition, also take into consideration the corresponding crystalline system of the solid.

We have the possibility of collecting a considerable amount of experimental material which, without doubt, will be used in constructing a theory of melting. However, it has happened that, despite the large number of liquids whose properties have been investigated under pressure, we have investigated only a very small number of melting curves. The reason for this is almost exclusively purely technical: the apparatus which we used for measuring the compressibilities of a large number of liquids would have been ruined if we had allowed the liquid in it to freeze; therefore, in the experiments we deliberately selected liquids which would not freeze under the conditions of the experiment. Thus we have a completely unsatisfactory body of experimental material on the question of the mutual change of the properties of the liquid and the solid on the melting curve. We, for example, cannot answer even such simple questions as: how, in general, does the volume of a liquid change along the melting curve—

...whether it decreases or increases, whether the reserve of energy of the liquid decreases or increases under the same conditions, etc.

The normal melting curve rises to higher temperatures with increasing pressure, which, according to the Clapeyron equation, is a consequence of the fact that the volume of the solid is almost always smaller than the volume of the liquid and that the latent heat of transition from the lower to the higher temperature phase is necessarily positive. The only exceptions are water, bismuth, and gallium. The abnormally falling melting curve for water, it turns out, occurs only in a certain region of pressures, since at pressures above \(2000\ \mathrm{kg}/\mathrm{cm}^2\) ice passes into another modification, denser than water, and, consequently, beginning with this pressure, the melting curve of water rises normally. It would seem that for bismuth and gallium we should expect an analogous picture; for this reason many attempts were made to find these new modifications. Recently I was at last fortunate enough to obtain the second modification of bismuth; the transition occurs at a pressure of \(25000\ \mathrm{kg}/\mathrm{cm}^2\) at room temperature. It has not yet been shown directly that the melting curve of this new modification rises, but the magnitude of the volume change leaves no doubt in this respect. It may be considered that this new bismuth will not possess the majority of the anomalous properties of ordinary bismuth, and that the general effect of pressure on bismuth is the tendency to restore its “normality,” as we have also in the case of water.

Attempts to obtain a new modification of gallium were made at pressures not exceeding \(12000\ \mathrm{kg}/\mathrm{cm}^2\); however, the smallness of the deviations of the gallium curve from linearity leads one to expect that the second modification will be obtained at still higher pressures than in the case of bismuth. Bearing in mind the example of bismuth, I think that theory may safely regard a rising melting curve as the normal curve.

Usually all melting curves have one more general property, which must be explained by theory, namely their concavity

\[ \left(\frac{d^2 t}{dp^2} < 0\right) \]

regardless of whether they rise or fall.

With regard to the theory of melting, two remarks may be made. First: if the melting temperature is plotted as a function of the atomic number of the element, we find far more irregularities than, for example, for such properties as atomic volume, or compressibility, or electrical conductivity. On such a curve there will be unexpected bends, for example, mercury between gold and thallium, or gallium between zinc and germanium. These anomalies can no longer be explained by the natural assumption that there are omitted solid modifications which may be discovered under sufficiently high pressure, such as, for example, the new modification of bismuth. The point is that these abnormally low melting points lie at too low temperatures, whereas the melting point of any...

an as yet undiscovered modification stable at high pressure must lie still lower. The situation can be saved only by the presence of a new liquid phase, stable under high pressure. But we do not yet know of a single substance having two liquid phases—unless, of course, liquid crystals are counted.

The second observation is that the melting of different substances does not obey the law of corresponding states, which is valid for the evaporation of liquids. If such a law existed, then the temperature at which the difference between the volumes of the liquid and the solid would constitute a definite fraction of the volume at the normal melting temperature would stand in a definite ratio to this normal temperature, and this ratio would be common to all substances. It can be asserted with certainty, however, that this is not so. One may nevertheless think that something like a law of corresponding states exists for individual groups of substances. The alkali metals, for example sodium, potassium, rubidium, and caesium, approximately satisfy such a condition for volume differences.

The alkali metals possess yet another remarkable property with respect to melting. The melting curves of sodium and potassium intersect at about \(9000\ \mathrm{kg}/\mathrm{cm}^2\), and the slope of the remaining curves in the experimentally accessible region is such that it is quite possible that at about \(30000\ \mathrm{kg}/\mathrm{cm}^2\) a complete reversal of the order of melting of all the alkali metals will occur, i.e. lithium will become the most easily fusible, and caesium, conversely, the most refractory. It seems to me that this circumstance should be compared with changes in the internal structure of atoms, which we have already considered with the aid of Schottky’s theorem: it is quite obvious that caesium has far more latent possibilities for rearrangement than lithium, owing to the considerably more complex electronic structure of its atom.

Schottky’s theorem requires special internal changes in the atom upon melting, in addition to the changes associated with temperature and pressure that occur in a homogeneous phase. Let us write equation (1) for melting:

\[ \Delta T = -\Delta E + 3p\Delta v, \]

but

\[ \Delta E = L - p\Delta v, \]

therefore,

\[ \Delta T = -L + 4p\Delta v. \]

With the same degree of accuracy with which one may assume that the kinetic energy of motion of the masses of the molecules is the same in both the liquid and the solid phase, this equation shows that at low pressures (neglecting the term \(4p\Delta v\)) the internal kinetic energy of the molecules in the liquid is less than in the solid by the amount of the latent heat, which signifies a “swelling” of the molecule in the liquid. But at high pressures the term \(4p\Delta v\) begins to act,

having the opposite sign, so that at high pressures the internal structure of the molecules of the liquid and of the solid tends to become identical. Experience shows that, indeed, for most liquids for which melting curves have been measured, the term \(4p\Delta v\) becomes greater than \(L\) at pressures below \(12000\ \mathrm{kg}/\mathrm{cm}^{2}\), so that an inflection is obtained. Of course, in order to find out what exactly is taking place, new, more precise methods must be devised for studying the kinetic energy of the motion of molecular masses in the case of the two-phase state; but in any event it is already clear at this stage that at high pressures there are appreciable differences as compared with low pressures.

2. Polymorphic transitions in solids. From the thermodynamic point of view, since Clapeyron’s equation and Schottky’s theorem are applicable, there is no difference between the transition from a liquid to a solid (crystalline) body and the transition between two solid bodies. From the experimental point of view, however, the phenomena associated with transitions between two solid phases are considerably more varied than in the case of transitions between a liquid and a solid. All melting phenomena fall under one heading—the melting curve rises, having a concave form; the difference in volumes between the liquid and the solid decreases with increasing temperature; the compressibility of the liquid phase is always greater than that of the solid, even for the anomalous case of ice. The transition curves between solids, however, sometimes rise, sometimes fall; they are sometimes convex, sometimes concave; the difference of volumes along the curve may either increase or decrease; and the compressibility of the phase having the larger volume may be either greater or smaller than the compressibility of the phase with the smaller volume. Transitions between solid phases reveal not only a very great variety from the standpoint of the thermodynamic parameters of the transition, but here there may also be very strong changes in the dynamic characteristics. The transition may take place rapidly or slowly; the rate of transition may change asymmetrically for small deviations of pressure and temperature from their equilibrium values in one direction or the other. The transition may admit either only supercooling, or only superheating, or both; the new modification may have either a quite definite orientation with respect to the original one or an arbitrary one; and, finally, sometimes the transition may proceed only if the original substance is a single crystal of a high degree of purity.

For the time being only one proposition, common to all transitions between solid phases, can be stated, namely: the transition never ends at a critical point. This, of course, means that one type of crystal lattice never passes continuously into another type. Such a continuous transition is impossible from a purely geometrical point of view: for example, a cubic lattice under gradual expansion, predominantly

in a direction along one of the axes of the cube, can pass into a tetragonal one. However, such a transition requires differences between the axes of the cube, and if at the outset the structure was truly cubic, then there could be no such differences. From the physical point of view such a continuous transition from one type of lattice to another must, of course, seem extremely improbable, and the fact that it has never yet been observed speaks in favor of our ideas about the structure of crystals, at least in this respect.

