ON THE NATURE OF METALS IN CONNECTION WITH THE STUDY OF THEIR PROPERTIES AT HIGH PRESSURES

P. W. Bridgman

For most investigations in the field of science it is characteristic that they are not satisfied merely with the discovery of new facts, however curious and unexpected these facts may be in themselves. Alongside the accumulation of factual data there exists a constant striving to explain them from a general point of view. The scientist’s interest, and the satisfaction he derives from discovering a number of new phenomena, increase enormously if he at the same time has a theory or hypothesis that leads him and enables him to foresee what kind of new experimental discoveries may follow. Conversely, the theoretical physicist’s interest in a theory rises greatly if, together with its development, there comes the discovery of new experimental facts that confirm or modify the theory as it develops.

Of course, in the history of physics there have been long and tedious periods when experimenters merely accumulated new data—true, data of paramount importance—without making any attempt to explain them. For fifty years spectroscopy experienced such a dull and dry history; but how wearying this was is clearly seen from the sigh of relief that greeted N. Bohr’s first successes in his endeavor to bring understanding into these facts, and from the onslaught made by experimenters with fresh forces upon this field as soon as the possibility of theoretical explanations was pointed out to them.

Investigations in the field of high pressures, with which I have been occupied for many years1, until recently were in almost the same position as spectroscopy before the appearance of N. Bohr. A large quantity of varied data had been collected for very many solid and liquid substances. There can be no question as to the importance of these data, since the change of dimensions caused by hydrostatic pressure is, perhaps, the simplest possible kind of change that can be experi—

to feel the substance. But, with the exception of a few problems, there was no theoretical illumination of the facts here, so that the fascination which sustained me was, to a considerable extent, the fascination of a collector. However, in the last few years the wave mechanics of solids has developed so rapidly that in many cases it proves possible to calculate and even to predict the effect of pressure for a whole series of simple phenomena, and this possibility has greatly increased my interest in experimental work. At the same time my experimental work entered a new phase owing to the fact that technical improvements made it possible to obtain pressures of \(50\,000\ \mathrm{kg}/\mathrm{cm}^{2}\) (several times greater than before) and to measure certain simple properties in this range of pressures[^2].

In this range of pressures there is a remarkable deviation from the linear dependence of certain physical properties, so that a new task for theory is to reproduce the complete curve instead of the straight-line segments representing the results of experiments in a more limited interval of pressures. Theory has proved capable of solving this problem in the simplest cases. I shall try first of all to give, for metals, a summary of the new experimental data that I have been able to obtain in the region of high pressures, and then to show how these data fit into the theoretical constructions of wave mechanics. We cannot make too great demands on a theory which is still at an early stage of development, but even now I believe that our theoretical concepts are incomparably fuller and broader than we dared hope 15 years ago.

EXPERIMENTAL TECHNIQUE

First of all I shall briefly describe the technique which has made it possible at the present time to extend the pressure interval so greatly. It is well known that it is impossible to increase the resistance of a hollow cylinder to internal pressure above a certain value, even if its walls are made infinitely thick. The reason is that, as the thickness increases, each new layer bears only a rapidly diminishing part of the total stress, which is concentrated near the inner surface of the wall. Even in an infinitely thick cylinder there finally arises a plastic flow, initially at the inner surface of the cylinder, when the internal pressure reaches a value of the order of the tensile yield limit; these plastic conditions rapidly spread through the entire thickness of the cylinder, as a result of which rupture occurs. This, consequently, is the first natural limit for the strength of a vessel with respect to internal pressure. This limit could be raised if the stress could be redistributed within the walls so that parts of the wall farther from the center could take on a more significant share of the stress. As is well known, this can be achieved by excit—

METALS AND THE STUDY OF THEIR PROPERTIES AT HIGH PRESSURES

...the application of an initial compressive stress inside the cylinder by means of external hoops, as in artillery guns, or in some other way. Thanks to these devices, the inner layers of the walls at first contribute negative components to the stresses, so that the internal pressure arises in parts of the walls somewhat removed from the center. As the pressure increases, the principal zone sustaining it proves to be farther from the center, so that at maximum pressure the stress distribution becomes more uniform, and the outer layers carry a large share of the load. The limit of the increase in strength achieved in this way is restricted by the occurrence of plastic flow at the inner surface under compression, because, if the hoops are pulled too tight, the stress at the inner surface reaches the yield point, and the cylinder is destroyed by a plastic flow directed toward the center before any working pressure has been applied. This establishes the second natural limit of the strength of a cylinder with respect to internal pressure. It is approximately twice the first, or about twice the limiting stress of plastic flow in tension. Until recently, the upper limit of the pressure I used was determined by this second limit. Measurements thus limited could be made, with great difficulties and inaccuracies, up to 20,000 kg/cm²; but most of my measurements were made at pressures up to 12,000 kg/cm², at which there is no danger of destroying the apparatus.

We could, obviously, obtain still higher pressures if, instead of producing within the cylinder a maximum compressive stress at the lowest working pressures, when it is least needed, we could neutralize the tension of the inner surface arising from the pressure by some compressive stress increasing in magnitude as the working pressure increases. This can be accomplished by applying to the outer surface of the cylinder a pressure increasing proportionally with the increase in internal pressure. This is the method I now use. It is clear that it introduces considerable complications into the apparatus, but it is worthwhile, since in this way it has proved possible to raise the limit of working pressure from 20,000 to 50,000 kg/cm².

