THE NATURE OF THE METALLIC STATE1

W. Shockley, New York

The nature of the interatomic forces in metals is, obviously, of great importance from the standpoint of metallurgy. These forces ultimately determine the properties of metals. Attempts to understand the nature of these forces, undertaken in terms of the usual chemical notions of atomic valences, have not proved especially fruitful. The reasons for this failure can now be explained satisfactorily on the basis of the latest advances in the theory of metals. During the last six years a quantitative theory of metals has for the first time been formulated, making it possible to calculate, from first principles, many of the most important properties of the simplest metals, such as, for example, specific heats, coefficients of expansion, heats of vaporization, compressibility, and electrical conductivity. The successes of this theory, due to its quantitative character, stem from the fact that mathematical methods were created that were specially adapted to the treatment of the problem of metals. At the same time, however, alongside the mathematical apparatus there arose new concepts and new physical pictures which, in a simpler form, embody the same basic features of the theory that yield quantitative results when the question is treated mathematically. The purpose of the present article is to describe the features of these new concepts and to demonstrate how they explain the properties of the alkali metals. Particular attention will be paid to sodium, since this metal has been most fully investigated from both the theoretical and the experimental sides.

There is no doubt that the metallurgist begins to feel somewhat uneasy when he finds that the physics of metals will be discussed as applied to the alkali metals. He knows that these elements in the metallic state are of negligible importance from the standpoint of metallurgy, and that handling them is extremely difficult and inconvenient even under laboratory conditions; and it seems to him that the physicist is displaying incomprehensible and unjustified obstinacy in choosing as the object of his investigations precisely these cases, so barren in practical respects.

In an analogous way, the physicist feels somewhat disappointed that metallurgy deals predominantly with such complicated cases as iron–carbon alloys, whose complexity he cannot at present hope to overcome. It is probably easier for the physicist to understand the metallurgist’s activity than conversely, for it is clear that in most cases the latter is not free in his choice of objects of investigation and must devote himself to the direction that yields immediate practical results. On the other hand, the physicist may be to a considerable extent free in his choice, but in doing so he naturally chooses those problems which he can hope to solve. In any new field this means turning to the simplest problems. In this article we shall try to show, in particular, why, from the point of view of the theoretical physicist, the alkali metals appear to be the simplest.

The instrument that proved so effective in solving problems of physics connected with individual atoms was Schrödinger’s wave equation. Initially Schrödinger’s equation was applied to the simplest systems, consisting of free atoms and ions, and later—to molecules. It was also used in application to ionic crystals, which constitute the simplest case of a solid, representing an assemblage of free ions. However, when Schrödinger’s equation was used as applied to metals, difficulties arose due to the presence in them of the so-called free electrons.

Free electrons give a metal its properties of high electrical conductivity and metallic luster. These electrons are free in the sense that they are always ready to be subject to the influence of an electric field and, when it is present, to move through the crystal, creating an electric current. By studying the behavior of this current in the case of rapidly varying electric fields existing in light waves, i.e., in other words, by studying the optical properties of metals, it proves possible experimentally to determine the number of free electrons in a metal. As it turned out, in the case of the alkali metals each atom corresponds to one free electron. Although the properties of free electrons were well studied and their presence presented no obstacle from the point of view of explaining the electrical properties of metals, they could not at first be fitted into a coherent scheme which could describe equally well the other properties of metals. Only in 1933 did the work of Wigner and Seitz open new paths in the quantitative consideration of the problem of metals, leading to the results set forth below1.

How do free electrons arise? To answer this question as applied to the case of metallic sodium, let us consider a free sodium atom. Such an atom consists of a nucleus having charge \(+11e\), surrounded by eleven electrons. The nucleus is approximately 4,000 times heavier than the eleven electrons surrounding it and may be regarded as a more or less immobile center. The eleven electrons of the sodium atom cannot be regarded as completely identical. They must be divided into two groups according to

...according to their arrangement relative to the nucleus. The first group, consisting in the case of sodium of one electron, is the group of valence electrons. (This electron, as we shall see below, becomes a free electron of the metal.) The second group consists of ten inner electrons. This subdivision is a direct consequence of firmly established quantum-mechanical laws, the very same laws that explain the structure of the periodic system of the elements, thereby providing one of the best confirmations of all the constructions of quantum mechanics. In the case of the sodium atom the valence electron moves in an orbit considerably farther from the nucleus than those along which the core electrons revolve, which are comparatively very tightly bound to the nucleus.