But, apart from the impossibility of a critical point, as it were all other possibilities are allowed, even those which at first sight appear to us anomalous. The very existence of certain types of transitions is highly significant. For example, there exist vertical transition lines, which means the absence of latent heat; in such transitions, however, there is a change of internal energy by the amount \(p\Delta v\). If the transition takes place at comparatively low pressures, when in both phases tensile forces are predominantly acting, this means that, despite the bringing together of atomic or molecular centers, work is performed against the forces of attraction. This is incomprehensible if the centers of attractive forces coincide with the geometrical centers of the molecules; however, it becomes comprehensible if the centers of attractive forces are essentially located on projections on the molecules. If the phase at low pressure is such that these projections on different molecules are in some order and are held by attractive forces, then it is easy to see that under the influence of high pressure these centers of projections may be brought out of order and thus perform work against the attractive forces. The volume will, however, decrease, since the projections will slide past one another and somehow interlock in the modification corresponding to high pressure. Such a picture is quite admissible in modern wave mechanics.

In such a transition the change in the internal structure of the atom, required by Schottky’s theorem, may be considerable; in this case \(L\) is equal to zero, consequently, \(\Delta T = 4p\Delta v\). Let us consider the transition in bismuth, occurring at \(25\,000\ \mathrm{kg}/\mathrm{cm}^2\); this transition is accompanied by a change of volume equal to \(8\%\),

\[ \Delta T = 4 \cdot 25\,000 \cdot 0.08 = 8\,000\ \mathrm{kg}/\mathrm{cm}^2 \cdot \mathrm{cm}^3 . \]

In each cubic centimeter there are \(2.8 \cdot 10^{22}\) atoms of bismuth; \(1\mathrm{eV} = 1.62 \cdot 10^{-18}\ \mathrm{kg}/\mathrm{cm}^2\). Hence

\[ \Delta T = \frac{8000 \cdot 10^{18}}{2.8 \cdot 10^{22} \cdot 1.62} = 0.18\ \mathrm{eV} \]

per atom. The sign of \(\Delta T\) is such that the atom under high pressure possesses less internal kinetic energy. Under normal conditions this will mean expansion of the atom. The result, however, is so paradoxical that, evidently, one of the first tasks of an exact theory will be to investigate to what extent our basic assumptions are valid, namely, that the kinetic energy of atomic motion is, on the whole, the same in both phases (i.e., the kinetic energy is constant

at constant temperature) and that one and the same simple relation connects the internal kinetic energy and the mean radius of an atom under high pressure with those of the free atom considered by elementary theory.

With a vertical tangent, the line of transition is usually not quite straight, but has a noticeable curvature. A simple thermodynamic argument shows that below the point where the tangent is vertical, the phase stable at lower temperatures has a greater heat capacity—an entirely anomalous situation.

An example of a transition with a maximum temperature has been found, and it is quite possible that there is at least one example of a transition with a minimum temperature, although many difficulties stand in the way of realizing it because of the viscous resistance to the transition at low temperatures. At the maximum point the transition proceeds without a change of volume, but with a non-vanishing latent heat. On the transition curve from the high-pressure side, the phase having the smaller volume has the greater compressibility—a result again sufficiently paradoxical. It turns out, however, that more than half of the solid transitions for which measurements have been made are paradoxical in this respect, i.e., for them the phase with the smaller volume has the greater compressibility. Such a situation is nevertheless entirely compatible with the picture set forth above, namely that in most crystals the centers of attractive forces are essentially located at projections and do not coincide with the centers of the molecules. The phase in which the centers of attraction are arranged so that they are in order, i.e., the phase of larger volume, should possess greater hardness or smaller compressibility in comparison with the phase in which the centers are brought out of order, i.e., with the phase of smaller volume. The chemical similarity of different substances is unlikely to manifest itself in the similarity of transition phenomena more strongly than in the similarity of any physical property; and indeed, we know very few examples of chemically similar groups having polymorphic similarity.

One of the best examples is the group NH$_4$Cl, NH$_4$Br, and NH$_4$I. All these substances have two structural modifications: a body-centered cell and a face-centered cell; the line of transition at pressures below 1000 kg/cm$^2$ is almost vertical, and the latent heats of the transitions are very small. In addition, within the group the transition pressure changes systematically. RbCl, RbBr, and RbI constitute another similar group, very similar to the group of ammonium halides; here too there are body-centered cells and face-centered cells, almost vertical transition lines, and successive changes of pressure within the group, which, however, in this group changes in the opposite direction compared with the ammonium family. The mean transition pressure for the rubidium family is 5000 kg/cm$^2$. Quite recently I found that the potassium family has transitions near 20,000 kg/cm$^2$; for the sodium family transitions below 50,000 kg/cm$^2$ are hardly possible.

Alkali-metal nitrates form a family with a large number of polymorphs. Here one can find many similar properties, but at the same time some members of this family exhibit sharp deviations. Moreover, there are also examples in which we cannot detect any similarity from the point of view of polymorphism, despite a very strong chemical kinship, as, for example, in \( \mathrm{CCl}_4 \) and \( \mathrm{CBr}_4 \).

Up to the present time theory has not advanced very far in explaining the causes of polymorphic transitions; still less is it able to predict under what conditions such a transition may occur, or to calculate the thermodynamic parameters of the transition. Indeed, it is difficult to expect anything much greater here until the fundamental problem has been solved—the elucidation of the reason why a substance crystallizes into a lattice.

In fact, it is impossible to discuss the problems of polymorphic transitions until we are in a position not only to understand why a lattice exists, but also to calculate the type of lattice. Nevertheless, some problems of polymorphic transitions have been investigated. The difference between a body-centered cell and a cell with centered faces was reduced to a difference in energy minima. Gund\(^9\), proceeding in this way, tried to show that, when the exponent in the expression for the repulsive forces passes through a certain critical value, the type of lattice changes. This, in his opinion, was to explain the difference in the crystalline systems of the halide compounds of cesium and of other alkali metals. However, Gund’s attempts were not crowned with success—the critical value of the exponent lay at values considerably higher than compressibility gave. Born and Mayer\(^ {10}\) modified Gund’s method and included in the discussion, in addition to the forces of attraction and repulsion, also the van der Waals forces arising from the polarization of the ions. The most stable type of lattice proves to be that of \( \mathrm{NaCl} \), but if the van der Waals forces are doubled—which at first glance seems purely empirical, but in fact has received a theoretical justification (Mayer)—then, for the cesium halide compounds, the lattice of the \( \mathrm{CsCl} \) type is the most stable. The only exception is \( \mathrm{CsF} \). Moreover, the difference in energies between the two types of lattices for rubidium salts proved to be of the same order of magnitude as the actually found value of the energy of transition under pressure. However, apparently, additional considerations will be required in order to explain why the latent heat of transition is equal to zero, whereas the change in energy is rather considerable. For potassium salts the energy values calculated by Born and Mayer lead one to expect that, under pressure, these salts can be forced to pass from lattices of the \( \mathrm{NaCl} \) type to lattices of the \( \mathrm{CsCl} \) type. I have in fact recently succeeded in detecting these transitions at a pressure of \(20\,000\ \mathrm{kg}/\mathrm{cm}^2\).

Even the simplest of all possible transitions—the transitions of alkali-metal halide compounds from lattices of the \( \mathrm{NaCl} \) type to lattices of the \( \mathrm{CsCl} \) type—are complicated by deviations. \( \mathrm{RbCl} \) underwe-

causes a transition with a very small change in volume1 at a pressure of 2000 kg/cm², i.e., at one half of the pressure required for the principal presumed transition from a lattice of the NaCl type to the CsCl type. It is unlikely that such a transition can take place, and the nature of the new lattice is very uncertain. It seems to me probable that the lattice type does not change in this transition, but that less significant rearrangements of the electron energy levels occur within the atoms.

Of course, for constructing the theory it would be extremely useful for us to know the structure of lattices stable under high pressure. Unfortunately, no experiments have yet been made using X-rays to investigate structure at high pressure, so that our ideas are based on those cases in which a structure stable under high pressure can be obtained at normal pressure by a suitable choice of temperature. Fortunately, we know a sufficient number of such substances. The results obtained are rather unexpected. It turns out that for most substances the form stable at higher pressures, i.e., possessing a smaller volume, is the form with lower symmetry. Spherical symmetry is a property of the atom at low pressure, when it has much free space at its disposal. When the volume of free space decreases, the atom has to assume a less symmetrical form. In the case of directed valences, for example in the case of water molecules, a complete proof was given that wave mechanics requires a spherically asymmetric solution.