External pressure on the vessel can be applied easily if the outer shell is given a conical form, similar to the form of a rubber stopper, and is pressed into an identical conical ring with a force increasing together with the internal working pressure. The simplest way of achieving this consists in making the force that moves the piston producing the internal pressure simultaneously press the cone into the outer supporting ring, so that the external pressure automatically increases in the same measure as the internal. This is shown schematically in Fig. 1. With such an apparatus many measurements were made up to 50,000 kg/cm². However

*

with such a method one has to apply pressure [[unclear: phrase]] which depends on the size of the vessel. These limitations, arising from the magnitudes of the force required for a given pressure in the largest volumes, made it necessary to change the method of applying the force to the reverse one and to move the conical vessel \(A\) in relation to the stationary holder, independently of the direction in which the piston is moved. The apparatus, built on this principle, is shown in Fig. 2.

Fig. 1. Schematic diagram of an apparatus for automatically producing pressure inside and outside a cylinder simultaneously.

Fig. 2. Diagram of an installation for obtaining pressure outside and inside the cylinder independently of one another. The pressure inside vessel \(A\) is produced by piston \(B\), set in motion by hydraulic press \(P_1\). The pressure on the external surface of \(A\) is produced by hydraulic press \(P_2\), which forces \(A\) into the massive cone \(C\). The upper and lower presses are each connected by three rods, of which, for simplicity in the drawing, only one is shown.

Up to this point we have discussed the design of a cylinder strong enough to withstand the internal working pressure, and have said nothing about the method of producing the pressure. Fortunately, heat-treated steel is much stronger in compression than in tension, so that up to the second stage of our experiments the pressure can be produced simply by a steel piston (hardened to the hardness of glass). For the third stage, however, the strength of steel in compression is already insufficient, and in order to produce a pressure comparable with the pressure that the vessel is capable of withstanding, we must either develop a special method for strengthening the piston, or make it of a material stronger than steel. Very fortunately, we can make use of the second and simpler solution to the problem. Certain grades of Carboloy, recently manufactured—cemented tungsten carbide and cobalt—have a compressive strength more than twice as high as that of steel.

output, [[unclear]] of the steel piston, and it proved possible to use a comparatively simple piston of Carboloy in order to obtain the highest pressures attained at the present time. I am most grateful in this connection to Dr. Zeo Z. Jeffries of the Westinghouse Electric Company, who generously supplied me with this material.

A pressure of 30,000 kg/cm², obtained in modern installations, cannot be increased significantly as a result of two different effects. This pressure (30,000 kg/cm²) is already very close to the strength limit of a piston made of Carboloy. To obtain a higher pressure it is necessary to seek a material still stronger than Carboloy, or to strengthen the piston.

On the other hand, the cylinder for pressure is also already close to its strength limit because of longitudinal tensile stresses caused by pressure on the curved surface, which I call the “pinch effect.” Indeed, in a whole series of cases such a longitudinal effect is observed—the cylinder splits into two parts along a plane perpendicular to its axis. This can be prevented by a simpler external reinforcement of the cylinder both from the side surface and from the bases, but this will require an even more cumbersome apparatus.

PHENOMENA [[unclear]] AT PRESSURES ABOVE 20,000 kg/cm²

Three phenomena can now be studied in the new pressure region, above 20,000 kg/cm². Two of them are connected with simple measurements of volume. Changes in volume can be easily determined from the position of the piston, and this in turn can be easily measured in an instrument of the special type of construction shown in Fig. 1, for pressures up to 50,000 kg/cm². The simplest of the volume effects is that which is observed in polymorphic transformations from one modification of a solid body into another. In the vast majority of cases such a polymorphic transformation is accompanied by a discontinuous change in volume, so that the presence of such transitions is very easily established by plotting the position of the piston as a function of pressure. The transformation point is determined by the discontinuity on such a curve. The magnitude of the change in volume is given by the size of the jump. The pressure at which this change occurs is, in turn, a function of temperature. Determination of the pressure at the transition point as a function of temperature and the magnitude of the change in volume gives the data necessary for a complete thermodynamic characterization of the transformation.

The second case is the change of the volume itself as a function of pressure, whence the coefficient of compression is immediately obtained. Changes in volume, as before, are determined by the motion of the piston as a function of pressure. In order, however, to make the final calculations, one must introduce certain corrections for the change in the dimensions of the instrument, since the compressibility over the entire pressure interval is quite

not so simple as to determine a polymorphic transformation from a discontinuity break, when most of the corrections vanish. Nevertheless, the corrections can be introduced with sufficient accuracy, and a whole series of investigations of the compressibility of metals and other substances has been carried out at pressures above \(20\,000\ \mathrm{kg}/\mathrm{cm}^{2}\).

The third phenomenon is electrical resistance. The resistance of many metals changes greatly under pressure. From the technical point of view, the measurement of resistance requires a more bulky and more complicated apparatus than the measurement of volume: insulated wires must be introduced into the pressure cylinder; the pressure must be transmitted by means of a non-conducting liquid. Thus the technical requirements for the apparatus become still more serious.

For measuring resistance the second model of the high-pressure apparatus was used. The pressures employed in this case were limited to \(30\,000\ \mathrm{kg}/\mathrm{cm}^{2}\). With the aid of this apparatus measurements were made of the resistance of 19 metals at two different temperatures.

1. Polymorphic transformations under pressure

The most interesting results are the polymorphic transformations under pressure. It is well known that many metals undergo transformations at atmospheric pressure when the temperature is raised, and these phenomena are very important in practical respects. If pressure is added to temperature as a second variable, the phenomena become still richer and more varied. Many new modifications become possible, and they are never encountered at atmospheric pressures under any temperature conditions. Unfortunately, if some of these new modifications have useful properties, it proves impossible to make use of them for practical application, since these transformations are reversible, and the metal returns to its normal state when the pressure is removed. In Fig. 3 are given equilibrium phase diagrams for nine metals, for which the characteristics of the transformation were obtained by the method of volume measurements. All these metals have comparatively low melting points, or at least all of them are soft from the mechanical point of view. There are various reasons why precisely these metals were investigated and why diagrams were obtained precisely for them.