One of the consequences of the laws of quantum mechanics is that expressions such as “moves in an orbit” lose their exact meaning when an electron is in question. We must not picture electrons as moving along completely definite trajectories. According to quantum mechanics, the motion of an electron is described by wave functions, which do not specify exactly the path followed by the electron, but give only the values of the probability of finding the electron at a given place. In Fig. 1 the outer ring, with light shading, depicts the wave function of the valence electron. In a three-dimensional representation this ring would give a spherical shell. The inner dark region depicts the comparatively very high mean charge density produced by the ten core electrons. A consequence of such an arrangement is that a comparatively small energy is required to remove the valence electron from the atom. If the valence electron has left the atom, the latter is converted into an ion consisting of the nucleus and the core electrons. The energy bound up with the atom is usually expressed in electron-volts: 1 electron-volt (eV) is the energy required to move one electron through a potential difference of 1 V. Expressed in these units, the energy required to detach the valence electron from a sodium atom, i.e. the ionization energy of the sodium atom, is 5.12 eV.

Fig. 1. Distribution of electrons in the sodium atom according to the wave-mechanical theory

The inner electrons are bound much more strongly. In order to remove one of them (the first), in the case of the sodium ion an energy of 47 eV is necessary. This energy is so large that, in considering the questions that interest us here, we may assume that nothing at all can happen to the ion, and regard the atom as an indivisible particle—the ion—around which the valence electron rotates.

Let us now consider the behavior of the valence electrons in a solid and see why they become free. In Fig. 2,a the arrangement of the atoms in metallic sodium in the 110 plane is shown. We see that the atoms are situated so close to one another,...

that the valence electron of each of them proves to be very close also to its neighbors—so close that it can leave its atom and pass to a neighboring one, and thus move farther through the lattice. This description is, of course, highly pictorial. The correct way to consider the problem is to apply Schrödinger’s equation to the valence electron moving in the field of the ions, and to investigate the mathematical form of the solution of this equation. However, when such a solution is obtained, it turns out that in a number of its most important properties it corresponds to the very simple physical picture given above.

Fig. 2. Arrangement of sodium atoms in the 110 plane:

a — relative dimensions of the atomic distribution, b — distribution of electrons predicted by theory

The valence electrons in the case of solid sodium prove to be so free that they can travel within the crystal and are not localized near particular atoms. Thus a very uniform distribution of negative electricity is obtained in the crystal, with positive ions embedded within the negative charge, as shown in Fig. 2, b. This distribution should not, however, be regarded as rigid; in its properties it is rather closer to a gas, whose particles are free electrons. We therefore picture a solid metal as a uniform electron gas in which ions float freely. At first glance such a description seems incorrect, since it appears to depict a very mobile and unstable system, which does not correspond to the properties of solid metals; but, as we shall see, it leads to results in excellent agreement with experiment as regards hardness, shear modulus, and compressibility over a very wide range of pressures.

ENERGY OF ALKALI METALS

Let us consider how the energy of a particular specimen of material changes when we subject it to pressure, compressing the elementary cell but not changing its shape. This energy is the kinetic energy of the motion of the electrons in the electron gas, plus the electrostatic interaction between these electrons and the ions. This does not include the energy of thermal motion, since the kinetic energy of the electrons is not a consequence of thermal excitation but is based on the quantum-mechanical nature of the electron gas. Calculations of the value of the energy \(E\) for sodium at different values of the volume \(V\) were made by Bardeen\(^2\). The results he obtained are shown in Fig. 3, where the independent variable is not the volume, but the value of the lattice constant, i.e., the length of the edge of the elementary cell. For the curve shown in Fig. 3 there is no quite exact analytical expression, but with sufficient approximation it can be expressed by the equation

\[ E=\frac{A}{V}+\frac{B}{V^{2/3}}-\frac{C}{V^{1/3}}, \tag{1} \]

where \(A\), \(B\), and \(C\) are constants determined from atomic theory.

Fig. 3. Energy (in kilocalories per gram-atom and in electron-volts per atom) as a function of the lattice constant:
\(a\) — theoretical curve, \(b\) — third-order approximation to the experimental curve according to the data of Table 2

At absolute zero temperature and pressure equal to zero, the stable state of the system is represented by the minimum of the curve. In any expression for energy the state corresponding to its zero value may be chosen arbitrarily. In the case of equation (1), the state chosen as such is that in which the electrons and ions are completely separated. We thus see that, to transfer the given substance from the state with the least energy to that in which the ions and electrons are separated, an amount of energy is required equal to \(6.13\) eV per atom, or \(141\) kg-cal per gram-atom. This energy exceeds the energy necessary for evaporating the substance in the form of neutral atoms by an amount equal to the ionization energy. From this we find for the heat of evaporation, or for the energy of formation, the value \(6.13 - 5.12 = 1.01\) eV per atom, or \(23\) kg-cal per gram-atom. The latter value has been obtained purely theoretically and is in good agreement with the experimental value of \(26\) kg-cal per gram-atom. The minimum of the theoretical curve should correspond to the equilibrium configuration of the metal. For this point we find the value of the lattice constant to be \(4.53\) Å; the experimental value of this quantity is \(4.25\) Å. From the curvature of the curve at this point we can

determine the compressibility \(K\); a comparison of the theoretical and experimental values of the compressibility is given in Table 1.