Any theory of polymorphic transitions must account quite distinctly for one very important property by which a transition between solid phases may differ from liquid–solid transitions. If a solid and a liquid are in equilibrium in direct contact with one another, and if this equilibrium is disturbed by a change in pressure or temperature, then it is restored by itself either through melting or through solidification, i.e., the system by itself arrives at a quite definite position of equilibrium, independent of the direction from which it is approached, provided, of course, that both phases are in contact. Usually this is explained by saying that equilibrium is a dynamic state, i.e., at all times some of the molecules pass from the liquid phase into the solid and, consequently, crystallize, while another part of the molecules passes from the solid phase into the liquid, i.e., melts. At equilibrium the rates of both molecular fluxes are the same. When equilibrium is disturbed, the rate of one flux becomes greater than that of the other, and this excess of rate acts in such a direction that equilibrium is immediately restored again. In the case of two solid bodies an entirely different picture may be observed—sometimes it is possible to change the temperature or pressure for two solid phases in contact, and no change whatsoever results from this

will occur. However, if the temperature or pressure is changed very appreciably, the process begins to proceed in such a direction as to restore the initial conditions. Solids, therefore, possess a “region of indifferent equilibrium.” The magnitude of this region can be measured with sufficient accuracy; it has turned out that it depends on temperature and pressure. There are no general regularities here. The very existence of a “region of indifferent equilibrium” and the character of its variation with temperature and pressure are completely independent of one another and are wholly determined by the given substance. In those cases where the “region of indifference” exists, the equilibrium between two phases cannot be a dynamic equilibrium of two streams flowing in opposite directions; apparently, the mechanism of the phenomenon here is more static. It is quite possible that we have here something like a potential barrier between two phases. This barrier can be crossed only in the event of a sufficiently considerable disturbance of the equilibrium conditions. It seems to me quite obvious that in such cases, regardless of the fact that motion takes place from one potential valley to another through the barrier lying between them, there can be no Maxwellian distribution of velocities, since otherwise there would always be molecules whose velocities are sufficiently great to overcome the barrier.

The enormous number of various polymorphic transitions discovered experimentally is now becoming increasingly comprehensible, since theory too provides an ever greater number of possibilities. It is now already recognized that lattices are held in equilibrium by various types of forces; there are ionic lattices, lattices held by affinity forces (valences) or by van der Waals forces, and molecular lattices.

The differences between the individual types just enumerated are not always sufficiently clear. Polymorphic transitions, one may think, correspond to a transition from one type of lattice to another, or to a partial change of type. In addition, we must expect transitions consisting in an internal rearrangement of electrons in atoms. Such changes are especially possible in atoms having unfilled inner orbitals. It may be that the transitions occurring in cadmium and in cerium belong precisely to this type. A complete theory of polymorphic transitions and crystallization will undoubtedly include one further consideration, which until now has remained almost unnoticed. It seems to me that general considerations, such as the condition of minimum energy, are insufficient here; one must be sure that all the intermediate stages required for the formation of a crystal can actually be realized. This is not always necessary, since experience shows that nature often somehow finds a possibility of satisfying such general principles. For example, in systems where the Maxwellian distribution of velocities operates, there will always be such molecules as can occupy a position corresponding to an energy minimum,

although for this they would have to pass through intermediate states, moving in the direction of higher energy. Such a situation occurs in polymorphic transitions—we judge this from the existence of the “indifference band,” of which we have already spoken. An analogous argument may also be applied to the formation of a crystal from a liquid or gaseous phase, or from a solution. Only recently have physicists begun to study carefully how a crystal grows. It turns out that at the contact surfaces we have conditions of growth and decomposition that differ substantially from the conditions in the inner parts of the crystal. These conditions of growth, as well as the conditions of the filled structure, must also be observed in the case where the crystal exists. Thermodynamic relations, expressed, for example, in the equality of two thermodynamic potentials, are only one side of the matter.

IX. Irreversible Transitions

All the transitions considered up to now have been reversible both thermodynamically and mechanically. However, there is one irreversible transition caused by high pressure which, it seems to me, is of enormous theoretical interest. I have in mind the transition from white to black phosphorus. First of all, this transition is remarkable for a very large increase in density—up to 46%—and for a sharp change in properties—from a good insulator to a conductor of electricity—as well as for a number of details in the conditions of the transition. The rate of this transition cannot be increased by the usual method—by introducing a nucleus of the second modification; some sort of preparatory process is required, proceeding, apparently, uniformly throughout the whole volume, with a slight increase in density and with an ever-increasing acceleration over the course of 10 or 15 minutes, until some critical state is reached. Then the whole system at once passes into the black modification. What this preparatory process actually consists of is at present completely unknown. It would be extremely surprising if phosphorus, nonmetallic in its ordinary form, turned out to be the only element that can pass irreversibly into a metallic state. I tried to detect an analogous transition in sulfur—a neighbor of phosphorus in the periodic system—but without success, although a pressure of 50,000 kg/cm² was applied at normal temperature. For carrying out similar experiments it would be highly desirable to extend the range of attainable pressures.

Ag₂O gives an example of a transition of an intermediate type between the completely irreversible transition of white phosphorus into black and the completely reversible transitions of most substances. If the volume of Ag₂O is measured as a function of pressure, first increasing the pressure to some maximum value and then decreasing it to the initial value, a wide hysteresis loop is obtained, similar in shape to the magnetization loops of many ferro-

magnetic substances. Small values of hysteresis occur quite often in transitions between solid phases, and are even rather natural, since there are impurities forming solid solutions, which entails the phenomenon of diffusion in the solid state. However, in the case of Ag$_2$O, apparently, we have something different, still not susceptible of explanation.

X. Discontinuities of continuity—transitions of the second kind, etc.

The phase transformations considered by us up to now, to which the Clapeyron equation is applicable, contained discontinuities in volumes and in energy reserves. Recently other types of transformations have begun to be studied, having discontinuities of continuity in the derivatives of the volume and the energy reserve, i.e. discontinuities in thermal expansion, compressibility, and heat capacity.

These transformations were first observed at low temperatures. Perhaps the best-studied case is that of NH$_4$Cl, investigated by Simon$^{12}$. Simon discovered a narrow temperature interval, measured by several degrees, lying near $-30^\circ$ C, in which colossal changes in heat capacity and thermal expansion occur. Somewhat later Keesom$^{13}$ discovered that the anomalous effects found by him in liquid helium, which at first he had been inclined to ascribe to two supposedly existing modifications of the liquid, essentially consisted in a discontinuity in the derivatives (but precisely the derivatives) of volume and energy. For observing these discontinuities one uses a curve in the $p,t$ plane. Ehrenfest$^{14}$ published a paper in which he considered such transitions thermodynamically, and proposed calling them “transitions of the second kind.” This proposal was accepted.

Shortly before the appearance of Ehrenfest’s work, I experimentally investigated the influence of pressure on the anomalies of NH$_4$Cl and NH$_4$Br$^{15}$ discovered by Simon. For NH$_4$Cl it turned out that the anomaly manifests itself in a sudden change of direction [sudden—within the limits of experimental accuracy] of the pressure–volume isotherm, which of course means a discontinuity in compressibility and thermal expansion. With increasing temperature the pressure at which the anomaly occurred shifted toward higher values, i.e. in the same way as the pressure shifts in ordinary transitions. At 9500 kg/cm$^2$ the temperature of the discontinuity is equal to $+30^\circ$ C. I showed that there is a purely geometrical relation connecting the discontinuity in compressibility and thermal expansion with the slope of the line on which this discontinuity occurs, namely:

$$ \frac{d\tau}{dp}=-\frac{\Delta\left(\dfrac{\partial v}{\partial p}\right)_{\tau}}{\Delta\left(\dfrac{\partial v}{\partial \tau}\right)_{p}}, $$

as indicated in Fig. 2.

Using the thermodynamic relations

\[ \left(\frac{\partial Q}{\partial p}\right)_{\tau} = -\tau\left(\frac{\partial v}{\partial \tau}\right)_{p}, \]

one can bring this equation to a form completely analogous to the Clapeyron equation

\[ \frac{d\tau}{dp} = \tau \frac{dv}{dQ}, \]

where \(\dfrac{dv}{dQ}\) is the ratio of the change in volume to the amount of heat absorbed in the transition corresponding to \(AB\) in the drawing. Ehrenfest also used the equivalent relation

\[ \frac{d\tau}{dp} = \tau \frac{ \Delta\left(\dfrac{\partial v}{\partial \tau}\right) }{ \Delta C_{p} }. \]

The behavior of \(\mathrm{NH_4Br}\) is somewhat different from that of \(\mathrm{NH_4Cl}\). The volume anomaly has the opposite sign, so that at high pressures it shifts toward low temperatures, not high ones. In view of the great technical difficulties associated with observing pressure at low temperatures, the anomaly was investigated only at \(-72^\circ\mathrm{C}\), where it occurs at \(1600\ \mathrm{kg}/\mathrm{cm}^2\). Moreover, this anomaly is still more abrupt than for \(\mathrm{NH_4Cl}\), so that, within the limits of experimental error, it could rather be attributed to a discontinuity of the volume itself than to that of the derivative, i.e. it has the properties of an ordinary transition.