Almost always there exists a considerable resistance of internal friction during the transition from one modification to another, and this frictional resistance may be so great that the transformation does not occur even if the pressure is significantly higher than that at which, according to thermodynamics, the new phase becomes capable of replacing the old one. The viscous resistance is the smaller, the higher the corresponding temperature of the metal, i.e. the closer it is to its melting point. It may therefore be expected that there exist many other examples of polymorphism under pressure, in addition to those depicted

on the diagram, namely for harder and more refractory metals. This is apparently valid, because in tests of metals in shear (and we may note here that shear stress overcomes internal friction much more easily than hydrostatic pressure does), clear indications are obtained that the more refractory metals as well, such as lanthanum, cerium, thorium, and vanadium, transform into other modifications under high pressure.

Fig. 3. Equilibrium diagrams of nine metals. The temperature axis is shifted for mercury and gallium.

Let us now consider the diagrams of several individual metals presented in Fig. 3. The two most interesting examples of polymorphism under pressure are bismuth and gallium. Both of these metals melt at atmospheric pressure with a decrease in volume, which is a highly anomalous fact; another well-known example is the transition from ice to water. At present it is already known that the anomalous relation of the volumes of liquid and solid water is more or less temporary and accidental. It is characteristic only of low pressures, since at pressures above 2000 kg/cm² ordinary ice is replaced by a series of more stable forms having a greater density than the corresponding liquid phase. By analogy, one may expect the same in the case of bismuth and gallium, but until recently all experimental searches for the expected new modifications had been unsuccessful. Apparently, the reason for the failure was that the pressure was not sufficiently high. At pressures above 13,000 kg/cm² for gallium and 25,000 kg/cm² for bismuth, the presumed new modifications have recently been discovered, so that at sufficiently high pressures

all substances reveal normal relations between the volumes of the solid and liquid phases. Bismuth forms at least three new modifications in the region of high pressures, and, what is very curious, its equilibrium diagram resembles the diagram for water.

Three alkaline-earth metals—calcium, strontium, and barium—all exhibit polymorphism under pressure, and their equilibrium diagrams have a certain similarity among themselves. A more detailed consideration of them shows, however, that this similarity does not extend as far as seems at first glance, because the structure of the normal modification of barium is cubic, whereas the lattice of calcium and strontium is cubic with centered faces. Since a cubic structure with centered faces is a close packing, the supposition suggests itself that barium under pressure should tend to acquire this structure, which calcium and strontium already possess at atmospheric pressure. The modifications of strontium and calcium at high pressures must therefore be something else. It is possible that in these cases there occurs a redistribution of the electrons in the atom to levels below the valence level, since the atomic packing in them is already close. In any case, it is obvious that the transformation under pressure in barium cannot have the same character as in the case of strontium and calcium.

The greatest interest for theoretical calculations, among all the indicated cases of phase transformations, should be presented by cesium, since this metal has a sufficiently simple structure for it to be possible to carry out calculations, which are insurmountably difficult in the case of most other metals, and to bring them to completion. In fact, Dr. J. Bardeen (John Bardeen) at Harvard succeeded in establishing by calculations not only that cesium should undergo a transformation under pressure, but also the approximate magnitude of the pressure at which the transformation occurs, as well as the magnitude of the change in volume. It is especially remarkable that J. Bardeen carried out his calculations earlier than the actual existence of the transformation was definitively established experimentally, and his calculations served as a stimulus for further experimental investigations. The elementary cell of the modification of cesium at atmospheric pressure is a centered cube. Calculations show that at high pressures it should transform into a cubic structure with centered faces, i.e. into a close-packed one, which indeed might have been expected in advance. However, generally speaking, the behavior of cesium is rather unusual. For many other substances the form stable at low pressure is the one that already has close packing, i.e. represents a close-packed arrangement of spheres.

It might have been supposed that the immediate task of a theoretical study of any crystalline substance is to determine, for each pressure and temperature at which it crystallizes, and, for example, after establishing the system, to determine those

of its physical properties, which may be found by calculation. If this were so, then predicting the magnitude of the pressure at which one may expect the appearance of new modifications would be one of the easiest theoretical problems; but it turns out that the theoretical prediction of the crystallographic system, at the present level of development of the theory, is one of the most difficult problems. At present the crystallographic system is usually assumed to be known from experimental data, and then, knowing the system, it is already much easier to calculate other properties of the substance, such as compressibility or electrical resistance. The reason for this is that differences in energies and, consequently, differences in thermodynamic potentials are usually very small in comparison with the total energy or thermodynamic potential, so that it is necessary to know the total energy with the highest degree of accuracy in order to determine its differences and, consequently, also the parameters of transformations even with moderate accuracy. With the exception of cesium, I do not know of a single successful calculation carried out for a metal. Even in the simplest of possible cases, in the case of transformations in the series of ionic compounds from NaCl to the CsCl-type structure, the calculations proved too difficult to yield good values for the pressure at which the transformation occurs. The prevailing forces in ionic compounds are the forces of simple electrostatic interaction according to the inverse-square law of the distances between charged particles; repulsive forces are superimposed on these when the ions approach one another too closely. These repulsive forces can be approximately calculated from phenomena of another order, but it turns out that the assumed magnitudes of the repulsive forces that make it possible to calculate, for example, the compressibility do not satisfy the calculations of the transformation parameters. Thus, in the August 1937 issue of Physical Review, in an article by May, it is indicated that, in order to obtain satisfactory values for the transformation temperatures of NH₄Cl and CsCl at atmospheric pressure, it is necessary to introduce into the theoretically calculated magnitudes of the forces arbitrary factors—3.5 in one case and 0.6 in the other. The phenomena of polymorphic transformations thus prove, from the theoretical point of view, to be among the most subtle and complex. This is confirmed on the experimental side by the fact that no other phenomenon reveals so many qualitative differences for substances that are chemically close to one another and have a great similarity in other properties.