Table 1

Let us now return to our picture of the metal and trace in it the origin of the three terms on the right-hand side of equation (1). The simplest of these is the second term, which arises as a result of calculating the energy of the electron gas. When the atoms are in a state of close packing, as shown schematically in Fig. 2, each of the electrons turns out to be moving in the field of positive ions, which attract it, and of negative electrons, from which it experiences repulsion. Since the numbers of electrons and ions are the same, these forces, to a first approximation, balance one another, and the electrons behave as if they were uncharged particles moving in empty space. Such a system of uncharged particles forms a gas obeying quite definite laws—not, however, the laws obeyed by ordinary gases, but laws well known in quantum mechanics and called the laws of a degenerate electron gas. According to these laws, the mean energy of one electron forming the gas is given by the expression

\[ \frac{3h^{2}}{40m}\left(\frac{3}{\pi V_{1}}\right)^{2/3} = 21.6\,\frac{1}{V_{1}^{2/3}}\,\mathrm{eV}, \tag{2} \]

where \(V_{1}\) is the volume per electron, expressed in cubic ångströms, \(h\) is Planck’s constant, and \(m\) is the mass of the electron. The energy corresponding to one gram-atom of electrons is equal to

\[ E_g = 359\,\frac{1}{V^{2/3}}\,\text{kg·cal}, \]

where here \(V\) denotes the volume occupied by one gram-atom and expressed in cubic centimeters.

This energy gives the second term in equation (1) for the energy of the metal. Knowing the energy, we can also determine the magnitude of the pressure

\[ p=-\frac{dE_g}{dV}=9.9\cdot 10^{6}V^{-5/3}\,\text{atm}, \tag{3} \]

where \(V\) again denotes the volume occupied by one gram-atom of the degenerate gas.

We see that the behavior of a degenerate gas differs very substantially from that of a classical gas in that it does not depend on temperature. Generally speaking, this is not quite exact: both the pressure and the energy of a degenerate gas depend on temperature. However, this dependence is so weak that in our treatment we may rightfully neglect these effects. We see further, from our equation (3), that the pressure always has a positive sign, i.e., that the electron gas always tends to expand the metal. Since the metal does not fly apart, there evidently exist forces acting in the opposite direction and balancing the pressure of the electron gas. In equation (1) these forces correspond to the third term. In their origin they are due to the electrostatic interaction which we neglected in our (corresponding to a first approximation) treatment of the electron gas.

Fig. 4. Division of the lattice into neutral polyhedral cells

The electrostatic energy of a metal has as its source interactions of three types: between ions, between ions and electrons, and between electrons. In Fig. 4 is shown a scheme for dividing the metal lattice into polyhedral elementary cells, by means of which we shall reduce the consideration of interactions of all the enumerated types to the consideration of only one electron–ion interaction, and moreover only within the limits of a single cell. The cells in Fig. 4 are constructed in such a way that each of them includes one ion and that part of the electron gas which is nearer to this ion than to any of the others. Each of the cells is electrically neutral, since the positive charge of the ion is compensated by the corresponding volume of electron gas. In addition, each such cell represents a fairly good approximation to a sphere. This can be seen from Fig. 5, where a three-dimensional image of an elementary cell is given.

Fig. 5. Three-dimensional image of an elementary cell

A neutral volume—of spherical shape—does not create an electric field outside itself. Therefore, if our polyhedra were actually spheres, they would not act electrostatically on one another, and we could calculate the electrostatic energy by considering only one sphere separately. In reality, the polyhedral cells do act electrostatically on one another. However, if these forces are calculated, it turns out that they are very small. This gives us the right to neglect them in our further ...

consideration¹) and calculate the electrostatic energy as the energy of neutral polyhedral elementary cells.

As the dimensions of the lattice decrease, the cells also decrease, and the electron gas proves to be more densely packed around the positive ions. Since positive and negative electricity attract one another, this means a decrease in potential energy. Further, since the interaction energy of two charges varies inversely as the first power of the distance between them, we find that this energy is inversely proportional to the lattice constant (since the lattice constant is a measure of how far, on average, the electron cloud is from the ion). As a consequence, since, to within a constant factor, the lattice constant is expressed in terms of the volume \(V\) as \(V^{1/3}\) in the energy equation, for the corresponding term we have \(-C V^{1/3}\). This term enters with a negative sign, since it is associated with attractive forces. The action of these forces is opposite to the pressure of the electron gas, and they are sufficiently large to overcome it. Thus this term represents the binding energy. Under certain simplifying assumptions, the theory gives for \(C\) the value \(673\ \mathrm{kg\cdot cal.}\)

The origin of the first term in the energy equation (1) cannot be explained by elementary reasoning. It is due to the wave nature of the electrons. An electron moving in the field of a sodium ion has a certain wave function characteristic of this state, to which, roughly speaking, a certain wavelength is inherent. Since the crystal is also characterized by a certain length, namely the lattice constant, we may expect the existence in the energy equation of a term depending on the ratio of these lengths. This term always has a positive sign and, consequently, corresponds to forces tending to expand the lattice.