Fig. 2. Geometrical relations satisfying a discontinuity gap in the derivative of volume, which is manifested in a temperature shift when the pressure is shifted.

Later I found a large number of anomalies in the behavior of solids in the temperature interval \(0\)—\(100^\circ\mathrm{C}\) and at pressures up to \(12000\ \mathrm{kg}/\mathrm{cm}^2\).^6 Part of these anomalies, again within the accuracy of the experiment, consists of sharp and reversible discontinuities of continuity in the first derivatives and, consequently, must be assigned to “transitions of the second kind.” This type of transition is common for many alloys. There are cases when the discontinuities are completely reversible, i.e. hysteresis phenomena are observed. There are also substances in which the anomaly has no sharp boundaries, but rather there is a region of anomalous curvature, which shifts to other pressures when the temperature is changed. An example of an anomaly of this type may also be \(\mathrm{NH_4Cl}\). This second anomaly, apparently, has no relation to the one already discussed. Another striking example is metallic chromium of high degree of purity.

The great variety of phenomena associated with such anomalies shows that the division of transitions into transitions of the “first,” “second,” “third,” etc. kind (since it is obvious that the classifi-

...classification and nomenclature of Ehrenfest can be continued indefinitely) at present is still a matter of convenience, more or less closely reflecting the accuracy of the experiment. In particular, this applies to \(\mathrm{NH}_4\mathrm{Br}\). In Simon’s experiments only rapid changes in thermal expansion were found, but under pressure at a temperature \(40^\circ\mathrm{C}\) lower the anomaly became so sharp that it was necessary to consider a discontinuity in volume. What appears, with a cruder measurement, as a discontinuity in the first derivative may, with a more exact measurement, turn out to be a discontinuity in the second derivative. It should be remembered that even an ordinary phase transformation would not prove to be discontinuous if we could make measurements with sufficient accuracy and observe, during the transition, the effect of the change in the ratio of surface energy to volume energy. It seems to me that the physical meaning of “transitions of the second kind” depends on the discovery of some characteristic physical process corresponding to this transition, analogous, for example, to the rearrangement of the lattice from one type to another, which characterizes ordinary transitions. So far as I know, this has not yet been done, but I see no grounds either for thinking that it has not been done, or for thinking that it will not be found that a transition of the second kind corresponds to some definite physical process. There may be several such processes; indeed, we know that transitions of the first kind may denote either the change of one amorphous phase into another, or the change of an amorphous phase into a crystalline one, or, finally, the change of one crystalline phase into another crystalline phase. It even seems to me very possible that we already understand the mechanism corresponding to at least one type of transition of the second kind: according to the data of Bragg and Williams, when certain alloys of gold and copper are heated, there occurs a change from a lattice type in which the gold and copper atoms are arranged regularly to a type in which the arrangement is chaotic. This change in the lattice type seems to be associated with a definite jump in the derivative.

There have nevertheless been some successes in explaining such anomalies. Probably the best known is Pauling’s explanation[^18] for the anomaly of \(\mathrm{NH}_4\mathrm{Cl}\). It consists in the fact that there exists a temperature range in which the molecules possess rotational motion. This is, to a certain extent, a self-catalyzing process, since as soon as a small part of the molecules has begun to rotate, it loosens the structure and facilitates an increase in the additional energy of rotation. Bragg uses an analogous autocatalysis to explain the changes in the gold—copper system, and it is possible that it is an important characteristic of transitions of the second kind. But the details of Pauling’s explanations have not yet been investigated; it is not clear whether we have a sharp jump in the derivative or whether the transition occurs more or less smoothly; nor has any explanation been given of the difference in the sign of the effect in \(\mathrm{NH}_4\mathrm{Cl}\) and \(\mathrm{NH}_4\mathrm{Br}\), although it would seem that there are no insurmountable difficulties here.

In general, one can point to a multitude of reasons causing such anomalies, just as we have seen that there are many types of polymorphic transitions. The number of these possibilities seems especially large if one remembers that a solid is not simply built from a large number of independent atoms, but that the whole body as a whole is a single structure, possessing an enormous number of energy equations and various types of wave functions, far more complex than the structure of an individual atom.

XI. Electrical Resistance

Let us now consider the influence of pressure on electrical resistance, in particular on the resistance of metals. Theory has not achieved particular success in this field; indeed, one can perhaps name only a single attempt, by Król[^19], who tried to construct an explanation of the processes of conductivity on the basis of modern theoretical achievements. The reason for this defect of theory should apparently be sought in the fact that a wave-mechanical treatment of the processes of electrical conductivity of metals gives an expression for the conductivity only as a third approximation in the calculation,2 and consequently here we are dealing with the influence of pressure on the factors of this third approximation. The difficulties become clear at once if one considers the simplest formula for the electrical conductivity \(\sigma\), given by the elementary Sommerfeld theory, in which the electrons are regarded as particles of a gas obeying Fermi statistics:

\[ \sigma = \frac{e^2 l n}{m \bar{v}}, \]

where \(l\) is the mean free path, \(\bar{v}\) is the limiting velocity of the Fermi distribution, and the remaining symbols have their usual meanings. The elementary theory considers \(\frac{1}{\bar{v}}\) proportional to the mean distance between atomic centers, so that from this point of view the influence of pressure on \(\frac{1}{\bar{v}}\) should be proportional to the compressibility. But from Schottky’s theorem we know that the mean kinetic energy of the electrons increases with pressure not simply in proportion to the compressibility, and therefore the relation between pressure and \(\frac{1}{\bar{v}}\) is hardly likely to be as simple as the elementary theory assumes. The calculation of the changes of \(l\) with pressure is, of course, very difficult, since Sommerfeld’s elementary theory leaves this quantity unknown, and it can be calculated only if one adopts a formula and substitutes numerical values for all the other quantities. Obvi-

it is evident that \(l\) will depend in some complicated way on the amplitude of atomic vibrations, interatomic distances, and on the frequency, i.e., on the stiffness of the restoring forces. The situation becomes still more complicated with regard to the pressure coefficient if, for the conductivity, one adopts the formula given by a more exact theory (see Bethe, Handb. d. Phys., p. 523). This formula contains four or five factors which may vary with pressure in a rather complicated manner.

Despite the lack of clarity in present-day theoretical views on the phenomena observed under pressure, it nevertheless seems to me that the experimental data are sufficiently important, and that we are entitled to demand a more satisfactory explanation from theory. These changes of resistance under pressure are very noticeable; they are almost always several times greater than the corresponding changes in volume. Potassium, for example, under a pressure of \(15\,000\ \mathrm{kg}/\mathrm{cm}^2\) has a volume equal to \(70\%\) of the initial volume, while its resistance at this pressure is only \(25\%\) of the initial value. The resistance of strontium under a pressure of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) is \(34\%\) greater than the initial value, while the volume is only \(9\%\) smaller. Among nonmetallic substances still more striking examples may be found: the conductivity of black phosphorus under a pressure of \(18\,000\ \mathrm{kg}/\mathrm{cm}^2\) is 100 times greater than the initial value; \(\mathrm{Ag}_2\mathrm{S}\) at \(3\,000\ \mathrm{kg}/\mathrm{cm}^2\) is 20 times a better conductor than in the normal state.