2. Compression of Volume

Let us now consider the second group of phenomena, namely, simple compression of volume. From the experimental point of view, of greatest interest is the very magnitude of the change in volume that can be achieved with the high pressures obtained today. At ordinary pressures, changes in the volume of solid bodies can be...

neglected in most practical cases, but at high pressures they become very appreciable. Indeed, in the pressure range in which I worked earlier, up to \(12\,000\ \mathrm{kg/cm^2}\), the changes in volume were considerably greater than upon cooling from room temperature to absolute zero at atmospheric pressure. Of course, at \(50\,000\ \mathrm{kg/cm^2}\) the changes should increase still further, but not in proportion to the pressure, because if the volume decreased with increasing pressure at a constant rate, it would already have had to reach zero and become negative. Thus, for example, if the compressibility of cesium remained constantly the same as at the beginning of compression, it would, so to speak, have squeezed itself out already at \(14\,000\ \mathrm{kg/cm^2}\). It is obvious that, in the general case, the volume curve as a function of pressure at high pressures must be convex toward the pressure axis, and it may be expected, at least for the most compressible substances, that its deviation from straightness must be very considerable. A theory satisfying these requirements must be able to reproduce not only the initial segment of the volume—pressure curve, but also the entire course of the volume curve as a function of pressure.

Experimental determinations of volume compression at \(50\,000\ \mathrm{kg/cm^2}\) present certain difficulties owing to the various corrections for distortion of the instruments mentioned above, so that up to the present sufficiently satisfactory measurements can still be made only for strongly compressible substances. At present I have results of measurements for the 11 most compressible metals and for three other substances. Table 1 gives the volumes of these metals as functions of pressure. The relative accuracy for the more compressible metals is, naturally, higher than for the less compressible ones.

TABLE 1

Compressibility of 11 low-melting metals

* In this interval polymorphic transformations occur.

These 11 metals include five alkali metals, which are at present of greatest interest to theoretical physicists, and three alkaline-earth metals, namely calcium, strontium, and barium, which also possess comparatively high compressibility. The alkali metals are the most strongly compressed of all metals in general. Their compressibility increases with increasing atomic weight, reaching a maximum for cesium, which at a pressure of \(45\,000\ \text{kg}/\text{cm}^2\) is compressed to more than twice less than its initial volume. From this same table one can get an idea of the deviation from a linear dependence, since it shows that the compression at \(40\,000\ \text{kg}/\text{cm}^2\) is greater than the compression at \(20\,000\ \text{kg}/\text{cm}^2\) not by a factor of two, but by considerably less. Among the alkali metals the greatest deviation from proportionality is observed for cesium, and the smallest for lithium. This proves to be a general rule, namely: the decrease in compressibility with increasing pressure is greatest for substances with the greatest absolute compressibility.

In the series of alkaline-earth metals the same tendency is observed—the compressibility increases with increasing atomic weight, and the relative decrease of compressibility with pressure also increases. It is necessary to note that the changes in volume of cesium and barium given in Table 1 also include a polymorphic transformation. In both cases the volume change at the transformation is very small, and passage through the transformation point has little effect on the compressibility. In both of these cases the change in the form of the lattice is, apparently, an insignificant episode, insofar as other properties are concerned. On the other hand, this follows from the fact that the difference in energy of the modification of cesium at low pressures and at high pressures amounts to only \(1/120\) of the energy of the cesium bonds.

Theoretical calculations of compressibility were undertaken with varying degrees of success several years ago. The first successful calculations were made for nonmetallic lattices of the NaCl type. A number of simpler and more essential properties of these lattices can be calculated on the basis of the simple assumption concerning the law of action of the force between ions situated at the vertices of the lattice. These forces are simple electrostatic forces acting according to the inverse-square law of the distance between positive and negative ions, with superposed repulsive forces that increase rapidly when the centers of the ions approach more closely than a certain critical distance, corresponding approximately to the ionic radii. In calculations made earlier, the repulsive forces were assumed to depend on distance in some power that was determined empirically. For most simple ionic lattices they are taken to be inversely proportional to the ninth power of the distance.

Modern wave mechanics introduces these forces in the form of an exponential term. With the aid of this simple expression one can calculate certain most important properties of the lattice, for example its

energy and, consequently, the compressibility. In this way, however, only the initial compressibility is obtained, and the results of the calculations do not at all agree with experiment in computations of the dependence of compressibility on pressure, which also indicates the deviation of the volume–pressure curve from straightness. It is necessary, therefore, to make the theoretical calculations more detailed in order to obtain the volumes of simple lattices as functions of pressure.

Passing to the calculation of the compressibility of metals, it must be said that the nature of the forces and the character of the structure are here very substantially changed, and that other methods of calculation must be used, taking the more detailed exact structure into account. The basic idea is not changed, namely: one seeks to determine the lattice energy as a function of the period. If the lattice period in the equilibrium state at atmospheric pressure and its energy at other values of the period are known, then the difference of energies between the equilibrium state and any other is also known. From this one directly obtains the magnitude of the pressure required to raise the lattice energy upon compressing it from the initial to the final distances between atoms, and then also the volume as a function of pressure. At ordinary temperatures these calculations must be corrected for various temperature effects. These corrections are not easy to introduce, so that usually one has to be satisfied with calculating the compressibility at absolute zero, when all corrections vanish. Strictly speaking, this is already good enough, since experiment shows that the compressibility does not depend strongly on temperature.