Wave-mechanical considerations similar to those needed in discussing the first term in the energy equation are necessary for clarifying the question of the degree of “freedom” of the electron gas in a metal. The circumstance that the electron gas is not completely free is manifested in the magnitude of the constant \(B\), whose value deviates from that inherent in an ideal degenerate gas, equal to \(359\ \mathrm{kg\cdot cal.}\)

We have already referred, in connection with Table 1 and Fig. 3, to the calculations carried out by Bardeen. In his work the energy was calculated as a function of volume, the minimum value of the energy proving to be

¹ From this reasoning it would seem to follow that, since there is no significant force between separate cells, nothing holds them together and the crystal should break up into separate cells. This, however, is not so, because in reality the electrons pass continuously from cell to cell, and if the lattice were to expand, no voids would form between the cells. In all cases, when considering a lattice in the equilibrium state, it is not so easy to establish exactly what kinds of forces of attraction act between the atoms. The usual method—the only one to which we shall adhere—is to show that the energy of the crystal as a whole is minimal for a definite arrangement of the atoms (the equilibrium arrangement). In this case the attractive forces arise as a consequence of the fact that, upon any change in the arrangement of the atoms, the energy of the system increases.

corresponding to the lattice constant of \(4.53\ \text{Å}\). If the metal is compressed as a result of the application of external pressure, the value of the energy changes. Analyzing the course of the change in energy that arises when the volume is changed from its equilibrium value \(V_0\), Bardeen, on the basis of the theory he developed, calculated the dependence of the compressibility on the relative compression \(\Delta V/V_0\), and obtained the results shown in Fig. 6 (dashed lines). The experimental curves obtained by Bridgman in the Harvard University laboratory using pressures up to \(40\,000\ \text{atm}\) are plotted in the same figure (solid curves). We see that the agreement between theory and experiment over a wide range of compression values is as good as for the equilibrium state. Carrying out complete calculations for other alkali metals is not feasible, but for them it is possible to compare theory and experiment by determining the constants \(A\), \(B\), and \(C\) in equation (1) from experimental data. Having found the values of these constants, we can use equation (1) to predict other experimental results. The method applied by Bardeen in this direction was as follows. The constants \(A\), \(B\), and \(C\) were determined on the basis of the assumptions: a) that equation (1) has a minimum at the observed atomic volume; b) that the value of the energy given by equation (1) at the minimum (i.e., the cohesion energy) coincides with the experimental value of this quantity; c) that the compressibility calculated from equation (1) (it is determined by the curvature of the curve at the minimum) coincides with its experimental value. These three conditions are sufficient to find \(A\), \(B\), and \(C\). The values obtained for them are given in Table 2.

Fig. 6. Compressibility as a function of compression. Solid curves—experimental, dashed—theoretical.

Table 2

Empirical values of the constants of equation (1), in \(\text{kg}\cdot\text{kcal}/\text{mol}\)

Next, the relative compression produced by a given external pressure was calculated from equation (1). The results of these calculations are shown in Fig. 7 together with Bridgman’s experimental data. As is easy to see, the agreement between these and the other results is very good. The degree of agreement of the results becomes especially striking and takes on the character of very weighty evidence for the validity of equation (1) if we recall that all the data used for the calculations referred to only one state of the metal, corresponding to a pressure equal to zero. This means that equation (1) is such a good approximation to the true relation between energy and volume that we can successfully use it for extrapolation from the normal state of the metal to states at very high pressures, when the degree of compression \(\left(\dfrac{\Delta V}{V_0}\right)\) is close to \(50\%\), as is the case for cesium.

Fig. 7. Degree of compression as a function of pressure. Solid curves are experimental, dashed curves are theoretical.

Further, it is necessary to note that in determining the constants \(A\), \(B\), and \(C\) we used the binding energy. Undoubtedly, since this is an experiment, the connection between this quantity and the data of Fig. 7 is very remote. Nevertheless, the binding energy is used in determining the constants of equation (1); when they are determined, equation (1) turns out to be in good agreement with experiment. Both the binding energy and the pressure produced by compression are, of course, due to one and the same cause—the interaction between ions and electrons. Equation (1) represents an attempt to relate the various properties of these forces to the relation between energy and volume. The success of this attempt over such a broad pressure interval is convincing evidence for the correctness of the new theory and of the physical picture underlying it.

The scheme of a metal of which we have spoken, representing it as positive ions floating in an electron gas, is not new in its general features. The classical theory of metals, developed by Drude and Lorentz, was also based on the idea of an electron gas. However, the behavior of the electron gas predicted by the classical theory did not correspond to reality in many respects: the pressure produced by it proved to be too small; the changes in the properties of this gas under the influence of temperature were incorrect (we shall return to this below); and the electrons were imagined as clustering around the ions, instead of

to create a uniformly distributed charge. Below we shall see that the uniform distribution of charge is one of the characteristic features of the new theory. An essential feature of the former theory—the idea that free electrons are present in the metal—is preserved in the new theory, but, since the quantitative side of the question is involved, the wave-mechanical model has a whole series of sharp differences from the classical one. The important role of these changes becomes obvious if one recalls the good agreement between the new theory and experiment demonstrated above in Table 1 and in Figs. 3, 6, and 7.