If we turn to the metals, the considerable diversity of their behavior immediately strikes the eye. Most ordinary metals under pressure decrease their resistance, and the rate of decrease is smaller at high pressures. However, approximately one fifth of the metals tested showed an increase of resistance with pressure. These include, as was to be expected, bismuth and arsenic, but also such metals as lithium and strontium, whose anomaly is considerably less intelligible. No at all significant correlation can be drawn between position in the periodic system of the elements and the positive sign of the coefficient. The coefficient of change of resistance with pressure for all metals having a positive sign increases with increasing pressure, i.e. the curve of resistance as a function of pressure is convex toward the pressure axis. Finally, there are also metals whose resistance first decreases, then passes through a minimum, and with a further increase of pressure rises. Such, for example, is cesium, for which the minimum corresponds to \(4\,000\ \mathrm{kg}/\mathrm{cm}^2\), rubidium with a minimum at \(17\,800\ \mathrm{kg}/\mathrm{cm}^2\), and barium with a minimum at \(9\,000\ \mathrm{kg}/\mathrm{cm}^2\). Proceeding from present-day conceptions of the structure of metals, it would be natural to expect a decrease of resistance with increasing pressure. Electrical resistance arises mainly from interference of electron waves when the atomic lattice departs from complete periodicity, and these departures are simply connected with the amplitude of atomic vibrations. Under high pressure the amplitudes of atomic vibrations decrease owing to the increase in

rigidity of the bonds and an increase in the characteristic temperature; therefore the scattering of electron waves becomes smaller, and hence the resistance also decreases. This picture, given by wave mechanics, is in many respects very similar to the theory of electrical resistance to which I arrived in my experiments on resistance at high pressures. According to this theory, resistance to the motion of electrons arose owing to the gaps that appear between atoms during thermal motion. However, the existence of positive coefficients apparently requires the existence of some additional mechanism. In Kroll’s theory such a mechanism is the distance between atomic centers. Owing to a certain scattering effect, the resistance increases as this interatomic distance decreases, since at small distances these irregularities in the structure become more pronounced. In Kroll’s theory this gave a term proportional to the compressibility, tending to increase the resistance with pressure*.

In my theory I likewise had to introduce a second mechanism to explain certain anomalous dependences, in particular the dependence of resistance on stress, if it is connected with the pressure coefficient. The second mechanism which I proposed was as follows: some electrons slip through channels between atoms; these channels narrow when the atoms, under the influence of hydrostatic pressure, come closer together. As we see, this explanation is sufficiently similar to Kroll’s explanation. Thus introducing two different mechanisms, we can explain different signs of the influence of pressure in different metals. If we suppose, moreover, that at low pressures one effect predominates, and at high pressures the other, then the change of sign and the minimum of resistance can also be explained. Proceeding precisely in this way, Kroll gives qualitative explanations of the resistance minimum of cesium; however, quantitatively, in many cases his data are far from satisfactory.

Considering the whole body of experimental material, the greater part of which was obtained during the last ten years, i.e. after I had developed my theory, I remain more and more dissatisfied with this picture of two different mechanisms, of which now one, now the other predominates.

The point is that the influence of pressure on the resistance of almost all metals (elements) can be qualitatively represented by one family of curves (Fig. 3). Here the resistance of any metal at constant temperature is plotted as a function of pressure. Each curve has a minimum. This minimum shifts toward higher pressures.

* Kroll’s expression also contains a term depending on the change of Poisson’s ratio with pressure. Kroll, not having experimental data, did not investigate this dependence. Let us note in parentheses that we have studied the influence of pressure on hardness, and consequently, from these data and the data on the change of compressibility, the change in Poisson’s ratio can be calculated. Generally speaking, Poisson’s ratio increases approximately in the same way as the incompressibility.

The position of the origin of the pressure axis depends on the metal: for metals which usually have a positive coefficient of change of resistance with pressure, the origin of pressure should be placed beyond the minimum, so that for such metals one may expect a negative coefficient if it were possible to produce sufficiently large negative pressures. For normal metals the origin of pressure lies, as indicated in the drawing, and the pressure corresponding to the minimum lies beyond present experimental possibilities. Only for certain metals does the minimum lie within the range of the pressures obtained. This diagram suggests that metals with both a negative and a positive coefficient present two sides of essentially one and the same phenomenon. Moreover, ultimately, at colossal pressure, a state will be reached in which the resistance will begin to increase with increasing pressure for all metals. Even more remarkable is the circumstance that the curvature of the resistance curve is always directed upward. Many phenomena, with increasing pressure, change in the opposite direction; i.e., for an increase of pressure by the same amount, they change less in the region of high pressures. This is precisely what should be expected. Indeed, the curve shown in the drawing indicates that there is, as it were, no limit to the possible increase of resistance, and, consequently, metals at infinite pressure tend to become insulators. It may be supposed that, with the infinite approach of the atomic centers, a new solution of the wave equation appears, according to which the outer electrons, ordinarily in the free state and responsible for conductivity, become as tightly bound as the inner electrons under ordinary conditions; i.e., the entire metal as a whole becomes a single complex nucleus.

Fig. 3. The resistance of all metals as a function of pressure and temperature can be represented by a family of such curves. The curves are arranged according to increasing temperature (i.e., \(T_2 > T_1\), etc.).

It would be extremely important to demonstrate experimentally the existence of a resistance minimum for elements situated in all parts of the periodic system. However, in practice this has been shown only for cesium, rubidium, and barium; for potassium and sodium, by means of extrapolation, confidence has been obtained in the existence of such a minimum. Extrapolation for other metals indicates that, probably in all cases, the minimum lies at a pressure above \(40\,000\ \mathrm{kg/cm^2}\). However, if instead of \(R\) one plots its logarithm (\(\log R\)), the existence of a minimum becomes obvious. The point is that the resistance curve for all metals with a negative coefficient must be convex, since the resistance cannot

be less than zero; but this convexity is very small. On the other hand, $\lg R$ can vary down to minus infinity, i.e., in this case no restrictions are imposed on the curvature, and the convexity may be considerable. Measurements show that the curve $\lg R$ for all metals with a negative coefficient is convex with respect to the pressure axis, i.e., it has precisely the direction required by the existence of a minimum.

Experiments show that the change of resistance with pressure depends little on temperature, or, what is the same thing, that the temperature coefficient of resistance depends little on pressure. The small observed changes are compatible with Fig. 3. One might perhaps have expected a small dependence of the temperature coefficient on pressure, since the temperature coefficient for all metals is approximately the same $\left(\dfrac{1}{\tau}\right)$, and there is hardly any reason to expect that a metal under pressure should differ more strongly from the same metal in the absence of pressure than two different metals do. The magnitude of the temperature coefficient, equal to $\dfrac{1}{\tau}$, is given by modern theory, so that this part of the phenomena must be automatically included in a theory of pressure phenomena based on the considerations set forth above.

Several anomalous metals are known that do not fit the scheme of Fig. 3. However, bismuth, which might have been considered anomalous, is normal in this respect. Of course, this may be accidental. It is quite probable that the anomalies of bismuth are connected with its crystalline structure, since pressure phenomena in liquid bismuth proceed quite normally. This view of bismuth becomes still more probable if one recalls the newly discovered modification of bismuth, stable under high pressure, with a normal ratio of the volumes of the solid and liquid phases. It would be very interesting to determine the sign of the coefficient of change of resistance with pressure for this new modification.

Comparatively very few measurements have been made of the influence of pressure on the resistance of liquid metals; and the theory of the resistance of these metals has scarcely been developed. It seems to me that this question should not be postponed. Qualitatively, liquid metals possess many properties in common with solid metals, especially with respect to the magnitude of the resistance itself. On the other hand, the theory of electrical resistance is based on the assumption of the existence of a lattice, and this assumption plays a fundamental role in the whole theory. However, the resistance of liquids makes it necessary to regard this assumption as unnecessary, and some conclusions must evidently be obtained from more general considerations. In this respect, the following circumstance is probably significant: the resistance of the liquid phase, apparently, stands in a certain constant relation to the resistance of the solid phase that is reversible at the freezing point, regardless of whether freezing occurs at low temperatures and pressures or whether

it is shifted by higher pressure toward higher temperatures. The ratio of the resistance of liquid potassium to the resistance of solid potassium is 1.56 at the normal melting temperature, \(62.5^\circ\), and atmospheric pressure; it is 1.55 at the reversible point of solidification at \(165^\circ\) and \(9700\ \mathrm{kg/cm^2}\).

It is possible that the increase of resistance with pressure is just as characteristic of liquids as it is of solids. In particular, for cesium, extrapolation gives a large measure of confidence that—although the measurements could not be carried out completely—the resistance of liquid cesium passes through a minimum at approximately the same pressure as the resistance of solid cesium. Lithium in the liquid state has a positive coefficient of change of resistance with pressure, and the curve of resistance as a function of pressure is convex with respect to the pressure axis, as in solids. Liquid bismuth, on the other hand, has a negative coefficient, although for solid bismuth the coefficient is also positive. It is possible therefore that there is a special mechanism responsible for the positive coefficient at low pressures, in addition to the general mechanism responsible for the limiting case of a positive coefficient for all metals at exceptionally high pressures.