Numerical values of the energy as a function of volume have up to the present time been obtained only for a small number of simple cases. It is easy to see that a rigorous mathematical solution of the problem presents insurmountable difficulties. It is known that in classical mechanics the problem of three bodies moving under the action of mutual forces of attraction could not be completely solved; here, however, we are dealing with the problem of \(10^{22}\) atomic nuclei per unit volume, each nucleus being accompanied by from 2 to 92 electrons. Obviously, the only hope lies in approximate calculations. With respect to the nuclei it is assumed that they sit immovably at the vertices of the lattice; in practice this means that the calculations are carried out for absolute zero. To the electrons, however, the equations of wave mechanics are applied. These equations include the forces of interaction with the nuclei; they are calculated by the methods of classical mechanics with respect to the positions of the centers of the nuclei. Thus the nuclei are not treated at all by wave mechanics; this is justified by the fact that they are many times heavier than the electrons. An exact solution of the wave equation even for electrons alone is also impossible. Strictly speaking, all the electrons must be considered at once, since each electron acts on all the others. Fortunately, this interaction is on the average very weak for specimens of matter of ordinary dimensions. This is evident from the fact that the specific properties of metals, for example, proceed-

ity or specific resistance do not change when the measurements are made on a specimen of \(2\) or \(1\ \mathrm{cm}^3\). Consequently, it becomes possible to assign to the vast multitude of electrons certain more or less fixed positions around individual nuclei. The aggregate of these positions constitutes the inner shell of the atom. Generally speaking, the interaction of these inner shells may be neglected, except in those cases when the atoms approach one another so closely that repulsive forces arise between them, preventing the mutual penetration of the shells. This, obviously, makes possible enormous simplifications in the calculations.

The electrons that remain to be taken into account after all these approximations are the outer, or so-called valence, electrons. Calculations for them can be somewhat simplified, but with the aid of approximations opposite to those admitted for the inner shells. The periodicity of the structure of the entire lattice enters in an essential way into the solution of the wave equation for these electrons. These electrons are the property of the whole lattice as a whole and cannot be associated with any particular atom. As might be expected, the practical difficulties in solving this problem depend on the number of electrons. It has proved possible to bring the solution to a satisfactory conclusion for the simplest cases of the alkali metals, which have only one valence electron and in which the inner part of the atom is so small that the interaction of the inner parts of neighboring atoms may be neglected. For other metals with one valence electron—copper, gold, and silver—the calculations can also be carried out with a certain degree of success, but here the difficulties increase considerably because of the interaction of the closed inner shells.

If one takes atoms with two outer electrons, the difficulties increase still more, and only very few properties of such metals have been calculated sufficiently satisfactorily. We probably lose a great deal from the fact that theoretical capabilities are not yet at such a level as to make satisfactory calculations available for metals of technical and industrial importance.

The solution of the wave equation for the outer electrons in the case of the alkali metals corresponds to an almost uniform distribution of electrons in the lattice, with the possible exception of positions in the immediate vicinity of the nucleus. Consequently, in a first approximation an alkali metal may be regarded as an aggregate of positive point charges at the vertices of the lattice, immersed in a homogeneous sea of negative electricity. The electrons are distributed, owing to their mutual repulsion, in such a way that on the average they fill all space uniformly. This means that, on the average, the immediate neighbor of each nucleus at any moment of time is only one electron. From this is derived an approximate method of solution, first proposed by Wigner and Seitz. If one imagines in spatial

in the lattice the straight lines drawn from each nucleus to its nearest neighbors in all directions, and the planes perpendicular to these lines at their midpoints; the entire lattice is thereby divided into polyhedra. The approximation consists in regarding each polyhedron as containing a single positive charge at its center, surrounded by a homogeneous volume charge of negative electricity of such a density that the total negative charge of the polyhedron is also equal to unity. The whole polyhedron is thus electrically neutral and, consequently, produces outside itself only a very weak field. It follows from this that the energy of the system may be calculated with good approximation, neglecting the interaction of the polyhedra, simply by summing the energies of the individual polyhedra. The energy of each of them is composed of the potential energy of the electron cloud plus the kinetic energy of the electrons. In calculating these energies, a further simplification consists in replacing the polyhedron by a sphere of equal volume. When the volume of the lattice decreases under pressure, both terms of the energy also change. The potential energy, in turn, consists of two parts: the energy of the negative cloud and of the nucleus, and the mutual energy of the parts of the cloud itself. Both parts, as is easy to see, vary inversely as the first power of the linear dimensions, i.e. as \(V^{-1/3}\), on the assumption that, when the lattice is compressed, the electron cloud is compressed in the same ratio.

The character of the dependence of the kinetic energy on the dimensions of the lattice can be derived from the fundamental relation connecting the equivalent wavelength of a free electron with its momentum,

\[ p = \frac{h}{\lambda}. \]

Since the momentum varies inversely in proportion to the change in wavelength, it follows that the kinetic energy varies inversely in proportion to the square of the wavelength.