ELASTICITY OF ALKALI METALS

The comparison of theory with experiment has so far been discussed only for the equilibrium state and homogeneous compression. We must now see how the theory explains elasticity, i.e. elastic resistance to shear.

Resistance to shear arises simply as a consequence of the electrostatic repulsion of the ions. The ions tend to move as far apart from one another as possible, but are held in their positions by the attraction of the electron gas. This prevents the crystal from being torn apart by the forces of repulsion. However, for any value of the volume of the crystal the ions, naturally, arrange themselves in such a way as to be as far apart from one another as possible, and the theory shows that this leads to the formation of a regular lattice, for example, a body-centered one. Fig. 8 shows, in a greatly exaggerated form, the action of shear on a regular lattice. It is clear from this figure that shear leads to a decrease in the average distance between positive ions and consequently to an increase in the energy of the crystal. Thus, in order to shear it we must do work and hence apply to the metal forces that create an elastic stress in the metal, which thereby exhibits its strength.

Fig. 8. Influence of electrostatic interaction on shear

A quantitative determination of the magnitude of these forces can be made with the aid of classical electrostatics. In these calculations, however, it is necessary to take into account the charge density of the electron gas. On the assumption that the electron gas is uniformly distributed (let us emphasize that this assumption is confirmed by the new wave-mechanical calculations, but not by the preceding classical theory), the shear moduli of various metals were calculated. The elasticity of anisotropic crystals is different in different directions, but for a cubic crystal, to which sodium belongs, its elastic properties are completely determined by specifying the modulus in two

in different directions. The first two columns of Table 3 contain the calculated and measured values of the modulus for sodium.

Table 3

Elastic constants of sodium (dyn/cm²)

The calculated values of the constants given in this table were obtained by K. Fuchs³ of the University of Bristol on the basis of the electrostatic considerations just set forth, and with allowance for a uniform distribution of the electron gas. If one uses a representation of a nonuniform distribution, similar, for example, to that shown in Fig. 2,a, then completely different values are obtained. As for the experimental data of Table 3, they were obtained by Kimby and Ziegel⁴ of Columbia University, and after Fuchs’s results had already been published. The satisfactory agreement between theory and experiment with respect to the elastic constants of sodium is yet another proof of the correctness of the physical ideas associated with the new theory.

The last column of Table 3 and Table 4 are given for completeness of information. Table 3 gives the values of the bulk modulus \(\dfrac{1}{K}\), which is the quantity reciprocal to the compressibility considered above. The quantities given in Table 4 completely describe the elastic properties of sodium, and from them, if necessary, one can find the so-called elastic constants \(C_{11}\), \(C_{12}\), and \(C_{44}\).

Table 4

Electrostatic part of the elastic constants
(the figures are given in dyn/cm², with \(a\) the lattice constant in angstroms)

Table 3 gives Fuchs’s formula for the modulus of elasticity in the case of a face-centered and a body-centered lattice, obtained with the aid of the classical theory with allowance for a uniform distribution of the electron gas.

The proofs given by us above in favor of the correctness of the new theory concerned the simplest properties of the alkali metals, namely, their energy and its changes under small elastic

deformations. We shall not discuss in detail the electrical, magnetic, and thermal properties of the alkali metals, which are just as well described by the new theory; instead we shall confine ourselves to a brief summary, considering that the material set forth above sufficiently attests to the correctness of the theory. The essential features of the physical picture of a metal according to the new theory are that the valence electrons of the atoms are distributed uniformly within the crystal, forming a pronounced electron gas. The properties of this gas are determined by certain quantum-mechanical laws, and its energy can be calculated. The positive ions float freely in this gas. The opposite tendencies of the electron gas—to expand—and of the positive ions—to restrain it from expanding—compensate one another and create an equilibrium state. The ions mutually repel one another and as a result arrange themselves in a regular lattice.

THE INVALIDITY OF THE PREVIOUS THEORY

The new picture of the structure of a metal considered above is in sharp contradiction with the previous one, connected with notions of “interatomic forces” in a metal. According to the latter, each atom in a solid retains its integrity and attracts or repels other atoms in accordance with the laws governing interatomic forces. In the new picture the situation proves to be more complicated. The atoms are no longer preserved as wholes, but give up their valence electrons, forming an electron gas. The interactions that arise in this case can no longer, of course, be regarded as “interatomic.” Instead of interatomic forces, forces of a new type appear, such as, for example, the pressure of the electron gas.

Above we saw that the new point of view successfully explains the results of experiment; this, however, is not a sufficient basis for rejecting the previous, simpler representations. We must therefore show directly that the previous picture proves untenable.