The question of the influence of pressure on the resistance of nonmetallic substances is also very interesting. It is assumed that a nonmetallic conductor, or semiconductor, differs from a metallic conductor in that in certain semiconductors the allowed energy bands are separated by wider intervals than in metals; one may perhaps expect that under the influence of pressure these energy bands will draw closer together, i.e. will give the semiconductor the character of a conductor. The few semiconductors that have been investigated have shown enormous negative coefficients of resistance, so that at high pressures their resistances do indeed approach the resistance of metals. Tellurium is usually only partly classified among metallic substances. Under a pressure of \(20\,000\ \mathrm{kg/cm^2}\) its resistance falls to one percent of the initial value. The curve of \(\lg R\) with respect to the pressure axis is convex, so that one may suppose the existence of a minimum of resistance at considerably higher pressures. Black phosphorus, of course, much more closely resembles metals than yellow or red phosphorus, since it conducts electricity appreciably—whereas yellow and red phosphorus are excellent insulators—but nevertheless it is significantly less metallic than tellurium. A nonmetallic property of black phosphorus is the negative sign of the temperature coefficient of resistance. The influence of pressure on the resistance of black phosphorus is also very great. At the beginning the \(\lg R\) curve is concave toward the pressure axis, which means an increasingly rapid approach to the metallic state with increasing pressure. This is not unnatural, since at atmospheric pressure black phosphorus is farther from the metallic state than tellurium. Around \(12\,000\ \mathrm{kg/cm^2}\) the curvature of the \(\lg R\) curve

for black phosphorus changes sign, i.e. becomes convex for pressures greater than \(12\,000\ \mathrm{kg/cm^2}\), which means that at high pressures black phosphorus has become, in its properties, a metal. In addition, between \(15\,000\ \mathrm{kg/cm^2}\) and \(20\,000\ \mathrm{kg/cm^2}\) the sign of the temperature coefficient also changes, so that in this respect as well black phosphorus becomes similar to metals under high pressure.

One of the most significant successes achieved by the new theories of conductivity, and one that has greatly contributed to their development, is the question of the resistance of alloys. If the effective mean free path is large, as is the case in the ideas of wave mechanics, then a single atom that does not fit the lattice may have a very noticeable effect on the resistance. This picture is completely (without a single exception) compatible with the following generalization of the experimental material on the resistance of alloys: the initial effect of adding a foreign substance to a pure metal will be a tendency to make the coefficient of change of resistance with pressure more positive (algebraically). The reason for this is purely geometrical: the relative distortion of the lattice near a foreign atom which is, say, somewhat too large, will become greater and greater as the other atoms in the lattice are located closer to one another. But an increase in the distortion of the lattice means an increase in resistance, and an increase in pressure amounts precisely to bringing the atoms closer together. Consequently, pressure in such substances tends to increase the resistance more rapidly than in pure ones, i.e. the coefficient of change of resistance with pressure is shifted in the positive direction.

The phenomena occurring in single crystals of metals with a non-cubic lattice have so far been little treated by theory. Here, however, there are very important phenomena, and it is precisely in this field that the nearest investigations may be expected. Essential, for example, is the question of why the resistance of a crystal is almost always greatest in the direction of the maximum interatomic distance, i.e. in the direction of maximum compressibility. There are also other characteristic features; for example, the effect of high pressure on the resistance of the three “normal” non-cubic metals—zinc, cadmium, and tin—amounts to a tendency to make the crystal more isotropic. The effect is the same also for the spatial arrangement of atoms. For the two anomalous metals—bismuth and antimony—the effect is the opposite: here too pressure tends to make the lattice spatially identical in the different directions, but at the same time it increases the anisotropy of the resistance.

Usually theory gains much of value from the consideration of anomalous cases, which play almost the same role as pathological cases in medicine. We have already pointed out that in bismuth and antimony, apparently, the phenomena caused by pressure are due to a different mechanism than in normal metals, such as calcium and

lithium. Recently I discovered yet another highly anomalous metal in a place in the periodic system where few could have expected deviations from the norm—chromium. The resistance of chromium at atmospheric pressure is an anomalous function of temperature: there is a resistance minimum at \(10^\circ\) and a maximum at \(0^\circ\) C; nevertheless the form of the curve strongly resembles the form of the curve giving the volume of water as a function of temperature. The resistance is also a noticeably anomalous function of pressure: the coefficient is negative throughout, but the curvature of the resistance isotherm as a function of pressure may have either sign; on the isotherm there may also be a point of inflection, i.e., a change of sign. The resistance isotherms intersect and diverge in a very peculiar way, which implies both positive and negative temperature coefficients at constant pressure. However, it is not only the resistance that is anomalous, but also the relation between volume and pressure. There is no clear correlation between these anomalies of resistance and of volume.

Another anomalous metal is bismuth, but it shows deviations only when chemically pure and only in the form of a single crystal. These deviations are associated with the direction perpendicular to the principal crystallographic axis. Apparently these anomalies reduce to the fact that there are separate pressure regions in which the relation between pressure and resistance is linear, but the coefficients for different regions are different. In passing from one region to a neighboring one, the resistance undergoes no discontinuities. Here, for resistance, we have a kind of second-order transition. In addition, other rather unusual properties are observed in single crystals of bismuth.*

XII. Thermoelectric Phenomena

Thermoelectricity constitutes a special domain of electrical phenomena. An exact theory of these phenomena is very complicated. Sommerfeld’s simplified theory, which treats the electrons as free, gives satisfactory results only for the alkali metals. For other metals discrepancies arise even in sign. A more exact theory considers how the electrons are distributed among the possible energy bands, and obtains different signs for the Thomson effect depending on whether the bands are completely filled or almost empty. So far as I know, no attempts have been made to derive the influence of pressure on the thermoelectromotive force on the basis of this exact theory.

* After this survey had already been written, a very promising theory of the dependence of the resistance of metals on external pressure appeared (N. Frank). This theory (to appear in Phys. Rev.) explains the resistance minimum for the alkali metals.

Houston^20 gave an expression for this dependence on the basis of Sommerfeld’s simplified theory; for copper this theory gives a discrepancy with experiment by a factor of 10. One may expect that calculations by the exact theory will be extremely complicated. This expectation is to a certain extent confirmed by experiment, since the results of the action of pressure on thermoe.m.f. are perhaps the most complicated of all the effects produced by pressure: we have very strong deviations from linearity, curvatures of both signs, inflection points, maxima and minima, intersections of curves, and reversal of sign. As the crudest approximation one may say that, for the majority of the twenty metals tested, the sign of the effect is such that the electrons absorb heat in passing from the compressed to the uncompressed metal. In this respect the electron gas is similar to an ordinary gas, which also absorbs heat on expansion.

However, there are so many exceptions to this general rule that it should not be overvalued. What is important here is the fact that the magnitude of the effect is considerable: the Peltier heat when an electron passes from a metal under a pressure of 12,000 kg/cm² into the same metal, but not under pressure, is on the average of the same order of magnitude as in the case of an electron passing from one metal to another. If one agrees with the assertion of the theory that the thermoe.m.f. depends essentially on the particular way in which the electrons are distributed in the energy bands, then we must conclude that pressures already attainable experimentally can appreciably change this distribution in the energy bands. Therefore, despite the difficulties—or, more precisely, precisely because of them—a detailed theoretical investigation of the dependence of thermoelectric phenomena on pressure may give a deeper understanding of the details of the electron distribution than any other, simpler phenomenon.

XIII. Thermal Conductivity

Thermal conductivity has also been investigated by the new theories, and one of the most important results of Sommerfeld’s simplified theory was the explanation of the Wiedemann–Franz relation, which, as is well known, was a major victory of the classical Drude–Lorentz theory. The classical theory encountered insurmountable difficulties with respect to heat capacity. Sommerfeld’s theory resolved them. The new theories are applicable only to that part of the thermal conductivity which is due to electrons. As far as I know, that small part of heat transfer which is due to atoms has not yet been reflected in the picture of thermal conductivity. Of course, experimentally we cannot separate the two parts in such a way that, in measuring the influence of pressure on thermal conductivity, we measure the influence of pressure on both mechanisms. It turns out that, in passing from metal to metal, there is no uniformity: the thermal conductivities of lead and tin increase with pressure, while those of copper, silver, and nickel—