For a piece of metal, the solution of the wave equation for the electrons is obtained in the form of a series of standing waves, the wavelengths of which are determined by the requirement that they fit exactly into the dimensions of the piece. If the piece changes its dimensions under pressure, or in some other way, the length of each wave must change proportionally in order to satisfy this requirement. This means that the momentum of the electron increases proportionally to the square of the decrease in the linear dimensions, or proportionally to \(V^{-2/3}\). The signs of these two terms, \(V^{-1/3}\) and \(V^{-2/3}\), are opposite, the potential energy being negative and the kinetic energy positive. At large distances from the nucleus one term predominates, at small distances the other. Their sum gives an energy curve passing through a minimum. The position of the minimum determines the lattice period, and the change of the energy near the minimum determines the compressibility. It turns out that quite good values of the lattice constant and initial

compressibility can be obtained in the case of the alkali metals by means of this simple scheme. However, for other metals with one valence electron (copper, silver, and gold) the agreement is by no means so good. The explanation lies in the fact that the inner part of the ion of these metals is of much greater size than in the alkali metals, so that here there arise mutual repulsive actions due to the relative impenetrability of the ions. These forces must also act in the case of the alkali metals, but to a lesser degree. Calculations show that the repulsive forces must vary inversely as the volume. Adding all three terms, we can then write the expression for the energy of the metal in the following form:

\[ W=\frac{a}{V}+\frac{b}{V^{-2/3}}-\frac{c}{V^{-1/3}}. \]

Using a formula of this type, J. Bardeen calculated the volumes of five alkali metals with satisfactory approximation within the limits of experimental error. In Fig. 4 the theoretical

Fig. 4. Comparison of experimental values of compressibility for the alkali metals with the theoretical values obtained by J. Bardeen.

and experimental curves are presented. It turns out that the most important factor is the increase in the kinetic energy of the electrons when the lattice is compressed. Other metals, besides those shown in Fig. 4, have not, however, so far been investigated sufficiently well.

Electrical Resistance at High Pressures

Let us now turn to the consideration of the third group of phenomena, namely, the electrical resistance of metals at high pressures. As has already been said, the apparatus required for measuring resistance is more complicated than that which was used for studying the first two effects, so that even at the present time it has proved possible to carry out measurements only up to \(30\,000\ \mathrm{kg/cm^2}\). Earlier measurements were made at \(12\,000\ \mathrm{kg/cm^2}\) and, in a few individual cases, almost up to \(20\,000\ \mathrm{kg/cm^2}\). These previous measurements showed that the resistance of most metals decreases under pressure, but in such a way that the rate of decrease diminishes as the pressure increases. There is, however, a fair number of metals whose resistance increases under pressure, and for them, quite unexpectedly, the rate of increase of the resistance grows as the pressure is raised. In addition, three cases were observed in which the resistance passes through a minimum with increasing pressure. In all cases the curve of the dependence of resistance on pressure is convex toward the pressure axis. Generally speaking, the change in resistance is greater for metals with a low melting temperature. Furthermore, the order of magnitude of the change in resistance under pressure is ten times greater than for volume.

In the new region of high pressures, measurements were first made for comparatively refractory metals: copper, silver, gold, and iron. The effects obtained were those that might have been expected—the smooth continuation of results previously obtained for lower pressures. The resistance curve continues to remain convex toward the pressure axis, but the possible minimum lies so far away that it is impossible to extrapolate the curve to it. More fusible, soft metals give more interesting results. A large effect may be expected for the alkali metals, and this effect should lend itself most readily to theoretical treatment. Lithium differs from the other alkali metals in that its resistance under pressure immediately begins to increase. Up to \(30\,000\ \mathrm{kg/cm^2}\) the increase continues at a continuously growing rate; at \(30\,000\ \mathrm{kg/cm^2}\) the resistance proves to be almost \(25\%\) greater than the initial value. The resistance of sodium, on the contrary, decreases rapidly, but with increasing curvature. At \(30\,000\ \mathrm{kg/cm^2}\) it is only about \(0.4\) of the initial value. Potassium is softer and has greater compressibility; therefore one may expect that it will exhibit a large effect. This proved to be true, and the resistance of potassium at \(30\,000\ \mathrm{kg/cm^2}\) becomes less than \(0.2\) of the initial value. But the most interesting feature of potassium is that its resistance passes through a minimum near \(25\,000\ \mathrm{kg/cm^2}\). Indications of this had already been present in earlier experiments, but at that time there was not yet the possibility of reaching

pressure necessary for the phase Tiⁿ⁺ is smaller. The resistance at the end of the first branch, as seen, did not fall, but then passes through a minimum in the region of pressure \(15\,000\ \mathrm{kg/cm^2}\), as was observed earlier.

In the new pressure region the resistance continues to increase, and at an ever increasing rate. Cesium, as was found earlier, has a minimum near \(5\,000\ \mathrm{kg/cm^2}\), and above this pressure its resistance begins to grow rapidly. One of the most interesting facts that could be expected in the region of high pressures is connected with the polymorphic transformation

Fig. 5. Resistance of cesium at two different temperatures as a function of pressure. Breaks in the curves correspond to polymorphic transformations.

of cesium at \(22\,000\ \mathrm{kg/cm^2}\). This new modification has, of course, a smaller volume than the modification stable at low pressures. Therefore, above all, one should have expected a decrease in resistance upon transition to the new modification, since in all known cases of phase transformations the resistance decreases with decreasing volume. On the other hand, one may suppose that the resistance of the new modification should decrease with increasing pressure, because only in certain metals does it increase. Both these assumptions proved to be incorrect. The resistance increases upon transition to the modification existing at high pressures, despite its smaller volume, and the resistance of the new phase continues to increase with increasing pressure. Figure 5 shows the dependence of the resistance of cesium on pressure.

rubidium; the resistance does not change noticeably during this polymorphic transformation, since the curve continues after the transformation region without noticeable changes in its character.