The inapplicability of the concept of interatomic forces to the case of metals follows from the fact that it attempts to explain both the resistance of a metal to a change in volume and its resistance to shear (which is not connected with a change in volume) by one and the same system of forces. This is explained by Fig. 9, where it is assumed that the interatomic forces act along the lines connecting the centers of the atoms. In the upper part of Fig. 9 a lattice is shown in the absence of external—

Fig. 9. Illustration of Cauchy’s ideas on central forces

forces; in the middle part, a lattice subjected to compression; and in the lower part, the case of shear. Both in compression and in shear the acting forces turn out to be the same. From this follows the supposition that (since the forces are identical) the bulk modulus can be expressed through the shear modulus. This question was worked out by Cauchy, who found that for a cubic crystal the bulk modulus \(\frac{1}{K}\) is related to the two shear moduli \(C_{44}\) and \(C_{11}-C_{12}\) by the equation

\[ \frac{1}{K}=\frac{1}{3}(C_{11}-C_{12})+C_{44}. \]

This result is quite general so long as the atoms interact by forces directed along the lines of their centers, and is not limited to the structure shown in Fig. 9, where only nearest neighbors interact. In Table 5 we give the data necessary for checking the Cauchy condition for several

Table 5

Check of the Cauchy condition (in units of \(10^{10}\) dyn/cm\(^2\))

metals and ionic compounds\(^5\). As is easy to see, for metals there are large discrepancies between the calculated and observed values of the bulk modulus; moreover, the calculated values turn out to be considerably smaller than the experimental ones. On the basis of the new ideas this discrepancy is easy to explain: in shear we are dealing only with the electrostatic repulsion of ions, whereas in compression, along with this, there act the pressure of the electron gas and other forces leading to an increase in the compression modulus. For comparison, the same table gives figures for ionic crystals, in which there is no electron gas. The forces acting between ions must have the typical “interatomic character,” and the Cauchy condition should be satisfied. For other compounds, where direct valence bonds exist—for example, for diamond—we should expect, on the contrary, discrepancies between calculated and experimental data.

ON THE INFLUENCE OF COLD WORKING AND THERMAL EFFECTS

Now that we may regard as firmly established the proposed picture of the metal (an electron gas with positive ions floating in it), let us apply it to a question of interest from the point of view of metallurgy, namely, let us see whether cold-

cold working, accompanied by densification, in any changes in the atoms of the metal. It has been suggested that cold working of a metal leads to a disruption of the bonds between atoms that are typical of a metal, and to the formation of homeopolar bonds. In other words, the question posed above may be formulated as follows: do such changes occur in atoms as a result of cold working that the forces acting between them before and after the working are forces of a different nature?

We must answer this question in the negative. Let us first consider the electron gas. It is obvious that, since the free electrons in a metal have the properties of a gas, it is impossible, by external actions, to impart to this gas changes that would not disappear with the cessation of the action. The electron gas may, for example, be compressed, as occurred in Bridgman’s experiments, but this compression does not change its character, and as soon as the pressure is removed the electron gas again occupies its normal volume. Next we must turn to the ions and determine whether anything can happen to them—both to each one separately and to the entire collective of ions. In speaking above of atoms, we indicated that their valence electrons can be removed from them with great ease, but that it is much more difficult to remove an electron from an ion. Even the very greatest pressures that Bridgman succeeded in obtaining are hardly sufficient to create an overlap of the outer shells of ions, and they are certainly far too small to create, between ions, a proximity sufficient to tear out electrons. Before this can be achieved, it is necessary to obtain pressures at least 20 times greater than Bridgman obtained; and even if they could be obtained in separate localized regions of concentrated stresses, the ions would return to the normal state as soon as the external cause of the stresses were eliminated. There is also another possibility for a change in the properties of ions. We know that atoms and ions can exist in excited states, and in these cases their properties differ sharply from those corresponding to the normal state. However, the excitation of a sodium ion requires pressures of the same order as secondary ionization. We may therefore conclude that no cold working can change the properties of the ions themselves, and we may still regard them as unchanged elements of positive charge floating in the electron gas.

There is another possibility for changing the properties of a metal as a result of the creation of stresses in it; this consists in a certain change in the arrangement of the ions into a new and stable system. A change of this kind apparently occurs in the case of cesium at pressures of about 20,000 atm, at the magnitude of pressure where the curve for this metal (see Fig. 7) suffers a discontinuity. It is assumed that at pressures greater than the indicated value the ions are arranged in a face-centered lattice, returning, when the pressure is lowered, to the body-centered system. On the basis—

On the basis of X-ray analysis data for samples of metals subjected to the strongest cold working, we find that the atoms are arranged in the same way as in a well-annealed metal, with the exception of the presence of small strains. These strains are insignificant in character and disappear as a result of annealing. Although these strains are too weak to change in any way the nature of the interionic forces, they undoubtedly entail the formation, on the surface of the metal, of a system of particles with internal stresses, which, in particular, may impede the motion of slip planes and thereby increase hardness. Theories that consider the properties of metals in this aspect are still only in an embryonic state,^6 and we shall not discuss them here in greater detail.