decrease. Moreover, for all metals except lead and tin, the change in thermal conductivity under pressure is less than the corresponding change in electrical conductivity, so that for all metals, excluding lead and tin, the change in the Wiedemann–Franz ratio with pressure is negative. The experiments here are very difficult. We were unable to measure the most compressible metals, for which the largest effects should be expected, and in practically every case we could not determine exactly how the deviation from linearity occurs. Theoretically, one may perhaps expect a zero coefficient for the change with pressure of the Wiedemann–Franz ratio, if one recalls that this ratio is almost unchanged for all metals and that one should always expect that a metal under high pressure is more similar to the same metal under atmospheric pressure than to another metal. Consequently, the Wiedemann–Franz ratio should change little with pressure. However, it should be remembered that for individual metals the temperature change of this ratio is not quite what theory requires, and pressure undoubtedly affects the characteristic temperature of the metal. Taking into account the insignificance of the effect of pressure on the change in the Wiedemann–Franz ratio for those metals that have been investigated, the experimental results can always be explained by a small change in the ratio of the parts due to electronic and atomic thermal conductivity. This change in the ratio of the parts may occur only in a small region and may change sign at high pressures—such, at any rate, are the indications of experiment. Therefore a complete explanation of the effect of pressure must include in the theory the atomic part of the thermal conductivity. The theory will have to be able to explain why in many cases the atomic part becomes smaller at high pressures. Bearing in mind that thermal conductivity is due to the scattering of elastic waves, a decrease in the atomic part of the thermal conductivity will perhaps indeed be found, since any deviation from complete regularity that leads to scattering will undoubtedly be intensified by high pressure. But it will probably be sufficiently difficult to explain why in most cases pressure decreases the scattering of electronic waves while increasing the scattering of elastic waves.

The influence of pressure on the thermal conductivity of nonmetallic solids or liquids can be measured much more easily; we have far more experimental material, and it is known with much greater accuracy than for metals. The thermal conductivity of organic liquids increases under pressure, and to a first approximation it may be assumed that the increase is the same for all liquids; at \(12\,000\ \mathrm{kg/cm^2}\) the thermal conductivity increases approximately twofold. The curve of the dependence of conductivity on pressure is concave toward the pressure axis, so that the effect of increasing pressure decreases at high pressures, as is to be expected. Experiment has shown that there is a rather close connection between the effect of pressure on the speed of sound (which can be calculated from data for compres—

...and on the coefficient of thermal conductivity; this relation gives an expression for the thermal conductivity of a liquid:

\[ K = 2\alpha v \delta^{-2}, \]

where \(K\) is the thermal conductivity, \(\alpha\) is the gas constant, equal to \(2.02 \cdot 10^{-16}\), \(v\) is the speed of sound, and \(\delta\) is the mean distance between the centers of molecules, calculated approximately under the assumption that the molecules in the liquid are arranged in cubic cells, as in a simple cubic crystal. This expression can also be obtained in an elementary way, assuming that the mean difference of thermal energy in the direction of the temperature gradient between adjacent molecules, equal to \(2\alpha\delta \dfrac{d\tau}{dx}\) (\(\tau\) is the temperature), is transported in the direction of the gradient with the speed of sound. Jeffreys \(^{22}\) showed that the same result can also be reached by a better route, namely by assuming that the disorder of the structure of a liquid is so great that an elastic wave is completely scattered over the minimum possible distance, i.e. the distance between atomic centers. The derivation shows that this same formula should also be applicable to amorphous solids; experiment has shown that this is true for rubber and some glasses.

The simple expression \(2\alpha v\delta^{-2}\) is applicable with an accuracy up to 10 or 15% to ordinary organic liquids, and also to water, whose thermal conductivity is 4 times greater than that of normal organic liquids. Therefore, strictly speaking, it is even somewhat surprising that the formula does not give the correct value for the influence of pressure on thermal conductivity—it gives an increase in thermal conductivity at \(12\,000\ \mathrm{kg/cm^2}\) on the average by 3–4 times, whereas in reality we have an increase of only 2 times. Any ordering of the arrangement of the molecules of a liquid under the influence of high pressure, by which the distance of complete scattering will increase, will increase the thermal conductivity by a new term and, consequently, will also increase the discrepancy. On the other hand, if under high pressure an atom really acquires a tendency to become a structural unit instead of the molecule, which corresponds to the data obtained from heat capacity, then scattering centers are formed inside the molecule, and we can explain the coefficient of pressure that is too small. Of course, an exact accounting of all these factors is difficult. In any case, it is obvious that an exact theory of the dependence of the thermal conductivity of a liquid on pressure does not yet exist and will not exist until a theory is developed that can explain the simplest changes of volume.

In developing a more exact theory of the thermal conductivity of a liquid, one must bear in mind the following important circumstance: the temperature coefficient of thermal conductivity of almost all liquids changes sign at high pressures; at atmospheric pressure a liquid conducts heat worse (scatters better) at high temperatures, whereas at high pressures a liquid conducts better at high temperatures. The pressure corresponding to the change of sign,

approximately corresponds to the pressure at which the expression

\[ \frac{\partial^2 v}{\partial t^2} \]

changes sign.

The influence of pressure on the thermal conductivity of simple crystals has been investigated only for NaCl. Here there are still many opportunities for experimental work.

XIV. Viscosity of Liquids

We have a very large body of experimental material on the question of the influence of pressure on the viscosity of liquids, predominantly organic ones. The general characteristic of this influence is its extremely large magnitude, considerably exceeding the magnitudes of all other effects caused by pressure. For example, for eugenol the viscosity at a pressure of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) exceeds the viscosity at normal pressure by \(10^7\) times. At high pressures the increase of viscosity is approximately exponential, i.e., the dependence of the logarithm of the viscosity on pressure is almost linear. The end of this straight line, corresponding to low pressures, usually has positive or negative curvature. There is a clearly expressed connection between the influence of pressure and the size of the molecule. For monatomic mercury the increase of viscosity at a pressure of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\) amounts to only \(33\%\), as compared with the \(10^7\) increase for eugenol. There is as yet no theory of viscosity that would correctly express the effects caused by pressure. Elementary theories were proposed which regarded viscosity as a function of volume alone. Within the range of pressures attainable at that time, i.e. pressures of the order of \(3\,000\ \mathrm{kg}/\mathrm{cm}^2\), these theories gave approximately correct answers. However, on going over to pressures of \(12\,000\ \mathrm{kg}/\mathrm{cm}^2\), very noticeable discrepancies were obtained. In Bridgman’s modified theory it is allowed that viscosity may be a function not only of volume alone; nevertheless, the temperature coefficient of viscosity at constant volume obtained from this theory differs from that observed by approximately a factor of 5,000. Perhaps the most successful of all the proposed theories is Andrade’s theory \(^{23}\). It gives the correct temperature coefficient of viscosity at atmospheric pressure and in most cases rather accurately reflects the changes caused by pressure up to several thousand kilograms per \(1\ \mathrm{cm}^2\). At higher pressures, however, the theory definitely does not agree with experiment. Andrade observes that the pressure starting from which his theory becomes incorrect corresponds to the pressure at which \(\frac{\partial^2 v}{\partial t^2}\) changes sign. Evidently, at this pressure some significant changes begin to occur in the structure of the liquid. From a purely qualitative point of view, it seems to me that these effects of high pressure should be understood as the coupling of individual molecules; from this point of view the strong influence of pressure on substances with

large molecules and the fact that viscosity is not a function of volume alone. These cohesion effects require that, in any motion of a mass of liquid, such as occurs, for example, in the measurement of viscosity, the molecules should preserve their individuality, since it is difficult to admit that the forces necessary to tear molecules apart would not be many times greater than the viscous forces, even at the very highest pressures.

The presumed loss by molecules of their individuality at high pressures—a possibility which we considered earlier in connection with other phenomena—is more plausible, since those other phenomena consist in purely volume changes. A more probable form of loss of individuality may, for example, correspond to the appearance of internal vibrations of atoms in molecules, which must inevitably be reflected in an increase of the heat capacity. Motion of this kind may, at high pressures, possess greater relative freedom than at low pressures.

XV. Conditions of rupture

Let us finally consider a problem connected somewhat more remotely with high-pressure phenomena. In the detailed discussions that have taken place in recent years of the so-called structurally sensitive and structurally insensitive properties of crystals, much attention has been paid to the tensile strength of crystals. It is known that the strength observed experimentally is always many times smaller than the calculated one. A discrepancy by a factor of 100 is common. Calculations of strength are made by admitting a law of force. The following method of calculation is always used: the distance between particles is found for which the total cohesive force is maximal, and then it is postulated that rupture will occur when the stretching under the influence of the external force reaches this same value of the distance. Of course, there are complications here; for example, transverse compression and longitudinal stretching are not the same, but the basic argument is not altered by this. The experimental fact that rupture occurs long before the stretching reaches the calculated value is explained by internal irregularities.