A greater effect could also be expected from the alkaline-earth metals. With regard to calcium and strontium it was already known that their resistance increases with pressure; in the new pressure range it continues to grow at a continuously increasing rate. At \(30\,000\ \mathrm{kg/cm^2}\) the resistance of strontium proves to be almost 3.5 times greater than its initial value; this is the case of the greatest increase in resistance among all those observed. For barium it is known from earlier experiments that its resistance passes through a flat minimum and then begins to increase. In this case, as in the case of cesium, the question of how the new modification existing in the region of high pressures will behave is also of interest. It turns out that at first the resistance decreases slightly upon the transition to this modification, thus obeying the rule of parallelism with volume, the only exception to which so far has been only the cesium considered above. However, the resistance curve continues to rise beyond the transformation point. The transformation effect has little influence on the general character of the resistance curve, as was also the case for cesium. Mercury is likewise a low-melting metal, and for it one may expect a greater effect. The freezing temperature of mercury is raised under a pressure of \(30\,000\ \mathrm{kg/cm^2}\) to \(100^\circ\mathrm{C}\), so that in this region measurements for solid mercury are easy to carry out. It turns out that the resistance of mercury decreases under pressure at a rate almost 50% greater than for lead. This result appears quite natural.

Interesting data may be expected as a result of measurements of the influence of pressure on the resistance in different directions of a large single crystal of a metal with a non-cubic lattice. The resistance of zinc decreases in all directions, but here there is observed a curious case of asymmetry in the pressure effect along different directions: at high pressures the ratio of the resistances along the principal crystallographic axes changes to the reverse. The resistance of a single crystal of tin decreases smoothly throughout the entire pressure range and in all directions, and shows no reversal phenomenon. A single crystal of bismuth represents the only example of a metal whose resistance first increases with pressure, reaches a maximum, and then decreases. This anomaly is observed only in certain directions. Tellurium is also of special interest because of its semimetallic nature. It was already known earlier that its resistance decreases very strongly with increasing pressure. Measurements in two different directions showed that even at high pressures the resistance continues to decrease rapidly and, finally, at \(30\,000\ \mathrm{kg/cm^2}\) the resistance becomes equal to only \(1/600\) of its initial value. No noticeable difference in the resist—

in different directions in the crystal was found. The most interesting feature of tellurium is the behavior of the temperature coefficient of its resistance. The resistance of all pure metals increases with increasing temperature, but tellurium reveals its nonmetallic nature by exhibiting a negative temperature coefficient of resistance. However, at high pressures the temperature coefficient of tellurium changes sign, becoming positive, as for all pure metals, so that with some justification one may say that

Fig. 6. Resistance of bismuth at 30° C as a function of pressure up to 30,000 kg/cm². The Roman numerals denote the resistance of the different polymorphic modifications.

the effect of pressure on tellurium is expressed in its conversion from a semimetallic to a metallic state.

Finally, bismuth may be of interest in connection with its two transformations at about 30,000 kg/cm². Usually bismuth behaves anomalously, since its resistance increases with pressure. It turned out that the resistance of bismuth increases in the high-pressure region up to the first transformation point, as shown in Fig. 6. At the transition point the resistance drops, as was to be expected from the course of the volume changes, and by a factor of 6, i.e., extraordinarily sharply. The resistance of the new modification decreases with increasing pressure. This is the first case that justifies our general considerations. However, at the second transformation point the resistance again rises sharply, contrary to the volume, namely by a factor of 2.6. In this respect bismuth III (or, more precisely, bismuth II)

constitutes an anomaly, but, on the other hand, bismuth III, like bismuth II, has a negative pressure coefficient of resistance.

Thus the experimental situation in the region of high pressures is rather complicated. At the present stage of development of the theory one can hope only for the treatment of the simplest phenomena observed experimentally. And even in this case, in treating the simplest effects, theoretical calculations must be far more complicated than those which have made it possible to calculate, with fair satisfaction, the energy, the lattice constant, and the compressibility. It is impossible here to give a complete account of the wave theory of electrical conductivity. Only the main points can be indicated. First of all it must be taken into account that electrons cannot have all possible velocities. A piece of metal may in a certain sense be regarded as a gigantic molecule. There must be some structure, in which the electrons in this piece are arranged, analogous to that according to which the electrons are grouped in the shells of an individual atom. One can form an idea of such an energy structure of the electrons in a metal by imagining a metal consisting of separate atoms. The shells of each individual atom pass continuously into energy zones of closely packed energy levels of the metal. As in an individual atom, the energy likewise has a structure in which the electrons are distributed both under the influence of external stimuli and in the case when they are not excited. Thus in a metal there is a structure of energy levels both for the case of excited electrons and for electrons in the unexcited state. The band theory of metals considers the energy structure of the electrons. The phenomenon of electrical resistance is connected chiefly with the properties of the bands. Under ordinary conditions the energy which the electrons possess, owing to their wave nature, is much greater than the energy which they have thanks to thermal motion. At absolute zero the electrons descend to the lowest of the possible energy levels, filling them completely, while above them there remain unoccupied higher energy levels. As the temperature is raised, only a small number of individual electrons, those nearest to the upper occupied energy levels, can, as a result of collisions of atoms, acquire sufficient energy to pass to the higher energy levels allowed for them by the structural form of the energy. This is the explanation of the fact that electrons contribute far less, by their energy, to the specific heat than was assumed in classical mechanics. The explanation of this fact was the first major success of the new point of view on the nature of metallic properties. When an external electric field is applied to a metal, the electrons can acquire energy from the field, i.e., move under the action of the field, forming a current, only when there are unoccupied levels in the region of the energy structure,