The arguments presented above were based on consideration of the alkali metals. It is sometimes pointed out that, since these metals do not exhibit an increase in hardness upon cold working, discussion of their properties in application to these questions is inappropriate. We cannot agree with such a view, above all for the following reason. Experiments with sodium carried out at room temperature, which is about 0.8 of its melting temperature on the absolute scale, are equivalent to corresponding experiments with copper at 800° or with iron at 1,200°. At such high temperatures the process of annealing of the metal proceeds so rapidly that it is impossible to observe any influence of cold working. Therefore it seems to us that carrying out experiments on cold working of alkali metals at the temperature of liquid nitrogen would give entirely different results.

In addition, we must recall that the theoretical considerations applied to the alkali metals are applicable to all others to the same extent. We may therefore consider that our picture of an electron gas with positive ions floating in it (which explains the experimental data in the pressure range much higher than those encountered in practice) excludes the possibility of any changes in the atoms as a result of cold working, i.e. of such changes as would affect the type of forces acting in the metal.

Up to now, temperature has not figured at all in our arguments. We have limited our consideration to the region of low temperatures, where the stable state of the system is the state of minimum energy. The question of the influence of temperature on metals constitutes such an extensive section of theory that we cannot dwell on it here in detail and are compelled to confine ourselves to consideration of the two most important circumstances.

When a metal is subjected to heating, it absorbs energy. As the theory asserts, this energy is almost entirely expended in setting the ions into vibration. At temperatures much lower than the melting temperature, the amplitude of these vibrations is small in comparison with the lattice constant, so that the ions remain close to their equilibrium positions. As we have already indicated above, in order to change an ion

very large energies are required, so large that any significant quantities of thermally excited ions cannot be expected at temperatures below \(200\,000^\circ\). Thus, with the exception of vibrations as a result of temperature changes, the ions remain unchanged (they do not, for example, expand). Because of the special properties of the degenerate electron gas, the electrons composing it undergo no noticeable changes when the temperature is varied within ordinary limits. Therefore, as was noted above, all the energy used in heating a metal is expended in setting the ions into vibration about their equilibrium positions. The theory of the heat capacity of metals is based on this.

The second question on which we shall dwell is thermal expansion. Its cause is the circumstance that a metal expands more readily than it is compressed. In Fig. 6 we saw that the compressibility of a metal decreases as the compression increases. This means that the more strongly a metal is compressed, the more difficult it becomes to compress it further. During the thermal vibrations performed by the positive ions, different parts of the metal continuously expand and contract in accordance with these vibrations. Since compression is more difficult than expansion, the resulting effect is that, on the average, expansion predominates. On a macroscopic scale this gives measurable changes in dimensions. The degree of compression and expansion increases with temperature, and thus the macroscopic expansion grows. Theories based on these ideas have proved capable of predicting the correct values of thermal expansion on the basis of data on compressibility (Fig. 6) and specific heat.

THEORY OF OTHER METALS

In the case of alkali metals the ions are so far apart that they interact only electrostatically. For many other metals this is not the case. The ions in these metals are packed so densely that their electron shells overlap, and this creates additional repulsive forces. The absence of these forces is one of the important properties of the alkali metals that simplify their study. As an example let us consider copper. Figure 10 presents a comparison of the 111 plane of copper with the 110 plane of sodium, i.e. the planes most densely occupied by ions in each case. We see that in the case of copper the ions are situated much closer to one another, so much so that the additional energy arising from the overlap of the electron shells must be taken into account. This energy can be calculated and compared with other kinds of interaction. In Table 6 (on p. 270) just such a comparison is made, showing the relative importance of the various types of forces that determine other constants.

It is easy to see that the agreement of the theory with experiment is satisfactory and that the elastic forces in the case of copper have their principal

the cause is the superposition of electron shells. Since in this case as well the principal forces belong to the interatomic type, we might

Table 6

Comparison of elastic constants (in units of \(10^{10}\) dyn/cm\(^2\))

expect that the Cauchy condition should be well satisfied for copper also. In Table 5, however, we saw that for copper the discrepancy proves to be stronger than for sodium. Theoretical consideration shows that the Cauchy conditions are fulfilled only in the case when the ions are in equilibrium under the action of their own forces, as, for example, occurs in the case of sodium chloride, but not when they are pressed toward one another owing to attraction between the ions and the electron gas, as in the present case.