Leaving entirely aside the question of whether or not these internal irregularities causing premature rupture exist, it does not seem to me that the true criterion of rupture has been taken in theoretical calculations. I think that the following simple phenomenon, observed at high pressures, makes this assertion absolutely convincing: if an ebonite ring, tightly fitted onto a steel mandrel, is immersed in a liquid and subjected to hydrostatic pressure, then at a pressure of several thousand kilograms the ring will burst, as though it had been torn apart by a conical rod. Indeed, such is the nature of the phenomenon. If there were no steel mandrel at all, the rubber ring would be noticeably compressed owing to the strong compressibility of ebonite; but the steel mandrel, having small compressibility, prevents this

natural compression, so that in essence we have a steel wedge entering the compressed ring. A simple calculation shows that although the ring at the moment of rupture is larger, owing to the presence of the steel mandrel it is nevertheless smaller than it was at the initial moment in the absence of pressure. All the stresses arising in the ring are essentially compressive stresses, and yet an evident rupture takes place. Obviously, under these circumstances the critical distance between particles cannot serve as the criterion of rupture.

It seems to me that a brief consideration shows that the requirement of a maximum of the cohesive forces at the critical extension is simply inappropriate, since such a criterion has been rather uncritically taken from the principle of action of certain types of testing machines. One may admit that the maximum force is a necessary condition for rupture, but not a sufficient one. It seems to me much more appropriate to introduce a condition of stability: rupture will undoubtedly occur if an extension is reached at which the structure has become unstable.

The condition of stability is also a sufficient condition, but apparently it is close to a necessary condition as well. This was the method used by Born in his article in Handbuch d. Phys., pp. 768—769. But this is the only published work in which the generally accepted criterion of rupture is called into question. Of course, conditions of stability are much more difficult to formulate; the usual theory of ionic crystals has still not satisfactorily dealt with this fundamental circumstance and can only indicate that the lattice is stable for certain special types of deformation. Therefore it will probably not be easy to derive exactly the rupture conditions required by the conditions of stability. But qualitatively, it seems to me, it is easy to see that the critical deformation required by this condition will be smaller than for the condition of maximum force, since the latter is also a condition of stability, but for the deformation of hydrostatic tension, which is a very unstable form of tension in general. Undoubtedly, the condition of stability for hydrostatic tension is only an upper limit. In view of the extraordinary complexity of the general conditions of stability, I do not think that a rigorous application of these conditions would yield any simple condition of rupture, such as the conditions of maximum extension, or maximum tensile strength, or maximum shearing force that are commonly used in practice. My supposition is confirmed by experience—one can cite types of rupture at high pressure showing that none of these conditions can be valid in the general case. Further details may be found in The Physics of High Pressures.

XVI. Conclusion

In conclusion, we shall permit ourselves to express several assumptions about those new phenomena which we may rightly expect under

pressures many times greater than those already achieved. There is no natural limit to pressure; there is likewise no limit to the amount of energy that can be imparted to matter by compression; in some stars there exist utterly staggering pressures of the order of billions of atmospheres, and we also know that sometimes, at such pressures, matter condenses to densities of the order of 100,000. Thus we have sufficient scope for speculation.

First of all I must make one remark, or rather a correction. Several years ago I published discussions on this same subject in two papers.^24 Several arguments in the first paper were based on experimental data on the compressibility of alkali metals, and in particular on the abnormal constancy of the compressibility of potassium at high pressures.

Recently I discovered that, in converting linear compressibility into volume compressibility, a considerable error had occurred; after correction all the anomalies disappeared, and therefore part of the reasoning in the first paper became invalid. The corrected data will soon be published, together with an extension of the pressure range from \(12\,000\ \mathrm{kg/cm^2}\) to \(20\,000\ \mathrm{kg/cm^2}\). A substantial part of the reasoning in those papers was the supposition that a solid body can, under the influence of high pressure, disintegrate into a gas consisting of nuclei and electrons. This argument was justified both thermodynamically and by means of Schottky’s theorem. At that time, however, we did not yet know the Pauli exclusion principle. Now it appears that such a disintegration is not entirely compatible with the exclusion principle, since it is not clearly apparent how an increase in pressure could so increase the number of energy levels near \(\frac{1}{2}kT\) that all the electrons would find places with such an average energy. Therefore the idea of disintegration into an ideal electron gas will probably have to be abandoned after all, although it is quite legitimate to ask oneself whether quantum conditions are necessarily valid under such exceptional circumstances.

In some parts of this review hints have already appeared of the probable behavior of matter at very high pressures. Thus, for example, it was suggested that at high pressure the smaller units of structure will begin to play a more significant role. The compressibility of isomers and the change in heat capacity with pressure indicate that in liquids under high pressure an atom begins in part to play the role that at low pressures belongs to a molecule; in metals, the course of thermal expansion and the resulting argument concerning entropy at high pressures compel us to transfer to the electrons some part of the functions of atoms. In semiconductors the enormous change in electrical conductivity under pressure undoubtedly signifies the liberation of electrons, which is in essence a similar fact. On the other hand, the probable ultimate increase in the resistance of metals means a stronger binding of the electrons.

The irreversible transition of yellow phosphorus into black phosphorus makes one expect other analogous transitions, if only the pressure is raised sufficiently. Can matter with a density of 100,000 exist at atmospheric pressure, if we were able, by exceptionally high pressure, to realize such a state? One of the properties of such a state is indicated by the uncertainty principle, which asserts that matter cannot, under such conditions, exist in the form of a regular spatial lattice. The reason for this is quite similar to the impossibility for hydrogen of existing in the form of a lattice of the NaCl type, consisting of electrons and protons. The mass of the electron is so small that the uncertainty principle does not permit the uncertainty in position that is required by a lattice with a density equal to that of solid hydrogen.

With increasing mass the uncertainty in position also increases, and ordinary atoms can occupy positions in spatial lattices of ordinary densities; but at a very high density even the mass of an ordinary atom may already impose limitations. Let us consider, for example, sodium, and suppose that it exists in the form of a cubic lattice of density 100,000. Then the distance between atomic centers will be \(7.3 \cdot 10^{-10}\) cm. Using the classical expression for the energy of thermal motion

\[ \frac{mv^2}{2} = \frac{3}{2} kT, \]

we obtain from the uncertainty principle

\[ \Delta l \simeq \frac{h}{(3kmT)^{1/2}} \simeq 3 \cdot 10^{-9}\ \text{cm} \]

for normal temperatures, i.e., a value 4 times greater than the interatomic distance.

If, instead of the classical expression for the energy, one adopts Sommerfeld’s expression, i.e., assumes that at such high pressures, owing to the enormous restoring forces and the high characteristic temperature, the atoms are entirely in the state of a degenerate Fermi gas with an upper limit of kinetic energy given by the formula

\[ V_0 = \frac{3}{10} \cdot \frac{h^2}{m} \left( \frac{3n}{4\pi} \right)^{\frac{2}{3}}, \]

then it is easy to calculate the corresponding uncertainty in position required by chaotic motion of such intensity. We obtain a value 2.1 times greater than the interatomic distance. Consequently, it is meaningless to try to arrange the atoms in a lattice. Probably, at such pressures the substance will exist in the form of a more or less amorphous jelly. Matter at such limiting densities must exist in a new state, as different from the ordinary states as, for example, an ordinary gas differs from Krukov’s “fourth state of matter.”

The value of \(\Delta l\) given by the uncertainty principle decreases with increasing temperature. Consequently, denser states are characterized by lower temperatures. Thus one part of my assumptions is sufficiently plausible, namely:

the pressure–temperature plane is diagonally crossed by a band rising from low pressures and low temperatures to high pressures and temperatures. Within this band the substance exists in states known to us; toward high temperatures it dissociates into a gas consisting of electrons and nuclei, as was first suggested by Saha; toward high pressures it turns into a jelly. Whether quantum principles can be applied in their present form to such a jelly will be possible to determine only by knowing how far they are applicable to the internal parts of nuclei. It must also be allowed as possible that this jelly consists of neutrons, electrons, and protons, compelled by the enormous pressure to form pairs. Such a system will be an insulator and in this respect will complete the tendency observed in metals at the extreme pressures accessible to experiment.

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2. In the first approximation the atomic nuclei are regarded as at rest; the second approximation allows elastic vibrations of the atoms independently of the electrons; the third approximation considers electrical resistance as the result of the interaction of atomic vibrations and the motion of electrons. See Bethe, Handbuch der Physik, vol. XXIV, 2, 2nd ed., p. 369. ↩