to which they move. According to the character of the energy structure we obtain different features of the electrical conductivity of metals. It may happen that the highest band in the structural form of the energy, occupied by electrons in general, is only half filled by them. This occurs in the case of alkali metals, which have only one valence electron. In such metals there are unoccupied energy levels adjacent to any electron, to which it can easily be shifted by an external exciting field. The electrons in such a metal readily respond to the action of an external field, and the metal is a good conductor. However, no metal can be an infinitely good conductor, owing to the continuous scattering of the energy absorbed by the electrons from the external field when they collide with atoms moving chaotically in thermal motion. Further, if each atom has two valence electrons, then the very upper occupied energy band is entirely filled with electrons in their natural, unexcited state. The only way in which the electrons of such a substance can acquire energy from an external field consists in a transition to other unoccupied bands. If the nearest unoccupied band is situated considerably above the occupied one, then the electrons will not be able to obtain from the external field an amount of energy sufficient for the jump, and the substance is an insulator. This is the case with many nonconductors. On the other hand, if the nearest band lies quite close to the one occupied by electrons or even partially overlaps it, then the outermost electrons can easily be excited by the field, and the substance will be a conductor. This is the case for divalent alkaline-earth metals.

When a substance is compressed, the energy structure must also change and, consequently, the resistance will change as well. In a detailed treatment of this question several factors must be taken into account. The rate at which the energy borrowed from the field is scattered by the atoms is affected by pressure. In the present case, results obtained before the advent of wave mechanics give an approximately correct solution: the rate of scattering is proportional to the amplitude of the atomic vibrations. As the pressure is increased, the natural frequency of the atom’s vibrations increases because of the greater intensity of the restoring forces in small volumes, and the amplitude decreases. This factor should cause a decrease in resistance with increasing pressure. This is perhaps the most essential effect for normal metals, and it is precisely for this reason that the resistance of most metals decreases under pressure.

There is also another important effect. It is quite natural that the obsolete image of the free electron cannot find a place in wave mechanics. It turns out, however, that if the electron is situated in the middle of an energy band, then the old picture describes the state of affairs fairly well; the electron can be assigned its normal mass, and its kinetic energy is equal

to the square of the momentum divided by twice the mass. But near the upper part of the band this simple relation between momentum and energy no longer holds. Here the energy does not change for some time, although the momentum may change, and then there immediately occurs an abrupt transition of the energy to a higher value without a change in momentum. In this region the electron is not “free,” as in the center of the band. The average behavior of electrons in the band is something intermediate between the behavior of a free and a non-free electron; in other words, the “effective” number of free electrons must be less than their total number. The ratio of the “effective” number of electrons to the total number must depend on the details of the band structure. This, in turn, also depends on the pressure, so that here we have the possibility of observing an effect of pressure which was not considered by the classical theory—namely, its influence on the “effective” number of free electrons. Detailed calculations show that this number may either increase or decrease under pressure; thus here we have a mechanism explaining the increase of resistance with pressure observed for certain metals.

There is still a third factor, arising from the interaction of the ions and also taken into account by J. Bardeen. Apparently this factor, too, may either increase or decrease with pressure. Detailed calculations of the relative magnitude of these effects in any particular case are very difficult, but they cannot be avoided, because there is no possibility of intuitively predicting whether, for example, the number of free electrons will increase or decrease for some definite metal. J. Bardeen and Weiner carried out calculations for lithium and sodium, taking into account changes in all three factors.

For lithium there was found to be such a decrease in the effective number of free electrons with increasing pressure that this effect more than covers the influence of the other factors. Thus in this case one may expect an increase of resistance with pressure, which was indeed found experimentally. The numerical agreement, however, is not satisfactory. For sodium, on the contrary, the calculations showed that the entire pressure effect as a whole is due to the change in the amplitude of the atomic vibrations, and it proved possible to calculate the resistance for all pressure values with good approximation.

More powerful methods of calculation make it possible to extend these calculations to other metals. At present there is no reason to think that the basic equations are incapable of describing the state of affairs: the whole difficulty lies in the difficulty of carrying out detailed calculations. One of the tasks now in prospect is to extend the calculations to the case of a single crystal, in order to take into account the change of mechanical and electrical properties in different directions. Up to now theoreticians are only trying to approach the solution of this question.

The Character of Modern Theory

In conclusion I wish briefly to characterize the features of modern theoretical constructions. It is quite true that, concerning many phenomena in modern theoretical physics, one can speak in purely descriptive terms; but I believe that, if one thinks carefully about the state of affairs, it will turn out that these descriptions are far from simple, and that without a prior, thorough mathematical analysis one cannot rely on intuition. Thus, we may speak of energy bands filled with electrons, and we may learn to apply this picture in such a way that many experimental results are explained by it. However, this picture cannot be treated with the same naïveté with which we were accustomed to treat the old mechanical models. We cannot, for example, ask: “Which particular electrons in a given energy band have the smallest store of energy, or in what part of the metal are they located?” This question has no meaning from the point of view of wave mechanics (although the natural simplification in our speech makes the question seem natural).

The conceptions now being developed by theoretical physics in order to explain new phenomena are only partly grounded in our previous experience; and for each new case it is necessary to find the appropriate image and to work it out by means of the most difficult mathematical analysis. At this stage in the development of theoretical concepts, those of us who were not formerly theorists can only wait and receive guidance from those who have carried out all these exhausting and complex calculations.

Literature

1. Most of the earlier works may be found in the book: P. W. Bridgman, Physics of High Pressures, trans. from English by M. P. Volarovich, GTTI, 1935.

2. P. W. Bridgmann, Phys. Rev., 48, 893, 1935; Proc. Am. Acad. Sci., 72, Nos. 2, 4, 5, 6, 1937–1938.

1. A report read in New York in February 1938 at the Institute of Metals, Trans. Am. Inst. Mining. a. Metall. Engin., 128, 15, 1938. Translation by A. A. Leont’eva, edited by Prof. M. P. Volarovich. ↩