Fig. 10. Electron distributions:

a — in sodium and b — in copper

Alkali metals are the simplest from the standpoint of theory, because their ions are so far removed from one another that the energy arising from the overlap of electron shells is very small, and also because they have one valence electron per atom. Next in simplicity after the alkali metals come copper, silver, and gold, since they, like the alkali metals, give up one electron per atom to the electron gas and, in addition, despite the considerable effect of shell overlap, the corresponding forces can be calculated comparatively simply. The latter is due to the fact that the ions of these metals consist of “closed” electron shells and their excitation energy is large. In the case of other metals each atom gives several electrons to the electron gas and, moreover, new circumstances appear, which we have not mentioned earlier. It turns out,

that the electron gas has certain “critical” values of concentration. For a face-centered lattice the theory predicts a critical concentration of 1.362 electrons per atom, and for a body-centered lattice—of 1.48 electrons per atom. These quantities form the basis for explaining the rules established by Hume-Rothery, which state that the α-phase (face-centered) ceases to exist at an electron concentration of 1.36 per atom, and that the β-phase (body-centered) begins to exist upon reaching a concentration of

\[ \frac{2}{3}=1.5 \]

(or approximately 1.48) electrons per atom. Along with changes in the concentration of electrons, changes in interionic forces also occur here. These forces have a very complex character in the elements of the transition series of the periodic system, such as, for example, manganese, iron, cobalt, nickel, molybdenum, and others. In the case of these ions the forces of interaction are a certain combination of the forces of the electron gas, repulsion of the ions—similar to that of which we spoke in application to copper—and directed valences, typical of the tetrahedral structure of carbon compounds. These forces play a large role in the case of the transition metals and are the cause both of their high mechanical strength and of the phenomena of ferromagnetism. Although, from the practical point of view, understanding and explaining the properties of the transition elements is incomparably more important than understanding and explaining the properties of the alkali metals, the presence of the above-mentioned strong interionic interactions makes this much more difficult. Nevertheless, many of the essential features of these forces are already understood, and often the theory can offer a plausible explanation of the observed phenomena.

In conclusion we may point out that the picture of the metallic state proposed here possesses sufficient flexibility. Thus, for example, metals and alloys differ from valence and ionic crystals in that they are capable of forming solid solutions over a wide range of concentrations. The theory set forth explains this by the fact that the atom of the dissolved element is ionized, gives up its valence electrons to the electron gas, and, in the form of an ion, replaces one ion of the base metal. If the atom of the dissolved metal is not too large and not too small in comparison with the replaced atom, its presence introduces only small distortions into the lattice. In a valence crystal the situation is entirely different. If the foreign atom does not possess the necessary valence, unbalanced valence electrons remain, and considerable disturbances arise in the neighborhood of the foreign atom. These disturbances cannot be eliminated, as in the case of a metal, by giving the excess valence electrons to the electron gas. The same flexibility of structure also explains the plasticity of metals. It is easy to understand how a displacement in the plane of arrangement of the ions can occur without producing destructive changes, as would happen in the presence of valence bonds or in the structure of an ionic crystal. Metals possessing a stronger interaction between

ions should therefore, in their properties, be closer to valence crystals, whereas the alkali metals and the noble metals prove to be more malleable and ductile, as is indeed the case. As we have indicated, among the ions of the transition elements—typical representatives of which are V, Cr, Mn, Fe, Ni, Mo, W—there exist the strongest interactions; correspondingly, these elements exhibit the greatest strength and the least malleability and ductility.

The present article does not claim to be a complete survey of the contemporary state of the theory of metals. Many important questions, such as, for example, the magnetic and thermal properties of metals, thermionic and photoelectric emission, and also the electrical conductivity of metals, have either not been touched upon at all or have been discussed only very superficially. The main emphasis has been placed on comparing theoretical and experimental data for the alkali metals. The aim thereby pursued was to clarify and establish the modern picture of the metal—a picture that has only recently proved its value by making it possible to obtain precise quantitative predictions in agreement with experiment. We believe that this picture will serve as the basis for further progress in the understanding of metals. It is the picture that represents a metal as an electron gas in which positive ions float.

LITERATURE

1. See, for example, F. Seitz and R. P. Johnson, J. Appl. Physics, 8, 84, 186, 246, 1937 (Russian translation: see Uspekhi Fizicheskikh Nauk, 23, 89, 293, 1940); N. F. Mott and H. Jones, Theory of the Properties of Metals and Alloys, Oxford, 1936; J. C. Slater, Rev. Mod. Phys., 6, 209, 1934; L. A. Du Bridge, Rev. Sci. Instr., 9, 1, 1938; W. Shockley, Bell Syst. Techn. Journ., July, 1939.
2. J. Bardeen, J. Chem. Phys., 6, 367, 372, 1938.
3. K. Fuchs, Proc. Roy. Soc., 157, 444, 1936; 153, 622, 1936; 151, 585, 1935.
4. S. L. Quimby and S. Siegel, Phys. Rev., 54, 293, 1938.
5. For data for Na see refs. 2 and 4; for Cu—ref. 3; for NaCl and CaF₂—in Landolt’s tables.
6. See W. L. Bragg, Proc. Roy. Soc., 168, 302, 1938.

1. J. Appl. Physics, 10, 543, 1939. Translated by N. S. Khlebnikov. ↩↩